Particles, Strings and Cosmology PASCOS 2004 (589 pages)

PARTICLES, STRINGS AND COSMOLOGY

PASC#S 2004

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Pran Nath and Michael T Vaughn Northeastern University, USA

l b World Scientific N E W JERSEY * LONDON - S I N G A P O R E * B E l J l N G S H A N G H A I * HONG K O N G * T A I P E I C H E N N A

PARTICLES, STRINGSAND COSMOLOGY

PASCOS 2004

George Alverson, Emanuela BarbarisPran

Editors

16-22 August2004Northeastern University, Boston

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Published by

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PASCOS 2004 Part I: Particles, Strings and Cosmology Proceedings of the Tenth International Symposium

Copyright 0 2005 by World Scientific Publishing Co. Pte. Ltd.

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ISBN 981-256-479-9 (Set) ISBN 981-256-391-1 (Part I) ISBN 981-256-429-2 (Part 11)

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INTERNATIONAL ADVISORY COMMITTEE

Jonathan Bagger (Johns Hopkins) John Bahcall (Institute for Advanced Study) John Ellis (CERN) Paul F'rampton (Chapel Hill) Margaret Geller (Harvard Smithsonian CfA) Sheldon Glashow (BU) David Gross (ITP, Santa Barbara) Jack Gunion (U California, Davis) Pran Nath (NU) Pierre Ramond (U Florida) Norman Ramsey (Harvard) D. P. Roy (TIFR) John Schwarz (Caltech) Joseph Silk (Oxford Yoji Totsuka (KEK, Japan) Michael T. Vaughn (NU) Alexander Vilenkin (Tufts) Albrecht Wagner (DESY) Kameshwar C. Wali (Syracuse) Steven Weinberg (U Texas, Austin) Frank Wilczek (MIT)

LOCAL ORGANIZING COMMITTEE

George Alverson Luis A. Anchordoqui Emanuela Barberis Darien Wood Haim Goldberg Tarek Ibrahim Tom Paul Stephen Reucroft Tomasz Taylor Michael T. Vaughn (Chair)

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PREFACE

The Tenth International Symposium on Particles, Strings and Cosmology (PASCOS-04) was held on the campus of Northeastern University, Boston, Massachusetts, during the period August 16-22, 2004. Two days of the symposium, August 18-19, were devoted to a Fest in celebration of the 65th birthday of Matthews University Distinguished Professor Pran Nath. The proceedings of the Nath Fest will be published in a separate volume.

There were over two hundred registered and walk-in participants at the symposium. Following the tradition with the Pascos series, the symposium was interdisciplinary in nature. and covered a wide range of topics in cosmology, particle physics, and string theory and their interconnections. Topics covered included relativistic astrophysics and cosmology, dark matter and dark energy, neutrino physics, electroweak physics, supersymmetry, string theory/M theory, branes and extra dimensions. Furthermore, there were a significant number of experimental talks related to DO and CDF experiments, BABAR and BELLE, 8-2, double beta decay, and neutrino experiments.

The scientific program of the symposium was organized with the active participation of the international organizing committee. Much help was provided by Professors Haim Goldberg and Darien Wood, and by Drs. Luis Anchordoqui and Thomas Paul, as well as the local organizing commitee.

The success and smooth organization of the symposium was due in large measure to the excellent work of the staff of the NU Physics Department, especially Barbara Najarian, Kathie Simmons, Alina Mak and Sara Sime- one, as well as people from the Office of Special Events a t Northeastern University. Thanks are also due to many students who helped during the conference. These include graduate students Alexander Barabanschikov, Daniel Feldman, Javier Gonzales, Peyman Khorsand, Thomas McCauley, Anastasios Psinas, Raza Syed and Zhen Wu, and undergraduates Matt Bouchard, Cory Fantasia, Kristen Flowers and Stephanie Ward.

PASCOS-04 was supported financially by the United States National Science Foundation, and by the United States Department of Energy. Spe-

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cia1 thanks are due Professor Jorge Jos6, Chair of the Physics Department, for making available the resources of the Physics Department in the organization of the symposium.

George Alverson, Emanuela Barberis, Pran Nath, Michael T Vaughn Boston, Massachusetts, USA

March 22, 2005

CONTENTS

Conference Organization

Preface

RELATIVISTIC ASTROPHYSICS AND COSMOLOGY

Joseph Silk Dark mat ter and galaxy formation: Challenges f o r the next decade

Margaret Geller Where the Dark Matter is ... and isn’t

Damien Easson Cosmic acceleration and modijied gravity models

Tirthabir Biswas Can we have a stringy origin behind Rn(T) cc Rn.r(T)?

S. Shankarnarayanan Planck scale eflects and the suppression of power o n the large scales in the primordial spectrum

Takayuki Hirayama Classical ghosts and the cosmological constant

Kenji Kadota Inflation model building in moduli space

Constantinos Skordis Getting around cosmic variance of the CMB temperature quadrupole

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Rudnei Ramos On the effective evolution for the inflaton 58

Masahide Yamaguchi Inflation models in supergravity with a running spectral index 63

Wan-il Park Modular cosmology, thermal inflation, baryogenesis and a prediction for particle accelerators 68

Torno Takahashi Accelerating universe and modified gravity 73

Luis Anchordoqui The Pierre Auger Observatory: Science prospects and performance at first light 78

DARK MATTER AND DARK ENERGY 83

Pierre Sikivie Cold dark matter flows and caustics

Rita Bernabei D A M A results and perspectives

Rupak Mahapatra First results from the CDMS at Soudan

Alexander Kaganovich New field theory effect at energy densities close to the dark energy density

Antonio L Maroto Branon dark matter: an introduction

NEUTRINO PHYSICS

Sandip Pakvasa Neutrino properties from high energy astrophysical neutrinos

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Steve King Neutrino mass models and leptogenesis

D. P. Roy Solar neutrino oscillations - an overview

Ion Stancu Status of the MiniBooNe experiment

Jose Maneira Solar neutrino results from SNO

Stefan Antusch Implications of running neutrino parameters for leptogenesis and for testing model predictions

Marco Cirelli Sterile neutrinos in cosmology, and supernovae

Roberto Floreanini Neutrino oscillations in random media

K. T. Mahanthappa Fennion masses and neutrino oscillations in SO(10) x s U ( 2 ) ~

Sergei Bashinsky Robust signatures of the relic neutrinos in CMB

Takahiro Kubota Radiative corrections to neutrino reactions off proton and deuteron

ACCELERATOR EXPERIMENTS

Markus Schumacher Searches for new phgsics at LEP

Rahul Malhotra Discovering the Higgs bosons of minimal supersymmetry with tau-leptons and a bottom quark

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Pushpalatha C Bhat Electroweak and top physics results from D0

Dmitri Tsybychev Status of searches fo r Higgs and physics beyond the standard model at CDF

Nick Hadley Searches fo r new phenomena with the D0 detector

Chiara Genta Searches for new physics at HERA

Roberto Chierici Electroweak physics at LEP2 and the fit to the standard model

Stefan0 Lacaprara Searching for dark matter at the LHC

ELECTROWEAK PHYSICS

Marco Fabbrichesi A heavy Higgs boson f rom flavor and electroweak symmetry unification

Max Chaves The standard model results from a scheme to protect the mass of the scalar bosons

Deog Ki Hong Opening the window for technicolor

Colin Froggatt The hierarchy problem and an exotic bound state

CP, CPT AND LORENTZ INVARIANCE VIOLATION

Shiro Suzuki Recent results f rom Belle

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Alfred Bart1 Effects of CP phases on the phenomenology of SUSY particles 347

Apostolos Pilaftsis Phenomenological implications of resonant leptogenesis

Roland E Allen Sfermions and gauginos in a Lorentz-violating theory

SUSY PHENOMENOLOGY

Michael Malinsky Fermion masses and mixings in next to minimal SUSY SO(l0) GUT

Ting Wang High scale study of possible Bd -+ q5Ks CP physics

Rishikesh Vaidya B 4 X , + y in supersymmetry without R-parity

Mario G6mez mSWGRA dark matter and the b quark mass

Bhasker Dutta Neutrino masses, mixings in a minimal SO(10) model and leptonic flavor violation

Daniel Larson Proton decay and the Planck scale

Shaaban Khalil Supersymmetric contributions to the CP asymmetry of the B 4 $Ks and B -+ q'Ks

Takeo Moroi Hadronic decay of the gravitino in the early universe and its implications t o inflation

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Arunansu Sil Inflation and leptogenesis in a SUSY left-right model and its conjecture about b-7 unification

Ian Jack Precision calculation of mass spectra in the MSSM

STRING THEORY, STRING PHENOMENOLOGY, EXTRA DIMENSIONS

Boris Kors How Stueckelberg extends the standard model and the MSSM

Thomas Dent Fermion masses and proton decay in a string inspired SU(4) x SU(2)2 x U ( l ) x model

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Filipe F'reire Can out-of-equilibrium plasma effects create embedded strings? 453

Louis Clavelli SUSY origin of gamma ray bursts 458

Mariano Quiros Tadpoles and symmetries in Higgs-Gauge unification theories 468

Emilian Dudas Vector/tensor supersymmetric duality in five dimensions and applications

Ilya Shapiro Anomaly-induced inflation

Scott Watson Stability moduli with string cosmology

Dmitri Burshtyn Probing orientifold behavior near NS branes

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RECENT THEORETICAL DEVELOPMENTS 505

Roman Jackiw Pure cotton kink in a funny place

M. Shifman Planar equivalence: from type 0 strings to QCD

Tom Kephart Knot energies and the glueball spectrum

Alfred0 Macias O n the experimental testing of the Dirac equation

George Siopsis Perturbative calculation of quasi-normal modes

Hung-Ming Tsai Resolution of the infrared divergence in QFT

A. A. Garcia Electromagnetic fields in stationary cyclic symmetric 2 + 1 gravity

PHOTOGRAPHS AND LIST OF PARTICIPANTS

Photographs

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List of Participants 569

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Relativistic Astrophysics and Cosmology


DARK MATTER AND GALAXY FORMATION: CHALLENGES FOR THE NEXT DECADE

JOSEPH SILK Depavtment of Physics, University of Oxford,

Denys Walkinson Building, Keble Road, Oxford OX1 4LN

E-mail: silkaastro. ox. ac.uk

The origin of the galaxies represents an important focus of current cosmological research, both observational and theoretical. Its resolution involves a comprehensive understanding of star formation, galaxy dynamics, the cosmology of the very early universe, and the nature of the dark matter. In this review, I will focus on those aspects of dark matter that are relevant for understanding galaxy formation, and describe the outlook for detecting the most elusive component, non-baryonic dark matter.

1. Introduction

Dark matter and galaxy formation are intimately related. This applies equally to baryonic and to nonbaryonic dark matter. In this talk, I will review the global budget for baryons and discuss the issue of dark baryons. I will describe the role of nonbaryonic dark matter in galaxy formation, and give an overview of the prospects for detection of cold dark matter. A con- fluence of data on the cosmic microwave background temperature fluctuations, large-scale galaxy redshift surveys, quasar absorption line structure of the intergalactic medium, and distant supernovae of Type Ia have led to unprecedented precision in specifying the cosmological parameters, including the matter and energy content of the universe. The universe is spatially flat, R = 1.02 f 0.02, and dominated by dark energy RA = 0.70 f 0.3 with equation of state w = 3 = -1.02f0.16, nonbaryonic dark matter amounting to R, = 0.27f0.07, and the baryon content = 0.0044+0.004. The latter number incorporates a value of the Hubble content Ho = 72 f Skrn~-~Mpc-A major assumption underlying the quoted errors is the adoption of priors. In particular, primordial gaussian adiabatic, scale-invariant density fluctuations are adopted. If, for example, an admixture of 30 per cent isocunrature fluctuations is included, consistency with CMB data is still obtained but

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the error bars are expanded by up to an order of magnitude l . Another assumption is that the fine-structure constant is actually constant. Allowing this to vary also gives further freedom, especially in the baryon density. A strong case for the dominance of dark matter in galaxy clusters was made as long ago as 1933. It is remarkable that our understanding of its nature has not advanced since then. Of course, modern observations have led to an increasingly sophisticated exploration of the distribution of dark matter, now confirmed to be a dominant component relative to baryonic matter over scales ranging from those of galaxy halos to that of the particle horizon.

2. Global Baryon Inventory

There are three methods for determining the baryon fraction in the high redshift universe. The traditional approach is via primordial nucleosynthesis of 4He, 2H and 7Li. The primary uncertainties lie in the systematic errors associated with ionisation corrections for 4He, and extrapolation to primordial values via corrections for synthesis of 4He and destruction of 'H and 7Li in stars. A unique value of f i b = 0.04 f 0.02 is generally consistent with recent data, although there is some tension between 2H, on the one hand, which in principle is the most sensitive baryometer and favours a higher at,, and both 4He and 7Li. This tension has recently been increased ' by the demonstration of a [Li/Fe] gradient of N - in extremely metal-poor halo stars with [g] < -2, indicative of a role for pregalactic stellar destruction of primordial Li, as well as by determinations of 2 = 0.05 - 0.08 that indicate a 10-15% spallation contribution to 7Li in this metallicity range. Hence more generous error bars may be preferred, at least until the role of systematic effects such as atmospheric depletion of depletion are fully understood. A completely independent probe of fib comes from me& suring the relative heights of the first 3 peaks in the acoustic temperature fluctuations of the cosmic microwave background. With the conventional priors, the data yields excellent agreement between the baryon abundance at z w 1000 and z w lo9. Relaxation of the priors increases the error bars, but the central value is relatively robust. Yet another independent measure of fib, this time at z w 3, comes from modelling the Lyman alpha forest of the intergalactic medium. This depends on the square root of the ionizing photon flux, in this redshift range due predominantly to quasars. The inferred value of fib is again 0.04, with an uncertainty of perhaps 50%. Finally at z N 0, one only has a reliable measure of the pri-

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mordial baryon fraction in galaxy clusters, which may be considered to be laboratories that have retained their primordial baryon fraction. The observed baryon fraction in massive clusters is about 15%, which is consistent with Slb = 0.04 for R, = 0.28, the WMAP-preferred value. Let us now evaluate the baryon fraction at the present epoch, both on galactic scales and in the general field environment. The following is an updated summary of the baryon budget recently presented by Fukugita and Peebles '. Stars in galactic spheroids account for about twice as much baryonic mass as do stars in disks. Disks dominate the (blue) light but spheroids have higher mass-to-light ratios. The total stellar contribution is about 15% of the total baryonic abundance of 0.04. Rich clusters only account for 5% of the galaxies in the universe, and so all of the hot diffuse gas in clusters, which account for 90% of cluster baryons, only accounts for about 5% of the total baryonic budget. Cold intergalactic gas at the current epoch is mapped out in Lyman alpha absorption towards quasars. Identified with the Lyman alpha forest observed at high redshift, the low redshift coun- terpart is sparser. Its detection is more difficult, requiring a UV telescope such as HST or FUSE. However it is found to dominate the known baryon fraction today, and amounts to about 30% of the total baryon fraction In summary: some fifty percent of the baryons in the local universe have been detected and mapped. There are indications, motivated as much by theory as by observations at this stage, that the remaining baryons are in the warm intergalactic medium (WIM) at a temperature of lo5 - 106K. Simulations of structure formation indicate that some intergalactic gas is shocked to a temperature of lo5 - 106K. Much of this gas has not yet fallen into galaxies. According to the simulations, up to about 30% of the baryons are heated by the present epoch and remain diffuse. This fraction is an upper limit because the simulations lack adequate resolution, and moreover the amount of shock-heated gas is controversial 5. Even more significantly, the theory of galaxy formation, as currently formulated, predicts that the WIM is metal-poor, in that those galaxies where most of the stellar mass resides, namely the massive galaxies, are energetically incapable of ejecting very much in the way of metal-enriched debris 6 . However, observations are confirming the existence of some WIM, in particular via detection of redshifted rest-frame UV OVI absorption towards quasars, extended soft x-ray emission near clusters 7, and OVII/OVIII x-ray absorption along lines of sight to AGN. The oxygen abundance exceeeds [O/H]i -1.5 at z - 2.5 '. In practice, too few lines of sight have so far been probed to say a great deal about the WIM mass fraction. In summary, something like 80 percent

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of the baryons at present have either been detected or are plausibly present with detection being imminent. One could conclude

flt,,ot,served = 0.032 f 0.005.

Clearly the case for 10 - 20% of the local baryons being unaccounted for and dark is possible but far from convincing given the WIM uncertainties. If the WIM is indeed the dominant gas reservoir, there are strong implications for feedback from galaxy formation, in order to account for the observed enrichment of the WIM. Strong enrichment is indeed found for the intracluster medium, and this most likely is a consequence of early galaxy outflows. However the generation of these outflows is not understood. One clearly needs to establish a more convincing case for the WIM bcfore pursuing the impact of massive gas outflows on the early evolution of the typical field galaxy. Nevertheless, since the possible inass in unaccounted-for dark baryons is on the order of the baryon mass in stars, it is clear that such a result would profoundly affect our theories of galaxy formation and evolution. Hence demonstrating that these baryons are not present in the Milky Way is a useful exercise.

3. Confirmation of baryonic shortfall

A detailed census of both the Milky Way and M31 confirms the lack of baryons in the amount predicted by primordial nucleosynthesis. The virial mass measured dynamically for the Milky Way from the HI rotation curve, dwarf galaxy orbits, and globular cluster peculiar velocities, amounts to - 1Ol2M~. This is valid to a galactocentric radius of 100kpc. The baryon mass, including stars and gas is (6 - 8) x 1O1OM0. However, the expected baryon fraction, both as observed at high redshift and in galaxy clusters, and especially as inferred from primordial nucleosynthesis and the CMB data, is about 17%. This is the initial baryon fraction when the Milky Way formed. A similar shortfall, amounting to a factor of about 2, is found for M31. There are two possibilities for the "missing" baryons. Either they are present in the galaxy halo and as yet undetected, or they have been ejected via energetic outflows early in the history of the galaxy. Intensive searches for compact halo objects have been performed via gravitational microlensing of several million stars in the Magellanic Clouds. The EROS and MACHO experiments set the following limits, for more than 5 years of data: no more than 20 percent of the dark halo mass can be in objects in the mass range N 10-8Mo to N 10Ma, with a detection claimed

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by the MACHO experiment that saturates this limit for objects of mass N 0.5Mo. The most plausible candidate for MACHOS of this mass are old halo white dwarfs. This requires a stellar initial mass function for the protogalaxy that forms the first stars with high efficiency in a narrow mass range (4 - 8Mo) . While this seems implausible, it cannot be ruled out by theoretical arguments, one possible signature being that of occa- sional Type Ia supernovae. However old white dwarfs are still emitting light, albeit weakly, at visible wavelengths, and proper motion searches for faint candidates have imposed strong limits on the halo white dwarf mass fraction of between - 2% relative to the local dark matter density l1 and N 0.2% 12. It seems reasonable to conclude that halo white dwarfs cannot account for more than a quarter of the unacounted-for baryons, and this is most likely an overestimate. One can imagine even less credible initial mass functions that would allow, say, ten percent of the dark halo to consist of primordial brown dwarfs, low mass primordial black holes, or even compact dense clouds of cold molecular gas. All of these possibilities have been studied as possible explanations for halo dark matter. Even if one's goal is only to account for halo baryonic dark matter, requiring even l0loMo to be in such a form stretches astrophysical credibility. But this cannot be ruled out. A more plausible direction for investigation is that the "missing" baryons have been ejected from the galaxy, in the form of a vigorous, early galactic wind. Such a wind, if it occurs presently, could involve very little mass outflow. Observations indicate that at the present epoch, vigorous winds are exceedingly rare, and are seen only in low mass, star- bursting galaxies. In the early galaxy, however, the star formation rate was much higher, and the situation could have been quite different with regards to mass loss. Evidence for early winds comes indirectly from the highly enriched intracluster medium, whose mass exceeds that in the stellar component of cluster galaxies by a factor of several. The substantial amount of metals in the intracluster gas, and even the presence of magnetic fields, are most likely accounted for via ejection in early galactic winds. At high redshift, the substantial population of the Lyman break galaxies (LBG) at z N 3 - 4 show broad linewidths displaced systematically to the blue by several hundred kilometres per second for the interstellar gas relative to the absorption lines of the stars 13. Moreover, stacked spectral energy distributions of LBGs seen in projection near background quasars show evidence of a proximity effect, with a - 1 Mpc hole (comoving) inferred from the lack of Lycr and CIV absorption 14. An energetic wind from galaxies with stellar mass similar to that of the Milky Way is inferred to have occurred,

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or at least, to provide the simplest explanation of these observations. Some of these galaxies most likely are massive, as their spatial clustering strongly favours their being the precursors of low redshift ellipticals 15. The principal counterargument comes from wind simulations. While it is unanimously agreed that dwarf galaxies, with masses below lo7 - 108Ma, and escape velocities below 5Okms-', are easily stripped of gas by supernova-driven winds, problems arise in driving winds from more massive galaxies. For disk galaxies, it is found that even for galaxies of mass 109-10'0Ma, the supernovae ejecta stream out in a hot wind but most of the interstellar gas remains in the disk 16. For forming galaxies, when the gas is more spherically distributed, ejection in a wind becomes inefficient for masses above about 10l0M@, according to the most recent multi-phase interstellar medium simulations 6 . These simulations adopt current supernovae rates and energetics per unit baryonic mass, along with a solar neighbourhoopd initial mass function, that is to say a rate of type I1 supernovae of 1051 energy input per 200Ma of gas that forms stars. This rate assumes a local fit to the initial mass function 17. However in addition to the observational indications, semi-analytical galaxy formation theory requires a wind to have ejected approximately half of the baryons from even the most massive galaxies. Otherwise, one finds that almost all of the gas that can cool within a Hubble time does cool and form stars, and the predicted luminosity function strongly disagrees with observations for luminosities above 2-3 times the galaxy characteristic luminosity, L, N 10l0Mo 18. Related model malfunctions include unacceptably recent and inefficient star formation for distant massive galaxies as studied in deep surveys 19.

4.

The numerical simulations of galactic outflows must cope with a variety of hydrodynamical and gravitational processes, including star formation, supernovae explosions, gas heating and cooling in a multi-phase interstellar medium, and gas escape from the galactic gravitational field. Hitherto, it has been necessary to severely approximate much of the relevant physics. For massive galaxies, winds are suppressed as the outflowing heated gas runs into surrounding, cold infalling gas, and most of the energy input is radiated away. Only about 2 percent of the initial supernovae energy is useful for expelling gas. The situation may not be as bleak as depicted by the simulations. One omission due to lack of resolution is the effect of both Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The former help

What could be wrong with the simulations?

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the hot medium break out of the galaxy and enhance the wind efficiency. The latter enhances entrainment of the cold gas into the hot supernova- heated medium and can help account for the observed enrichment of the intergalactic medium. While the situation with regard to outflows may be alleviated in this fashion for low and intermediate mass galaxies, more drastic measures are required for massive galaxies. These may include any of the following: use of a top-heavy initial stellar function to enhance the supernova rate or appeal to an increased frequency of hpernow at early epochs relative to supernovae, or finally, recourse to outflows from active galactic nuclei. Any or all of these may occur. If indeed substantial mass loss via a wind occurs, then a plausible ansatz is that &foutjlow - M* as observed in nearby starbursts, where the mass injection rate into the hot x-ray emitting diffuse gas is comparable to the star formation rate 20.

This means that about as much gas is ejected as is retained in stars. Such a conclusion is consistent with the observed baryon fraction in the Milky Way and M31, the two best-studied moderately massive galaxies. One can also understand the heavy element abundance observed both in the intercluster medium and in the warm (T N 106K) intergalactic medium detected in OVI absorption. While the baryon fraction is probably not a major problem for consensus cosmology, I now turn to the issue of cold dark matter, and its relation to structure formation.

5. Galaxy formation and CDM: the good, the bad and the

There are some noteworthy success stories for cold dark matter (CDM). First and foremost is its success in predicting the initial candidates for structure formation that culminated in the discovery of the cosmic microwave background temperature fluctuations. The amplitude of the Sachs-Wolfe effect was predicted to within a factor of 2, under the assumption, inspired qualitatively by inflation, but quantitatively by the theory of structure formation via gravitational instability in the expanding universe, of adiabatic scale-invariant initial density fluctuations. A direct confrontation with this theory was first met with the detection and mapping of the acoustic peaks. These are the hallmarks of galaxy formation, first predicted some three decades previously, and demonstrate the imprint of the density fluctuation initial conditions on the last scattering surface of the CMB at z - 1000. Another dramatic demonstration of the essential validity of CDM has come from the simulations of the large-scale structure of the universe. The initial

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conditions, including gaussianity, are specified, growth occurs by gravitational instability, and the sole requirements on dark matter are that it be weakly interacting and cold. Thus was born CDM, and the CDM scenario works so well that we cannot easily distinguish the artificial universe from the actual universe mapped via redshift surveys. More to the point, perhaps, is that the simulations are used to generate mock galaxy catalogues and maps that yield precise values of the cosmological parameters, in combination with the CMB maps. Dark matter-dominated halos of galaxies are another generic success of CDM, as mapped out by rotation curves. However the detailed predicted properties of halos do not seem to be well- matched to observations. There is considerable scatter in the predictions of high resolution simulations for the structure of galaxy halos. Neverthe- less, the predicted dark matter cusps ( p cx rWQ with 1 < a! < 1.5) are not found in most low surface brightness dwarfs, nor is the predicted dark matter concentration (C = r20o/rS N 5 - 10, where 7-200 is the radius at density contrast 200 and rs is the halo scale length) consistent with the dark matter distribution in barred galaxies, possibly including our own galaxy, nor finally is the predicted number of satellites similar to the observed satellite frequency. In general, many observed halos seem to have softer cores, lower concentrations, and less clumpiness than predicted by the simulations. However it has been argued that inclination, triaxiality and non-circular orbits make the dwarf situation unclear 22, quite apart from the fact that dwarf galaxy formation is not understood. Of course the same may be said for bars. The situation for early-type galaxies is at least as controversial. Indeed, for very round ellipticals, at least in projection, in low density environments and not especially luminous, studies of the distribution and kinematics of planetary nebulae suggest that mass traces light to N 5 effective radii 21. However, the opposite conclusion is inferred for massive early-type galaxies, which display evidence for as much as a 50% contribution of dark matter within N 1 effective radius 23. All of these issues have been debated. For example, reformation of bars by gas infall can avoid the problem of bar spin-down by dynamical friction, and astrophysical processes, discussed below, can render the dwarf satellites optically invisible. Hence it is difficult to be definitive about any possible contradiction between theory and observation. Certainly, on the baryonic front, the most accepted problem is the loss of angular momentum by the contracting and cooling baryons in the dark halo. The resulting disks are far too small. These various difficulties for galaxy formation theory have stimulated a variety of responses.

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5.1. Resurrection via modifging fundamental physics

Suppose that one changes the nature of the dark matter. Increasing the scattering cross-section helps alleviate several of the problems, such as CUS-

piness and clumpiness. However the resulting dark halos are too spherical. Another approach modifies the law of gravity. Indeed, one may be able to dispense entirely with dark matter. These approaches seem rather drastic, however, and I believe that one should argue that all alternatives should be fully explored before tinkering with fundamental physics.

5.2. Resurrection via astmphgsics

The obvious addition is stellar feedback. This can heat the baryons, and help reduce the loss of angular momentum. If the feedback is strong, mass loss is a likely outcome. The observed baryon fraction and the galaxy luminosity function for the most luminous galaxies both point to a possible loss of half the baryons during the galaxy formation process 24. How- ever to eject up to half the baryons may require more than normal stellar feedback, at least for galaxies comparable to, or more massive than, the Milky Way. One can appeal to a topheavy IMF that would yield up to an order-of-magnitude more supernovae per unit mass of baryons, to an augmented fraction of hypernovae relative to supernovae, or to outflow generated by Eddington luminosity-limited accretion onto a supermassive black hole. Outflows may also be effective at reducing the dark matter concentration, at least for dwarfs 25. Production of a soft core is best achieved for a massive galaxy by dynamical heating, as has been studied for the case of a rapidly rotating central baryonic bar 26 , although a contrary view is expressed in 27. Such bars are likely to be generic to galaxy formation via mergers, and if gaseous would leave little in the way of stellar tracers. Dynamical feedback also occurs via tidal evolution, and this can account for both the frequence and distribution of dwarf galaxies 2a.

6. Observing cold dark matter

The best way forward is to directly measure the halo properties by observing cold dark matter directly or indirectly. Direct detection is sensitive both to the local density of CDM and to its local phase space density. There is a candidate, motivated by supersymmetry, the LSP, usually considered to be massive with m, N 100GeV, the SUSY breaking scale, and generically known as the neutralino or WIMP. However light LSPs, such as the axino,

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are also possible, and there is even a LSP with purely gravitational interactions, the gravitino. However, in general, the WIMP undergoes elastic interactions with ordinary matter and is therefore potentially detectable via laboratory experiments. Early universe freezeout yields a mass estimate; more specifically, the annihilation cross-section is inferred to be of order R;la,,,k, and depends, via SUSY, on the WIMP mass. The corresponding elastic cross-section is model-dependent, but most models spans the range to 10-6pb for a relic abundance R,h2 x 0.1.

6.1. Direct detection

Scattering of WIMP particles leads to nuclear recoils that can be measured by three different techniques: scintillation, phonon production, and ionization. The various experiments currently underway use different combinations of these techniques. Only one experiment, now running for 7 years, has reported a positive result, using NaI scintillation and a claimed detection of annual modulation, to yield a model-dependent detection of m, = 50(flO)GeV with a cross-section of 7 ( f l ) x 10-6pb. However other experiments, including Edelweiss, ZEPLIN and CDMSZ, report a lower upper bound in the cross-section, with the more recent limit being ux < 4 x 10-7pb at 60 GeV 29.

6.2. Indimct detection

Annihilations currently occur in the dark halo, although the annihilation time-scale N (nh<ov>,,,,)-l N 1026(Tf/GeV)3/2s, where Tf is the freeze- out temperature. The annihilation products are potentially observable in the form of high energy y, e + , p and v, and are enhanced by the effects of halo clumpiness. There are tentative indications of possible detections of e+ and y. A positron feature & is seen above lOGeV that cannot easily be attributed to secondary production of e+. A modest clumpiness boost is required for the measured flux to lie in the range allowed by annihilation models combined with cosmic ray diffusion 30. Both the high galactic lat- itude gamma ray background and the unresolved diffuse gamma ray flux towards the galactic centre have relatively hard spectra that seems to be inconsistent with cosmic ray spallation and the ensuing 7ro decays. One possible explanation is in terms of population of hitherto unresolved discrete gamma ray sources, such as blazars in the extragalactic case or low mass x-ray binaries in the galactic case 33. Similar boost factors, of 10-100, from dark matter clumpiness are required to that invoked for positron annihila-

13

tion, if both the extragalactic and galactic diffuse gainma ray components have a WIMP annihilation origin.

6.3. A mdical suggestion

The Integral SPI detector has measured a substantial diffuse flux of electron-positron annihilation line emission at 511 keV from throughout the galactic bulge. Some photons s-l are generated over a region that extends up to 3 kpc from the galactic centre. There is no indication of any positron annihilation emission from any bulge source, such as might be connected with decays of Type I1 supernova-ejected radioactive 26A1 or e+ - e- jets from x-ray binaries. This therefore has led to consideration of CDM annihilation as a possible explanation 31. The principal novelty of such a hypothesis arises with the mass required for the annihilating particle. It must have a mass of -10 MeV, as a much heavier particle would annihilate via pion production and produce an excessive flux of diffuse gamma rays from ro decays. From the measured flux and angular distribution, one immediately infers the required cross-section and radial profile, namely gann - lOP5pb and px 0: r- ' l2. The profile is close to what is expected from CDM models, as inferred from rotation curve and microlensing modelling (actually, the derived CDM profiles are disputed for the Milky Way but a profile softer than NFW is inferred for barred galaxies and for LSB dwarfs). The required cross-section is very low, however, compared with the freezeout value at kT- mx/20, namely a,,, x (0.2/Rx)pb. One can reconcile the observed low annihilation cross-section required for the 5llkeV flux by assuming that the relativistic freeze-out limit is S-wave suppressed, so that oann 0: (3) (z)' This naturally reduces the low temperature value of the halo annihilation cross-section relative to the freeze-out value by a factor (v/c)~ - There is a price to pay however for the low mass, namely the introduc-

mass to the Z boson if m, is at the SUSY breaking scale. A mediating mu - 0.1 - lGeV could have observable consequences, for example with regard to the magnetic moment of the muon, and these are being investigated. One should also eliminate possible astrophysical sources of the 511 keV line. The most promising of these is the population of low mass x-ray binaries, which have a bulge distribution and are known to occasionally have high energy jets and outflows. However there has hitherto been no association of 511 keV emission with any class of discrete sources.

tion of a new light gauge boson mu 0: m, 1/2 , ordinarily comparable in

14

6.4. An equally radical suggestion

Three atmospheric Cerenkov radiation telescopes have recently reported the detection of TeV photons from the Galactic Centre. HESS has the most significant detection. The supermassive black hole associated with the SagA* radio source is measured to have a mass of 3x1O6Mo, and x- ray measurements indicate a low accretion rate. Hence a source of y-rays powered by accretion is unlikely. One could appeal to a high energy cosmic ray accelerator associated with the central black hole. However the low observed accretion rate may (weakly) argue against this. An acceleration power in TeV electrons or EeV protons of 1036-1039 ergs s-', respectively, is required, where the bolometric luminosity is only ergs s-l (or lO-'L~dd). An annihilation explanation requires WIMPs of mass at least 10-20 TeV. In this case, the observed hard spectrum is naturally explained 35,36. However there are difficulties that arise in reconciling the WMAP-constrained value of R, with the cross-section required to account for the HESS luminosity of 103%-1 above 200 GeV with half-width of 6 arc-minutes. To arrive at the required relic density for a 20 TeV neutralino mass, one has to fine-tune the particle physics annihilation channels via co-annihilations. The R, constraint prefers a cross-section around 1 pb. The natural value of the cross-section at 20 TeV tends to be lower than 1 pb, because of the unitarity scaling that sets in at large masses, and this results in WIMP overproduction: R, is too high. However, for a typical NFW profile, the inferred cross-section to account for the observed gamma ray flux at 10 TeV is about 10pb, and is even larger for a softer core. In this case, the inferred relic density is too low, only R, - 0.03. To reconcile these conflicting requirements is not straightforward. The simplest option is to relax the relic density constraint. Suppose that the 20 TeV WIMPs are subdominant. One can now tolerate a larger cross-section. Particle physics fine-tuning is required via co-annihilations, but this is rarely an unsurmountable problem. Although it appears to be very unnatural that the LSP mass would be any heavier than a few TeV, with a high degree of fine-tuning, co-annihilations can allow for much heavier LSPs. Even in this case, however, it would seem very unlikely that the LSP mass could be any heavier than 20 TeV, at least in the simplest classes of models. The following scenario might then apply. One would have two types of stable particle dark matter, as appropriate to N=2 SUSY 34. The light particle (m, N 10MeV) would be the principal dark matter component, and annihilate via e+e- to produce the 511 keV flux. The subdominant

15

particle, with mass -10-20 TeV, would account for the HESS flux. An alternative is the following. Suppose we settle for the lower cross-section as inferred from the relic WIMP density. Theory certainly has an easier time arriving at this goal. Then we need to boost the annihilation flw at the centre of the galaxy. It is unlikely we can appeal to the usual CDM clumpiness boost factor, because any clumps would be tidally disrupted. It is then appealing to reconsider the possibility of a spike of dark matter around the central SMBH within its zone of influence, a parsec or so. This occurs naturally for adiabatic formation of the SMBH, via the response of the CDM halo, and yields, in principle, an observable gamma ray signal from generic CDM annihilation models 32. A spike formed in a pregalactic SMBH would survive infall of the SMBH by dynamical friction to the centre of the Milky Way galaxy. This works best if the SMBH forms by baryonic accretion rather than by black hole mergers, although only major mergers are potentially catastrophic for a spike 37. The survival of a spike seems not unlikely because (a) there is no theoretical understanding of the "final parsec" problem of merging black holes, (b) minihalo mergers in hierarchical galaxy formation yield too few close-in SMBH candidates for successful mergers to prevail in the final system, and (c) forming the very massive SMBHs seen at z 2 6 requires an accretion formation mechanism given the limited time available. The adiabatic spike, which has profile p 0: r-7 with y > 2, dominates accretion and would yield the HESS point-like source but be unobservable at INTEGRAL/SPI resolution.

7. The future

Baryon dark matter will most likely be mapped out within five years. The intergalactic medium is the major repository where large uncertainty remains. The warm intergalactic medium can be studied via highly ionised oxygen, both in UV absorption and in x-ray emission. This most likely will require dedicated experiments that are being planned. Of course to distrib- ute the oxygen and other elements into the WIM/ICM requires a greatly improved understanding of galactic outflows. Considerable improvements will be needed in the accuracy and resolution of simulations of galactic outflows. Can the escape rate of gas be of the same order as the star formation rate in massive young galaxies? It will require improvement in the input physics of star formation as well as in the numerical sophistication of the codes before this question can be fully considered. Advances on the nonbaryonic matter front seem equally likely. Of course, here there is a big

16

assumption, that the elusive dark matter particle is a WIMP. Were it to be a light gravitino or an axion, almost all of the searches would be frustrated. Nevertheless there are more than a score of dedicated searches underway for direct and indirect detection of non-baryonic dark matter. These include searches for annihilation products, including positrons and antimatter (PAMELA, AMSZ), high energy neutrinos from the sun (ANTARES, ICECUBE), and gamma rays (GLASST, HESS, VERJTAS). It will be necessary with all of these searches to correlate complementary signals and corroborate astrophysical detections with accelerator evidence of existence of the relevant particle. Such evidence may be beyond the reach of the LHC, but a future linear collider should be able to provide the clean signature needed to identify the SUSY LSP, provided that the WIMP mass is below 1TeV. If the WIMP mass is greater, then ACT (gamma ray tel* scopes) may become the unique hope for detection. Other "smoking guns" include detection of gamma ray line emission and confirmation of annihilation signals associated with nearby dwarf galaxies and with the Galactic Centre, where primordial concentrations of dark matter should exist, by both spectral and spatial resolution.

Acknowledgments

I thank my colleagues, including R. Bandyopadhyay, C. Boehm, P. Ferreira, D. Hooper, H. Mathis,. J. Taylor and H. Zhm, for many discussions of relevant topics. I am also indebted to Professor Piet van der Kruit for hosting me as Blaauw Visiting Professor at the Kapteyn Institute in Groningen, where this review was completed.

References 1. Bucher, M. et al., PRL, 93, 081301 (2004) 2. Lambert, D., in Mitchell Symposium on Observational Cosmology, astro-

ph/0410418 (2004) 3. Fukugita, M. and Peebles, P., astro-ph/0406095 (2004) 4. Stocke, J., Shull, J. and Penton, S., in STScI Symposium, Planets to Cosmol-

ogy, astro-ph/0407352 (2004) 5. Birnboim, Y. and Dekel, A., MNRAS, 345,344 (2003) 6. Springel, V. and Hernquist, L., MNRAS, 339, 289 (2003) 7. Zappacosta, L. et al., A&A, 394, 7 (2002) 8. Simcoe, R., Sargent, W. and Rauch, M., ApJ, 606, 92 (2004) 9. Afonso, C. et al., A&A, 404, 145 (2003) 10. Alcock, C. et al., ApJ, 542, 281 (2000) 11. Creze, M. et al., A&A, in press, astro-ph/0403543 (2004)

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12. Spagna et al., A., A&A, in press, astro-ph/0410215 (2004) 13. Steidel, C. et al, Astr0phys.J. 604, 534 (2004 14. Adelberger, K. et al, ApJ, 584, 45 (2003) 15. Adelberger, K. et al, ApJ, in press, astro-ph/0410165 (2004) 16. Mac Low, M. and Ferrara, A., ApJ, 513, 142 (1999) 17. Kroupa, P., Science, 295, 82 (2002) 18. Benson, A. et al., MNRAS, 351, 121 (2003) 19. Thomas, D. et al., ApJ, in press, astro-ph/0410209 (2004) 20. Summers, L. et al, MNRAS, 351,l (2004) 21. Romanowsky, A. et al., Science, 301, 1696 (2003) 22. Hayashi, E. et al., ApJL, submitted, astro-ph/0408132 (2004) 23. Treu, T. and Koopmans, L., ApJ, in press, astro-ph/0401373, (2004.) 24. Silk,J., MNRAS, 343, 249 (2003) 25. Read, J. and Gilmore, G., MNRAS, in press, astro-ph/0409565 (2004) 26. Holley-Bockelmann, K., Weinberg, M. and Neal Katz, K., MNRAS, submit-

27. Sellwood, J., ApJ, 587, 638 (2003) 28. Kravtsov, A., Gnedin, 0. and Klypin, A., ApJ, 609, 482 (2004) 29. CDMS Collaboration, PRL, in press, astreph/0405033 (2004) 30. Baltz, E. et al, PRD, 65, 065311 (2002) 31. Boehm, C. et al, PRL, 92, 1301 (2004) 32. Bertone, G., Sigl, G. and Silk, J., MNRAS, 337, 98 (2002) 33. Ullio, P. et al., PRD, 66, 123502 (2004) 34. Boehm, C., Fayet, P. and Silk, J., PRD, 69, 101302 (2004) 35. Bergstrom, L. et al., preprint astro-ph/0410359 (2004) 36. Horns, D., Phys. Lett. B, submitted, astro-ph/0408192 (2004) 37. Ullio, P., Zhm, H. and Kamionkowski, M., PRD, 64, 3504 (2201)

ted, astro-ph/0306374 (2004)

WHERE THE DARK MATTER IS ... AND ISN'T

MARGARET J. GELLER Smithsonian Astrophysical Observatory

60 Garden St. Cambridge, MA 02138

E-mail: mgellerOcfa.harvard.edu

Observations of the cosmic microwave background combined with large redshift surveys suggest that 0, N 0.3. Direct dynamical measurements combined with estimated of the universal luminosity density suggest R, = 0.1 - 0.2. The a p parent discrepancy may result from variations in the dark matter fraction with mass and scale. Traditional techniques already indicate that these variations are present. Gravitational lensing maps combined with large redshift surveys promise to measure the dark matter distribution. The Hectopsec Survey of a field in the Deep Lens Survey is a first approach to this kind of measurement.

1. Introduction

Since the discovery of dark matter in the universe (Zwicky 1933), this ubiquitous component of the universe has remained a puzzle. Dynamical measurements of the masses of systems of galaxies combined with estimates of the mean luminosity density of the universe long provided the only route to an estimate of the mean matter density of the universe. As large redshift surveys came of age, these surveys became a tool for estimating the mass density.

Recently the powerful combination of large redshift surveys and measurement of fluctuations in the cosmic microwave background (CMB) has led to an apparently tight constraint on the universal mean matter density, 0,. WMAP(Wi1kinson Microwave Anisotropy Probe) and the two large redshift surveys 2DF and SDSS (Two-Degree Field and Sloan Digital Sky Survey, respectively) give R, = 0.27 f 0.04 and Q, = 0.30 f 0.04, in essential agreement (Spergel et al. 2003; Tegmark et al. 2004).

In some contrast with the WMAP results, estimates of R, derived from the more traditional combination of dynamical estimates and luminosity densities yield R, in the range 0.1-0.2 (see Rines et al. 2004 for a

18

19

compilation). Even ten years ago the apparent disagreement between the WMAP

results and dynamical estimates would have been regarded as negligible. Now, however, the discrepancy calls for a deeper investigation. Although constraints on R, have tightened, there has been much less progress in determining and understanding the relative distributions of dark and light- emitting matter in the universe. Recent investigation indicate that the mass-to-light ratios of systems of galaxies are a function of the system mass (Lin et al. 2003, 2004; Ramella et al. 2004). There is also accumulating evidence for a trend of mass-to-light ratio with radius in clusters of galaxies (Rines et al. 2004).

Perhaps the ultimate tool for measuring the distribution of dark matter in the universe is weak gravitational lensing. Until recently weak gravitational lensing studies were limited to measuring the masses of the richest clusters of galaxies. Now a number of wide-field weak lensing survey are underway. The first of these, the Deep Lensing Survey (Wittman et al. 2002), covers 28 square degrees in seven separate four-square-degree regions. Weak lensing surveys, combined with large redshift surveys of the foreground lensing galaxies, provide the enticing opportunity to measure the masses of condensations ranging all the way from individual galaxies to the large-scale structure of the universe without making any restrictive assumptions about the internal dynamics of the system.

I discuss some recent explorations of the relationship between dark and light-emitting matter in the universe. Section 2 is a discussion of recent work on clusters of galaxies which reveals a relationship between mass- to-light ratio and mass. Section 3 discusses the dependence of mass-to- light ratio on radius within individual systems. Section 4 introduces the Hectospec Survey of one of the fields in the Deep Lens Survey.

2. Dynamical Mass-to-Light Ratios of Clusters and Groups of Galaxies

The classical approach to the determination of R, depends on two steps (1) a computation of the mass-to-light ratio, M/L, for systems of galaxies ranging from groups to clusters and (2) determination of the luminosity density, L, from an extensive redshift survey. Then R, = M/L * L. There are at least two important assumptions which underlie this approach: (1) the mass-to-light ratios of systems are characteristic of the universe as a whole and (2) the luminosity density is determined from a region large

20

enough to sample the universal density. Although there have been indications of variation in mass-to-light ratio

with mass, the tendency has been to attribute these effects to systematic variation in the galaxy population with mass. The fraction of bluer, star-forming galaxies increases as the mass decreases. Another limitation on interpretation of these early indications of a more complex relation between the dark and light-emitting matter has been the absence of large-area uniform photometric surveys in near-infra-red bands insensitive to the star formation rate.

The 2MASS project now provides a uniform photometric catalog for systems of galaxies in the nearby universe. In an elegant series of papers Lin et al. (2003, 2004) have used these data to demonstrate a remarkably tight relationship between the K-band mass-blight ratio of systems of galaxies and mass. They show that the mass-to-light ratio increases with mass. This dependence is a window on galaxy formation in these systems. Either galaxy formation is less efficient or galaxy disruption is more efficient in more massive systems.

Figure 1: Increase of mass-to-light ratio with mass; M/L oc M0.56*0.05.

Lin et al. (2003, 2004) use x-ray observations to determine the masses of their systems. Ramella et a1 (2004) obtain masses by applying the virial

21

theorem to complete redshift surveys within the virial radius. Their systems cover a larger mass range than explored by Lin et al (2003, 2004) and confirm the relationship between M/LK and M (Figure 1). The values in Figure 1 are based on a Hubble constant H = lOOh km/s/Mpc with h = 1.

To use the results of Figure 1 to arrive at an estimate of R,, we need the abundance of systems as a function of mass (mass function). This function is poorly known particularly at the low mass end, but the abundance decreases strongly as the mass increases. The observed x-ray temperature function of systems of galaxies shows that groups are a hundred times as abundant as massive clusters (Ikebe et al. 2002). Analyses of large optical catalogs of groups of galaxies suggests that the number density of groups of mass M is approximately n(M) 0: M-2.3 (Pisani et a1 2003). The median mass-to-light ratio is thus weighted toward the least massive systems and falls in the range 30-50 with h = 1.

Although increasingly well known, the luminosity density of the local universe is remains uncertain by - 30%. A K-band mass-to-light ratio of 90 (h = 1) (at the high mass end of Figure 1) gives R, = 0.3. A more appropriate weighting of the low mass systems yields R, - 0.1.

3. The Behavior of Mass-to-Light Ratios Within Clusters

Massive clusters of galaxies are still forming today. They are surrounded by infurl regions. Galaxies and groups of galaxies within this infall region are destined for accretion by the cluster. The infall regions generally extend to radii of 10 Mpc. In these extended regions the systems axe neither in equilibrium nor in the linear regime. It is thus a great challenge to measure mass profiles and the related mass-to-light ratio on the scale of the infall region.

There are only two techniques available for measuring the mass distribution in the infall region: weak lensing and caustic mass estimation (Diaferio 1999). Weak lensing maps which reach into the infall region exist for a few clusters in the redshift range 0.2-0.3. Although there are a substantial number of redshift measurements for these clusters, the samples are not complete and thus the mass-to-light ratio at large radius is very uncertain.

At lower redshift, weak lensing studies place very poor constraints on mass distributions because the surface number density of background sources is small. On the other hand it is now possible to acquire very large, complete redshift surveys of these systems (Rines et al. 2003). The

22

2MASS survey provides uniform photometry and the caustic technique provides mass estimates which agree remarkably well with independent measurements from the x-ray where the techniques overlap (Rines et d. 2003).

On scales larger than 1.5 Mpc (approximately the virial radius), the light contained in galaxies is less clustered than the mass in rich clusters. The mass-to-light ratio inside the virial radius is a factor of 1.8+/-0.3 larger than that outside the virial radius (Rines et al. 2004) This difference could result from changing fractions of baryonic to total matter or from variations in the efficiency of galaxy formation or disruption with environment.

Remarkably, the dependence of the mass-to-light ratio with radius is in concert with the dependence on cluster mass discussed in the Section 2. Figure 1 shows that the K-band mass-to-light ratio increases with cluster mass or nearly equivalently, velocity dispersion. The velocity dispersion of a cluster decreases steadily with radius from the virial region throughout the infall region. In essence, the infall region is full of groups of galaxies directly comparable with the low mass systems in Figure 1. At the same effective velocity dispersion, the determinations of characteristic mass-to- light ratios for individual systems agree with the determination from the infall regions.

The agreement of these techniques poses a puzzle. They indicate that the dark matter is more concentrated than the light we observe from galaxies. The average mass-to-light ratio calculated from cluster infall regions, 38f6 (h = 1) suggest an R, - 0.1 (Rines et al. 2004). The typical system, a group of galaxies, gives a similar result (Section 2). These results are difficult to reconcile with independent methods that suggest larger S 2 , - 0.3. These results indicate that there are serious gaps in our understanding of the relative distributions of dark and light-emitting matter.

4. Weak Gravitational Lensing and Redshift Surveys

Weak gravitational lensing is a potentially powerful tool for measuring the relative distribution of mass and light on the universe in scales ranging all the way from individual galaxies to large-scale structure. It has only recently become possible to realize the power of this new tool for elucidat- ing the mass distribution in the universe. Wide-field instruments on large telescopes in good sites are first enabling the deep, high quality imaging necessary to measure the - 1% tangential distortions of background galaxies (sources) which reveal the mass distribution of the foreground structures (lenses) generally marked by galaxies.

23

The Deep Lens Survey (Wittman et al. 2002) is an ambitious ground- based ultra-deep survey of seven four square degree fields in four bands B,V,R,z' to 29/29/29/28 mag per square arcsecond surface brightness. One of these fields, centered at 09:19:32.4+30:00:00 (52000) is complete. The spectacular weak lensing map in Figure 2 shows the measured projected mass density in this region; all of the bright regions of the map signal mass concentrations along the line-of-sight. Distortions of galaxies with 22 < mR < 25.5 ( z = 1.2 is the angulardiameter distance weighted redshift of the background galaxies) reveal the matter distribution marked primarily by foreground galaxies with redshifts in the range 0.3-0.7, accessible with Hectospec, a 300-fiber robotic instrument mounted on the 6.5-meter MMT on Mt. Hopkins, Arizona (Fabricant et al. 1998).

Figure 2: Lensing map of the 2x2" field centered at 09:19:32.4+30:00:00. The bright regions indicate mass peaks along the line of sight. The brightest region in the map is a rich cluster identified by Abell(l958) (I. Dell'Antonio, Brown University)

24

Combining a foreground redshift survey with a lensing map like the one in Figure 2 is a newly enabled approach to the measuring the relative distributions of mass and light over an enormous range of scales: 50 kpc to tens of Mpc.

Figure 3 shows preliminary results from a redshift survey of the region in Figure 2. There are 3015 galaxies with measured redshifts so far. Appli- cation of a friends-of friends group-finding algorithm (Ramella, Pisani, and Geller 1997) to the redshift survey picks out systems of galaxies without any reference to the lensing map. In other words the group finding algorithm applied to a redshift survey and the lensing map are independent methods of identifying mass condensations in the same region of the universe. The solid points in Figure 3 shows that the group-finding algorithm easily identifies the massive cluster, A781.

Figure 3: Positions of 3015 galaxies with measured redshifts in the field of Figure 2. The solid dots are galaxies in the cluster A781 identified by an objective group-finding algorithm (Massimo Ramella, Trieste Observa- tory).

The lensing map identifies about fifty individual mass peaks. For five mass peaks at a similar redshift z N 0.3, Figure 4 shows the correlation between the line-of-sight velocity dispersion and the lensing mass. The square of velocity dispersion of the system is a proxy for the mass. Figure

25

4 shows that, as expected, the lensing mass is roughly proportional to the square of velocity dispersion. This comparison of systems spanning an order of magnitude in mass is the first of its kind. Combined with photometry of the galaxies, these studies will yield mass-to-light ratios over a large range of scales. A measurement of mass-to-light ratio as a function of mass for systems at moderate redshift will test the role of galaxy and cluster evolution in determining the relationship.

1 0 0 ' ' ' ' 1 1 I 10" 10"

solar masses

Figure 4: Relationship between the line-of-sight velocity dispersion for systems in the Hectospec redshift survey and the lensing mass for four systems at a redshift of - 0.3 (M. Kurtz, SAO)

Although the DLS-Hectosurvey is in its early stages, other applications of weak lensing to the determination of the cosmological mean mass density have produced results which range between the dynamical measures in sections 2 and 3 and the WMAP results (Spergel et al. 2003; Tegmark et al. 2004; Rines et al. 2004).

5. Conclusion

A variety of techniques anchored in optical and infra-red observations of systems of galaxies indicate that there are serious gaps in our understanding of the relative distribution of dark and light-emitting matter in the

26

universe. These puzzles may be resolved by a better understanding of the processes of galaxy formation and evolution, but they may also indicate deeper inconsistencies in our understanding of structure formation and the nature of the dark matter.

One issue is that the mean mass density derived from optical and infrared estimates appears to be significantly lower than the results derived from CMB measurements coupled with redshift surveys, Another is that the best measurements on intermediate spatial scales from 1.5-10 Mpc indicate that the dark matter is more concentrated that the light. The use of wide-field weak lensing maps coupled with dense redshift surveys offers prospects for progress on the observational front.

Acknowledgments

I have enjoyed working with Ian Dell’Antonio, Dan Fabricant, Michael Kurtz, Massimo Ramella and Ken Rines on the projects described here. I thank them for their support, skill, and scientific insight. My research is supported by the Smithsonian Institution.

References 1. G. Abell, Ap.J . Suppl., 3, 211 (1958). 2. A. Diaferio, MNRAS, 309, 610 (1999). 3. D.G. Fabricant, E.N. Hertz, A.H. Szentgyorgyi, R.G. Fata, J.B. Roll, and J.M.

Zajac, SPIE, 3355, 285 (1998) 4. Y. Ikebe, Y., T.H. Reiprich, H. Bhringer, Y. Tanaka, Y., and T. Kitayama,

A&A, 383, 7731 (2002). 5. Y. Lin, J.J. Mohr, and S.A. Stanford, ApJ, 591, 749 (2003). 6. Y. Lin, J.J. Mohr, and S.A. Stanford, ApJ, 610, 745 (2004). 7. A. Pisani, M. Ramella, and M.J. Geller, AJ , 126, 1677 (2003). 8. M.Ramella, W. Boschin, M.J. Geller, A. Mahdavi, and K.Rines, AJ , in press

(2004). 9. M. Ramella, A. Pisani, and M.J. Geller, A.J., 113, 483 (1997). 10. K. Rines, M.J. Geller, A. Diaferio, M.J. Kurtz, and T.H. Jarrett, A.J. , 128,

1078 (2004). 11. K. Rines, M.J. Geller, M.J. Kurtz, and A. Diaferio, AJ , 126, 2152 (2003). 12. D. Spergel et al., A.J.Suppl., 148, 175 (2003). 13. M. Tegmark et al., Phys. Rev. D, 65, 3501 (2004). 14. D. Wittman, SPIE, 4836, 33 (2002). 15. Zwicky, F. Helvetica Physica Acta, 6 , 110 (1933).

COSMIC ACCELERATION AND MODIFIED GRAVITY MODELS *

DAMIEN A. EASSON~ Department of Physics, Syracwe University

201 Physics Building, Syracuse, N Y 132&-1130, USA E-mail: [email protected]

There is now overwhelming observational evidence that our Universe is accelerating in its expansion. In these notes we discuss how modified gravitational theories can provide an explanation for this observed late-time cosmic acceleration. Specific low- curvature corrections to the Einstein-Hilbert action are examined. Many models generically contain unstable de Sitter solutions as well as late time accelerating attractor solutions.

1. Introduction

It appears that the universe is accelerating in its expansion ’. One possible explanation for this late-time acceleration arises in low-energy modifications of General Relativity 273. Modifications of GR have been used to obtain early-time inflation (for example, “Starobinsky inflation” *) and to try to eliminate curvature singularities in cosmological and black hole spacetimes 5. Unlike these high curvature examples, in order to explain late-time cosmic acceleration, we are interested in modifications that become important at low curvatures.

1.1. The CDTT Model

The first examples of low-curvature modifications where presented by Car- roll, Duwuri, Trodden and Turner (CDTT) and involved simple inverse- powers of the Ricci Scalar. As a toy model consider the modification,

*Talk presented at PASCOS 2004, Northeastern University, August 16-22. +Work partially supported by NSF-PHY-0094122 and funds from Syracuse University.

27

28

O . l4 0.12.

0.1.1

0.08.j

0.064

Here p is a new scale in the theory with dimensions of mass. Assuming a flat, F'riedmann-Robertson-Walker (FRW) metric:

ds2 = -dt2 + a2( t )dx2 , (1 .2)

the modified Friedmann equation (in terms of the Hubble parameter H = u/a) is

' r -',.,,~~.\\

I 1

'..\ '.,,

3H2 - " ( 2 H H + 15H2H + 2H2 + 6H4 12(H + 2 ~ 9 3

0.04'

0.02.

where pm is the matter energy density and we have assumed a perfect-fluid energy-momentum tensor

T p u = ( ~ m + Pm)upuv + P m g p u . (1 .4)

It is a trivial task to map this theory to an Einstein frame with Einstein- Hilbert action minimally coupled to a scalar field cp with potential

'-.., . '\.

This potential is plotted in Fig. ( 1 ) . From the plot of V(cp) it is clear

Figure 1. Plot of the potential V(9) in Eq. (1.5).

that this model exhibits three different possible cosmological behaviors, depending on the initial conditions cpi and cp: (where the prime denotes differentiation with respect to the cosmic time coordinate of the Einstein frame). The field cp can either roll back down the potential toward a future singularity at cp = 0, come to rest at the top of the potential (corresponding

29

to an unstable de Sitter phase) or roll off to infinity leading to late-time, power-law cosmic acceleration with equation of state parameter weff = -213. These basic features are not affected by the addition of matter. By choosing p N 10-33eV, the modifications to Einstein gravity only become important recently making this theory a candidate to explain the observed acceleration of the Universe.

This simple model is easily generalized to corrections of the form -P’(~+’)/R~. These generalizations maintain the desirable late-time accelerating solutions with behavior analogous to dark energy with equation of state parameter

2 ( n + 2) weff = -’ + 3(2n + 1)(n + 1) *

It is, therefore, a simple matter to construct models of this type that obey the observational constraints on the equation of state parameter -1.45 < w < -.74 (at the 95% confidence level) 6.

2. New Modified Gravity Models

We now discuss a few simple examples of more general modified gravitational models ’. Consider the action

A generic feature of the models we discuss here is an unstable de Sitter solution. Due to time constraints we focus on one new modified gravity example here.

2.1. Inverse powers of P R,,RP”

Consider a modification of the form f(P) = -p6/P. The unstable de Sitter solution occurs for RIP,’ = (16)1/3p2. The modified Friedmann equation

30

takes the form

(24H2H6 + 216H4H5 + p 6 H 4 + 864H6H4 1

8(3H4 + 3H2H + fi2)3 + l lp6H2h3 + 1944H8H3 + 2p6Hfi2fi + 33p6H4H2 + 2592Hl0H2 + 30p6H6H + 6p6H3HH + l944Hl2H + 6p6Hs + 648H14 + 4p6H5H) = 0 . (2.9)

and power-law attractors are identified by substituting a power-law ansatz H(t) = p / t and taking the late-time limit. By defining TJ = -&/H and requiring TJ to be a constant TJO = p we find the condition

62104 - 30~: + 41~: - 2 3 ~ 0 + 5 = 0 . (2.10)

which gives two late-time power-law attractors with p = 2 f &/2. One of these corresponds to an accelerator with p N 3.22, while the other is not an accelerating solution with p N .77. Both attractor solutions along with the de Sitter solution are easily identified in the phase space portrait given in Fig. (2).

-0.E - 0 . 6 - 0 . 4 - 0 . 2

H

- dH/dt

Figure 2. Phase portrait for f(R) cx l/R,,,RP'. The unstable de Sitter solution is at the point (0 , l ) . The two late-time power-law attractors are clearly visible for p N .77 (dashed lines) and p 21 3.22 (solid lines).

31

3. Conclusions

We have shown that a late-time period of cosmic acceleration emerges naturally in certain modified gravitational theories. The theories described in

involve inverse powers of linear combinations of general curvature invariants. Clearly, the modified gravitational theories presented in these notes are only a small handful of a much richer set of possible theories. It is an intriguing possibility that the observed cosmic acceleration of our universe may be a consequence of such modified gravitational theories.

Acknowledgments

I am very grateful to my collaborators on the work presented here, S. M. Carroll, A. De Felice, V. Duwuri, M. Trodden and M. S. Turner. I thank Northeastern University for their hospitality during PASCOS 2004. This work is supported in part by the National Science Foundation under grant PHY-0094122 and by funds from Syracuse University.

References 1. A. G. Riess et al. Astron. J. 116, 1009 (1998); S. Perlmutter et al. Astrophys. J.

517, 565 (1999); J. L. Tonry et al., Astrophys. J. 594, 1 (2003); C. L. Bennett et al., Astrophys. J. Suppl. 148, 1 (2003); C. B. Netterfield et al. Astrophys. J. 571, 604 (2002); N. W. Halverson et al., Astrophys. J. 568, 38 (2002).

2. C. Deffayet, G. R. Dvali and G. Gabadadze, Phys. Rev. D 65, 044023 (2002); K. Freese and M. Lewis, Phys. Lett. B 540, 1 (2002); M. Ahmed, S. Dodelson, P. B. Greene and R. Sorkin, Phys. Rev. D 69, 103523 (2004); N. Arkani-Hamed, S. Dimopoulos, G. Dvali and G. Gabadadze, arXiv:hepth/0209227; G. Dvali and M. S. Turner, arXiv:astro-ph/O301510; G. Dvali, arXiv:hepth/0402130; G. All* mandi, A. Borowiec and M. Francaviglia, arXiv:hepth/0407090.

3. S. M. Carroll, V. Duwuri, M. Trodden and M. S. Turner, arXiv:astr~+ph/0306438. 4. A. A. Starobinsky, Phys. Lett. B 91, 99 (1980). 5. V. P. Frolov, M. A. Markov and V. F. Mukhanov, Phys. Lett. B 216, 272 (1989).

V. P. Frolov, M. A. Markov and V. F. Mukhanov, Phys. Rev. D 41, 383 (1990). V. Mukhanov and R. H. Brandenberger, Phys. Rev. Lett. 68, 1969 (1992); M. Trod- den, V. F. Mukhanov and R. H. Brandenberger, Phys. Lett. B 316, 483 (1993); R. H. Brandenberger, V. Mukhanov and A. Sornborger, Phys. Rev. D 48, 1629 (1993); R. H. Brandenberger, R. Easther and J. Maia, JHEP 9808, 007 (1998); D. A. Easson and R. H. Brandenberger, JHEP 9909, 003 (1999); D. A. Easson and R. H. Brandenberger, JHEP 0106,024 (2001); D. A. Easson, JHEP 0302,037 (2003); D. A. Easson, Phys. Rev. D 68, 043514 (2003).

6. A. Melchiorri, L. Mersini, C. J. Odman and M. Trodden, Phys. Rev. D 68, 043509 (2003) [arXiv:astro-ph/0211522]; D. N. Spergel et al., Astrophys. J. Suppl. 148, 175 (2003).

7. S. M. Carroll, A. De Felice, V. Duwuri, D. A. Easson, M. Trodden and M. S. Turner, to appear on arXiv.

CAN WE HAVE A STRINGY ORIGIN BEHIND nA(T) OC nM(T)?

TIRTHABIR BISWAS*AND ANUPAM MAZUMDARt Centre for High Energy Physics

Physics Department, McGill University 3600, University Street, Montreal H3A 2T8, Qutbec, Canada

Inspired by the current observations that the ratio of the abundance of dark energy RA, and the matter density, Rm, is such that Rm/RA N 0.37, we provide a string inspired phenomenological model where we explain this order one ratio, the smallness of the cosmological constant, and also the recent cosmic acceleration. We observe that any effective theory motivated by a higher dimensional physics provides radion/dilaton couplings to the standard model and the dark matter component with different strengths. Provided radion/dilaton is a dynamical field we show that n A ( t ) tracks Rm(t) and dominates very recently.

The cosmological constant problem is one of the most difficult problems of theoretical physics. The observed energy content of the Universe, N 4 x (GeV)4, is one hundred and twenty orders of magnitude smaller than the theoretical prediction, M; N (GeV)4, for the cosmological constant energy density alone (supersymmetry only ameliorates the mismatch). Thus the puzzle is “why the bare cosmological constant is so small and moreover, stable under quantum corrections’’ l. There is also a kind of a coincidence problem, sometimes dubbed as the why now problem. Re- cent observations from supernova ’ and the cosmic microwave background (CMB) anisotropy measurements suggest that the majority of the energy density - 70% is in the form of dark energy, whose constituent is largely unknown, but usually believed to be the cosmological constant with an equation of state w = -1, which is also responsible for the current acceleration. In this respect not only the cosmological constant is small, but it is very close to today’s matter/radiation density! A related question would

* tirthoOhep.physics.mcgil1.ca t anupammOnordita.dk

32

33

be why the cosmological constant is dominating right now and for example not during the big bang nucleosynthesis (BBN)?

Both the smallness and the why now questions can be answered in part if we believe that the physics of the dark energy is somehow related to the rest of the energy density of our Universe. Attempts have been made to construct such tracking mechanisms in single and multi fields, such as dynamical quintessence 4 , k-essence type models, non-Abelian vacuum structure with non-vanishing winding modes 6 , modified F’riedmann equation at late times 7, or large distances *, rolling dilaton with dilatonic dark matter 9, or due to inflationary back-reaction lo.

In this paper we propose a simple model which can arise naturally from compactifications of higher dimensional theories. For simplicity we assume that all the moduli/dilaton fields have been stabilized at higher scales except a single linear combination, &, which has a runaway potential, see ll. In an effective four dimensional theory the Q field will couple to the SM degrees of freedom as well as the dark matter. We will show that by virtue of this coupling to the matter fields “the effective potential” for Q can have a local minimum which always tracks the matter density of the Universe. Further observe that the Q field is expected to have different couplings to different species of fermions (f), scalars (s) and gauge bosons (r) by virtue of its running below the string compactification scale 12. Depending on the strength of this coupling our universe will either accelerate or not.

Let us imagine that our world was originally higher dimensional, such that the three spatial dimensions, along with the origin of the SM are all due to some interesting compactification of the extra spatial dimensions. After dimensional reduction, in the Einstein frame we typically obtain a generic scalar-tensor action l3 of the form S = Jd4z fi[L,, + Cfemions + Cgauge + Lscalars] , where

Keeping in mind that in M-theory/supergravity compactifications one often encounters exponential potentials l4 (from fluxes, curvature of internal manifold, etc.) and couplings for the various modulii/dilaton 15, we assume

(2) V ( Q ) = h e - 2 P Q and &(Q) = e2piQ

34

where i = {f , r, s}. Observe, due to the couplings, the various masses - mie2p'Q and fine structure constants - a0e-2prQ now depend on Q. Observational data on variation of masses and fine structure constants, as well fifth force experiments constraint the p's for SM particles to be very small < - l 3 , l 6 , and hence forth we will assume this.

The dark energy density (d) is identified as

ih = Q 2 / 2 + V ( Q ) , (3)

although Q effectively obtains an additional contribution to its potential via its coupling to matter:

V&(Q) = &e-28Q + e2p'Qpi, (4) i=s,f

The "bare density", pi e-2piQ&, is defined such that it doesn't depend on Q; here Fi is the observed energy density of the ith component. Since, F2 - E2 - B 2 , which on an average is zero, radiation from gauge bosons do not contribute to an effective potential for Q. Given all the components, we can write down the F'riedmann equation for a Robertson Walker metric,

The evolution equations for Q, pi read as,

Q + 3HQ = -2(B&e-28Q + C pie2'"'pi) , pi + 3H(pi + p i ) = 0 . (6)

with an equation of state, pi = wipi. From Eq. (6 ) we obtain the standard result pi = poi (a/a0)-3(1+wi).

We will first work within the approximation that there are only two dominating components in the Universe, the dark energy determined by Q and the matter component. Observe, if p's and B have the same sign, then the effective potential for the Q field, see Eqs. (4), provides a dynamical stabilizing mechanism because the two overall exponents differ in signs. In this case the Q field will track the minimum of the potential Eq. (4) as it slowly shifts due to the redshifting of the matter contributions.

In this adiabatic approximation, which is satisfied if p ( p i + a) > 3(1+ 4 ) / 4 (this is a generalization of the result derived in l7 for pi = 0), the evolution of Q can be obtained in terms p.

i=s,f

35

and also solve the Hubble equation (5) for the scale factor:

In a matter (m) dominated epoch with w, = 0, then we obtain an interesting relationship in order to have an accelerated expansion pm/p > 1/2.

Consider now the current cold dark matter (c) dominated universe: R, - 0.27, gives us via Eq. (8), pc/p - 2.7, indicating an accelerated expansion. In other words p c / p - 2.7 explains the small value of & - 2.7jTc - 10-120M,4, or equivalently, Rc/fid N 0.37. One can also compute the equation of state for the dark energy using (5,7,8):

Indeed for relevant values of p, p, this will be very close to -1. In particular for wc = 0, pc//3 - 2.7 and p > 1 (from adiabaticity condition), we have WQ < -0.93, in remarkable agreement with the current observations 2!

During the radiation epoch, the evolution equations will be slightly complicated as one now has three relevant components. (6) yields

Thus Q tracks the minimum formed between V ( Q ) and matter-potential exactly as before, so that the dark energy always tracks the matter density & = (p/B)&. From (11) as long as pr > (1 +p/p)&, we are in a radiation era, after which point the universe enters a matter dominated phase.

From the above analysis it may appear that the moment we enter the matter dominated era, the Universe starts to accelerate, but this is not correct. The baryonic (b) and cold dark matter (CDM) redshift differently since p, >> pb indicating that baryons, pb, were dominating over the CDM not so long ago. In fact we can compute approximately when the CDM began to dominate over the baryons, which will also roughly correspond to the beginning of an accelerated expansion, 1 +.Zacc = (~c0 /Rbo) ' (Pc+P) /3Pc1 . Taking p,/p - 2.7, as obtained above, we find t,,, M 1.8! Note that the above interesting result is determined solely by the ratios of Rdo/flco and

However in this setup we end up with more baryons than CDM at early epochs, say during the CMB formation. This issue can be addressed by realizing that one can have two competing dark matter candidates; one with &, weakly coupled to Q , say pw - pb, and the other & strongly coupled to

R b o /ace.

36

dark energy with p, as before. In this scenario the “why now” problem is answered by simply postulating that today is the epoch when the strongly coupled dark matter is catching up with the weakly coupled matter, jjm (= P,,, + h). Note, since pv = pb, ?iy redshifts according to the baryons, and hence maintains a constant ratio with &, throughout the evolution until very recently when the ratio increases slightly due to the emergence of &. To estimate how much this ratio changes and its relationship with the acceleration epoch, one again has to look at the three component evolution, where in (11) r + c and ( 5 ) now becomes

-2pPd f 2pmPm f 2pcPc = 0 (12) Notice, since p, >> pm in (12) the third term starts to dominate over the second long before PC - P m 7 so that the dark energy starts to track jj, long before today. From (11,r + c) it is then clear that the universe enters the acceleration phase approximately when p , + fid = (1 + pc//3)& becomes equal to &, so that

(13) [ ( ~ . + 0 ) / 3 ~ ~ 1 1 + zacc = ((1 + ~ c / D ) f l c o / % n o )

Now, since a photon from CMB doesn’t for the most part see j jc , we obtain from WMAP analysis &,,/Rm, = 0.17. However, the current baryon- dark matter ratio and hence the current baryonic abundance is sensitive to fl,,/flm, which depends on zaccel and pc/p. For z,,, = 0.5, pc f p - 3 gives flboh2 = 0.014 while pc /B N 9 gives fleoh2 = 0.017. These are to be contrasted with the CMB estimate of fib, h2 = 0.0224. Thus our model might be able to explain why the baryonic abundances coming from BBN considerations, 0.009 < flboh2 < 0.0223 18, based on current observational data tend to give slightly lower values than CMB. BBN constraints also clearly ‘imply that z,,, has to be quite recent. For example, for z,,, = 1, pc/D - 3 yields fib, h2 = 0.01 which is just within the BBN range.

Thus we find that in the set up where only one dark matter component is strongly coupled to dark energy one is able to avoid observational constraints coming from fifth force experiments and variation of physical constants, but on the other hand can explain why the dark energy density is (and has been) close to the matter density as well as why its observed equation of state is close to -1. The coincidence problem is explained by assuming approximate equality of the strongly coupled dark matter to the weakly coupled matter in the near future (the universe starts to accelerate a little before that). This mechanism thus provides an alternative to the usual quintessence models which has to invoke an unnaturally small mass scale N mezl to explain why the universe accelerates recently.

37

Several interesting questions remain, such as the origin of the two types of CDM and their role in galaxy formation. It is also important to go beyond the two component approximation and perform a numerical simulation to determine the dynamics exactly. These questions we leave for future investigation.

This work is supported in part by the NSERC. We would like to thank Luca Amendola, Guy Moore and Andrew Liddle for discussions.

References 1. S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). 2. S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astro-

phys. J. 517, 565 (1999). A. G. Riess et al. [Supernova Search Team Collab- oration], -4stron. J. 116, 1009 (1998) A. G. Riess et al. [Supernova Search Team Collaboration], [arXiv:astro-ph/0402512].

3. D. N. Spergel et al., Astrophys. J. Suppl. 148, 175 (2003). 4. C. Wetterich, Nucl. Phys. B 302, 668 (1988); R. R. Caldwell, R. Dave, and

P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998). 5. C. Armendariz-Picon, T. Damour, and V. Mukhanov, Phys. lett. B 458, 209

6. P. Jaikumar, and A. Mazumdar, Phys. Rev. lett. 90, 191301 (2003). 7. K. Freese, and M. Lewis, Phys. lett. B 450, 1 (2002). 8. G. Dvali, G. Gabadadze, and M. Porrati, Phys. Rev. D 63, 065007 (2001). 9. L. Amendola, Phys. Rev. D 62, 043511 (2000); L. Amendola, and C. Quer-

cellini, Phys. Rev. D 68, 023514 (2003); L. Amendola, and D. T-valentini, Phys. Rev. D 64, 043509 (2001); L. Amendola, M. gasperini, and F. Piazza, astro-ph/0407573.

10. R. H. Brandenberger, arXiv:hep-th/0210165; R. H. Brandenberger, arXiv:hep-th/0004016; R. H. Brandenberger, and A. Mazumdar, arXiv:hep- th/0402205.

11. M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory. Vol. 2: Cambridge University Press, Cambridge (1987).

12. T. Damour, and A. M. Polyakov, Nucl. Phys. B 423, 532 (1994). 13. C. M. Will, Living Rev. Rel. 4, 4 (2001). 14. M. Bremer, M. J. Duff, H. Lu, C.N. Pope and K.S. Stelle Nucl. Phys. B 543

(1999) 321, hep-th/9807051; T. Biswas, and P. Jaikumar, Phys. Rev. D 70 (2004) 044011, hep-th/0310172; JHEP 0408 (2004) 053, hep-th/0407063

15. T. Applequist, A. Chodos, and P.G.O. Freund Modern Kalwa-Klein Theo- ries Addison-Wesley Publishing Co. Inc. (1987); M.J. Duff, C.N. Pope and B.E.W. Nilsson Phys. Rep. 130,l (1986); A. Frey, and A. Mazumdar, Phys. Rev. D 67, 046006 (2003).

16. C. J. Martins, et. al., astro-ph/0302295. 17. D. Wands, E.J. copeland, and A.R. Liddle Phys. Rev. D 57, 4686 (1998). 18. S. Sarkar, astro-ph/0205116.

(1999).

PLANCK SCALE EFFECTS AND THE SUPPRESSION OF POWER ON THE LARGE SCALES IN THE PRIMORDIAL

SPECTRUM

S. SHANKARANARAYANAN HEP Group, ICTP, Strada costiem 11, 34100 Trieste, Italy.

E-mail: shankiOictp.trieste.it

L. SRIRAMKUMAR Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad, India.

E-mail: sriramQmri. ernet.in

The enormous red-shifting of the modes during the inflationary epoch suggests that physics at the very high energy scales may modify the primordial perturbation spectrum. Therefore, the measurements of the anisotropies in the Cosmic Microwave Background (CMB) could provide us with clues to understanding physics beyond the Planck scale. In this proceeding, we study the Planck scale effects on the primordial spectrum in the power-law inflation using a model which preserves local Lorentz invariance. While our model reproduces the standard spectrum on small scales, it naturally predicts a suppression of power on the large scales-a feature that seems to be necessary to explain deficit of power in the lower multipoles of the CMB.

1. Introduction

In the inflationary scenario1, the perturbations corresponding to comoving length scales of cosmological interest today would have emerged from quantum fluctuations at the beginning of inflation with physical wavelengths smaller than the Planck length. Hence, in principle, quantum gravitational effects should have left their signatures on the primordial spectrum. This opens up the interesting possibility of probing trans-Planckian (TP) physics using the CMB ’.

The first year results of WMAP3 data show that the power in the quadrupole arid the octopole moments of the CMB are lower than as expected in the best fitting ACDM models. The deficit of power in the lower rnultipoles can not be explained within the context of the standard inflationary models (unless these models are fine-tuned4) and suggests a possible

38

39

signature of TP physics. Most of the earlier efforts in incorporating TP physics into the standard

field theory have involved models which break local Lorentz invariance2. However, theoretically, there exists no apriori reason to believe that Lorentz invariance may be broken at the scales of inflation. More importantly, recent observations of synchrotron emission from the Crab nebula seem to suggest that Lorentz invariance may be preserved to very high energies5. In such a situation, in order to study the TP effects on the primordial perturbation spectrum, it becomes important that we also consider models which preserve Lorentz invariance even as they contain a fundamental scale. In this proceeding, we consider one such model, evaluate the resulting spectrum of density perturbations in power-law inflation and also discuss its implications for the CMB angular power spectrum.

2. The model and its application to power-law inflation

In the inflationary scenario, the primordial perturbation spectrum per logarithmic interval, viz. [k3 PQ (k)] , is given by’

r d ( l n k ) [k3 P Q ( ~ ) ] = G t ( 2 , 2 ) , (1) 0

where G:(2,2’) denotes the Wightman function corresponding to a massless and minimally coupled, quantum scalar field (say, $) evolving in the inflating background and the spectrum is to be evaluated at Hubble exit. Therefore, in order to understand the effects of Planck-scale physics on the perturbation spectrum, we need to understand as to how quantum gravitational effects will modify the propagator of a massless scalar field in the inflationary background.

Motivated by the Pauli-Villars regularization procedure, we assume that the massless Wightman function, viz. G: (2 ,2’) , is modified due to TP effects (in a locally Lorentz invariant manner) to7

where GLc (2 ,2’ ) is the Wightman function of a massive scalar field of mass k, and k, denotes the cut-off scale which we shall assume to be three to five orders of magnitude above the Hubble scale during inflation. Then, following Eq. (l), we can define the resulting modified perturbation spectrum,

40

viz. [ k 3 P , ( I ~ ) ] ~ , as follows:

d(lI1Ic) [ k 3 Pq(k)], = G& ( 2 , 2 ) = G: (Z,2) - GLc ( 2 , Z ) . (3)

Let us now consider the massless and massive scalar fields to be propagating in a power-law inflationary background described by the line element

7 0

ds2 = a2(q) (dq2 - d x 2 ) , (4)

where q is the conformal time, a(q) = (-Nq)(p+l) with /3 5 -2 and 3-1 denotes the energy scale associated with inflation. If we assume that both the fields are in the Bunch-Davies vacuum, then the modified power spectrum [ Ic3 Pa (Ic)] can be expressed as

where p k and ,iik denote the normal modes of the massless and the massive fields satisfying the differential equations

pz + [ k 2 - aN/a] pk = 0,

@; + [k2 + (k, a)2 - a“/a] P k = 0, (6) (7)

respectively. On comparing Eqs. (1) and (5), it is clear that, in our model, the TP corrections to the standard spectrum arise as a result of the contribution due to the massive modes. Before we proceed further with the evaluation of the corrections, we need to stress the following point: In the standard inflationary scenario, it is well-known that the amplitude of the spectrum corresponding to the massive modes decays at super-Hubble scales. In our model, the TP corrections to the standard spectrum are due to the massive modes. Hence, within the standard inflationary picture, the amplitude of these corrections would be expected to decay at super-Hubble scales. However, as the massive modes we have considered are supposed to represent TP corrections to the standard massless modes, in what follows, we shall assume that the mechanism that ‘freezes’ the amplitude of the standard spectrum at super-Hubble scales will also ‘freeze’ the amplitude of the corrections at their value at Hubble exit.

The mode functions for the massless field in power-law inflation can be expressed in terms of Hankel functions and the standard power-spectrum, evaluated at Hubble exit, is given by’

41

where C is a constant of order unity. Unlike the massless case, the exact solution to the massive modes i i k is not known in power-law inflation. How- ever, it can be shown that, for k, >> 7-l, the WKB solutions are valid for all (kq) over a range of values of p and k of our interest7. On using the WKB solutions, we obtain the modified power-spectrum (5) at Hubble exit to be

where C is another constant order unity.

(i) Fig. (la) contains the plots of the modified and the standard spectrum. It is evident from the figure that the modified spectrum exhibits a suppression of power at the large length scales while it remains scale invariant at the small length scales. (ii) Naively, one would expect that TP effects will leave their imprints only at the ultra-violet end of the spectrum. However, we find that TP effects lead to a modification of the spectrum at the infra-red end. This can be attributed to the fact that the longer wavelength modes leave the Hubble radius at earlier epochs thereby carrying the signatures of the TP effects. (iii) The modified spectrum (9) has some similarities to the power spectrum that has been obtained recently in non-commutative inflation6. (iv) Though the modified spectrum we have obtained exhibits a suppression of power around the expected values of k, the extent of the suppression is far less than that is required to fit the CMB observations. In order to illustrate this feature, in Fig. ( lb), we have plotted the relative power spectrum (i.e. the ratio of the modified spectrum to the standard spectrum) of our model and the fit to the WMAP data proposed by Contaldi et al.a.

The following points are note-worthy regarding the above result:

3. Discussion

In this proceeding! we have studied the TP effects on the spectrum of primordial perturbations in power-law inflation using an approach that preserves local Lorentz invariance. We assumed that the TP effects modify the standard propagator in a particular manner. We find that the resulting modified spectrum remains scale invariant at the ultra-violet end, but,

~

Vontaldi et aL4 proposed the following form for the primordial spectrum:

[k3 P*(k)lCpKL = A, k ( n s - l ) l-exp- ( k / / ~ , ) ~ , where A, and ny are the amplitude [ 1 and index of the standard spectrum, k, = 5 x Mpc-' and y ~r 3.35

42

-55 -50 45 -35 -30 -25

(b)

Figure 1. (a) Plots of the standard power spectrum (solid curve) and the modified powerspectrum (dashed curve), normalized to Ti? for /3 = -2.04. (b) Plots of the relative powerof our model (dashed curve) and Contaldi et al.'s spectrum (solid curve). In both theseplots, the dashed curve corresponds to (H/kc) = 1Q~3—value chosen so that the WKBapproximation is valid around k ~ 10~4 Mpc"1—and H = 1014 GeV = 10s2 Mpc"1.

interestingly, it exhibits a suppression of power at the infra-red end—a fea-ture that seems to be necessary to account for the deficit of power in thelower multipoles of the CMB3'4. However, the amount of suppression pre-dicted by our model in power-law inflation turns out to be far less thanthat seems to be required to fit the WMAP data. Nevertheless, the loss ofpower at large scales suggests that the power spectrum we have obtainedmay fit the WMAP data better than the standard ACDM model. It willbe interesting to analyze the implications of our model for WMAP data inthe context of slow-roll inflation.

References

1. A. R. Liddle and D. H. Lyth. Cosmological inflation and large-scale structure.Cambridge University Press, England, 1999.

2. R. Brandenberger, J. Martin, Mod. Phys. Lett. A, 16:999, 2001; J. Martin,R. Brandenberger, Phys. Rev. D, 63:123501, 2001; J. C. Niemeyer, Phys.Rev. D, 63:123502, 2001; N. Kaloper et al. Phys. Rev. D, 66:123510, 2002;S. Shankaranarayanan, Class. Quant. Grav., 20:75, 2003; U. H. Danielsson,Phys. Rev. D, 66:023511, 2002.

3. H. V. Peiris et al. Astrophys. J. Suppl., 148:213, 2003.4. C. Contaldi, M. Peloso, L. Kofman, and A. Linde, JCAP, 0307:002, 2003.5. T. Jacobson, S. Liberati, and D. Mattingly. Nature, 424:1019, 2003.6. S. Tsujikawa, R. Maartens, R. Brandenberger, Phys. Lett. B, 574:141, 2003.7. S. Shankaranarayanan and L. Sriramkumar. hep-th/0403236.

CLASSICAL GHOSTS AND THE COSMOLOGICAL CONSTANT*

T. HIRAYAMA~ Department of Physics, University of Toronto,

Toronto, Ontario, M5SlA7, Canada E-mail: [email protected]

For a large region of parameter space involving the cosmological constant and mass parameters, we discuss spacetime solutions that are effectively Minkowskian on large time and distance scales. A negative energy fluid is involved, resulting in a scale factor oscillating about a constant, with a frequency determined by the size of a negative cosmological constant. Gravity itself induces a coupling between the ghost-like and normal fields, and we find that this results in stochastic rather than unstable behavior. Ghosts could also allow for the existence of Lorentz invariant fluctuating solutions of finite energy density.

One of the cosmological constant problem is that why the cosmological constant is so tiny (equivalently why the spacetime is so flat) compared with other physical scales, electroweak, SUSY breaking or Planck scale. General Relativity does not provide any dynamics or physical reasons why the cosmological constant is tiny. And as Weinberg discussed2, the zeroness of the cosmological constant is an additional constraint so that we generally need a fine tuning to make it zero. Therefore we might not expect this problem is solved within the content of general relativity and a quantum field theory. Thus in this talk we would like consider the following possibility: a high energy dynamics is responsible for the cosmological constant problem. That is, supposing the scale of the cosmological constant is around the Planck scale, the Planck scale cosmological constant induces Planck scale dynamics, generating rapid fluctuations around flat space in the metric. These rapid fluctuations are then effectively hidden from a low

'Based on work with Bob Holdom [I].

of Canada. Work partially supported by the National Sciences and Engineering Research Council

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energy observer and the spacetime averaged over the fluctuations appears to be Minkowskian.

In order to discuss this possibility, we need to model Planck scale dynamics. At around the Planck scale a gravity action contains higher derivative terms such as R2 and (R,,)2 with order one coefficients in Planck unit. These higher derivative terms induce new degrees of freedom3 which are a massive graviton and massive scalar field. (For special coefficients one or both new degrees of freedom have infinite masses and disappear.) One of the universal features is that the kinetic term of massive graviton is negative, i.e. ghostlike particle. Thus we can use the following action for describing Planck scale dynamics,

where 4, (4,) is a ghostlike (normal) particle with mass m, (mg). We can calculate Einstein’s equations of this action and obtain a classical solution. Here we further simplify the equations of motion by introducing the equation of state (w,, w,). This could be a dangerous simplification since we treat the negative energy mode as a decoupled fluid and we know a ghost system often shows instability because of interaction between negative modes and positive energy normal modes. I will discuss the stability problem Later and show this system is stable.

A solution after simplification is

A(t) = 1 + (1 - cos(wt))~ + O ( E ~ ) , w = /-, (2)

pg( t ) = - ~ , A ( t ) - ~ ( ’ + ~ g ) / ~ , p n ( t ) = ~ , A ( t ) - ~ ( ’ + ~ g ) / ~ , (3) 3 1+Wn 2 = 1 + - ( 1 + w,)E, T: = -A

T9” 2 wg-wn’ (4)

( 5 )

where A(t) is the scale factor in FRW metric and pg (p,) is the energy density of a ghostlike (normal) particle. What is important here is that for a general initial value of the energy densities, i.e. E # 0, the cosmological constant induces a fluctuation to the scale factor and the spacetime becomes flat for a low energy observer when the cosmological constant is negative and wg > w,. This solution is quite counter-intuitive, since an infrared property, their effective flatness, shows up in a high energy dynamics.

Without any fine-tuning we have obtained the metric what we wanted, but we need to answer some questions. Those are (1) stability question

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and (2) what is the low energy effective theory? The first question arises since we are discussing a ghost system which is believed to be an unstable system. The second question comes from that there is no nontrivial finite energy Lorentz invariant excitation in a system with normal particles.

We first discuss the first question. There are two kinds of processes which may cause instability. One is quantum processes and the other is classical processes. If gravity is the only one which mediates interaction between a ghost and a normal particle, then the quantum process due to virtual graviton exchange4 will be the most dangerous graph. But if the ghost mass is of order the Planck mass then the virtual graviton is far offshell in the deep Planckian region, and there is no guarantee that the perturbative quantum description holds. We assume that it does not.

The classical stability can be studied numerically and we have shown that it is classically stable1. We shortly describe key reasons. Let us consider plane waves with spatial momentum p, and pg respectively, i.e. 4, = o(t) cos(&t + pn . x + 6,) and dg = ~ ( t ) cos(p:t + p, . x + 0,). We look at the instability that occurs for equal masses m, = m, = m between momentum modes that are in resonance p i = p:. While in resonance, the amplitudes a(t) and ~ ( t ) have an exponentially increasing component and exponentially decreasing component. The ratio of two components depends on the relative phase 0, - 0, and the amplitudes finally blow up as long as they contain an exponentially increasing component. However this situation changes in our case. We are thinking the unequal mass case (m, > m, because of wg > w,) and gravity influences the resonance condition pi + mi = pi + m i . The physical 3-momenta of two modes are dependent on the scale factor, so that in terms of the comoving wavevector of each mode we have p; + pi/A2(t) and pi + pi/A2(t). Thus a changing scale factor causes two modes (p,, p,) that were in resonance to fall out of resonance and so the pair (p,, p,) which is in resonance is continuously changing in time. Because of this and the fact that the phases 0, and 0, are randomly distributed, the system does not show instability, but develops a stochastic behavior. Figure 1 is an example in which we have included two modes and the scale factor shows a stochastic behavior.

Now let us discuss the second question. Naively the low energy effective theory is not general relativity, since the excitation of ghost and normal particles breaks Lorentz invariance although the spacetime appears to be Minkowskian on large scales. For a classical field theory without negative energy modes, the only Lorentz invariant configurations of finite energy density have constant fields. However, as discussed below, the negative

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1.2 -

1.15 -

Figure 1. An example of a scale factor A(t) for two pairs of positive and negative energy modes. The dashed lines show the values of A(t ) for which a resonance occurs. m, = 1, m, = .8 and A = .0001 in Planck units. A(t ) also has a small amplitude, high (order 1) frequency component, but it is not visible.

energy modes allow the existence of less trivial Lorentz invariant configurations, which are the Lorentz invariant analogs of rotationally invariant states of fluctuating thermal systems. We start from our solution which consists of a pair of modes, one positive and one negative energy, and each with a particular 4momentum. A Lorentz transform of such a solution will also be a solution and we consider the superposition of all the Lorentz transformed solutions. If we had only positive energy modes, we end up with an infinite energy configuration after this superposition because a boost can create arbitrary high energy particles. However the negative energy particle supplies negative energy and we can have a finite energy configuration after the superposition. This configuration does not have a particular direction of 4-momentum and thus will be Lorentz invariant. This configuration is actually an Lorentz analog of a rotationally invariant thermally fluctuating system in which each particle breaks rotational invariance, but as a sum of many particle rotational invariance is restored. The idea of this kinds of Lorentz invariant gas is not new and it is discussed in’. Once we accept the existence of this non trivial Lorentz invariant configuration, the low energy effective theory should be described by general relativity with zero cosmological constant and the usual expansion of universe follows from low energy perturbations.

47

In this talk we have tried to demonstrate some links between Lorentz invariant classical fluctuations, the existence of negative energy modes, and the cosmological constant problem.

References

1. T. Hirayama and B. Holdom, arXiv:hep-th/0408223. 2. S. Weinberg, Phys. Rev. Lett. 59 (1987) 2607. 3. K.S. Stelle, Gen. Rel. Grav. 9 (1978) 353. 4. J. M. Cline, S. Y. Jeon and G. D. Moore, arXiv:hep-ph/0311312. 5. T. H. Boyer, Phys. Rev. D 11 (1975) 790.

INFLATION MODEL BUILDING IN MODULI SPACE*

KENJI KADOTA~ NASA/Ferrnilab Theoretical Astrophysics Center,

Ferrnilab, Batawia, IL 6051 0, USA

EWAN D. STEWART* Department of Physacs, KAIST, Daejeon 305-701, South Korea

A self-consistent modular cosmology scenario and its testability in view of future CMB experiments are discussed. Particular attention is drawn to the enhanced symmetric points in moduli space which play crucial roles in our scenario. The running and moreover the running of running for the cosmic perturbation spectrum are also analyzed.

1. Inflation model building

The important questions we would like to answerer in inflation model building are

What is the inflaton field? What are its properties?

To try to answer these questions, we need to consider what is a natural particle theory framework to discuss the dynamics of the early Universe. (i) What fields were there an the early Universe? The most promising candidate to describe the physics of the early universe is string theory, and string theory has many flat directions whose potentials vanish in exact supersymmetry. The fields parameterizing such flat directions are called moduli, and we shall discuss if a moduli field can realize a

*Based on the talk presented by KK at PASCOSO4 workshop. +Work partially supported by NASA grant NAG5-10842. t Work partially supported by Astrophysical Research Center for the Structure and Evo- lution of the Cosmos funded by the Korea Science and Engineering Foundation and the Korean Ministry of Science, and by the Korea Research Foundation grants KRF-2000- 015-DP0080 and KRF PBRG 2002-070-COOO22.

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successful inflationary scenario or not. (ii) What does a modulus potential look like? The properties of moduli fields are heavily dependent on the way supersymmetry is broken. In the following, we discuss a hidden sector symmetry breaking scenario, in which the generic form of the moduli potential becomes

where M, is the supersymmetry breaking scale and F is a dimensionless function with of order unity coefficients. In particular, for gravity mediated supersymmetry breaking, M , M 10lO-llGeV and the mass of modulus 4 consequently becomes m+ = M,"/M, M 102-3GeV. Note, as is usually the case for supergravity inflation, the slow-roll parameter V"/V is of order unity, which is a generic problem for supergravity inflation unless we choose a special form of K a e r potential.

2. Can a consistent inflationary scenario be realized in this

The detailed discussion of a possible self-consistent modular cosmology scenario based on this natural simple particle theory setup is given in [l]. We shall concentrate on the predicted cosmic perturbations for this scenario in this article. Note that we will consider a single modulus field which is complex. This does not mean we add an additional degree of freedom (namely angular component in addition to the radial component) because all scalar fields are complex in supersymmetry. The overview of the dynamics of the inflaton modulus is the following.

natural context?

Near a maximum, the potential of Eq.(l) has the forma

where VO - M," and m4 - M:. Our scenario starts with an eternally inflating universe consisting of an ensemble of eternally inflating extrema throughout the field space of string theory, which avoids the Brustein- Steinhardt problem '. Our local universe is a region where the field rolled down from its maximum to escape from the eternally inflating region, and

aThis form relies on the maximum being a point of enhanced symmetry. See [3] for details.

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the observable cosmic perturbations are produced while the field rolls down to its minimum. For a single complex modulus

the total curvature perturbations arise from radial and angular component fluctuations

aN aN & = SN = -S4 + - 6 8 . a4 ae (4)

Before presenting the detailed results in Sec 2.2, simple estimates tell us that the contribution of radial component fluctuations to the final curvature perturbations is

and that for the angular component is

A crucial point here is that the angular component perturbations dominate for a large enough $$.

2.1. Points of enhanced symmetry

The existence of points of enhanced symmetry is a robust and unique feature of moduli space. At those special points, some fields (matter, gauge, or moduli) become massless and become massive away from it, being Higgsed by the modulus that parameterizes the distance from the point of enhanced symmetry. At such a point of enhanced symmetry, the couplings of the modulus to those light degrees of freedom will cause the moduli mass to renormalize as a function of 4. This can turn the mass squared of the modulus positive for small 4, shifting the maximum of the potential out to some finite value 4 = q!~*. The modulus maximum now becomes a rim and the field starts rolling down from there. The expression for the angular component fluctuation given in Eq.(6) had a steep spectrum because 4 in the expression H / 4 changes rapidly due to non-slow-roll. It now, taking account of the loop corrections in the potential, goes like Hf40 where cpo is radius of the rim maximum of the potential, making the spectrum flat. The remaining problem is if large aN/ae is probable, or, if the initial angle 0 corresponding to large aN/aO is probable. It is indeed probable, if we

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consider that the regions corresponding to those initial angles leading to large a N / M would undergo greater expansion and hence occupy a larger volume at late times.

It turns out, as discussed in detail in [3], that the renormalized potential induced by the effects of light degrees of freedom at the saddle points identified as enhanced symmetric points will dynamically select the desirable initial angle for the inflaton modulus.

2.2. Observable predictions

In [3], we performed detailed analytic and numerical calculations for our modular cosmology scenario without assuming slow-roll conditions. The perturbation spectrum in our model has negligible deviations from scale invariance over a wide range of k with running becoming significant at (very) small (but possibly observable) scales. The form of the spectral index turned out to be a simple polynomial form

n - 1 = Aka (7) Its running and running of running

= a2Aka dn d2n dlnk (dln k)2 -- - aAka, ~

illustrate our theoretical expectations that the usually assumed hierarchy In”l << 172’1 << In - 11 << 1 is valid only for a limiting region of parameter space, a << 1, where the running would in any case be small, while for a wider range of a the running is significant but so is the running of the running, In”I - In‘l - In - 11.

It is also worth noting that Eq. (7) indicates that the running and the running of running are expected to be most significant toward smaller scales, i.e. negligible n’ at large scales does not necessarily guarantee the absence of running in the spectrum, which cannot be taken into account if we ignore n”. Thus it is crucial for our observations to probe the smallest possible scales to search for a signal of running.

3. Discussion

There are several works which try to explain the apparent discrepancy between the natural energy scale of the moduli potential (- 101O-llGeV) and the energy scale of inflation (- 1015GeV) obtained in some simple inflation models. They tend to try to find a non-trivial mechanism to scale up the

52

energy scale of the modulus potential to the GUT scale to match the amplitude of fluctuations with observations. We, however, instead stuck with the value of - 10lO-llGeV and saw if our natural particle theory setup leads to the observationally consistent inflationary scenario. We discussed the cosmic perturbations in our scenario, which gives one of the most stringent constraints on an inflation model, with a particular emphasis on the possible running of running which in general cannot be ignored for the consistency of perturbation calculations.

In addition to the problems discussed in this article, another well-known and long-standing problem is the cosmological moduli problem. We proposed a baryogenesis scenario following thermal inflation in [5] to make this aspect of modular cosmology self-consistent too. Besides these phenomene logical aspects of modular cosmology, more fundamental problems, such as moduli stabilization, are also under intense investigation 6 . The ubiquitous existence of moduli is a generic prediction of string/M-theory, and the realization of a successful modular cosmology scenario would be worth further study.

References 1. K. Kadota and E.D. Stewart, JHEP 0307, 013 (2003). 2. R. Brustein and P.J. Steinhardt, Phys. Lett. B 302 196 (1993). 3. K. Kadota and E.D. Stewart,JHEP 0312, 008 (2003). 4. M. Sasaki and E.D. Stewart, Prog. Theor. Phys. 95, 71 (1996). 5. D. Jeong, K. Kadota, W. Park and E.D. Stewart, hep-ph/0406136. 6. See for example: S. Kachru, R. Kallosh, A. Linde and S.P. Trivedi, Phys.Rew.

D68,046005 (2003); S. Kachru, R. Kallosh, A. Linde, J. Maldacena, L. McAl- lister and S.P. Trivedi JCAP 0310, 013 (2003).

GETTING AROUND COSMIC VARIANCE OF THE CMB TEMPERATURE QUADRUPOLE'

CONSTANTINOS SKORDIS~ Astrophysics, Keble Road,

Oxford OX1 3RH, England E-mail: [email protected]

I present a new method for inferring the Cosmic Microwave Background(CMB) temperature quadrupole with accuracy better than cosmic variance. The method relies on measurements of the large angle CMB polarisation spectrum generated by a reionieation epoch, exploiting the fact that CMB polarisation is sourced by the local quadrupole. It is generic enough and can be used to explicitly distinguish models that generate a low quadrupole from those that don't.

1. Introduction

The first release of Wilkinson Microwave Anisotropy Probe (WMAP) data has given firm evidence for a concordance ACDM model : a flat universe, consisting of 26% Cold Dark Matter (CDM), 4% baryons and 70% cosmological constant, and with a gaussian, nearly scale invariant and predominantly adiabatic spectrum of initial fluctuations.

The WMAP data however, also show evidence of "anomalies", particularly with the large scales, hinting that ACDM might not be the end of the story. One of these "anomalies" is the CMB temperature quadrupole CF, appears to be low compared with expectations based on ACDM cosmology. This confirmed the Cosmic Background Explorer (COBE) measurement but more cleanly, as the lower detector noise and wider frequency range to pin down galactic foreground emission renders the measurement more robust.

One possibility is that there is no problem with ACDM and that the quadrupole is low by chance, which in this case can occur with - 5%

'Work done with Joseph Silk. Based in part on Ref.1 +Supported by a Leverhulme foundation grant

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probability (see below). The alternative would be that it is actually low and we are seeing new physics 5 . Given the not so small probability of occurring by chance, measuring the quadrupole directly can never tell us whether it is low or not with statistical significance. One would have to use other methods, such as the one presented here.

2. Is the quadrupole low?

How precisely can the quadrupole be measured? Let @(A) be the temperature fluctuation of the CMB about the mean

temperature in direction A. This can be expanded in spherical harmonics as

The expansion coefficients hem are random variables which I assume that they obey Gaussian statistics, i.e. their probability distribution function is proportional to e- lecm12/(2Ct) , where Ce is their variance which depends only on e.

The angular power spectrum Ce is related to 81, by

It obeys x2 statistics with 21 + 1 degrees of freedom. This means that its mean value is equal to the variance, i.e. ( C e ) = Ce, and that its variance is given by

(3) 2

V W [ C ~ ] = -(co2. 2e+ 1

The above expression defines cosmic variance, a fundamental uncertainty in any measurement of the angular power spectrum C e . At the quadrupole it is equal to 40%.

Table 1. Measurements of the quadrupole with corresponding probabilities.

measurement I C ? ( P K ) ~ Probability

WMAP 123.4 0.01

Tegmark et al.9 201.6 0.03

Efstathiou'O 250 0.05

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Table 1 shows the probabilities that the quadrupole is as low as or lower than corresponding measurements from the WMAP data, using as a fiducial model the best fit adiabatic model of Ref. 6. One can conclude from those probabilities that the quadrupole cannot be said to be low (see for example Refs. 7, S ) , particularly because its value is sensitive to the galactic cut and the estimator usedl0.

One has to use different tests of the quadrupole.

3. The quadrupole and polarisation

An idea put forward by Kamionkowski and Loeb l1 is to use the polarisation spectrum produced by clusters through the Sunyaev-Zel’dovich effect 12.

Since the polarisation spectrum depends on the local quadrupole at the cluster, one can get information on the quadrupole CT(z) at redshift z, by taking large samples of clusters and averaging. One hopes that the averaging method will recover a quadrupole close to the cosmological value. Further calculations involving clusters have been carried out in Ref. 13.

Lets consider now an alternative method, which uses the CMB polarisation signal generated by a period of reionization instead.

0.1

Figure 1. The local quadrupole today(so1id) and at reionization(dotted).

The quadrupole today is given as an integral over a transfer function AF(lc, t o ) (shown in Fig.1) and an initial power spectrum Pq(k), where lc is a wavenumber and t o is the conformal time today, as

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Similarly the E-mode polarisation angular power spectrum is given by

where z = k(to - t ) , ~ ( t ) = st”, gdt is the optical depth to time t , j , is a spherical Bessel function and n ( k , t ) is the polarisation source.

We now need a relation between the local quadrupole today, A f ( k , to) , and the polarisation spectrum. It turns out that one can relate the two through the local quadrupole at the start of reionization, A f ( k , t,)(shown in Fig.1, where t , is the conformal time at the start of reionization. The local quadrupole today and the polarisation source are given by

AT(k,to) = eATrA:(k,t ,) + 6” dt e&T(t) S2(k,t), (6)

and t

n ( k , t ) = e&[Tr-T(t)lA?(k, t,) + e-&[T(t)l &’ e&T(t’) &(k, t‘) (7) l-? respectively, and where S2(k, t ) and Su(k, t ) are source terms explained in Ref. 1.

Thus the quadrupole is parametrically linked to the polarisation spectrum through the initial condition AF(k, t,) in the last two equations. This means that by measuring the polarisation spectrum over a wide range of multipoles (typically C < 30) we can get a measure on the quadrupole today.

4. Estimating the variance

Consider a zero noise polarisation experiment, which is limited only by cosmic variance. To get an estimate on the variance of CT using the above method, we can parametrise the amplitude of the local quadrupole at reionization by a parameter q, by letting AT(k,t,) -+ qAf(k, t , ) in (6) and (7). The required spectra were numerically computed using a modified version of DASh 14. The next step is to calculate the Fisher matrix F for q, the optical depth and the spectral index. The other cosmological parameters are not needed as they will be very well determined by small scale measurements of the CMB temperature and polarisation. The variance of q is then given by (F- l )qq and for the fiducial model is found to be 0.002. This gives a variance on CT of 6% (see Ref. 1 for more details) which is much lower than the cosmic variance value of 40%. Changing the fiducial model is not expected to change this value much.

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5. Conclusion

I have presented a method to get around cosmic variance for the CMB temperature quadrupole. The method exploits the connection between the quadrupole and polarisation. In particular the quadrupole today is parametrically linked to the polarisation spectrum generated by reionization, via the local quadrupole at the start of reionization. This method would be useful when large scale polarisation data become available as it has the potential to answer the question whether the quadrupole is low or not.

References

1. C.Skordis & J.Silk, astro-ph/0402474. 2. C.L. Bennett et al., Astrophys. J . Supp. 148, 1 (2003). 3. C. L. Bennett et al., -4strophys. J. Lett. 464, L1 (1996). 4. C. L. Bennett et al. Astrophys. J. Supp. 148, 1 (2003). 5. M. Bastero-Gil et al., Phys.Rev. D68, 123514 (2003); R. Bean & 0. Dor6,

Phys. Rev. D69, 083503 (2004); J . M. Cline et al., JCAP 0309, 010 (2003); C. Contaldi et al., JCAP 0307, 002 (2003); S. DeDeo et al., Phys.Rev. D67, 103509 (2003); G. Efstathiou, Mon. Not. R. Astron. SOC. L95, 343 (2003); B. Feng & X. Zhang, Phys. Lett. B570, 145 (2003; A. Lasenby & C. Doran, astro-ph/0307311; M. Linguori et al.,astro-ph/0405544; J. P. Luminet et al., Nature 425,593 (2003); T. Moroi & T. Takahashi, Phys.Rev.Lett. 92, 091301 (2004); Y-S. Piao et al., Phys.Rev. D69, 103520 (2004); Y-S. Piao et al., hep-th/0312139; S. Tsujikawa et al., astro-ph/0311015; S. Tsujikawa et al., Phys.Lett. B574, 141 (2003); J. Weeks et al., astro-ph/0312312; J.Weller & -4. M. Lewis Mon.Not.R.Astron.Soc. 346, 987 (2003).

6. M. Bucher et d i , Phys.Rev.Lett. 93, 081301 (2004). 7. E. Gaztaiiaga et al., Mon. Not. R. Astron. SOC. 346, 47 (2003). 8. G. Efstathiou, Mon. Not. R. -4stron. SOC. 346, L26 (2003). 9. M. Tegmark, -4. Oliveira-Costa and A. J. S. Hamilton, Phys. Rev. D 68,

123523 (2003). 10. G. Efstathiou, astro-ph/0310207. 11. M. Kamionkowski & A. Loeb, Phys.Rev. D 56, 4511, (1997). 12. R. A. Sunyaev and Ya. B. Zel'dovich, Comments Astrophys. Space Phys. 4,

173 (1972); ibid Mon. Not. R. Astron. SOC. 190, 413 (1980). 13. N. Set0 and M. Sasaki, Phys. Rev. D 62, 123004 (2000); A. Cooray

and D. Baumann, Phys.Rev. D 67, 063505 (2003); J. Portsmouth, astro- ph/0402173.

14. M. Kaplinghat et al., Astrophys. J . 578, 665 (2002).

ON THE EFFECTIVE EVOLUTION FOR THE INFLATON*

RUDNEI 0. RAMOS Departamento de Fisica Tedrica, Universidade do Estado do Rio de Janeiro,

20550-013 Rio de Janeiro, RJ, Brazil E-mail: [email protected]

The dynamics of the inflaton field is studied in the context of its interaction with bosonic and fermionic fields modeled by a minimal SUSY like model.

Despite the many studies on the dynamics of inflation, much still remains to be done in the context of fully understanding the many aspects concerning the microscopic dynamics underlying many of the models for inflation, particularly in those cases where the inflaton is coupled to several other fields, like in hybrid inflation models and supersymmetric (SUSY) model extensions. In most of these models it is a fact that large regions of parameter space still remain unexplored and that can be feasible to inflation and new phenomena. This is also true when we consider the common approximations used to study the different aspects of inflation, which most of the time have been restricted to particular perturbative and linear regimes. This is understandable since when nonperturbative and nonlinear effects may become important standard techniques may not always apply, which have slowed down progress in that direction. At the same time methods and techniques developed in quantum field theory devoted to the description of nonequilibrium dynamics have become essential to study these new phenomena, like in those cases where there can be non-negligible particle and radiation production, e.g. in the description of the preheating phase after isentropic inflation or the study of the emergence of non-isentropic inflation (or warm inflation) scenarios, where the scalar inflaton field dissipates non-negligible amounts of radiation during inflation 1 * 2 , 3 7 4 .

Of special interest as concerned to non-isentropic inflationary scenarios is the process of how the inflaton can dissipate its energy during inflation.

*Work partially supported by FAPERJ and CNPq.

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We have recently identified an efficient mechanism for that in the context of a nonlinear and nonperturbative regime for the inflaton dynamics and elaborated on its details in [ 5, 6 , 71. We consider the inflaton field g!J in interaction with other scalar and fermion fields, with standard interaction terms of the form

for field masses satisfying m, > min(2mqd,mb) and m# < min(mq,,,m,) so there are kinematically allowed decay channels of the scalar x into the I j d , $d fermions. The decaying of the inflaton field in this model is an indirect effect, interpreted in terms of the effective theory for 4 (after integrating over the x , $)d, $ d , $I,, $, fields), which shows that the inflaton does not interact for instance with vacuum like x excitations but rather with the collective x excitations which can decay into light fermions. These are scattering like processes through which the inflaton transfers its energy (or radiates) and that can be efficient in nonlinear and nonperturbative regimes and can happen even deep inside the inflationary phase, as shown in [ 71.

The role of the spinors coupled to g!J in (1) is to mimic SUSY, keeping the quantum corrections AV&(4) to the effective potential for the inflaton under control, so preserving the flatness of its potential. In fact, as shown in [ 61, a minimal SUSY model that reproduces the above interactions and decay mechanism has the superpotential W = + g@X2 + f X 3 + mX2 + h X Y 2 , where @, X , Y are chiral superfields. Here, even for SUSY breaking, for # 0 and H # 0, we still can have Al&(d) << &(g!J), where Vo(4) is the tree level potential for the inflaton, that we will take to be a quartic potential with self-coupling A.

The effective equation of motion (EOM) that emerges for a homogeneous classical inflaton field, g!J = cp(t), from (1) and with parameters satisfying the adiabatic (or slow) dynamics for the inflaton, X N 0(10-13), g N g' N h 2 0(1OW1) and cp N O(mpl), was shown in Refs. [ 5, 71 to be given by

(Fi(t)+3H(t)@(t)+ 9 + Z 7 ( t ) R ( t ) + 4 g 4 c p ( t ) ~ d t ~ ( t f ~ @ ( t ' ) K , ( t , t')X2) dcp t o

where we are working in a FRW background metric, R(t) is the scalar of curvature, with coupling of the inflaton to the gravitacional field and K,(t, t') is a nonlocal (dissipative) kernel that results from the interaction of the inflaton with the scalar x within the relevant scattering like term at one-loop order,

60

where w x ( q , t ) = [qz//a(t)2 + M i ( t ) ] " 2 , M i ( t ) = mt + g;p(t)2 + (s - 1/6)R(t) and rx(q,t) N h2M,(t) / [87rw,(q, t )] for m, >> mQa.

In Refs. [ 6 , 71 we have studied the dynamics of the inflaton through the full numerical solution of (2) (which is numerically implementable, since rx > H and so the highly oscillatory nonlocal kernel is effectively damped). We also have shown that approximating the kernel as a nonexpanding one (in the Minkowski approximation of Refs. [ 4,5]) is a very good approximation for the exact numerical dynamics as well (which is expected since for our parameters M , >> H and so curvature effects are subleading). Finally we also have shown that a Markovian (local) approximation for (2) is as well an excellent approximation to describe the evolution for the inflaton (which again is expected, since during inflation and parameters we consider, d/cp, H < rx and so the dynamics is effectively adiabatic).

A representative example of the effects of dissipation in the inflaton's EOM as a result of its interactions to other fields, in the relevant region of parameters for our mechanism of dissipation to work, is shown in Fig. 1.

O.O t 1 -0.21 " ' I . ' " " " ' I

0 20 40 60 80 100 120 140

t x 10i3GeV

Figure 1. ~ ( 0 ) = mPl, +(O) = 0 and a(0) = 1.

Evolution for ~ ( t ) for 9 = h = 0.5, = 0, A = mx = 1013GeV,

Figure 1 is obtained by numerically solving Eq. (2) in the Markovian approximation simultaneously with the acceleration equation for the scale

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factor. For comparison we also show the result for q( t ) when the dissipation due to the nonlocal term in Eq. (2) is absent, for the case of a quartic effective potential, Ve, = Xq4/4 (dotted line). In the absence of the nonlocal term in Eq. (2), inflation for the quartic potential with the parameters of Fig. 1 ends when qend N 0.47mp1, or by the time tend N GeV-l. At this time and well after the inflaton starts oscillating around its minimum value of the potential, in the presence of the effective dissipation term in Eq. (2) the inflaton is still in the inflationary regime (the solid curve in Fig. l), which ends by the time w 1.1 x GeV-l, when Vend N 0.17mpl. Till the end of inflation the dynamical regime for the inflaton is overdamped, dominated by the nonlocal dissipative kernel. Despite the noticeable change in behavior due to field dissipation, the overall amount of radiation energy density produced is only a fraction of the inflaton’s energy density. For the parameters of Fig. 1, the radiation energy density reaches a peak value pT/pv N lop2 at an early time, decaying next till reaching an approximate constant fraction value of N We have also checked that the adiabatic approximation used to derive Eq. (2), wx/wz << 1 and the Markovian approximation for the nonlocal kernel, @/(qFx) << 1 (see Ref. [ 7]), are both very robust, breaking down, for the parameters used in Fig. 1, at a time N 3 x G e V 1 and therefore well after the end of inflation.

Figure 2 compares how the number of e-folds of inflation, N,, changes when we vary either the initial inflaton’s amplitude or couplings g, h for the interactions terms in Eq. (1) and again contrast the results with those obtained for the quartic potential for the inflaton, in the absence of the interactions Lint. For q ( 0 ) = 4.4mp1, which for the inflaton’s self-interaction X = results in N , 21 60 in the absence of dissipation effects in the inflaton’s EOM (dotted line in both plots shown in Fig. 2), the inner plot in Fig. 2 shows the number of e-folds when we vary g (taking also h = 9). We observe that the interaction terms (1) start to influence the inflaton’s evolution in an appreciable way for g = h 2 0.2, with number of e-folds quickly raising up as the couplings are increased. Increasing the number of fields x or fermions $d has also similar effect of increasing rather quickly the number of e-folds or the duration of the inflationary phase.

The results discussed above show that in typical multi-field inflation models there are parameter regions feasible to inflation for which nonlinear and nonperturbative effects can become important and that can lead to important changes in the dynamics for the inflaton, with the emergence of effective strong dissipative effects that alone can sustain inflation long enough and with observationa! effects on density perturbations 697. We

62

Figure 2. The number of e-folds Ne in terms of cp(O), considering the effective evolution (solid line) and without dissipation (dotted line). Parameters are the same as in Fig. 1. Inner plot shows Ne as a function of the coupling constant, with fixed 6 = 0, X = m, = 1013GeV, cp(0) = 4.4mp1, d(0) = 0.

should note that the appearance of strong dissipative effects in our mechanism is not related to a direct decay for the inflaton field, but it is a consequence of decaying modes for fields coupled to the inflaton that results in an effective dissipation in the inflaton’s EOM, whose magnitude can be expressive for nonlinear and nonperturbative regimes. These dissipative mechanisms discussed here have found several uses in the recent literature, like in alleviating many of the problems associated with typical inflation models (the 77 problem, graceful exit, quantum-to-classical transition, large inflaton amplitude, initial conditions), in the study of baryogenesis during nonisentropic inflation, among other studies.

References 1. A. Berera and L. Z. Fang, Phys. Rev. Lett. 74, 1912 (1995). 2. H. P. de Oliveira and R. 0. Ramos, Phys. Rev. D57, 741 (1998); 3. A. Berera, M. Gleiser and R. 0. Ramos, Phys. Rev. D58, 123508 (1998);

Phys. Rev. Lett. 83, 264 (1999). 4. A. Berera and R. 0. Ramos, Phys. Rev. D63, 103509 (2001). 5. A. Berera and R. 0. Ramos, Phys. Lett. B567, 294 (2003). 6. A. Berera and R. 0. Ramos, hepph/0308211. 7. A. Berera and R. 0. Ramos, hep-ph/0406339.

INFLATION MODELS IN SUPERGRAVITY WITH A RUNNING SPECTRAL INDEX

MASAHIDE YAMAGUCHI Department of Physics and Mathematics, Aoyama Gakuin University,

Kanagawa 889-8558, Japan

M. KAWASAKI ICRR, University of Tokyo, Kashiwa 877-8582, Japan

JUN’ICHI YOKOYAMA Department of Earth and Space Science, Graduate School of Science, Osaka

University, To yonaka 560- 004 3, Japan

The first year Wilkinson Microwave Anisotropy Probe data favors primordial adiabatic fluctuation with a running spectral index with ng > 1 on a large scale and ns < 1 on a smaller scale. The model building of inflation that predicts perturbations with such a spectrum is a challenge. We give sensible particle physics models in supergravity that accommodate the desired running of the spectral index and discuss a method to discriminate those models.

1. Introduction

The first year data release of the Wilkinson Microwave Anisotropy Probe (WMAP) has opened a new era of high-precision cosmology. It has not only confirmed the “concordance” values of the cosmological parameters with much smaller uncertainties than before but also extracted important information on the primordial spectrum of density perturbations. It has been reported that their result favors purely adiabatic fluctuations with a remarkable feature that the spectral index runs from n, > 1 on a large scale to n, < 1 on a smaller scale. More specifically they obtain n, = 1.13 f 0.08 and dn,/d In k = -0.055?E:~~~ on the scale ko = 0.002 Mpc-l.

It is not straightforward to make a model of inflation that predicts perturbations with such a spectrum. One possibility to realize such a spectral feature is a hybrid inflation model of Linde and Riotto, in which supergravity effect becomes dominant in the early stage of inflation and one-loop

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effect plays an important role in the final stage of inflation. Another is smooth hybrid inflation in supergravity, in which nonrenormalizable terms and the term originated from supergravity effects compete during inflation. Unfortunately, however, although the spectral index crosses unity in both models, its variation is too mild to reproduce the WMAP result for a sensible set of model parameters. In order to circumvent this problem we consider a double inflation model combining (smooth) hybrid inflation and new inflation in supergravity. In our model, first of all, (smooth) hybrid inflation proceeds and the desired spectrum with a running index from n, > 1 to n, < 1 is generated then. At the same time the initial condition for subsequent new inflation is naturally prepared, which continues about 40 e-folds to expand the comoving scale with the desired spectral shape of density fluctuation to larger scales observable with WMAP. Fur- thermore, new inflation predicts sufficiently a low reheating temperature to avoid overproduction of gravitinos.

In the next section, we give our models of inflation and discuss that hybrid and smooth hybrid inflation3 in supergravity can reproduce the results of WMAP. In the final section, we give our discussion and conclusion.

2. Hybrid new inflation and smooth hybrid new inflation in supergravity

2.1. Hybrid inflation in supergravity

First of all, we consider hybrid inflation in supergravity of Linde and Riotto and show that it can reproduce the results of WMAP. The superpotential is given by

W H = As$@ - p2s, (1) where S is a gauge-singlet superfield, while 9 and $ are a conjugate pair of superfields transforming as nontrivial representations of some gauge group, and X and p are positive parameters much smaller than unity. The R- invariant Kahler potential is given by

KH = ISI2 + [ @ I 2 + l?FI2. (2)

Identifying a real part of the complex field S, u = ReS/&, with the inflaton, the effective potential of the inflaton u during hybrid inflation reads

65

Note that the dynamics of the scalar field is dominated by the nonrenormalizable term for u > ,/m = ffd and by the radiative correction for

The amplitude, the spectral index, and its variation of curvature fluc- 0 < U d .

tuation are given by

2 4 ns - 1 = - 6 ~ + 2 r ) S 2r) = 3a - -, u2

When we take X = 0.53, ,LL = 3.2 x and u = 0.27, the WMAP results are reproduced. But the e-folds of hybrid inflation after the comoving scale with the observed spectral shape has crossed the Hubble radius is only about 10.4. Hence we must invoke another inflation to push the relevant scale to the scale IcO = 0.002 Mpc-l.

2.2. Smooth hybrid inflation in supergravity

Next we consider smooth hybrid inflation in supergravity. The superpotential is given by

where 9 and % are a conjugate pair of superfields transforming a s rioiitrivial representations of some gauge group, under which a superfield S is singlet. ,LL is the energy scale of hybrid inflation, and A4 is a cutoff scale of the nonrenormalizable term. The R-invariant Kahler potential is given by

KsH = )SI2 + ) @ I 2 + )GI2. (8)

Identifying a real part of the complex field S, u = ReS/fi, with the inflaton, the effective potential of the inflaton u during smooth hybrid inflation is given by

The dynamics is determined by the first term for cr < a d and the last term for u > (Td with D d = (16/27)1/8(pM)1’4.

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The amplitude, the spectral index, and its variation of curvature fluctuation are given by

-1 R = - ( 2 + . 3 ) P2

&7r

-- - 1 6 q - 24c2 - 2[ Z -26 = - dn, dlnk

When we take M = 1.6, p = 2.7 x and cr = 0.28, the WMAP results are reproduced. But, also in this case, the e-folds of smooth hybrid inflation after the comoving scale with the observed spectral shape has crossed the Hubble radius is only about 10.0. Hence we must invoke another inflation to push the relevant scale to the scale ICo = 0.002 Mpc-l.

2.3. New inflation in supergravity

The superpotential of new inflation is given by

(13) @n+l w N [ @ ] u2@ - - n + l '

where we introduce a chiral superfield CP with an R charge 2/(n + l), but assume that the U( 1 ) ~ symmetry is dynamically broken to a discrete 2 2 , R

symmetry at a scale 21 << p. Here g is a coupling constant of order of unity and we also assume that both g and u are real and positive for simplicity. The R-invariant Kahler potential is given by

where CN is a constant smaller than unity. Then, the potential of new inflaton is approximated as

where we identified the real part of CP with the inflator1 #) = f iRe@.

2.4. Smooth hybrid new inflation in supergravity

We investigate full dynamics of smooth hybrid new inflation. Full dynamics of hybrid new inflation can be obtained in the same way.

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In the smooth hybrid inflation stage, the interaction terms between S and Qr yields

Hence at the end of smooth hybrid inflation, u = uc, 4 and x = &Im@ have a minimum at

For example, if we take CN = 0.1 and g = 1, N N = 40 implies 21 = 2 . 7 ~ 1 0 - ~ . In this case, R - 3.8 x lop4 at the comoving scale t, - 650 kpc, which may be helpful for early star formation which is required for early re-ionization and from the age estimate of high-redshift quasars using the cosmological chemical clock.

3. Discussion and Conclusion

In this talk we showed that hybrid inflation and smooth hybrid inflation in supergravity can explain the running feature of density fluctuations as preferred by the WMAP data. However, both inflations do not last enough, which requires another inflation to push the scales with the desired spectral shape of density fluctuations to cosmologically observable scales. As another inflation, we consider new inflation because it predicts a sufficiently low reheating temperature to avoid overproduction of gravitinos. Further- more, in our model, the initial value of new inflation is dynamically set during (smooth) hybrid inflation, which evades the severe initial value problem of new inflation.

Though hybrid inflation has a problem of initial conditions, this problem can be solved by considering hybrid inflation with a chaotic initial condition (chaotic hybrid inflati~n)~.

Finally, cosmic strings are formed after hybrid inflation while they are not formed after smooth hybrid inflation. The search for cosmic strings by future observations will be able to discriminate two models.

References 1. M. Kawasaki, M. Yamaguchi, and J. Yokoyama, Phys. Rev. D 68, 023508

(2003). 2. M. Yamaguchi and J. Yokoyama, Phys. Rev. D 68, 123530 (2003). 3. M. Yamaguchi and J. Yokoyama, Phys. Rev. D 70, 023513 (2004). 4. See also references in the above three references.

MODULAR COSMOLOGY, THERMAL INFLATION, BARYOGENESIS AND

A PREDICTION FOR PARTICLE ACCELERATORS

WAN-IL PARK Department of Physics, KAIST, Daejeon 305-701, South Korea

E-mail: [email protected]. ac.kr

Talk based on hepphf0406136'

We propose a simple model in which the MSSM plus neutrino mass term (LH,,)2 is supplemented by a minimal flaton sector to drive the thermal inflation, and make two crucial assumptions: the flaton vacuum expectation value generates the p-term of the MSSM and m i + mLU < 0. The second assumption is particularly interesting in that it violates a well known constraint, implying that there exists a nearby deep non-MSSM vacuum, and provides a clear signature of our model which can be tested at future particle accelerators. We show that our model leads to thermal inflation followed by Affleck-Dine leptogenensis along the LH, flat direction.

1. Motivation

In string theory inspired scenarios of cosmology, over-production of unwanted relics at relatively low energy scale has been an obstacle to successful scenario of cosmology. Among unwanted relics produced after inflation, string moduli might be the most dangerous one due to their coherent oscillation with amplitude of order Plank scale after inflation at energy scale N Jq2. Because some moduli fields are supposed to have vacuum expectation value of order Plank scale, their oscillation indicates production of huge amount of moduli particles with very long life time such that moduli could decay after BBN or be stable depending on SUSY-breaking scenario. Therefore, if there is no following dilution after coherent oscillations of moduli, we are faced on a disaster in cosmology.

Thermal inflation3 was proposed as a compelling solution of this moduli problem. For example, let's consider following equation,

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where VO is set to make cosmological constant be zero. At high temperature, a flaton, 4 is held origin due to thermal effect. In this case, if VO dominate the energy density of universe, universe inflates. It is expected to occur at relatively low energy scale of order 107-8 GeV and end around electroweak scale as flaton rolls out when temperature drops below the critical temperature. In spite of its small number of e-folding, thermal inflation can resolve the moduli problem through huge amount of entropy release by flaton decay.

However, this thermal inflation scenario is incompatible with baryogenesis models proposed so far. Because those models occur either before thermal inflation or just before flaton decay in an inefficient way, such that primordially generated asymmetry is diluted out. After flaton decay, temperature is too low for any baryogenesis model to work. Therefore, new model is required and we suggest a model.

2. Model

Our model is based on MSSM superpotential

W = AzeabQ:o:H:uy + Ayc,bQ:aHjdy + AyEabLqHjej + pcabH:Hj (2)

where i is the generation index, a is the SU(2) index and a is the SU(3) index. Because the origin of MSSM p-term is unknown, we assume its absence initially. Insteady of it, we include

Ap42EabHtHi, (3)

as. was suggested long ago in Ref. 4, such that MSSM yterm can be reproduced when flaton, 4 get right size of vacuum expectation value. To give mass to neutrino, we include

1 . . -A: 2 cabL:H:ecdLTH,d. (4)

Finally, flaton self-interaction term, as a minimal form for simplicity5 , 1

( 5 )

is included to stabilize flaton field. Even though we composed this superpotential in phenomenological rea-

sons, the same type of potential can be obtained by imposing discrete 2 4 symmetry with assigning proper charges to each field content. Thus, our

70

superpotential is given by

w = XzEabQ4QHbua U J + XyEabQPaH2dy + Xij€,bL?Hjej 1 " 1

Xp42€abH,"H2 + ZXy€abLqHt€,dLFH,d + ~ X 4 4 ~ (6)

with assumption, [ X u [ >> IX,I N 1x41 to give right size of neutrino mass and the scale of vacuum expectation value of flaton.

We impose following key assumptions on our model to check dynamics of field configuration in a definite way.

All the field is held at origin initially due to thermal effect m42 < 0 and m H U 2 + mL2 < 0 LH, rolls away earlier than 4

The first assumption sets the initial condition of dynamics. The second one makes corresponding field unstable below certain temperature. The last one is required to use Affleck-Dine mechanism6 for baryogenesis. Affleck-Dine mechanism is considered due to its efficient way of generation of asymmetry such that it may survive dilution due to decay of flaton.

Note that the m H U 2 + m L 2 < 0 violates a well known ~ o n s t r a i n t ~ ? ~ . This is key assumption of our model and it is seemingly problematic because it indicates the existence of non-MSSM minima. But it turns out that, in wide range of parameter space, our model is phenomenologically consistent under this assumptiong.

As an ansatz, we assume just LH,, H,Hd and 4 develop nonzero values. This is because, during thermal inflation, p-term is absent such that some supersymmetric flat directions containing H , or H d could be unstable. By considering just single generation for simplicity, we obtain following potential;

v = V&b + V F + V D

= h + m&ulh,12 + m&d]hd12 + mi1112 + m$4I2

1 1 1 + ApXp42h,hd + ~A,XUl2h: + -A4X444 + C.C. [ 4

+ I X p 4 2 h d + XuZ2h,12 + IX,42h,12 + IXuZh:12 + I2X,4huhd + X44312

where & is adjusted to give zero cosmological constant, g2 = (g: + g;)/4, and all the other supersymmetry breaking parameters are of the order of the electroweak scale. The first two lines in Eq. (7) is vacuum supersymmetry

71

breaking terms, and the other lines are F-terms and D-term potential in order.

3. Dynamics

As temperature drops below certain value during thermal inflation, LH, rolls away. It is stabilized by (X,lhi(2 and ~ A , X , Z 2 h ~ + C.C. fixes its initial phase. We assume LH, settles down to its minimum position before flaton rolls away for simplicity. As flaton rolls away, ending thermal inflation, Hd start to develop nonzero value simultyaneousely due to h e a r terms, A,X,42h,hd + C.C. and X,42hd ( ~ , ~ 2 h , ) * . Nonzero value of Hd is critical to stabilize dangerous field direction and generation of lepton asymmetry. As 4 reaches its vacuum expectation value, IX,q52h,l brings back LH, into origin and the phase dependent cross term, (X,f$2hd)* X,12h, rotates its phase generating lepton asymmetry.

2

4. Preserving L-asymmetry

The generated asymmetry has to be preserved to explain observation. Preservation requires damping of Affleck-Dine field. This means that the amplitude of lepton number violating operator in our model should be reduced such that field configuration is brought into symmetric region of potential to conserve its angular momentum.

Energy transfer of homogeneous mode to inhomogeneous modes of field(preheating) and thermal friction are expected to act as the sources of damping.

Our aim of this work is to show just the generation of asymmetry in a phenomenologically consistent way. Therefore, to complete this scenario, complete analysis of damping of LH, is required as a future work.

Finally, the generated asymmetry through the dynamics of LH, field is converted to baryon asymmetry through sphaleron process which is ac- tivated due to partial reheating by decay of Affleck-Dine field. Therefore, the expected B-asymmetry is obtained through following equation;

s n4m4 It turns out, in our model,

is required to get observed asymmetry, n B / s - 10-l'.

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5. Conclusion

We proposed a baryogenesis model compatible with thermal inflation in a fairly minimal way. Our model is unique in the context of gravity mediated SUSY-breaking and thermal inflation. In this model, flaton generated p- term and triggered the generation of lepton asymmetry. The assumption, r n ~ , ~ + r n L 2 < 0 indicates the existence of deeper non-MSSM minima, but cosmological evolution leads us to our vacuum and it can be tested at future particle accelerators as a clear signature of our model.

Acknowledgements

WIP thank Patrick Greene for collaboration at an earlier stage of this project. WIP were supported in part by the Astrophysical Research Center for the Structure and Evolution of the Cosmos funded by the Korea Science and Engineering Foundation and the Korean Ministry of Science, the Korea Research Foundation grant KRF PBRG 2002-070-COOO22, and Brain Korea 21.

References 1. D. Jeong, K. Kadota, W. Park and E. D. Stewart, [arXiv:hepph/0406136]. 2. G. D. Coughlan, W. Fischler, E. W. Kolb, S. &by and G. G. Ross, Phys. Lett.

B131, 59 (1983); J. R. Ellis, D. V. Nanopoulos and M. Quiros, Phys. Lett. B174, 176 (1986); T. Banks, D. B. Kaplan and A. E. Nelson, Phys. Rev. D49, 779 (1994), [arXiv:hep-ph/9308292]; B. de Carlos, J. A. Casas, F. Quevedo and E. Roulet, Phys. Lett. B318, 447 (1993), [arXiv:hep-ph/9308325].

3. D. H. Lyth and E. D. Stewart, Phys. Rev. Lett. 75, 201 (1995), [arXiv:hep ph/9502417]; Phys. Rev. D53, 1784 (1996), [arXiv:hep-ph/951020].

4. J. E. Kim and H. P. Nilles, Phys. Lett. B138, 150 (1984). 5. E.D. Stewart, M. Kawasaki and T. Yanagida, Phys. Rev. D54, 6032 (1996),

[arXiv:hepph/9603324]. 6. I. Affleck and M. Dine, Nucl. Phys. B249, 361 (1985); M. Dine, L. Randall and

S. Thomas, Phys. Rev. Lett. 75, 398 (1995), [arXiv:hep-ph/9503303]; Nucl. Phys. B458, 291 (1996), [arXiv:hep-ph/9507453].

7. J.A. Casas, A. Lleyda and C. Munoz, Nucl. Phys. B471,3 (1996), [arXiv:hep ph/9507294],

8. H. Komatsu, Phys. Lett. B215, 323 (1988). 9. A. Kusenko, P. Langacker and G. Segre, Phys. Rev. D54, 5824 (1996),

[arXiv:hep-ph/9602414].

ACCELERATING UNIVERSE AND MODIFIED GRAVITY

TOM0 TAKAHASHI Institute for Cosmic Ray Research,

Unaversity of Tokyo, Kashiwa 277-8582, Japan

We discuss models of the accelerating universe where gravity is modified at infrared region. In particular, we consider the so-called Dvali-Gabadadze-Porrati (DGP) brane world model and an extension of the model. Interestingly, in some cases of the extended model, an enhanced big rip can occur. Some implications to observations of these models are also discussed.

1. Introduction

There are mounting evidences that the universe is accelerating. To explain the present cosmic acceleration, we can introduce a component whose equation of state parameter w, = px/px is less than -1/3. This kind of component is usually called “dark energy.” Although a lot of models of dark energy have been proposed so far and many works have been done from phenomenological point of view, we still do not know its nature.

In fact, the universe can be accelerated without assuming the “conventional” dark energy if gravity theory is modified at infrared region. The pioneering work along this direction was done in brane world context I . In the model (now it is called the DGP model), we assume a 3-brane in the 5 dimensional Minkowski spacetime, then consider the following action;

2 d 5 x f i R s + s d 4 x 6 C m - !% 2 / d 4 x f i R 4 (I)

where M5 and M4 are the five and four dimensional Planck mass, 95 and 94 are the determinant of the metric in five and four dimension respectively. Cm is ordinary matter on a 3-brane. With this setup, gravity theory is modified at around the crossover scale L = M:/2M: where the gravity force becomes four dimensional form l / r2 to five dimensional form l/r3. The Friedmann equation in this model is also modified as

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74

where p is the total energy density, H = u /a is the Hubble parameter and a is the scale factor. If we assume that the crossover scale is around the present Hubble scale Ho, the second term in the left hand side can be larger than the right hand side at late time. Hence, the Hubble parameter can be written as H - 1/L at late time. Thus the situation becomes very similar to the case of the cosmological constant, then the accelerating universe is realized. From observations of type Ia supernovae (SNeIa), we can obtain the constraint on the crossover scale was obtained as 1.1H;’ < T, < 1.4H,j-l assuming a flat universe 3,4.

As for the “conventional” dark energy case, if the equation of state of dark energy is assumed to be constant in time, constraint on w, can be obtained as -1.31 < w, < -0.82 ’. Thus we cannot exclude the region where w, < -1 from current observations. If the equation of state is less than -1, the secalled “big rip” where the scale factor of the universe diverges in finite time can happen 6 . In the original DGP model, the big rip cannot occur, however if we consider an extension of this model, the big rip can happen in some cases. Moreover the scale factor can diverge much more singularly than that in the case with the big rip, which we call “bigger rip” or the enhanced big rip ‘. We discuss of the extended model in the next section.

2. An Extension of the DGP model

Now we discuss an extension of the DGP model. We assume that the crossover scale L depends on time, i.e., the five dimensional Planck mass M5 depends on time. In particular, we consider the time-dependence of L(t) as

L(t) = L(to)T(t)P

where the power satisfies p L 1 and

(3)

in which trip is the time of the Rip.

H - l/L(t). Thus that we arrive at: Similarly to the DGP case, at late time, the Hubble parameter becomes

1 TP

H(t) = HrJ- (5)

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Defining y = -dT/dt = (trip - to)-’ gives:

and hence, for p = 1, which is similar to dark energy with a constant w < -1 equation of state:

a ( t ) = T - h (7)

while for the Bigger Rip case p > 1 one finds

Here we see that the scale factor diverges more singularly in T for p > 1, hence the designation of Bigger Rip. In particular we study the values p = 2,3, . . . as alternative to the “dark energy” case p = 1. Inverting Eq. (8) gives:

T = [i + ( p - i ) y ~ ( t ” ) ~ n a ( t ) l - h (9)

In this case there is strictly no dark energy, certainly not with a constant equation of state, but we can mimic it with a fictitious energy density p~ by noticing that H 2 - T-2p and writing

pL N [I + ( p - l ) y ~ ( t o ) lna(t)l* (10)

If we use Eqs.(8) and (10) in conservation of energy

we find a time-dependent w ~ ( t ) for the “fictitious” dark energy

2 mL(t0) W L ( t ) = -1 - -

3 1 + ( p - l)yL(to) lna(t)

so the effective wL(t) has the limiting values wL(t0) = -1 - i p (yL( t0 ) ) and

Now we discuss constraints on model parameters from SNe data. To discuss the constraint, we have to include other component such as cold dark matter (CDM) and baryon. Including all components, we can write the F’riedmann equation as

WL(tr ip) = -1.

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If we define the density parameter R, = pn/pcrit = p m o ( l + z ) ~ , we can rewrite Eq. ( 1 3 ) as

where o k and f l ~ are defined as R k = - k / H $ and f i ~ = 1/4L2H$. Thus at the present time, we have the relation among the density parameters,

R k + ( K + d m ) 2 = 1 . ( 1 5 )

To obtain a constraint from the SNeIa observations, we assume that redshift dependence of RL is as in Eq. ( 1 0 ) and that L is dependent on time as Eq. (3); thus we can say that we consider a new component p~ which is defined as Eq. (10).

Now we discuss the constraint on this model from SNeIa data using recent result '. In Fig. 1, we show contours of 95 and 99 % C.L. in f i~-R, plane for yL0 = 0 (which is the constant L case), 0.5 and 1 with p = 2. In the figure, we also plot the line for the flat universe. Notice that the line is different from the standard case because we have the modified Friedmann equation in this model. To obtain the constraint, we marginalize the Hubble parameter dependence by minimizing x 2 €or the fit.

3. Summary

We discussed models of modified gravity where the present cosmic acceleration of the universe can be realized. In particular, we considered an extension of the so-called DGP model. Interestingly, in some cases, the big rip or the enhanced big rip can occur, in which the scale factor of the universe diverges in finite time. In such a scenario, gravitationally bounded system can be disintegrated before the time of the rip '. From cosmological observations such as SNe, we can obtain constraints on the crossover scale and the time to the rip.

Acknowledgments

The author would like to thank the Japan Society for Promotion of Science (JSPS) for finantial support.

References 1. G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B 485, 208 (2000).

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6 o-8 0.6 I +...-. Oa4 1 *.;Y 0.2 ,/.. . . . *. 5-

0 '------- - 0 0.2 0.4 0.6 0.8 1

%-I

Figure 1. Constraint from SNeIa observation in R L - R ~ plane for ?Lo = 0 (bottom), ?Lo = 0.5 (middle) and 7~50 = 2 (top). Contours are for 95 % (dotted line) and 99 % (dashed line) C.L. constraints respectively. The solid line indicates parameters which give a flat universe.

2. C. Deffayet, Phys. Lett. B 502, 199 (2001). 3. C. Deffayet, G. R. Dvali and G. Gabadadze, Phys. Rev. D 65, 044023 (2002);

Z. H. Zhu, arXiv:astro-ph/O404201; J. S. Alcaniz and N. Pires, Phys. Rev. D 70, 047303 (2004).

4. P. H. Frampton and T. Takahashi, arXiv:astr~-ph/O405333. 5 . A. Melchiorri, arXiv:astreph/0406652. 6. R. R. Caldwell, Phys. Lett. B 545, 23 (2002); P. H. Frampton and T. Taka-

hashi, Phys. Lett. B 557, 135 (2003); R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003).

7. A. G. Riess et al. [Supernova Search Team Collaboration], Astrophys. J. 607, 665 (2004).

THE PIERRE AUGER OBSERVATORY: SCIENCE PROSPECTS AND PERFORMANCE AT FIRST LIGHT

LUIS A. ANCHORDOQUI

[for the AUGER Collaboration] Department of Physics

Northeastern University, Boston M A 02115

E-mail: 1.anchordoquiOneu. edu

The Pierre Auger Observatory is a major international effort aiming at high- statistics study of highest energy cosmic rays. A general description of the experimental set-up and overall performance of the detector at first light are presented.

The Pierre Auger Observatory (PAO) is designed to study cosmic rays with energies above about 1O1O GeV with the aim of uncovering their origins and nature.l Such events are too rare to be directly detected, but the direction, energy, and to some extent the chemical composition of the primary particle can be inferred from the cascade of secondary particles induced when the primary impinges on the upper atmosphere? These cascades, or air showers, have been studied in the past by measuring the nitrogen fluorescence they produce in the atm~sphere,~ or by directly sampling shower particles at ground level.4 The PA0 is a “hybrid” detector, exploiting both of these methods by employing an array of water Cerenkov detectors over- looked by fluorescence telescopes; on clear, dark nights air showers are simultaneously observed by both types of detectors, facilitating powerful reconstruction methods and control of the systematic errors which have plagued cosmic ray experiments to date. Additionally, since the center-f- mass energy in the collision of the primary with the atmosphere is above about 100 TeV (i.e., exceeding the contemporary and forthcoming collider reach by 2 orders of magnitude), PA0 will also be capable of probing new physics beyond the electroweak scale.5

The Observatory will be covering two sites, in the Southern (Pampa Amarilla) and Northern hemispheres. Each site consist of 1600 stations

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spaced 1.5 km apart from each other, with 4 fluorescence eyes placed on the boundaries of the surface array. The energy threshold is defined by the 1.5 km spacing of the detector stations: a loxo GeV vertical shower will hit on average 6 stations which is enough to fully reconstruct the extensive air shower. The installation of the Southern site is now well underway, with detectors looking at the yet poorly covered part of the sky in which the direction of the center of the Milky Way is visible.

Comms and GPS Antennas

12

Solar panels

Phototube -

m3 of water-

box

Tyvek Liner

- Batteries

Figure 1. Schematic view of a water Cerenkov detector.

Each ground-based detector is a cylindrical opaque tank of 3.6 m diameter and 1.55 m high, where particles produce light by Cerenkov radiation, see Fig. 1. The filtered water is contained in a bag which diffusely reflects the light collected by three photomultipliers (PMT’s) installed on the top. The large diameter PMT’s (M 20 cm) are mounted facing down and look at the water through sealed polyethylene windows that are integral part of the internal liner.

Due to the size of the array, the detectors have to be able to function independently. The stations operate on battery-backed solar power and time synchronisation relies on a comercial Global Positioning System (GPS) receiver.6 A specially designed wireless LAN radio system is used to provide communication between the surface detectors and the central ~ t a t i o n . ~ Each tank forms an autonomous unit, recording signals from the ambient cosmic ray flux, independent of the signals registered by any other tank in the surface detector array. A combination of signals clustered in space and

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time is used to identify a shower.

Figure 2. can be seen the aperture system, the photomultiplier camera and the spherical mirror.

Schematic view of a fluorescence telescope at Los Leones. From left to right

A single fluorescence detector unit (or eye) comprises 6 telescopes, each located in independent bays, overlooking separate volumes of air. A schematic cross-sectional view of one fluorescence telescope is shown in Fig. 2. A circular diaphragm, positioned at the centre of curvature of the spherical mirror, defines the aperture of the Schmidt optical system! Ultra- violet transmitting filters are installed in the entrance aperture. Just inside the ultra-violet filter is a ring of (Schmidt) corrector elements. Light is focused by a large 3.5 m x 3.5 m spherical mirror onto a 440 PMT camera, which accommodates the 30" azimuth x 28.6" elevation field of view. Each camera pixel has a field of view of approximately 1.5".

In the hybrid mode (- 10% of the time) the detector is expected to have energy resolution of 13% (lo) at lo9 GeV improving to 5.5% at 1011 GeV, and an angular resolution of about 0.5". For the ground array alone these numbers become 10% and lo, again for primary energy > 10l1 GeV. Esti- mating event rates is a risky business because above 10l1 GeV the fluxes are essentially not known.g However, for zenith angle less than 60" the total aperture of the Southern surface array is - 7350 km2 sr, and thus extrapolation from AGASA measurementsg implies that PA0 should detect of the order of 2500 events above lo1" GeV and of a 50 to 100 events above

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loll GeV every year. Moreover, if events above 60 degrees can be analyzed effectively, the aperture will increase by about 50% .

PA0 also provides a promising way of detecting ultra-high energy neutrinos by looking for deeplydeveloping, large zenith angle (> 60”) or horizontal air showers.1° At these large angles, hadronic showers have traversed the equivalent of several times the depth of the vertical atmosphere and their electromagnetic component has extinguished far away from the detector. Only very high energy core-produced muons survive past 2 equivalent vertical atmospheres. Therefore, the shape of a hadronic (background in this case) shower front is very flat and very prompt in time. In contrast, a neutrino shower appears pretty much as a “normal” shower. It is therefore straightforward to distinguish neutrino induced events from background hadronic showers. Moreover, if full flavor mixing is confirmed, tau neutrinos could be as abundant as other species and so very low v, fluxes could be detected very efficiently by PAO’s detectors by looking at the interaction in the Earth crust of quasi horizontal v, inducing a horizontal cascade at the detector.”

The first PA0 site is now operational in Malargiie, Argentina, and is in the process of growing to its final size of 3000 km2. At the time of writing, 12 telescopes and about 400 water tanks were operational. The first analyses of data from the PA0 are currently underway. Figure 3 shows the arrival directions of all events recorded from January to July 2004. The pixels have a size of 1.8 degrees and the map was smoothed with a Gaussian beam of 5 degrees.12 On 21 May 2004, one of the larger events recorded by the surface array triggered 34 stations. A preliminary estimate yields an energy - 10l1 GeV and a zenith angle of about 60”. First physics results will be made public in the Summer of 2005 at the 29th International Cosmic Ray Conference.

Acknowledgments

I would like to thank all my collaborators in the Pierre Auger Observa- tory. Special thanks goes to Tere Dova, Jean-Christophe Hamilton, An- toine Letessier-Selvon, Tom McCauley, Tom Paul, Steve RRucroft, and John Swain for assistance in the preparation of this talk. This work has been partially supported by the US National Science Foundation (NSF), under grant No. PHY-0140407.

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Gaussian smoothing : 5.0 deg.

0 0052 - 0 086

Figure 3. z are related to the number of events per pixel n according to: n = 330 z + 1.716.

All event (from January to July 2004) skymap in Galactic coordinates. Units

References 1.

2.

3. 4. 5.

6. 7.

8. 9.

J. Abraham et al. [AUGER Collaboration], Nucl. Instrum. Meth. A 523, 50 (2004). L. Anchordoqui, T. Paul, S. Reucroft and J. Swain, Int. J. Mod. Phys. A 18, 2229 (2003). R. M. Baltrusaitis et al., Nucl. Instrum. Meth. A 240, 410 (1985). N. Chiba et al., Nucl. Instrum. Meth. A 311, 338 (1992). L. A. Anchordoqui, arXiv:hep-ph/0306078. See also, L. Anchordoqui, H. Gold- berg and C. Nunez, arXiv:hep-ph/0408284; L. Smolin, in these Proceedings. C. L. Pryke and J. Lloyd-Evans, Nucl. Instrum. Meth. A 354, 560 (1995). P. D. J.Clark and D. Nitz, Proc. of 27th International Cosmic Ray Conference, Copernicus Gesellschaft, 765 (2001). A. Corder0 et al., GAP-1996-039. M. Takeda et al. [AGASA Collaboration], Astropart. Phys. 19, 447 (2003); R. U. Abbasi et al. [HiRes Collaboration],-Phys. Rev. Lett. 92,151101 (2004).

10. K. S. Capelle, J. W. Cronin, G. Parente and E. Zas, Astropart. Phys. 8, 321 (1998).

11. X. Bertou, P. Billoir, 0. Deligny, C. Lachaud and A. Letessier-Selvon, As- tropart. Phys. 17, 183 (2002); J. L. Feng, P. Fisher, F. Wilczek and T. M. Yu, Phys. Rev. Lett. 88, 161102 (2002).

12. E. Armengaud et al., GAP-2003-105.

The references to internal reports written by the AUGER Collaboration (GAP notes) are accesible via www . auger. org/admin/GAP-NOTES.

Dark Matter and Dark Energy


COLD DARK MATTER FLOWS AND CAUSTICS *

P. SIKIVIE Institute f o r Fundamental Theory

Department of Physics University of Florida

Gaineswille, FL 3261 1-8440 E-mail: [email protected]

The late infall of cold dark matter onto an isolated galaxy, such as our own, produces discrete flows and caustics in its halo. The set of caustics includes simple fold catastrophes located on topological spheres surrounding the galaxy, and a series of caustic rings in or near the galactic plane. The caustic rings are closed tubes whose crosssection is an elliptic umbilic catastrophe. The self-similar model of galactic halo formation predicts that the caustic ring radii a, follow the approximate law a, N l/n. In a study of 32 extended and well-measured external galactic rotation curves evidence was found for this law. Also, the locations of ten sharp rises in the rotation curve of the Milky Way fit the prediction of the self-similar model at the 3% level. Moreover, a triangular feature in the IRAS map of the Galactic plane is consistent with the imprint of a ring caustic upon the baryonic matter. These observations imply that the dark matter in our neighborhood is dominated by a single flow. Estimates of that flow's density and velocity vector are given.

1. Introduction

There are compelling reasons to believe that the dark matter of the universe is constituted in large part of non-baryonic collisionless particles with very small primordial velocity dispersion, such as axions and/or weakly interacting massive particles (WIMPS) '. Generically, such particles are called cold dark matter (CDM). Knowledge of the distribution of CDM in galactic halos, and in our own halo in particular, is of paramount importance to understanding galactic structure and predicting signals in experimental searches for dark matter.

One should expect this dark matter to form caustics. A caustic is a place in physical space where the density is very large because the sheet

'This work is supported in partby the U.S. Department of Energy under grant DEFG05- 86ER--40272.

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on which the dark matter particles lie in phase-space has a fold there. Caustics are commonplace in the propagation of light. Examples include the sharp luminous lines at the bottom of a swimming pool on a breezy sunny day, rainbows, the twinkling of stars, and the shimmering of the sea. Caustics occur generically when two conditions are satisfied. First, the flow must be collisionless. Second the flow must have low velocity dispersion. Light propagation is collisionless, and the flow of light from a point source has zero velocity dispersion. Thus caustics are common in light. Caustics in ordinary matter are very unusual because ordinary matter is not normally collisionless. But CDM is collisionless and has very small primordial velocity dispersion. This leads us to expect that caustics are common in the distribution of CDM.

The primordial velocity dispersion of the cold dark matter candidates is indeed very small, of order

bva(t) - 3 . 1 0 - l ~ (10;~")~ - (q for axions, and

GeV

for WIMPS. Here to is the present age of the universe and ma and mw are respectively the masses of the axion and WIMP. The small velocity dispersion means that the dark matter particles lie on a thin 3-dim. sheet in 6-dim. phase-space. The thickness of the sheet is 6v. The sheet cannot break and hence its evolution is constrained by topology.

2. The Phase-Space Structure of Galactic Halos

Where a galaxy forms, the sheet wraps up in phase-space, turning clockwise in any two dimensional cut (x,k) of that space. x is the physical space coordinate in an arbitrary direction and x its associated velocity. The outcome of this process is a discrete set of flows at any physical point in a galactic halo 2. Two flows are associated with particles falling through the galaxy for the first time (n = l), two other flows are associated with particles falling through the galaxy for the second time (n = 2), and so on. Scattering in the gravitational wells of inhomogeneities in the galaxy (e.g. molecular clouds and globular clusters) are ineffective in thermalizing the flows with low values of n. The flows are seen in N-body simulations

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of galactic halo formation when care is taken to enhance the numerical resolution in the relevant regions of phase-space.

A commonly raised objection to the above picture is that, before the dark matter falls onto a large galaxy such as our own, it has already clustered on smaller scales, making dwarf halos and other types of clumps, in a process called "hierarchical clustering". However, the effect of hierarchical clustering is only to produce an effective velocity dispersion for the infalling dark matter, i.e. a thickening of the phase space sheet. This effective velocity dispersion is at most equal to the velocity dispersion, of order 10 km/s, of dwarf halos and on average should be much less than that. Because the effective velocity dispersion of the infalling dark matter is much less than the 300 km/s velocity dispersion of the Galaxy as a whole, the phase space sheet folds in qualitatively the same way as in the zero velocity dispersion case. The flows and caustics remain.

Caustics appear wherever the projection of the phase-space sheet onto physical space has a fold 4,5. Generically, caustics are surfaces in physical space. On one side of the caustic surface there are two more flows than on the other. At the surface, the dark matter density is very large. It diverges there in the limit of zero velocity dispersion. There are two types of caustics in the halos of galaxies, inner and outer. The outer caustics are simple fold (Az ) catastrophes located on topological spheres surrounding the galaxy. They occur near where a given outflow reaches its furthest distance from the galactic center before falling back in. The inner caustics are rings '. They occur near where the particles with the most angular momentum in a given inflow reach their distance of closest approach to the galactic center before going back out. A caustic ring is a closed tube whose cross-section is an elliptic umbilzc (0-4) catastrophe '. The existence of these caustics and their topological properties are independent of any assumptions of symmetry.

Primordial peculiar velocities are expected to be the same for baryonic and dark matter particles because they are caused by gravitational forces. Later the velocities of baryons and CDM differ because baryons collide with each other whereas CDM is collisionless. However, because angular momentum is conserved, the net angular momenta of the dark matter and baryonic components of a galaxy are aligned. Since the caustic rings are located near where the particles with the most angular momentum in a given infall are at their closest approach to the galactic center, they lie close to the galactic plane.

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A specific proposal was made for the radii a, of caustic rings 4:

{a, : n = 1,2, ...} Y (39, 19.5, 13, 10, 8, ...) kpc x (k) (%) (3) 2 2 0 p

where w,,t is the rotation velocity of the galaxy and j,, is a parameter with a specific value for each halo. For large n, a, 0: l /n. Eq.(3) is predicted by the self-similar infall model 677 of galactic halo formation. j,, is then the maximum of the dimensionless angular momentum j-distribution 7. The self-similar model depends upon a parameter E '. In CDM theories of large scale structure formation, c is expected to be in the range 0.2 to 0.35 7. Eq.(3) is for c = 0.3. However, in the range 0.2 < E < 0.35, the ratios a,/al are almost independent of 6. When j,, values are quoted below, E = 0.3 and h = 0.7 will be assumed.

that including angular momentum in the self-similar infall model results in a depletion of the inner halo and hence

It was pointed out in ref.

A c 0

U > a, n

.- u .-

1.

0 0 a, >

Y .- -

L

Figure 1. exterior galaxies to test the hypothesis that the caustic ring radii are given by Eq.(3).

Composite rotation curve constructed in ref. It combines data on 32

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an effective core radius. The average amount of angular momentum of the Milky Way halo was estimated by requiring that approximately half of the rotation velocity squared at our location is due to dark matter, the other half being due to ordinary matter. This yields 3 - 0.2 where is the average of the j-distribution for our halo. j and j,, are related if some assumption is made about the shape of the j-distribution. For example, if the j-distribution is taken to be that of a rigidly rotating sphere, one has j,, = ,I. Hence j,, - 0.25 for our halo.

Since caustic rings lie close to the galactic plane, they cause bumps in the rotation curve, at the locations of the rings. In ref. a set of 32 extended well-measured rotation curves was analyzed and statistical evidence was found for the n = 1 and n = 2 caustic rings, distributed according to Eq.(3). In this analysis, each of the 32 individual galactic rotation curves was rescaled according to

4 7

220 km/s (4)

where ur0t is the measured rotation velocity. To isolate the outer halo- dominated portion of the rotation curves, all data with rescaled radii i; < 10 kpc were removed. Each rotation curve was then fitted to a line or a quadratic polynomial. The residual deviations were normalized to the rms deviation in each fit and then binned together. The result is the composite rotation shown in Fig. 1 for the case where the individual rotation curves were fitted to quadratic polynomials. The composite rotation curve has two peaks, near 20 kpc and 40 kpc, with statistical significance of 3u and 2 . 6 ~ respectively. It implies that the j,, distribution is peaked near 0.27. The rotation curve of NGC3198, one of the best measured, by itself shows three faint bumps which are consistent with Eq.(3) and three faint bumps which are consistent with Eq.(3) and j,, = 0.28 '. Also our earlier estimate of jm, for the Milky Way halo is close to the peak value of 0.27.

3. Evidence for Ring Caustics in the Milky Way

Eq.(3) with j,, = 0.25 implies that our halo has caustic rings with radii near 40 kpcln, where n is an integer. Here we point to evidence in support of this extraordinary claim.

Galactic rotation curves are obtained from HI and CO surveys of the Galactic plane. A list of surveys performed to date is given in ref. lo.

Everything else being equal, CO surveys have far better angular resolution than HI surveys because their wavelength is nearly two orders of magnitude

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smaller (0.26 cm vs. 21 cm). The most detailed inner Galactic rotation curve appears to be that obtained l1 from the Massachusetts-Stony Brook North Galactic Plane CO survey 12. It is reproduced in Fig. 2. It shows highly significant rises between 3 and 8.5 kpc. Eq.(3) predicts ten caustic rings between 3 and 8.5 kpc. Allowing for ambiguities in identifying rises, the number of rises in the rotation curve between 3 and 8.5 kpc is in fact approximately ten. Below 3 kpc the predicted rises are so closely spaced that they are unlikely to be resolved in the data. The rises are marked as slanted line segments in Fig. 2.

The effect of a caustic ring in the plane of a galaxy upon its rotation curve was analyzed in ref. 5. The caustic ring produces a rise in the rotation curve which starts at r1 = a,, where a, is the caustic ring radius, and which ends a r2 = a, + p,, where p , is the caustic ring width. The ring widths depend in a complicated way on the velocity distribution of the infalling dark matter at last turnaround and are not predicted by the model. They also need not be constant along the ring.

Figure 2. North Galactic rotation curve from ref. ll. The locations of rises mentioned are indicated by line segments parallel to the rises but shifted downwards. The caustic ring radii for the fit described in the text are shown as vertical line segments. The position of the triangular feature in the IRAS map of the galactic plane near 80' longitude is shown by the short horizontal line segment. It coincides with a rise in the rotation curve.

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In the past, rises (or bumps) in galactic rotation curves have been interpreted as due to the presence of spiral arms 13. Spiral arms may in fact cause some of the rises in rotation curves. However there may be other valid explanations. Two properties of the high resolution rotation curve of Fig. 2 favor the interpretation that its rises are caused by caustic rings of dark matter. First, there are of order ten rises in the range of radii covered (3 to 8.5 kpc). This agrees qualitatively with the predicted number of caustic rings, whereas only three spiral arms are known in that range: Scutum, Sagittarius and Local. Second, the rises are sharp transitions in the rotation curve, both where they start (r1) and where they end (Q). Sharp transitions are consistent with caustic rings because the latter have divergent density at r1 = a and 1-2 = a + p in the limit of vanishing velocity dispersion. Finally, there are bumps and rises in rotation curves measured at galactocentric distances much larger than the disk radius, where no spiral arms are seen. In particular, the features found in the composite rotation curve of Fig. 1 are at distances 20 kpc and 40 kpc when scaled to our own galaxy.

The self-similar infall model prediction for the caustic ring radii, Eq.(3), was fitted to the eight rises between 3 and 7 kpc by minimizing rmsd = [$C(1- 3 2 ] 4 with respect to j,,, for t = 0.30 . The fit yields j,, =

0.263 and rmsd = 3.1% . The corresponding caustic ring radii a, are indicated by short vertical line segments at the bottom of Fig. 2.

Caustic rings of dark matter produce gravitational forces on the baryonic matter. They may reveal themselves in maps of the sky by the gas and dust that they have accreted. Looking tangentially to a ring caustic from a vantage point in the plane of the ring, one may recognize the tri- cusp shape of the 0 - 4 catastrophe. I searched for such features. The IRAS map of the galactic disk in the direction of galactic coordinates (1, b) = ( B O O , 0') shows a triangular shape which is strikingly reminiscent of the cross-section of a ring caustic. The relevant IRAS maps are posted at http://www.phys.ufl.edu/-sikivie/triangle/ . They were downloaded from the Skyview Virtual Observatory (http://skyview.gsfc.nasa.gov/). The vertices of the triangle are at (1 , b) = (83.5', 0.4'), (77.8", 3.4') and (77.8", -2.6'). The shape is correctly oriented with respect to the galactic plane and the galactic center. To an extraordinary degree of accuracy it is an isosceles triangle with axis of symmetry parallel to the galactic plane, as is expected for a caustic ring whose transverse dimensions are small com-

14

n=7

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pared to its radius. Moreover its position is consistent with the position of a rise in the rotation curve, the one between 8.28 and 8.43 kpc (n = 5 ) . The caustic ring radius implied by the image is 8.31 kpc and its dimensions are p - 130 pc and q - 200 pc, in the directions parallel and perpendicular to the galactic plane respectively. It therefore predicts a rise which starts at 8.31 kpc and ends at 8.44 kpc, just where a rise is observed. The probability that the coincidence in position of the triangular shape with a rise in the rotation curve is fortuitous is less than

In principle, the feature at (80",0") should be matched by another in the opposite tangent direction to the nearby ring caustic, at approximately (-80", 0"). Although there is a plausible feature there, it is much less compelling than the one in the (+80", 0") direction. There are several reasons why it may not appear as strongly. One is that the (+80", 0") feature is in the middle of the Local spiral arm, whose stellar activity enhances the local gas and dust emissivity, whereas the (-80", 0") feature is not so favorably located. Another is that the ring caustic in the (+8Oo,O0) direction has unusually small dimensions. This may make it more visible by increasing its contrast with the background. In the (-80", 0") direction, the nearby ring caustic may have larger transverse dimensions.

4. The Big Flow

Our proximity to a caustic ring means that the corresponding flows, i.e. the flows in which the caustic occurs, contribute very importantly to the local dark matter density. Using the results of refs. 435,7, we can estimate their densities and velocity vectors. Let us assume, for illustrative purposes, that we are in the plane of the nearby caustic and that its outward cusp is 55 pc away from us, i.e. a5 + p5 = 8.445 kpc. The densities and velocity vectors on Earth of the n = 5 flows are then:

where i , 4 and 2 are the local unit vectors in galactocentric cylindrical coordinates. 4 is in the direction of galactic rotation. The velocities are given in the (non-rotating) rest frame of the Galaxy. Because of an ambiguity, it is not presently possible to say whether d* are the densities of the flows with velocity & or iF. The large size of d+ is due to our proximity to the outward cusp of the nearby caustic. Its exact value is sensitive to our distance to the cusp. We do not know that distance well enough to estimate d+ with accuracy. However we can say that d+ is very large, of order the

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value given in Eq.(5), perhaps even larger. If we are inside the tube of the nearby caustic, there are two additional flows on Earth, aside from those given in Eq.(5). A list of local densities and velocity vectors for the n # 5 flows can be found in ref. 14.

Eq.(5) has dramatic implications for dark matter searches. Previous estimates of the local dark matter density, based on isothermal halo profiles, range from 5 to 7.5 5. The present analysis implies that a single flow (&) has of order three times (or more) that much local density.

The sharpness of the rises in the rotation curve and of the triangular feature in the IRAS map implies an upper limit on the velocity dispersion ~ W D M of the infalling dark matter. Caustic ring singularities are spread over a distance of order Sa N 'FM where v is the velocity of the particles in the caustic, ~ W D M is their velocity dispersion, and R is their turnaround radius. The sharpness of the IRAS feature implies that its edges are spread over ha 2 20 pc. Assuming that the feature is due to the n = 5 ring caustic, R 21 180 kpc and v N 480 km/s. Therefore ~ V D M <, 53 m/s.

The caustic ring model may explain the puzzling persistence of galactic disk warps 15. These may be due to outer caustic rings lying somewhat outside the galactic plane and attracting visible matter. The resulting disk warps would not damp away, as is the case in more conventional explanations of the origin of the warps, but would persist on cosmological time scales.

The caustic ring model, and more specifically the prediction Eq.(5) of the locally dominant flow associated with the nearby ring, has important consequences for axion dark matter searches 16, the annual modulation 17918719720714 and signal anisotropy 21719 in WIMP searches, the search for y-rays from dark matter annihilation 2 2 7 2 3 , and the search for gravitational lensing by dark matter caustics 2 4 7 2 5 . The model makes predictions for each of these approaches to the dark matter problem.

References 1. E.W. Kolb arid M.S. Turner, The Early Universe, Addison-Wesley, 1990;

M. Srednicki, Editor Particle Physics and Cosmology: Dark Matter, Nort- Holland, 1990.

2. P. Sikivie and J.R. Ipser, Phys. Lett. B291 (1992) 288. 3. D. Stiff and L. Widrow, astro-ph/0301301. 4. P. Sikivie, Phys. Lett. B432 (1998) 139. 5. P. Sikivie, Phys. Rev. D60 (1999) 063501. 6. J.A. Filmore and P. Goldreich, Ap.J. 281 (1984) 1; E. Bertschinger, Ap. J.

Suppl. 58 (1985) 39.

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7. P. Sikivie, I. Tkachev and Y. Wang, Phys. Rev. Lett. 75 (1995) 2911; Phys. Rev. D56 (1997) 1863.

8. W. Kinney and P. Sikivie, Phys. Rev. D61 (2000) 087305. 9. P. Sikivie, Phys. Lett. 567 (2003) 1.

10. J. Binney and M. Merrifield, Galactic Astronomy, Princeton University Press, 1998, pp 550, 553.

11. D.P. Clemens, Ap.J. 295 (1985) 422. 12. D.B. Sanders et al., Ap.J.S. 60 (1986) 1; D.P. Clemens et al., Ap.J.S. 60

(1986) 297. 13. C. Yuan, Ap. J. 158 (1969) 871; W.B. Burton and W.W. Shane, Proceedings

of the 38th IAU Symposium The Spiral Structure of our Galaxy, edited by W. Becker and G.I. Kontopoulos, Dordrecht, Reidel, p. 397; W.W. Shane, Astron. and Astroph. 16 (1972) 118.

14. F . 3 . Ling, P. Sikivie and S. Wick, astrc+ph/0405231, to appear in Phys. Rev. D.

15. R.W. Nelson and S. Tremaine, MNRAS 275 (1995) 897; J. Binney, I.-G. Jiang and S. Dutta, MNRAS 297 (1998) 1237.

16. C. Hagmann et al., Phys. Rev. Lett. 80 (1998) 2043; I. Ogawa, S. Matsuki and K. Yamamoto, Phys. Rev. D53 (1996) 1740.

17. P. Sikivie, Proceedings of the Second International Workshop on The Iden- tification of Dark Matter, edited by N. Spooner and V. Kudryavtsev, World Scientific 1999, p. 68.

18. J. Vergados, Phys. Rev. D63 (2001) 063511; A. Green, Phys. Rev. D63 (2001) 103003; G. Gelmini and P. Gondolo, Phys. Rev. D64 (2001) 023504.

19. D. Stiff, L.M. Widrow and J. Frieman, astro-ph/0106048. 20. P. Sikivie and S. Wick, Phys. Rev. D66 (2002) 023504; 21. C. Copi, J. Heo and L. Krauss, Phys. Lett. B461 (1999) 43. 22. L. Bergstrom, J. Edsjo and C. Gunnarsson, Phys. Rev. D63 (2001) 083515. 23. C. Hogan, Phys. Rev. D64 (2001) 063515. 24. C. Hogan, Ap. J. 527 (1999) 42. 25. C. Charmousis, V. Onemli, Z. Qiu and P. Sikivie, Phys. Rev. D67 (2003)

103502.

DAMA RESULTS AND PERSPECTIVES

R. BERNABEI, P. BELLI, F. CAPPELLA, F. MONTECCHIA*, and F. NOZZOLI

Dipartamento d i Fisica, Universitci d i Roma “Tor Vergata” and INFN, Sezione d i Roma2, I-00133 Rome, Italy

A. INCICCHITTI and D. PROSPER1 Dipartamento d i Fisica, Universitci d i Roma “La Sapienza”

and INFN, Sezione d i Roma, I-00185 Rome, Italy

R. CERULLI INFN - Labomtori Nazionali del G m n Sasso, 1-67010 Assergi (Aq), Italy

C.J. DAI, H.H. KUANG, J.M. MA and Z.P. YE+ IHEP, Chinese Academy, P.O. Box 918/3, Beijing 100039, China

DAMA is an observatory for rare processes based on the development and use of various kinds of radiopure scintillators. Several low background set-ups have been realized with time passing and many rare processes have been investigated. In particular, the DAMA/NaI set-up (= 100 kg highly radiopure NaI(T1)) has investigated the model independent annual modulation signature for Dark Matter particles in the galactic halo. With the total exposure of 107731 kg x day, collected during seven annual cycles a model independent evidence at 6.3 u C.L. has been achieved. At present the second generation DAMA/LIBRA is in operation deep underground.

1. Introduction

DAMA is an observatory for rare processes based on the development and use of various kinds of radiopure scintillators. Several low background set- ups have been realized; the main ones are: i) DAMA/NaI (= 100 kg of radiopure NaI(Tl)), which took data underground over seven annual cy-

‘also: Universiti ”Campus BieMedico” di Roma, 00155, Rome, Italy talso: University of Zhao Qing, Guang Dong, China

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cles and was put out of operation in July 2002 1,2,314,51s17181g,10,11,12,13; ii) DAMA/LXe (N 6.5 kg liquid Xenon) 14; iii) DAMA/R&D, which is devoted to tests on prototypes and small scale experiments 15; iv) the new second generation set-up DAMA/LIBRA (N 250 kg; more radio-pure NaI(T1)) in operation since March 2003. These set-ups have investigated many rare processes. Moreover, in the framework of devoted R&D for higher radiop ure detectors and PMTs, sample measurements are regularly carried out by means of the low background DAMA/Ge detector, installed deep underground since 2 10 years and, in some cases, by means of Ispra facilities.

In the following we will focus on the N 100 kg radiopure NaI(T1) set- up, DAMA/NaI, and on its results on the annual modulation signature. The set-up and its performances have been described elsewherel7l0l2. We just remind that the experimental set-up, located deep underground, was equipped with a neutron shield made by Cd foils and 10/40 cm thick polyethylene/paraffin moderator; a !Y 1 m of concrete almost completely surrounds the installation acting as a further neutron moderator.

DAMA/NaI has been a well competitive set-up because of its: i) high intrinsic sensitivity; ii) standard and well defined operating procedures; iii) well known technology; iv) proved possibility of an effective control of the experimental conditions during several years of running; v) high duty cycle; vi) possibility to deeply investigate the WIMP model independent signature; vii) sensitivity to both spin-independent (SI) and spin-dependent (SD) couplings; viii) favoured sensitivity in some of the possible particle and astrophysical models; ix) sensitivity both to low (’3Na) and to high (lZ7I) mass candidates; x) high benefits/cost; etc. It has been designed and realized in order to investigate the presence of a Dark Matter particle component in the galactic halo by means of the model independent WIMP annual modulation signature. The cumulative model independent result obtained over seven annual cycles (for a total exposure of 107731 kg x day) has been published elsewhere’ together with some of the many possible corollary model dependent quests for the candidate particle. We invite the reader to refer to ref. ’ and references therein for a detailed discussion of many of the related arguments, while in the following we just summarize some of them.

The model independent annual modulation signature (originally suggested in 16) is very distinctive since it requires the simultaneous satisfac- tion of all the following requirements: the rate must contain a component modulated according to a cosine function (1) with one year period, T, (2) and a phase, t o , that peaks around N Znd June (3); this modulation must

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only be found in a well-defined low energy range, where WIMP induced recoils can be present (4); it must apply to those events in which just one detector of many actually "fires" (single-hit events), since the WIMP multi-scattering probability is negligible (5); the modulation amplitude in the region of maximal sensitivity is expected to be 57% (6), but this latter rough limit would be larger in case of some of the possible scenarios such as e.g. those depicted in l7>l8. To mimic such a signature spurious effects or side reactions should be able both to account for the whole observed modulation amplitude and to contemporaneously satisfy all the requirements; no one has been found or suggested by anyone over about a decade.

The presence of a model independent positive evidence in the data of DAMA/NaI has been firstly reported by the DAMA collaboration in l9

and published also in ref. 6 , confirmed in ref. 7 7 8 , further confirmed in ref. 9 1 1 0 1 1 1 1 1 2 , 1 3 and conclusively confirmed, at end of experiment, in 2003 2.

Corollary model dependent quests for a candidate particle have been carried out in some of the many possible model frameworks and have been improved with time. In particular, some scenarios either for mixed SI/SD coupled WIMPs or for purely SI coupled WIMPs or for purely SD coupled WIMPs have been considered in some of the many possible model frameworks as well as the case of WIMPs with preferred inelastic scattering.

2. The final model independent result over 7 annual cycles

A model independent approach on the data of the seven annual cycles (total exposure: 107731 kg x day) offers an immediate evidence of the presence of an annual modulation of the rate of the single-hit events in the lowest energy region as shown in Fig. 1 - left, where the time behaviour of the measured (2-6) keV single-hit events residual rates is reported (for more see ref. '). The data favour the presence of a modulated cosine- like behaviour ( A . cosw(t - to)) at 6.3 (T C.L. and their fit for the (2- 6) keV larger statistics energy interval offers modulation amplitude equal to (0.0200 f 0.0032) cpd/kg/keV, to = (140 f 22) days and T = 5 = (1.00 f 0.01) year, all parameters kept free in the fit. The period and phase agree with those expected in the case of a WIMP induced effect (T = 1 year and t o roughly at N 152.5th day of the year). The x2 test on the (2-6) keV residual rate disfavours the hypothesis of unmodulated behaviour giving a probability of 7 . lop4 ( x 2 / d . o . f . = 71/37).

The same data have also been investigated by a Fourier analysis (performed according to ref. 2o including also the treatment of the experimental errors and of the time binning), obtaining the result shown in Fig. 1 - right,

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2-6 keV ~~

4

2

500 loo0 1500 ZOO0 2500 Time(&y) ‘I a w aw a16 u

Frrpueaey(d’ Figure 1. On the left: model independent experimental residual rate for single-hit events in the ( 2 4 ) keV energy interval as a function of the time over 7 annual cycles (total exposure 107731 kg x day); end of data taking July 2002. The experimental points present the errors as vertical bars and the associated time bin width as horizontal bars. The superimposed curve represents the cosinusoidal function behaviour expected for a WIMP signal with a period equal to 1 year and phase exactly at 2nd June; the modulation amplitude has been obtained by best fit. See ref. 2. On the right: power spectrum of the measured (2-6) keV single-hit residuals calculated including also the treatment of the experimental errors and of the time binning. As it can be seen, the principal mode corresponds to a frequency of 2.737. d-l, that is to a period of N 1 year.

where a clear peak corresponding to a period of 1 year is evident. In Fig. 2 the model independent experimental single-hit residual rate

from the total exposure of 107731 kg x day is presented, as in a single annual cycle, for two different energy intervals; as it can be seen the modulation is clearly present in the (2-6) keV energy region, while it is absent in all the higher energy regions (for more see ref. 2).

6-14 keV - 0.1 2-6 keV I % h 0.1

%

Figure 2.

++ -0.05

-0.1

Model independent experimental single-hit residual rate from the total

3 0 0 4 0 0 5 0 0 6 0 0 a -Osl 300 400 500 600 Time (day) Time (day)

, exposure of 107731 kg x day as in a single annual cycle in the (2-6) keV energy interval (on the left) and in the (6-14) keV energy interval (on the right). The experimental points present the errors as vertical bars and the associated time bin width as horizontal bars. The initial time is taken at August 7th. Fitting the data with a cosinusoidal function with period of 1 year and phase at 152.5 days, the following amplitudes are obtained: (0.0195 f 0.0031) cpd/kg/keV and -(0.0009 f 0.0019) cpd/kg/keV, respectively. Thus, a clear modulation is present in the lowest energy region, while it is absent just above.

Finally, a suitable statistical analysis has shown that the modulation amplitudes are statistically well distributed in all the crystals, in all the

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data taking periods and considered energy bins. More arguments can be found elsewhere '. 5 0.1-

v1

4 -0.05

'I -OJ 300 400 500 600

Time (day) Figure 3. Model independent experimental residual rates over seven annual cycles for si&e-hit events (open circles) - class of events to which WIMPs events belo& - and over the last two annual cycles for multiple-hits events (filled triangles) - class of events to which WIMPs events cannot belong - in the (2-6) keV cumulative energy interval obtained by considering all the data as collected in a single annual cycle and using in both cases the same identical hardware/software procedures. The initial time is taken on August 7th. See text.

A careful investigation of all the known possible sources of systematic and side reactions has been regularly carried out and published at time of each data release while detailed quantitative discussions can be found in ',lo. No systematic effect or side reaction able to account for the observed modulation aniplitude and to niimic a WIMP induced effect has been found. As a further relevant investigation, the multiple-hits events also collected during the DAMA/NaI-6 and 7 running periods (when each detector was equipped with its own Transient Digitizer with a dedicated renewed electronics ') have been studied and analysed by using the same identical hardware and the same identical software procedures as for the case of the single- hit events (see Fig. 3). The multiple-hits events class - on the contrary of the single-hit one - does not include events induced by WIMPs since the probability that a WIMP scatters off more than one detector is negligible. The fitted modulation amplitudes are: A = (0.0195 f 0.0031) cpd/kg/keV and A = -(3.9 f 7.9) . cpd/kg/keV for single-hit and multiple-hits residual rates, respectively. Thus, evidence of annual modulation is present in the single-hit residuals (events class to which the WIMP-induced recoils belong), while it is absent in the multiple-hits residual rate (event class to which only background events belong). Since the same identical hardware and the same identical software procedures have been used to analyse the two classes of events, the obtained result offers an additional strong support for the presence of Dark Matter particles in the galactic halo further excluding any side effect either from hardware or from software procedures or from background.

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In conclusion, the presence of an annual modulation in the residual rate of the single-hit events in the lowest energy interval (2 - 6) keV, satisfying all the features expected for a WIMP component in the galactic halo is supported by the data of the seven annual cycles at 6.3 u C.L.. This is the experimental result of DAMA/NaI. It is model independent; no other experiment whose result can be directly compared with this one is available so far in the field of Dark Matter investigation.

3. Some corollary model dependent quests for a candidate

On the basis of the obtained model independent result, corollary investigations can also be pursued on the nature and coupling of the WIMP candidate. This latter investigation is instead model dependent and - considering the large uncertainties which exist on the astrophysical, nuclear and particle physics assumptions and on the parameters needed in the calculations - has no general meaning (as it is also the case of exclusion plots, of expected recoil energy behaviours and of the WIMP parameters evaluated in indirect search experiments). Thus, it should be handled in the most general way as we have preliminarily pointed out with time passing in the past 6 , 7 , 8 , 9 , 1 0 , 1 1 , 1 2 , 1 3 1 2 . Some specific details can be found elsewhere2 and references therein. Here we only remind that the results summarised here are not exhaustive of the many possible scenarios which at present level of knowledge cannot be disentangled. Some of the open questions are: i) which is the right nature for the WIMP particle; ii) which is its right couplings with ordinary matter (mixed SI&SD, purely SI, purely SD or preferred inelastic) iii) which are the right form factors and related parameters for each target nucleus; iv) which is the right spin factor for each target nucleus (some nuclei are disfavoured to some kinds of interactions; for example, in case of an interaction with SD component even a nucleus sensitive in principle to SD interaction could be blinded by the spin factor if unfavoured by the B value"); v) which are the right scaling laws (let us consider as an example that even in a MSSM framework with purely SI interaction the scenario could be drastically modified as discussed recently in ref. 21); vi) which is the right halo model and related parameters; vii) which are the right values of the experimental parameters within their uncertainties; etc. As an example, we remind that not only large differences in the measured rate can be expected when using target nuclei sensitive to

"We remind that tg0 = an/ap is the ratio between the WIMP-neutron and the WIMP- proton effective SD coupling strengths, a, and apr respectively ' ' 9 ' ; 0 is defined in the [O,n) interval.

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the SD component of the interaction (such as e.g. 23Na and 1271) with r e spect to those largely insensitive to such a coupling (such as e.g. natGe and natSi), but also when using different target nuclei although all - in principle - sensitive to such a coupling (compare e.g. the Xenon and Tellurium cases with the Sodium and Iodine cases).

In the following some of the results discussed for some of the many possible model dependent quests for a WIMP candidate are briefly reminded; for the details see ref. 2 . The results summarised here are obviously not exhaustive of the many scenarios possible at present level of knowledge; moreover, some other ideas have been recently presented e.g. in 18t21.

For simplicity, here the results of these corollary quests for a candidate particle are presented in terms of allowed regions obtained as superposition of the configurations corresponding to likelihood function values distant more than 40 from the null hypothesis (absence of modulation) in each of the several (but still a limited number) of the possible model frameworks considered here. Obviously, larger sensitivities than those reported in the following figures would be reached when including the effect of other existing uncertainties on the astrophysical, nuclear and particle Physics assumptions and related parameters; similarly, the set of the best fit values would also be enlarged as well.

As well known, DAMA/NaI is intrinsically sensitive both to low and high WIMP mass having both a light (the 23Na) and a heavy (the 1271) target-nucleus; in previous corollary quests WIMP masses above 30 GeV (25 GeV in ref. 6, have been presented 739711J2913 for few (of the many possible) model frameworks. However, that bound holds only for neutralino when supersymmetric schemes based on GUT assumptions are adopted to analyse the LEP data 22. Thus, since other candidates are possible and also other scenarios can be considered for the neutralino itself as recently pointed out b, the present model dependent lower bound quoted by LEP for the neutralino in the supersymmetric schemes based on GUT assumptions (37 GeV 2 5 ) is simply marked in the following figures.

Fig. 4 shows some of the obtained allowed regions; details and descrip tions of the symbols are given elsewhere2. We just remind that the C.L.’s, we quote for the allowed regions, already account for compatibility with the measured differential energy spectrum and ~ since - with the measured upper bound on recoils.

fact, when the assumption on the gauginemass unification at GUT scale is released neutralino masses down to N 6 GeV are allowed 23,24.

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s ' a Y

bPl0 -2 up

4 10

-6 i n

-. -" 0 loo 200 300 100 200 300

d) G(keV)

Figure 4. Some examples: a): Case of a WIMP with mixed SI&SD interaction in the model frameworks given in '. Coloured areas: example of slices (of the allowed volume) in the plane (us1 vs (USD for some of the possible mw and 0 values. b): Case of a WIMP with dominant SI interaction for the model frameworks given in '. Region allowed in the plane (mw, ( 0 ~ 1 ) . c): Case of a WIMP with dominant SD interaction in the model frameworks given in '. An example of the region allowed in the plane (mw, [ O S D ) ; here 0 = 2.435, ZO coupling, (0 is defined in the [O, . rr ) range). d): Case of a WIMP with preferred inelastic interaction in the model frameworks given in '; examples of slices (coloured areas) of the allowed volumes ([up, 6, mw) for some mw values. Inclusion of other existing uncertainties on parameters and models would further extend the regions; for example, the use of more favourable form factors and/or of more favourable spin factors than the considered ones would move them towards lower cross sections. The vertical dotted line in b) and c) represents a bound in case of a neutralino candidate when supersymmetric schemes based on GUT assumptions are adopted to analyse the LEP data; the low mass region is allowed for neutralino when other schemes are considered and for every other WIMP candidate. While the area at WIMP masses above 200 GeV is allowed only for few configurations, the lower one is allowed by most configurations (the colored region gathers only those above the vertical line). For details and more see '.

In Fig. 5 the theoretical expectations in the purely SI coupling for the case of the neutralino candidate in MSSM with gaugino mass unification at GUT scale released 23724 are shown together with the corresponding DAMA/NaI allowed region (Fig. 4-b).

Specific arguments on some claimed model dependent comparisons can be found elsewhere2. They already account, as a matter of fact, also e.g. for the more recent model dependent CDMS(-11) claim 26 (based on a statistics of 19.4 kg day and on a discrimination technique), where DAMA/Nal is

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Figure 5. Theoretical expectations of versus mw in the purely SI coupling for the case of the neutralino candidate in MSSM with gaugino mass unification at GUT scale released; the curve encloses the DAMA/NaI allowed region of Fig. 4 b . Taken from ref.24.

not correctly and completely quoted and the more recent result of the 7 annual cycles is quoted but not accounted for (as also done in some talks at this Conference). In addition, in the particular scenario of 26, uncertainties from the model (from astrophysics, nuclear and particle physics) as well as some experimental ones, are not accounted at all and the existing interactions and scenarios to which CDMS is largely insensitive - on the contrary of DAMA/NaI - are ignored. 4. Conclusions

DAMA/NaI has been a pioneer experiment running at the Gran Sass0 Na- tional Laboratory of INFN for several years and investigating as first the WIMP annual modulation signature with suitable sensitivity and control of the running parameters. During seven independent experiments of one year each one, it has pointed out the presence of a modulation satisfying the many peculiarities of an effect induced by Dark Matter particles, reaching a significant evidence. As a corollary result, it has also pointed out the complexity of the quest for a candidate particle mainly because of the present poor knowledge on the many astrophysical, nuclear and particle physics aspects.

At present after a new devoted R&D effort, the second generation DAMA/LIBRA (-250 kg NaI(T1)) set-up has been realized and is in data taking since March 2003.

A third generation R&D is in progress toward a possible ton NaI(T1) set-up, we proposed in 1996 27.

References 1. R. Bernabei et al., Nuovo Czmento A 112 (1999) 545. 2. R. Bernabei et al., La Rivista del Nuovo Cimento 26 (2003) 1-73 (astm-

ph/0307403).

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3. R. Bernabei et al., Phys. Lett. B 389 (1996) 757. 4. R. Bernabei et al., Nuovo Cimento A 112 (1999) 1541. 5. R. Bernabei et al., Phys. Lett. B 408 (1997) 439; P. Belli et al., Phys. Lett.

B 460 (1999) 236; R. Bernabei et al., Phys. Rev. Lett. 83 (1999) 4918; P. Belli et al., Phys. Rev. C 60 (1999) 065501; R. Bernabei et al., Phys. Lett. B 515 (2001) 6; F. Cappella et al., Eur. Phys. J. direct C 14 (2002) 1.

6. R. Bernabei et al., Phys. Lett. B 424 (1998) 195. 7. R. Bernabei et al., Phys. Lett. B 450 (1999) 448. 8. P. Belli et al., Phys. Rev. D 61 (2000) 023512. 9. R. Bernabei et al., Phys. Lett. B 480 (2000) 23.

10. R. Bernabei et al., Eur. Phys. J. C 18 (2000) 283. 11. R. Bernabei et al., Phys. Lett. B 509 (2001) 197. 12. R. Bernabei el al., Eur. Phys. J. C 23 (2002) 61. 13. P. Belli et al., Phys. Rev. D 66 (2002) 043503. 14. R. Bernabei et al., Nuovo Cimento A 103 (1990) 767; Nuovo Cimento C 19

(1996) 537; Astrop. Phys. 5 (1996) 217; Phys. Lett. B 387 (1996) 222 and Phys. Lett. B 389 (1996) 783 (err.); Phys. Lett. B 436 (1998) 379; Phys. Lett. B 465 (1999) 315; Phys. Lett. B 493 (2000) 12; New Journal of Physics 2 (2000) 15.1; Phys. Rev. D 61 (2000) 117301; Eur. Phys. J. direct C 11 (2001) 1; Nucl. Instrum. Methods A 482 (2002) 728; Phys. Lett. B 527 (2002) 182; Phys. Lett. B 546 (2002) 23; in “Beyond the Desert 03”, Springer (2004) 541.

15. R. Bernabei et al., Astrop. Phys. 7 (1997) 73; R. Bernabei et al., Nuovo Cimento A 110 (1997) 189; P. Belli et al., Nucl. Phys. B 563 (1999) 97; P. Belli et al., Astrop. Phys. 10 (1999) 115; R. Bernabei et al., Nucl. Phys. A 705 (2002) 29; P. Belli et al., Nucl. Instrum. Methods A 498 (2003) 352; R. Cerulli et al., Nucl. Instrum. Methods A 525 (2004) 535.

16. K.A. Drukier et al., Phys. Rev. D 33 (1986) 3495. K. Freese et al., Phys. Rev. D 37 (1988) 3388.

17. D. Smith and N. Weiner, Phys. Rev. D 64 (2001) 043502. 18. K. Freese et al. astro-ph/0309279. 19. P. Belli, talk at TAUP 97, LNGS (1997); R. Bernabei et al., Nucl. Phys. B

(Proc. Suppl.) 70 (1999) 79 20. W.H. Press and G. B. Rybicki, Astrophys. J. 338 (1989) 277; J.D. Scargle,

Astrophys. J. 263 (1982) 835 21. G. Prezeau et al., Phys. Rev. Lett. 91 (2003) 231301. 22. D.E. Groom et al., Eur. Phys. J. C15 (2000) 1 23. A. Bottino et al., Phys. Rev. D67 (2003) 063519; A. Bottino et al, h e p

ph/0304080; D. Hooper and T. Plehn, MADPH-02-1308, CERN-TH/SOOB- 29, [hep-ph/0212226]; G. BBlanger, F. Boudjema, A. Pukhov and S. Rosier- Lees, hepph/0212227

24. A. Bottino et al., Phys. Rev. D 69 (2004) 037302. 25. K. Hagiwara et al., Phys. Rev. D66 (2002) 010001

27. See e.g.: R. Bernabei et al., Astrop. Phys. 4 (1995) 45; R. Bernabei, in The identification of Dark Matter, World Sc. pub. 574( 1997).

26. CDMS toll., a~tro-ph/0405033.

FIRST RESULTS FROM THE CRYOGENIC DARK MATTER SEARCH AT SOUDAN

R. MAHAPATRA* Department of Physics, University of California,

Santa Barbam, CA 93106, USA E-mad: mpakQhep.ucsb.edu

We report the first results from a search for weakly interacting massive particles (WIMPs) in the Cryogenic Dark Matter Search (CDMS) experiment at the Soudan Underground Laboratory. Four Ge and two Si detectors were operated for 52.6 live days, providing 19.4 kg-d of Ge net exposure after cuts for recoil energies between 10-100 keV. A blind analysis was performed using only calibration data to define the energy threshold and selection criteria for nuclear-recoil candidates. Using a standard dark matter halo distribution and nuclear-physics WIMP model, these data set the world’s lowest exclusion limits on the coherent WIMP-nucleon scalar cross-section for all WIMP masses above 13 GeV/c2, ruling out a significant range of neutralino supersymmetric models. The best sensitivity at the minimum of the limit curve corresponds to a 90% C.L. cross-section exclusion of 4 x cm2 at a WIMP mass of 60 GeV/c2.

1. INTRODUCTION

Nonluminous, nonbaryonic, weakly interacting massive particlesl~~ (WIMPs) may constitute most of the matter in the universe3. Super- symmetry provides a natural WIMP candidate in the form of the lightest superpartner4y5. WIMPs are expected to be in a roughly isothermal Galatic halo. They would interact elastically with nuclei, generating a recoil energy of a few tens of keV, at a rate < 1 event kg-12*49576.

The Cryogenic Dark Matter Search (CDMS) collaboration is operating a new apparatus7i8 to search for WIMPs in the Soudan Underground Labo- ratory. This CDMS I1 experiment uses Ge (each 250 g) and Si (each 100 g) ZIP (Z-dependent Ionization and Phonon) detectorsg surrounded by substantial shielding deep underground to reduce backgrounds from radioac-

*FOR THE CDMS COLLABORATION

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tivity and cosmic-ray interactions. Simultaneous measurement of ionization and athermal phonon signals in the Ge and Si detectors allows excellent rejection of the remaining gamma and beta backgrounds. These background particles predominantly scatter off electrons in the detectors, while WIMPS (and neutrons) scatter off nuclei. The ZIP detectors allow discrimination between electron and nuclear recoils through two effects. First, for a given energy, recoiling electrons are more ionizing than recoiling nuclei, resulting in a higher ratio of ionization to phonon signal, called “ionization yield”. Second, athermal phonon signals due to nuclear recoils have longer rise times and longer delay than those due to electron recoils. For recoils within a few pm of a detector’s surface (primarily from low-energy electrons), the charge collection is incomplete”, making discrimination based on ionization yield less effective. But these events can be effectively rejected by timing cuts on the phonon pulse because they have, on the average, even faster phonon signals than those from bulk electron recoils”. These effects are in qualitative agreement with our understanding of the complex phonon and semiconductor physics involved12.

The detectors are surrounded by 0.5 cm of copper, 22.5 cm of lead, and 50 cm of polyethylene, which reduce backgrounds from external photons and neutrons. A 5-cm thick plastic scintillator muon veto encloses the shielding. An overburden of 780 m of rock, or 2090 meters water equivalent, reduces the muon flux by a factor of 5 x lo4.

All materials surrounding the detectors were screened to minimize radioactive decays which could produce neutrons. Neutrons resulting from radioactive decays outside the shield are moderated sufficiently to produce recoil energies below our detector threshold. Neutrons produced in the shield by high-energy cosmic-ray muons are tagged by the veto scintillator with an efficiency > 99%. The dominant unvetoed neutron background is expected to arise from neutrons produced by cosmic-ray muon interactions in the walls of the cavern. Events due to neutrons can be distinguished in part from those due to WIMPS because neutrons often scatter in more than one detector and interact at about the same rate in Ge and Si, whereas, WIMPS would not multiple-scatter, and coherent scalar WIMP interactions would occur m 6x more often in Ge than in Si detectors.

2. ANALYSIS

Between October 11,2003 and January 11,2004, we obtained 52.6 live days with the six close-stacked ZIP detectors of “Tower 1” (labeled Zl(Ge),

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Z2(Ge), Z3(Ge), Z4(Si), Z5(Ge) and Z6(Si) from top to bottom), after excluding time for calibrations, cryogen transfers, maintenance, and periods of increased noise. The trigger rate was B 0.1 Hz, with a recoil-energy threshold of w 2 keV (4 keV for Zl). Tower 1 was operated previously in an identical configuration at the shallow Stanford Underground Facility (SUF) 13.

Energy calibrations were performed repeatedly during the run. The excellent agreement between data and Monte Carlo simulations as well as the observation of the 10.4 keV Ga line from neutron activation of Ge indicate that the energy calibration is accurate and stable to within a few percent. Observation of the predicted energy spectrum from a 252Cf neutron source confirm the energy scale for nuclear recoils’.

We performed a blind analysis, in which the nuclear-recoil region for the WIMP-search data was not inspected until all cuts and analysis thresholds were defined using in situ gamma and neutron calibrations (see Figure 1). A combination of ionization-yield and phonon-timing cuts reject virtually all calibration electron recoils while accepting most of the nuclear recoils. The phonon timing cuts are based on both the phonon rise time and the phonon start time relative to the ionization signal (see Figure 2). Additional

remove events outside a fiducial volume (85%), close in time to hits in the veto system, with increased noise, or with recoil energy outside of our analysis range of 10-100 keV (20-100 keV for Zl). Figure 3 shows a conservative esitmate of the combined efficiency of all cuts on a WIMP signal. The cuts yield a spectrum-averaged effective exposure of 19.4 kg- days between 10-100 keV for a 60 GeV/c2 WIMP.

Analysis shows that about half the surface electron recoils with interactions in only a single detector (“singles”) were due to beta decays of con- taminants on surfaces, while the other half were from gamma rays. Gamma rates are M 50% higher at Soudan than they were at SUF, consistent with the higher Rn levels at Soudan and the absence of a l-cm-thick ancient Pb liner which surrounded the detectors at SUF. Total surface-event rates at Soudan are also somewhat higher than at SUF, consistent with the increased component due to gammas.

Based on 133Ba calibration sets and including systematic errors, we expected 0.4 f 0.3 electron recoil events in Z1 to be misidentified as nuclear recoils and a total of 0.3 f 0.2 electron recoil events to be misidentified in the other Ge detectors. Monte Carlo simulations predicted 0.58 f 0.08 neutrons (some unvetoed) produced from muon interactions outside the shielding, including uncertainties on neutron production.

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1.50 - I I I

0.5 -

n- "0 20 40 60 80 100

Recoil Energy (keV)

Figure 1. Ionization yield versus recoil energy for calibration data with 133Ba gamma source and 252Cf neutron source for detectors 22, 23 and 25 in Tower 1 showing the f 2 a gamma band (solid curves) and the f2a nuclear-recoil band (dashed curves) for 25, the detector with the worst noise of the three. Events with ionization yield < 0.75 (grey) are shown only if they pass the phonon-timing cuts. The vertical line is the 10 keV analysis threshold for these three detectors.

3. RESULTS

This blind analysis of the first Soudan CDMS I1 WIMP-search data set revealed no nuclear-recoil events in 52.6 kg-d raw exposure in our Ge detectors. As shown in Figure 4, these data set an upper limit on the WIMP- nucleon cross-section of 4 x cm2 at the 90% C. L. at a WIMP mass of 60 GeV/c2 for coherent scalar (spin-independent) interactions and a standard WIMP halo.

After unblinding the nuclear-recoil region, we found that an inferior pulse-fitting algorithm had been inadvertantly used to analyze most of the WIMP-search data. The limit based on the blind analysis correctly ac-

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.

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Ionization Yield

Figure 2. Phonon start time versus ionization yield for '33Ba gamma-calibration events (diamonds) and 252Cf neutron-calibration events (dots) in the energy range 20-40 keV in detector 25 in Tower 1. Lines indicate typical timing and ionication-yield cuts, resulting in high efficiency for nuclear recoils and a low rate of misidentified electron-recoil events.

counts for the resulting reduced efficiency. A second, non-blind analysis, using the intended pulse-fitting algorithm and the same blind cuts, re- sulted in a 5% higher WIMP detection efficiency (see Figure 3) and one nuclear-recoil candidate (at 64 keV in Z5), consistent with the expected surface-event misidentification quoted above.

At 60 GeV/c2, these limits are a factor of four below the best previous limits set by EDELWEISS1'. These data confirm that events detected by CDMS at SUF13 and those detected by EDELWEISS were not WIMP signals. Under the assumptions of a standard halo model, our new limits are clearly incompatible with the DAMA (1-4) signal region14; our data are for coherent scalar WIMP interactions (for DAMA regions under other assumptions see Ref [17]). Our new limits significantly constrain supersymmetric models under some theoretical frameworks5.

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50.4 c a, 0 .-

E0.2-

-

I I

Figure 3. Efficiency of combined cuts as a function of recoil energy for the blind analysis (solid) and for the second, non-blind analysis (dashed). The step at 20 keV is due to Zl’s 20 keV analysis threshold.

4. SYSTEMATIC EFFECTS

Several efforts have been made in the recent years to reconcile the evidence of the DAMA collaboration with the lack of signal in the CDMS and EDEL- WEISS searches, by studying effects from possible variations in the dark matter halo distribution and WIMP-nucleon interaction. The observed rate of events in any of these dark matter search experiments is converted to physical WIMP properties such as mass and interaction cross-section, by taking into account the dark matter density based on the assumed halo model and the WIMP-nucleon interaction type.

The Dark matter community has evolved a standard method for report- ing such results6. It is first assumed that the galactic dark matter halo is of a particular form, called the “standard” halo which is an isothermal sphere of WIMP gas with a certain velocity dispersion and average density6. This halo model allows us to interpret our observed spectrum (or limit) in terms of a relationship between a WIMP’S mass and its cross-section for interaction with our target material. It is also desirable to express the results in a manner that is independent of the particular choice of target material, i e . , in terms of WIMP-nucleon interaction parameters. To do this, we need to assume some particular form for the WIMP-nucleon interaction.

Conceptually, the simplest type of WIMP-nucleon elastic interaction is the one that is independent of the relative orientations of the WIMP and nucleon spins. This is refered to as a “spin-independent” (SI) interaction. Alternatively, for a WIMP with spin, we may have a WIMP-nucleon “spin-

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0 Y

10’ 10“ WIMP Mass [GeV]

Figure 4. New 90% C. L. upper limits on the WIMP-nucleon scalar cross-section from CDMS I1 at Soudan with no candidate events (solid curve). These limits constrain super symmetric models5 (shaded regions). The DAMA (1-4) 3a signal region14 is shown as a close contour. Also shown are limits from CDMS at SUF13 (dots), EDELWEISS15 (x’s), and the limit from the second, non-blind analysis of CDMS I1 at Soudan with 1 nuclear-recoil candidate event (dashes). All curves16 are normalized following6, as described in7V0.

dependent” (SD) interaction in which the interaction amplitude changes sign when either spin is reversed. In the completely non-relativistic limit (appropriate for halo velocities R 10-3c), SD and SI are the only two forms of interaction between a nucleon and a WIMP of any spin18.

4.1. EFFECTS DUE TO HALO MODEL

A range of halo modelslg, ranging from spherically symmetric to triaxial to discontinuous, have been explored to study the ratio of the observed DAMA modulation to the CDMS/EDELWEISS rate ratio as a function of

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WIMP mass. Some variations, such as local dark matter density, affects all experiments in similar ways, since they all see the same flux. However, variations in the velocity dispersion can affect different experiments in different ways, due to the different target materials. For a 60 GeV/c2 WIMP, the systematic effect can be on the order of 50% for extreme halo models such as streams of dark matter particles striking earth being ejected from a nearby galaxy. Variation due to halo models appear not be high enough to reconcile the positive signal observed by DAMA with the null signals from CDMS and EDELWEISSIQ.

4.2. EFFECTS DUE T O WIMP INTERACTION T Y P E

For publishing WIMP-nucleon exclusion limit plots, CDMS assumes the WIMP-nucleon interaction is a coherent scalar interaction, where the WIMP does not resolve the individual nucleons. WIMPS can in general have spin-dependent interactions, where the WIMP couples to individual nucleons. This spin-dependent interaction can also have in general different couplings to protons and neutrons. The primary target nucleus of CDMS detectors, 72Ge and 2sSi, do not have any net spin and hence have no sensitivity to SD interactions. Exclusive SD WIMP interaction could have beeen a systematic effect which would have reconciled the CDMS null signal with the DAMA observation, since DAMA uses NaI as their target nucleus, which has net spin from 23Na and 1271.

However, a recent exploration2' of the SD sensitivity of the CDMS 11, due to the presence of 8% 73Ge and 5% 29Si (isotopes with net spin), has shown that CDMS I1 places the strongest limits cm2 for a 60 GeV/c2 WIMP) on WIMP-neutron SD interaction, inconsistent with DAMA. If the SD interaction i s exclusively due to WIMP-proton couplings, even though CDMS I1 has sensitivity comparable to DAMA, it does not rule out the DAMA signal.

4.3. EFFECTS DUE T O QUENCHING FACTORS

The goal of direct detection experiments is to be sensitive to recoil energies down to 10 keV, at event rates well below l/kg/day. For this, the detector response to low-energy nuclear recoils must be well known. Depending on the detection process (ionization, scintillation or heat), this response may differ significantly from that inferred from calibration with electron or gamma-ray sources. To account for this a parameter called the quenching factor is typically used, defined as the ratio of the signal amplitudes induced

113

by a nuclear recoil and an electron recoil of the same energy. This factor depends mainly on the detector material, the energy of the recoil and the detection process. Systematic uncertainties in understanding the quenching factor for a particular experiment will lead to an error in the deduced cross- section.

The main difficulty for a precise and efficient measurement of the quenching factor is the production of recoils with known recoil energies so that one can measure the corresponding visible electron equivalent energy with an experimental apparatus and calculate the quenching factor for that detector. Such measurements have been performed at neutron scattering multidetector facilities, where well collimated monoenergetic neutron beams provide the necessary neutron recoil energy13. However, in any specific detector implementation, one has to make the assumption that the heat quenching factor is 1, so that one can interpret the ionization quenching measurement. This assumption is reasonable if the detector can be approximated as a closed system where all the excitation energy is ther- malized within a time scale less than that of the read-out of the heat signal and no energy is stored in crystal defects or other long-lived processes.

CDMS has done the absolute calibration of the energy scale for eletron- recoils with an earlier version of the current ZIP detector by measuring the ionization and phonon energy with a gamma-ray source at different bias voltages applied across the crysta121, where the heat quenching has been established to be indeed unity. The relative quenching factor for corresonding nuclear recoil with a neutron source has been done in sit% many times as part of our calibration process (see Figure 1). The relative ionization yield for nuclear-recoil events vs electron recoil events shows up at the expected 1/3 with the expected variation (see Figure 1) with recoil energy22. This confirms that our knowledge of the quenching factor for nuclear-recoils is accurate to within M 10% for true nuclear recoil energies betwen 10-100 keV.

5. ACKNOWLEDGEMENTS

This work is supported by the National Science Foundation under Grant No. AST-9978911, by the Department of Energy under contracts DEAC03- 76SF00098, DE-FG03-90ER40569, DE-FG03-91ER40618, and by Fermilab, operated by the Universities Research Association, Inc., under Contract No. DE-AC02-76CH03000 with the Department of Energy. The ZIP detectors were fabricated in the Stanford Nanofabrication Facility operated under

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NSF. We are grateful to the Minnesota Department of Natural Resources and the staff of the Soudan Underground Laboratory for their assistance.

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man and E. Witten, Phys. Rev. D31, 3059 (1985). 2. J. R. Primack, D. Sockel and B. Sadoulet, Ann. Rev. Nucl. Part. Sci. 38, 751

(1988). 3. L. Bergstrom, Rep. Prog. Phys. 63, 793 (2000); S. Eidelman et al., Phys. Lett.

B592, l(2004). 4. G. Jungman, M. Kaminowski, and K. Griset, Phys. Rep. 267, 195 (1996); J.

Ellis et al., Phys. Rev D67, 123502 (2003). 5. Y. G. Kim et al., J. High Energy Phys. 0212, 034 (2002); E. A. Baltz and P.

Gondolo, Phys. Rev. D67, 063503 (2003); A. Bottino et al., Phys. Rev. D69, 037302 (2004).

6. J . D. Lwein and P. F. Smith, Astropart. Phys. 6, 87 (1996). 7. D. S. Akerib et al., (CDMS Collab.) submitted to Phys. Rev. Lett.. 8. D. S. Akerib et al., (CDMS Collab.) in preparation for submission to Phys.

Rev. D. 9. K. D. Irwin et al., Rev. Sci. Instr. 66, 5322 (1995); T. Saab et al., AIP Proc.

605, 497 (2002). 10. T. Shutt et al., Nucl. Instr. Meth. A 444, 340 (2000). 11. R. M. Clarke et al., Appl. Phys. Lett. 76, 2958 (2000); V. Mandic et al., Nucl.

12. B. Cabrera et al., J. of Low Temp. Phys. 93 (3/4), 365 (1993). 13. D. S. Akerib et al., (CDMS Collab.) Phys. Rev. D68 82002 (2003). 14. R. Bernabei et al., Phys. Lett. B480, 23 (2000). 15. A. Benoit et al., Phys. Lett. B545, 43 (2002); G. Gerbier et al. (EDELWEISS

Collab.), Proceedings of the 8th TA UP Conference, Univ. of Washington, Seat- tle, Washington, Sep. 5-9, 2003.

Instr. Meth. A520, 171 (2004).

16. R. J. Gaitskell and V. Mandic, http://dmtools.brown.edu. 17. R. Bernabei e t al., Riv. Nuovo Cim. 26, 1 (2003). 18. A. Kurylov and M. Kamionkowski, hep-ph/0307185 19. C. J . Copi and L. M. Krauss, Phys. Rev. D67, 103507 (2003). 20. C. Savage, P. Gondolo and k. Freese, astro-ph/U408346 21. T. Shutt et al., Phys. Rev. Lett. 69, 3425 (1992). 22. J. Lindhard, V. Nielsen, M. Schaxf€, and P. V. Thomsen, Mat. Fys. Medd.

23. E. Simon et al. , astro-ph/0212491 K. Dan. Vidensk. Selsk. 33, 10 (1963).

NEW FIELD THEORY EFFECT AT ENERGY DENSITIES CLOSE TO

THE DARK ENERGY DENSITY

E. I. GUENDELMAN AND A. B. KAGANOVICH Physics Depadment, Ben Gupion University of the Negev,

Beer Sheva 8.4105, Israel E-mail: guendelabgumail. bgu. ac.il ; alexk@bgumail. bgu.ac.il

An alternative gravity and matter fields theory is studied where the consistency condition of equations of motion yields strong correlation between states of "primordial'' fermion fields and local value of the dark energy. The same "primordial" fermion field at different densities can be either in states of regular fermionic matter or in states presumably corresponding to the dark fermionic matter. In regime of the fermion densities typical for normal particle physics, each of the primordial fermions splits into three generations identified with regular fermions. When restricting ourselves to the first two fermion generations, the theory reproduces general relativity and regular particle theory. When fermion energy density is comparable with vacuum energy density, the theory allows new type of states. The possibility of such Cosmo-Low Energy Physics (CLEP) states is demonstrated by means of solutions of the field theory equations describing FRW universe filled by homogeneous scalar field and uniformly distributed nonrelativistic neutrinos. Pri- mordial neutrinos in CLEP state are drawn into cosmological expansion by means of dynamically changing their own parameters. Some of the features of the CLEP state in the late time universe: the mass of the primordial neutrino increases as a3I2 (a = a(t) is the scale factor); its energy density scales as a sort of dark energy and approaches constant as a -+ 00; this cold dark matter possesses negative pressure and its equation of state approaches that of the cosmological constant as a 4 00; the total energy density of such universe is less than it would be in the universe free of fermionic matter at all. The latter means that nonrelativistic neutrinos are able to produce expanding bubbles of the CLEP state playing the role of a true "cosmological vacuum" sorrounded by a "regular" vacuum.

1. Two Measures Theory (TMT) and scale invariant model

In the series of papers'-8 we have studied a generally coordinate invariant theory where the action has the form S = J LlCPd4x+ J L z J - g d 4 x including two Lagrangians LI and LZ and two measures of integration: the usual one 6 and the new one CP. The latter is built of four scalar fields qa (u = 1,2,3,4), CP = & ~ u Q 4 & a b C d a ~ ( p a d u ( ~ ~ ~ q c ~ ~ q d and it is a scalar den-

115

116

sity too. The basic assumption is that L1 and L2 are functions of the matter fields, the dilaton field, the metric, the connection (or spin-connection ) but not of the "measure fields" va. Important feature of TMT that is responsible for many interesting and desirable results of the field theory models studied so far1-8 consists of the assumption that all fields, including also metric, connection (or vierbein and spin-connection) and the measure fields pa are independent dynamical variables. Varying pa, we get equations that yield L1 = M4 = const if Q, # 0. Here M is an integration constant with the dimension of mass.

In TMT there is no a need5,6 to postulate the existence of three species for each type of fermions (like three neutrinos, three charged leptons, etc.) but rather this is achieved as a dynamical effect of TMT in normal particle physics conditions. The matter content of our model includes the dilaton scalar field 4, two so-called primordial fermion fields (the neutral primordial lepton N , for short - primordial neutrino, and the charged primordial lepton E ) and electromagnetic field A,. Generalization to the non-Abelian gauge models including Higgs fields and quarks is straightforward6.

Keeping the general structure of the TMT action it is convenient to represent it in the form:

- J d 4 z e f a @ / M p [(a + hN.\/-S)pNE'N + (@ + h ~ f i ) p ~ E E ] (1)

where 3i (i = N , E ) is the general notation for the primordial fermion fields N and E l VI and V2 are constants, F,p = 8,Ap - 8oA,, p~ and pE are the mass parameters, VLN) = 8 + i W z d U c d ,

!pEdIscd + ieA,; R(w, e ) = e a p e b U R p y a b ( w ) is the scalar curvature, eg and wib are the vierbein and spin-connection; g p u = ege;qab and R,vab(W) = 8pWvab+WLaLdvcb-(p * v); constants b, k , hi are dimensionless parameters.

TiE) = 3, + +

The action (1) is invariant under the global scale transformations

e t ---f ee12et, w $ , + w$,l pa --f xaqa where I I X a = e28 - MP A , + A,, 4 -+. 4 - -el qi -+ e - 8 / 4 9 i , !Pi -+ e - 8 / 4 T i . (2) a!

117

One can show that except for a few special choices providing positivity of the energy, Eq.(l) describes the most general TMT action satisfying the formulated above symmetries.

The measure @ degrees of freedom appear in all equations of motion only through dependence on the scalar field C = @/6. In particular, the gravitational and all matter fields equations of motion include noncanonical terms proportional to aP<. With the set of the new variables

main the same) which we call the Einstein frame, the spin-connections become those of the Einstein-Cartan space-time and the noncanonical terms proportional to aP< disappear from all equations of motion. The appearance of a nonzero integration constant M4 in the mentioned above equation L1 = M4 spontaneously breaks the scale invariance (2 ) .

The gravitational equations in the Einstein frame take the standard GR form GP,(ijaa) = ;T$f where TELf = $,,@,, - ~ i j , u~aa$ ,a$ ,p + ijPUV,ff (4; <) +Tgm)+T$can) + T j f y r n a n C a n ) . Here GPu(ijap) is the Einstein tensor in the Riemannian space-time with the metric ijpv; Kff($;<) = (< + b)-2[b (M4e-2"4/Mp + 6) - b]; Ti";") and TLCcan) are the canonical energy momentum tensors respectively for the electromagnetic field and for (primordial) fermions N' and E' in curved space-time including also

is the noncanonical standard electromagnetic interaction of El. TIL$ contribution of the primordial fermions into the energy momentum tensor: * p o n c a 4 = -jPU xi Fi(<)Fi!Pi = -ijPuALKm) where i = N', E', and

The structure of T$'nOnCan) shows that it is originated by fermions but behaves as a sort of variable cosmological constant. This is why we will refer to it as dynamical fermionic A term. becomes negligible in gravitational experiments with observable matter. However it may be very important for some astrophysics and cosmology problems. The dilaton $ field equation has canonical form of the scalar field equation with the potential V,ff($; <) and with the constant of the Yukawa coupling to the primordial fermions N Pi(<). Equations for the primordial leptons take the standard form where the standard electromagnetic interaction presents also. All the novelty consists of the form of the < depending "masses" mi(<) of the primordial fermions:

Qi = e-aa4/Mp(c+k)l/a@i (4 and A, re- EaP = ea"@/Mp(< + b)1/2eaP, ( ( '+b)3 /4

(f noncan)

Fi(<) ~ i 2 - ~ / ~ ( < + k ) - 2 ( < + b)-ll2[C2 + (3hi - k)< + 2b(hi - k ) + M i ] .

The scalar field < is defined by the constraint which is nothing but the

118

consistency condition of equations of motion:

Generically the constraint (4) determines C as a very complicated function of 4 and q9:. However in a few most important situations the theory allows exact solutions of great i n t e r e ~ t ~ , ~ .

The detailed analysis of the c o n ~ t r a i n t ~ , ~ shows that in a typical particle physics situation, say detection of a single primordial fermion, the scalar field < in the space-time region occupied by the fermion can get only one of the three different almost constant values. Then according to Eq.(3) the primordial fermion is able to be exactly in three mass eigenstates which we identify with the three fermion generations. For two of them, which are realized as Fi(<) = 0, the gravity and matter fields equations turn into those of GR and regular particle physics. Since the classical tests of GR deal with fermionic matter of the first two generations, one shoud identify the above two generations with the first two ones. Notice that the Yukawa coupling constant of these fermion generations to the dilaton q!~ turns out to be zero automatically that solves the 5-th force problem.

In the case of the complete absence of massive fermions, the constraint determines < as the function of 4: < = b - 2%/(& + sM4e-2a4/Mp). The effective potential of the +field is then

2. New Field Theory Effect at Cosmo-Low Energy

Let us study a toy model8 where in addition to the homogeneous scalar field 6, the spatially flat universe is filled also with uniformly distributed nonrelativistic neutrinos as a model of dark matter. Spreading of the neutrino wave packets during their free motion lasting a long time yields extremely small values of q'9' = utu (u is the large component of the Dirac spinor a'). There is a solution where the decaying fermion contribution utu N 9 to the constraint is compensated by approaching < * -k. After averaging over typical cosmological scales (resulting in the Hubble low), the constraint (4) reads

Densities

119

where F N ( < ) ( c = - ~ = p,,(hv - k ) (b - k)lI2 (C + k)-2 + O((< + k)-') and niN) is a constant determined by the total number of the primordial neutrinos in the state with C -, - k , that we call Cosmo-Low Energy Physics (CLEP) state . We assume that VI > 0, VZ > 0, b > 0, k < 0, h N < 0, h N - k < 0, b + k < 0. Asymptotically, as a ( t ) --t 00, we obtain for the pressure and density of the uniformly distributed neutrino in the CLEP state

which is typical for the dark energy sector including both a cosmological constant and an exponential &potential. The total energy density and the total pressure (including both &field and neutrinos in CLEP state ) behave as those of a dark energy described by the scalar field with the potential 'dark (tota"l(4) = + &M4e-2a'#'/Mp. The remarkable result consists

in the fact that if bVi > h, which is needed for positivity of ty;, Eq.(5), then v$)(4) > vdark (4) This means that (for the same value of 4') the universe in "the CLEP state" has a lower energy density than the one in the "absent of fermions" state. One should emphasize that this result does not imply at all that pclep is negative.

For a particular value Q = m, the cosmological equations allow the analytic solution +(t) = $0 + g ln (Mpt ) , a( t ) o( t1 I3ext , where X = [&Mp(b-k)]-'(G +JkJh)1/2 The mass of the neutrino in such CLEP state increases exponentially in time: m N 1 C L E p N a3/2(t) - t'/Ze?zXt.

(total)

References 1. E.I. Guendelman and A.B. Kaganovich, Phys. Rev. D53, 7020 (1996); ibid.

D55, 5970 (1997); ibid. D56, 3548 (1997); ibid. D57, 7200 (1998); ibid. D60, 065004 (1999); Mod. Phys. Lett. A12, 2421 (1997); ibid. A13, 1583 (1998).

2. E.I. Guendelman, Phys. Lett. B412, 42 (1997); gr-qc/0303048; E.I. Guendel- man and E. Spallucci, hepth/0311102.

3. E.I. Guendelman, Mod. Phys. Lett. A14, 1043 (1999); Class. Quant. Gmv. 17, 361 (2000); gr-qc/0004011; Mod. Phys. Lett. A14, 1397 (1999); gr-qc/9901067; hepth/0106085; Found. Phys. 31, 1019 (2001).

4. A.B. Kaganovich, Phys. Rev. D63, 025022 (2001). 5. E.I. Guendelman and A.B. Kaganovich, Int. J . Mod. Phys. A17, 417 (2002). 6. E.I. Guendelman and A.B. Kaganovich, Mod. Phys. Lett. A17, 1227 (2002) 7. E.I. Guendelman and 0. Katz, Class. Quant. Grav. 20, 1715 (2003). 8. E.I. Guendelman and A.B. Kaganovich, gr-qc/0312006.

BRANON DARK MATTER: AN INTRODUCTION*

J.A.R. CEMBRANOS Departamento de Estadistica e Investigacidn Operativa 111, and

Departamento de Fisica Tedrica, Universidad Complutense de Madrid, 28040 Madrid, Spain

A. DOBADO AND A.L. MAROTO Departamento de Fisica Tedrica,

Universidad Complutense de Madrid, 28040 Madrid, Spain

This is a brief introduction to branon physics and its role in the dark matter problem. We pay special attention to the phenomenological consequences, both in high-energy particle physics experiments and in astrophysical and cosmological observations.

1. Introduction

Most of the works done in the context of the brane-world scenario consider our world brane as a rigid object which is placed at a given position in the extra dimensions. However, rigid objects are incompatible with Relativity and we should consider instead branes as dynamicd objects which can move and fluctuate along the extra dimensions. In such a case, apart from the Kaluza-Klein (KK) modes of the gravitons propagating in the bulk space, new fields appear on the brane which parametrize its position in the extra dimensions. This fields are the branons which we will study in this work.

2. Branon dynamics

Let us consider our four-dimensional space-time M4 to be embedded in a D-dimensional bulk space whose coordinates will be denoted by (x", y"), where xp, with p = 0,1 ,2 ,3 , correspond to the ordinary four dimensional space-time and ym, with m = 4 , 5 , . . . , D- 1, are coordinates of the compact

*This work is supported by DGICYT (Spain) under project numbers FPA 2000-0956 and BFM 2002-01003

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121

extra space of typical size RB. For simplicity we will assume that the bulk metric tensor takes the following form:

where the warp factor is normalized as W(0) = 1. The position of the brane in the bulk can be parametrized as Y M = (d', Y"(x) ) , and we assume for simplicity that the ground state of the brane corresponds to Y"(x) = 0.

In the simplest case in which the metric is not warped along the extra dimensions, i.e. W ( y ) = 1, the transverse brane fluctuations are massless and they can be parametrized by the Goldstone boson fields ~"(z), (Y = 4 , 5 , . . . D - 1, associated to the spontaneous breaking of the extra-space traslational symmetry. In that case we can choose the y coordinates so that the branon fields are proportional to the extra-space coordinates: na(z) = f2bEYm(z) , where the proportionality constant is related to the brane tension T = f4.

In the general case, the curvature generated by the warp factor explicitly breaks the traslational invariance in the extra space. Therefore branons acquire a mass matrix which is given precisely by the bulk Riemann tensor evaluated at the brane position: M& = gDYRDav~ly=~ .

The dynamics of branons can be obtained from the Nambu-Goto action. In addition, it is also possible to get their couplings to the ordinary particles just by replacing the space-time by the induced metric in the Standard Model (SM) action. Thus we get up to quadratic terms in the branon fields 2,3,4.

We can see that branons interact with the SM particles through their energy-momentum tensor. The couplings are controlled by the brane tension scale f . For large f , branons are therefore weakly interacting particles. In the case of a three-brane, branons are pseudoscalar particles. Parity on the brane then requires that branons always couple to SM particles by pairs, which ensures that they are stable particles. This fact means that branons are natural dark matter candidates '.

122

1w

80

560 (1 a

40

20

O 50 100 150 200

ai

Figure 1: Collider limits on branon parameters from single-photon and single-Z processes at LEP (L3) Limits from monojet and single-photon processes at Tevatron-I ’ (right)

(left).

3. Limits from colliders

Collider experiments can be used to set bounds on the parameters of branon physics, i.e. the brane tension scale f and the branon mass M . The L3 collaboration at LEP experiment has recently obtained very stringent limits from the analysis of single-photon processes in e+e- collisions (see Fig. 1). In addition, we have also estimated the limits coming from mono-jet and singlephoton processes at Tevatron (see Fig. 1).

4. Cosmological and astrophysical limits

The potential WIMP nature of branons means that these new particles are natural dark matter candidates. In the relic branon abundance has been calculated in two cases: either relativistic branons at freezeout (hot-warm) or non-relativistic (cold), and assuming that the evolution of the universe is standard for T < f (see Fig. 2). Furthermore, if the maximum temperature reached in the universe is smaller than the branon freeze-out temperature, but larger than the explicit symmetry breaking scale, then branons can be considered as massless particles decoupled from the rest of matter and radiation. In such a case, branons can act as nonthermal relics establishing a connection between the coincidence problem and the existence of large extra dimensions *.

If branons make up the galactic halo, they could be detected by direct search experiments from the energy transfer in elastic collisions with nuclei of a suitable target. From Fig. 3 we see that if branons constitute the

123

10

10

f GeV)

10 -l

10 -5

10 +3 -1 3 7

10 -9 10 10 10 10

M ( G a l

Figure 2: Relic abundance in the f - M plane for a model with one branon of mass M. The two lines on the left correspond to the R B , ~ ~ = 0.0076 and Re,h* = 0.129 - 0.095 curves for hot-warm relics, whereas the right line corresponds to the latter limits for cold relics (see for details). The lower area is excluded by single-photon processes at LEP-I1 ' together with monojet signal at Tevatron-I 7. The astrophysical constraints are less restrictive and they mainly come from supernova cooling by branon emission 5 .

dominant dark matter component, they could not be detected by present experiments such as DAMA, ZEPLIN 1 or EDELWEISS. However, they could be observed by future detectors such as CRESST 11, CDMS or G E NIUS 5 .

Branons could also be detected indirectly: their annihilations in the galactic halo can give rise to pairs of photons or e+e- which could be detected by y-ray telescopes such as MAGIC or GLAST or antimatter detectors (see for an estimation of positron and photon fluxes from branon annihilation in AMS). Annihilation of branons trapped in the center of the sun or the earth can give rise to high-energy neutrinos which could be detectable by high-energy neutrino telescopes such as AMANDA, Icecube or ANTARES. These searches complement those in high-energy particle colliders (both in e+e- and hadron colliders 6 3 7 ) in which real (see Fig. 1) and virtual branon effects could be measured '. Finally, quantum fluctuations of branon fields during inflation can give rise to CMB anisotropies through their direct contribution to the induced metric (work is in progress in these directions).

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1 o2 ,3 I (Gel 5

Figure 3: Elastic branon-nucleon cross section u,, in terms of the branon mass. The

thick (red) line corresponds to the R B , ~ ~ = 0.129 - 0.095 curve for cold branons in Fig. 2 from N = 1 to N = 7. The shaded areas are the LEP-I1 and Tevatron-I exclusion regions. The solid lines correspond to the current limits on the spin-independent cross section from direct detection experiments. The discontinuous lines are the projected limits for future experiments. Limits obtained from g .

References

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2. R. Sundrum, Phys. Rev. D59, 085009 (1999); A. Dobado and A.L. Maroto Nucl. Phys. B592, 203 (2001)

3. M. Bando et al., Phys. Rev. Lett. 83, 3601 (1999) 4. J.A.R. Cembranos, A. Dobado and A.L. Maroto, Phys. Rev. D65, 026005

(2002) and hep-ph/0107155 5. J.A.R. Cembranos, A. Dobado and A.L. Maroto, Phys. Rev. Lett. 90, 241301

(2003); T. Kugo and K. Yoshioka, Nucl. Phys. B594, 301 (2001); J.A.R. Cembranos, A. Dobado and A.L. Maroto, Phys. Rev. D68, 103505 (2003); hep-ph/0307015; hep-ph/0402142; hep-ph/0405165 and hep-ph/0406076; AMS Collaboration, AMS Internal Note 2003-08-02

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(2004) 9. R. Gaitskell and V. Mandic, http://dmtools.berkeley.edu

Neutrino Physics


NEUTRINO PROPERTIES FROM HIGH ENERGY ASTROPHYSICAL NEUTRINOS

SANDIP PAKVASA

It is shown how high energy neutrino beams from very distant sources can be utilized to learn about some properties of neutrinos such as lifetimes, mass hierarchy, etc. Furthermore, even mixing elements such as Ue3 and the CPV phase in the neutrino mixing matrix can be measured in principle. Pseudo-Dirac mass differences as small as 10-'8eV2 can be probed as well.

1. Introduction

We make two basic assumptions which are reasonable. The first one is that distant neutrino sources (e.g. AGN's and GRB's) exist; and furthermore with detectable fluxes at high energies (upto and beyond PeV). The second one is that in the not too far future, very large volume, well instrumented detectors of sizes of order of KM3 and beyond will exist and be operating; and furthermore will have (a) reasonably good energy resolution and (b) good angular resolution (w 1' for muons).

2. Neutrinos from Astrophysical Sources

If these two assumptions are valid, then there are a number of uses these detectors can be put to1. In this talk I want to focus on those that enable us to determine some properties of neutrinos: namely, probe neutrino lifetimes to 104s/eV (an improvement of lo8 over current bounds), pseudo-Dirac mass splittings to a level of 10-I8eV2 (an improvement of a factor of lo6 over current bounds) and potentially even measure quantities such as Ues and the phase 6 in the MNSP matrix2.

3. Astrophysical neutrino flavor content

In the absence of neutrino oscillations we expect a very small v, component in neutrinos from astrophysical sources. Fkom the most discussed and the most likely astrophysical high energy neutrino sources3 we expect nearly equal numbers of particles and anti-particles, half as many V L S as

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uhs and virtually no v:s. This comes about simply because the neutrinos are thought to originate in decays of pions (and kaons) and subsequent decays of muons. Most astrophysical targets are fairly tenous even compared to the Earth’s atmosphere, and would allow for full muon decay in flight. There are some predictions for flavor independent fluxes from cosmic defects and exotic objects such as evaporating black holes. Observation of tau neutrinos from these would have great importance. A conservative estimate4 shows that the prompt v, flux is very small and the emitted flux is close to the ratio 1 : 2 : 0. The flux ratio of ve : up : v, = 1 : 2 : 0 is certainly valid for those AGN models in which the neutrinos are produced in beam dumps of photons or protons on matter, in which mostly pion and kaon decay(fol1owed by the decay of muons) supply the bulk of the neutrino flux.

Depending on the amount of prompt v-flux due to the production and decay of heavy flavors, there could be a small non-zero vr component present.

4. Effect of Oscillations

The current knowledge of neutrino masses and mixings can be summarized as follows5. The mixing matrix elements are given to a good approximation with the solar mixing angle given by about 32’’ the atmospheric angle by about 45O and Ue3 < 0.17 limited by the CHOOZ bound. The mass spectrum has two possibilities; normal or inverted, and with the mass differences given by Smi, - 2.10V3eV2 and dmi, - 7.10W5eV2. Since dm2L/4E for the distances to GRB’s and AGN’s (even for energies upto and beyond PeV) is very large (> lo7) the oscillations have always averaged out and the conversion(or survival) probability is given by

i

Assuming no significant matter effects enroute, it is easy to show that the mixing matrix in Eq. (1) leads to a propagation matrix P, which, for any value of the solar mixing angle, converts a flux ratio of ve : up : v, = 1 : 2 : 0 into one of 1 : 1 : 1. Hence the flavor mix expected at arrival is simply an equal mixture of ve,vp and v, as was observed long If this universal flavor mix is confirmed by future observations, our current knowledge of neutrino masses and mixings is reinforced and conventional wisdom about the beam dump nature of the production process is confirmed as well. However, it would much more exciting to find deviations from it,

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and learn something new. How can this come about? Below is a shopping list of a variety of ways in which this could come to pass.

5. Deviations from Canonical Flavor Mix

There are quite a few ways in which the flavor mix can be changed from the simple universal mix.

The first and simplest is that initial flavor mix is NOT 1 : 2 : 0. This can happen when there are strong magnetic fields causing muons to lose energy before they decay, and there exist models for neutrino production in AGN's in which this does happen7. In this case the v:s have much lower energies compared to vhs and effectively the initial flavor mix is 0 : 1 : 0 and averaged out oscillations convert this into 1/2 : 1 : 1 on arrival.

The possibility that the mass differences between neutrino mass eigenstates are zero in vacuum (and become non-zero only in the presence of matter) has been raised recently8. If this is true, then the final flavor mix should be the same as initial, namely: 1 : 2 : 0.

Neutrino decay is another important possible way for the flavor mix to deviate significantly from the democratic mixg .We now know that neutrinos have non-zero masses and non-trivial mixings, based on the evidence for neutrino mixings and oscillations from the data on atmospheric, solar and reactor neutrinos.

If this is true, then in general, the heavier neutrinos are expected to decay into the lighter ones via flavor changing processes". The only questions are (a) whether the lifetimes are short enough to be phenomenologically interesting (or are they too long?) and (b) what are the dominant decay modes.

Since we are interested in decay modes which are likely to have rates (or lead to lifetimes) which are phenomenologically interesting, we can rule out several classes of decay modes immediately. For example, the very strong constraints on radiative decay modes and on three body modes such as u + 3u render them as being uninteresting.

The only decay modes which can have interestingly fast decays rates are two body modes such as ui + vj + x where x is a very light or massless particle, e.g. a Majoron. In general, the Majoron is a mixture of the Gelmini- Roncadellill and Chikasige-Mohapatra-Peccei12 type Majorons. The effective interaction is of the form:

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giving rise to decay:

v, -+ Dp (m vp) + z (3)

where 2 is a massless, spinless particle; va and vp are mass eigenstates which may be mixtures of flavor and sterile neutrinos. Explicit models of this kind which can give rise to fast neutrino decays have been discussed13. These models are unconstrained by p and T decays which do not arise due to the AL = 2 nature of the coupling. The couplings of v,, and ue are constrained by the limits on multi-body r, K decays, and on p-e university violation in r and K decays14, but these bounds allow fast neutrino decays.

There are very interesting cosmological implications of such couplings. The details depend on the spectrum of neutrinos and the scalars in the model. For example, if all the neutrinos are heavier than the scalar; the relic neutrino density vanishes today, and the neutrino mass bounds from CMB and large scale structure are no longer operative, whereas future measurements in the laboratory might find a non-zero result for a neutrino mass 15. If the scalars are heavier than the neutrinos, there are signatures such as shifts of the nth multipole peak (for large n) in the CMB 16. There are other implications as well, such as the number of relativistic degrees of freedom(or effective number of neutrinos) being different at the BBN and the CMB eras. The additional degrees of freedom should be detectable in future CMB measurements.

Direct limits on such decay modes are also very weak. Current bounds on such decay modes are as follows. For the mass eigenstate v1, the limit is about

TI 2 lo5 sec/eV (4)

based on observation of ijes from SN1987A l7 (assuming CPT invariance). For u2, strong limits can be deduced from the non-observation of solar anti-neutrinos in KamLAND18, a more general bound is obtained from an analysis of solar neutrino datalg leads to a bound given by:

~2 2 10-~ sec/eV ( 5 )

For v3, in case of normal hierarchy, one can derive a bound from the atmospheric neutrino observations of upcoming neutrinos20 :

73 2 10-l' sec/eV (6)

The strongest lifetime limit is thus too weak to eliminate the possibility of astrophysical neutrino decay by a factor about lo7 x (L/lOO Mpc) ~ ( 1 0

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TeV/E). It was noted that the disappearance of all states except ul would prepare a beam that could in principle be used to measure elements of the neutrino mixing matrix, namely the ratios Uzl : Uil : U:121. The possibility of measuring neutrino lifetimes over long baselines was mentioned in Ref.22, and some predictions for decay in four-neutrino models were given in Ref.23. The particular values and small uncertainties on the neutrino mixing parameters allow for the first time very distinctive signatures of the effects of neutrino decay on the detected flavor ratios. The expected increase in neutrino lifetime sensitivity (and corresponding anomalous neutrino couplings) by several orders of magnitude makes for a very interesting test of physics beyond the Standard Model; a discovery would mean physics much more exotic than neutrino mass and mixing alone. Neutrino decay because of its unique signature cannot be mimicked by either different neutrino flavor ratios at the source or other non-standard neutrino interactions.

A characteristic feature of decay is its strong energy dependence: exp(-Lm/E.r), where T is the rest-frame lifetime. For simplicity, consider the case that decays are always complete, i.e., that these exponential factors vanish.

The simplest case (and the most generic expectation) is a normal hierarchy in which both u3 and u2 decay, leaving only the lightest stable eigenstate ul. In this case the flavor ratio is Uzl : Uil : U:121. Thus if Ue3 = 0

due : duw : qLT N 5 : 1 : 1, (7)

where we used the neutrino mixing parameters given aboveg. Note that this is an extreme deviation of the flavor ratio from that in the absence of decays. It is difficult to imagine other mechanisms that would lead to such I high ratio of v, to up. In the case of inverted hierarchy, u3 is the lightest and hence stable state, and so9

due : dhIw : (bur = u,"3 : Ui3 : u;3 = 0 : 1 : 1.

If Ue3 = 0 and Oatm = 45', each mass eigenstate has equal up and ur components. Therefore, decay cannot break the equality between the q5+ and & fluxes and thus the (hue : I#+ ratio contains all the useful information. The effect of a non-zero Ue3 on the no-decay case of 1 : 1 : 1 is negligible.

When Ue3 is not zero, and the hierarchy is normal, it is possible to obtain information on the values of Ue3 as well as the CPV phase SZ4. The flavor ratio e / p varies from 5 to 15 (as Ue3 goes from 0 to 0.2) for cosS = +1 but

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from 5 to 3 for cosh = -1. The ratio r / p varies from 1 to 5 (cosh = +1) or 1 to 0.2 (cosh = -1) for the same range of Ue3.

If the decays are not complete and if the daughter does not carry the full energy of the parent neutrino; the resulting flavor mix is somewhat different but any case it is still quite distinct from the simple 1 : 1 : 1 mixg.

If the path of neutrinos takes them thru regions with significant magnetic fields and the neutrino magnetic moments are large enough, the flavor mix can be affectedz5. The main effect of the passage thru magnetic field is the conversion of a given helicity into an equal mixture of both helicity states. This is also true in passage thru random magnetic fieldsz6.

If the neutrino are Dirac particles, and all magnetic moments are comparable, then the effect of the spin-flip is to simply reduce the overall flux of all flavors by half, the other half becoming the sterile Dirac partners. If the neutrinos are Majorana particles, the flavor composition remains 1 : 1 : 1 when it starts from 1 : 1 : 1, and the absolute flux remains unchanged.

What happens when large magnetic fields are present in or near the neutrino production region? In case of Dirac neutrinos, there is no difference and the outcoming flavor ratio remains 1 : 1 : 1, with the absolute fluxes reduced by half. In case of Majorana neutrinos, since the initial flavor mix is no longer universal but is v, : vfi : v, x 1 : 2 : 0, this is modified but it turns out that the final(post-oscillation) flavor mix is still 1 : 1 : 1 !

Other neutrino properties can also affect the neutrino flavor mix and modify it from the canonical 1 : 1 : 1. If neutrinos have flavor(and equivalence principle) violating couplings to gravity (FVG) , or Lorentz invariance violating couplings; then there can be resonance effects which make for one way transitions(ana1ogues of MSW transitions) e.g. vp + v, but not vice ver~a~' .~*. In case of FVG for example, this can give rise to an anisotropic deviation of the vf i /v , ratio from 1, becoming less than 1 for events coming from the direction towards the Great Attractor, while remaining 1 in other directions2'.

Another possibility that can give rise to deviations of the flavor mix from the canonical 1 : 1 : 1 is the idea of neutrinos of varying mass(MaVaNs). In this proposalz9, by having the dark energy and neutrinos(a sterile one to be specific) couple, and track each other; it is possible to relate the small scale 2 x eV required for the dark energy to the small neutrino mass, and furthermore the neutrino mass depends inversely on neutrino density, and hence on the epoch. As a result, if this sterile neutrino mixes with a flavor neutrino, the mass difference varies along the path, with potential resonance enhancement of the transition probability into the sterile neutrino, and

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thus change the flavor mix30. For example, if only one resonance is crossed enroute, it can lead to a conversion of the lightest (mostly) flavor state into the (mostly) sterile state, thus changing the flavor mix to 1 - Uzl : 1 - U;l : 1 - U:,., 1/3 : 1 : 1, in case of inverted hierarchy and similarly M 2 : 1 : 1 in case of normal hierarchy.

Complete quantum decoherence would give rise to a flavor mix given by 1 : 1 : 1, which is identical to the case of averaged out oscillations as we saw above. The distinction is that complete decoherence always leads to this result; whereas averaged out oscillations lead to this result only in the special case of the initial flavor mix being 1 : 2 : 0. To find evidence for decoherence, therefore, requires a source which has a different flavor mix . One possible practical example is a source which emits vLs by decay of neutrons, and hence no U L S at all, with an initial flavor mix of 1 : 0 : 0. In this case decoherence leads to the universal 1 : 1 : 1 mix whereas the averaged out oscillations lead to 3 : 1 : 131. The two cases can be easily distinguished from each other.

If each of the three neutrino mass eigenstates is actually a doublet with very small mass difference (smaller than 10-6eV), then there are no current experiments that could have detected this. Such a possibility was raised long It turns out that the only way to detect such small mass differences (10-12eV2 > bm2 > 10-18eV2) is by measuring flavor mixes of the high energy neutrinos from cosmic sources. Relic supernova neutrino signals and AGN neutrinos are sensitive to mass difference squared down to 10-20eV2 33 .

Let (vl , uz , v3 ; v1 , vz , v;) denote the six mass eigenstates where Y+ and v- are a nearly degenerate pair. A 6x6 mixing matrix rotates the mass basis into the flavor basis. In general, for six Majorana neutrinos, there would be fifteen rotation angles and fifteen phases. However, for pseudo- Dirac neutrinos, Kobayashi and Lim34 have given an elegant proof that the 6x6 matrix VKL takes the very simple form (to lowest order in 6rn2/rn2:

+ + + - -

where the 3 x 3 matrix U is just the usual mixing matrix determined by the atmospheric and solar observations, the 3 x 3 matrix UR is an unknown unitary matrix and VI and VZ are the diagonal matrices V1 = diag (1,1,1)/fi, and Vz=diag(e-i@l, e--i@2, e-2@3)/fi, with the q& being arbitrary phases.

As a result, the three active neutrino states are described in terms of

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the six mass eigenstates as:

The flavors deviate from the democratic value of

6P, = -- -xl + - x 2 3 4 "3 3 where xi = sin2 (6m:L/4E).The flavor ratios deviate from 1 : 1 : 1 when one or two of the pseudo-Dirac oscillation modes is accessible. In the ultimate limit where LIE is so large that all three oscillating factors have averaged to i, the flavor ratios return to 1 : 1 : 1, with only a net suppression of the measurable flux, by a factor of 1/2.

6. Cosmology with Neutrinos

If the oscillation phases can indeed be measured for the very small mass differences by the deviations of the flavor mix from 1 : 1 : 1 as discussed above, the following possibility is raised. It is a fascinating fact that non- averaged oscillation phases, 64j = 6m:t/4p, and hence the factors xj, are rich in cosmological i n f ~ r m a t i o n ~ ~ ? ~ ~ . Integrating the phase backwards in propagation time, with the momentum blue-shifted, one obtains

where I is given by

1 I+& & I=/I 2 d w 3 R m + (1 - 0,) '

Z, is the red-shift of the emitting source, and H t l is the Hubble time, known to 10% 36. This result holds for a flat universe, where R, + RA = 1, with R, and RA the matter and vacuum energy densities in units of the critical density. The integral I is the fraction of the Hubble time available for neutrino transit. For the presently preferred values R, = 0.3 and RA = 0.7, the asymptotic (z, -+ 00) value of the integral is 0.53. This limit

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is approached rapidly: at z, = l (2 ) the integral is 77% (91%) saturated. For cosmologically distant (ze > 1) sources such as gamma-ray bursts, non- averaged oscillation data would, in principle, allow one to deduce 6m2 to about 20%, without even knowing the source red-shifts. Known values of R, and RA might allow one to infer the source redshifts z,, or vice-versa.

This would be the first measurement of a cosmological parameter with particles other than photons. An advantage of measuring cosmological parameters with neutrinos is the fact that flavor mixing is a microscopic phenomena and hence presumably free of ambiguities such as source evolution or standard candle assumption^^^^^^. Another method of measuring cosmological parameters with neutrinos is given in Ref.38.

7. Experimental Flavor Identification

It is obvious from the above discussion that flavor identification is crucial for the purpose at hand. In a water cerenkov detector flavors can be identified as follows.

The up flux can be measured by the p's produced by the charged current interactions and the resulting p tracks in the detector which are long at these energies. V L S produce showers by both CC and NC interactions. The total rate for showers includes those produced by NC interactions of uhs and u:s as well and those have to be (and can be) subtracted off to get the real flux of u:s. Double-bang and lollipop events are signatures unique to tau neutrinos, made possible by the fact that tau leptons decay before they lose a significant fraction of their energy. A double-bang event consists of a hadronic shower initiated by a charged-current interaction of the ur followed by a second energetic shower from the decay of the resulting tau lepton4. A lollipop event consists of the second of the double-bang showers along with the reconstructed tau lepton track (the first bang may be detected or not). In principle, with a sufficient number of events, a fairly good estimate of the flavor r z 5 u, : up : u, can be reconstructed, as has been discussed recently. Peviations of the flavor ratios from 1 : 1 : 1 due to possible decays are so extreme that they should be readily identifiable3g. Future high energy neutrino telescopes, such as Icecube40, will not have perfect ability to separately measure the neutrino flux in each flavor. However, the situation is salvagable. In the limit of up - v, symmetry the fluxes for up and v, are always in the ratio 1 : 1, with or without decay. This is useful since the u, flux is the hardest to measure.

Even when the tau events are not at all identifiable, the relative number

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of shower events to track events can be related to the most interesting quantity for testing decay scenarios, i.e., the u, to up ratio. The precision of the upcoming experiments should be good enough to test the extreme flavor ratios produced by decays. If electromagnetic and hadronic showers can be separated, then the precision will be even better39 .Comparing, for example, the standard flavor ratios of 1 : 1 : 1 to the possible 5 : 1 : 1 generated by decay, the more numerous electron neutrino flux will result in a substantial increase in the relative number of shower events.The measurement will be limited by the energy resolution of the detector and the ability to reduce the atmospheric neutrino background which drops rapidly with energy and should be negligibly small at and above the PeV scale.

8. Discussion and Conclusions

The flux ratios we discuss are energy-independent because we have assumed that the ratios at production are energy-independent, that all oscillations are averaged out, and that all possible decays are complete. In the standard scenario with only oscillations, the final flux ratios are & : &p : g5v7 = 1 : 1 : 1. In the cases with decay, we have found rather different possible flux ratios, for example 5 : 1 : 1 in the normal hierarchy and 0 : 1 : 1 in the inverted hierarchy. These deviations from 1 : 1 : 1 are so extreme that they should be readily measurable.

If we are very fortunate41, we may be able to observe a reasonable number of events from several sources (of known distance) and/or over a sufficient range in energy. Then the resulting dependence of the flux ratio (ve /vp) on L/E as it evolves from say 5 (or 0) to 1, can be clear evidence of decay and further can pin down the actual lifetime instead of just placing a bound.

To summarize, we suggest that if future measurements of the flavor mix at earth of high energy astrophysical neutrinos find it to be

+"e/+vp/d"T = a / W ; (13) then

(i) a M 1 (the most boring case) confirms our knowledge of the MNSP2 matrix and our prejudice about the production mechanism;

(ii) a M 1/2 indicates that the source emits pure uhs and the mixing is conventional;

(iii) a M 3 from a unique direction, e.g. the Cygnus region, would be evidence in favour of a pure fie production as has been suggested

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recently42; (iv) a > 1 indicates that neutrinos are decaying with normal hierarchy;

and (v) a << 1 would mean that neutrino decays are occuring with inverted

hierarchy; (vi) Values of a which cover a broader range (3 to 15) and deviation

of the p / ~ ratio from l(between 0.2 to 5 ) can yield valuable information about Ue. and cos6. Deviations of a which are less extreme(between 0.7 and 1.5) can also probe very small pseudeDirac 6m2 (smaller than 10-I2eV2).

Incidentally, in the last three cases, the results have absolutely no dependence on the initial flavor mix, and so are completely free of any dependence on the production model. So either one learns about the production mechanism and the initial flavor mix, as in the first three cases, or one learns only about the neutrino properties, as in the last three cases. To summarise, the measurement of neutrino flavor mix at neutrino telescopes is absolutely essential to uncover new and interesting physics of neutrinos. In any case, it should be evident that the construction of very large neutrino detectors is a “no lose” proposition.

9. Acknowledgements

This talk is based on work in collaboration with John Beacom, Nicole Bell, Dan Hooper, John Learned and Tom Weiler. I thank them for a most enjoyable collaboration. I would like to thank the organisers of PASCOS 2004 for the invitation to give this talk as well as their hospitality and for providing a most stimulating atmosphere during the meeting. This work was supported in part by U.S.D.O.E. under grant DEFG03-94ER40833.

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NEUTRINO MASS MODELS AND LEPTOGENESIS

S . F. KING School of Physics and Astronomy, University of Southampton,

Southampton, SO1 7 lBJ, United Kingdom, E-mail: sjkQhep.phys.soton.ac.uk

In this talk we show how a natural neutrino mass hierarchy can follow from the type I see-saw mechanism, and a natural neutrino mass degeneracy from the type I1 see- saw mechanism, where the bi-large mixing angles can arise from either the neutrino or charged lepton sector. We also show that in such type I1 models the leptogenesis asymmetry parameter becomes proportional to the neutrino mass scale, in sharp contrast to the type I case, which leads to an upper bound on the neutrino mass scale, allowing lighter right-handed neutrinos and hence making leptogenesis more consistent with the gravitino constraints in supersymmetric models.

1. Introduction

The discovery of neutrino mass and mixing at the end of the last century implies that the Standard Model is incomplete and needs to be extended, but how '? In attempting to answer this question, it is useful to being by classifying models in terms of the mechanisms responsible for small neutrino mass, and large lepton mixing, as a first step towards finding the Next Standard Model. Amongst the most elegant mechanisms for small neutrino mass is the see-saw mechanism '. However the see-saw mechanism by itself does not provide an explanation for bi-large lepton mixing for either hierarchical or denegerate neutrinos.

In this talk we discuss model independent approaches to accounting for bi-large mixing in a natural way, based on the see-saw mechanism, which are valid for both hierarchical or denegenerate neutrino mass spectra. For the case of hierarchical neutrino masses arising from the type I see-saw mechanism, it is shown how the neutrino mass hierarchy and bi-large mixing angles could originate from the sequential dominance of right-handed neutrinos 3. It is then shown how to obtain partially degenerate neutrinos in a natural way by including a type I1 contribution proportional to the unit mass matrix, with the neutrino mass splittings and mixing angles controlled by type I contributions and sequential dominance '. The bi-large

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mixing angles could originate either from the neutrino or the charged lepton sector '. For a review see '. We also discuss leptogenesis in such type I1 models. The leptogenesis asymmetry parameter becomes proportional to the neutrino mass scale, in sharp contrast to the type I case, which leads to an upper bound on the neutrino mass scale, allowing lighter right-handed neutrinos and hence making leptogenesis more consistent with the gravitino constraints in supersymmetric models '.

2. See-saw Mechanism

The most commonly discussed version of the see-saw mechanism is sometimes called the type I see-saw mechanism 2 . The type I see-saw mechanism is illustrated diagramatically in Fig. l(a).

Figure 1. Diagram (a) shows the contribution from the exchange of a heavy right- handed neutrino as in the type I see-saw mechanism. Diagram (b) illustrates the contribution from an induced vev of the triplet A. At low energy, they can be viewed as contributions to the effective neutrino mass operator from integrating out the heavy fields v i and Ao, respectively.

In models with a left-right symmetric particle content like minimal left-right symmetric models, Pati-Salam models or grand unified theories (GUTS) based on SO( lo), the type I see-saw mechanism is often generalized to a type I1 see-saw (see e.g. 8 ) , where an additional direct mass term mFL for the light neutrinos is present.

With such an additional direct mass term, the general neutrino mass

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matrix is given by

Under the assumption that the mass eigenvalues M R ~ of MRR are very large compared to the components of mEL and mLR, the mass matrix can approximately be diagonalized yielding effective Majorana masses

I1 I mtL M mLL + mLL

with

(3) mLL = -mLR M& mtR vT

for the light neutrinos. The direct mass term mEL can also provide a naturally small contribution to the light neutrino masses if it stems e.g. from a see-saw suppressed induced vev. The type I1 contribution may be induced via the exchange of heavy Higgs triplets of s U ( 2 ) ~ as illustrated diagramatically in Fig. 1 (b).

3. A Natural Neutrino Mass Hierarchy

In this section we discuss an elegant and natural way of accounting for a neutrino mass hierarchy and two large mixing angles in the type I see-saw mechanism. The starting point is to assume that one of the right-handed neutrinos contributes dominantly to the see-saw mechanism and determines the atmospheric neutrino mass and mixing. A second right-handed neutrino contributes sub-dominantly and determines the solar neutrino mass and mixing. The third right-handed neutrino is effectively decoupled from the see-saw mechanism.

The above Sequential Dominance mechanism is most simply described assuming three right-handed neutrinos in the basis where the right-handed neutrino mass matrix is diagonal although it can also be developed in other bases. In this basis we write the input see-saw matrices as

x o o a d p l \ l a R = ( O y o ) , 0 0 2 m L R = ( ! ; ; ) (4)

Each right-handed neutrino in the basis of Eq.4 couples to a particular column of mER. There is no mass ordering of X, Y, 2 implied in Eq.4. The dominant right-handed neutrino may be taken to be the one with mass Y

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without loss of generality. Sequential dominance occurs when the right- handed neutrinos dominate sequentially 3 :

le2L l f 2 L lef I IXYI IX‘Y’I >> - >> - Y X z ’ ( 5 )

where x, y E a, b, c and x’, y‘ E p , q, r . This leads to a full neutrino mass hierarchy mi >> mi >> m;. Ignoring phases, in the case that d = 0, corresponding to a Yukawa 11 texture zero in Eq.4, we have:

(el2 + If l2 Y m3 = 5’ y’ 1.12 mi - O ( y ) , m2 M - xs;2 ’

where s12 = sin012 is given below. Note that each neutrino mass is generated by a separate right-handed neutrino, and the sequential dominance condition naturally results in a neutrino mass hierarchy ml << m2 << m3.

The neutrino mixing angles are given to leading order in m2/m3 by 3 :

Physically these results show that in sequential dominance the atmospheric neutrino mass m3 and mixing 823 is determined by the couplings of the dominant right-handed neutrino of mass Y . The angle 613 is generically of order 013 - O(m2lm3) - 0.2. However the coefficient in Eq.7 can be arbitrarily small, since to leading order as b + -c, 6’13 + 0, but 0 1 2

remains large. The solar neutrino mass m2 and mixing 012 is determined by the couplings of the sub-dominant right-handed neutrino of mass X . From Eq.7, the solar angle only depends on the sub-dominant couplings and the simple requirement for large solar angle is a - b - c. The third right-handed neutrino of mass 2 is effectively decoupled from the see-saw mechanism and leads to the vanishingly small mass ml = 0.

4. A Natural Neutrino Mass Degeneracy

We now show that it is possible to obtain a (partially) degenerate neutrino mass spectrum by essentially adding a type I1 direct neutrino mass contribution proportional to the unit matrix: Thus we shall consider a type I1 extension 4, where the mass matrix of the light neutrinos has the form:

(8)

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Assuming here that the type I mass matrix miL is real, the full neutrino mass matrix is diagonalised by the same matrix that diagonalises the type I matrix:

(mKL)diag = mI1 V V ~ + vmiLvT = mIr n + diag(mi, mi, mi). (9)

In this case the neutrino mass scale is controlled by the type I1 mass scale m”, while the neutrino mass splittings are determined by the type I mass eigenvalues:

ml M lmlI -mil, m2 = lmI1 -mil, m3 M lmI1 -mil. (10)

Sequential dominance in the type I sector naturally predicts mi << mi << mi. Hence the very small mass splittings required for degenerate neutrinos can be achieved naturally by sequential dominance. The predictions for the mixings are determined from the type I mass matrix (the type I1 unit matrix is irrelevant). In particular the atmospheric and solar angles are given by the sequential dominance estimates in Eq.7 and are independent of the type I1 neutrino mass scale m”. However the angle 613 in Eq.7 is now of order 613 - O(mi/m!). This is now much smaller than the type I result 613 - O(m2/m3) since the neutrino mass splittings, controlled by the type I masses, are very small for partially degenerate neutrinos.

5. Mixing Angles From the Charged Leptons?

In this section we show how bi-large mixing could originate from the charged lepton sector using a generalization of sequential right-handed neutrino dominance to all right-handed leptons 5 . We write the mass matrices for the charged leptons mE as

mE = (I: q’ 4: e’ 1:) b’ .

In our notation, each right-handed charged lepton couples to a column in mE. For the charged leptons, the sequential dominance conditions are 5 :

14, Ib’L I4 14, k’l, If’l Idl, l9’Llr’l . (12)

They imply the desired hierarchy for the charged lepton masses mT >> m, >> me and small right-handed mixing of UeR. We assume zero mixing from the neutrino sector. A natural possibility for obtaining a small 613 is 5

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In leading order in Id' I / I f ' 1 and I e' 1 / I f '1 , for the mixing angles 012 , 023 and 613, we obtain (again ignoring phases here)

(14) a' 512 a' + c12 b' s12 e' - c12d' b' l C' f' tan012 = - tan023 M , tan013 M

613 only depends on d'/ f ' and el / f ' from the Yukawa couplings to the sub-dominant right-handed muon and on 012. We find that in the limit Id'[, le'l << I f ' l , the two large mixing angles 012 and 023 approximately depend only on a'lc' and b'lc' from the right-handed tau Yukawa couplings. Both mixing angles are large if a', b' and c' are of the same order.

6. Model Building Applications

6.1. Effective Two Right-Handed Neutrino Models

In sequential dominance we have seen that one of the right-handed neutrinos effectively decouples from the see-saw mechanism. If the decoupled right-handed neutrino is also the heaviest then it would be expected to play no part in phenomenology. In this case sequential dominance reduces to effectively two right-handed neutrino models 3. Recently there have been several studies based on the "minimal see-saw" involving two right-handed neutrinos g 1 and it is worth bearing in mind that such models could naturally arise as the limiting case of sequential dominance.

6.2. GUT and Family Symmetry Models

There are many models in the literature based on sequential dominance. A Pati-Salam model with U(1) family symmetry was considered in lo. Single right-handed neutrino dominance has also been applied to SO(10) GUT models involving a U ( 2 ) family symmetry l l . Sequential dominance with S U ( 3 ) family symmetry and SO(10) GUTS has been considered in 12. Type I1 up-gradable models based on sequential dominance of the ISD type with SO(3) family symmetry have been considered in 514. For GUT models the renormalisation group corrections need to be taken into account, although for a natural hierarchy such corrections are only a few per cent 13.

6.3. Sneutrino Inflation Models

Sequential dominance has recently also been applied to sneutrino inflation 14, 15. Requiring a low reheat temperature after inflation, in order to solve the gravitino problem, forces the sneutrino inflaton to couple very weakly to

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ordinary matter and its superpartner almost to decouple from the see-saw mechanism. This decoupling of a right-handed neutrino from the see-saw mechanism is a characteristic of sequential dominance.

7. Leptogenesis

Neutrino mass allows the possibility that the baryon asymmetry of the universe is generated by out-of-equilibrium decay of lepton-number v i e lating Majorana right-handed (s)neutrinos, whose decays result in a net lepton number which is subsequently converted to a net baryon number by sphaleron transitions. This mechanism is known as leptogenesis 16. In models which give a natural neutrino mass hierarchy, if the dominant right-handed neutrino is the lightest one then the washout parameter 6 1 - O(m3), which is rather too large compared to the optimal value of around eV, while if the dominant right-handed neutrino is either the intermediate one or the heaviest one then one finds 6 1 - O(m2) or arbitrary 6 1 , which can be closer to the desired value 17. a

We now discuss the consequences of the neutrino mass scale for l e p togenesis via the out-of-equilibrium decay of the lightest right-handed (s)neutrinos in type I1 see-saw models ’. In ’ we calculated the type I1 contributions to the decay asymmetries for minimal scenarios based on the Standard Model (SM) and on the Minimal Supersymmetric Standard Model (MSSM), where the additional direct mass term for the neutrinos stems from the induced vev of a triplet Higgs. The diagrams are shown in Figure 2. The result we obtained for the supersymmetric case is new and we corrected the previous result in the scenario based on the Standard Model. 19.

We subsequently derived a general upper bound on the decay asymmetry and found that it increases with the neutrino mass scale:

It leads to a lower bound on the mass of the lightest right-handed neutrino, which is significantly below the type I bound for partially degenerate

aIf a texture zero in the 11 entry of the neutrino Yukawa matrix is assumed, then an indirect link between the phase relevant for leptogenesis and the phase 6 measurable in neutrino oscillation experiments is possible ’*.

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neutrinos. It is worth emphasizing that these results are in sharp contrast to the type I see-saw mechanism where an upper bound on the neutrino mass scale is predicted. Here we find no upper limit on the neutrino mass scale which may be increased arbitrarily. Indeed we find that the lower bound on the mass of the lightest right-handed neutrino decreases as the physical neutrino mass scale increases. This allows a lower reheat temperature, making thermal leptogenesis more consistent with the gravitino constraints in supersymmetric models.

Figure 2. Loop diagrams in the MSSM which contribute to the decay v i + L:H,a for the caSe of a type I1 see-saw mechanism where the direct mass_ term f2r the neutrinos stems from the induced vev of a Higgs triplet. In diagram (f), A1 and A2 are the mass eigenstates corresponding to the superpartners of the SU(2)~-triplet scalar fields A and A. The SM diagrams are the ones where no superpartners (marked by a tilde) are involved and where Hu is renamed to the SM H i m .

8. Conclusion

In this talk we have discussed how a natural neutrino mass hierarchy can follow from the type I see-saw mechanism, and a natural neutrino mass degeneracy from the type I1 see-saw mechanism, where the bi-large mixing angles can arise from either the neutrino or charged lepton sector. The key to achieving naturalness is the idea of sequential dominance of right- handed neutrinos, namely that in the see-saw mechanism one of the right- handed neutrinos dominates and couples with approximately equal strength to the T and p families, leading to an approximately maximal atmospheric

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mixing angle. A second right-handed neutrino then plays the leading subdominant role, and couples with approximately equal strength to all three families, leading to a large solar mixing angle. We have shown that this can lead either to a hierchical neutrino mass spectrum or, if a type I1 contribution proportional to the unit matrix is considered, to approximately degenerate neutrino masses. We have pointed out that in such type I1 models the leptogenesis asymmetry parameter becomes proportional to the neutrino mass scale, in sharp contrast to the type I case, which leads to an upper bound on the neutrino mass scale, allowing lighter right-handed neutrinos and hence making leptogenesis more consistent with the gravitino constraints in supersymmetric models.

References

1. For a review see for example: S. F. King, Rept. Prog. Phys. 67 (2004) 107 [arXiv:hepph/0310204], and references therein.

2. P. Minkowski, Phys. Lett. B 67 (1977) 421; T. Yanagida in Pmc. of the Work- shop on Unified Theory and Baryon Number of the Universe, KEK, Japan, 1979; S.L.Glashow, Cargese Lectures (1979); M. Gell-Mann, P. b o n d and R. Slansky in Sanibel Talk, CALT-68-709, Feb 1979, and in Supergravity (North Holland, Amsterdam 1979); R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912; J. Schechter and J. W. F. Valle, Phys. Rev. D 22 (1980) 2227.

3. S. F. King, Phys. Lett. B 439 (1998) 350 [arXiv:hep-ph/9806440]; S. F. King, Nucl. Phys. B 562 (1999) 57 [arXiv:hep-ph/9904210]; S. F. King, Nucl. Phys. B 576 (2000) 85 [arXiv:hep-ph/9912492]; S. F. King, JHEP 0209 (2002) 011 [arXiv:hepph/0204360].

4. S. Antusch and S. F. King, arXiv:hep-ph/0402121. 5. S. Antusch and S. F. King, Phys. Lett. B 591 (2004) 104 [arXiv:hep

ph/0403053]. 6. S. Antusch and S. F. King, arXiv:hep-ph/0405272. 7. S. Antusch and S. F. King, Phys. Lett. B 597 (2004) 199 [arXiv:hep

ph/0405093]. 8. G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B 181 (1981) 287; R. N.

Mohapatra and G. SenjanoviC, Phys. Rev. D23 (1981), 165; C. Wetterich, Nucl. Phys. B187 (1981), 343.

9. P. H. Frampton, S. L. Glashow and T. Yanagida, Phys. Lett. B 548 (2002) 119 [arXiv:hep-ph/0208157]; M. Raidal and A. Strumia, Phys. Lett. B 553 (2003) 72 [arXiv:hepph/0210021]; S. F. King, Phys. Rev. D 67 (2003) 113010 [arXiv:hepph/0211228]. A. Ibarra and G. G. Ross, [arXiv:hep-ph/0312138].

10. S. F. King and M. Oliveira, Phys. Rev. D 63 (2001) 095004 [arXiv:hep ph/0009287]; T. Blazek, S. F. King and J. K. Parry, JHEP 0305 (2003) 016 [arXiv:hepph/0303192].

11. R. Barbieri, P. Creminelli and A. Romanino, Nucl. Phys. B 559 (1999) 17

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[arXiv:hep-ph/9903460]; S. Raby, Phys. Lett. B 561 (2003) 119 [arXiv:hep ph/0302027].

12. S. F. King and G. G. Ross, Phys. Lett. B 520 (2001) 243 [arXiv:hep ph/0108112]; G. G. Ross and L. Velasco-Sevilla, Nucl. Phys. B 653 (2003) 3 [arXiv:hep-ph/0208218]; S. F. King and G. G. Ross, Phys. Lett. B 574 (2003) 239 [arXiv:hep-ph/0307190].

13. S. F. King and N. N. Singh, Nucl. Phys. B 591 (2000) 3 [arXiv:hep ph/0006229]; S. Antusch, these proceedings, arXiv:hep-ph/0409229.

14. J. R. Ellis, M. Raidal and T. Yanagida, Phys. Lett. B 581 (2004) 9 [arXiv:hepph/0303242]; P. H. Chankowski, J. R. Ellis, S. Pokorski, M. Raidal and K. Turzynski, arXiv:hep-ph/0403180.

15. S. Antusch, M. Bastero-Gil, S. F. King and Q. Shafi, “Sneutrino Hybrid Inflation” (in preparation).

16. M. Fukugita and T. Yanagida, Phys. Lett. B174 (1986) 45. 17. M. Hirsch and S. F. King, Phys. Rev. D 64 (2001) 113005 [arXiv:hep-

ph/0107014]. 18. S. F. King, Phys. Rev. D 67 (2003) 113010 [arXiv:hegph/0211228]. 19. T. Hambye and G. Senjanovic, Phys. Lett. B 582 (2004) 73 [arXiv:hep

ph/0307237].

SOLAR NEUTRINO OSCILLATION - AN OVERVIEW

D.P. ROY Tata Institute of Fundamental Research,

Homi Bhabha Road, Mumbai 400 005, India

After a brief summary of the neutrino oscillation formalism and the solar neutrino sources and experiments I discuss the matter effect on solar neutrino oscillation. Then I discuss how the resulting alternative solutions are experimentally resolved in favour of the LMA solution, with particular emphasis on the SK, SNO and KL data.

The last four years have been widely described as the golden years of solar neutrino physics, thanks to three pioneering experiments - Su- perKamiokande (SK), Solar Neutrino Observatory (SNO) and KamLAND (KL). They have provided for the first time a unique solution to the solar neutrino anomaly in terms of neutrino oscillation with unambiguous mass and mixing parameters. I shall give an overview of the subject with particular emphasis on the role of these experiments. After a brief summary of the neutrino oscillation formalism and the solar neutrino sources and experiments I shall discuss the matter effect on solar neutrino oscillation. We shall see how it leads to four alternative solutions to the solar neutrino anomaly and then their experimental resolution in favour of the so called Large Mixing Angle (LMA) solution over the last four years.

Neutrino Mixing and Oscillation: It was already noted by Pontecorvo back in the sixties that if the neutrinos have non-zero mass then there will in general be mixing between the flavour and the mass eigenstates, which will lead to neutrino oscillation [l]. We assume for simplicity two neutrino flavours, in which case mixing can be described by one angle 8, i.e.

In fact it provides a very good approximation to the three neutrino mixing scenario for solar neutrino oscillation, with the second neutrino representing

150

151

a mixture of the vp and u, flavours. Now each mass eigenstate propagates with its own phase

(2) ,-i(Et-Pe) II e- - im2e

2E 1

where we have made the relativistic approximation, E = p + m2/2p, since neutrino masses are much smaller than their kinetic energy. Thus a ue produced at the origin will propagate as

Decomposing the u1,2 back into ue,p after a distance C one sees that the up terms do not cancel, which implies neutrino oscillation. In fact the coefficient of the up term represents the probability of u, -+ up oscillation, i.e.

= sin2 20sin2 ( - e ) Am2 , (4 )

where the first factor represents the amplitude and the second factor the phase of oscillation, with Am2 = mf - mg. Converting to convenient units for Am2(eV2), e(m) and E (MeV) one gets

Pv,--ryp (e) = sin2 20 sin2( 1.3Am2e/E), (5)

which corresponds to an oscillation wavelength 7r A = - .-- 2.4E/Am2.

1.3 Am2 - Thus for large mixing angle (sin228 - 1) one expects the following

pattern of oscillation probability from eqs. (5) and (6) , where the factor of in the last case comes from averaging over the phase factor.

(7) e < A = x/2 > > A

P”. +”@ 0 sin2 28 - 1 4 sin2 20 - $ Note that the corresponding survival probability is given by the remainder, i.e.

Pee Pv,+ve = 1 - Pve+v, * (8)

Now the typical energy of u, coming from a nuclear reactor or the sun is - 1 MeV. The typical distance between the source and the detector is a

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few hundred Km ([ - lo5 m) for the long base line KamLAND reactor experiment, while [ - 10l1 m for solar neutrino experiments. Thus one sees from eqs. (6) and (7) that the KamLAND and the solar neutrino experiments can probe neutrino mass down to Am2 - eV2 and 10-l' eV2 respectively, which is far beyond the reach of any other method of mass measurement.

Solar Neutrino Sources and Experiments: The main sources of solar neutrinos are the p p chains of nuclear reaction, taking place at the solar core, which convert protons into 4He (a particle). They are

(I) pp+2H+e++ve,2H+p-+ 3He+y, 3He+3He -+ 4 H e + 2 p ; or (11) 3He + 4He + 7Be + y, 7Be + e- -+ 7Li + ve, 7Li + p -+ 2 4He; or

(111) 7Be + p -+ 8B + y, 8B -+ 'Be* + e+ + ye, 'Be* -+ 2 4He.

While most of this conversion takes place by the shortest path (I) a small fraction (15%) follows a detour via 7 B e (11); and a tiny fraction (0.1%) of the latter follows a still longer detour via ' B (111). The resulting neutrinos are (I) the low energy p p neutrino, (11) the intermediate energy Be neutrino and (111) the relatively high energy B neutrino, with decreasing order of flux. The standard solar model (SSM) prediction for these fluxes is shown in Fig. 1 from BP 2000 [2]. It also shows the neutrino energy ranges covered by the different solar neutrino experiments.

The Gallium [3] and the Chlorine [4] experiments are based on the charged current reactions

v, + 71Ga --f e- + 71Ge and ve + 37Cl -+ e- + 37Ar. (9)

The produced 71Ge and 37Ar are periodically extracted by radiochemical method, from which the incident neutrino fluxes are estimated. The value of measured flux R relative to the SSM prediction gives the v, survival probability Pee. The SK is a real-time water Cerenkov experiment [5], based on the elastic scattering of neutrino on electron. The elastic scattering is dominated by the charged current reaction ue + e- - ve + e-, but it also has a limited sensitivity to the neutral current reaction ve,p + e- - ve,p + e-. Thus

cc N C

(10) 1 6

(I - Pee) N Pee + -(I - Pee). f f N G Ree = Pee + ,NC+CC

This experiment can also measure the energy and direction of the incident neutrino from those of the outgoing electron. The SNO is a Cerenkov

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SuperK, SNO I Chlorine , .

X 3

10"

10"

10'0

10'

10'

107

10'

10'

10'

10'

10'

10

Neutrino Energy (MeV)

Figure 1. energy ranges of the solar neutrino experiments [2].

The SSM prediction for the solar neutrino fluxes is shown along with the

experiment with a heavy water target, which can detect both the charged and neutral current events [6-91

(11) cc

u e + d - p + p + e - ,

(12) N C

~ e , p + d + Ue,p + p + 72.

In the 1st phase of the experiment the NC events were detected via neutron capture on deuteron, n + d -+ t + y [7]. In the 2nd phase salt was added to the heavy water target to enhance the NC detection efficiency via neutron capture on chlorine, n + 35C1 -+ 36Cl + y [8]. In the 3rd phase of this experiment going on now 3 H e gas filled counters are inserted into the heavy water target to detect NC events via n + 3 H e ---t t + p [9].

Table 1 shows the energy thresholds of the above four experiments along with the compositions of the corresponding solar neutrino spectra. It also shows the corresponding survival probability Pee measured by the rates of the charged current reactions relative to the SSM prediction [2]. For the SK experiment the survival probability calculated from Ree via eq. (10) is shown in parentheses. It shows that the survival probability is slightly

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above 1/2 for low energy i/e, falling to 1/3 at high energy. To understandits magnitude and the energy dependence we have to consider the effect ofsolar matter on neutrino oscillation.

Table 1. The ve survival probability Pee measured by the CC event rates of varioussolar neutrino experiments relative to the SSM prediction. For SK the Pee obtainedafter NC correction is shown in bracket. The energy threshold and composition ofneutrino beam are also shown for each experiment.

ExperimentR

Eth (MeV)Composition

Gallium0.55 ± 0.03

0.2PP (55%),

Be (25%), B (10%)

Chlorine0.33 ± 0.03

0.8B (75%),Be (15%)

SK0.465 ± 0.015(0.36 ±0.015)

5B (100%)

SNO-I0.35 ± 0.03

5B (100%)

Matter Enhancement (Resonant Conversion): Propagation throughsolar matter gives an induced mass to ve, which can have profound effecton neutrino oscillation. This is known as MSW effect after its authors [10].It arises from the charged current interaction of ve with solar electrons,while the neutral current interaction has no net effect since it is flavourindependent. Adding this interaction energy density to the free particlewave equation gives

= f,+^^V^V a.)dt V"/

where Hint = V2GNe(0) for ve(^^), with G and Ne denoting Fermi couplingand solar electron density. The quantity on the rhs numerator can beregarded as an effective mass or energy, i.e.

M'2 =c s

—s c7lf 0

0 mlc —ss c \ 0

• 2V2EGNe -scAm2

—scAm2 s2m?(14)

where s,c denote sin 0, cos 6.To get a simple picture let us assume for the moment that sin<9 <C 1,

so that the nondiagonal elements are small. Then we can identify the twodiagonal elements with the eigenvalues and the corresponding eigenstates"1,2 with the flavour eigenstates i/e>/i respectively. Fig. 2 shows the twoeigenvalues Ai j 2 against the solar electron density. At the solar surface

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( N , = 0) the 1st eigenvalue (m:) is smaller than the 2nd (m;). But XI increases steadily with N, and becomes much larger than A2 at the solar core. The cross-over occurs at a critical density

Am2 N,“ - 2 4 G E ‘OS 2e’

corresponding to Mi: = Mi:. Note however that the two eigenvalues actually never cross. There is a minimum gap between them given by the nondiagonal element, i.e.

r = Am2 sin2 28. (16)

This means that a u, produced at the solar core will come out as u2,

provided the transition probability between the two energy levels remains small.

Figure 2. tions of the solar electron density.

Schematic diagram of the effective mass (energy) eigenvalues of v ~ , ~ as func-

It is easy to show that this remarkable result does not depend on the sin8 << 1 assumption. Consider the effective mixing angle OM in matter,

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which diagonalises the above matrix, i.e.

sin 28 I cos 28 - 2fiGEN,/Am21 . (17) - - 2Mi; tan28M =

- Mi? I The electron density at the solar core is N," >> N,", so that the 2nd term in the denominator of eq. (17) is much larger than the 1st. This means OM << 1 at the solar core for any vacuum mixing angle 8; so that the u, produced there is dominated by the v1 component. At the critical density the denominator of eq. (17) vanishes, which corresponds to maximal mixing between the two components, again for any value of 9. This is why it is called matter enhanced (or resonant) conversion. It comes out from the sun as v2 with

P,, = sin2 8, (18)

provided the transition probability between the two levels remains small through out the propagation and in particular in the critical density region. This transition probability is given by the Landau-Zenner formula, i.e.

where X represents the oscillation wavelength in matter (which should not be confused with the eigenvalues X ~ J ) . If the solar electron density varies so slowly that the resulting variation in the 1st eigenvalue over an oscillation wavelength is small compared to the gap between the two, then y << 1 and the transition rate is exponentially suppressed. This is called the adiabatic condition. Thus the two conditions for the solar v, to emerge as v2 can be written as

Am2 cos 28 2fiGN,0

Am2 sin2 28 ' 2 cos 28(dN,/dCN,), ' where the 1st inequality ensures N," > N," and the second one is the adiabatic condition.

Fig. 3 shows the triangular region in the Am2 - sin2 28 plot satisfying the above two conditions for a typical neutrino energy of E = 1 MeV. The horizontal side follows from the 1st inequality, which gives a practically constant upper limit of Am2 since cos 28 N 1. The 2nd inequality (adiabatic condition) gives a lower limit on sin228. Moreover since this condition implies a lower limit on the product Am2sin228, it corresponds to the diagonal line on the log-log plot. The vertical side is simply given by the physical boundary, corresponding to maximal mixing. This is the so-called

157

MSW triangle. The indicated survival probabilities out side the triangle follow from the vacuum oscillation formulae (7) and (8) , while that inside is given by eq. (18). Thus Pee < 1/2 inside the MSW triangle and > 1/2 outside it, except for the oscillation maximum at the bottom (l N X/2) where Pee goes down to cos228. Finally the earth matter effect gives a small but positive u, regeneration probability, which implies a day-night a symmetry - i.e. the sun shines a little brighter at night in the ve beam. After day-night averaging one expects a ve regeneration probability

where p is matter density in the earth in gm/cc and Ye the average number of electrons per nucleon. For favourable values of Am2 and 8 bP,.,, can go upto 0.15 as indicated in the Figure. It is important to note that the positions of the MSW triangle, the earth regeneration region and the vacuum oscillation maximum depend only on the ratio Am2/E, as one can see from the relevant formulae. Thus their positions on the right hand scale hold at all energies.

10-l

-1 5

lo-* 10-I sin2 2 e

Figure 3. The positions of the MSW triangle, the earth regeneration effect and the vacuum oscillation maximum at E = 1 MeV are shown along with those of the SMA, LMA, LOW and VAC solutions. While the former positions scale with E the latter ones are independent of it.

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Four Alternative Solutions: Fig. 3 marks four regions in the mass and mixing parameter space, which can explain the magnitude and energy dependence of the survival probability Pee shown in Table 1. They correspond to the so-called Large Mixing Angle (LMA), Small Mixing Angle (SMA), Low Mass (LOW) and Vacuum Oscillation (VAC) solutions. For the LMA and SMA solutions (Am2 - eV2) the low energy Ga experiment ( E << 1 MeV) falls above the MSW triangle in Am2/E, while the SK and SNO experiments ( E >> 1 MeV) fall inside it. Therefore the solar matter effect can explain the observed decrease of the survival probability with increasing energy. For the LOW solution the low energy Ga experiment is pushed up to the region indicated by the dashed line, where it gets an additional contribution to the survival probability from the earth regeneration effect. Finally the VAC solution explains the energy dependence of the survival probability via the energy dependence of the oscillation phase in eq. (5). Fig. 4 shows the predicted survival probabilities of the four solutions as functions of neutrino energy. The LMA and LOW solutions predict mild and monotonic energy dependence, while the SMA and VAC solutions predict very strong and nonmonotonic energy dependence.

E, (in MeV)

Figure 4. VAC solutions.

The predicted ve survival probabilities (rates) for the SMA, LMA, LOW and

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Experimental Resolution in Favour of LMA: The survival rates in Table 1 show a slight preference for a nonmonotonic energy dependence, since the intermediate energy Chlorine experiment shows a little lower rate than SK and SNO. Therefore the SMA and VAC were the favoured solutions in the early days. However the situation changed drastically with the measurement of the energy spectrum by SK [S ] , shown in Fig. 5. It shows very little energy dependence in clear disagreement with the predictions of the SMA and VAC solutions in Fig. 4. In particular the SMA solution in Fig. 4 can not reconcile a low survival rate of Pee N 1/3 with an energy independent spectrum, in conflict with the SK data. The first charged current data from SNO [6] agreed with the low survival rate as well as the energy independent spectrum of the SK data. Thus the global analyses of the solar neutrino data at this stage ruled out the SMA and VAC solutions in favour of LMA and LOW [ll] .

2 0.7 0)

3 0.6 Q

s

0.5

0.4

0.3

Figure 5. The energy (in)dependence of the SK spectrum 151.

Then came the first neutral current data from SNO [7]. Being flavour independent the neutral current reaction is unaffected by neutrino oscill& tion. Hence it can be used to measure the Boron neutrino flux, for which

160

the SSM has a large uncertainty (see Fig. 1). The SNO neutral current measurement of this flux was in agreement with the SSM prediction and significantly more precise than the latter. Fig. 6 shows the results of global fit with and without the SNO (NC) data 1121. The left and the middle panels show two different methods of using the NC data and give very similar results. They strongly favour the LMA solution while barely allowing LOW at 3c7 level. The reason is that the earth regeneration contribution is too small to account for the large survival rate of the Ga compared to those of the SK and SNO (CC) experiments. Without the $NO (NC) data it was possible to effectively push up the survival rate of the latter experiments from 0.35 to 0.45 by exploiting the large uncertainty in the Boron neutrino flux, and hence accommodate the LOW solution as shown in the right panel (see also Fig. 4). However this was no longer possible after the SNO (NC) data [12,13] due to the improved precision of the Boron neutrino flux.

b) CI+Ga+SK

c) C1 +Ga+ SK

Figure 6. (right pannel) the first SNO neutral current data [12].

Results of global solar neutrino data fits with (left and middle) and without

Confirmation and Sharpening of the LMA Solution: Independent confirmation of the LMA solution came from the reactor antineutrino data

161

of the KamLAND experiment [14], assuming CPT invariance. It is a 1 kiloton liquid scintillator experiment detecting Ve from the Japanese nuclear reactors via

y;, + p --+ e+ + n. (22)

It also measures the incident Fe energy E via the visible scintillation energy produced by the positron and its annihilation with a target electron, i.e.

Evis = E + me + mp - m, = E - 0.8 MeV. (23)

The mean base line distance of the detector from the reactors is (C) - 180 km, which means it is sensitive to the Am2 2 eV2 region as mentioned earlier. Thus the experiment was designed to probe the LMA region. It was shown in [15] that if the survival rate seen at KL is < 0.9, then it would rule out the LOW solution at 3u level. The first KL result from 162ty data showed a survival rate R = 0.611 f .085 f .041 [14]. This was in perfect agreement with the LMA prediction and ruled out LOW at 5u level. Moreover the observed spectral distortion of the KL data, taken together with global solar neutrino data, confined the LMA solution to two subregions around Am2 = 7 and 14 x eV2, as shown in Fig. 7 [16]. They correspond to the 1st and 2nd oscillation minima ((C) = X and 2X) for Evis N 4.5 MeV, where the observed spectrum touches the no-oscillation prediction (see Fig. 9). The best fit point lies in the lower region called LMA-I, while LMA-I1 is allowed only at 99% CL [16,17].

This was followed by the data from the second (salt) phase of SNO [ B ] , with better NC detection efficiency. Combining the data from the two phases in a global analysis helped to constrain the mass and mixing parameters further [18,19]. Fig. 8 shows the results of global fits with phase-1, phase-2 and the combined SNO data [HI. The combined fit is seen to allow the LMA-I1 region only at 3u and disallow maximal mixing at 5u. The most important issue at this point was a definitive resolution of the LMA-I and I1 ambiguity. It was shown through a simulation study in [18] that if the KL spectrum from 1 kty data continues to favour the LMA-I region, then combining this with the global solar neutrino data will rule out LMA-I1 at > 3u level.

Recently the KL experiment has published their 766 ty data, whose best fit point is indeed in the LMA-I region [20]. Combining this with the global solar neutrino data rules out EMA-I1 at > 3a while sharpening the LMA-I region further [21,22]. Fig. 9 compares the KL spectrum with the no-oscillation and oscillation best fit predictions. Fig. 10 shows the result

162

10-5' I lo-' 1 on

Figure 7. The 90,95,99 and 99.73% CL contours of global fit to solar neutrino plus the 162 ty KL data. The 99.73% CL (30) contour of the solar fit is shown for comparison P61.

of combined fit to the KL and global solar neutrino data [21]. The 90% CL contour of the global solar fit is shown for comparison. In comparing the two 90% CL contours we see that the mass parameter is mainly determined by the KL data, while the mixing angle is determined mainly by the solar neutrino data. The best fit values along with la errors are

Am2 = (8.4 f .6) x 10W5 eV2, sin2 0 = 0.28 f .03. (24)

The best fit value of Am2 corresponds to the 1st oscillation minimum (A = ( e ) = 180km) occurring at Evis N 5 MeV, in agreement with Fig. 9. It is evident from eq. (24) and Fig. 10 that the solar neutrino oscillation has finally entered the arena of precision physics. Concluding Remarks: Oscillation analysis of global solar neutrino plus KL data has been extended to three neutrino flavours [21]. The resulting mam and mixing parameters agree very well with those of the two-flavour analysis discussed above. Moreover dropping one of the solar neutrino experiments from the global analysis is also found to make little difference to the result. Thus the results are stable and robust. As regards future prospects, the precision of Am2 will improve further with accumulation

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CI + Ga+ Sk spec + SNO (D20)

t L 0.2 0.3 0.4 0.5

t 1

0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5

sin%,,

Figure 8. 162 ty KL data with phase-I, phase-I1 and combined SNO data [18].

The 90,95,99 and 99.73% CL contours from global fits to solar neutrino plus

of more KL data. One expects some improvement in the precision of the mixing angle from the 3rd phase of SNO, but not from KL. The main reason for this is the occurrence of an oscillation minimum in the middle of the Ve spectrum N 5 MeV). This means that the coefficient of sin2 28 in the oscillation probability (eq. 5 ) is very small over this energy range. An interesting suggestion made in [23] is to reduce the base line length by half (- A/2), which would mean an oscillation maximum in the middle of the I&

spectrum instead. A KL type experiment at a reduced base line length of - 70 km has been estimated to improve the precision of sin2 6' significantly. It is my pleasure to thank the organisers of PASCOS'O4 for their in-

vitation and kind hospitality. Let me also take this opportunity to thank my teammates Abhijit Bandyopadhyay, Sandhya Choubey and Srubabati Goswami.

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E, (in MeV)

Figure 9. best fit predictions [21].

The 766 ty KL spectrum compared with the no-oscillation and oscillation

i i i

3.OxlO~~,, 0,2 0.3 0.4 0.5

sin%,,

Figure 10. ty KL data. The 90% CL contour of the solar fit is shown for comparison [21].

The 90,95,99 and 99.73% contours from global fit to solar neutrino plus 766

165

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1. B. Pontecorvo, Zh. Eksp. Teor. Fiz. 53 (1967) 1717. 2. J.N. Bahcall, M.H. Pinsonneault and S. Basu, Astrophys J. 555

3. SAGE: J.N. Abdurashitov et al., JETP 95 (2002) 181; (2001) 990; http://www.sns.ias.edu/-jnb.

GALLEX: W. Hampel et al., Phys. Lett. B447 (1999) 127; GNO: C. Cattadori, Talk at Neutrino 2004, Paris (2004).

4. B.T. Cleveland et al., Astro Phys. J . 496 (1998) 505. 5. SK Collaboration: S. Fukuda et al., Phys. Lett. B539 (2002) 179. 6. SNO: Q.R. Ahmad et al., Phys. Rev. Lett. 87 (2001) 071301. 7. SNO: Q.R. Ahmad et al., Phys. Rev. Lett. 89 (2002) 011301 &

8. SNO: S.N. Ahmed et. al., arXiv: nucl-ex/0309004. 9. SNO: H. Robertson, Talk at TAUP 2003, Seattle (2003).

011302.

10. L. Wolfenstein, Phys. Rev. D17 (1978) 2369; S.P. Mikheyev and A.Y. Smirnov, Sov. J. Nucl. Phys. 42 (1985) 913 pad . Fiz. 42 (1985) 14411.

11. A. Bandyopadhyay, S. Choubey, S. Goswami and K. Kar, Phys. Lett. B519 (2001) 83; G.L. Fogli, E. Lisi, D. Montanino and A. Palazzo, Phys. Rev. D64 (2001) 093007; J.N. Bahcall, M.C. Gonzalez-Garcia and C. Pena-Garay, JHEP 0108 (2001) 014; P.I. Krastev and A.Y. Smirnov, Phys. Rev. D65 (2002) 073002.

12. A. Bandyopadhyay, S. Choubey, S. Goswami and D.P. Roy, Phys. Lett. B540 (2002) 14.

13. V. Barger, D. Marfatia, K. Whisnant and B.P. Wood, Phys. Lett. B537 (2002) 179; J.N. Bahcall, M.C. Gonzalez-Garcia and C. Pena- Garay, JHEP 0207 (2002) 054; P.C. de Hollanda and A.Y. Smirnov, Phys. Rev. D66 (2002) 113005; A. Strumia, C. Cattadori, N. Ferrari and F. Vissani, Phys. Lett. B541 (2002) 327.

14. KamLAND: K. Eguchi et al., Phys. Rev. Lett. 90 (2003) 021802. 15. A. Bandyopadhyay, S. Choubey, R. Gandhi, S. Goswami and D.P.

Roy, J. Phys. G29 (2003) 2465. 16. A. Bandyopadhyay, S. Choubey, R. Gandhi, S. Goswami and D.P.

Roy, Phys. Lett. B559 (2003) 121. 17. G.L. Fogli et al., Phys. Rev. D67 (2003) 073002; M. Maltoni, T.

Schwetz and J.W. Valle, Phys. Rev. D67 (2003) 093003. J.N. Bah- call, M.C. Gonzalez-Garcia and C. Pena-Garay, JHEP 0302 (2003) 009; P.C. de Holanda and A.Y. Smirnov, JCAP 0302 (2003) 001.

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18. A. Bandyopadhyay, S. Choubey, S. Goswami, S.T. Petcov and D.P. Roy, Phys. Lett. B583 (2004) 134.

19. G.L. Fogli, E. Lisi, A. Marrone and A. Palazzo, Phys. Lett. B583 (2004) 149; P.C. de Hollanda and A.Y. Smirnov, Astropart. Phys. 21 (2004) 287.

20. KamLAND: T. Araki et al., arXiv: hepex/0406035. 21. A. Bandyopadhyay, S. Choubey, S. Goswami, S.T. Petcov and D.P.

Roy, arXiv:hepph/0406328. 22. J.N. Bahcall, M.C. Gonzalez-Garcia and C. Pena-Garay, JHEP 0406

(2004) 016; 0. Miranda, M. Tartola and J.W. Valle, arXiv; h e p ph/0406280.

23. A. Bandyopadhyay, S. Choubey and S. Goswami, Phys. Rev. D67 (2003) 113011.

STATUS OF THE MINIBOONE EXPERIMENT

ION STANCU* Department of Physics and Astronomy

University of Alabama Tuscaloosa, AL 35487, USA E-mail: ion.stancuOua. edu

The MiniBooNE E89S1 experiment at Fermilab has been designed to confirm or dismiss the LSND observation by looking for v, appearance in a vp beam. The experiment began taking beam data in September 2002. Here we describe the experiment, the first neutrino candidate events, and our expected sensitivity to a neutrino oscillation signal.

1. Introduction

Neutrino oscillations appear to be a widely-accepted phenomenon which successfully explains the solar electron neutrino deficit, as well as the atmospheric muon neutrino deficit. Moreover, the same deficits have been observed in artificial neutrino sources, as reported by the KamLAND2 and K2K3 experiments, respectively. The mass squared differences involved in these phenomena are Am:ol NN 7 x 10-'eV2/c4 and Amit,,, NN 3 x eV2/c4, while the corresponding mixing angles appear to be nearly maximal. In addition, the LSND experiment, which ran at the Los Alamos National Laboratory from August 1993 until December 1998, has also reported evidence for neutrino oscillations in two channels: the decay-at-rest channel i jp + f i e , and the decay-in-flight channel vfi + v,. The LSND final result4, combining all the data, yielded an excess of 87.9 f 22.4 f 6.0 events after background subtraction, which corresponds to an oscillation probability of (0.264 f 0.067 f 0.045)%. The allowed values in the (sin2 28, Am2) parameter space corresponding to this result are shown in Figure 1. Also shown are the 90% confidence level excluded regions for the Bugey' and KARMEN6 experiments. Despite the fact that the KARMEN data appears

*representing the MiniBooNE collaboration.

167

168

"E 10 a

-.* - 2

10 ' ''ud ' ' " " " ' ' ' " " " I ' ' " "

1 o - ~ 1 o - ~ 10-2 lo-' 1

s in24

\ i Pale shading. 99% 4

I-

. NU E M . I@' PROTONS ON IARGEI N U W BEW. 10" PROTONS ON TARGET

90% Conf

Figure 1. expected oscillation sensitivity after 2 years of running.

LSND allowed regions in the (sin' 28, Am') parameter space and MiniBooNE

to exclude a significant fraction of the LSND-favoured region, a combined analysis of the two data sets showed that practically this entire region is compatible with both experiments at the 90% confidence level7.

The Booster xeutrino Experiment (BooNE) at Fermilab is a natural follow-up to the LSND experiment, and has been designed to confirm or dismiss the evidence for neutrino oscillations reported by the Los Alamos measurement. The first phase of the project, a single detector known as MiniBooNE (E-898), has become fully operational in September 2002. The experiment has two initial goals: (i) extend the sensitivity for vp -+ ve oscillations by one order of magnitude in Am2 over previous searches; (ii) obtain several hundreds of events per year if the LSND signals are indeed due to neutrino oscillations. Moreover, should neutrino oscillations be observed, MiniBooNE can test for CP violation in the lepton sector by switching to an antineutrino ( C p ) beam, while the full BooNE project would add a second detector and carefully parameterize the v, + ue and up -+ De mixings.

2. The Neutrino Beam

The MiniBooNE neutrino beam is initiated by a primary beam of 8 GeV protons from the Fermilab Booster incident on a 71-cm-long Be target

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within a magnetic horn focusing system, followed by a 50-m-long pion decay volume. The proton beam is delivered to the experiment at a rate of up to 5Hz and an intensity of about 5 x 10l2 per spill. Each spill is made up of 84 buckets of beam every 18.8ns for a total duration of 1 . 6 ~ s - which allows for a very low cosmic-ray background in the detector. The Booster can reliably deliver protons for about 2/3 of a calendar year, which allows the experiment to receive up to 5 x 1020 protons on target (POT) per year.

The magnetic horn focuses secondary pions and kaons from the primary interactions. It operates at a current of 170kA for a pulse duration of 140ps, producing a toroidal magnetic field that focuses s+ and defocuses s- (or vice-versa). Therefore, a fairly pure vp or G,, beam can be produced, depending on the horn polarity - as shown in Figure 2 below.

Figure 2. Calculated MiniBooNE neutrino fiuzes at the location of the detector.

The neutrino flux at the detector will be determined using a variety of methods. Detailed simulations of the neutrino production processes have been performed and are ongoing. These simulations have been tuned using existing hadron production data, and will be complemented by data from the HARP experiment8 at CERN - which ran with the MiniBooNE Be target. A measurement of the up charged-current rate in the detector will check the up flux, as well as determine the energy distribution of the

170

muons in the decay region - which contribute to the intrinsic u, background via p+ --f e+v,ijp. The u, background from K+ -+ nOe+v, decays will be determined by measuring the high-transverse-momentum muons from K+ -+ p+v, decays.

3. The Detector

The MiniBooNE detector is located 500 meters from the neutrino source. It consists of a spherical tank of radius 6.1 m, lined with 1280 8-inch photomultiplier tubes (PMTs) supported on an inner structure of 5.75 m radius. These PMTs point inward and provide 10% photocathode surface coverage. The PMT support structure also provides an optical barrier to create an outer veto region, viewed by 240 8-inch PMTs. The tank is filled with 800 t of mineral oil, which provides an inner fiducial region of about 500 t.

The MiniBooNE mineral oil (Exxon/Mobil Marcol7) has an attenuation length of approximately 26 meters at 450 nm, a density of 0.836 g/cm3 and an index of refraction of 1.46. This oil produces some scintillation light, so both prompt Cherenkov and delayed scintillation light will be produced for particles with > 0.68. The total amount of light provides a good energy measurement for particles above and below the Cherenkov threshold.

A circular room located above the tank vault houses the electronics, data acquisition (DAQ), oil circulation, and calibration systems. The entire structure is covered with a mound of earth to provide some cosmic-ray shielding. Each PMT is attached to one Teflon-jacketed cable which provides the high voltage (HV) and returns the signal as well. The PMT cables are routed out of the tank into the DAQ system, where the signal is picked off the HV cable, amplified, and digitized. The DAQ hardware consists of custom-built cards in 13 VME crates which are read out via MVME2304 single-board computers (SBC). The data is zero suppressed by the SBCs and shipped via ethernet to a single Intel-based computer running Linux. This computer assembles the data and ships it to the Fermilab computer center where it is written to tape.

The trigger consists of an additional VME crate housing custom-built cards that collect PMT multiplicity and beam information to form event triggers. The primary trigger is a “beam-on-target” signal from the accelerator, which initiates a data readout in a 20ps window around the 1.6 ps beam spill (regardless of PMT multiplicity). The DAQ hardware and software have sufficient data buffering capabilities to create a virtually dead-time-free system.

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Calibration for the detector is obtained through three different systems. A pulsed laser provides light to four different light-scattering flasks hanging in the inner region of the tank. This light is used to determine the time and gain calibrations of each of the inner PMTs. An array of seven scintillator cubes hanging in the main region of the tank are used to tag a sample of stopping muons in the tank which allows energy calibration with muon tracks of known length. A muon tracker system consisting of 2 horizontal planes of scintillator is installed above the tank. The tracker allows muons of a well-known direction to be tagged and studied for direction calibration. In addition to these systems, the ubiquitous cosmic-ray muons provide a constant source of data with which to study and calibrate the detector. In particular, the stopped cosmic-ray muons provide, through their subsequent decays, an invaluable source of electrons as cross-checks of the Monte Car10 (MC) simulations, energy scale measure, and energy resolution. Preliminary studies indicate an electron energy resolution of 14% at the Michel electron endpoint energy (52.8MeV).

4. The MiniBooNE Neutrino Oscillations Search

The up -+ ve oscillation search will be conducted by measuring the event rate for the ve induced reaction, veC -+ e - X , and comparing to the rate expected from background processes. If the LSND result is indeed due to neutrino oscillations, approximately 1000 ve-induced events are expected due to vb + ve oscillations in two calendar years of running (1021 POT). The three main backgrounds to this search are: the intrinsic v, background in the beam from p and K decay in the decay pipe, mis-identification of p events (vpC -+ p - X ) , and mis-identification of no events (v,C -+ vpnoX). The number of events for signal and backgrounds are listed in

Table 1. Estimated number of neutrino oscillation signal and background events after 2 years of data taking with neutrinos (1021 POT). Also shown are the number of events from other neutrino reactions in MiniBooNE.

Process Reaction I Number of events

Intrinsic Ve background ueC + e - X Mis-ID p- background v p c + p - x

500,000 50,000

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Table 1. These backgrounds will be measured. As described in Section 2, the number of events from intrinsic ve that are produced from p+ decays in the target region will be determined from vb charged-current scattering in the detector. Also, the number of intrinsic ve from K+ decays will be determined from the p detected with the off-axis LMC spectrometer. The number of p- and no events mis-identified as e- events will be measured via the large number of correctly identified vpC -+ p - X and v@C -+ v,.lroX events, respectively.

5. Preliminary Data

The MiniBooNE detector and beam have been fully operational for over a year and a half now, and data taking has been proceeding quite smoothly. The detector has been calibrated with laser calibration events, the energy scale and resolution have been determined from cosmic-ray muons and Michel electrons, and approximately 378,000 clean neutrino events have been recorded with 3.55 x 1020 POT. Figure 3 below illustrates the simple selection criteria which reduce the beam off backgrounds to a level: a veto shield multiplicity cut Nveto < 6 eliminates the cosmic-ray muons, while a tank multiplicity cut Ntank > 200 eliminates the Michel electrons from the decay of the stopped cosmic-ray muons. The beam pulse width of about 1.6 ps is clearly seen.

Figure 3. after the simple selection criteria which reduce the beam off background.

MiniBooNE event distribution in and around the beam window before and

The ve appearance search will be a blind analysis. Consequently, despite the fact that MiniBooNE can clearly identify the beam-induced events, one is allowed to either look at all information for some events or some information for all events, but not all information for all events. Meanwhile, in

173

parallel to continuing understanding the detector response and tuning the MC simulations, the experiment is concentrating on other physics analyses which are not only interesting in their own right, but they are also necessary for the vp + v, oscillations analysis: they will check the data/MC agreement, the reliability of the reconstruction and particle-identification algorithms, and provide understanding of the beam-induced backgrounds.

MiniBooNE is clearly identifying charged-current quasi-elastic v,C -+ p X events. These events have a relatively high abundance (about 40%), a simple topology, and can be identified with a relatively high efficiency and purity (of approximately 30% and 90%, respectively). The muons are forward peaked along the incident neutrino beam direction, as clearly seen in Fig. 4 below. The predicted MC distribution is shown superimposed in the same figure, with our current (conservative) estimates for the error bars; they include errors in the neutrino fluxes and cross sections (as denoted by the red-shaded rectangles), as well as different systematic variations of the underlying optical parameters of the detector medium (yellow-contour rectangles). Both distributions have been normalized to unit area. The visible energy distribution is also in good agreement with the MC expectations, as illustrated in the same figure.

Figure 4. MiniBooNE reconstructed event direction with respect to the incident neutrino beam and visible (electron-equivalent) energy distributions for u,,C charged-current quasi-elastic events.

From a simple kinematic reconstruction one can use the reconstructed muon energy and direction to calculate the incident neutrino energy and also the momentum transfer. These quantities are shown in Fig. 5 below,

174

along with the MC predictions with a relative normalization. The lower- than-predicted data values at low Q2 are currently under investigation and can be influenced by a variety of factors, such as nuclear effects, nuclear form factors, etc. This effect is also related to the lower-than-predicted number of events in the most forward direction (first bin in the cos8, distribution of Fig. 4).

0.161

0.14

0.06

0.04

0.02k OO

* MC' QS a Shape Errors

M C 9, a Shape Errors + optcal Mcdel Variations

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

E p (GeV) 4' (GeV')

Figure 5. vpC charged-current quasi-elastic events.

MiniBooNE reconstructed incident neutrino energy and Q2 distributions for

MiniBooNE is also clearly identifying and reconstructing neutral current xo events from either coherent production u,C --+ v ,xoX, or resonant production u,(p/n) -+ v,A and the subsequent A decay. These events have a characteristic two ring topology (from the ?yo + yy decay), and the invariant 7ro is reconstructed from the reconstructed energies of the two photons, El and E2, and their relative angle, 012:

m2, = 2 & ~ ~ ( 1 - c0sel2).

The distribution of the reconstructed invariant mass is shown in Fig. 6 below, which yields a mass resolution of about 21MeV. The events contributing to this sample have the standard Nveto < 6 and Ntank > 200 selection criteria applied, the event vertex must reconstruct at least 50 cm away from the surface of the PMTs, and each ring must have at least an electron-equivalent energy of 40 MeV.

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_I MC signal +background _ _ _ _ _ __. MC background

Data (statistical errors only)

PRELIMINARY NO. no’s = 7208f 144

Mass = 0.1391 f 0.0005 GeV/c*

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 9.55 no mass (GeV/c )

Figure 6. the expected background from MC simulations, which is also peaked.

MiniBooNE reconstructed KO mass distribution. The dashed curve denotes

6. Conclusions

The MiniBooNE detector and beam line have been operating smoothly for over a year and a half now and we are in the process of analyzing different physics channels, while continuously improving our understanding of the detector response and MC simulations. Despite the fact that the total number of protons on target is a factor of 2.5 below the original design intensity, we are convinced that continuing modifications and improvements to the Fermilab Booster will bring the neutrino beam to the required levels in the near future.

The current plan is to run in the up mode until MiniBooNE collects 5 x 1020 protons on target, with the possibility of changing to the Pp mode afterwards and also 25 m absorber running. The future MiniBooNE sched- ule is dependent on the number of protons delivered per year to the experiment. First oscillations results are expected by 2005, and if the LSND signal is confirmed, an initial determination of the oscillation parameters can be made. A second detector (BooNE) will then be built at a different distance in order to obtain the highest precision measurement of the oscillation parameters. The neutrino flux goes as r-2 to very good approximation, so that a simple ratio of events in the two detectors as a function

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of energy will cancel most of the systematic uncertainties and will allow Am2 to be measured to about f0.02eV2/c4.

References

1. The MiniBooNE collaboration consists of the following institutions: Univer- sity of Alabama, Bucknell University, University of Cincinnati, University of Colorado, Columbia University, Embry Riddle Aeronautical University, Fermi National Accelerator Laboratory, Indiana University, Los Alamos Na- tional Laboratory, Louisiana State University, University of Michigan and Princeton University.

2. K. Eguchi et al. (KamLAND Collab.), Phys. Rev. Lett. 90, 021802 (2003). 3. M. H. Ahn et al. (K2K Collaboration), Phys. Rev. Lett. 90, 041801 (2003). 4. A. Aguilar et al. (LSND Collaboration), Phys. Rev. D64, 112007 (2001). 5. B. Achkar et al., Nucl. Phys. B434, 503 (1995). 6. B -4rmbruster et al. (KARMEN Collab.), Phys. Rev. D65, 112001 (2002). 7. E. D. Church et al. (LSND Collaboration), Phys. Rev. D66, 013001 (2002). 8. “Proposal to study hadron production for the neutrino factory and for the at-

mospheric neutrino flux”, M. G. Catanesi et al., CERN-SPSC/99-35 (1999).

SOLAR NEUTRINO RESULTS FROM SNO

J. C. MANEIRA ON BEHALF OF THE SNO COLLABORATION. Queen’s University

Physics Dept., Stirling Hall Kingston, Ontario K7L 3N6, Canada E-mail: [email protected]. queensu. ca

The Sudbury Neutrino Observatory (SNO) is an underground heavy-water Cherenkov detector designed to detect sB solar neutrinos through neutral (NC) and charged (CC) current interactions on deuterons and elastic scattering on electrons. The results from the pure DzO phase of the experiment confirmed solar model predictions and gave strong evidence for flavor change. In the second phase, 2 tonnes of NaCl were added to the heavy water, in order to enhance the detection of neutral current interactions. This allowed for precision, energy-unconstrained measurements of the solar neutrino flux that exclude maximal flavor mixing at a level of 5a. The talk focused on the characterization of the detector response and the implications of the salt phase results on Neutrino Oscillation Physics. A status report was also given on the installation and commissioning work for the third phase of SNO, in which 40 3He gas counters are deployed in the DzO volume in order to detect the NC neutrons independently from the CC events.

1. Introduction

The discrepancy between measured1~273i476i5 and predicted7v8 solar neutrino fluxes, that lasted for over thirty years, is known as the solar neutrino problem. A possible solution is that the ye’s produced in the Sun core undergo a flavor change on their way to the Earth, and an elegant mechanism for this is supplied by matter-enhanced oscillationsg~lO.

The Sudbury Neutrino Observatory15 is an underground heavy water Cherenkov detector located in the INCO, Ltd. Creighton mine near Sud- bury, Canada, at a depth of about 2 km (6010 m of water equivalent), specifically designed to test the flavor change hypothesis by separately measuring the flux of electron and all flavor solar neutrinos. The active target of the detector, consists of 1000 tonnes of heavy water(DzO), and is contained in a 12 m diameter transparent acrylic vessel. The Cherenkov photons produced in the D20 are detected by 9456 8 inch photomultiplier tubes (PMTs) mounted on a 17.8 m diameter stainless steel geodesic sphere and coupled

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to light concentrators that raise the effective coverage to 55 %. Ultr&pure light water surrounds the acrylic vessel and the PMT support structure, in order to shield the heavy water from high energy y rays originating in the PMTs and neutrons from the surrounding rock. An additional 91 PMTs are mounted facing outward on the support structure to serve as a cosmic muon veto.

SNO detects 'B solar neutrinos through the processes:

ve + d + p + p + e- v, + d + p + n + v, v, -t e- + v, + e-

(CC), (NC) 7

(ES). The charged current (CC) reaction is sensitive exclusively to ve's in

the energy range of solar neutrinos. The neutral current reaction (NC) is equally sensitive to all active neutrino flavors (x = e,p ,T) , but its energy and direction information are lost. The elastic scattering (ES) reaction is sensitive to all electron flavors as well, but with reduced sensitivity to vp and v,.

The counting of the free neutrons from the NC reaction provides a model-independent measurement of the total flux of active 'B solar neutrinos with an energy above the kinematic threshold of 2.2 MeV. In the first phase of SNO, with a pure heavy water target, the neutron was detected by observing the 6.25 MeV y ray following its capture on a deuteron. In the second phase of SNO, 2 tonnes of salt were added to the DzO and the NC rate measurement was made by observing an 8.6 MeV y ray cascade following the neutron capture on 35Cl. In the third phase, the neutrons will be detected by an array of 40 3He counters, independently from CC and ES events. After a brief review of the pure D20 phase result^'^^'^^^^ in subsection 1.1, the paper will describe the calibration and first results from the salt phase14 in section 2. The status and prospects for the third phase of SNO will be presented in section 3.

1.1. Previous Flux Results from the Pure DzO phase

The pure DzO data set was collected in 306.4 live-time days and contained 2928 events, after removal of instrumental and non-Cherenkov like backgrounds and application of the energy and volume cuts. The events were selected above a kinetic energy threshold of 5 MeV and below a fiducial radius of 550 cm.

The data are resolved into contributions from CC, ES, neutrons (NC+background) and low energy Cherenkov background events with the

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extended maximum likelihood method using probability density functions in kinetic energy, cosine of the angle with respect to the Sun, and radial position. The background rates were constrained to the measured levels. Under the assumption of an undistorted 'B energy spectrum, the solar neutrino fluxes measured with each reaction were found to be, in units of lo6 cm-'s-':

@cc = 1.76f:::;(stat.) f O.OS(syst.) @ E S = 2.39?::it(.~t~t.) f 0.12(syst.)

+o 44 +0.46 @Arc = 5.09-,:43 (stat.) -0.43 (syst.)

When the requirement of an undistorted spectrum is relaxed and only the direction and position variables are used in the signal extraction, the NC flux is found to be:

= 6.42 f 1.57(stat.)+::~~(syst.)

These results showed the presence of a non v, active flavor component in the solar neutrino flux, different from zero at a 5.30 level, demonstrating flavor change. In addition, the NC measurement of the total active 'B solar neutrino flux is in good agreement with solar model predictions.

2. The Salt Phase of SNO

In order to enhance the precision of the NC flux measurement, 2 tomes of NaCl (salt) were added to the D20 in May 2001. Neutron capture on 35Cl has a high cross section and results in a multiple y ray cascade, with a total energy of 8.6 MeV; higher than the 6.25 MeV y ray from 'H capture. Hence, not only is the neutron capture efficiency higher in the salt phase, but also the neutron events are shifted to higher energies, further away from the low energy background region. In addition, the higher y multiplicity allows the use PMT hit pattern isotropy to statistically separate between neutrons and single electron events (CC and ES).

The salt data set presented here was collected between July 26, 2001 and October 10, 2002 and corresponds to 254.2 live days.

2.1. Detector Response

After removal of instrumental and non-Cherenkov like backgrounds, the times and positions of hit PMTs are used to calculate estimators for the vertex position, the particle direction and energy and the event isotropy.

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The detector response was calibrated by using a source deployment system capable of reaching most positions in two perpendicular vertical planes inside the acrylic vessel, as well as in vertical axes in the light water volume.

The energy estimator uses an energy normalization obtained with the 16N y source and an optical model Calibrated with a laser diffusing source ("laserball") to estimate the event energy based on its number of hit PMTs (nhit), reconstructed position and direction. The laserball calibration measured an increase in optical attenuation in the D2O volume throughout the salt phase, as shown in Figure la) and the 16N calibration showed a steady drop in the detector response (about -2%yr-'). As shown in Figure lb), this drift was predicted by Monte Carlo simulations using time-varying D20 attenuation, so a correction was applied to the energy estimator. The uncertainty in the energy scale was measured with the 16N calibration to be 1.1%.

1 MO Inverse Attenuation vs. Wavelength I I Consted Nhk Mean n SNO Jullan Dale I

Figure 1. Optical and energy calibration. Left:a) Inverse attenuation length of D20 in function of wavelength, for different calibration periods in the salt phase, as measured with the laserball source. Right:b) Number of hit PMTs (corrected to the center of the detector) in function of date, for data (dark points) and Monte Carlo simulations(1ight points) as measured with the 16N source.

The neutron detection efficiency was calibrated with a 252Cf spontaneous fission source and its measured distribution in function of radius, for the analysis kinetic energy threshold of 5.5 MeV, is shown in Figure 2a). The volume-weighted detection efficiency within the analysis fiducial radius of 5.5 m was 0.399 f 0.012, about a factor of three higher than in the pure

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D2O phase. The angular distributions of the PMT hit pattern of each event was

expanded in terms of Legendre polynomials and the combination p14 = 01 + p4 was chosen as an isotropy parameter. Since single Cherenkov ring electron events from CC interactions are less isotropic than multiple ring y cascades from NC neutrons, 014 distributions were used to statistically separate NC and CC events. Figure 2 shows a comparison of data and Monte Carlo ,014 from the 252C f (neutrons, multiple 7s) and 16N (single 7) sources. The uncertainty in the mean value is 0.87%.

- Salt phase

---+--- ROphase

' '3; ' ' 'A ' ' '5; ' ' '€02 -Cf Source Radial Position (cm)

Figure 2. a) Neutron detection efficiency in the salt phase (squares) and in the pure DzO phase (circles), as measured with a z5zCf source. b) Distributions of the isotropy parameter 014 for data and Monte Carlo from the 252Cf and 16N sources.

2.2. Backgrounds

The dominant backgrounds to the solar neutrino analysis window are deuteron photo-disintegration neutrons due to uranium and thorium chain activity in the DzO. The average levels of this activity throughout the salt dataset were measured by ex-situ and in-situ methods, and used to calculate the number of expected neutrons, leading to 74.8?;",: events in the salt dataset. After adding other contributions (like fission, 24Na activation), the total amounts to 113f 26 background neutrons. The total contribution from low energy Cherenkov backgrounds was estimated to be lower than 14.7 events (68% C.L.), significantly lower than in the pure D2O data analysis due to the higher energy threshold. A class of events characterized by a nearly isotropic light distribution and a reconstructed position close to the acrylic vessel was identified and an upper limit of 5.4 events (68% C.L.)

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was set for this background. Neutrons produced at the acrylic vessel or in the H2O can propagate into the fiducial volume, but the enhanced neutron capture efficiency in salt makes the external neutrons readily apparent, so an additional radial distribution function was included in the fits to extract the rate of this component.

2.3. Solar Neutrino Results

The maximum likelihood method was applied to extract the fluxes of each neutrino interaction and the external neutron background rate from the salt data set. The likelihood function was constructed with probability density functions of volume-weighted vertex radius ( p = (Rfi t /600)3) , event angle with respect to the Sun (cosOsUn) and isotropy (&). As can be seen from the distributions shown in Figure 3, the isotropy parameter allows a good separation of the CC and NC classes of events, making possible an energy- unconstrained and model-independent analysis.

The fitted number of events yield the following fluxes, in units of 106cm-2s-1:

Q'cc = 1.59f~:~~(stat . )_o,o~(syst . ) +0.06

QES = 2.21.I;:;&tat.> f O.lO(syst.)

@ N C = 5.21 f 0.27(stat.) f 0.38(syst.)

The systematic uncertainties are listed in Table I1 or reference 14.

earlier results yields, in units of 106cm-2s-1: Adding the constraint of undistorted 8B spectrum for comparison with

QCC = 1.70*0.07(stat.)+;:~~(syst.) @'ES = 2.13f~:~~(stat.)-,,,, +0.15 (syst.)

@ N C = 4.90 f 0 . 2 4 ( ~ t ~ t . ) f o o : i ~ ( ~ y ~ t . )

in agreement with the pure D20 results shown in Section 1.1. A combined x2 analysis of the shape-unconstrained fluxes from the salt

phase, the day and night energy spectra from the pure D20 phase and results from other solar neutrino experiments was carried in order to constrain the allowed regions of the neutrino mixing parameters Am2 and tan28. The 8B flux is a free parameter ( f ~ ) in the fit and the hep flux is fixed to 9.3 x 103cm-2s-1. This analysis selects the LMA region, as shown in Figure 4a). A global analysis including the results from the reactor neutrino experiment KamLANDl' further reduces the allowed region (Figure 4b)), and the best fit point is Am2 = 7.12;:; x 8 = 32.5fi:i,

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Figure 3. Distribution of (a) p14, (b)cosOsun and (bottom plot) volumeweighted radius ( p ) for the selected events. Also shown are the Monte Carlo predictions for CC, ES, NC + internal neutrons and external neutron events scaled to the fit results. The dashed lines represent the summed components.

f~ = 1.02 (lo errors). At 99% C.L., only the lower band of the LMA region is allowed, and maximal mixing is disfavored at a level of 5 . 4 ~ .

3. Status report on the third phase of SNO

The main goal of the third phase of SNO is the measurement of the neutral current rate by an array of detectors designed to capture neutrons, in order to have event-by-event separation of NC and CC/ES neutrino interactions, leading to different and smaller systematics than in the previous two phases. The neutral current detector (NCD) array consists in 36 sets ("strings") of proportional counters filled with 3He-CF4([85:15] a t 2.5 atm) gas and 4 strings filled with 4He-CF4 gas. Figure 5 shows the string array installed in the D2O volume in a l m x l m grid. The diameter of the counters is 5 cm and the length is 9-11 m, depending on the attachment position in the

184

10.’ > 2

U “E

lo4

10 -’

i

-95% CL

99% CL

-99.73% CL

1 10.‘ t a n 2 9

1 tan%

Figure 4. LAND.

Global neutrino oscillation contours. a) Solar global, b) Solar global + Kam-

bottom of the AV. The total length of active counters is 398 m. The counter wall is 0.36 mm thick and built of low background (<lo ppt U, Th) nickel.

The detection reaction for the free neutron is:

3He + n + p + 3H + 0.76 MeV

and the counters detect the proton and/or triton ionization of the 3He-CF4 gas. The NCD electronics acquires the ionization pulses in two different ways: low rate (2 Hz) digitizing scopes read a 15,000 point full waveform and shaper/ADC channels can measure integrated charge at high rates (kHz). Rejection of a/@ radioactivity background is done through a combination of pulse shape and total pulse energy.

Prior to the NCD installation, the salt was removed from the D20 and the subsequent short calibration phase showed that the optical attenuation length and the neutron capture efficiency returned to the levels of the first pure D2O phase and the energy response returned to the level of the beginning of the salt phase, as expected. The NCD installation started in November 2003, lasted for 5 months and consisted of the tasks of tempo- rary removal of calibration equipment, underground laser welding of the counters and actual deployment of the 40 strings. Presently both the PMT and NCD systems of the SNO detector are being re-calibrated and com- missioned. Production data taking is expected to start soon and to last for about 2 years.

As opposed to the first two phases of SNO, in which the correlation coefficient between the NC and CC rates was -0.52, the NC and CC rate

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Figure 5. DzO volume.

Schematic view of the SNO detector with the NCD array installed in the

measurements in the NCD phase will be essentially independent. This will allow for a factor of 2 improvement in the precision of the CC/NC ratio and also an improved 812 value, in a global oscillation analysis.

4. Summary and Prospects

The salt phase of SNO has provided a high precision measurement of the total active *B solar neutrino flux in a model-independent way, without any assumption on the energy dependence of the neutrino flavor transformation. This has further demonstrated the phenomenon of neutrino flavor transformation and excluded maximal mixing in a scenario of 2-flavor neutrino oscillations.

The analysis of the full salt phase data set is close to completion and a detailed paper including updated fluxes, CC energy spectrum, day-night results and oscillation analysis will be published soon. The collaboration

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has also recently completed the installation of the NCD array and is moving forward to the third phase of SNO, from which further improved solar neutrino measurements are expected.

Acknowledgments

This work is supported by Canada: NSERC, Industry Canada, NRC, Northern Ontario Heritage Fund, Inco, AECL, Ontario Power Generation, HPCVL, CFI; USA: DOE; UK:PPARC. We thank the SNO technical staff for their strong contributions.

References 1. B.T. Cleveland et al., Astrophys. J. 496, 505 (1998). 2. Y. Fukuda et al., Phys.Rev. Lett. 77, 1683 (1996). 3. V. Gavrin, 4th Int. Workshop on Low Energy and Solar Neutrinos; Paris, May

4. W. Hampel et al., Phys. Lett. B447, 127 (1999). 5. T. Kirsten, X X t h Int. Conf. on Neutrino Physics and Astrophysics, Munich,

May 25-30, 2002; to be published in Nucl. Phys. B Proc. Suppl. 6. S. Fukuda et al., Phys. Lett. B539, 179 (2002) 7. J.N. Bahcall, M. H. Pinsonneault, and S. Basu, Astrophys. J. 555,990 (2001). 8. AS. Brun, S. Turck-Chihze, and J.P. Zahn, Astrophys. J. 525, 1032 (1999);

S. Turck-Chihze et al., Ap. J. Lett., v. 555 July 1, 2001. 9. L. Wolfenstein, Phys. Rev. D17, 23692374 (1978) 10. S.P. Mikheyev, A.Y. Smirnov, Sov. Jour. Nucl. Phys. 42, 913-917 (1985) 11. Q.R. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett. 87, 071301 (2001). 12. Q.R. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett. 89, 011301 (2002). 13. Q.R. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett. 89, 011302 (2002). 14. S.N. Ahmed et al. (SNO Collaboration), e-Print Archive:nucl-ex/0309004,

15. SNO Collaboration, Nucl. Instr. and Meth. A449, 172 (2000). 16. K. Eguchi et al., Phys. Rev. Lett. 90, 021902(2003)

19-21, 2003

submitted to Phys. Rev. Lett.

IMPLICATIONS OF RUNNING NEUTRINO PARAMETERS FOR LEPTOGENESIS AND FOR

TESTING MODEL PREDICTIONS

STEFAN ANTUSCH Department of Physics and Astronomy, University of Southampton,

Southampton, SO1 7 lBJ, United Kingdom, E-mail: [email protected]. soton. ac. uk

The running of neutrino parameters in see-saw models and its implications for lep togenesis and for testing predictions of mass models with future precision experiments are discussed using analytical approximations as well as numerical results.

1. Introduction

Fermion mass models and the leptogenesis' mechanism for explaining the baryon asymmetry of our universe typically operate at high energies close to the unification scale M u or at the see-saw scales, i.e. the masses the heavy right-handed neutrinos. Our knowledge about the neutrino masses and mixings on the other hand mainly stems from experiments on neutrino oscillations, performed at low energy. In order to compare the high-energy predictions with the low energy experimental data, the renormalization group (RG) running of the relevant quantities has to be taken into account. In see-saw models for neutrino masses (type I and type II), this requires solving the R G E s ~ - ~ for the effective neutrino mass matrix for the various effective theories which arise from successively integrating out the heavy degrees of freedom, in particular the heavy right-handed neutrinos.

2. Implications for Leptogenesis

For the leptogenesis mechanism, the relevant scale is the mass M I of the lightest right-handed neutrino, or, in the type I1 case, possibly also the mass scale M A of the lightest SU(2)~-triplet. In the energy range between the leptogenesis scale and the electroweak scale MEW, we can consider the running of the effective neutrino mass operator, which is produced from integrating out the heavy right-handed neutrinos and/or triplets.

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2.1. Enhancement of the Decay Asymmetries

The decay a ~ y m m e t r i e s ~ ? ~ ? ~ for type I leptogenesis as well as for type I1 leptogenesis (via the lightest right-handed neutrino) can be written as €1 N

is an effective mass for leptogenesis. Y, is the neutrino Yukawa matrix and we have considered the case of hierarchical right-handed neutrino masses and M A >> M I . In the SM or for a moderate tanP in the MSSM, the RG running from high to low energy leads mainly to a scaling of the neutrino mass matrix m, (see Fig. 1). Including the RG effects thus leads to an enhancement of the decay asymmetry for leptogenesislOill by a factor of roughly 20% in the MSSM and 30% - 50% in the SM.ll

-* ’EW (m,B,AU), where (m:kU) := (Yyt y” 11 1 Cf, Im [(Y,’)fl(Y,’),1(mv)fgl

1.3 1.6

1.25 1.5

9 1.4

2 1.15 s 1.3

- - v

g 1.2

p 1.1 $ 1.2

B B

B

1.05 1.1

1 1 2 4 6 8 1 0 1 2 2 4 6 8 1 0 1 2

1ogWGeV) log(p/GeV)

(a) (b) Figure 1. Scaling of the neutrino mass eigenvalues m, from the RG evolution of the effective neutrino mass matrix in the MSSM (Fig. l(a)) and in the SM (Fig. l(b)).” In the MSSM with large tanP, we have shown the evolution of m3 for a normal mass ordering and with CP phases set to 0. For the plots, mt = 178 GeV has been used.

2.2. Correction to the Bound on the Neutrino Mass Scale

From the requirement of successful thermal type I leptogenesis, a bound on the absolute neutrino mass scale can be derived.12 Among the significant corrections to this bound, included in recent calculations, are the effects from RG running between low energy and M I . With an increased decay asymmetry E~ as discussed in section 2.1, the produced baryon asymmetry increases as well. However, scattering precesses which tend to wash out the produced baryon asymmetry are enhanced as well, which typically over- compensates the correction to the bound from the enhanced decay asymmetry and makes the bound on the neutrino mass scale more r e ~ t r i c t i v e . ~ ~ ? ~ ~ ’ ~ ~

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2.3. Implications for Resonant Leptogenesis

Resonant leptogenesis15 relies on a small splitting between the masses of the lightest right-handed neutrinos, M I and M2, of the order of their decay widths. Given a model for neutrino masses with such a small mass splitting defined at Mu, it can be affected significantly by the RG evolution of the mass matrix of the heavy right-handed neutrinos from Mu to MI x M2.

On the other hand, one can also have exactly degenerate masses MI = M2 at high energy and generate the required mass splitting radiatively.16

3. Implications for Testing Model Predictions by F'uture Precision Experiments

Future reactor and long-baseline experiments have the potential to measure the neutrino mixing angles to a high precision. For testing the predictions of mass models using such precision measurements, the RG corrections will be important, in particular, if the observed mixing angles turn out to be close to theoretically especially interesting values. For conservative estimates of the RG effects, analytical f ~ r m i i l a e ~ ~ * ' ~ ~ ~ ~ for the running of the neutrino parameters below the see-saw scales can be used. For an accurate determination of the RG effects in specific models, the model dependent running above and between the see-saw scales can contribute significantly and often even dominates the RG effects.l7y6 Formulae which allow an analytic understanding of the running above the see-saw scales are in preparation.20

3.1. Radiative Generation of 013

One important parameter is the mixing angle 813. The knowledge of its value will allow to discriminate between many fermion mass models and furthermore, only if 1313 is not too small, future experiments on neutrino oscillations have the potential to measure leptonic CP violation. Do we expect 813 very close to zero at low energy? Even if 813 = 0 is predicted by some model at high energy, RG running will generate 613 # 0 at low energy. From a conservative estimate using the analytical formulae below the see-saw scales, it has been shownll that the RG corrections are often comparable to, or even exceed the expected sensitivity of future experiments. Note that for 013, small values of CP phases (as predicted e.g. by certain type I1 see-saw models21) can protect against large RG corrections.'l Ra- diative generation of 813 from running above the see-saw scales has been analyzed in Ref. 22.

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3.2. Modification of Complementarity Relations for 012

With the present neutrino data, complementarity relations such as 612 + 6c = 7r/4 (with 6c being the Cabibbo angle) axe allowed and will be tested by future experiments to a high accuracy. RG corrections can lead to significant modifications of such relations for 1912 .23

3.3. Corrections to Maximal Mixing

The present best-fit value for 623 is close to maximal. Typically, mass models predict a deviation of 623 from maximality, which is within reach of future long baseline experiment^.^^ If 623 turns out to be close to maximal, this would point towards a symmetry which fixes maximal mixing at high energy Mu. However, even if 823 = 1r/4 is predicted by some model at high energy, RG corrections from the running between Mu and low energy generate a deviation of the low energy value for 623 from m a ~ i m a l i t y . ~ ~ . ~ ~ In many cases, even for hierarchical neutrino masses, this deviation is comparable to, or exceeds the sensitivity of future experiments (see Fig. 2).

After ten years RG induced Asin2&q )

0.3 0.4 0.5 0.6 0.7 True value. of sin2&

(a) Experimental sensitivity

" I

10 20 30 40 50 ma

(b) RG corrections

Figure 2. Fig. 2(a) shows the expected sensitivity of future long-baseline experiments (combined MINOS, ICARUS, OPERA, JPARC-SK and MuMI; see Ref. 24 for details) for excluding maximal mixing sin2 623 = 0.5 in 10 years at lu, 2u and 30 (from light to dark shading). The dashed line shows the currently allowed region for 023 at 3 ~ . Fig. 2(b) (from Ref. 24; see also Ref. 11 for details) shows a conservative estimate (ignoring Yv- effects) for the RG corrections to maximal 623 from the running between Mu and MEW in the MSSM. The contour lines correspond to A sin2 623 = 0.02, 0.05, 0.08 and 0.1.

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4. Summary and Conclusions

Facing the high expected sensitivities of future experiments on the neutrino parameters, RG corrections are increasingly relevant for testing predictions of mass models. They are particularly important, if the neutrino mass spectrum turns out to be non-hierarchical or if experiments find the lepton mixing angles close to specific values such as 0 for 613, 1r/4 for 623 or compatible with complementarity relations such as 4 2 + 6c = n/4 (with 6c being the Cabibbo angle) for 612. For leptogenesis, the scaling of the neutrino masses by RG effects enhances the decay asymmetries for type 1/11 leptogenesis, effects washout parameters and finally lowers the bound on the absolute neutrino mass scale from the requirement of successful thermal type I leptogenesis.

References 1. M. Fukugita and T. Yanagida, Phys. Lett. €3174 (1986), 45. 2. P. H. Chankowski and Z. Pluciennik, Phys. Lett. B316 (1993) 312. 3. K. S. Babu, C. N. Leung and J. Pantaleone, Phys. Lett. B319 (1993) 191. 4. S. Antusch, M. Drees, J. Kersten, M. Lindner and M. Ratz, Phys. Lett. B519

(2001) 238, Phys. Lett. B525 (2002) 130. 5. S. Antusch and M. Ratz, JHEP 0207 (2002) 059; 6. S. Antusch, J. Kersten, M. Lindner and M. Ratz, Phys. Lett. B538 (2002) 87. 7. L. Covi, E. Roulet, and F. Vissani, Phys. Lett. B384 (1996), 169. 8. T. Hambye and G. Senjanovic, Phys. Lett. B582 (2004) 73. 9. S. Antusch and S. F. King, Phys. Lett. B597 (2004) 199. 10. R. Barbieri et al., Nucl. Phys. B575 (2000) 61. 11. S. Antusch, J. Kersten, M. Lindner, M. Ratz, Nucl. Phys. B674 (2003) 401. 12. W. Buchmiiller, P. Di Bari, M. Pliimacher, Nucl. Phys. B665 (2003), 445. 13. G . F. Giudice et al., Nucl. Phys. B685 (2004) 89. 14. W. Buchmuller, P. Di Bari, and M. Plumacher, (2004), hep-ph/0401240. 15. see also: A. Pilaftsis, these proceedings, (2004). 16. R. Gonzalez Felipe, F. R. Joaquim and B. M. Nobre, hep-ph/0311029. 17. S. F. King and N. N. Singh, Nucl. Phys. B591 (2000) 3; 18. P. H. Chankowski, W. Krolikowski, S. Pokorski, Phys. Lett. B473 (2000) 109. 19. J. A. Casas et al., Nucl. Phys. B573 (2000), 652. 20. S. Antusch, J. Kersten, M. Lindner, M. Ratz, M. Schmidt, in preparation. 21. S. Antusch and S. F. King, hep-ph/0402121. 22. J. w. Mei and Z. z. Xing, hep-ph/0404081. 23. H. Minakata and A. Y. Smirnov, hep-ph/0405088. 24. S. Antusch, P. Huber, J. Kersten, T. Schwetz, W. Winter, hep-ph/0404268.

STERILE NEUTRINOS IN COSMOLOGY AND SUPERNOVAE

M. CIRELLI' Yale University, New Haven, CT, USA

E-mail: marw. cirelliOyale.edu

I present some selected parts of a recent analysis in which we explore (and combine together) all possible effects due to sterile neutrinos; I focus here on cosmology (BBN, CMB, LSS) and supernovre.

1. Introduction

The study of sterile neutrinos (namely: additional light fermionic particles that are neutral under all Standard Model gauge forces, but can have a non negligible role in our world through their mixing with e , p, 7 neutrinos) has recently acquired a different flavor: the established solar and atmospheric anomalies seem produced by oscillations among the three active neutrinos so that their explanation in terms of oscillation into a us state is believed to be now ruled out as the dominant mechanism. This means that the relevant questions concerning sterile neutrinos nowadays have become: which is the subdominant role still possible for sterile neutrinos (in solar and atmospheric neutrinos, but also elsewhere)? which are the most sensitive experiments (in astrophysics, cosmology or man-made set-ups) in which sterile neutrinos can be discovered? which are the signatures of their presence? The investigation on us requires therefore a more extensive and deep approach, which must (i) include the established u, - up,., up - u, mixings, (ii) consider any possible u ~ , ~ , ~ - us mixing pattern and (iii) discuss all possible u sources and contexts. These considerations motivate the analysis performed in

Are sterile neutrinos still interesting? Yes, for at least two different sets and presented here, restricting to cosmology and SNe.

Work supported by DOE grant DEFG02-92ER-40704.

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of reasons. First (topdown), sterile fermions that are naturally light or cleverly lightened arise in many theories that try to figure out what is beyond the Standard Model. The discovery of a new light particle would be of fundamental importance and deserves to be investigated per se. Second (bottom-up), sterile neutrinos are repeatedly pointed as a possible explanation for several puzzling situations in particle physics, astronomy and cosmology. Any one of these puzzles, in general, points to specific sterile neutrinos (i.e. with a specific mixing pattern with the active ones) and it is therefore worthwhile to explore them in an extensive way.

2. Four neutrino mixing formalism

The extra sterile neutrino can mix with a mixing angle 8, with an arbitrary combination of active neutrinos. We allow all possible values of the mass m4. Such a formalism is completely general and covers of course all the possible mixing patterns. We need to choose, however, some intuitive limiting cases to present the results in the following: Mixing with a flavor eigenstate; the sterile neutrino oscillates into a well defined active flavor (v, or up or vT). Mixing with a mass eigenstate; the sterile neutrino oscillates into a matter eigenstate (v1 or v2 or v3), which consists of mixed flavors.

3. Sterile effects in cosmology

We have access to several windows during the history of the Universe. BBN: BigBang Nucleosynthesis occurs at T - (1 + 0.1) MeV and describes the era when the light elements were synthesized, out of a bath of nucleons, e+e-, photons, and neutrinos, which play a crucial role. For this reason BBN can probe sterile neutrinos effectively. We follow the e v e lution with the temperature during the whole period of BBN of a 4 x 4 neutrino density matrix, for each possible choice of the sterile mixing parameters. Starting (conservatively) from a zero initial abundance of the sterile neutrino, at a certain point oscillations start producing it.This essentially happens when the plasma effects/thermal masses for the (active) neutrinos cease to be dominant compared to the vacuum masses, as the Universe expands and cools; at what point precisely (and how efficiently) the production occurs is something which is determined by the sterile mixing parameters. In the meanwhile, other cosmological processes occur, as standard: at T - fewMeV neutrinos freeze-out; at T - 1MeV e+e- annihilate etc ... The production of sterile neutrinos can enter this game

194

by (A) enlarging the total energy density and thus increasing the Hubble parameter H i.e. the expansion rate; (B) affecting the rp+n,rn--rp rates directly, if the v,, Ve population is depleted by oscillations. At the end of the process, we can compute how the light elements abundances are modified and compare them to the observations. LSS: Neutrinos can also be studied looking at the distribution of galaxies. The connection lies at the time of the formation of the anisotropies in the primordial plasma that were the seeds for the formation of the Large Scale Structures (which took place much time later). The point is that the neutrinos, relativistically traveling through the plasma (from which they were decoupled) until their mass was of the order of the temperature, had the effect of smoothing the anisotropies, i.e. they caused a suppression in the power spectrum of the galaxies that is measured today. In formulz, the effect is usually expressed in terms of the quantity R,, which is related to the sum of the neutrino masses by RVh2 = Tr[m. ~]/93.5 eV, where m is the 4 x 4 neutrino mass matrix and Q is the 4 x 4 neutrino density matrix (at late cosmological times), introduced and computed above. We compare the outcome for this quantity, in presence of sterile neutrinos, with the present determinations and the predicted improvements. CMB: Finally, neutrinos can be studied through the pattern of the CMB anisotropies measured by WMAP (and other experiments). They affect the CMB anisotropies in various ways; the all important quantity is their contribution pvtOt = T~[Q] to the total relativistic energy density, straight- forwardly parameterized in terms of an effective number NFMB.

We shade the regions that correspond to Yp 20.26 (i.e. NuHe > 3.8) or OVh2 > and are therefore ‘strongly disfavoured’ or ‘excluded’ (depending on how conservatively one estimates systematic uncertainties) within minimal cosmology. The other lines indicate the sensitivity that future experiments might reach (N , > 3.2 or R,h2 >

Any specific model of a sterile neutrino identifies a preferred point (or area) in one of these spaces or in a suitable combination of them, and should therefore be tested on them.” To single out a particularly interesting case, it turns out that the entire region of sterile neutrino parameters proposed to solve the LSND anomaly is ruled out by the BBN constraint: basically, for every value of its mixing parameters the LSND sterile neutrino

Results are collected in Fig. 1.

~ ~~ ~

aOf course, allowing non-standard modifications to BBN has the power to relax the bounds to some extent.

195

V& or VJV, 10

I

LB'

$ IB'

$ 1°-' I F '

I S $

l B 6 I O - ~ 10-J I O - ~ 10.' IB' 10-l I

Figure 1. Cosmological effects of sterile neutrino oscillations.

completely thermalizes and implies an unacceptable modification of the 4He primordial abundance.

4. Sterile effects in Supernovae

Supernovae can be powerful laboratories for neutrino physics, and therefore in particular for the physics of sterile neutrinos.

We must follow the fate of the neutrinos emitted from neutrino-spheres along their travel through the star mantle, the vacuum and (possibly) the Earth. The existence of the sterile neutrino can introduce modifications at each of these steps, via matter or vacuum conversions, in different ways for each possible choice of the sterile mixing parameters. We follow the process evolving the 4 x 4 neutrino density matrix from production to detection. In the end, we collect the modified neutrino fluxes and deduce the changes to some relevant observables; in particular, we focus on the final flux of fie, which are best detected through fiep -+ e+n at the Cerenkov detectors.

Results are collected in fig. 2, where we plot the reduction of the Pe event rate due to sterile mixing and (for a few specific choices of mixing parameters) the distortion of the ve spectrum. The shaded region is excluded by the observation of the SN1987a events. Other large areas will be tested in the occasion of the next galactic SN explosion.

196

V

10'

1

2 .: 10-2 N-.

a

10-4

10-6 1 lo2 lo4 lo6 an2 e,

V h .

10-6 10-4 10-2 1

tan2 e,

E, in MeV

v./v, "./"I

Figure 2. Sterile effects in supernova?.

References

1. M. Cirelli, G. Marandella, A. Strumia, F. Vissani, "Probing oscillations into sterile neutrinos with cosmology, astrophysics and experiments", hep-ph/0403158, Nucl. Phys. B in press. All other references therein.

NEUTRINO OSCILLATIONS IN RANDOM MEDIA

F. BENATTI AND R. FLOREANINI Dipartimento d i Fisica Teorica, Universitd di Trieste €9 Istituto Nationale di

Fisica Nucleare, Setione d i Trieste, Trieste, Italy

The consistent treatment of the propagation of neutrinos in randomly fluctuating media requires the use of quantum dynamical semigroups: they allow taking into account matter-induced, decoherence phenomena that go beyond the standard MSW effect. Within this framework, a widely adopted density fluctuation pattern is found to be inconsistent: it leads to transition probabilities that take unphysical negative values.

1. Introduction

A neutrino that propagates into matter constitutes a concrete realization of an open system, i .e. a subsystem immersed in an external e n ~ i r o n m e n t . ' ~ ~ ~ ~ In such situations, one is not interested in analyzing in detail the fast dynamics of the medium; rather, one concentrates on the study of the effective time-evolution of the subsystem, obtained by eliminating (ie. integrating over) the matter degrees of freedom: it differs from that in vacuum by environment-induced effects leading to decoherence and dissipation.

Although the interaction of the neutrino with ordinary matter is weak, the associated scattering phenomena can significantly modify the oscillation pattern. Mean field effects have been identified and analyzed long ago and manifest themselves through the sacalled MSW niechanism4i5, while the role of matter fluctuations have been first discussed in [6-81 and recently reconsidered in [9-111.

These analysis are however based on a rather simple model of fluctuations in the medium, in which all time correlations are assumed to be exactly proportional to a &function; this is a highly idealized description of the environment, never attainable in practice. Instead, a more physically realistic treatment involves the use of an exponentially damped form for the environment correlations, as adopted in the following. Limiting for simplicity the discussion to the case of two neutrino flavours, one then finds that matter effects can be fully described in terms of a limited number of

197

198

phenomenological parameters12. Not all matter fluctuation patterns are in general allowed though: re-

markably, the request of physical consistency of the generalized neutrino dynamics puts constraints on the form of the effective neutrinematter interaction. In particular, a widely adopted pure density fluctuation model is found to be physically inconsistent, since within it, certain transition probabilities take unacceptable negative values. This result may have important implications in the choice of the correct phenomenological model for interpreting the data from present and future neutrino experiments.

2. Master Equation

The action of the medium on travelling neutrinos can be most simply modeled in terms of a classical, random field. Indeed, any external environment is amenable to such an effective description, provided the characteristic decay time of its correlations is sufficiently small with respect to the typical evolution time of the immersed subsystem. Since for relativistic neutrinos this time scale can be roughly identified with the oscillation length, an environment that fluctuates on time scales much shorter than this is effectively seen as a stochastic medium by a travelling neutrino. It has been recently pointed out that the interior of the sun can be precisely treated in this way13, as likely the earth mantel.

The states of a neutrino in a medium need to be described by a density operator p; with respect to the flavour basis {lye), IvIL)}, p is represented by an herinitian 2 x 2 matrix, with non-negative eigerivalues and unit trace. After averaging over the matter degrees of freedom, one can show that its evolution in time is governed by the following e q u a t i ~ n l ~ * ~ ~ :

The first piece on the r.h.s. of (1) is of standard hamiltonian form and generates a unitary evolution. It involves the effective hamiltonian in matter, H , which contains three contributions. The first one represents the standard vacuum effective hamiltonian, Ho = w d.3, where w = Am2/4E, Am2

199

being the square mass difference of the two mass eigenstates and E the average neutrino energy, while the unit vector n’ = (sin 20,0, - cos 20) contains the dependence on the mixing angle, 3 = ((TI, u2,ug) being the vector of Pauli matrices. On the other hand, the presence of matter gives rise to the additional two pieces HI and Ha. As mentioned earlier, we shall consider environments that can be described by classical random fields. They can be conveniently organized in the hermitian matrix V ( t ) = ?(t) Z, where the components V1 ( t ) , Vz(t), V3(t) form a real, stationary Gaussian stochastic field g(t); they are assumed to have in general a nonzero constant mean and translationally invariant correlations:

W i j ( t - s ) ( K ( t ) & ( s ) ) - ( % ( t ) ) ( & ( ~ ) ) , Z , j = l , 2 , 3 . (4)

Then, one explicitly finds: HI = (c(t)) . 3, H2 = x;,j,k=l € i j k cij (Tk,

where the coefficients Czj = xi==, Jr d t Wik(t) U k j ( - t ) involve the noise correlations in (4) and the matrix U ( t ) , defined through the transformation

@Ho ui e - i t H ~ = C,”=, Ui j ( t ) uj. The second contribution L [ p ] in (2) is a time-independent , trace-preserving linear map involving the symmetric coefficient matrix Dij = Cij + Cji . It introduces irreversibility, inducing in general dissipation and loss of quantum coherence. Altogether, equation (1) generates a semigroup of linear maps, yt : p(0) I--) p ( t ) f r t [p (0 ) ] , for which composition is defined only forward in time: Yt o f̂s = y t+s , with t , s 2 O;1i2>3 this is a very general physical requirement that should be satisfied by all open system dynamicsa

3. Transition Probability

The typical observable that is accessible to the experiment is the probability PVe+,@(t) for having a transition to a neutrino of type vp at time t , assuming that the neutrino has been generated as v, at t = 0. In the language of density matrices, it is given by:

PVe-+U, (4 = Tr [P”, (4 PV,] 7 (5)

where pue(t) is the solution of (1) with the initial condition given by the matrix pu, (0) = pV, = Ive) ( v e l , while pV, = 1 - pue.

Wij = 0, the presence of matter in (1) is signaled solely by the hamiltonian contribution H I . In ordinary

In absence of fluctuation, ie.

aNotice that the procedure of averaging transition probabilities over random matter profiles as performed in [15] is not compatible with this basic evolution law, and therefore it is a pure phenomenological devise that falls short of any theoretical support.

200

situations, when only electron neutrinos are affected by the presence of the medium, the stochastic field results oriented along the third axis, whence H1 = A o ~ , where A (V3(t)) = G ~ n ~ / f i , GF is the Fermi constant, while ne represents the electron number density in the medium. In this case, the evolution (1) reproduces the standard (MSW) matter effects, and the transition probability takes the familiar expression

Pu,+u, ( t ) = sin2 28M sin2 WMt , (6) 1/2

in terms of a modified frequency WM = w sin2 28 + (1 - A / A R ) ~ cos2 281 , AR = L J C O S ~ ~ being the value of A at resonance, and mixing angle, sin 2 0 ~ = ( W / W M ) sin 28.

The situation drastically changes in presence of a fluctuating medium. In a typical bath at finite temperature, the correlation functions assume an exponentially damped form, so that one can generically write:

[

(7) W. .(t - s ) = w. . e-Xt3It-sl 23 23

with wij and Xi3 2 0, time-independent, real coefficients. In this respect, as remarked before, assuming Wij (t - s) exactly proportional to 6 ( t - s) as in [6-111 is an highly idealized situation, hardly justified from the physical point of view; indeed, it can be attained only in the infinite temperature limit.

A simple instance of a random medium is the one adopted in [6-111: ordinary matter with pure density fluctuations; here again only electron neutrinos are affected, and, as in the MSW case, only the stochastic field V3(t) can be taken to be nonzero. Then, the only nonvanishing correlation strength and decay constant are "33 E w and A33 = A , respectively. The noise contributions in (1) can be explicitly computed; one finds that only the entries 0 2 3 and 0 3 3 of the coefficient matrix in (3) are nonvanishing,

a = = ~ W / X , b = -023 = w(w/x2) sin20 , (8)

while the hamiltonian contribution H1 is proportional to 03 (the standard MSW piece) and H2 to 01: 01 = w(1 + w/X2)sin28, s22 = 0, a3 = -w( l - A / A R ) cos 28. Surprisingly, the dynamics generated by (l), with these coefficients appears to be physically unacceptable, since it gives rise to negative transition probabilities. As mentioned at the beginning, any density matrix must be a positive operator ( i e . its eigenvalues should be non-negative) in order to represent a physical state: its eigenvalues have the physical meaning of probabilities. Therefore, a time evolution that does not preserve this property results inconsistent, since an initial state

20 1

would not be mapped to another state at a later time. In the present case, the existence of negative eigenvalues can be ascertain by examining certain experimentally relevant transition probabilities: they involve the combination of neutrino propagation in vacuum with that in the medium.

Consider a neutrino, created as v,, that propagates for time t’ in vacuum, then enters the random medium in which stays for a time t , and is finally detected after having travelled again in vacuum for a further time t”. The probability Pv,-,v,(~) of finding a neutrino of type vp at the final time T = t’ + t + t” can be explicitly computed using (1) and (5).

When the vacuum evolution times t’ and t” are chosen to be equal and very short, such that sinwt‘ = b/(2usin28) = w/4X, one explicitly finds:

[ (cos2rt + (B/r) s i n ~ t ) ] , (9) 1 2

P v , + v , ( ~ ) = - 1 - e-at

where B = [b2 + u2/4]1/2 and r = [w2 - B2I1l2; this expression indeed assumes unphysical negative values for sufficiently small times: Pv,+,, (7) N

(u/2 - B)t. In order to avoid this serious inconsistency, a more general model for the fluctuating field p ( t ) needs to be considered, where all its components are nonvanishing12: this is possible only in presence of flavour changing effective neutrinematter interactions.

References 1. R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applacations,

Lect. Notes Phys. 286, (Springer, Berlin, 1987) 2. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems,

(Oxford University Press, Oxford, 2002) 3. Dissipative Quantum Dynamics, F. Benatti and R. Floreanini eds., Lect.

Notes Phys. 612, (Springer, Berlin, 2003) 4. L. Wolfenstein, Phys. Rev. D 17 (1978) 2369; ibid. 20 (1979) 2634 5. S.P. Mikheyev and A.Yu Smirnov, Sov. J. Nucl. Phys. 42 (1985) 913; Nuovo

Cim. 9C (1986) 17 6. F.N. Loreti and A.B. Balantekin, Phys. Rev. D 50 (1994) 4762 7. H. Nunokawa et al., Nucl. Phys. B472 (1996) 495 8. E. Torrente-Lujan, Phys. Rev. D 59 (1999) 073001 9. C.P. Burgess et al., Astrphys. 3. 588 (2003) L65

10. A.B. Balantekin and H. Yuksel, Phys. Rev. D 68 (2003) 013006 11. M.M. Guzzo, P.C. de Holanda and N. Reggiani, Phys. Lett. B569 (2003) 45 12. F. Benatti and R. Floreanini, Dissipative neutrino oscillations in randomly

fluctuating matter, preprint, 2004 13. C.P. Burgess et al., Mon. Not. Roy. Astron. SOC. 348 (2004) 609 14. F. Benatti, R. Floreanini and R. Romano, J. Phys. A 35 (2002) 4955 15. F.N. Loreti et al., Phys. Rev. D 52 (1995) 6664

FERMION MASSES AND NEUTRINO OSCILLATIONS IN SO(10) x SU(2)F

MU-CHUN CHEN Physics Department, Brookhaven National Lab, Upton, NY 1197’3, USA

E-mail: [email protected]. bnl.gou

K.T. MAHANTHAPPA Physics Department, University of Colorado, Boulder, CO 80309, USA

E-mail: [email protected]

We present in this talk a model based on SO(10) x S u ( 2 ) ~ having symmetric mass textures with 5 zeros constructed by us recently. The symmetric mass textures arising from the left-right symmetry breaking chain of SO(10) give rise to good predictions for the masses, mixing angles and CP violation measures in the quark and lepton sectors (including the neutrinos), all in agreement with the most upto-date experimental data within 1 u. Various lepton flavor violating decays in our model are also investigated. Unlike in models with lop-sided textures, our prediction for the decay rate of p + ey is much suppressed and yet it is large enough to be probed by the next generation of experiments. The observed baryonic asymmetry in the Universe can be accommodated in our model utilizing soft leptogenesis.

1. Introduction

SO(10) has long been thought to be an attractive candidate for a grand unified theory (GUT) for a number of reasons: First of all, it unifies all the 15 known fermions with the right-handed neutrino for each family into one 16- dimensional spinor representation. The seesaw mechanism then arises very naturally, and the small yet non-zero neutrino masses can thus be explained. Since a complete quark-lepton symmetry is achieved, it has the promise for explaining the pattern of fermion masses and mixing. Recent atmospheric neutrino oscillation data from Super-Kamiokande indicates non-zero neutrino masses. This in turn gives very strong support to the viability of SO(10) as a GUT group. Models based on SO(10) combined with discrete or continuous family symmetry have been constructed to understand the flavor problem’. Most of the models utilize “lopsided mass textures which

202

203

usually require more parameters and therefore are less constrained. The right-handed neutrino Majorana mass operators in most of these models are made out of 1 6 ~ x 1 6 ~ which breaks the R-parity at a very high scale. The aim of this talk, based on Ref. [2-41, is to present a realistic model based on supersymmetric SO(10) combined with SU(2) family symmetry which successfully predicts the low energy fermion masses and mixings. Since we utilize symmetric mass textures and 126dimensional Higgs representations for the right-handed neutrino Majorana mass operator, our model is more constrained in addition to having R-parity conserved. We also investigate several lepton flavor violating (LFV) processes in our model as well as soft leptogenesis5.

2. The Model

There are so far no fundamental understandings of the origin of flavor have been found. A less ambitious aim is to reduce the number of parameters by imposing texture assumptions. We concentrate on symmetric mass matrices as they are more predictive and can arise naturally if SO(10) is broken to the SM with the left-right symmetry at the intermediate scale. Naively one would expect that there are six texture zeros for symmetric quark mass matrices because there are six non-zero quark masses. It has been shown that this does not work and in order to obtain viable predictions, there can at most be five texture zeros. We consider the following combination for the up- and down-type quark Yukawa matrices with five zeros, which reads, after removing all the non-physical phases by rephasing various matter fields:

O O a

a c l YUrYLR = (0 beie c ) d

(2)

The above texture combination can be realized by utilizing an s U ( 2 ) ~ family symmetry. In order to specify the superpotential uniquely, we invoke 2 2 x 2 2 x 2 2 discrete symmetry. The matter fields are

$a N (16,2)-++ (a = 1,2), $3 N (16,l)'"

where a = 1,2 and the subscripts refer to family indices; the superscripts +/- refer to ( . ~ 2 ) ~ charges. The Higgs fields which break SO(10) and give

204

rise to mass matrices upon acquiring VEV's are

(10 , l ) : T:++, TF+-, T--+ , T;--, T:-- +++ --++- ____

(126,l) : c 7 C 1 7 C 2 . .

Higgs representations 10 and 126 give rise to Yukawa couplings to the matter fields which are symmetric under the interchange of family indices. SO(10) is broken through the left-right symmetry breaking chain, and symmetric mass matrices and the following intra-family relations arise,

Mu - Y,'b" (lo+) + Y a y (@) Md - Y,'b" (lo-) + Y a y (m-) Me N Y,'b" (lo-) - 3YaY (126-)

(3)

(4)

(5)

(6) MuLR N Y,'b" (lo+) - 3Yar 126 (126 * ) The SU(2) family symmetry is broken in two steps and the mass hierarchy is produced using the Froggatt-Nielsen mechanism: SU(2) 3 U(1) 3 nothing where M is the UV-cutoff of the effective theory above which the family symmetry is exact, and eM and ;A4 are the VEV's accompanying the flavon fields given by

(L2) : 4;-, 4&+, a-+- (1'3) : S;,-, SGT-, C++- .

The vacuum alignment in the flavon sector is given by

The various aspects of VEV's of Higgs and flavon fields are given in Ref. [2-41.

The superpotential of our model is

w = WDirac + WURR (7)

205

The mass matrices then can be read from the superpotential to be

0 0 (lo;)€' 0 0 T2E'

MZL,WLR = ( 0 (lo,+)€ (lo:)€) = ( 0 r;E ; ) Mu (9) (lo;) €' (10;) € (lo:) T2€'

where M u ( lor) , MD (lo;), ~2 (10;) / ( lot ) , T4 (10:) / (10;) and p G (m-) / (101). The right-handed neutrino mass matrix is

with MR = (126:). Here the superscripts +/ - / O refer to the sign of the hypercharge. It is to be noted that there is a factor of -3 difference between the (22) elements of mass matrices Md and Me. This is due to the CG coefficients associated with 126; as a consequence, we obtain the phenomenologically viable Georgi-Jarlskog relation. We then parameterize the Yukawa matrices as given in Eq. (1) and (2).

206

We use the following as inputs at MZ = 91.187 GeV:

mu = 2.21 MeV(2.33+::;:) m, = 682 MeV(677f;;) mt = 181 GeV(181'13) me = 0.486 MeV(0.486847) m, = 103 MeV(102.75) m, = 1.74 GeV(1.7467)

IV,, I = 0.225(0.221- 0.227)

lVubl = 0.00368(0.0029 - 0.0045) IVcb( = 0.0392(0.039 - 0.044)

where the values extrapolated from experimental data are given inside the parentheses. Note that the masses given above are defined in the modified minimal subtraction (m) scheme and are evaluated at Mz. These values correspond to the following set of input parameters at the GUT scale, MGUT = 1.03 x 10l6 GeV, and t anp = 10:

a = 0.00250, b = 3.26 x

c = 0.0346, d = 0.650 e = 0.74

e = 4.036 x f = 0.0195 h = 0.06878, = -1.52 91 = ~2 = g3 = 0.746

the one-loop renormalization group equations for the MSSM spectrum with three right-handed neutrinos are solved numerically down to the effective right-handed neutrino mass scale, MR, At MR, the seesaw mechanism is implemented. With the constraints lmv3 I >> Imv2 1, lmv, I and maximal mixing in the atmospheric sector, the up-type mass texture leads us to choose the following effective neutrino mass matrix

207

with n = 1.15, and from the seesaw formula we obtain

a2 61 = -

r

62 = - b2te2i6

r

(13)

(14)

Y (15) -a(beie(l + t1*15) - c) + bcteie

r 63 =

where r = ( ~ ~ t + a ~ t ~ . ~ ~ ( 2 + t ~ . ~ ~ ) - 2 4 - 1 +c+ct1.15)). A generic feature of mass matrices of the type given in Eq.(12) is that they give rise to bi-large mixing pattern. And the value of IUe3I2 is proportional to the ratio of Am& to Am:,,.

We then solve the two-loop RGE’s for the MSSM spectrum down to the SUSY breaking scale, taken to be mt(mt) = 176.4 GeV, and then the SM RGE’s from mt(mt) to the weak scale, M z . We assume that tanP =_

v,/vd = 10, with v: + vz = (246/fi GeV)2. At the weak scale Mz, the predictions for ai = gs/47~ are

01 = 0.01663, a 2 = 0.03374, a3 = 0.1242 .

These values compare very well with the values extrapolated to M z from the experimental data, (a l , a2, ag) = (0.01696,0.03371,0.1214 f 0.0031). The predictions at the weak scale M z for the charged fermion masses, CKM matrix elements and strengths of CP violation, are summarized in Table. The predictions for the charged fermion masses, the CKM matrix elements and the CP violation measures. The predictions of our model in this updated fit are in good agreement with all experimental data within lu, including much improved measurements in B Physics that give rise to precise values for the CKM matrix elements and for the unitarity triangle. Note that we have taken the SUSY threshold correction to ma to be -18%.

The allowed region for the neutrino oscillation parameters has been reduced significantly after Neutrino 2004. Using the most-up-to-date best fit values for the mass square difference in the atmospheric sector Am:,, = 2.33 x eV2 and the mass square difference for the LMA solution Am: = 8.14 x eV2 as input parameters, we determine t = 0.344 and MR = 6.97 x 1012GeV, which yield (61,62,63) = (0.00120, 0.000703ei (1.47) , 0.0210ei (0.175)). We obtain the following predictions in the neutrino sector: The three mass eigenvalues are give by

(mvl, mv2, mvs) = (0.00262,0.00939,0.0492) eV . (16)

208

ms lmd

ms mb

lvud I lvcdl

IVCSI

l&dl

I K S I l&bl

J S P

sin 2 a

sin 28

7

P

9

-

-

experimental results predictions at Mz

extrapolated to M z

17 N 25 25

93.4+ii:zMeV 86.OMeV

3.00 f O.11GeV 3.03GeV

0.9739 - 0.9751 0.974

0.221 - 0.227 0.225

0.9730 - 0.9744 0.973

0.0048 - 0.014 0.00801

0.037 - 0.043 0.0386

0.9990 - 0.9992 0.999

(2.88 f 0.33) x 2.87 x 10-5

-0.16 f 0.26 -0.048

0.736 f 0.049 0.740 60' f 14O 64O

0.20 f 0.09 0.173

0.33 f 0.05 0.366

The prediction for the MNS matrix is

0.852 0.511 0.116

0.304 0.652 0.695 (17)

which translates into the mixing angles in the atmospheric, solar and reactor sectors,

sin2 813 = IVev3 l 2 = 0.0134 .

The prediction of our model for the strengths of CP violation in the lepton sector are

JAP = I ~ { U ~ I U ; ~ U & U ~ ~ } = -0.00941

(a31,(Y21) = (0.934, -1.49) . (21)

(22)

Table 1. The predictions for the charged fermion masses, the CKM matrixelements and the CP violation measures.

209

Using the predictions for the neutrino masses, mixing angles and the two Majorana phases, a31 and C Q ~ , the matrix element for the neutrinoless double p decay can be calculated and is given by I < m > I = 3.1 x eV, with the present experimental upper bound being 0.35 eV. Masses of the heavy right-handed neutrinos are

MI = 1.09 x lo7 GeV

M2 = 4.53 x lo9 GeV M3 = 6.97 x 10l2 GeV .

The prediction for the sin2 813 value is 0.0134, in agreement with the current bound 0.015 at la. Because our prediction for sin2 813 is very close to the present sensitivity of the experiment, the validity of our model can be tested in the foreseeable future.

3. Lepton Flavor Violating Decays and Soft Leptogenesis

Non-zero neutrino masses imply lepton flavor violation. If neutrino masses are induced by the seesaw mechanism, new Yukawa coupling involving the RH neutrinos can induce flavor violation. Observable decay rates can be obtained if the relevant scale for these LFV operators is the SUSY scale. We consider LFV decays resulting from the non-vanishing off-diagonal matrix elements in the slepton mass matrix induced by the RG corrections between MGUT and MR. In this case, the branching ratios for the decay of Ci -+ Cj + y is

(26) In our model, Br(p -+ ey) < Br(7 -+ ey) < BT(T + py) is predicted. Our predictions for the branching ratio of the decay p -+ ey arising from the RG effects induced by neutrino Dirac Yukawa couplings as a function of the gaugino mass M112 is given in Fig. 1. In contrast to the predictions of models with lop-sided textures, in which the off-diagonal elements in (23) sector of Me are of order O(1) leading to an enhancement in the decay branching ratio and the need of some new mechanism to suppress the decay rate of p -+ ey, the predictions of our model for LFV processes, Ci -+ Cjy, p-e conversion as well as p -+ 3e, are well below the most stringent bounds up-to-date. Our predictions for many processes are nontheless within the reach of the next generation of LFV searches. This is especially true for p - e conversion and p -+ ey. More details are contained in Ref. [5].

210

Soft leptogenesis (SFTL) utilizes the soft SUSY breaking sector, and the asymmetry in the lepton number is generated in the decay of the superpartner of the RH neutrinos. The lepton number asymmetry is then converted to the baryonic asymmetry by the sphaleron effects. The source of CP violation in the lepton number asymmetry in SFTL is due to the CP violation in the mixing which occurs when the following relation Irn(Arl /MlB) # 0 ( A and B are the tri-linear A-term and B-term) is satisfied. The total lepton number asymmetry integrated over time, e, is defined as the ratio of difference to the sum of the decay widths for 5~~ and SL1 into final states of the slepton doublet z and the Higgs doublet H , or the lepton doublet L and the higgsino E or their conjugates,

where f denotes the final states (z H ) , ( L fi) and 7 denotes their conjugate, (zt H t ) , ( E E) . This leads to a total amount of baryon asymmetry in our model due to soft leptogenesis is,

-

In Fig. 2, we show the predictions for the asymmetry, n B / s , as a function of B' for different values of Im(A). With B' - 1 TeV and Irn(A) - 1 TeV, sufficient baryonic asymmetry can be generated. More details are contained in Ref. [5].

4. Conclusion

To conclude, the observed fermion mass hierarchy and mixing have been successfully accommodated in our model utilizing the two-step breaking in s U ( 2 ) ~ . Due to the SO(10) and s U ( 2 ) ~ symmetries and the resulting symmetric mass textures, the number of parameters in the Yukawa sector has been significantly reduced. With 11 parameters, our model gives rise to values for 12 masses, 6 mixing angles and 4 CP violating phases, all in agreements with available experimental data within 1 m. In contrast to the predictions of models with lop-sided textures, the predictions of our model for LFV processes, Ci -+ Cjy, 1.1 - e conversion as well as p -+ 3e, are well below the most stringent bounds up-to-date, and yet many of them are within the reach of the next generation of LFV searches. The observed baryonic asymmetry in the Universe can be accommodated in our model utilizing soft leptogenesis.

21 1

References

1. M.-C. Chen and K.T. Mahanthappa, Int. J. Mod. Phys. A18, 5819 (2003). 2. M.-C. Chen and K.T. Mahanthappa, Phys. Rev. D62, 113007 (2000). 3. M.-C. Chen and K.T. Mahanthappa, Phys. Rev. D65, 053010 (2002). 4. M.-C. Chen and K.T. Mahanthappa, Phys. Rev. D68, 017301 (2003). 5. M.-C. Chen and K.T. Mahanthappa, hep-ph/0409096.

fiitiire reach

100 150 200 250 300 350 400 450 500 M1/2 (GeV)

Figure 1. The branching ratio of p -+ e7 as a function of the gaugino mass MI/^, for various values of scalar masses, mo and Ao: (Sl): mo = A0 = 100 GeV; (S11): mo = 100 GeV,Ao = 1 TeV; (S2): mo = A0 = 500 GeV; (S3): mo = do = 1 TeV.

500 1000 1500 2000 2500 3000 8' (GeV)

Figure 2. and 1 TeV.

The prediction for n B / s as a function of B' for IIm(A)I = 10 TeV, 5 TeV

ROBUST SIGNATURES OF THE RELIC NEUTRINOS IN CMB

SERGE1 BASHINSKY International Centre for Thwreticd Physacs

Stmda Costiem 11, 31014 Tkieste, Italy E-mail: bashinskyOictp.trieste.it

When the perturbations forming the acoustic peaks of the cosmic microwave background (CMB) reentered the universe horizon and interacted gravitationally with all the matter, neutrinos presumably comprised 41% of the universe energy. CMB experiments have already reached a capacity to probe this background of relic neutrinos. I discuss the imprints left by neutrinos on CMB anisotropy and polarization at the onset of the acoustic oscillations, the underlying physics, robustness or degeneracy of the imprints with changes of free cosmological parameters, and examples of non-minimal physics for decoupled relativistic species with detectable signatures in CMB.

1. Introduction

The counts of solar, atmospheric, reactor, and accelerator-produced neutrinos provide a solid evidence of physics beyond the Standard Model. They can be explained by oscillations among neutrino states of different masses, subject to tightening upper cosmological bounds [l]. These bounds, however, rely on the assumption that the neutrino sector hides no additional surprises. Specifically, that a neutrino per photon ratio has been fixed since the decoupling of the active neutrinos and the subsequent e+e- annihilation. Or that neutrinos do not couple to unseen fields [2, 31.

These assumptions are verifiable by cosmological observations. Pertur- bations in the cosmic microwave background (CMB) carry signatures of any species which contributed to the universe density when the perturbations reentered the horizon. Some of these signatures are robust and reveal the internal interactions of species which are decoupled from the visible sector. The big bang nucleosynthesis (BBN) yield of light elements, notably He4, is also sensitive to all the dominant species in the radiation epoch, but at high redshifts z - lo9 - lo8 (- lOOkeV). The CMB peaks, on the other hand, probe the late radiation era z - lo4 - lo3 (- 1 eV).

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The data of the first WMAP release places the effective number of relic neutrinos in a 2-u range 1 5 N , 5 7 [4], mildly impoved by complimentary probes. The range will be narrowed by an order of magnitude and more as the CMB sky is mapped at smaller angles with PLANCK and next missions: at 1-u, to [5: 61 A N , % 0.2 by PLANCK and up to A N , = 0.05 by more advanced proposed experiments.

2. Imprints of neutrino gravity on CMB

The CMB fluctuations could not escape the gravity of relic neutrinos during the radiation era, when neutrinos were among the dominant species. The neutrino fraction R, = pv/pr of the total radiation energy density pr = pr + p, = pr/(l - Rv) is 41% in the standard scenario. With the present cosmological bounds my 5 0.14 to 0.6 eV [l], the standard neutrinos were all highly relativistics in the radiation era, prior to matter-radiation density equality at

zeq -k 1 = (pc+b,O/Pr,O)(l - &). (1) [Here, c+b labels cold dark matter (CDM) plus baryons and “0” denotes the present values. zeq w 3.2 x lo3 in the standard scenario.] They become non-relativistic below the redshift zm, = m,/(%T,,o) N 200rn,/(0.1 eV), that is, only after the CMB decoupling (zyd % 1080).

The growth of cosmological structure in the matter era, not discussed below, is affected by neutrino mass density. It carries signatures of the mass- sensitive neutrino free streaming scale [7], underlying the cosmological m, bounds. This may also have an eventually detectable effect on CMB via gravitational lensing [8].

2.1. Background effects

According to Friedmann equation, higher neutrino density speeds up the cosmological expansion in the radiation era. For f i e d density fl,+bh2 (a Pc+b,O), the faster expansion would reduce the size of the CMB acoustic horizon. In addition, a larger fraction of radiation density prior to the CMB last scattering would enhance the first acoustic peak by a stronger ISW effect from the increased proximity of the radiation-matter transition (1) and by lesser suppression of the peak by matter perturbations [9].

Although these, notably the horizon, effects are frequently mentioned in the literature, they can not serve as neutrino probes. Indeed, our knowledge of other densities, in particular CDM, is likewise derived from the observed

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CMB spectrum. Even if p, is not equal to the assumed value, the angular acoustic horizon scale and zeq remain essentially unchanged when P b / P r ,

hence the acoustic speed, has the standard value while the densities &+b,

Pdark (and p~ for non-flat models) are rescaled in proportion to pr = pr+pv. Moreover, the observed matter power P,(lc/h) is also almost un-

changed. [Small changes are induced by variations in p b / p c , in signatures of neutrino masses (to be detected), and in the impact of neutrino perturbations, Sec. 2.2.1 There is a simple explanation: In the compared scenarios most of the characteristic cosmological time and distance intervals differ by the same factor. This conformal rescaling preserves measured angles and redshifts.

Several consequences of the common density rescaling can still be o b served. First, a different value of the Hubble constant H: cx Call,Kpa,~. Constraints on Ho, however, tend to be weak.

Second, the scale of Silk damping Asilk depends on the mean time of photon collisionless flight T~ = (anegye) - l , not affected by the density rescaling. (We use comoving scales and conformal times.) A higher ratio ASilk/7.tFI-' - ( ~ ~ 7 . t ) ' / ~ in a model with greater p, leads to increased damping of small-scale CMB anisotropies and helps to constrain N,, (Table I11 in [6 ] ) . However, the density of free electrons n, around the photon decoupling, when only hydrogen was ionized and ne E x,nH = xe(1- Yp)pb/mH, depends (for known p b ) on the primordial helium abundance Yp. With its present uncertainties, Yp becomes a free parameter for achieving AN, 5 0.1. A detailed analysis [6] shows that rescaling 1 - Yp 0: p;I2 yields almost degenerate x , ( z ) , hence, equivalent CMB decoupling and unchanged kXlk / f iH- ' .

2.2. Perturbation effects

The conformal degeneracy of CMB and matter spectra in a faster expanding background is broken by the gravitational impact of neutrino perturbations. Perturbations of photons (coupled to baryons) evolve in the perturbed metric as

Here @ and @ parameterize the metric perturbations ds2 = a2[-(1 + 2@)d? + ( 1 - 2 8 ) d x 2 ] , overdots denote the derivatives with respect to the conformal time T , 7.t = b/a, R b = 3pb/(4p7), and d, = 6p,/(p, + p, ) - 3 8 = 3& = 3(6T,/T7-8) (in the Newtonian gauge) is a general-relativistic

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generalization” of 6n,/n,.

is easily solved [6] in real space with the Green’s function approach [lo]: For the radiation era, when the CMB peaks enter the horizon, eq. (2)

where we assume adiabatic initial conditions. After Fourier transformation to k-space, the delta-function term describes the famous acoustic oscillations A,cos(h/&) in the photon fluid with the speed of sound c, = l/& Whenever the metric is perturbed at the acoustic horizon, i e . (a + 9)lzl=7,fi # 0, the small-scale singularity of the second term in eq. (3) contributes a sine component to the subhorizon oscillations in Fourier space; equivalently, shzfts the oscillation phase.

It can be proven (Appendix B of [S]) that the metric in the Green’s function (3) can be perturbed beyond the acoustic horizon if only some perturbations physically propagate faster than the acoustic speed cs. Among the standard cosmological species, only free-streaming relativistic neutrinos support a faster speed, the speed of light. (Perturbations in early quintessence do as well. They induce an even larger phase shift per equal density [9].) This phase shift results in a non-degenerate shift of all the acoustic peaks in Cl, for either temperature or polarization spectra. For Nu = 3, the peak positions change as dl,,,k/dN, M -4.

In addition, neutrino perturbations somewhat suppress the magnitude of the acoustic oscillations while enhance the matter power [ll, 61. With unknown primordial power, a detection of this effect requires precise measurement of the matter spectrum and is difficult. Quintessence perturbations cause an opposite change of the CMB to matter power ratio [9]. If CMB finds a non-standard Nu, this may, in principle, discriminate between contributions of sterile particles and early quintessence.

3. Probing particle physics

An apparent contribution to Nu may come from sterile neutrinos - light neutral particles which mix with the active neutrinos. Motivations to consider such states and their status in view of the latest oscillation data and other constraints were reviewed in detail in Cirelli’s talk [12].

aUp to terms vanishing on superhorizon and subhorizon scales, d , is a unique [9] perturbation variable which a) reduces to bn,/n, on subhorizon scales, b) freezes beyond the Hubble horizon for unperturbed nb/n7, and c) is insensitive to dynamics of decoupled species with infinitesimal density and no physical gravitational impact.

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N , might also be enhanced by yet unknown light particles which neither couple nor mix to the visible sector and fell off the thermal bath early, say, above the SUSY breaking scale. Since the radiation entropy was then shared by a large number of relativistic particles, the N , enhancement may be small and pass the current BBN constraints.

Since the phase-shift of CMB peaks induced by neutrinos is absent when neutrinos do not free-stream, CMB will probe a growing class of models where neutrinos recouple after BBN. For example, recoupling is expected if the small neutrino masses are generated by a coupling luRhr$/M ( 1 and h are the standard lepton and Higgs doublets and 4 is a new field) after low-energy spontaneous symmetry breaking 141 + d4 [13]. Other neutrino mass models leading to recoupling have been considered [2, 3, 141. If neutrinos recouple to a light field 64, they annihilate as soon as become non- relativistic [2, 31. Then neutrino mass may not be bounded by the standard cosmological limits 111. A changed phase of the acoustic peaks is a clean signature of neutrino recoupling [13].

Today we are not pressed to prefer any of the above scenarios. Yet, neither dark matter nor dark energy entered the cosmological scene with strong theoretical motivation. The upcoming CMB data will testify whether or not the radiation epoch conceals its own surprises.

References 1. For a review of current cosmological constraints on my see

S. Hannestad, hep-ph/0409108. 2. J. F. Beacom, N. F. Bell and S. Dodelson, Phys. Rev. Lett. 93, 121302 (2004). 3. Z. Chacko, L. J. Hall, S. J. Oliver and M. Perelstein, hepph/0405067. 4. P. Crotty, J. Lesgourgues and S. Pastor, Phys. Rev. D 67, 123005 (2003);

E. Pierpaoli, MNRAS 342, L63 (2003); S. Hannestad, JCAP 0305, 4 (2003); V. Barger et al., Phys. Lett. B 566, 8 (2003).

5. R. E. Lopez et al., Phys. Rev. Lett. 82, 3952 (1999); R. Bowen et al., Mon. Not. Roy. Astron. SOC. 334, 760 (2002).

6. S. Bashinsky and U. Seljak, Phys. Rev. D 69, 083002 (2004). 7. J. R. Bond and A. S. Szalay, Astrophys. J. 274, 443 (1983). 8. M. Kaplinghat, L. Knox and Y . S. Song, Phys. Rev. Lett. 91, 241301 (2003). 9. S. Bashinsky, astro-ph/0405157. 10. S. Bashinsky and E. Bertschinger, Phys. Rev. Lett. 87, 081301 (2001);

Phys. Rev. D 65, 123008 (2002). 11. W. Hu and N. Sugiyama, Astrophys. J. 471, 542 (1996). 12. Contribution by M. Cirelli; M. Cirelli et al., hepph/0403158. 13. Z. Chacko, L. J. Hall, T. Okui and S. J. Oliver, hep-ph/0312267. 14. N. Arkani-Hamed and Y . Grossman, Phys. Lett. B 459, 179 (1999);

T. Okui, hepph/0405083.

RADIATIVE CORRECTIONS TO NEUTRINO REACTIONS OFF PROTON AND DEUTERON *

T. KUBOTA Graduate School of Science, Osaka University

Toyonaka, Osaka 560-0043, Japan

M. FUKUGITA Institute for Cosmic Ray Research, University of Tokyo

Kashiwa 877-8582, Japan

Radiative corrections are calculated for antineutrino proton quasielastic scattering, neutrino deuteron scattering, and the asymmetry of polarised neutron beta decay from which GA/Gv is determined. A particular emphasis is given to the constant parts that are usually absorbed into the coupling constants, and thereby those that appear in the processes that concern us are unambiguously tied among each other.

1. Introduction

Neutrino experiments have now entered the era of precision measurements with the accuracy reaching the level that radiative corrections cannot be ignored. In this talk we report our recent work '-' on the radiative corrections to neutrino scattering, iie + p - e+ + n as measured in KamLAND and ve + d - e- + p + p , v, + d - v, + p + n that are measured at SNO, and those to the asymmetry in polarised neutron beta decay.

2. Radiative Corrections to GA/Gv

The study of radiative corrections to weak processes has a long history, and the wisdom acquired for neutron beta decay will be transcribed in the calculation of neutrino scattering processes. With the local four-Fermi

*Presented by T.K. The work is supported in part by Grants in Aid of the Ministry of Education.

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interactions, the GA/Gv ratio (= gA) enters the tree-level cross sections as

a ( g e + p - e + + n ) m (f;+3gi) (1)

for neutrino-proton quasielastic scattering and

a( y e + d -+ e- + p + p ) C( g;, a(% + d - v, + p + 72) m g;

(CC) (2) [. = e , P , 7 ] (NC) (3)

for scattering off deuteron. Here fv = 1 is retained to trace the contribution of the weak vector current.

We separate the radiative corrections to charged current processes into the inner and outer parts as was done for neutron beta decay. The outer corrections depend on the e* velocity in the final state and are free from UV and IR divergences. They are independent of the details of strong interactions and the calculation is straightforward. The inner part is velocity- independent but is plagued by the UV-divergence that is not cancelled within four-Fermi theory and depends on the details of the structure of nucleons. The inner corrections amount to replacing f; and g i by

(4) g = f;(1+6:), g; = g i ( 1 +6, G T ).

The evaluation of 6; and SET requires not only renormalisable Weinberg- Salam theory but receives complications from hadron structure. For the Fermi transitions bE is already obtaineda-', but the evaluation of 6gT for the Gamow-Teller transitions has eluded the literature for long time.

The corrections to (3) differ from those to (1) and (2). There is no outer correction. The corrections are incorporated by the replacement of g i by

g i - g i ( 1 + AET). ( 5 )

The prime purpose of our work is to give SET and AET, so that the gA factor that appears in NC processes is related with that in the CC processes. Subsidiarily we show that the radiative correction to the polarised neutron beta decay asymmetry (from which we determine gA) is described by the same factors as those that appear in (4) and in the neutron beta decay rate. This is expected, but we do not find any proofs. So we carried out explicit calculations. The G A / G V ratio measured in this process is

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3. Calculational Strategy

Let us begin with the charged current processes (1) and (2). Following the procedure for the neutron beta decay rate7, we divide the integration region of the exchanged gauge boson into

(i) 0 < lkI2 < M 2 , and (ii) M 2 < 1kI2 < 00. (7)

The mass scale M is supposed to be greater than the proton mass (m,), but to be much smaller than the W and 2 boson masses (mw, mz), ie., mp << M << mw,mz. We use the four-Fermi interactions for nucleons in region (i), thereby dealing with the nucleons as point-like and only photons are exchanged between nucleons and charged leptons. In region (ii) we employ Weinberg-Salam theory for quarks and photons and Z bosons are exchanged. The mass scale M is the UV-cutoff for the four-Fermi theory, but is also regarded as the scale for the onset of the asymptotic behaviour to which Weinberg-Salam theory applies.

With four-Fermi theory, we end up with UV-divergences, ie., logM2 terms in the calculation for (i). These divergences are classified into two types, the one eliminated by renormalisable gauge theories and the other that is rendered finite only by considering the structure of nucleons. It is known 5-6 that such classification is possible for the Fermi transitions by using CVC and current algebra techniques. We showed in Ref. 1 that a parallel classification is possible for the Gamow-Teller transitions on the basis of CVC, PCAC and current algebra (see also Ref. 10).

In the first type logM2 terms are universal, their coefficients being independent of the details of strong interactions. These logM2 terms are cancelled when the integrals in (i) and (ii) are added. The logM2 terms that appear axial-vector vector interference terms cannot be cancelled. So, logM2 terms in (i) are tamed by introducing form factors at the electromagnetic and weak vertices in Feynman integrals. Weak magnetism cannot be ignored' at the weak vertices, because the mass scale of the form factor is on the order of mp and the loop integral over the weak magnetism form factor gives the same order of magnitude as does the V - A contribution.

4. Antineutrino Quasielastic Scattering off Proton

We write the differential cross section as

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Here E and P are the energy and velocity of the final positron, 8 is the angle between incident antineutrino and the positron and Gv = GFcosec is the vector coupling constant to nucleons.

After calculations we find that A(P) and B(P) are written

A(P) = (1 + &out(E)} (C + 33;) 1 (9)

(10) B(P) = { 1 + ZOUt@)} (C - 3;) 7

where the inner corrections included in $ and g; are given in (4) with

6F In = e2 8n2 { 410g (%) + log (2) + cF} ,

6:' = & { 41og (%) + log (2) + 1 + cGT}. (12)

The effect of the nucleons structure appears only in CF and CGT. We computed these two numbers by introducing form factors. The contributions from the weak magnetism are nonnegligible at the weak vertex; it even dominates CGT. Our results are CF = 2.160 and CGT = 3.281, and hence 6: = 0.0237 and SET = 0.0262 for M M 1 GeV .

One sees in (10) that the energy-dependent outer corrections are fac- tored out. One of them, bout(E), has been known'l; the other 6out(E) is new1. The outer corrections for (2) are given in Ref. 12.

5. Asymmetry in Polarised Neutron Beta Decay

The GAIGv ratio is determined from the asymmetry parameter A. The corrections are again separated into the inner and outer parts, as2

fvg* - g; f; + 3g; .

A = 2 (1 + C ( E ) } - (13)

The energy-dependent factor C(E) is given in Ref. 2. The important point is that exactly the same inner corrections appear in f; and g;, as given by (4) with (11) and (12), so that i j ~ from asymmetry can be used to predict the neuron decay rate without further corrections.

6. The Deuteron Neutral Current Process

Since the outer correction is absent, all we need is the evaluation of the inner part using Weinberg-Salam theory on the quark level. This type of

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calculation was done by Marciano and Sirlin13. The effective interaction of quarks and neutrinos at low energy with an iso-singlet target is given by

where Q1, and $ are the neutrino and the quark doublet, and &Ah' - 1 and K ( ~ ; ~ ) - 1 are the radiative corrections, which are found in Ref. 13.

If we sandwich (14) between the deuteron and two-nucleon states to evaluate the cross section (3), only the axial current in (14) contributes to the 3S1 -+ 'SO transition. The effective interaction for the nucleon doublet l c f ~ reads

The term of K ( ~ ; ~ ) does not contribute to (3). Eq. (15) indicates that gA is renormalised multiplicatively by &$). Writing pNC (v;h) = (1 + AET)1'2, the

radiative correction is the replacement (5). We compute &Ah' = 1.00955 for Higgs boson mass r n ~ = 1.5rnz and 1.00862 for r n ~ = 5mz.

7. Summary

We computed radiative corrections to antineutrino proton quasielastic scattering, neutrino deuteron scattering, and the asymmetry of polarised neutron beta decay, with an emphasis given to the constant parts that are usually absorbed into the coupling constants. Hereby couplings that appear in the processes that concern us are unambiguously tied. For instance, the NC to CC ratio for neutrino-deuteron reactions receives the overall correction (1 + AgT)/(l + 62') = 0.992 f 0.001 up to the outer correction for CC which is accounted for separately.

References

1. M. Fukugita and T. Kubota, Actu Physim Polonica B 35, 1687 (2004). 2. M. Fukugita and T. Kubota, Phys. Lett. B 598, 67 (2004). 3. M. Fukugita and T. Kubota, work in progress 4. T. Kinoshita and A. Sirlin, Phys. Rev 113, 1652 (1959). 5. E.S. Abers, R.E. Norton and D.A. Dicus, Phys. Rev. Lett. 18, 676 (1967); 6. A. Sirlin, Phys. Rev. 164, 1767 (1967). 7. A. Sirlin, Nucl. Phys. B71, 29 (1974).

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8. W.J. Marciano and A. Sirlin, Phys. Rev. Lett. 56, 22 (1986). 9. I.S. Towner, Nucl. Phys. A540, 478 (1992). 10. A. Sirlin, Nucl. Phys. B196, 83 (1982), 11. P. Vogel, Phys. Rev. D29, 1918 (1984); S.A. Fayans, Sou. J. Nucl. Phys. 42,

12. I S . Towner, Phys. Rev. C58, 1288 (1998); A. Kurylov, M. J. Ramsey-Musolf

13. W.J. Marciano and A. Sirlin, Phys. Rev. D22, 2695 (1980).

590 (1985).

and P. Vogel, Phys. Rev. D65, 055501 (2002); ibid 67, 035502 (2003).

Accelerator Experiments


SEARCHES FOR NEW PHYSICS AT LEP

MARKUS SCHUMACHER ON BEHALF OF THE FOUR LEP COLLABORATIONS

Physikalisches Institut, Universitat Bonn, Nussallee 12,

53115 Bonn, Germany E-mail: [email protected] bonn. de

The search for phenomena arising from new physics was one of the central research topics of the LEP2 physics program. Below we discuss the search for the Standard Model Higgs boson, neutral Higgs bosons of the MSSM, searches for Supersym- metric particles, mass limit for the Neutralino LSP and effects from extra spatial dimensions.

1. Introduction

The Standard Model (SM) of particle physics from the point of view of high energy collider experiments is still in a pretty good shape despite the evidence e.g. for neutrinos masses, dark matter and dark energy reported as this conference. Still, there are crucial open questions like: What creates mass and is responsible for electroweak symmetry breaking? Is there a hierarchy between the Planck and the electroweak scale and how it is caused and stabilised? The huge data set collected at LEP recorded at center-of mass energies fi up to 209 GeV, s L x 2.6 fb-' at fi > 183 GeV and L x 35.2 fb-' at ,/3 > 207.5 GeV, tried to unravel the answers to the above questions by looking for various topologies of new physics. This summary focuses on discussing results from direct searches for neutral Higgs bosons in the SM and its Minimal Supersymmetric Extension (MSSM), searches for supersymmetric partner particles, determination of a lower limit for the mass of the lightest supersymmetric particle (LSP) under the assumption of R-parity conservation, and searches for new phenomena arising from extra spatial dimensions. Preference is given to new and LEP combined results. Details of the analysis and an overview of almost all searches performed at LEP can be found at ', here only the results are reported. All limits quoted are obtained at the 95% confidence level.

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2. Higgs boson Searches

2.1. Searches for the Standard Model Higgs boson

The final findings in the search for the SM Higgs bosons from combining the data from all four LEP experiments are summarised in 2 . In the SM the Higgs boson profile is completely determined by its mass. Therefore the cross section limit can be turned unambiguously into a mass limit. The mass spectrum after tight selection cuts, the background confidence level 1 - CLb and the normalised signal+background confidence level CL, = cL$+b/cLb as a function of hypothetical Higgs boson mass are shown in Figure 1. No excess beyond the level of 2.3 sigma is observed in the whole mass region. In the region of mH above 115 GeV the approximate value of 1 - c & = 0.09 translates into a 1.7 standard deviation from the background only hypothesis. The final Higgs boson mass limit obtained from the full LEP data sample is 114.4 GeV, while the median expected limit in the background only hypothesis is 115.3 GeV.

N $ 7

mH= (QV/C~ m,(GeV/cS m,(GeV/cS

Figure 1. Final LEP combined results for the SM Higgs boson. Left: mass spectrum after tight cuts, middle: 1-CLb background only hypothesis confidence level, right: CL. normalised signal+background hypothesis confidence level.

2.2. Searches for neutral Higgs bosons of the MSSM

The Higgs sector of the MSSM consists of two Higgs doublets which after electroweak symmetry breaking yield five physical Higgs bosons: three neutral and two charged. If the SUSY breaking parameters are chosen to be real two of the neutrals are CP-even, h and H , and one is CP-odd, A. In case of complex parameters the CP symmetry is not conserved at loop level and the Higgs mass eigenstates, H I , HZ and H3 ordered by mass, have no well defined CP-parity. The LEP collaborations have searched for production

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via Higgstrahlung (e+e- + Zh, Z H , ZHi) and associated pairproduction (e+e- -+ hA, H A , HiHj ) with various final state topologies.

The exclusion region in the CP-conserving mh - m a benchmark scenario 31 which maximises the theoretically allowed range for mh and yields conservative exlcusions in t a n p at LEP, is shown in Figure 2 ‘. The absolute mass limits are obtained in the vincinity of the “mass diagonal”, where mh M mA, and depend only marginally on the value of mt. Masses mh < 92.9 (94.8 expected) GeV and mA < 93.4 (95.1 expected) GeV are excluded. The tanp exclusion depends strongly on mt. For mt = 179.3 GeV the range between 0.9 and 1.5 (0.8 and 1.6 expected) is excluded. However for mt M 183 GeV the exclusion in tanP vanishes. Exclusion limits for other benchmark scenarios can be found in and 5 .

Q E. a u

10

I

0 20 40 60 80 100 120 j40 0 20 40 60 80 100 120 j 4 0 mho (GeV/cL) m,,. (GeV/c’)

Figure 2. Preliminary LEP combined exlusions in the mh - m u benchmark scenario.

The exclusion obtained in the CP-violating CPX-scenario is shown in Figure 3 ‘. No absolute limit on the mass of the lightest Higgs bosons H I is obtained. A top quark mass dependent lower bound can be set on the value of tanp: for mt = 174.3 (179.3 and 183.0) GeV this bound is 2.9 (2.6 and 2.5). In addition searches have been performed for any kind of decay modes of neutral Higgs bosons (e.g. invisible, yy, WW), for their associated production with b-quarks and for the production of charged and doubly charged Higgs bosons. Model independent upper cross section limits have been derived and await their final LEP combination.

3. Searches for SUSY Particles

Supersymmetry (SUSY) is one of the most favoured extensions of the SM in order to solve several theoretical shortcomings of the SM as e.g. the hier-

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Figure 3. Preliminary LEP combined exclusions in the CPX benchmark scenario.

archy problem. Here we only discuss the case of R-parity conserving SUSY. This implies that SUSY particles are produced in pairs and that the lightest supersymmetric particle (LSP), which is assumed to be the lightest neutralino x(: in this review, is stable leading to a signature of missing energy in the detector. The main strategy at LEP was to search for pairproduction of the next-to-lightest supersymmetric particle (NLSP), which then decay predominantly to the LSP and the corresponding SM partner particle. e.g. chargino NLSP with dominant decay mode x+ 4 W*x! or slepton NLSP with dominant decay mode t! + ex(:. The signature is missing energy, n jets plus m leptons eventually accompanied by additional photons. The topology depends on the mass of the NLSP and the mass difference A M between the NLSP and the LSP. For large mass of the NLSP the dominant backgrounds are W-boson and Z-boson pair production, whereas for small AM the main background are so-called two photon interactions.

3.1. Searches for Charged Sleptons and Charginos

Sleptons would be pair-produced at LEP. The right-handed sleptons are lighter than the left-handed and their production cross section is smaller. Mass limits are therefore quoted for the right-handed states. In the case of smuons the production cross sections depends almost only on the smuon mass. In the case of selectrons the negative interference with the t-channel contribution usually yields lower cross sections for small neutralino masses. For stau leptons the mixing between right-handed and left-handed states

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to the mass eigenstates can lead to a case where the lighter stau decouples from the Z boson. The dominant decay mode is into a neutralino LSP and the corresponding lepton. The number of events selected by the four LEP experiments is in good agreement with the expectation from SM background processes, which allows to set upper limits on the production cross section and lower limits on the slepton masses, which are shown in Figure 4 (left) 7. For an LSP mass of 40 GeV right-handed selectrons (smuons) can be exluded below masses of 99.9 (96.6) GeV. Right-handed staus can be exluded up to 93.2 GeV and the limit is only lowered to 92.6 GeV in the case of stau mixing leading to a decoupling of the lighter stau from the Z boson ?.

Pairproduction of chargions at LEP would proceed via s-channel exchange of a photon or Z boson and via t-channel exchange of an electron- sneutrino, which interfere destructively. The cross section depends not only on the chargino mass, but in addition on the mass of the electron-sneutrino and on the composition of the chargino in terms of higgsino and gaugino components. For heavy sfermions the dominant decay mode is the so-called three body decay into neutralino LSP and pair of SM fermions arising from the decay of a W boson. For small sfermion masses two body decays into charged slepton plus neutrino or sneutrino plus charged lepton are also possible. For nearly degenerate chargino and sneutrino masses this leeds to almost invisible and hence undetectable final states. This situation is called the “sneutrino corridor”. Again no devation from the SM background expectation was observed, which allowes to set limits on the mass of the chargino of 103.5 GeV under the assumptions that the sneutrino is heavier than 300 GeV and that the mass difference between chargino and LSP exceeds 3 GeV ’. For smaller mass differences dedicated selections have been peformed: (i) the so called ISR analysis covering the mass range between two times the pion mass and 3 GeV and (ii) analysis looking for signatures of the finite lifetime of the chargino, e.g. kink tracks, secondary vertices, for small mass differences below two times the pion mass. The mass limits obtained in case of a higgsino-like scenario is shown in Figure 4 (right) Q. A chargino mass below 92.4 GeV (91.9 GeV) is excluded in the higgsino-like (gaugino-like) scenario ’.

3.2. Mass limits for the Lightest Supersymmetric Particle

The above discussed searches for charged sleptons and charginos in con- junction with other searches for production of supersymmetric particles

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ds = 183-208 GeV h

"el00 5 u, ZTg0

60

40

20 Excluded at 95% CL

50 60 70 80 90 100 Mg(GeV/cz)

ds = 183-208 GeV "JX <> h

"el00 5 u, ZTg0

60

40

20 Excluded at 95% CL

50 60 70 80 90 100 Mg(GeV/cz)

70 80 90 10

1

10

Figure 4. Left: Final LEP combined mass limits for charged sleptons depending on LSP mass. Right: Preliminary LEP combined chargino mass limit in the Higgsinelike scenario.

performed at LEP can be used to derive an absolute lower limit on the mass of the neutralino LSP. The excluded area in the Mxy vs tan p plane in the cMSSM, which depends on the common gaugino mass m1/2, the Higgs mass mixing paramter p, the common trilinear coupling At, the common sfermion mass mo, the mass of the CP-odd Higgs boson M A and the ratio of the vacuum expectation values tanj3 is shown in Figure 5 (left) lo.

The contribution of the Higgs searches and slepton searches especially in the so-called "sneutrino corridor" is indicated. In case of no mixing in the stau-sector the lower limit on the LSP mass from the LEP combined data is determined to be 47 GeV lo . ALEPH has perfomed a careful scan of the cMSSM parameter space, studying the effect of stau mixing, and developed specific searches for neutralino and chargino production with subsequent decays into staus to recover sensitivity. Their finding is that the LSP mass limit decreases by roughly one GeV, when / A , I is restricted to values below 4 TeV and the results from Higgs boson searches are included ll.

In mSUGRA models which depend on m112, tanj3, sign(p) , A and the common scalar mass mo, which then relates the sfermion and H i e s sector, the limit obtained from the combined LEP data is stronger and amounts to MLSP > 50.3 GeV for mt = 175 GeV (see figure 5 (right)) 12. The limit is lowered by roughly one GeV for an increase in mt by one GeV. It should however be noted that no limits on MLSP can be obtained from collider data in the general MSSM without unification of gaugino and sfermion masses.

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Figure 5. (left) and preliminary LEP combined LSP mass limit in mSUGRA (right).

Final LEP combined lower limit on the neutrlino LSP mass in the cMSSM

4. Searches for phenomena from extra dimensions

The introduction of new spatial dimension seems to be an alternative scenario to solve or avoid the hierarchy problem. In the simplest models only gravity may propagate in the extra dimensions and the Planck mass in the full theory taking into account all spatial dimension may be at the TeV scale. In the ADD model l3 the hierarchy between the 3- and 3+6 dimensional Planck mass is achieved by 6 large extra dimensions compaticfied on a torus of size R. In the simple RS model l4 the hierarchy is created by one small extra dimension with a warped metric. Both models can accomodate the fundamental Plack scale to be at the order of TeV.

4.1. Searches for direct gmviton production

In the ADD model l3 graviton production in association with a photon is the most promising channel to look for direct observation of effects from extra spatial dimesions. The experimental signature is one isolated photon and missing energy with the dominant background from e+e- ---$ uuy. A fit of the expected background and signal distributions for energy and angle of the photon to the observed data spectra allows to set a lower limit on the 3 + &dimensional Planck mass M D depending on the number of extra dimensions 6. The preliminary limits on M D obtained from the combination of the ALEPH, DELPHI and L3 data up to the highest center- of-mass energies and the corresponding upper limit on the size of the new

232

dimensions R is shown in Figure 6 and Table 1 l5

1

F

B

0

4 ALEPH DELPHI L3 I Preliminary

.5 .5

1

5

2 3 4 5 6 2 3 4 5 6 Number of Extra Dimensions

Figure 6. Preliminary LEP combined limits on the Planck scale M D in 3+6 dimensions and the size R of the extra dimensions. In the left figure the solid area is excluded by LEP, the hatched area by DO and the bars with arrows by CDF.

Table 1. dimensions and the size R of the extra dimensions

Preliminary LEP combined limits on the Planck scale M D in 3 + 6

6 I 2 1 3 I 4 I 5 I 6

4.2. Searches for the Radion

Radions are particles associated with the stabilisation mechanism of the brane distance. The radion in the RS-model l4 decays dominantly into gluons and can mix with Higgs boson from the SM sector. The production cross sections and branching ratios depend on the masses of the radion-like scalar, the higgs-like scalar, the mixing parameter E and the energy scale on the SM brane Aw. The OPAL collaboration has used their results from SM, flavour-blind and decaymode independent Higgs boson searches to restrict the parameter space of the model 16. The lower mass limits on the two scalar masses in the (A, vs. 5 ) plane are shown in Figure 7 (left). A lower limit on the mass of the higgs-like state of 58 GeV for hw > 246 GeV, any mixing and the mass of the radion-like scalar between 1 MeV and 1 TeV (see Figure 7 (right)). No absolute limit on the mass of the radion-like scalar can be obtained.

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EXpeCted Observed

Figure 7. Left: lower limit on the mass of the Higgs-like scalar (upper row) and on the mass of the radion-like scalar (lower row). Right: dependence of the lower limit on the mass of the Higgs-like scalar on hw.

The L3 collaboration has searched for another type of particles associated with brane fluctuations called branons, which provide a dark matter candidate (see l7 for details).

5. Conclusion

Unfortuanetly no hints for any kind of new physics have been observed in the huge LEP data set. A plentitude of dedicated searches for new topological signals and signal from e.g. Higgs bosons, Supersymmetric particles, effects from Extra Dimensions, Technicolour particles, Compositeness, Lep- toquarks has been performed. The overall good agreement of the observations with the expectation from SM processes allowed to set in a first step almost model independet upper limits on the production cross section for various final state topologies, which are useful for limiting the parameter space of current and future models of physics beyond the SM. In a second step this cross section limits can been turned into limits on particle masses and model parameter in specific scenarios of new physics. The LEP data allowed to rule out a significant part of parameter space of e.g.SUSY models. However one should very carefully notice the assumptions under which this limits have been obtained; e.g. in the CP-violating scenario of the MSSM no absolute limit on the mass of the lightest neutral Higgs boson can be obtained from the LEP data and also no absolutely valid limit on the neutralino LSP mass can be derived from collider data in the general

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MSSM. It remains the task of experiments at present and future colliders as TEVATRON, LHC and ILC to discover new, hopefully completely unexpected, phenomena.

Acknowledgments

I would like to thank C. Rembser and K. Desch for their help in preparing this talk. The help of the LEP Higgs, LEP SUSY and LEP Exotica working groups in providing the information and for useful discussions is greatly acknowledged.

References 1. http://lephiggs.web.cern.ch/LEPHIGGS/www/Welcome.html

http: //lepsusy.web .cern.ch/lepsusy/Welcome.html http://lepexotica.web.cern,ch/LEPEXOTICA/Welcome.html

2. R. Barate et al. [ALEPH, DELPHI, L3, OPAL Collaborations], Phys. Lett. B 565 (2003) 61.

3. M. Carena, S. Heinemeyer, C. E. M. Wagner and G. Weiglein, “Suggestions for improved benchmark scenarios for Higgs-boson searches at LEP2,”arXiv:hep- ph/9912223.

4. LEPHIGGSWG, ALEPH, DELPHI, L3 and OPAL LHWG-Note 200401. 5. G. Abbiendi et al. [OPAL Collaboration], Eur. Phys. J. C 37 (2004) 49. 6. M. Carena, J. R. Ellis, A. Pilaftsis and C. E. Wagner, Nucl. Phys. B 586

(2000) 92. 7. LEPSUSYWG, ALEPH, DELPHI, L3 and OPAL experiments (ADLO), note

LEPSUSYWG/04-01.1 8. LEPSUSYWG, ADLO, note LEPSUSYWG/Ol-03.1 9. LEPSUSYWG, ADLO, note LEPSUSYWG/02-04.1 10. LEPSUSYWG, ADLO, note LEPSUSYWG/0407.1 11. A. Heister et al. [ALEPH Collaboration], Phys. Lett. B 583 (2004) 247. 12. LEPSUSYWG, ADLO, note LEPSUSYWG/02-06.2 13. N.Arkhani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429 (1998)

14. L. Randall and R. Sundrum, Phys. Lett. B 83 (1999) 3370 and Phys. Lett.

15. LEP Exotica WG, ADLO, note LEP Exotica 200403. 16. G. Abbiendi et al. [OPAL Collaboration], “Search for radions at LEP2,”

17. Antonio L. Maroto, “Branon dark matter”, these proceedings.

263.

B 83 (1999) 4690.

arXiv: hep-ex/0410035.

P. Achard et al. [L3 Collaboration], Phys. Lett. B 597 (2004) 145.

DISCOVERING THE HIGGS BOSONS OF MINIMAL

BOTTOM QUARK SUPERSYMMETRY WITH TAU-LEPTONS AND A

RAHUL MALHOTRA The University of Texas at Austin

Department of Physics 1 University Station C1600

E-mail: [email protected] Austin, TX 78718, USA

We investigate the prospects for the discovery at the CERN Large Hadron Col- lider of a neutral Higgs boson produced with one bottom quark followed by Higgs decay into a pair of tau leptons. We work within the framework of the minimal supersymmetric model. The dominant physics background from the production of br+r-, jr+r- ( j = g, u, d, s, c), bbW+W-, W + 2 j and b6g is calculated with realistic acceptance cuts and efficiencies. Promising results are found for the CP-odd pseudoscalar (Ao) and the heavier CP-even scalar ( H o ) Higgs bosons with masses up to 1 TeV.

1. Introduction

In the minimal supersymmetric standard model (MSSM) ', the Higgs sector has Yukawa interactions with two doublets 41 and $2 that couple to fermions with weak isospin -1/2 and +1/2 respectively '. After spontaneous symmetry breaking, there remain five physical Higgs bosons: a pair of singly charged Higgs bosons H* two neutral CP-even scalars H o (heavier) and ho (lighter), and a neutral CP-odd pseudoscalar Ao. The Higgs potential is constrained by supersymmetry such that all tree-level Higgs boson masses and couplings are determined by just two independent parameters, commonly chosen to be the mass of the CP-odd pseudoscalar ( M A ) and the ratio of vacuum expectation values of neutral Higgs fields ( tanp = v2/v1).

At the CERN Large Hadron Collider (LHC), gluon fusion (gg -+ c$l 4 = h o , H o , or A') is the major source of neutral Higgs bosons in the MSSM for tanP less than about 5. If t anP is larger than 7, neutral Higgs bosons are dominantly produced from bottom quark fusion bb + 4 3*495,6*7. Since the Yukawa couplings of 4bb are enhanced by 1/ cosp, the production rate

235

236

of neutral Higgs bosons, especially the Ao or the Ho, is enhanced at large tan 0.

For a Higgs boson produced along with a single bottom quark at high transverse momentum ( p ~ ) , the leading-order subprocess is bg --f bq5 879110111. If two high p~ bottom quarks are required in association with a Higgs boson, the leading order subprocess should be gg -+ bbq5 3912913914715.

Recently, it has been suggested that the search at the LHC for a Higgs boson produced along with a single bottom quark with large p~ should be more promising than the production of a Higgs boson associated with two high p~ bottom quarks lo.

This has already been shown to be the case for the p+p- decay mode of the Higgs 16. For large tano, the T+T- decay mode 17J8 is also a promising discovery channel for the Ao and the H o in the MSSM. This is because the branching fraction for Higgs decay into tau leptons is greater by a factor of (m,/m,)2 - 286. The downside is that unlike muons, tau leptons can only be observed indirectly in the detector via their hadronic or leptonic decay products.

In this article, we present the prospects of discovering the MSSM neutral Higgs bosons produced with a bottom quark via Higgs decays into tau pairs. We calculate the Higgs signal and the dominant Standard Model (SM) backgrounds with realistic cuts and efficiencies and evaluate the 5 ~ 7 discovery contour at the LHC in the M A - t an0 plane.

2. The production cross sections and branching fractions

We calculate the cross section at the LHC for p p + b#+X (4 = Ho, ho, Ao) via bg --f b4 with the parton distribution functions of CTEQ6Ll 19. The factorization scale is chosen to be M H / ~ 6*20. In this article, unless explicitly specified, b represent a bottom quark (b) or an anti-bottom quark (6). The bottom quark mass in the 4b& Yukawa coupling is chosen to be the NLO running mass m b ( p ~ ) 21, which is calculated with mb(po1e) = 4.7 GeV and the NLO evolution of the strong coupling 22. We have also taken the renormalization scale to be M H / ~ . This choice of scale effectively reproduces the effects of next-to-leading order (NLO) lo. Therefore, we take the K factor to be one for the Higgs signal.

The cross section for p p + bd -+ h+~- + X can be thought of as the Higgs production cross section ~ ( p p --f b 4 + X ) multiplied by the branching fraction of the Higgs decay into muon pairs B(q5 -+ T+T-). When the b i mode dominates Higgs decays, the branching fraction of 4 + T+T- is about

237

m:/(3mi(m6) + m:) where mb(m,#,) is the running mass at the scale m4. This results in a branching fraction for A’ + T+T- of - 0.1 for M A = 100 GeV. For tanP 2 10 and M A 2 125 GeV, the cross section of bAo or that of bHo is enhanced by approximately tan2 P; the tau branching fraction is sustained by the large decay width of the Higgs into bottom quarks.

3. Tau Decay and Identification

Tau leptons can decay either purely leptonically, T- * l-fip,, with a branching ratio of around 18% for each mode 1 = e,p, or they can decay into low-multiplicity hadronic states and a v, with a branching ratio M 64% 23. More than 95% of all hadronic tau decays are l-prong or 3-prong, i.e. they contain either one or three long lived charged particles. Therefore, for a T+T- pair, the most likely scenario is one decaying leptonically and the other hadronically. This combination has a combined branching ratio of 46%. Also, the presence of an isolated lepton in the final state is useful in triggering the event and reducing backgrounds. Hence, we use this “lepton + T-jet” signature in our study.

We model hadronic tau decays as the sum of two-body decays into w,, pv, and alv, with branching ratios given in the literature 23. The tau is assumed to be energetic enough that all its decay products emerge in approximately the same direction as the tau itself. This manifests itself in the so-called “collinear approximation” which we use for both leptonic and hadronic decays. The approximation is confirmed to be accurate by comparison with an exact matrix element simulation for tau decay.

The ATLAS collaboration has studied identification efficiencies of T-jets in detail 24. We use an overall efficiency of 26% over 1- and 3-prong decays with a corresponding cut, Pt(h) > 40 GeV for the hadron h = ?r, p, al. This also corresponds to a mistag efficiency of 1/400 for non-7 (i.e. QCD) jets. Rejection of jets from b quarks is higher, with only 1 in 700 being mistagged as TS. The transverse momentum cut on the lepton from tau decay is weaker, with Pt(1) > 20 GeV. Both the hadron and lepton are required to be in the central rapidity region Jr]J < 2.5.

4. The Physics Background

From the above discussion, the signal we are looking for is: bjet + T-jet + lepton + Ptmiss + X .

The dominant physics backgrounds to this final state come from: (i) Drell-Yan processes: p p +. jZ*/y* + X ~ T + T - + X , j =

238

u, d , s, c, b, g . Nearly 60 - 70% of the DY contribution arises from the s u b process bg -+ br+r-.

(ii) W + 2 j processes: pp -+ W + 2 j with the subsequent decay W -+

lvl; 1 = e , p. Here, one jet is tagged or mistagged as a b quark and the other mistagged as a T-jet.

(iii) Top Production: gg -+ tf -+ bbW+W- and qij -+ tt -+ bbW+W-. This can contribute in several ways depending on subsequent W decays. In order of highest to least importance the channels are: (a) One W decays into Zq while the other provides a rv, with the r decaying hadronically. (b) One W decays into lvl while the other decays into jets W -+ jj. We now have four possible jets in the final state i.e. 2 b’s and 2 j and one of them is tagged as a b quark while one of the others is mistagged as a r-jet. (c) Lastly one can have both W ’ s decaying into rv, with one tau decaying leptonically and the other hadronically. This is the smallest contribution from tt.

Due the huge cross-section for p p -+ QQg; Q = b, c production at the LHC, it is also pertinent to check if heavy quark semi-leptonic decays such as b -+ clv do not overwhelm the signal. We find that this background is effectively cut to less than 10% of the dominant background at all times by an isolation cut on the lepton Iq(2,j)l > 0.3, the large rejection factor for non-r jets and the requirement P p s > 20 GeV.

For an integrated luminosity ( L ) of 30 fb-l, we require P~(b,j) > 15 GeV and Iq(b,j)l < 2.5. The b-tagging efficiency (q,) is taken to be SO%, the probability that a c-jet is mistagged as a b-jet ( 6 , ) is 10% and the probability that any other jet is mistagged as a b-jet ( ~ j ) is taken to be 1%. For L = 300 fb-’, € b = 50% and PT(b,j) > 30 GeV.

In order to improve the signal significance, we also apply the following cuts: 1.8 < 4(l ,~- je t ) < 3.4 and Transverse mass m ~ ( Z , ~ - j e t ) < 25 GeV. Using the definition of transverse mass given in 25 we find that the latter is found to be the most effective in controlling the W + 2 j and tf backgrounds.

We have also applied a K factor of 1.3 for the DY background 26 and a K factor of 0.9 for W + 2 j 27 to include NLO effects. In order to further cut down the tf background, we apply a veto on events with more than 2 jets in addition to the b and T jets. This is very effective because, in tt+ X production, nearly 50% of events have at least one gluon from initial or final state radiation that passes PT > 15 GeV and 1711 < 2.5 28. Such events are then vetoed and we are able to use a K factor of 1, instead of 2 that would be required otherwise. We are also able to reduce contributions from top production where one W -+ j j decay occurs [(iii)(b) above].

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We have employed the programs MADGRAPH 29 and HELAS 30 to evaluate matrix elements for both signal and background processes.

5. Higgs Mass Reconstruction

The Higgs mass can be reconstructed indirectly, using the collinear a p proximation for r decay products and the missing transverse momentum 2-vector, PFiss. Taking 21, Xh to be the energy fractions carried away by the lepton and hadron respectively arising from r decay, we have:

1 1 (- - 1)Ph + (- - 1)Pk = P Y i S S 21 xh

This yields two equations for x1 and Xh, which can be solved to reconstruct the original T 4-momenta p , = pl/xl, ph/xh. So, m$ = (pl/xl + ~ h / s h ) ~ .

Naturally, physically we must have 0 < x1,xh < 1, and this provides a further cut to reduce the background. We further limit 0 < xi < 0.75 as this has negligible affect on the signal, but reduces the W + 2 j and tf backgrounds.

Measurement errors in lepton and T-jet momenta as well as missing transverse momentum give rise to a spread in the reconstructed mass about the true value. Based on ATLAS specifications we model these effects by Gaussian smearing of momenta:

AE 5.2 0.16 -- - - 63 - 63.009 E E L ! ?

for jets (with individual terms added in quadrature) and

- 2% AE E -- (3)

for charged leptons. We find that in more than 95% of the cases, the reconstructed mass

lies within 15% of the actual mass. Therefore we apply a mass cut, requiring the reconstructed mass to lie in the mass window 7nb f Am,, where Am, = 0.15m.4. This cut is actually rather conservative as for larger Higgs masses, more than 90% of the reconstructed masses are within 5 - 10% of m,. Therefore, while we use 15% throughout for simplicity, significant improvements in our discovery contour might be possible by narrowing Am, depending on m4.

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6. The Discovery Potential at the LHC

Based on the cuts defined so far, in Figure 1 we show the signal and background cross sections for luminosity L = 30 fb-l. The signal is shown for t a n p = 10 and 50, with a common mass for scalar quarks, scalar leptons and the gluino m,- = m, = p = 1 TeV. All tagging efficiencies and K factors discussed above are included. F'rom this figure we note that the cross sec-

n

$1 U

1 o3

1 o2

1 o1

1 oo

10-l

1 o-2

10"

Signal and Background I I I I I I -I

J --. --.

I I I I I I I

100 200 300 400 500 600 700 800 MA (GeV)

Figure 1. The signal cross section at the LHC for luminosity L = 30 fb-', as a function of M A , for me = mg = p = 1 TeV and tb = tang = 10,50. Also shown are the background cross sections in the m a s window of MA f Am, as discussed in the text for the SM contributions. We have applied K factors, acceptance cuts, and efficiencies of b, T tagging and mistagging.

tion of the Higgs signal with tan@ - 50 can be much larger than that of the physics background after acceptance cuts. The Drell-Yan processes make the major contributions to the physics background for Higgs mass 5 180 GeV, but tf contributions become dominant for higher masses. The W + 2 j

24 1

contribution is very effectively controlled by the b tagging requirement. Figure 2 shows the 5a discovery contours for the MSSM Higgs bosons

where the discovery region is the part of the parameter space above the contour. We define the signal to be observable if the lower limit on the

LHC Discovery Region

c c

60

50

40

30

20

10

0 100 200 300 400 500 600 700 800

Figure 2. The 50. discovery contours at the LHC for an integrated luminosity (L) of 30 f t - l , 100 f t - l , 300 f t - l in the M A versus tanp plane. The signal includes 4 = Ao and ho for M A < 125 GeV and 4 = Ao and Ho for M A > 125 GeV. However, for tanp < 10, q5 = Ao only. The discovery region is the part of the parameter space above the contours.

signal plus background is larger than the corresponding upper limit on the background 31t32, namely,

242

which corresponds to

us > L [1+ 2&/N] (5)

Here L is the integrated luminosity, u, is the cross section of the Higgs signal, and Ub is the background cross section. Both cross sections are taken to be within a bin of width &Am, centered at m+. In this convention, N = 2.5 corresponds to a 5u signal. We take the integrated luminosity L to be 30 fb-' and 300 fb-' 24.

For t a n 0 2 10, MA and m H are almost degenerate when MA 2 125 GeV, while M A and mh are very close to each other for M A 5 125 GeV in the MSSM. Therefore, when computing the discovery reach, we add the cross sections of the A' and the h' for MA < 125 GeV and those of the Ao and the H' for M A 2 125 GeV 33 We have chosen Msusy = me = m- 9 = mi = p = 1 TeV. If Msusy is smaller, the discovery region of A', H' + T+T- will be slightly reduced for M A 2 250 GeV, because the Higgs bosons can decay into SUSY particles 34 and the branching fraction of 4 + T+T- is suppressed. For M A 5 125 GeV, the discovery region of H' + T+T- is slightly enlarged for a smaller Msusy, but the observable region of ho + T+T- is slightly reduced because the lighter top squarks make the H' and the h' lighter; also the H'bb coupling is enhanced while the h'bb coupling is reduced 33.

We find that the discovery contour even dips below tanP = 10 for 100 GeV < M A < 300 - 400 GeV depending on luminosity. However, below t a n p = 10 our approximation of mass degeneracy of MSSM Higgs bosons breaks down. Therefore below t a n 0 = 10 we include only one Higgs boson ( M A ) in our calculations.

7. Conclusions

The tau pair decay mode is a promising channel for the discovery of the neutral Higgs bosons in the minimal supersymmetric model at the LHC. The Ao and the H' should be observable in a large region of parameter space with t a n 0 2 10. In particular, Fig. 2 shows that the associated final state of bq5 -+ 2 r r + ~ - could discover the A' and the H o at the LHC with an integrated luminosity of 30 fb-l if M A 5 800 GeV. At higher luminosities of 100-300 fb-', the discovery region in MA is easily expanded up to MA 5 1 TeV for t a n 0 - 50.

Excellent b tagging, .r-jet tagging abilities and jet energy resolution at the CMS and the ATLAS detectors will be important for Higgs searches in

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this channel. The discovery of the associated final states of b4 + ~ T + T - will provide information about the Yukawa couplings of b64 and an opportunity to measure tanP. The discovery of both 4 4 T+T- and 4 + p+p- will allow us to understand the Higgs Yukawa couplings with the leptons.

Acknowledgments

This work is being extended in collaboration with D. Dicus of University of Texas at Austin and C. Kao and P. Williams of the University of Oklahoma. This research was supported in part by the US. Department of Energy under Grant No. DE-FG03-93ER40757.

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ELECTROWEAK AND TOP PHYSICS RESULTS FFtOM D 0

PUSHPALATHA c BHAT+ Fermi National Accelerator Laboratory

Batavia, ZL 60510, USA

For the D0 Collaboration

The collider Run I1 at Fermilab that started in March 2001 with upgraded accelerator complex and detectors is progressing extremely well. An integrated luminosity of 670 pb-' was delivered to the CDF and D 0 experiments each, by the end of August 2004. Additional planned upgrades to the accelerators will result in an integrated luminosity of 4 - 8 fb-' for each experiment by the end of 2009. I present some preliminary electroweak and top quark physics measurements made by the D 0 collaboration analyzing data sets corresponding to integrated luminosity in the range of 150-250 pb-'.

1.1. Introduction

The CDF and D 0 experiments, at the Fermilab proton-antiproton collider, have entered a new era in the exploration of physics near the electroweak scale. Already about four times more data than collected in Run I has been accumulated and an order of magnitude more will become available over the next several years, greatly expanding the scope for exploration and discovery with direct searches and with precision measurements of the top and electroweak physics. The excellent performance of the Tevatron and the CDF and D 0 experiments during Run I (1992-96) made possible the discovery of the top quark in 1995 [l], which was widely acknowledged as a major triumph of the Standard Model (SM). Other highlights of Run I results include precision measurements of the top quark mass, top-antitop production cross section, W boson mass, rates of a number of SM processes, and searches for possible new physics. The collider Run I1 started in March 2001 after major upgrades to the accelerator complex and the detectors, and with the higher center of mass energy of & =1.96 TeV. Additional Run I1 upgrades [2] underway throughout the accelerator complex are expected to result in 4-8 fb-' of data delivered to each of the experiments by 2009. This large data set will allow refined measurements of the top quark and its dynamics and the exploration of the tantalizing possibility that the top quark plays a role in electroweak symmetry

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breaking. Precision measurements of electroweak processes and a wealth of other physics are guaranteed and the potential for discoveries is great.

Both the CDF and D 0 detectors underwent major upgrades in preparation for Run 11. The Run I1 DO detector [3] has three major sub-systems - a new central tracking system, comprising a Silicon Microstrip Tracker (SMT) and a Central Fiber Tracker (CFT) inside a 2T solenoidal magnetic field, a Uranium/liquid- Argon calorimeter surrounding the central tracker and an upgraded outer muon spectrometer. Fig. 1 shows a photograph of the assembled detector in the collision hall and Fig. 2 shows the schematic drawings of the full detector and the central tracking system. The SMT has six concentric barrels of four layers each, nested around the beam pipe, interspersed with twelve radial disks and, additionally, 4 large radial disks at large z along the beam axis, collectively providing a coverage in the pseudorapidity range of lq1<3, (where q =-ln(tan 8/2) and 8 is the polar angle). The CFT has eight coaxial barrels, each supporting two doublets of overlapping scintillating fibers, one doublet parallel to the beam axis and the other alternating by f3' relative to the beam axis. The calorimeter has three parts - a central calorimeter (CC) up to lql-1.1 and two end calorimeters (EC) extending the coverage to lql-4.2, each housed in a separate cryostat. To aid in particle identification, each calorimeter consists of an inner electromagnetic (EM) section, a fine hadronic (FH) section and a coarse hadronic (CH) section. The absorber in the EM and FH sections is depleted Uranium; in the CH section, it is a mixture of stainless steel and copper. Scintillators between the CC and EC cryostats provide sampling of showers for 1.1 <)+I .4. The muon system consists of a layer of drift chambers for tracking and scintillation trigger counters in front of magnetized iron toroids, with a magnetic field of 1.8 T and two similar layers behind the toroids. Tracking in the muon system for Iql<l uses 10 cm wide drift tubes while 1 cm mini drift tubes are used in the forward muon system for l<(q(<2.

The online trigger system that selects events for storage and offline analysis has three levels. The first two levels are implemented in hardware and firmware and the third level is primarily implemented in software running on a parallel processing farm of commodity CPUs. Candidates for tracks, electrons, jets, muons and missing energy are formed using information from individual detectors at Level 1, then processed and combined with increasing sophistication, in the higher level systems. Data at Level-3 are written to mass storage at the rate of 50 to 60 Hz.

The accelerator performance has improved significantly in the past year as a direct result of many reliability and operational improvements and upgrades [2]. The integrated luminosity delivered by the Tevatron so far, during Run 11, to the experiments is about 670 pb-' of which D 0 has recorded about 460 pb-' with a fully operational Run I1 detector. It is expected that 2 fi' will be delivered by

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mid-2006, - 4 fb-’ by 2007 and in excess of 8 fb-’ by 2009. Of the 460 pb” data collected so far, the electroweak and top physics results reported here come from analyses of data corresponding to 150 - 250 pb-’.

Figure 1 Photograph of the D 0 detector m the collision hall, prior to the start of Run I.

Figure 2 Schematic drawings of (Left) the full detector showing the three major sub-systems the Central Tracker, the central and end-cap Calonmeters and the Muon system and (nght) the components of the central trachng system See text for details.

1.2. Measurements with Electroweak Bosons

Rigorous studies of the electroweak gauge bosons, W and Z, and the processes containing them are of paramount importance for a variety of reasons. The W

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boson mass is an important standard model parameter and, together with the top quark mass, constrains the mass of the SM Higgs boson. The production rates of the W and Z bosons and the di-boson states WW, WZ and ZZ provide stringent tests of the electroweak Standard Model and Quantum Chromodynamics (QCD). A comparison of the leptonic branching fractions B(W+Cv), where C=e,p, or T, tests the universality of leptonic couplings to the weak current. The angular distributions of leptons from W boson decay provides constraints on ratios of parton distribution functions (PDF) of the proton and will yield better understanding of the next-to-leading order (NLO) corrections to the production models. The W and Z bosons are present in the final states of many interesting physics processes that are the prime objects of study, such as the decay products of the top quark and in associated production with the, yet to be discovered, Higgs boson. The SM processes involving the production of electroweak bosons also form important backgrounds to searches for new physics. Large W and Z boson samples provide excellent calibration tools for the detector energy scale and to benchmark the performance of the detector, trigger, and event reconstruction algorithms. The W boson production can also be used to calibrate the luminosity measurement for the experiments.

1.2.1. Wand Z Boson Production Cross Sections

Hadronic decays of the W and Z bosons are difficult to measure due to large backgrounds from QCD jet production. Therefore, W and Z bosons are mainly identified and measured in the leptonic decay channels. The W and Z production cross sections times their branching fraction into leptonic channels (aw.B(W+!'v) and oz.B(Z+!+C-), where C is a charged lepton) can be measured with great precision. These measurements provide tests of the QCD predictions for W and Z production. The ratio, R, of these cross section measurements, can be used to measure indirectly the width of the W boson.

The W boson decays are identified by a high transverse energy charged lepton and large transverse missing energy due to the neutrino, while the Z boson events are characterized by two high transverse energy charged leptons. In this paper, we present results from W decays in the electron and muon channels and Z decays in electron, muon and tau channels. The tau channel is particularly interesting because it can be gainfully exploited in searches for the Higgs boson and Supersymmetry (SUSY).

Electrons are identified as electromagnetic (EM) clusters in the calorimeter using a simple cone algorithm. To suppress backgrounds from QCD jet events (jets faking electrons), electron candidates are required to have a large fraction of their energy (typically >90%) deposited in the EM section of the calorimeter and satisfj certain isolation and cluster shape criteria. Electron candidates are classified as tight electrons if a track in the central detector is matched spatially

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to the EM cluster and if the track momentum is close to the transverse energy of the EM cluster. The W event sample is selected by requiring one central (lql4.0) electron with transverse energy ET>25 GeV and missing transverse energy &>25 GeV. The Z boson sample is selected requiring two tight central electrons with ET>25 GeV. The transverse mass distribution of the W candidate events and the invariant mass of the two electrons from the Z candidates are shown in Fig. 3, compared with simulations using PYTHIA [4] for signal event generation and a parameterized Monte Car10 model of the detector response.

-.* P f v . saoc-" 2 .

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Figure 3. (Left) Transverse mass distribution of W bosons decaying into an electron and a neutrino. mght) Invariant mass of Z bosons in the di-electron channel. (IL.dt = 177 pb-').

Muons are identified as tracks in the outer muon system, matched to tracks in the central tracking system. Muons from the decays of heavy-flavor hadrons form a significant background to the vector boson events in the muon channel. This background is significantly reduced by requiring the muons to be isolated spatially from other objects in the event. Contamination from cosmic ray muons are reduced with criteria on the timing of the signal in the muon chambers relative to the p7j interaction time and the impact parameter of the track at the vertex.

The W sample in the muon channel is selected by requiring an isolated muon with transverse momentum pr >15 GeV and lq1<2.0. The Z sample is selected by requiring two oppositely charged, isolated, muons with pr >15 GeV and 1111 < 1.8. To reduce the background from semi-leptonic b decays, a number of isolation criteria are applied. These include (1) the sum of m of other tracks

within a cone of radius R=0.5 (where R = d-, 4 being the azimuth) of the candidate muon be less than 3.5 GeV and (2) the total energy deposition in the calorimeter cells, within a radius of R=0.4, excluding that of the muon, be less than 2.5 GeV. The invariant mass distribution of the 14352 di-

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muon candidates coming primarily from Z bosons is shown in Fig.4. The major physics backgrounds for this channel come from b-pairs (estimated to be (0.5&0.3)'?40,Z+rr (0.5+0.1)%, and WW+jets (0.2f0. I)%.

Figure 4. The invariant mass of the di-muons from the 2 candidate sample. (jL.dt = 148 pb-').

Identification of T leptons is challenging at a hadron collider. Unlike electrons or muons, the taus decay very close to where they are formed (with a decay length about 4 times smaller than that of the b-quarks) and have to be reconstructed using their decay products. The T identification used in the current analysis is based on the use of neural networks to discriminate 1-prong and 3-prong decays of taus from other sources, and is optimized for the Z+TT process.

The Z+TT event selection requires a T candidate (identified as .c+hadrons or r+ev,v, ) back to back with a single isolated muon (from ~+pv,,vJ. Most of the neural network variables are ratios of the energies of various objects (and their sums) to the ET of the T, in order to minimize the dependence on ET. The isolated muon is required to have a pr >12 GeV and the T candidate pT > 5-10 GeV, depending on the type of decay. The 1946 events selected have an estimated background of 55% from b-quark pairs, W+jets and Z+pp. Fig. 5 shows the invariant mass distribution of the p and T tracks for the background (estimated from the like-sign data sample) and for data (from opposite sign sample and background subtracted) compared with Z+TT monte carlo.

The cross section times the branching fraction measurements for all channels discussed above are summarized in Table 1. In the 2 channels, they are after subtraction of the virtual photon exchange process. The main systematic uncertainty in the measurements is from the luminosity measurement and is about 6%, and is shown separately from other systematic errors. The PDF uncertainties are typically about 2% and estimated for CTEQ6M PDFs. The

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contribution from lepton identification is -2%. This error is expected to reduce

Channel

c 18[

Number of Bkg. It dt 0. B (pb) Candidates (%) (nh-')

Figure 5. Invariant mass distribution of the p and 'I tracks for background sample using like-sign tracks (left) and for Z candidate data with opposite sign tracks (right) for the Z+rr process shown normalized to Z+rr monte carlo. The data sample corresponds to IL.dt = 207 pb'.

W j e v W j p v

. , \r- I

116569 3.1 5 1 77 2865f8.3 (stat) +74.7( sys)f 1 86.2( lum) 8305 11.8 17 3226+128(stat) +lOO(sys) +323(lum)

Zje'e- z+J.l&J.l- Z+r+t-

4712 1.8 177 264.9+3.9(stat) +9.9(sys) +17.2(lum) 14352 1.4 148 291.3+3.0(stat) +6.9(sys) *18.9(lum) 1946 55 207 2%+l hfstat\+17(~9\ +1 fdliim\

The measured cross sections are in good agreement with theoretical calculations as shown in Fig. 6. The ratio of the leptonic W and Z cross sections is much more precisely determined than the individual cross sections because the largest systematic uncertainty due to luminosity measurement cancels. This ratio, in the electron channel, is measured to be R = 10.82~0.16(stat)+0.28(sys).

Table. 1W and Z boson inclusive cross sections in various leptonicdecay channels.

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6

Figure 6. W and Z boson cross section measurements in all leptonic channels from Run I1 (preliminary) and Run I, from CDF and DO, compared with " L O (next-to-next leading order) theoretical predictions.

1.2.2.Di-boson Production Cross Sections

Studies of multiple electroweak boson production at the Tevatron can help probe the gauge structure of the SM and search for anomalous couplings. Tri- linear gauge couplings larger than those predicted by the SM may signal new physics. The di-boson production processes are also important backgrounds in many new particle searches and therefore need to be understood and measured well.

The WW production has been studied in ee, ep and pp channels, using data sets corresponding to luminosities in the range of 220-250 pb-I. In the ee and pp channels, one of the important backgrounds is due to Z bosons. The dominant background in the ee channel is from W+jet events where a hadronic jet can fake an electron. The ep channel is the cleanest of all, with a signal to background ratio of about 4. Combining all three channels, we expect 16.1k0.1 events from WW production and 8.05k0.70 events from background sources. With 25 events found in data, a cross section of 13.8T::i(stat) f I.O(sys)f 0.9(lum) has been measured.

In a search for associated production of WZ events in the trilepton channel (containing electrons and muons), in about 200 pb-' of data, we expect to find 1.02f0.07 signal events and 0.39H.02 background events after the selection cuts. One tri-muon candidate event remains in the data. The analysis yields an upper limit of 15.1 pb on the WZ cross section at 95% C.L. The SM prediction is about 3.7 pb.

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The associated production of a photon with a W or Z has been studied using W and Z decays in electron and muon channels. Photons are identified as EM clusters in the calorimeters with isolation and cluster shape criteria as in the case of electrons. Photons are distinguished from electrons by the absence of tracks in the central tracker pointing to and consistent with the EM cluster. The sum of prof the tracks within a cone of R=0.4 is required to be less than 3 GeV. The pr of the photon is required to be larger than 8 GeV. The dominant background in both channels comes from W/Z +jet events where the jet fakes a photon. This background is determined by measuring the rate for a jet to fake a photon in a sample of QCD multi-jet events and applying it to samples of W/Z +jet events. In the Wy analysis, using 162 pb-' of data for the electron channel and 82 pb-' of data for the muon channel, 146 and 77 events are observed, with estimated backgrounds of 87. 1f7.5 and 37k10, respectively. In the Zy analysis using 117 pb-' of data in the electron channel and 144 pb-' in the muon channel 33 and 68 events are observed, with estimated backgrounds of 4.7rt0.7 and 10.1k1.3 events, respectively. Combining the two channels, the cross sections are measured to be CT( p p +Wy+X+!vy+X)=19.3f6.7(stat+sys)k1.2(lum) pb and CT( p p +Zy+X+ !!y+X)=3.9f0.5(stat+sys)~0.3(lum) pb.

1.3. Results on Top Quark Production and Properties

The dominant production mechanism for top quarks at a hadron collider is pair production of top and anti-top quarks through the strong interaction processes of quark-antiquark annihilation and gluon-gluon fksion. At the Tevatron, at 6 =1.96 TeV, the relative contributions of these two processes are about 85% and 15%, respectively. The top quarks can also be produced singly, through electroweak processes, but with a rate calculated to be about half of that for pair production. According to the SM, each top quark decays into a W boson and a b-quark, nearly 100% of the time. The t i events are classified into dilepton, lepton +jets and all-jets channels depending on whether one or both of the W bosons decay leptonically or hadronically. The all-jets events have the largest branching fraction (-45%) but suffer from huge backgrounds from other QCD multijet events. The lepton+jets channels each have a branching fraction of 14.5% for each kind of charged lepton, and the cleanest dilepton channel, ep, has -2.5% while ee and pp have each -1.2%. In the following sections, we describe the recent results on the top quark cross section measurements in various channels and the measurement of its mass, using a Run I1 data sample corresponding to L d t = 140 - 160pb'.

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1.3. I Top-antitop Pair Production Cross Section

Preliminary measurements of the cross section from dilepton and lepton+jets channels are reported here.

The signature for tl decays in dilepton channels is the presence of two high ET,

central, isolated leptons, two jets initiated by the b-quarks (one or both may be tagged using the SMT) and a large E, arising from the two neutrinos. The dominant physics backgrounds are from leptonic decays of the Z, the Drell-Yan process, and vector boson pair production. The instrumental backgrounds arise from mis-measured E, in Z + l t decays and from fake leptons in W+jets events. Candidates were selected by requiring two isolated leptons (electron or muon), with ET>15 GeV in ep channel, ET>20 GeV in ee channel and 1q1<1.1 or 1.5<1q1<2.5 for electrons and PT >15 GeV and lq1<2.0 for muons. At least two jets with ET>20 GeV and E, >25 GeV for the ep channel and E, >35 GeV in the case of ee and pp channel, are required. Events with dilepton invariant mass consistent with a Z are removed in the ee and pp channels. To further increase the signal to background ratio, a cut on HT (the scalar sum of the transverse energies of the leading lepton and the jets in the event) is imposed. HT is required to be larger than 140 GeV in the ep channel and 120 GeV in the pp channel. Combining the ee, ep and pp channels yields 17 observed events, with an expected background of 4.8f0.7 events and signal of 6.0f0.5 events (for mt=175 GeV and an assumed t i cross section of 7 pb). This yields a cross section measurement of, (T - = 14.3-, (stat)-, (sys) f 0.9(lwn) pb . A separate analysis has also been performed in the ep channel requiring one of the jets be b-tagged with a reconstructed secondary vertex. This selection is virtually background-free (< 0.1 event); 5 events are observed with an expected signal of 3.2 events, and this results in a measured cross section of (T - = 11.1-, (stat) f 1.4(sys) k 0.6(lum)pb.

+5 1 +2 6

tt

+5 8

tt

In the lepton+jets channels, where the signature is one high pr lepton, large E, and four or more jets (2 of which are b-jets), two analysis approaches have been employed. The analyses use *jets (p+ jets) events with E; (p;) >20 GeV,

1q1<1.1(2.0), E,>20(17) GeV, four jets with ET>15 GeV and lq1<2.5. As pioneered by DO in Run I [ 5 ] , in the so-called "topological" analysis, four variables with good discriminating power between the signal and background are combined into a likelihood discriminant. The variables used here are the event aplanarity, sphericity, a third variable that measures the centrality of the event and a fourth variable that measures the extent to which the jets are clustered together. The discriminant distribution for the data events and results

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of fits to signal and backgrounds for the e+jets channel are shown in Fig. 7 (plot on the left). The measured cross section from combining both channels is, 6,- = 7.2-,,, (stat)'::' (sys) f 0.5(lum) pb . +2.6

An analysis has also been performed using two different b-tagging algorithms: (1) based on the reconstruction of a secondary vertex (SVT) and (2) based on measuring the impact parameter of tracks with respect to the primary interaction vertex (CSIP). The jet multiplicity distributions for data for the single SVT tagged events are shown in Fig. 7 (right plot), and compared with signal and background contributions. These analyses provide the most precise ti cross section measurements from the Run 11 D 0 data and are, CT - = 8.2 f 1.3(stat)': 6" (sys) f 0.5(lum)pb from the SVT analysis and

CT - = 7.2-, (stat)': 3 (sys) f 0.5(lum)pb from the CSIP analysis. tt

tt + I 3

]st multiplicity

Figure 7. &eft) Distribution of the likelihood discriminant of data events, compared with expected signal and main backgrounds (W+jets and QCD) in the electron+jefs channel from the topological analysis. (Right) Jet multiplicity distribution of single btagged events in the lepton+jefs channel. The data bins with jet multiplicity of 23 jets per event show an excess of events above background consistent with expected top-antitop events.

The results of cross section measurements in a variety of final states are summarized in Fig. 8.

1.3.2 Top Quark Mass Measurement

The top quark mass was measured with impressive precision in Run I [6]. The latest CDF and D 0 combined result for the top mass using Run I data is 178.0k4.3 GeV. That precision will be well surpassed in Run I1 because of the huge statistics of t i events that will be accumulated. However, for this to

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Figure 8. Top pair production cross section measurements using a variety of decay channels. The band shows the predicted theoretical cross section with uncertainties. The luminosity of the data sample used in each case and the cross section with statistical and systematic errors are listed to the right of the plot. The uncertainty from luminosity is about 6%.

happen detailed understanding of the detector energy scale and control of other systematic effects are essential. Preliminary results from two analyses of the lepton+jets events using about 160 pb-' of data are reported here.

The data sample used is the same as for the topological cross section analysis in the lepton+jets channel. A further requirement of ET>20 GeV is imposed on the three leading jets. Kinematic fits to the top-antitop hypothesis are performed for candidate events to extract top quark mass information for each event. Since there are 12 possible permutations to assign the four jets in the decay hypothesis, a fitted top quark mass and the fit x2 is obtained for each permutation. Events that have at least one fit with xz<10 are retained. To further reduce backgrounds from mis-identified electrons, I E; I + I E, I %5 GeV is required. A total of 101 &jets and 90 p+jets events pass these criteria; the expected number of signal events is about 30 in each case.

The top quark mass is extracted from this sample using two methods. The first method, called the template method, compares the observed fitted mass distribution (using the lowest x2 solution) with background and signal fitted mass templates (for different possible top masses) to extract the top quark mass. The second method, which is an ideogram technique, extracts the mass

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information from a combined likelihood for the sample, taking into account all information in the kinematic fit and the probability for an event to be signal or background, as measured by the event likelihood discriminant similar to that described in section 1.3.1. The top quark mass is measured to be mt = 170 f 6.5(stat)I:; (sys) GeV with the Template method and

mt = 177.5 k 5.8(stat) f 7.l(sys)GeV with the Ideogram method. The dominant systematic uncertainty comes from jet energy scale corrections.

1.3.3 Single Top Production

The single top quark production cross section is directly proportional to lvtb12 (in the absence of any new physics and anomalous couplings of the top quark) and hence can provide a direct measurement of the CKM matrix element lvtbl and a verification of the unitarity of the CKM matrix.

Single top production through the electroweak processes of virtual W exchange @p + tb + x) (s-channel) and W-gluon fusion @p + tqb + x) (t-channel) has not yet been observed. A search has been performed using a sample of 164 pb-'. Events with one high pT electron (muon) with pr >15 GeV and lql<l.l (lql<2.0), ET>15 GeV and between 2 and 4 jets with ET>15 GeV, and (q1<3.4 and with the leading jet having ET >25 GeV and )q)<2.5. In addition, at least one jet is required to be b-tagged. The backgrounds arise from W/Z+jets production, t i and QCD mulitjet events. A 95% C.L. upper limit on the cross section of ot<23 pb has been extracted for the sum of the cross sections in the s- and t- channels, assuming their ratios to be as predicted by the SM. It is expected that with about 1 fb" of data, observation of the single top production at a 50 level will be possible.

1.4. Summary andprospects

Fermilab Run I1 is progressing very well; the accelerator upgrades for Run I1 are providing continually improving luminosity performance, with 670 pb-' delivered to each experiment so far. With a data set about four times larger than that collected in Run I already in hand, the era of precision measurements has begun at the Tevatron collider experiments. The D 0 detector is performing well and many important physics results are emerging from the Run I1 data. The preliminary results presented here on W and Z boson and di-boson production cross sections in leptonic channels are consistent with Standard Model expectations. The top pair production cross section has been measured using difepton and fepton+jets channels which are in agreement with the SM prediction of 6.7:;; pb . Measurements of the top quark mass using lepton+jets

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events are consistent with measurements from Run I. A limit on the single top production cross section has been set. As yet, there are no signs of new physics.

In a couple of years, about 2 fb-’ of data will be available. With this data set, the W mass will be measured within an uncertainty of 30 MeV and the top quark mass will be known to within 2 GeV. These two measurements, coupled with the precision electroweak results, will provide a tighter constraint on the mass of the SM Higgs boson. The error on the top pair production cross section, which is now about 30% will be reduced to about 10%. Single top quark production will likely be observed, which itself would be a major achievement, and lVtbl will be measured to about 10% precision. These and other electroweak and top physics measurements will not only provide a rich harvest of Standard Model physics but also might bring into view some hints of new physics beyond the Standard Model.

1.5. Acknowledgements

I would like to thank my D 0 collaborators, in particular Dmitri Denisov, Ia Iashvili, Aurelio Juste, Lisa Shabalina and Marco Verzocchi for their invaluable input during the preparation of this talk. Fermilab is operated by the Universities Research Association Inc., under contract number DE-AC02-76CH03000 with the U.S. Department of Energy.

References

1 . CDF Collaboration, F. Abe et al., Phys. Rev. Lett., 74, 2626 (1995); DO) Collaboration, S. Abachi, et al., Phys. Rev. Lett., 74, 2632 (1995); For a comprehensive review of top quark physics at the Tevatron, see P.C. Bhat, H.B. Prosper and S.S. Snyder, Int. J. Mod. Phys. A13,5113 (1998).

2. P.C. Bhat and W.J. Spalding, FERMILAB-CONF-04-280-AD, hep- ed04100046, To be published in the Proceedings of the Topical Conference on Hadron Collider Physics, HCP2004, Michigan State University, East Lansing, MI, June 14-18,2004 (AIP, NY, 2004).

3. The DO Collaboration, V. Abazov et al., in preparation for submission to Nucl. Instrum. Methods Phys. Res. A; T. LeCompte and H.T. Diehl, Ann. Rev. Nucl. Part. Sci. 50,71 (2000).

4. T. Sjostrand et al., Comput. Phys. Commun., 135,238 (2001). 5 . The DO Collaboration, Phys. Rev. D., 58, 052001 (1998); Phys. Rev. D., 67,

012004 (2003). 6. The D 0 Collaboration, Nature 429, 638 (2004) and references therein;

“Combination of CDF and D 0 results on the top quark mass,” The CDF and D 0 Collaborations and Tevatron Electroweak Group, hep-ed04040 10.

STATUS OF SEARCHES FOR HIGGS AND PHYSICS BEYOND THE STANDARD MODEL AT CDF

D. TSYBYCHEV * Department of Physics

University of Florida, Gainesville, FL, 3261 1-8440 E-mail: [email protected]

FOR CDF COLLABORATION

This article presents selected experimental results on searches for Higgs and physics beyond the standard model (BSM) at the Collider Detector at Fermilab (CDF). The results are based on about 350 pb-l of proton-antiproton collision data at & = 1.96 TeV, collected during Run 11 of the Tevatron. No evidence of signal was found and limits on the production cross section of various physics processes BSM are derived.

1. Introduction

A wide range of predicted phenomena in the Standard Model (SM) of particles physics has been remarkably confirmed by many experiments over the past decades. However, the central piece of the SM, Higgs boson, has not been observed so far. Furthermore, there are many fundamental questions that SM does not address. For example, the origin of the electro-weak symmetry breaking, hierarchy of scales, unification of gauge interactions and the nature of gravity are left unexplained in SM. In addition, recent cosmological observations indicate that SM particles account only for - 4% of the matter of the Universe.

Several extensions to the SM are proposed to resolve these issues. These extensions include supersymmetry (SUSY), grand unified theories (GUT), technicolor (TC), extra dimensions (ED), etc. The phenomenology of these extensions is very rich, with some signatures very different from SM processes. However, masses, production cross-sections and decay branching ratios of particles in such theories are unknown and highly model-dependent,

*Currently at Department of Physics, State University of New York, Stony Brook, NY, 11794-3800

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which eventually complicates the prediction of experimental signatures. Ex- amples of such signatures from physics BSM can be large missing transverse energy ($T ), same sign leptons, multiple photons, heavy flavor jets or some combination of several of those. Thus two different approaches can be adopted. The most general searches look for any deviations from the SM. In this approach all measurable signatures and kinematical phase space are scanned and the results can be interpreted based on some model. However those searches are not fully efficient for a particular model. An alternative approach is to search for specific signals suggested by particular model in which some theoretical assumptions are also made to reduce the number of free parameters.

At CDF we search for evidence of SM Higgs and physics BSM in p@ collisions at 4 = 1.96 TeV. In this paper we report on recent results for searches based on about 350 pb-' collected since the start of Run I1 of the Tevatron in March 2001.

2. Searches for Higgs Boson

The Higgs boson is crucial to our understanding of the electro-weak symmetry breaking and mass generation of the gauge bosons and fermions. In addition, the mass of the Higgs boson may indicate the scale of new physics. The search for the SM Higgs boson is one of the major goals of the Tevatron Run 11 program.

2.1. Searches for Standard Model Higgs Boson

At Tevatron the dominant Higgs boson production mechanism is through gluon-gluon fusion. For light Higgs (MH < 135 GeV/c2) the dominant decay channel is to bb, which results in a signature of the two heavy flavor jets. However, this final state is dominated by the large SM QCD multi-jet production. Therefore, for the light Higgs CDF concentrates on channels where it is produced in association with W or Z bosons. On the other hand, for Higgs masses greater than 135 GeV/c2, the Higgs boson is predicted to decay predominantly to W+W- pairs, in which case gg + H process is considered.

W * H + l*vbb channel using 162 pb-', by looking for a peak in the dijet invariant mass distribution in the two jet events. Candidate events were selected by requiring an energetic isolated lepton (ET > 20 GeV for electrons or p~ > 20 Gev/c for muons) in the central detector (171 < l ) , two jets with ET > 15 GeV and

A search for light Higgs was performed in p p

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A L3 Q

I 10 m * c 0

w 1

W

.-

& > 20 GeV. At least on of the two jets was required to be heavy-flavor tagged. The expected number of SM background events was 60.55 f 4.43, coming from Wbb, WcZ, Wc, tfprocesses and events with light flavored jets misidentified as bjets. After applying all selection cuts, 62 events were observed. The 95% C.L. upper limit on the cross section times branching ratio as a function of the Higgs mass is shown in the Figure 1.

I " " l " " l " " l " " I " " / " " l " ' ~ I ' " ' -

- ._.,,.. -...-. : .. -*- - - . I . ._ . . ._ . . . .

--*-.....-..._ H + WW'" 3 lvlv I

WH -+ lvbb ~ ~ = i a4 pb-' L,= 1 62 pb-' -

Figure 1. The 95 % C.L. upper limit on the SM production crass section times branching ratio as a function of the Higgs boson mass from W * H - > l*v& and H + WW analyses.

A search was also performed for Higgs boson decaying to W+W- pairs in dileptons and & final state with 200 pb-' of data. The candidate events were selected by requiring two oppositely charged, high-pT, isolated leptons (electrons or muons) and large missing FT (& > 25 GeV). The & was required not to be parallel to either of leptons, in order to ensure that it was well measured. Events consistent with the Z boson mass were removed. In addition, we vetoed events with reconstructed jets above ET > 15 GeV, to suppress diboson production background. Finally, to discriminate possible Higgs signal from WW background we exploit the fact the Higgs boson is fundamental spin-zero particle, which results in smaller invariant mass of

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the two leptons mil. Therefore, events with small ma were selected and azimuthal angular separation between the leptons was fitted using maximum likelihood method, to extract 95% C.L. limit on production cross section times branching ratio.

Figure 1 summarizes our current sensitivity for SM Higgs in both H + W+W- and WH channels. Currently, no limit on the Higgs mass can be set as these searches are statistics limited.

2.2. Search for MSSM Neutral Higgs Boson at large tanp

The Minimum Supersymmetric extension to the Standard Model predicts the existence of five Higgs particles, three neutral and two charged. The coupling of the neutral pseudo-scalar Higgs boson ( A ) to the third generation fermion can be enhanced by a factor of tan0 (the ratio of the vacuum expectation values of the two Higgs fields and it is a free parameter in the theory) relative to the SM. Thus, the production cross section for A will scale as tan2P. While the branching ratio of A 4 bb dominates (-go%), a search in this channel is very difficult due to overhelming multi-jet background. Therefore, if A is light, the A + rr decay mode ( 8%) becomes interesting. This mode is an important part of the MSSM Higgs search at CDF.

CDF has searched for the A + 77 at high tanp in about 200 pb-' of data, collected with a set of dedicated r-triggers. The signal consists of a tau pair, where one of the taus decays leptonicaly, while the other decays hadronicaly. After initial selection of the events with two reconstructed tau candidates, the multi-jet background was suppressed by requiring large H T , which is defined as a scalar sum of momenta of tau-decay products and &

In addition, the $T was required not to point in direction opposite to the tau decay products. After all of the selection cuts were applied, 236 events were observed, and the expected number of the SM background events was 263.6 f 30.1. The dominant source of the background was from 2 -+ rr. To further discriminate signal from background and set a limit on production cross section a mass-like variable mvis was constructed using the four momentum of the lepton, the four-momentum of visible decay products of hadronicaly decaying tau and $T . Figure 2 shows mvis for remaining events. Maximum binned-likelihood fit was performed to the mvis to extract Higgs signal. Observed limit on the production cross section times branching ratio at 95% C.L. for the neutral pseudo-scalar Higgs is shown in Figure 3 along with expected limit from pseudo-experiments. Observed

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limit is about order of magnitude higher than theoretically predicted cross section. The sensitivity of this analysis is expected to improve with higher accumulated statistics.

Higgs -+ T r Search, Mass of Lepton, T, &

. & 30

20

Higgs -+ z z Search, 95OA CL Upper Limit

CDF Run II Preliminary 195 pb 3 20

\ -0bSeNed

..., Expected

'. ........... .. ..... ... .. . ....

' 120 140 160 180 200 I , , , I . , , l l l l l l l l l l

m, (GeVIc')

Figure 2. Observed partially recon- Figure 3. Observed and expected up- structed ditau m a s and contribution per limit at 95% C.L. on the u(pp -+

from various SM background processes. A)xBR(A -+ T T ) .

2.3. Searches for Doubly Charged Higgs Boson

The existence of the doubly charged Higgs boson H** is predicted in several left-right (L-R) symmetry breaking models with Higgs triplets. In some L-R SUSY models Higgs boson can be produced in a mass range accessible at Tevatron (-100 GeV/c2-1 TeV/c2). The H** may be produced at the Tevatron in pairs via Z/y' exchange, or singly via WW fusion or W Drell-Yan. The experimental signature of the H** will be a pair of same sign leptons 6 , if the Yukawa couplings (ha)) of H** to charged leptons is in the range < hlp < 0.5 (which corresponds to narrow resonances that decay promptly). Otherwise, H** appears as a stable particle and decays outside the detector volume.

The search for a pair of long-lived H** was done with 206 pb-l of data and concentrated on selecting events with isolated muon and an additional high momentum track with p~ > 20 GeV/c. The fact that energy loss due to ionization is proportional to the square of the particle charge is further exploited, and both muon and track in this analysis were required to be highly ionizing in the drift chamber. Zero events were observed, which was consistent with expectation from SM background processes. The experimental limit on the H** production cross section at 95% C.L along with theoretical cross section are shown in Figure 4.

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The search for the prompt H** was performed in Hi* --+ e*e*, H** +. p*p* and H** 4 e*p* channels, using a data sample of 240 p b - l . This search was performed in the mass window of &lo% around a given H** mass (- 30 of the detector resolution). No events were observed after applying the selection cuts in the mass region m H i i > 80 GeV/c2 (mHii > 100 GeV/c2 for electron pair channel), which is consistent with the SM background expectation. Figure 5 shows the H** mass limits for different hlp couplings in the mass-coupling plane.

- NLOTheoiy ~ - Limit 195*h c L I

Y - - - L - L _-__ 100 110 120 130 140

Figure 4. Experimental limit on the H** production cross section at 95% C.L compared with the thoretical expctations.

:/i 8

I) 110 120 13

Figure 5. The H** lower mass limits versus lepton couplings (hllr) from prompt H* f analysis, assuming exclusive decay to a given dilepton pair.

3. Search for Scalar Bottom Quarks from Gluino Decays

Supersymmetry solves the fine tuning problem associated with the SM Higgs mass and provides a framework for unification of the fundamental interactions. SUSY is a larger space-time symmetry that relates bosons and fermions, so that every SM particle has a supersymmetric partner with spin difference 1/2. In addition a new conserved quantum number is introduced to avoid Lepton(L) and Baryon ( B ) number violation. R-parity is a multiplicative quantum number, defined as R = ( - l )3(B-L)+2S where S is the spin of the particle. For the SM particles R = 1, whereas R = -1 for the SUSY partners. If R-parity is conserved SUSY particles would be produced in pairs in collider experiments, and decay to stable final state particles: the Lightest Supersymmetric Particle (LSP) and SM quarks and

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leptons. The electrically neutral LSP is a candidate for cold dark matter. The SUSY partners of the left-handed and right-handed fermions, can mix to form mass eigenstates. For scenarios with large tan/?, the mixing in the third generation of supersymmetric particles can be substantial. In some SUSY models ', the lighter scalar-bottom quark ( b l ) mass eigenstate can be significantly lighter than other SUSY particles.

CDF performed a search for 61 from gluino (ij) decays in a R-parity conserving SUSY scenario with r n ~ > rngl using 156 pb-' of data. In this scenario, gluinos are pair produced and they decay into ij * 616 with a 100% branching ratio, followed by the sequential decay of the b 1 6jjy1 where j j y is the lightest neutralino and the LSP. Since neutralinos escape detection, this gives a signature of four bjets and & . Two analyses were performed using exclusive single b tagged events and inclusive double btagged events. Final selection cuts require & > 80 GeV and no reconstructed leptons. In both analyses, backgrounds are dominated by top quark pair production and W / Z boson production in association with jets. In the single-tag analysis 16.4f3 .6 events were expected and 21 were observed in data. Whereas, in double-tag analysis 2.6 f 0.7 events were expected and 4 observed. No evidence of new physics was observed and limits were set in gluino-sbottom mass plane as shown in Figure 6.

Gluino+ 6,b , 95% C.L. Exclusion Limit, 156pb-' 280 BRm+bS3=,00X CDF Run II Preliminary G- A .

180 200 220 240 260 280

2 240 220

E 200

$ 180 (sxc . sing 8 lag)

160

140

120

100 180 200 220 240 260 280

Gluino mass [ GeV/.2]

Figure 6. sbottom quarks from gluino decay.

The 95 C.L. exclusion region in the mi and mhl plane from the search for

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4. Searches for Extra Dimensions

Current string theory proposes that as many as seven new dimensions may exist and the geometry of these extra dimensions are responsible for why gravity is so weak. In recent years a number of models with extra dimensions were proposed g to solve the hierarchy problem. In particular Randall-Sundrum (Rs) model lo proposes a non-factorisable geometry in 5 - 0 space, with a constant negative curvature. The extra dimension is warped by an exponential factor, which determines the masses and couplings of Kaluza-Klein (KK) states. The spectrum of the KK states is discrete and they appear as narrow resonances. The properties of the RS model are determined by the ratio of k /Mpl , where k is a curvature scale of the theory and Mpl is the Plank mass.

The CDF has searched for a RS graviton signal of the first excited KK state in diphoton channel, using 345 pb-' data sample. The analysis required two photons with p~ > 15 GeV/c in the central region of the detector and with invariant mass greater than 30 GeV/c2. The dominant SM backgrounds to this search are the diphoton production and milti-jet production, where jets fragment into no and pass photon selection cuts. The diphoton mass spectrum observed in data agrees well with SM prediction and is shown in Figure 7. The limits on a RS graviton particle of the first excited state was placed using a 3a mass window around a given graviton mass point in a range between 200 GeV/c2 and 900 GeV/c2. Figure 8 shows the excluded region in the RS graviton mass and warp factor plane. Also shown the CDF results for the RS graviton searches from the dimuon and dielectron channels.

5. Searches for Leptoquarks

The remarkable symmetry between quarks an leptons in the SM suggests that some more fundamental theory may exist, which allows interactions between them. Such interactions are mediated by a new type of particle, a leptoquark (LQ). Leptoquarks are hypothetical color-triplet bosons carrying both lepton and baryon quantum numbers that are predicted in many extensions of the standard model (e.g. grand unification models, technicolor, and supersymmetry with R-parity violation) ll. The Yukawa coupling of the leptoquark to a lepton and quark and the branching ratio to a charged lepton, denoted by p, are model dependent. Usually it is assumed that leptoquarks couple to only one generation to accommodate experimental constraints on flavor-changing neutral currents, which allows

Dlphoton RS Gravlton Search

L CDF Run I1 Prellmlnarv. 345 6'1

Figure 7. The observed diphoton mass spectrum compared with the SM prediction.

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RS Gravhon Searches, 95% C.L. Exclusion Regions 0.1

0.09

0.08

0.07

.O.W

E 0.05

0.04

0.03

0.02

o'o~OO 300 400 500 600 M

Figure 8. The limits on the Randall- Sundrum graviton model displayed as a function of graviton mass and warp factor for searches in the diphoton and dileptons channels.

one to classify leptoquarks as first-, second-, or third-generation. In pji collisions, leptoquarks can be produced in pairs via the strong interaction through gg fusion or qq annihilation.

Several analyses were carried out at CDF to search for first (second) generation leptoquarks: in L Q m + ee(pp)qij (p = 1) and in L Q G + e(p)vqq (p = 0.5) channels in 200 pb-' data sample. In addition the search was performed in L Q m +. vVqQ (0 = 0) channel, which is sensitive to the leptoquarks of all three generations. The main sources of the SM backgrounds in these searches are represented by production of a vector boson (2 or W ) accompanied by jets and top quark production. No evidence for the existence of leptoquarks was observed. Upper limit on leptoquark pair-production cross sections were placed at 95% C.L. and lower limits on their masses were derived. The results are summarized in Table 1.

6. Summary

We reviewed some of the most recent results from the CDF on the searches for SM Higgs and other new phenomena at the Tevatron. No evidence for new physics has been observed so far and limits on different models were derived. The new results already have surpassed many Run 1 results. The sensitivity will improve as CDF detector continues to take data and many

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Table 1. Lower limits on the first- and second-generation leptoquark mass at 95% C.L. from CDF Run 11.

Generation p 95% C.L. lower mass limit (GeV/c2)

1 0.5 176

0 117

1 242

2 0.5 175

new results should be expected soon in the data set of about 0.5 fb-l. The Tevatron remains the energy frontier of collider searches for new physics and provides the best opportunity for finding evidence for physics BSM before LHC starts.

References 1. CDF Collaboration, FERMILAB-PUB-96/390-E. 2. A. Stange, W. Marciano and S. Willenbrock, Phys. Rev. D49, 1354 (1994). 3. R. N. Mohapatra and J. C. Pati, Phys. Rev. D11, 566 (1975). 4. C. S. Aulakh, A. Melfo and G. Senjanovic, Phys. Rev. D57, 4174 (1998). 5. R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980). 6. J. F. Gunion et al. Phys. Rev. D40, 1546 (1989). 7. D. Acosta et al. [CDF Collaboration], arXiv:hepex/0406073; Submitted to

Phys. Rev. Lett. 8. A. Bartl, W. Majerotto and W. Porod, 2. Phys. C64, 499 (1994) [Erratum C68, 518 (1995)l.

9. For a review, see J. Hewett and M. Spiropulu, Annu. Rev. Nucl. Part. Sci. 52, 397 (2002).

10. L. Randall, R. Sundrum, Phys. Rev. Lett. 83 3370 (1999). 11. W. Buchmiiller, R. Ruck1 and D. Wyler, Phys. Lett. Bl91, 442 (1987) [Er-

ratum B448, 320 (1999)].

SEARCHES FOR NEW PHENOMENA WITH THE DO DETECTOR

N. J. HADLEY The University of Maryland

Physics Department College Park, MD 20742, USA

FOR THE DO COLLABORATION

We report preliminary results on searches for extra dimensions, new gauge bosons, and SUSY particles with the D 0 detector at the Fermilab Tevatron collider. No evidence for such effects was found and we place limits on size of extra dimensions and the masses of new particles.

1. Introduction

The DO collaboration has been taking data with an upgraded detector at the Fermilab Tevatron collider at a center of mass energy of 1.96 TeV since 2001. The Tevatron is the current world’s highest energy accelerator and a natural place to search for new physics. The D 0 collaboration has done many such searches whose results can be found in publications or linked from the DO public web pages [l]. In this paper, we report preliminary results on searches for extra dimensions, new gauge bosons, and SUSY particles with the D 0 detector at the Fermilab Tevatron collider. No evidence for such effects was found and we place limits on size of extra dimensions and the masses of new particles. These topics were selected to match the themes of this conference. The limits quoted are current as of August 2004.

We will first briefly describe the upgraded D 0 detector and the current performance of the Tevatron. We will then present our results on Searches for Extra Dimensions and new gauge bosons. We will conclude with the results of a search for SUSY particles in the channels involving decay to three leptons.

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2. DO Detector and the Fermilab Tevatron

Both the DO detector and the Fermilab Tevatron collider have been sub- stantially upgraded since 1996 when the previous running period (Run I) ended. In anticipation of significantly higher luminosities and a change in the bunch cross time from 4 ps to 396 ns, the DO detector added a 2 T solenoid, a silicon vertex detector, and a scintillating fiber tracker. In addition, the trigger and data acquisition systems were modified, and forward muon system upgraded. Details of the upgrades can be found in Ref. [2].

The Fermilab Tevatron collider was upgraded with a new main injector and an antiproton recycler. The proton antiproton collision energy was raised to 1.96 TeV from 1.8 TeV. The luminosity goal for the current run is 2 x cmP2 s-l. The current performance of the Tevatron and the DO detector are much improved over the somewhat disappointing start to the run in 2001. The luminosities goals for this year (FY2004) have been exceeded, and DO has more than doubled its integrated the amount of data on tape in the last 12 months. The current best instantaneous luminosity is 1.03 x cm-2 s-l. DO now has about 470 fb-' on tape taken with a complete detector in the period April 2002 through August 2004. Data taking efficiencies typically average between 85% and 90% .

3. Search for Extra Dimensions

One of the most intriguing ideas to explain the weakness of gravity and the problem of the enormous difference in scales between the electroweak scale and the Planck scale are models that include extra dimensions. In these models, gravity is as strong as the other forces, but is the only force allowed to propagate in the extra dimensions. To an observer restricted to our three dimensional world, gravity then appears weak. These extra dimensions affect the force of gravity at small distances, leading to differences from Standard Model predictions that can be tested by experiment. These new theories predict deviations both in the inverse distance squared force law of Newtonian gravity and in the production of pairs of fermions and, in some models, bosons at high energy colliders. These models can then be searched for in the disparate realms of table top gravity experiments and high energy colliders.

At the Fermilab Tevatron, the effects of extra dimensions would appear in the high mass spectra of pairs of electrons, muons, and photons. (The effects would also appear in tau pair and quark pair production , but those channels have lower sensitivity due to higher backgrounds.) Here high mass

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implies masses well above the Z mass. We have studied the dielectron, dimuon, and diphoton mass distributions in our current processed data set of 200 pb-I and see no deviation from the predictions of the Standard Model. A representative mass distribution, in this case, for di-electrons and di-photons is given in Figure 1.

diEM Mass, GeV , L .... ~ ..................................................... ~ ~ .._. ~ ~ ........................... :

Figure 1. The mass distribution for the combined dielecton and diphoton samples. Points with the error bars are the data; light filled histogram represents the instrumental background from jets misidentified as EM objects; solid line shows the fit to the sum of the instrumental background and the SM predictions from Drell-Yan and direct diphoton backgrounds. The dashed line shows the effect of an hypothesized ED signal.

Since we see no excess of events in these distributions, we proceed to set limits on extra dimensions in various models.

3.1. Large Extra Dimensions

In the model of Arkani-Hamed, Dimopoulos and Dvali, there is a number, n, of extra dimensions in which gravity is allowed to propagate [3]. The case n=l is ruled out by observations of the solar system, since the size of the extra dimension in that case is expected to be of order lo8 km. However, for n> 2, the expected size of the extra dimensions is less than 1 mm, and does not contradict existing gravitational experiments. The effects of the extra dimensions can be parameterized by a single variable, qG = F / M i , where F is a dimensionless parameter and M s is the fundamental (3+n) di-

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mension Planck scale. Different models use different definitions of F. In theGRW [4] model, F = 1. In the model of Hewett [5], F = ±2A/7r = ±2/?rwhere we have taken the parameter A expected to be of order one to beexactly one. In the HLZ model [6], the value of F depends on the numberof extra dimensions. In our analysis, we follow the method of reference [7]which was used in an earlier the D0 publication [8] on this subject basedon our 1993-1996 data. We analyze the data in terms of the dielectron ordiphoton invariant mass and the cosine of the scattering angle in the centerof mass frame, as this choices of variables gives optimum sensitivity to theeffects of extra dimensions. We use a parameterized Monte Carlo generatorand include the effects of Standard Model Drell-Yan contributions as wellas backgrounds from mis-identified QCD and direct photon events. We nor-malize our Monte Carlo to the data at low masses (below 150 GeV) wherethe effects of extra dimensions should be negligible. This enables us to nor-malize both the Drell-Yan background dominated by Z boson productionand QCD background since they have very different shapes as a function ofinvariant mass. For the signal, we include the the effects of Kaluza-Kleingraviton exchange diagrams and (if relevant) their interference with Stan-dard Model processes in dilepton and diphoton production. We assume aconstant AT-factor of 1.3.

In approximately 200 pb"1 of data, we see no evidence for deviationsfrom the Standard model, and present limits on MS in Table 1.

Table 1. Lower limits at 95% CL on the fundamental Planckscale, MS, in TeV.

GRW [4]

1.36

HLZ [6]n=2 n=3 n=4 n=5 n=6 n=71.56 1.61 1.36 1.23 1.14 1.08

Hewett [5]A =1 / A = -1

1.22 / 1.10

If we combine our results with those in our previous publication [8], weobtain the results shown in table 2. We note that these results are the moststringent limits on large extra dimensions obtained to date for n > 3. Forn = 2, the table top gravity experiments obtain MS > 1.7 TeV which isslightly better than our result of 1.67 TeV.

Table 2. Combined lower limits at 95% CL on the fundamen-tal Planck scale, MS, in TeV.

GRW [4]

1.43

HLZ [6]n=2 n=3 n=4 n=5 n=6 n=71.67 1.70 1.43 1.29 1.20 1.14

Hewett [5]A =1 / A = -1

1.28 / 1.16

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GRW [4]

1.09

We have performed a similar analysis in the dimuon channel where once again we see no deviation from the Standard Model, this time in the dimuon mass distribution. Our corresponding limits on M s for large extra dimensions are given in table 3. The muon limits are based on 250 pb-’ of data and are slighly worse than the dielectron and diphoton limits due to the better resolution in those channels.

HLZ [6] Hewett [5] X = 1 / X = -1

0.97 / 0.95 n=2 n=3 n=4 n=5 n=6 n=7 1.00 1.29 1.09 0.98 0.91 0.86

3.2. TeV-’ Extra Dimensions

We have also performed the first dedicated search at a hadron collider for TeV-l Extra Dimensions. In TeV-l Extra Dimensions [9,10], the Standard Model gauge bosons that propagate in the extra dimensions are equivalent to towers of Kaluza-Klein states with masses, M, = J M i + n2/R2 where n = 1 ,2 , . . . and R = Mgl is the size of the extra dimension. In these models, from our dielectron data, we set limits of R < 1.75 x m or M c > 1.13 TeV at the 95% CL. Our data set size is approximately 200 pb-l. Note that there are better indirect limits from an analysis [ll] of LEP experimental data of order M c > 6.6 TeV at 95% CL.

3.3. Search for Randall-Sundrum Extra Dimensions and Z’ Bosons

In the Randall-Sundrum model [12] of extra dimensions, there is a graviton whose coupling to the Standard Model fields is determined by a dimensionless model parameter, k/MpL. k /MpL is expected to be of the order of 0.01 - 0.1. These gravitons could be resonantly produced at the Teva- tron and we search for them in our dielectron and diphoton data in the same manner one would search for a Z’. Based on approximately 200 pb-l of data, our limits for Randall-Sundrum graviton production are given in figure 2.

For comparison, our Z’ limits at 95% CL, based on essentially the same data set, are 780 GeV and 680 GeV in the dielectron and dimuon channels respectively. These limits apply to Z’ bosons with Standard Model couplings.

Table 3. Dimuon Channel lower limits at95% CL on the fun-damental Planck scale, Ms, in TeV.

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0 Mass (GeV)

Figure 2. Upper bounds at the 95% CL on the dimensionless coupling k / M p L in the Randall-Sundrum model. Red dots show the mass points for which the cross section limits have been obtained in the analysis. The lower boundary of the exclusion region is a result of polynomial interpolation between these points. Note that for masses above 500 GeV, signal windows overlap, thus ensuring no gaps in the mass-spectrum coverage.

4. Search for SUSY Particles in the Trilepton Final State

As an example of the DO collaboration searches for SUSY particles, we present here the result of a search for associated production of the lightest chargino and the next to lightest neutralino in trilepton final state. We combine the results of four different analyses, taking proper account for overlaps. The four channels are: two electrons + a third lepton (249 pb-l), electron + muon + a third lepton (221 pb-l), two muons + a third lepton (235 pb-I), and two like-sign muons (147 pb-l). Here the “third lepton” is any isolated track to increase the efficiency while electron or muon implies particle identification cuts. After all cuts, there is no significant excess in any individual channel or in all channels combined where we see three events with an expected background of 2.93 events. Once again we set limits, this time on the production of the lightest chargino and the next to lightest neutralino. We interpret our results in terms of an MSugra model. Our results are shown in figure 3. Here Run I means our 1993-1996 data

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run. Note that although we do not set limits in the “interesting” region for minimal MSugra at this time, we should reach this region when we have analyzed about 25% more data. We have about a factor of two more data on tape.

n pa 2.2 2

1 1.8 % 1.6

1.4

Y =

tanp = 3 e l A,=O

+‘I 1.2

0.8 P>O 0.6 0.4 0.2

95 100 105 110 115 120 125 130

Figure 3. Limits on the total cross section for associated chargino and neutralino p r e duction with leptonic final states in Run I (top line) and in this analysis (second from top) in comparison with the expected limit (dashed line). Three model lines are plotted as a reference. The top line corresponds to the signal cross section times leptonic branching fraction predicted for models with heavy squark masses and low slepton masses. The middle line corresponds to the signal expectation for low slepton masses in mSUGRA and the bottom line describes the signal expectation for large mO with the chargino and the neutralino decaying via virtual gauge bosons. Chargino masses below 103 GeV are excluded by direct searches at LEP.

5. Conclusion

We present preliminary results on searches for extra dimensions, new gauge bosons, and associated production of the lightest chargino and the next to lightest neutralinowith the DO detector at the Fermilab Tevatron collider. No evidence for such effects was found and we place limits on size of extra dimensions and the masses of new particles.

These results were based typically on data sets of approximately 200 pb-l. We expect to present results based on more than double this

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amount in the winter conferences of 2005. Interested readers should check the DO public web pages [I] for updates.

Acknowledgments

The author would like to thank the conference organizers for an interesting meeting and to congratulate Professor Pran Nath on the celebration of his work and birthday.

We thank the staffs at Fermilab and collaborating institutions, and acknowledge support from the Department of Energy and National Science Foundation (USA), Commissariat B 1’Energie Atomique and CNRS/Institut National de Physique Nucl6aire et de Physique des Particules (France), Ministry of Education and Science, Agency for Atomic Energy and RF Pres- ident Grants Program (Russia), CAPES, CNPq, FAPERJ, FAPESP and FUNDUNESP (Brazil), Departments of Atomic Energy and Science and Technology (India), Colciencias (Colombia), CONACyT (Mexico), KRF (Korea), CONICET and UBACyT (Argentina), The Foundation for Funda- mental Research on Matter (The Netherlands), PPARC (United Kingdom), Ministry of Education (Czech Republic), Natural Sciences and Engineering Research Council and WestGrid Project (Canada), BMBF and DFG (Ger- many), A.P. Sloan Foundation, Civilian Research and Development Foun- dation, Research Corporation, Texas Advanced Research Program, and the Alexander von Humboldt Foundation.

References 1. Current public results from the DO collaboration can be found linked

from the DO web page, www-do.fnal.gov, or directly from www- d0.fnal.gov/Run2Physics/WWW/results.htm

2. T. LeCompte and H.T.Dieh1, Ann. Rev. Nucl. Part. Sci. 50 , 71 (2000). 3. N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B429 , 263 (1998);

I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B436, 257 (1998); N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Rev. D 59, 086004 (1999); N. Arkani-Hamed, S. Dimopoulos, J. March-Russell, SLAC-PUB-7949, e-Print Archive: hep-th/9809124.

4. G. Giudice, R. Rattazzi, and J. Wells, Nucl. Phys. B544, 3 (1999), and revised version hepph/9811291.

5. J. Hewett, Phys. Rev. Lett. 82, 4765 (1999). 6. T. Han, J. Lykken, and R. Zhang, Phys. Rev. D 59, 105006 (1999), and revised

version hep-ph/9811350. 7. K. Cheung and G. Landsberg, Phys. Rev. D 62, 076003 (2000). 8. B. Abbott et al. (DO Collaboration) Phys. Rev. Lett. 86, 1156 (2001). See

also V. Abazov et al. ( D 0 Collaboration) Phys. Rev. Lett. 90, 251802 (2003).

277

9. K. Dienes, E. Dudas, and T. Gherghetta, Nucl. Phys. B537, 47 (1999). 10. A. Pomarol and M. Quirbs, Phys. Lett. B438, 255 (1998); M. Masip and A.

Pomarol, Phys. Rev. D 60, 096005 (1999); I. Antoniadis, K. Benakli, and M. Quirbs, Phys. Lett. B460, 176 (1999).

11. K. Cheung and G. Landsberg, Phys. Rev. D 65, 076003 (2002). 12. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999).

SEARCHES FOR NEW PHYSICS AT H E M

C . GENTA ON BEHALF OF THE H1 AND ZEUS COLLABORATIONS

University of Florence and INFN, via G. Sansone 1, I50019 Sesto Fiorentino, Florence, Italy

E-mail: gentaafi. infn. it

The H1 and ZEUS experiments collected about 110 pb-l of integrated luminosity in e+p collision and about 16 pb-’ in e-p collision during HERA I data taking (1994-2000). Data were analysed searching for contact interactions, extra- dimensions, R-parity violating SUSY, lepton-flavor violating and flavor-changing neutral current interactions. Possible deviations from Standard Model were furthermore investigated studying events with isolated leptons and high missing transverse momentum, or with two or more isolated high transverse momentum objects (photons, leptons, jets or neutrinos). A summary of the results is given. The first results of HERA I1 (2003-2004 running period) are also shown.

1. Introduction

The H1 and ZEUS collaborations collected during the HERA I running about 110 pb-l of integrated luminosity in e+p collisions and 16 pb-’ in e-p . These data were analysed to investigate different scenarios beyond the Standard Model (BSM). In 2003, after a shutdown, a new data taking period with higher luminosity and polarized electrona beam started. In these proceedings the searches for new physics performed by the H1 and ZEUS collaborations will be reviewed and the first results using HERA I1 data will be presented.

2. Contact Interactions

The contact interaction (CI) formalism is a convenient way to study interactions which have a characteristic energy scale much higher than the scales associated with the SM fields. H1 and ZEUS investigated ‘72 a number of possible extensions of the Standard Model that can be treated as

aIn the following both electron and positron will be referred as “electron”.

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279

contact interactions. No deviation from SM expectation was found by both collaborations.

Compositeness: The most general chiral invariant Lagrangian for Neutral Current (NC) vector-like four-fermion contact interactions can be expressed as:

g i , j = L , R

where ei and qj are the currents associated to the electron and quark fields and $" represent the four possible coupling coefficients. L and R are the fermion left and right helicities. General models that assume a lepton or quark substructure can be described using CI coefficients in the form rlij = eij g , where A is the compositeness scale and ei j can take the values 0, fl. Depending on the compositeness models, ZEUS and H1 lower limits on A range from about 1.7 to 6.2 TeV and from 1.6 to 5.5 TeV respectively.

High Mass Leptoquarks and R-parity violating Squarks: Lepto- quarks (LQs), which are predicted by Grand Unification Theories (GUT), are bosons with a fractional charge carrying both lepton (1,) and baryon (B) number. According to the Buchmuller-Ruckl-Wyler (BRW) classification, there are 14 types of leptoquarks, 7 scalar S$3 and 7 vector where x is the chirality of the incoming electron and T3 the third component of the isospin. In this c a e the contact interaction coefficients are proportional to the square of the ratio between the LQ Yukawa coupling A, and the LQ

Fig. 1 (left) shows ZEUS data compared to the 95% C.L. exclusion limits for different kinds of scalar and vector LQs. H1 limits on MLQ/X are in the range 0.3 - 1.4 TeV. ZEUS finds similar results (0.27 - 1.23 TeV). Leptoquark limits on X for S,$ and $,2 correspond to the limits on and Xi,, for up-type and down-type squarks respectively, assuming the branching ratios / 3 ~ + ~ ~ = 1.

Large Extra Dimensions: In the large extra dimensions models the electroweak bosons are confined in the 4-dimensional space while the gravity can propagate through n extra dimensions confined to a volume of radius R. The Planck scale Mp is therefore related to the fundamental scale M s by: Mp - RnMi+n. If R - 1 mm and n = 2, Ms can be of the order of TeV. The graviton exchange contribution in electron-quark scattering can be described as a contact interaction with an effective coupling strength

mass MLQ: r$j = a"%, 9 x2 where aij depends on the leptoquark species.

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Figure 1. In the left pictures ZEUS data are compared with 95% C.L. exclusion limits for the ratio between the leptoquark mass and the Yukawa coupling MLQ/X as a function of the transferred four-momentum Q 2 , for different LQ types. Results are normalized to the SM expectations. The insets show the comparison in the Q2 < lo4 GeV2 region with a linear scale. On the right, H1 data normalized to the SM expectations are compared with curves corresponding to 95% C.L. exclusion limits on the gravitational scale M s , obtained from e+p and e-p scattering data and from the combined data.

770 = 4, where X = f l . In Fig. 1 (right) the NC cross-section measured by H1 is compared to the curves corresponding to the 95% C.L. limits on the M s scale for positive and negative A. ZEUS sets similar limits: M s > 0.78 (0.79) for X = 1 (-1).

Ms

Quark Radius: The fermion substructure can be studied also introducing a finite distribution of the fermion charge. For point-like electrons ZEUS puts a limit of 0.85. cm on the quark radius.

cm and H1 puts a limit of 1.0.

3. R-parity violating Supersymmetries

In supersymmetric (SUSY) models each SM particle has Rp = 1 and its supersymmetric partner has Rp = -1, where the R-parity is defined as Rp = (-l)3B+L+2S, S being the spin, B the baryon number and L the lepton number of the particle. Since HERA is not competitive with LEP and TeVatron in the search for Rp conserving interactions, in the following only Rp violating SUSY models will be considered. The Rp violating squark production at HERA can be described by the superpotential X i j k L i Q j D k

where X i j k is the Yukawa coupling, i, j and k are the generation indices

28 1

-1 2

;<

10 -

10 - Mw> 30 GeV impased

m,, (GeV)

Figure 2. On the left: exclusion limits (95% C.L.) on Xijl for j = 1,2 as a function of the squark mass from a scan of the MSSM parameters as described in the figures. Indirect limits from neutrinoless double beta decay experiments (PBOY) and atomic parity violation (APV) are also shown. On the right: excluded regions (95% C.L.) in mSUGRA with Xij, = 0.3 for tan = 6. The dashed lines indicate the curves of constant squark (CL, i1) mass. The limits from TeVatron are given by the dotted lines.

and Li, Qj and Dk are the superfields of the left-handed lepton, left-handed quark and right handed down quark respectively and of their supersymmetric partners c, & and &. Down-type squarks can be dominantly produced as a resonance in e - p scattering while up-type squarks in e+p collisions. In addition to their Rp violating decays, squarks decay also via Rp conserving gauge couplings into a quark and a gaugino. Neutralinos, charginos and gluinos are unstable and decay via R p violating coupling into SM particles. H1 analysed 98-00 data taking into account all the different decay channels, leading to 9 different event topologies in the final state. No evidence of such kind of interactions was found, therefore limits on the squark production have been set. In order to investigate the dependence of the limits on the MSSM parameters, a scan on a range of values for M2 and p and for tanp = 2,6 was performed. In Fig. 2 (left) the limits on A&, versus the squark mass are shown for j = 1 , 2 and tanp = 6. For an electromagnetic strength A' - 0.3 up- (down-)type squarks with mass below 275 GeV (280 GeV) can be excluded at 95% C.L. H1 calculated also the limits in the minimal supergravity (mSUGRA) model framework. Fig. 2 (right), for example, shows the obtained limits in the m112, mo plane with Xi j l = 0.3 and tanp = 6. In this case H1 limits extend well beyond the DO limits

282

- ZEUS

Id -21

10!80 200 220 240 260 280

Figure 3. as a function of the s-top mass for the different vaiues of the SUSY parameter space. Low energy limit from APV is also reported. On the right: exclusion limits at 95% C.L. in the (Mg , M E ) plane for A131 = 0.3 from a scan of the MSSM parameter space as indicated in the figure.

On the left: limits on

and masses of G and 2. below 275 GeV can be nearly fully excluded, while f masses up to 270 GeV can be ruled out. The ZEUS collaboration searched for Rp violating s-particles focusing on the t" production and his subsequent decays into bX+ followed by the Rp violating decay x+ + e+ and two jets. No evidence of signal was found in 99-00 e+p data so limits at 95% C.L. were set on as shown in Fig. 3 (left). The H1 collaboration performed also an analysis on bosonic stop decays

f -+ 6W using all HERA I e+p data 5 . Since no signal was found, limits on stop production were set in the framework of the MSSM. Fig. 3 (right) shows the H1 excluded region in the ( M i , Mi) plane for Finally H1 studied the neutralino production via the Rp violating exchange of a selectron using 99-00 e+p data (65.4 pb-l) '. In this case a combination of Rp violating SUSY and Gauge Mediated Supersymmetry Breaking (GMSB) scenario was considered. The gravitino is the lightest supersymmetric particle (LSP) while the neutralino xy is assumed to be the next-to- lightest SUSY particle (NLSP) and to decay into a gravitino and a photon. Therefore H1 searched for events with a photon, a jet and missing transverse momentum. One event was found for an expected background of 2.55 f 1.30 events. In Fig. 4 (left) H1 limits in the (rn~,rn,~) plane are shown for different values of

= 0.3.

283

H1 e'p preliminary

F g 110 ,. E

120

I W

00

m

m,: (GeW M,(GeV)

Figure 4. On the left: excluded regions at 95% C.L. in the plane spanned by selectron and neutralino mass for different values of X i J The upper left region is not theoretically accessible within the GMSB scenario while the lower right region is not analysed because the neutralino is required to be the NLSP. On the right: upper limits on Xegl, assuming A,, = X e q l r are shown for Vt LQ. Dashed, dotted and dashed-dotted lines are the existing limits from low energy experiments.

4. Lepton-flavor Violation

The recent discovery of neutrino oscillations showed that lepton flavor is not conserved in the neutrino sector. The lepton-flavor violation (LFV) induced by the neutrino oscillation in the charged lepton sector is not observable at the present colliders due to the small rates. The search for such interactions is nevertheless an interesting field of study since it permits to test many BSM theories that predict LFV at a fundamental level. Lepton-flavor violating interactions can be investigated at HERA7y8 looking at events where the electron in the final state is substituted by a p or a r. To give a quantitative description of the results H1 and ZEUS used the BRW model with LQs coupling to different generations of lepton?. In the p channel the background from SM processes is very low and the signal can be detected with high efficiency, but there are already strong constraints from low energy experiments such as rare muon or meson decays and electron conversion in nuclei. The T channel is more challenging because in - 65% of the cases the r decays into hadrons producing a jet that has to be discriminated from QCD jets. In this channel HERA gives a significant contribution on setting limits on lepton-flavor violating LQ production. Since both collaborations did not observe any deviation from the SM, 95% C.L. limits were set on the Yukawa coupling Xegl . In Fig. 4 ZEUS limits on Yukawa coupling, Xeql

284

(assuming A,, = Aegl) versus the LQ mass are compared to limits from low energy experiments (from mesons and r rare decays) for one vector LQ. ZEUS limits are better or competitive up to 250 GeV especially when a higher generation quark in the final state is involved. For LQ with mass much greater than the center of mass energy of HERA (300 and 318 GeV) the LFV interaction can be described using the CI formalism. In this case ZEUS sets limits on the factor e- in the range 0.8 - 20 Aeqi A,, ,

5. Isolated Leptons

H1 found in e+p data an excess of events characterized by isolated leptons, high missing PT and with high transverse momentum of the hadronic system, P;. In Table 1 H1 and ZEUS results on isolated lepton search is summarized 9 ~ 1 0 3 1 1 . In particular H1 found 6 events with isolated muons or

Table 1. Summary of isolated lepton search performed by H1 and ZEUS.

Electrons Muons Taus PT" Data SM Data SM Data SM

P x >25GeV 4 1.49 6 1.44 0 0.53 H1 P> >40GeV 3 0.54 3 0.55 0 0.22

P x >25GeV 2 2.90 5 2.75 2 0.20 ZEUS P$ > 40GeV 0 0.94 0 0.95 1 0.07

electrons with P; > 40 GeV, whereas only about one event was expected from the SM predictions, mainly from the process ep + eWX. ZEUS did not confirm such excess but found two isolated r candidates. The number of events expected from the SM with an isolated r decaying into hadrons and P; > 25 GeV is 0.20. The H1 excess could be interpreted by the flavor-changing neutral-current (FCNC) top production shown in Fig. 5 (left). Both experiments performed an analysis to search for single top production using optimized cuts and adding the hadronic channel ZEUS did not find evidence of the signal while H1 finds an excess of events in the lepton channel compatible with top production via anomalous FCNC coupling: 5 events to be compared to a SM expectation of 1.31 f 0.22. ZEUS calculated the limits taking into account both tuy and tuZ contributions at leading order, while H1 used NLO cross-section neglecting the Z contribution. In Fig. 5 H1 and ZEUS limits are compared to the ones derived by LEP and CDF. As it can be seen, HERA is more sensitive in the case of single top production via anomalous ktUy couplings. The ZEUS excess in the r channel is unlikely to be explained with the hypothesis of top quark production: the observed events would correspond to a cross section

285

Ktuy

Figure 5. On the left: anomalous single top production via a FCNC interaction at HERA. On the right: ZEUS exclusion limits at 95% C.L. in the ktU7 - v t U Z plane for different values of Mtop assuming ktc7 = v t c Z = 0. The CDF, L3 and H1 limits are also shown.

already excluded by the analysis of the other lepton channels. Moreover the T candidate with higher P$ has a wrong sign of the charge. A possible interpretation of such kind of events is the stop production via Rp violating Yukawa coupling and its subsequent decays t' + TCTb and i! + ?u,b with T + T ~ O and CT + uTgo. However also in this case the T should have the same charge of the incoming lepton beam, which is true only for one of the two events surviving the selection cuts.

6. Multi-leptons

The cross section for di-lepton production is accurately predicted from the SM, therefore a deviation from this prediction would be a clear sign of new physics. The H1 and ZEUS searched for events with two or more leptons with high invariant mass l4,l5,l6. The overall agreement with the SM is good in both the analyses, however H1 found 6 events with two or three electrons with an invariant mass M above 100 GeV while only 0.53 were expected. In Table 2 the number of di- and tri-electron events with reconstructed M > 100 GeV found by ZEUS and H1 is reported. ZEUS finds 2 di-muon events with M > 100 GeV compatible with the SM expectations (2.16), while H1 finds one event with a SM prediction of 0.08.

286

H1 ZEUS

Table 2. Summary of the di-electron and tri-electron events with high invariant mass found by H1 and ZEUS.

2e 3e Data SM Data SM

3 0.30f0.04 3 0.23f0.04 2 0.77f0.08 0 0.37f0.04

7. Generic Searches

H1 performed a model-independent search for deviation from the SM using the whole HERA I statistics 18. Events with at least two objects (electrons, muons, jets or neutrinos) with high transverse momentum were considered and classified into exclusive event classes. The invariant mass and the sum of the transverse momentum of the objects were studied and subdivided into regions which have at least twice the resolution as size. A statistical estimator p which quantifies the discrepancy from the SM was then calculated. The more interesting regions are the ones with the smallest value of p . In Fig. 6 the distribution of - logP for the C Pt variable is shown where P is the probability of finding, in a given event class, a deviation as large as the one observed in the data. The pvjet channel is the event class which gives the largest deviation. This discrepancy corresponds to the excess described in Sec. 5.

8. First HERA I1 Results

The H1 collaboration has updated the analysis described in Secs. 5 , 6 and 7 with HERA I1 data (45 pb-l). New events with high-momentum isolated electrons and high missing transverse energy have been observed. In Ta- ble 3 H1 results using all 1994-2004 data, corresponding to an integrated luminosity of 163 pb-l are shown.

ti1 Genera sear& - r P, results

-log P

Figure 6. from MC experiments.

The - log P values for the data event classes and the expected distribution

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Electrons Muons

Table 3. or muons and missing transverse momentum in the full HERA data set

Summary of the H1 results of searches for events with isolated electrons

Combined

Total P z > 25 GeV

Data SM Data SM Data SM 18 15.4f2.1 9 4.1 f0.7 27 19.5f2.8 8 2.6f0.5 6 2.5 f0.5 14 5.1 f 1.0

9. Conclusions

The H1 and ZEUS experiments searched for hints of many BSM processes like leptoquarks, contact interactions, resonant Rp violating squark production, excited fermions, isolated leptons with high missing momentum and events with two or more leptons, using HERA I data. The latest results have been briefly reviewed. In most cases such results are compatible with SM expectations. Where no evidence of signal was found, HERA limits are competitive with the ones from LEP and TeVatron experiments, especially on particles that can be produced as a resonance in the s-channel. First results of HERA 11 data taking have also been shown. Open questions, especially in the high pt lepton sector, will be hopefully solved by HERA 11.

References 1. H1 Coll., Phys. Lett. B 568 (2003) 35 and references therein. 2. ZEUS Coll., Phys. Lett. B 591 (2004) 23 and references therein. 3. H1 Coll., Eur. Phys. J. C 36 (2004) 425 and references therein. 4. ZEUS Coll., ICHEP04 paper 254 and references therein. 5. H1 Coll., Accepted by Phys Lett B, hepex/0405070 and references therein. 6. H1 Coll., EPS03 paper 676 and references therein. 7. ZEUS Coll., ICHEPO4 paper 255 and references therein. 8. H1 Coll., ICHEP04 paper 255 9. H1 Coll., Phys. Lett. B 561 (2003) 241. 10. ZEUS Coll., Phys. Lett. B 559 (2003) 153, addendum: DESY-03-188 (2003). 11. ZEUS Coll., Phys. Lett. B 583 (2004) 41. 12. H1 Coll., ICHEP04 paper 188. 13. H1 Coll., Eur. Phys. J. C 33 (2004) 9. 14. H1 Coll., Eur. Phys. J. C 31 (2003) 17. 15. H1 Coll., Phys. Lett. B 583 (2004) 28. 16. ZEUS Coll., ICHEPO2 paper 910. 17. ZEUS Coll., ICHEP04 paper 230. 18. H1 Coll., Submitted to Phys. Lett. B, hep-ex/0408044. 19. H1 Coll., ICHEPO4 paper 765.

ELECTROWEAK PHYSICS AT LEP2 AND THE FIT TO THE STANDARD MODEL

R. CHIERICI CERN,

121 1 CH Geneva, Switzerland. E-mail: Roberto. ChiericiQcern.ch

In this article the most recent results from electroweak precision physics at LEP2 are reviewed and interpreted. Limits on the Higgs mass and constraints on the validity of the Standard Model (SM) are also derived via a fit to all eledroweak results.

1. Introduction

The LEP experiments have collected about 700 pb-’ of integrated luminosity each during the LEP2 phase, with an e+e- collision energy ranging from 161 GeV up to 209 GeV. This corresponds to a total statistics of approximately 40000 W-pair produced at LEP. In what follows the most recent experimental results on four-fermion cross-section measurements and their interpretation and on the W mass will be shortly reviewed. All results have to be considered as preliminary, if not stated otherwise. The derived constraints on the SM and the Higgs mass from all electroweak results will also be presented.

2. Four-fermion cross-sections and derived quantities

2.1. W-pair production

W-pair production in e+e- annihilations is defined as the subset of CC03 diagrams ’ leading to four-fermion final states. Their production at LEP is investigated through all possible final states, according to the W decays. One can therefore distinguish fully hadronic decays, where both Ws decay into hadrons, the so-called semileptonic where one of the W decays hadronically and the other into a lepton and a neutrino, and the fully leptonic one. The main backgrounds are given by two fermion production, mainly qq(y ) for hadronic and “(7) for leptonic channels, and four fermion backgrounds.

288

289

The purest channels (up to p=95%) are the semileptonic ones because of the clean signature represented by the highly energetic and isolated lepton. Figure 1 shows the LEP combined value of the measured cross-sections as a function of the centre-of-mass energy and compared with the most recent theoretical predictions, which include O ( a ) electroweak corrections in double-pole approximation (RACOONWW 2, YFSWW 3) . The LEP combination properly takes into account the correlation in energy and experiment of the systematics.

0 ’ I 160 180 200

4s (GeV)

MeasurcJ (I ‘‘ ’ Eacoc-WW

P R E L Z W

183 GeV - 1034*0023

189 GeV 0 986 + 0 013 192 GeV 100310029

1 YE GeV 1006+0019

200 GeV 0 986 -f 0 018

202 GeV 0 998 6 0 024

205 GeV 0 982 + 0 018

207 GeV 1 W3jOO15

LEP combined 0.995 f 0.009 ?/n61-32 0/31

1 1 0 9 1

Figure 1. Left: LEP combined W-pair production cross-section, compared to the predictions of RACOONWW and YFSWW. The shaded area represents the uncertainty on the theoretical predictions, estimated as f2% for &< 170 GeV and ranging from 0.7 to 0.4% above 170 GeV. Right: ratios of LEP combined W-pair cross-section measurements to the expectations according to RACOONWW. The band represents a constant relative errors of 0.5% on the cross-section predictions.

The agreement with the expectations is excellent; from the data it is possible to determine the ratio between the measured and the predicted WW cross-section values, Rww. This is shown in the right plot of figure 1, per energy and combined, for the RACOONWW calculation (which agrees at the 0.2% level with YFSWW). The band indicates the theory error of 0.5% (its dependence with energy is ignored in the plot), which starts to be comparable with the experimental error of less than 1%. The systematic part of the experimental error on Rww is dominant and its major component is given by the uncertainty on the fragmentation modelling in the qqqq signal and the background.

290

2.2. Measurements derived from the WW cross-section

From the partial cross-sections in the different W decay channels it is possible to determine the branching fractions to hadrons and leptons. If lepton universality is assumed, the LEP combined value for the W branching ratio to hadrons yields:

BR(W -+ hadrons) = (67.49 f 0.28)%

in agreement with the expectations of 67.5%. Within the Standard Model this also determines the leptonic W branching fraction that can be expressed in terms of the CKM quark mixing matrix elements not involving the top quark:

where a S ( M & ) is the strong coupling constant. Taking as(M&) = 0.119 f 0.002 and using the experimental knowledge of the sum IVud12+ IVzLs12 + IVub12+IVcd12+IVcb12 = 1.0476f0.0074, the above result can be interpreted as a measurement of IV,, I which is the least well determined of these matrix elements:

lVcsl = 0.976 f 0.014

where the uncertainties on the SM parameters are included in the error. If lepton universality is not assumed, from the different leptonic channel

the branching fractions of the W to leptons can be determined. The LEP combined values are shown in figure 2, with a negative correlation of 19.1% (13.2%) between the TV and ev (pv) branching fractions, while between the electron and muon decay channels there is a positive correlation of 10.9%.

In the combined values the W branching ratio in TU is significantly higher than the others lepton. This can be quantified via the tweby-two comparison of these branching fractions:

BR(W -+ pF,)/BR(W .--) eFe) = 0.994 f 0.020 BR(W + TF',)/BR(W -+ eFe) = 1.070 f 0.029

BR(W + rFT)/BR(W --t pF,) = 1.076 f 0.028

where the errors include correlations as well. From the total WW and single-W (next section) cross-section and the

W angular distributions, limits on anomalous trilinear gauge couplings can be determined. The set of constrained parameters at LEP follows what

291

Summer 2004 - LEP Preliminary

W Leptonic Branching Ratios ALEPH

L3 OPAL

LEP W+ev ALEPH

13 OPAL

LEP w+p ALEPH

13 OPAL

1 ; .

LEP W - m

LEP W+lv

I 02/0BnoM

-.-. -*- t 1081 * 029

10 i8. t 032 104011: 035

10.66 jI 0.17 4 1091 3 0 3 6

1003c 031 10.61 f 0.35

i '

-*-

1 0 . 6 0 ~ 0.15 4 I i 15 1 0'38

1189- 0 4 5 11 1811: 0.48

~~

11.41 5 0.22

10.84 t 0.09 X2/ndf = 6.8 I 9

X21ndf=15/11 1 ,.

1'0 4'1 15 ' Br(W+lv) [%]

Figure 2. LEP combined branching ratios of W into leptons.

proposed in 5 , and derives from all independent couplings describing the triple gauge boson interaction in the assumption of electromagnetic gauge invariance and conservation of C and P. The resulting three parameters are fitted to the data; the preliminary LEP results of single parameter fits, where the non-fitted parameters are fixed to their SM values, are:

where the error include statistical and systematic contributions and in brackets the SM expected value is shown. No deviation from the SM are seen. The measurements are dominated by the statistical error, the most important systematic contribution comes from radiative corrections, which introduce important distortions in the shape of the W angular distributions '.

292

2.3. Non- WW production

Four-fermion production in the non-CC03 part of the four-fermion phase space is also investigated at LEP2. This is important because many of these channels are backgrounds to others or to searches for new physics, and also as a probe of the SM. The different signals are defined with cuts in the phase space, especially when the diagrammatic definition is not straightforward (like for NC02 contribution, identifying 2-pair production). In this way single boson production (Weu, Zee) or Zy' production can be defined 7. Also three boson final state are studied ( m y , Zyy), particularly relevant for setting limits on anomalous quartic gauge couplings.

h n

M Q

0

v

1

0.5

0 180 190 200 180 190 200 210

4s (GeV) 4s (GeV)

Figure 3. (right) as a function of the centre of mass energy compared with the expectations.

Z-pair production cross-section (left) and single-W production crosssection

Figure 3 shows two examples of four-fermion cross-sections compared with the expectations. In analogy with the W-pair production, ratios between the observed cross-section values and the predicted ones can be defined, yielding:

Rzz = 0.952 f 0.052

Rwev = 1.002 f 0.075 Rzee = 0.963 f 0.065

No discrepancies from the expectations are observed and one can notice that, also in the case of other portions of the four-fermion phase space, the

293

final experimental resolution approaches the values of the theory uncertainties, typically ranging from 2 to 5%.

3. The W mass measurement

The W mass at LEP2 is measured through the direct reconstruction of the W boson from its decay products. The mass resolution is greatly improved by applying kinematical constrained fits to impose the energy-momentum conservation in the event. The mass distributions from the data are then fitted analytically or by making use of the Monte Carlo itself to extract the value of the W mass. In these fits the W width is fixed to the SM value, whereas the width is extracted in a two-parameter fit. Given the very large statistics accumulated at LEP2, the mw measurement is now dominated, in the LEP combination, by the systematic error. Particularly relevant is the contribution of the so-called Final State Interactions (FSI) in the fully hadronic channel, mainly indicating possible colour reconnection through gluon exchange between quarks in the final state, or Bose-Einstein effects which tend to produce pions close in the phase space. The typical radius in which FSI happen is larger than the average decay distance between the two Ws and this could, in principle, correlate the two decays. The large uncertainty introduced by these effects reflects our inability in modelling the underlying physics. Dedicated analyses in the hadronic part, less sensitive to those effects (which mainly involve the soft part of the particle spectrum), are currently under study; the possibility to trade some of the statistical power of the analyses for a significant reduction of the LEP correlated systematic error could be very beneficial for the combination of the results.

Table 1 presents the breakdown of the systematic errors on mW for the semileptonic and the hadronic channel separately and combined. In the LEP combination the correlation in energy, experiment and channel are properly taken into account. Detector systematics include uncertainties in the jet and lepton energy scales and resolution. The ‘Other’ category refers to errors, all of which are uncorrelated between experiments, arising from simulation statistics, background estimation, four-fermion treatment, fitting method and event selection.

In figure 4 the present LEP combined value of mw and of the W width is reported. The weight in the combination of the hadronic channel is about 9%. As a crosscheck of possible presence of hidden systematics in the hadronic sector, the mass difference between the leptonic channels and the hadronic one can be determined. The combined LEP value yields

294

Winter 2003 - LEP Preliminary

I [ Iso6.?oaoJ 80 176+0 077

OPAL [1996-1999] 80.49MO oci5

LEP 80.412fO 042

LEP working group

80.0 81.0

Mw[GeVl

Winter 2003 - LEP Preliminary

1.5 2.0 2.5

r,rCcvl

Figure 4. Combined LEP W mass and W width.

(22h43) MeV, in agreement with zero.

4. Constraints on the Standard Model

The only undetermined parameter of the Standard Model is the mass of the Higgs boson. No evidence of Higgs production at LEP was seen. The LEP limit from direct searches of the boson was of 114.4 GeV at the 95% confidence level. However, with the precise electroweak measurements

Table 1. bined LEP W mass results.

Source bniw(syst) (MeV)

Error decomposition for the com-

QQLV PQQQ Comb. ISR/FSR 8 8 8

Hadronisation 19 18 18 Detector Systematics 12 8 11

LEP Beam Energy 17 17 17

Colour Reconnection - 90 9

B-E Correlations - 35 3 Other 4 5 4

Total Systematic 29 101 30

Statistical 33 36 30

Total 44 107 42

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performed at LEP2 it is possible to constraint the SM Higgs mass. This can be done because the SM parameter themselves depend on the Higgs mass via electroweak radiative corrections.

In the electroweak fit all experimental observables and the SM input parameters are constraints in the fit and subject to their experimental and theoretical uncertainties. Aside the Higgs mass itself, the dominant uncertainties concern the top and the W mass and the hadronic contribution to the fine structure constant. Figure 5 , left, shows the x2 distribution of the fit to all data as a function of the Higgs mass. The band indicates the region excluded by direct searches. The most probable value falls very close to where the direct limit is. Such indirect determination of mH gives:

mH = 114f~~GeV/c2

The central value of the Higgs mass from the fit largely depends on the top mass (one standard deviation in mt corresponds to a 30% shift in mH), confirming the importance of improving the precision on the knowledge of the top mass in the coming years.

The preference for light SM Higgs masses is confirmed by the right-hand plot of figure 5 , which shows the 68% CL contour in the mw-mt plane for the direct measurement and for the indirect fit using all electroweak measurements excluding mw and mt.

130 150 170 190 210

Figure 5. and indirect data (right).

SM Higgs x2 curve from LEP data (left) and mw-mt contours from direct

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The figure also shows level curves for different values of the Higgs mass. It can be noticed that the direct and indirect data agree in preferring a light Higgs.

5. Conclusions and outlook

The LEP2 era has allowed very precise electroweak measurements in the four-fermion sector. W-pair production physics has allowed to test the non- abelian structure of the SM and to measure cross-sections with an accuracy which is now demanding theory predictions with radiative corrections at the loop level.

Full reconstruction of W decays allows to directly determine the W mass, whose error has been greatly reduced at LEP. The branching fractions of the W boson into hadrons and leptons agree with the SM expectations, though an excess of decay into 7 leptons with respect to e or p is visible from data.

At LEP the four-fermion production in other regions of the phase space has also been extensively studied providing, among others, important input in the description of neutral current processes.

LEP collaborations axe still active: amongst the different goals to be achieved, the need to further reduce the systematic error on the W mass and to complete the combinations of total and differential four-fermion cross- sections. The SM Higgs has not been discovered yet, but with the excellent electroweak measurements performed so far its mass can be constrained; the indications are towards a light SM Higgs that should either soon become a precision measurement itself, or leave space for signals of new physics.

References 1. M.W. Griinewald et al., in Reports of the working groups on precision calcu-

lations for LEP2 Physics, CERN 2000-009, (2000), hepph/0005309. 2. A. Denner, S. Dittmaier, M. Roth and D. Wackeroth, Nucl. Phys. B587 (2000)

67 and references therein. 3. S. Jadach, W. Placzek, M. Skrzypek, B.F.L. Ward, Z. Wqs, Comput. Phys.

Commun. 140 (2001) 432 and references therein. 4. Particle Data Group Collaboration, K. Hagiwara et al., Phys. Rev. D66 (2002)

010001. 5. G. Gouriaris et al., in Physics at LEP2, CERN 96-01, (1996), eds. G. Altarelli,

T Sjostrand, F. Zwirner, Vol 1, p. 525 6. R. Chierici, F. Cossutti, Eur. Phys. 3. C23 (2002) 65. 7. The LEP ElectroWeak Working Group, hep-ex/0312023 8. The LEP Higgs Working Group, CERN-EP/2003-011

SEARCH FOR DARK MATTER AT LHC

S. LACAPRARA I. N . F. N . and Padova University


on behalf of the CMS collaboration

A review of the potential of the LHC experiments to discover and measure supersymmetric Dark Matter is presented. Different methods for sparticle mass spectrum reconstruction in the framework of R-Parity conserved mSUGRA model are discussed.

1. Introduction

The evidence for Dark Matter in the universe is today well established by a variety of indirect observations, mostly coming from Cosmology. Recently, the WMAP [l] collaboration has improved significantly the detection accuracy of Cosmic Microwave Background. The outcome of this new measurement is twofold: the universe mass density is very close to the critical one, and a large fraction of it is Cold Dark Matter (CDM). The CDM contribution to the total Universe density (Ci2,h2) is constrained, at 95% C.L., in the interval:

0.093 < Omh2 < 0.0119 (1)

in units of critical density pc = 1.86 . Until today, nonetheless, no direct evidence of Dark Matter has been

found. The only exception is the result presented by the DAMA collaboration [2], which has not been confirmed by other experiments [3]. Many candidates for CDM have been proposed within different theoretical frameworks. The mass of the candidates varies from extremely light particles, such as massive neutrinos or axions, to super heavy metastable particles.

One of the most popular and promising candidates comes from Super- symmetric (SUSY) extensions [4] to the Standard Model (SM). In R-parity conserving SUSY, the Lightest Supersymmetric Particle (LSP) is stable,

g/cm3.

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and its relic abundance can explain at least a fraction of CDM. Supersym- metric models have the desirable feature of being consistent with all high precision electroweak measurements performed at LEP and SLC while solving one of the most important pending issues in the SM, the so called mass hierarchy problem [4].

The Large Hadron Collider (LHC), being built at CERN, will deliver p p collisions at a center-of-mass energy of 14 TeV starting in 2007. The unprecedented energy and luminosity achievable at LHC will place this accelerator in a very promising position for SUSY discovery.

In this report the minimal Supergravity (mSUGRA) model [5], a strongly constrained version of the more general Minimal Supersymmet- ric Standard Model [6] is considered. The five free parameters of the model are: the universal scalar and gaugino masses mo and m1/2, the ratio of Higgs doublet VEV’s tanp, the sign of the Higgs mixing parameter signp and the trilinear SUSY-breaking parameter Ao. In the mSUGRA model the LSP is the lightest neutralino (xy), which, given the R-parity conservation, is stable and hence is a candidate for Dark Matter. The next-tdightest sparticle is usually a slepton (typically the ?). The heaviest sparticles are the gluino,

The constraint from the Dark Matter measurements reduces strongly the allowed regions in the mSUGRA parameter space. There are four regions with different features which are compatible with the measured CDM, as shown qualitatively in Fig. 1.

Bulk region: the relic xy annihilates via a t-channel ? exchange into a r pair: the limit on the CDM density yields a limit on the annihilation cross section, which, in turn, limits the possible mo, rnlp values to be smaller than 150 GeV/c2 and 300 GeV/c?, respectively.

Co-annihilation region: if xy and ? are almost degenerate in mass, then the direct ? -+ rxy decay is forbidden. The ? relic density is thus enhanced, meaning that the co-annihilation process xy? -+ r y must have played a significant role in the early universe. The LSP is thus allowed to have larger mass, given the larger disappearance rate.

Rapid annihilation funnels: at large tanp, and for mo N m1/2, the heaviest Higgs boson H, as well as the pseudo-symmetric one, A, have mass nearly half that of the LSP, allowing resonant annihilation. The larger LSP annihilation rate allows for larger LSP masses to be compatible with the measured CDM.

Focus points: a combination of large mo and ad hoc values of tan p results

and the first generations squarks, tj.

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Figure 1. Pictorial view of the four re- Figure 2. Total mSUGRA cross section gions in mSUGRA mo, mll2 parameter in mo, ml12 plane (solid). The contribu- space compatible with CDM constraint. tion from the ij+g cross section is shown

as a dashed line.

in LSP with large (- 100%) higgsino components, enhancing the offshell LSP annihilation via s-channel Higgs production.

A set of benchmark points in the mSUGRA parameter space has been proposed in Ref. [7], in order to define a common playground to test SUSY discovery potential. Most of the work presented in this report focuses on point B, which lies in the bulk region. Perspectives and problems for other space parameter points are also briefly discussed.

The rest of this report is organized as follows: the LHC discovery potential for SUSY is briefly described in Section 2. Possible techniques for sparticle mass reconstruction in some favourable SUSY benchmark points are presented in Section 3, and the implication for Dark Matter evaluation is discussed in Section 4.

2. Discovering SUSY

At the LHC, sparticle production would occur mainly via strong pp --+ Gg, Gij, ,@ interactions, with large cross section. Gluinos and squarks would then give rise to long decay chains. The sparticle production cross section can be as high as O( 10) pb, for low jij and ij masses, corresponding to low mo and m1/2. For larger masses the production cross section decreases, as shown in Fig. 2.

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Because of R-parity conservation, sparticles are always produced in pairs. At the end of the decay chain, hence, there are always at least two x?, which escape detection, leading to large missing transverse energy (Et). Moreover, g and Q decay preferentially into quarks, resulting in typical multi-jets plus missing-Et signature. Since third-generation Q (6 and {I) are typically lighter than other Q, their presence in the decay chain is enhanced. Subsequent k and decays provide high b-jets and W multiplicity. Further down the decay chain, heavy neutralinos and charginos can be produced, which in turn decay into sleptons, and then into leptons (and

A SUSY event is hence characterized by a very rich experimental signature, including large missing Et, multiple jets, b jets and isolated leptons, easy to trigger on. The trigger threshold on missing Et and jet Et are determined by the need of keeping the total rate, dominated by QCD events, at an affordable level. As an example the CMS High Level Trigger (HLT) thresholds are listed in Table 1. Even with these simple selections, the efficiency is estimated to be N 60 + 70% in the initial LHC low luminosity scenario (L = 2 nb-ls-') and for Ma N 400 GeV/c2. At high luminosity (L = 10 nb-ls-l), and for higher Q masses (Ma N 2 TeV/c?), the efficiency is estimated to be in range 75 + 90%. More details can be found in Ref. [8].

X 3 -

Table 1. CMS High Level Trigger thresholds for missing Et plus jet and multi jets selection, for low and high luminosity scenarios.

Luminosity Trigger Threshold (GeV or GeV/c)

Low 1-jet * EP 180 * 123

L = 2 nb-ls-l 4-jets 113

High E P 239

L = 10 nb-ls-' 4-jets 185

The CMS discovery potential, corresponding to an integrated luminosity of SL = 100 ft-l, is shown in Fig. 3. Many different topologies have been considered, all including missing Et plus jets, with additional leptonic signatures: zero, one, two opposite sign (0s) as well as same sign (SS) and three leptons (here e, p) . The highest reachable mass for Q and g is expected to be ME,^ N 2.5 TeV/c2. With full integrated luminosity L = 300 fb-', it increases up to N 3 TeV/c2. The highest sensitivity is expected for the inclusive missing Et plus jets channel.

30 1

800

600-

400

200-

-

Ht!Iltt lt$+f ,

?t{ + ' t - t t#

t t 'It t

,#' + tlk'l

- I))+% t t

+ +>'#* .*.* . i++

10

m, (GV)

Figure 3. The CMS reach with 100 ft-' Figure 4. Di-lepton invariant mass disintegrated luminosity is shown for the inclu- tribution for opposite-sign-same-flavor sive channel with E F plus various lepton leptons, after subtracting the opposite- signatures. sign-opposite-flavor background distribu-

tion, shown for the ATLAS detector for J L = 100 ft-1.

3. Sparticles spectroscopy

Assuming the existence in Nature of Supersymmetry and its discovery at the LHC, it would be very important trying to understand which SUSY model is correct among the many available. The determination of the sparticle mass spectrum is one of the most important goal for the post-discovery phase.

A study is presented here for the benchmark point B, corresponding to the following values of the mSUGRA parameters: rno = 100 GeV/c2, ml12 = 250 GeV/c2, tanP = 10, p > 0 and A0 = 0. In this scenario, the masses of all sparticles are expected to be light, between 100 and 800 GeV/c2 and the sparticle production cross section very large (0 - 58 pb). Long decay chains are possible, and can be used to evaluate the sparticle masses, through the so-called end-point reconstruction

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[9,10,11]. The following decay chain is considered:

The final state contains b jets, same-flavor-opposite-sign (SFOS) isolated leptons and missing energy. This rich signature can be rendered almost Standard Model background free with simple requirements. The leptons in the final state come from the x: decay (via slepton), which has a branching ratio BR(x$ -+ xylZ), strongly dependent on tanp. At point B (tanp = 10) the x$ branching ratio into electrons and muons is about IS%, while the rest is accounted for decays into 7'5, which leads to different and more complex experimental problems. At larger tan p the situation worsen; e.g. at the benchmark point G (tanp = 20) BR(x: -+ X ~ Z Z ) x 2.3% and at point I ( tanp = 35) it is almost negligible.

The kinematic of the three body decay is characterized by a sharp upper edge in the di-lepton invariant mass distribution, which depends on the mass of the sparticles involved according to the following formula:

(3) MIT" = - 1 /(mz! - m:) (m; - mzn> .

mi The small background, coming mostly from other SUSY events, can be

further reduced by subtracting the distribution of events with opposite- flavor-opposite-sign leptons (OFOS) . The expected background-subtracted distribution, obtained by ATLAS for 100 fb-', is shown in Fig. 4.

Other kinematical edges can be reconstructed using also the jet energy, as described on Ref. [ll]. For instance, the largest invariant masses of the first and second lepton with a jet Mql1, Mq12 are bounded from above by a value which, as M T " , depends on the masses of the sparticles involved in the decay. More precisely, the upper edge, as for M Y " , depends on mass dzfferences rather than simply on masses. Other edges are present in the Mqll distribution.

The same procedure can be applied also for decay chains involving squarks other than L. The final state is less background free due to the lack of b-tagged jets: this is partially compensated by an anti b-jet veto and also by the higher statistics which can be collected.

Sparticle masses can be evaluated by solving the set of equations corresponding to the measured edges, which give rise to an over constrained system. Since the kinematic edges are sensitive to mass differences, the mass values determined from the equation system suffer from very strong correlations. This is why all masses can be determined with high precision

303

(- 1%) in term of just one of them (e.g. the Adx:). The value of Adx:, however, is affected by about N 10% uncertainty which reflects on all other mass values. In order to fix the mass scale of all sparticles, an external determination of Mxy, e.g. coming from a Linear Collider, would be of great help.

As already mentioned, final states including r will play a major role in case of larger tanP. The r decay, producing always at least one escap ing neutrino, spoils the possibility to fully reconstruct the di-r invariant mass and to determine the edge position. Recent studies [ll], however, demonstrated that it is possible to reconstruct the di-r final state, using r hadronic decays. The SM background is small, and the SUSY one can be subtracted by looking at the same-sign T pairs. The invariant mass distribution obtained is sensitive to the edge position, although the edge itself is not as visible as in the e, p case, as shown in Fig. 5. Additional work is needed to fully exploit this final state; this preliminary result, however, is encouraging.

Near the kinematical edge of the di-lepton invariant mass distribution, it is possible to reconstruct the xi four-momentum using the following formula, which assumes the knowledge of Adx,:

Selecting events near the edge, and adding the most energetic b jet, it is possible to reconstruct directly the G mass peak. With full LHC integrated luminosity, it would be even possible to resolve the masses of and 62.

Adding the other b jet to the momentum, also the g mass peak can be reconstructed. The same can be done with decay chains involving squarks different from b: in this case, however, it appears very difficult to resolve individual squark masses, which are expected to be almost degenerate.

4. Dark Matter evaluation

The determination of the sparticle mass spectrum alone cannot give direct information about the SUSY contribution to the Dark Matter. Assum- ing that Supersymmetry is indeed described by the mSUGRA symmetry breaking mechanism, it is possible to estimate Supersymmetric Dark Mat- ter with a model dependent analysis [12]. The knowledge of the sparticle mass spectrum can be used to determine the best set of mSUGRA parameters by performing a direct fit of the model parameters to the measured

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300 i?

i : t i :

0

Figure 5. Mr+r- distribution after subtraction Figure 6. Distribution of R,h2 as of same-sign background (full circles). Also shown obtained with the global mSUGRA the distributions for xi -+ TT signal (full line), fit for many simulated experiments, x: decays (gray), and undecayed T'S from xi as described in the text. (dashed).

observables. The parameters determined in this way allow a precise determination of the LSP mass, more precise than the direct measurement, though strongly model dependent. The LSP relic density can be determined as well. The LSP contribution to the Dark Matter (Qxh2) can be evaluated by:

where nxp and mxp are the neutralino relic density and mass, respectively. The impact of the experimental uncertainties on the evaluation of Dark

Matter has been studied by generating many simulated experiments for a given mSUGRA scenario. For each experiment , all end-point measurements have been generated, sampling the probability density functions of the measured observable, which include uncertainties coming from either statistical and systematic effect, such as lepton and jet energy scale. The points in the mSUGRA parameter space which minimize the global x 2 have been chosen and the properties of LSP Dark Matter have been computed. The distribution of the LSP Dark Matter shows the expected precision which can be achieved using this technique.

Results presented in Fig. 6 show that a precision of about 3% can be obtained, with full statistic accumulated by each experiment at the end of LHC (SL = 300 fb-l), including also systematic uncertainties com-

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ing from relic density calculation. With smaller integrated luminosity (SL = 100 fb-l), a tail in the R,h2 distribution is present, arising from the lack of 61 and b2 mass determination.

5. Conclusion

If Supersymmetry exists in nature, LHC will offer the optimal conditions to discover it: in lucky scenarios, SUSY will show itself very quickly with spectacular signatures.

A “golden” scenario in the “bulk region” (benchmark point B) would allow LHC to provide a good evaluation of the LSP contribution to the Dark Matter, with a precision similar to that achieved with direct cosmological measurements.

Other benchmark points in the bulk region, like point G, can be treated in the same way as point B. Given the smaller sparticle production cross section and the lower x: branching ratio into e and p, however, the full LHC statistic will be needed. Points with larger tanP values, like point I, having only r final states, need further study.

In the “co-annihilation tail”, the situation is somehow similar, with points for which the r final state dominates (D,J,L) and others which can be treated like point B, although larger statistic is needed (A,C,H).

Finally, “focus points” and “rapid annihilation funnels” provide very heavy and g, beyond the reach of LHC. In these cases, the only SUSY particle to be produced at LHC would be the lightest Higgs boson.

The work presented in this report makes a strong assumption on the Supersymmetric model, since the mSUGRA model with R-parity conservation is assumed throughout the studies. Different SUSY breaking scenarios, including unconstrained ones, are not considered.

Moreover, the studies presented here have been performed with fast simulation of the detector response. Careful studies aiming to evaluate the impact of b-jet tagging efficiency, jet energy calibration, missing Et resolution, detector alignment and so on, need to be carried out with full detector simulation.

It might well be that LHC will contribute, by opening a new field in particle physics, in achieving a deeper understanding of cosmology. The last word is left to real data.

“Data! Data! Data! I canZ make bricks without clay!” - sir Arthur Conan Doyle

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Acknowledgments

I wish t o thank the conference organization committee for the friendly welcome. I would also thank Nancy Marinelli for her careful reading of this report.

References 1. C. L. Bennett et al., Astrophys. J. Suppl. 148 (2003) 1 [arXiv:astro-

2. R. Bernabei, these proceedings. 3. D. Cline, these proceedings. 4. For a review, see e.g. S. P. Martin, A Supersymmetry primer, hepph/9709356

(1999) and references therein. 5. P. Nath, R. Arnowitt, and A. H. Chamseddine, Applied N = 1 Supergravity

(World Scientific, Singapore, 1984). 6. H. E. Haber and G. L. Kane, Phys. Rep. 117 (1985) 75. 7. M. Battaglia et al, Eur. Phys. J. C 33 (2004) 273 [arXiv:hepph/0306219].

8. The CMS collaboration, Data Aquisition & High-Level Trigger TDR,

9. M. Chiorboli and A. Tricomi, Squark and gluino reconstruction with the CMS detector, CMS Rapid Note, CMS-RN-2003/002, (2003)

10. S. Abdullin et al, J. Phys. G 28 469-594 (2002) 11. B. K. Gjelsten et al, A detailed analysis of the measurement of SUSY masses

with the ATLAS detector at the LHC, ATLAS physics Note, ATL-PHYS- 2004007, (2004).

12. G. Polesello and D. R. Tovey, JHEP 0405 (2004) 071 [arXiv:hep ph/O403047].

ph/0302207].

2072 (1995).

CERN/LHCC 2002-26, CMS TDR 6.2 (2002).

Electroweak Physics


A HEAVY HIGGS BOSON FROM FLAVOR AND ELECTROWEAK SYMMETRY UNIFICATION*

MARC0 FABBFUCHESI INFN, Sezione di %este and

Scuola Internazionale Superiore d i Studi Avanzati via Beirut 4 , 1-34014 R e s t e , Italy

We present a unified picture of flavor and electroweak symmetry breaking based on a nonlinear sigma model spontaneously broken at the TeV scale. Flavor and Higgs bosons arise as pseudo-Goldstone modes. Explicit collective symmetry breaking yields stable vacuum expectation values and masses protected at one loop by the little-Higgs mechanism. The coupling to the fermions generates well-definite mass textures-according to a U( 1) global flavor symmetry-that correctly reproduce the mass hierarchies and mixings of quarks and leptons. The model is more constrained than usual little-Higgs models because of bounds on weak and flavor physics. The main experimental signatures testable at the LHC are a rather large mass mho = 317 f 80 GeV for the (lightest) Higgs boson.

Naturalness of the standard model seems to demand new physics at or around the 1 TeV scale. On the other hand, precision measurements do not show any departure from standard physics up to roughly 10 TeV. This little hierarchy problem is solved in little-Higgs models by introducing new particles near 1 TeV in a sufficiently hidden fashion as not to show in the precision tests.

The Higgs sector of the standard model also contains the physics of flavor in the Yukawa lagrangian, that is, fermion masses and mixing matrices of the three generations. What happens if we try to include flavor (horizontal) symmetry breaking in the little-Higgs models? At first sight, this seems impossible because of the much higher scale of the order of lo4 TeV (mainly coming from bounds on flavor changing neutral currents) at which flavor symmetry breaking should take place. However, the bounds depend on the specific realization of the symmetry breaking and they are

'This work is supported by the European TMR Networks HPRN-CT-2000-00148 and 00152.

309

310

not necessarily so strong if the flavor symmetry is global and there are no flavor charged gauge bosons 2 .

It is indeed possible to take closely related breaking scales for the electroweak and flavor symmetries and thus unify the two in a single little- Higgs model. This unification, beside solving the little hierarchy problem provides-along the lines of the Froggart-Nielsen mechanism 3-stable textures of a well-defined type that correctly reproduce the mass hierarchies and mixings of quarks and leptons.

The model gives a characteristic spectrum testable at the LHC of new particles, in addition to those of the standard model, and a lightest Higgs boson mass that is more constrained than in the usual, electroweak only, little-Higgs models.

In order to have a single, unified model rj la little Higgs describing the entire flavor structure as well as the electroweak symmetry breaking, the Higgs boson and the flavons must be the pseudo-Goldstone bosons of the same spontaneously broken global symmetry. These pseudo-Goldstone bosons-we shall call them flhiggs-should transform under both flavor and electroweak symmetries.

We consider a SU(10) global symmetry spontaneously broken to SO(10) at the scale f. Fifty-four generators of SU(10) are broken giving 54 real Goldstone bosons we parametrize in a non-linear sigma model fashion as

C(z) = exp [il-q.)/.f] Co , (1)

with II(z) = taxa(z ) , where ta are the broken generators of SU(lO), xa(z) the fluctuations around the vacuum CO given by

Within SU( 10) we identify seven subgroups

SU(10) I) U(1)F x [SU(3) x U(1)1& x [ W ) X l 2 , (3)

where the U ( ~ ) F is the global flavor symmetry while the [SU(3) x U(l)]& are two copies of an extended electroweak gauge symmetry. The standard model weak group SU(2) must be extended because it is otherwise impossible to have different vacuum expectation values for the weak and the flavor symmetries by means of fhiggs fields in the fundamental representations. The groups [U(l)xI2 are two copies of an extra gauge symmetry we need

311

in order to separate standard fermions from the exotic fermions the model requires because of the enlarged S U ( 3 ) w symmetry that turns the fermion weak doublets into triplets.

The breaking of SU(10) into SO(10) also breaks the subgroups [SU(3) x U ( l ) ] & x [U(1)xI2 and only a diagonal combination survives. On the contrary, the flavor symmetry U ( ~ ) F survives the breaking. The breaking in the gauge sector [SU(3) x U ( l ) ] & x [ U ( l ) x I 2 -+ [SU(3) x U ( l ) ] w x U ( l ) x leaves 10 gauge bosons massive. The remaining 44 Goldstone bosons can be labeled according to representations of the U ( 1 ) ~ x [SU(3) x U ( l ) ] w x U ( 1 ) x symmetry as two complex fields @ I and @2 that transform as triplets of [ S U ( 3 ) ] w , have the same U ( 1 ) w and opposite U ( ~ ) F charges (they are not charged under the exotic gauge symmetry U ( 1 ) x ) ; two complex fields @3

and @4 that transform as triplets of [ S U ( 3 ) ] w , have the same U ( l ) w (equal to that of @ 1 , 2 ) and opposite U ( 1 ) x charge (they are not charged under the flavor symmetry U ( ~ ) F ) ; a sextet of complex fields Z i j ; and four complex fields s, s1, s2, and sg, which are flavor singlets.

All these fields are still Goldstone bosons with no potential; their potential arises after the explicit breaking of the symmetry to which we now turn.

The effective lagrangian is given by the kinetic term

82

Lo = Jn 2 (DYq(D,C)* , (4)

the covariant derivative of which couples the pseudo-Goldstone bosons to the gauge fields WG, Bi, and X i , of the SU(3),i , U(l),i and U(l),i respectively. The index i runs over the two copies of each group. We denote the gauge couplings of SU(3),i by gi and those of U(l),i by 9:.

The lagrangian in eq. (4) gives mass to the zij and sg fields. On the other hand, each term of index i preserves a SU(3) symmetry so that only when taken together they can give a contribution to the potential of the flhiggs fields.

At this point the fields s, s1 and s2 are still massless. To give them a mass, we introduce plaquette terms-terms made out of components of the C field that preserve enough symmetry not to induce masses for the flhiggs fields.

In the breaking of [SU(3) x U(1)]& + [SU(3) x U ( l ) ] w nine gauge bosons become massive with masses of order O ( f ) . Their presence is the major constrain on the scale f and, accordingly, the naturalness of the model, as discussed for the littlest-Higgs model in '.

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The effective potential for the flhiggs fields is given by the tree-level contribution coming from the plaquettes and the one-loop Coleman-Weinberg effective potential arising from the gauge interactions.

After integrating out the massive states by means of their equations of motion, the potential of the four pseudo-Goldstone bosons @i, the flhiggs, is made of only quartic terms. Quadratic and quartic terms that are necessary to induce vacuum expectation values for the flhiggs fields are not generated in the bosonic sector discussed so far. In order to introduce them we couple the pseudo-Goldstone bosons to right-handed neutrinos with masses at the scale f . This is done again along the lines of the little-Higgs collective symmetry breaking. In this way the flhiggs bosons receive a mass term only from diagrams in which all the approximate global symmetries of the Yukawa lagrangian are broken. Because of this collective breaking, the one-loop contributions to the flhiggs masses are only logarithmic divergent.

We want to find vacuum expectation values for the flhiggs fields @i in this potential that breaks the symmetry [SU(3) x U(l)]w x U(1)F x U ( l ) x down to the electric charge group u(1)Q. Such a vacuum is given by the field configurations

After symmetry breaking, the model is described at low-energy by the standard model with the addition of a massive neutral gauge boson 2'. The bound on the mass of 2'

The number of degrees of freedom of 4 complex triplets is 24. Of these 9 are eaten by the gauge fields, while 1-the would-be Goldstone boson of the spontaneous breaking of the U ( ~ ) F global symmetry-is eliminated, after introducing the fermions in the model, by an anomaly. Therefore, the scalar sector contains 12 scalar bosons, ten of which are neutral, two charged.

To obtain an estimate of these masses, we vary the numerical value of the coefficients in the potential by a Gaussian distribution around the natural value 1 with a spread of 20% (that is CT = 0.2). For each solution we verify that all bounds on flavor changing neutral currents are satisfied. This is possible at such a low energy scale because the relevant effective operators induced by the exchange of the flavor-charged flhiggs fields are suppressed by powers of the fermion masses over f '.

The lightest neutral scalar boson (what would be called the Higgs boson

requires V F X 1260 TeV.

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in the standard model) turns out to have a mass mho = 317 f 80 GeV. This a rather heavy-that is, larger than 200 GeV-Higgs mass is due to the value of VF we were forced to take in order to satisfy the bound on the mass of 2’. It is a value that only partially overlaps with the 95% CL of the overall fit of the electroweak precison data tha t gives mho < 237 GeV 6. It provides us with a very testable prediction of the model.

The coupling of the flhiggs to the fermions gives rise to mass matrices with well-defined textures generated by the flavor charge assignment and the powers of parameter k = v$/f2 which accordingly enter the Yukawa lagrangian ’. For k of the order of the Cabibbo angle (and therefore f II 3 TeV), these mass matrices reproduce- for assigned values of the Yukawa coefficients of O(1)-in a satisfactory manner the experimental data on mass hierarchies and mixing angles. Notice that, on the other hand, the problem of the absolute value of the neutrino Yukawa couplings with respect to the others is only partially addressed by the low-energy see-saw taking place in the model.

References

1. N. Arkani-Hamed, A. G. Cohen, E. Katz and A. E. Nelson, JHEP 0207, 034 (2002) [arXiv:hep-ph/0206021]; I. Low, W. Skiba and D. Smith, Phys. Rev. D 66, 072001 (2002) [arXiv:hep-ph/0207243]; D. E. Kaplan and M. Schmaltz, JHEP 0310, 039 (2003) [arXiv:hep-ph/0302049]. S. Chang and J. G. Wacker, Phys. Rev. D 69, 035002 (2004) [arXiv:hep-ph/0303001]. W. Skiba and J. Terning, Phys. Rev. D 68, 075001 (2003) [arXiv:hep-ph/0305302]. S. Chang, JHEP 0312, 057 (2003) [arXiv:hep-ph/0306034].

2. F. Bazzocchi, S. Bertolini, M. Fabbrichesi and M. Piai, Phys. Rev. D 68, 096007 (2003) [arXiv:hep-ph/0306184]; Phys. Rev. D 69, 036002 (2004) [arXiv:hep-ph/0309182]; F. Bazzocchi, arXiv:hep-ph/0401105.

3. H. Harari, H. Haut and J. Weyers, Phys. Lett. B 78, 459 (1978). C. D. Frog- gatt and H. B. Nielsen, Nucl. Phys. B 147, 277 (1979). T. Maehara and T. Yanagida, Prog. Theor. Phys. 61, 1434 (1979). G. B. Gelmini, J. M. Ger- ard, T. Yanagida and G. Zoupanos, Phys. Lett. B 135, 103 (1984).

4. T. Han, H. E. Logan, B. McElrath and L. T. Wang, Phys. Rev. D 67, 095004 (2003) [arXiv:hep-ph/0301040]; C. Csaki, J. Hubisz, G. D. Kribs, P. Meade and J. Terning, Phys. Rev. D 67, 115002 (2003) [arXiv:hep-ph/0211124].

5. F. Abe et al. [CDF Collaboration], Phys. Rev. Lett. 79, 2192 (1997). 6. The LEP Electroweak Working Group, http: //lepewwg . web. cern. ch-

/LEPEWWG/plots/winter2004/. 7. F. Bazzocchi and M. Fabbrichesi, to appear.

THE STANDARD MODEL RESULTS FROM A SCHEME TO PROTECT THE MASS OF THE SCALAR BOSONS

M. CHAVES Universidad de Costa Rica,

San Jose, Costa Rica E-mail: mchavesOcariari.ucr.ac. cr

The masses of phenomenological scalar bosons are not protected against large contributions due to quantum loops involving the hypothetical very high energy theory that we assume is the true theory. Here I present a generalization of Yang- Mills theories that treats vector and scalar bosons on the same footing and results in Ward identities and current conservations that protect the masses of the scalars. It is remarkable that that the Standard Model obtains immediately upon choosing an appropriate gauge symmetry and allowing for one scalar field and one vector field to possess large vacuum expectation values. The Standard Model then appears as the unitary gauge of the chosen gauge symmetry. Due to the vectorial VEV there is a small breaking of the Lorentz invariance.

INTRODUCTION. At present we understand the Standard Model as an effective quantum field theory of some ultimate very high energy theory. We can understand the masslessness of the vector bosons because their mass is protected by gauge invariance, and the masslessness of the fermions is protected by chiral symmetry. But scalar bosons are not protected and should obtain, through quantum loops, a large mass. Particularly upsetting is the instability of the mass of the scalar bosons due to quadratic divergences of the form s d4k/k2 in the self-energy of the Higgs. Supersymmetry can achieve this protection, at least in the limit of small SUSY breaking, but it still has to be experimentally observed.

In the present model we present an alternative to SUSY to protect the masslessness of the scalar bosons. It is a modification of the Yang-Mills concept.

THE BOSONS. In a usual nonabelian gauge field theory one has to choose a Lie group as symmetry group. I will use S U ( N ) as an example. Let A;, a = 1 , . . . , N 2 - 1, be the fields to be associated with the generators T". These fields transform as AZT" UA;TT"U-l, where U is an element

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of the fundamental representation; that is, they transform in the adjoint.

tulate the gauge transformation Treating the A; as dynamical fields obeying a gauge principle, we pos-

A, + UA,U-l - (8,U)U-'. (1)

D, = 8, + A,, A, = igAE(z)Ta, (2)

Then if the covariant derivative is

where g is a coupling constant, it follows then that under the gauge transformation and for any differentiable function f (x),

Dpf = 8,f + Apf + a, f + UA,U-l f - (8pU)U-1 f

= U(8, + A,)U-l f = UD,U-l. = d,UU-l f + UA,U-'f - (8,U)U-' f (3)

The Lagrangian of the YMT is the obviously invariant

CYMT = di@ + ([D,, DvI[D', Dull (4) 29

where $J is the fermion field matrix. Expanding:

CYMT = di(B + d)$J + (8[,AU] + [A,, A,])', 29

( 5 )

where the trace with the tilde G is over the S U ( N ) group generators. We shall use a trace without tilde when it applies to Dirac matrices. Notice that in (4) it seems that there are partial derivatives acting to the right, but that is actually not true.

We now generalize the covariant derivative in the spinorial representation with the following Theorem. Let D, = 8, + B,, where B, is a vector field (either abelian or nonabelian). Then:

F,,FPU = ((d[,BU]) + [B,,BUl)' (6) 1 1 8 2

= - tr' g' - - t rg4.

Using the Theorem we can write the kinetic energy of a YMT in the spinorial representation of the Lorentz group:

1 - -tr (d~,A,] + [A, ,A,]) 29'

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The proof has been given in previous papers.

scalar part 'p)

We proceed to define the extended fields Y (with a vector part A P &

'Y' 4 + = Y ~ A , + y5'p (8 )

T + uYu-l - (gu)v-'.

where A, = igAfT" and 'p = -g'paTa. We require these dynamical fields to obey the gauge transformation law

(9)

D=B+'Y ' , (10)

The covariant derivative is defined to be

and it is easy to show that it has to transform as D --t UDU-l. Finally, we define the gauge invariant Lagrangian of a Yang-Mills theory with extended fields as:

where the @ are the fermion fields. The expansion of this Lagrangian results in:

L=$(i8 + 4)@ + ((d[,A,] + [A,, (12) 29

- g&y5v"Ta@ + -$ ((8,~ + [A,, d2) The first line is a Yang-Mills theory, the next term is Yukawa and the next is the usual kinetic energy of the scalar bosons in a Yang-Mills theory.

A GUTS EXAMPLE. As an example of this type of theory, let us construct a grand unified version. In SU(5) GUTs, the vector bosons are in a 24 and the fermions in a 5* and a 10. The Higgs bosons for the large mass generation are in another 24 and for the small mass in a 5. A total of 5 irreps. For our example we make the choice of using SU(6) a5 a grand unification group. All the bosons go in the adjoint, a 35, so that the covariant derivative looks like

D = g + Y = g + ( 3 5 ) .

In SU(5) GUTs the vector bosons occupy a 24 distributed as follows:

(13)

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But our 35 is a 6 x 6 matrix of bosons, each component both vector and scalar. We can generate mass to the vectors (but not to the scalars!) through the term

if we assume a scalar field has a nonzero VEV. The group SU(6) has five diagonal generators Ta, a = 1 , . . . , 5 that give the quantum numbers of the particles by taking the commutator [T",T]. We choose the following unnormalized forms for them, consistent with the assignments of matrix (13):

T5 = diag( 1, 1, 1,-1,-1,-1) } UUO)

The first four are completely determined by the quantum numbers of the Standard Model. The hypercharge quantum number comes from the T4. We call the quantum number due to the T5 ultracharge.

Take the scalar field of the fourth generator p4 = v + (P'~, and study mass generation in the adjoint. The fields left without a large mass are:

where p is the scalar part (only) of an extended field. This result is promising because the Higgs doublet from the Standard Model (underlined) a p pears naturally. The problem is that there are too many massless scalars left.

The group SU(6) has several maximal subalgebras but only one leads to acceptable results. There is no space to list the problems with other maximal subalgebras; we simply concentrate on the only one that works:

SU(6) 3 SU(3) x SU(2) x Uy(1) x Uu(1). (16)

This is a bona bona fide maximal subalgebra, although it is no usually listed because of its two U(1)'s. Masses from VEV's are generated through the

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Lagrangian term in ( l l ) ,

The new field that has a VEV has to be a vector because otherwise the scalar bosons receive no mass. Thus there is going to be some degree of breaking of the Lorentz symmetry. Analysis shows it is not much for a large VEV. Some superstring theories also involve this type of symmetry breaking. V. Kostelecky, S. Samuel and others have been studying this the past few years.

We have little option as to which vector can have the VEV: it has to be the one associated with ultracharge generator T5 or we would inadvertently give very large masses to phenomenological particles. The mass generated goes as the square of the VEV so it makes no difference if the VEV belongs to a space or timelike component of the vector. After taking A: = w p +A: (for some specific p ) and generating new masses the boson matrix looks like

' r r r x x r r r x x r r r x x x x x r r x x x r r

T x x x c o w

X

ij where the x means that all components, vector and scalar, of the extended field there have acquired large masses. The meaning of the box in the matrix above is explained next.

We must now go to the unitary gauge, which in this case means to appropriately fix the degrees of freedom of the SU(6) gauge group that are not used in SU(3) x SU(2) x Uy(1) x Uu(1). Here we have to be careful because we cannot rotate any of the fields enclosed in the box in the matrix above. The reason is that in this model the Higgs doublet is part of the choice of gauge. Any such rotation would mix the Higgs doublet with vectors and we would obtain a form of the Standard Model that, while totally equivalent, is not the one we are used to. The only rotations allowed to us are the ones which do not involve the box: they are precisely those of SU(5) which are not part of SU(3) x SU(2) x Uy(1). They consist of 12 degrees of freedom, and we can use them to rotate away the 12 unphysical scalar bosons that form the scalar part of the extended fields that make up the SU(3) x SU(2) x Uy(1) group left. As a result we get the 12

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vector bosons and the Higgs doublet of the Standard Model. Finally, the reader can verify that the extended field that corresponds to T5 does not interacting with any of the other particles.

THE FERMIONS. In SU(6) there is a 15 that has the branching rule 15 = 5 + 10. Notice that this differs from the usual SU(5) assignation which is for 5 * ~ and 1 0 ~ . The fermions for the 5 have to be charge- conjugated in order to have the correct quantum numbers and thus this irrep becomes a 5R. As a result the fermion multiplet $J has mixed chirality. On the other hand, the resulting boson matrix has both vectos and scalars It is an interesting fact that the different chiralities mesh nicely with the scalar and vectors fields of the boson matrix in the term qT+ = q(d+@)+ and give precisely the terms of the Lagrangian of the Standard Model, with the right chiralities and quantum numbers.

GOOD FALLOUT. In this model the triplet-doublet problem of GUTs and SUSY GUTs does not appear at all, as the Higgs 5 is naturally split in two, with the color triplet acquiring a large mass. In GUTs it is not possible to do this because the large mass generation occurs in the scalar 24 and so it does not affect the color triplet which is in another irrep, the scalar 5, while in this model both occur in the boson 35.

Using only two terms in the Lagrangian and two irreps one can obtain the Standard Model in a particular gauge of SU(6) , but with the advantage that scalars and vectors enter on an equal footing. This gives the scalars stability in their mass since one has charge conservation and Ward identities to maintain masslessness in the quantum loop corrections. The good results basically come from the fact that SU(6) is a larger symmetry than SU(5) , and it can be used to make improvements. Notice too that quartic scalar couplings cannot exist.

OTHER COMMENTS. There is no mechanism to generate nonzero VEVs; this has to come from outside this model. A small breaking of Lorentz symmetry is predicted, its magnitude depending on the size of the vector’s VEV.

References 1. M. Chaves, “Yang-Mills theories using only extended fields (vectorial and

scalar) as gauge fields”, in “Proceedings of the Tenth Marcel Grossmann Meet- ing on General Relativity”, ed. M. Novello, s. Perez-Bergliaffa & R. Ruffini (World Scientific 2005), and references therein.

OPENING THE WINDOW FOR TECHNICOLOR

DEOG KI HONG* Center for Theoretical Physics,

Massachusetts Institute of Technology, Cambridge, MA 02139, USA

E-mail: dkhong@pusan. ac. kr, dlchong@lns. mat. edu

STEPHEN D. H. HSU Department of Physics,

University of Oregon, Eugene OR 97403-5203, U.S.A. E-mail: [email protected]

FRANCESCO SANNINO The Niels Bohr Institute,

Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark E-mail: [email protected]

Recently a new class of technicolor models are proposed, using technifermions of symmetric second-rank tensor. In the models, one can make reasonable estimates of physical quantities like the Higgs mass and the size of oblique corrections, using a correspondence to super Yang-Mills theory in the Corrigan-Ramond limit. The models predict a surprisingly light Higgs of mass, r n ~ = 150 - 500 GeV and have naturally small S parameter.

The standard model for the interaction of elementary particles has so far passed all experimental tests. Its gauge structure, SU(3) , x s U ( 2 ) ~ x U ( l ) y is extremely well tested and its flavor structure is measured precisely. Hence, we now firmly believe that the standard model is the correct theory for elementary particles at the shortest distance we have ever explored, though the Higgs, introduced in the standard model to account for the electroweak symmetry breaking, is yet to be found.

*On leave from Department of Physics, Pusan National University, Pusan 609-735, Korea

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The Higgs, which is the only undiscovered particle in the standard model, poses therefore pressing challenges for both theorists and experi- mentalists. As a scalar particle, its mass is extremely sensitive to the scale of new physics or the ultraviolet cutoff, A, of the standard model. By naive dimensional analysis the Higgs mass is given as

m$ = cA2,

where the dimensionless constant c is O( 1) and depends logarithmically on A. If the scale of new physics is higher than 100 TeV, it requires a severe fine-tuning, c 5 lod4 to get a light Higgs, mH < 1 TeV, as needed for perturbative unitarity of the standard model.

The fine-tuning problem associated with the Higgs, known as the hierarchy problem, has been one of the most fundamental problems in particle physics since the seventies. The earliest attempt to solve the hierarchy problem was to introduce a new strong interaction, Technicolor, that breaks the electroweak symmetry dynamically at ATC N 1 TeV where the new interaction becomes strong. The Higgs is then a composite particle, made of strongly bound technifermions. Another attempt was made soon after by supersymmetrizing the standard model2. The supersymmetric extension of the standard model was accepted quickly due to the fact that it is perturbative, consistent with experimental data 3 , and furthermore it indicates gauge coupling unification '. (See for a drastically different view on the fine-tuning problem .)

On the other hand, technicolor models have been largely abandoned, though several interesting ideas were introduced in recent years '. It is often claimed in the literature that technicolor is ruled out by electroweak precision data. However, the real killer of technicolor is not the electroweak precision data but rather our ignorance of strong dynamics and thus inability to make a systematic and precise estimate of physical quantities like the Higgs mass or the oblique corrections to the electroweak observables. Lacking a reliable means to solve strongly interacting systems, analyses have been made in analogy with QCD, and use the experimental hadronic data to study technicolor models. For instance, if one uses for technicolor a scaled-up version of QCD, the S-parameter of the oblique corrections will be given as

S M 0.11 NTC N o , (2)

where NTC is the number of technicolors and ND is the number of s U ( 2 ) ~ technifermion doublets. Since experimentally the S parameter from physics

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beyond standard model lo is measured to be S = -0.13 f 0.10, the QCD- like technicolor model is roughly ( NTC NO + 1) 0 away from the electroweak precision data, indicating that technicolor models with many electroweak doublets are presumably ruled out.

However, the simple QCD-like technicolor models are already ruled out by the constraint on the flavor-changing neutral currents when one tries to explain the fermion masses with additional strong interactions (ETC) l l .

The smallness of the flavor-changing neutral currents as in the mass difference in KL and Ks requires the ETC scale to be larger than lo3 TeV for QCD-like models, which would then lead to too small fermion masses. To solve this problem, technicolor models with a quasi IR-fixed point, called walking technicolor 12, were suggested, where technifermion bilinear o p erators have anomalous dimension, (̂m 21 1, allowing lower ETC scales AETC N 1OOTeV. Furthermore, contributions to the S parameter will be somewhat reduced due to walking 13.

The p function may be expanded in powers of coupling a.

We need b > 0 to be asymptotically free and c < 0 for a IR fixed point. Then IR fixed point is approximately a* N -b/c. To break electroweak symmetry, the critical coupling for chiral symmetry breaking, which in the ladder approximation is a, N n/(3Cz(R)), should be smaller than the coupling at the IR fixed point. (C2(R) is the eigenvalue of the quadratic Casimir operator for fermions in a representation R.) For a SU(N) technicolor gauge theory with N f technifermions we have listed b, c, and the critical couplings for the chiral symmetry breaking in Table 1. We see that the critical number of flavors for a S U ( N ) technicolor gauge theory to have a walking coupling constant is N f N 4N if technifermions are in the fundamental representation, while N f = 2 if in the second rank symmetric tensor representation (See Table 1). The most economic way to have a walking coupling is therefore to introduce a technifermion doublet in the second-rank symmetric tensor representation or S-type for SU(N)TC with N 5 5 14.

Unlike previous technicolor models, new technicolor models 14J5, where technifermions are in the second-rank symmetric tensor representation, allow systematic estimates of the Higgs mass l5 and other physical observables 16. This is possible because in the large N Corrigan-Ramond limit 17, it is mapped into super Yang-Mills when N j = 1 18. Not only one can

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Table 1. S U ( N ) T ~ technicolor with N f technifermions in either fundamental rep resentation or in second rank tensor representation. The upper sign is for the symmetric tensor and the lower sign is for the antisymmetric tensor.

11N - 2N

-3(N - 1 / N ) N -6(1 l / N ) ( N f 2)2N

2 w [ 3 ( ~ f 2 ) ~ F 111 N f N 4N Nf = 2 for S-type and N 5 5

export the exact results established in super Yang-Mills to nonsupersym- metric theories by considering 1 / N corrections l9 but one can also make relevant predictions about, previously unknown, nonperturbative aspects of super Yang-Mills 20.

In the large N limit, the Higgs particle is identified with the scalar fermion-antifermion state whose pseudoscalar partner in ordinary QCD is the 7'. The low lying bosonic sector contains precisely a scalar and a pseudoscalar meson. In the supersymmetric limit we can relate the masses to the fermion condensate (Tq) e ( $ i p j } q i i , j l ) 19:

with (qq) = 3NA3 and A the invariant scale of the theory:

16n2 -8T2 A3 = p3 ( 3 N g 2 ( p ) ) [ N g 2 ( p 2 ) ] ' (4)

We have also defined & = a [ 9 / ( 3 2 ~ ~ ) ] ' / ~ . The unknown 0 ( 1 ) numerical parameter & is the coefficient of the Kahler term in the Veneziano- Yankielowicz effective Lagrangian describing the lowest composite chiral superfield. By analogy with QCD, we take & N 1 - 3. Then, the Higgs mass is estimated as m H = M 11 150 - 500 GeV . Here we have chosen A = ATC N 250 GeV. The 1/N corrections to the Higgs mass can be made systematically 19715. Similarly, the oblique corrections are calculated precisely in the large N limit 16 .

In conclusion, the newly proposed technicolor models have overcome the typical barrier of technicolor theory, which is the inability to calculate precisely physical quantities like the Higgs mass or S parameters. This is possible due to their correspondence to Super Yang-Mill theories in the large NTC limit. Somewhat surprisingly the new technicolor models naturally produce light composite Higgs bosons. The models are also nearly

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conformal with the minimal number of flavors, making the S parameter naturally small.

Acknowledgments

The work of D.K.H. is supported by Korea Research Foundation Grant (KRF-2003-041-COOO73) and also in part by funds provided by the U.S. De- partment of Energy (D.O.E.) under cooperative research agreement #DF- FC02-94ER40818. The work of S.H. was supported in part under DOE contract DEFG06-85ER40224.

References 1. L. Susskind, Phys. Rev. D 20, 2619 (1979). S. Weinberg, Phys. Rev. D 13,

974 (1976); D 19, 1277 (1979). 2. S. Dimopoulos and H. Georgi, Nucl. Phys. B 193, 150 (1981); S. Dimopoulos,

S. Raby and F. Wilczek, Phys. Rev. D 24, 1681 (1981). 3. J. Ellis, “Summary of ICHEP 2004,” arXiv:hep-ph/0409360. 4. J. R. Ellis, S. Kelley and D. V. Nanopoulos, Phys. Lett. B 249, 441 (1990); 5. C. Giunti, C. W. Kim and U. W. Lee, Mod. Phys. Lett. A 6, 1745 (1991);

U. Amaldi, W. de Boer and H. Furstenau, Phys. Lett. B 260, 447 (1991). 6. N. Arkani-Hamed and S. Dimopoulos, arXiv:hep-th/0405159. 7. See for a recent review C. T. Hill and E. H. Simmons, Phys. Rept. 381, 235

(2003) [Erratum-ibid. 390, 553 (2004)j. 8. D. C. Kennedy and B. W. Lynn, Nucl. Phys. B 322, 1 (1989); M. E. Peskin

and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990). 9. M. E. Peskin and T. Takeuchi, Phys. Rev. D 46, 381 (1992). 10. S. Eidelman et aZ.,Phys. Lett. B 592, 1 (2004). 11. S. Dimopoulos and L. Susskind, Nucl. Phys. B 155, 237 (1979); E. Eichten

and K. D. Lane, Phys. Lett. B 90, 125 (1980). 12. K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev. Lett. 56, 1335

(1986); T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett. 57, 957 (1986); B. Holdom, Phys. Rev. D 24, 1441 (1981); A. G. Cohen and H. Georgi, Nucl. Phys. B 314, 7 (1989).

13. R. Sundrum and S. D. H. Hsu, Nucl. Phys. B 391, 127 (1993);T. Appelquist and F. Sannino, Phys. Rev. D 59, 067702 (1999).

14. F. Sannino and K. Tuominen, arXiv:hep-ph/0405209. 15. D. K. Hong, S. D. H. Hsu and F. Sannino, Phys. Lett. B 597, 89 (2004). 16. D. K. Hong, S. D. H. Hsu, F. Sannino, and K. Tuominen, in preparation. 17. E. Corrigan and P. Ramond, Phys. Lett. B 87, 73 (1979). 18. A. Armoni, M. Shifman and G. Veneziano, arXiv:hepth/0403071; Nucl.

19. F. Sannino and M. Shifman, arXiv:hepth/0309252. 20. A. Feo, P. Merlatti and F. Sannino, arXiv:hep-th/0408214. To appear in

Phys. B 667, 170 (2003);Phys. Rev. Lett. 91 (2003) 191601.

Physics Review D.

THE HIERARCHY PROBLEM AND AN EXOTIC BOUND STATE

C. D. FROGGATT Department of Physics and Astronomy, University of Glasgow,

Glasgow G12 SQQ, Scotland

The Multiple Point Principle, according to which there exist many vacuum states with the same energy density, is put forward as a fine-tuning mechanism. By assuming the existence of three degenerate vacua, we derive the hierarchical ratio between the fundamental (Planck) and electroweak scales in the Standard Model. In one of these phases, 6 top quarks and 6 anti-top quarks bind so strongly by Higgs exchange as to become tachyonic and form a condensate. The third degenerate vacuum is taken to have a Higgs fieid expectation value of the order of the fundamental scale.

1. Introduction

The hierarchy problem refers to the long-standing puzzle of why the electroweak scale is so very small compared to the fundamental scale Pfundamenta l , which we shall identify with the Planck scale PPlanck. In particular, radiative corrections to the Standard Model (SM) Higgs mass diverge quadratically with the SM cut-off scale A; this is the so-called technical hierarchy problem. For example the top quark loop contribution to the SM Higgs mass is given by:

This leads to a fine-tuning problem for A > 1 TeV, when the top quark loop contribution exceeds the physical SM Higgs mass. For a SM cut-off at the Planck scale A = pplalzck - lo1’ GeV, the quadratic divergencies have to be cancelled to more than 30 decimals at each order in perturbation theory.

The most popular resolution of this technical hierarchy problem is to introduce Supersymmetry or some other new physics (e.g. technicolor or a little Higgs model) at the TeV scale. Although SUSY stabilizes the hierarchy between the electroweak and Planck (or GUT) scales, it does not

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explain why the hierarchy exists in the first place. An alternative way to resolve the hierarchy problem is to accept the necessity for fine-tuning and to explicitly introduce a fine-tuning mechanism. The most well-known example is the anthropic We shall discuss here another fine-tuning mechanism: the Multiple Point Principle. We shall apply it to the pure SM, with no new physics below the Planck scale except presumably for a minor modification at the see-saw scale to generate neutrino masses.

2. Multiple Point Principle and the Large Scale Ratio

According to the Multiple Point Principle3, Nature chooses the values of coupling constants in such a way as to ensure the existence of several degenerate vacuum states, each having approximately zero value for the cosmological constant. This fine-tuning of the coupling constants is similar to the fine-tuning of the intensive variables temperature and pressure at the triple point of water, due to the co-existence of the three degenerate phases: ice, water and wpour. The triple point of water is an easily reproducible situation and occurs for a wide range of the fixed extensive quantities: the volume, energy and number of moles in the system.

We do not really know what is the dynamics underlying the Multiple Point Principle, but it is natural to speculate that by analogy it arises from the existence of fixed, but fine-tuned extensive quantities in the Universe, such as

and J

where 4(x) is the SM Higgs field. Such fixed extensive quantities, having the form of reparameterisation invariant integrals over space-time4 Ii = S d 4 x ~ C i ( x ) , can be imposed by inserting &functions in the Feynman path integral, similar to the energy fixing 6-function in the partition function for a microcanonical ensemble in statistical rne~han ics~?~ . Then the coefficient or coupling constant multiplying Ii in the effective action is constrained to lie in a very narrow range, analogous to the inverse temperature in the canonical ensemble for a macroscopic system with a fixed energy. The coupling constant acts as a Lagrange multiplier, which has to adjust itself to ensure that the extensive quantity Ii takes on its correct fixed value. There is then a generic possibility that, for a large range of values for li, the Universe has to contain two or more degenerate phases in different space-time regions. The imposition of a fixed value for an extensive quantity is a non-local condition, which seems to imply some mildly

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non-local physics, such as wormholes or baby universes7, must underlie the multiple point principle3y5i6. However we emphasize that the multiple point principle really has the status of a postulated new principle.

We now wish to apply this fine-tuning principle to the problem of the huge scale ratio between the Planck scale and the electroweak scale: pPlanck/pweak - It is helpful at this point to recall that another l a~ge scale ratio, pplanck/hQCD - lo2', is generally considered to be a natural consequence of the SM renormalisation group equation (RGE) for the QCD fine structure constant:

Then taking f23(/.Lplanck) 2 1/50, which corresponds to an order of unity value for the coupling constant at the Planck scale g3(pPlanck) N 1/2, the RGE gives pPlanck/AQCD = e x p ( 2 ~ / 7 a 3 ( p p l ~ ~ ~ k ) ) N exp(45). A full understanding would of course require a derivation of the value g3(pplanck) N

1/2 from physics beyond the SM, such as is done in the family replicated gauge group model8.

our proposed explanation for the large scale ratio p f undarnental/pweak

is similarly based on the use of the RGE for the running top quark Yukawa coupling gt(p) in the SM:

Here g3, g2 and g1 are the S U ( 3 ) x SU(2) x U(1) running gauge coupling constants, which we shall consider as given. The multiple point principle is used to fine-tune the boundary values of gt(p) at both the fundamental and weak scales, due to the existence of 3 degenerate SM vacua. Note that we do use the physical top quark mass as an input. These boundary values, gt(pfundamental) and gt(pweak), then fix the amount of running needed from the RGE (4) and hence the required scale ratio pfundamental/pweak-

3. Two degenerate minima in the SM effective potential

In order to fine-tune the value of gt(pfundamental) using the multiple point principle, we postulate the existence of a second degenerate v a c u ~ m ~ ~ ~ , in which the SM Higgs field 4 has a vacuum expectation value of order pfundamental. This requires that the renormalisation group improved effective potential Vefp(4) should have a second minimum near the fundamental

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scale, where the potential should essentially vanish in order to be degenerate with the usual electroweak scale minimum.

For large values of the SM Higgs field 4 - p f undamental >> Pweak, the renormalisation group improved effective potential is well approximated by

and the degeneracy condition means that A(p fundamenta l ) should vanish to high accuracy. The effective potential Vef f must also have a minimum and so its derivative should vanish. Therefore the vacuum degeneracy requirement means that the Higgs self-coupling constant and its beta function should vanish near the fundamental scale:

A(pfundamenta1) = @A(pfzlndamental) = 0 (6)

This leads to the fine-tuning condition5

relating the top quark Yukawa coupling gt(p) and the electroweak gauge coupling constants gI (p ) and gz(p) at p = pfundamenta l . we must now input the experimental values of the electroweak gauge coupling constants, which we evaluate at the Planck scale using the SM renormalisation group equations, and obtain our prediction:

gt ( p f undarnental) 2 0.39. (8 )

However we note that this value of g t (p fundamen ta l ) , determined from the right hand side of Eq. (7), is rather insensitive to the scale, varying by approximately 10% between p = 246 GeV and p = lo1' GeV.

4. Three degenerate vacua and the exotic bound state

We now want to fine-tune the value of g t (pWeak) using the multiple point principle. In order to achieve this, it is necessary to have 2 degenerate vacua which only deviate by their physics at the electroweak scale. So what could the third degenerate SM vacuum be? Different phases are most easily obtained by having different amounts of some Bose-Einstein Condensate. We are therefore led to consider a condensate of a bound state made out of some SM particles. We actually propose8?'' a new exotic strongly bound state made out of 6 top quarks and 6 anti-top quarks - a dodecaquark! The reason that such a bound state was not considered previously is that

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its binding is based on the collective effect of attraction between several quarks due to Higgs exchange.

The virtual exchange of the Higgs particle between two quarks, two anti-quarks or a quark anti-quark pair yields an attractive force in each case. For top quarks Higgs excahnge provides a strong force, since we know phenomenologically that gt(p) - 1. So let us consider putting more and more t and t quarks together in the lowest energy relative S-wave states. The Higgs exchange binding energy for the whole system becomes proportional to the number of pairs of constituents, rather than to the number of constituents. So a priori, by combining sufficiently many constituents, the total binding energy could exceed the constituent mass of the system! However we can put a maximum of 6t + 61 quarks into the ground state S-wave. So we shall now estimate the binding energy of such a 12 particle bound state.

As a first step we consider the binding energy El of one of them to the remaining 11 constituents treated as just one particle analogous to the nucleus in the hydrogen atom. We assume that the radius of the system turns out to be reasonably small, compared to the Compton wavelength of the Higgs particle, and use the well-known Bohr formula for the binding energy of a one-electron atom with atomic number 2 = 11 to obtain the crude estimate:

Here gt is the top quark Yukawa coupling constant, in a normalisation in which the top quark mass is given by mt = gt 174 GeV.

The non-relativistic binding energy Ehnding of the 12 particle system is then obtained by multiplying by 12 and dividing by 2 to avoid double- counting the pairwise binding contributions. This estimate only takes account of the t-channel exchange of a Higgs particle between the constituents. A simple estimate of the u-channel Higgs exchange contributions increases the binding energy by a further factor of (16/11)2, giving:

We have so far neglected the attraction due to the exchange of gauge particles. So let us estimate the main effect coming from gluon exchangelo with a QCD fine structure constant cx,(Mz) = gz(Mz)/47r = 0.118, corre-

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sponding to an effective gluon t - i coupling constant squared of: 4 4

3 e;t = 3g: N -1 .5 N 2.0

For definiteness, consider a t quark in the bound state; it interacts with 6 ? quarks and 5 t quarks. The 6 ? quarks form a colour singlet and so their combined interaction with the considered t quark vanishes. On the other hand the 5 t quarks combine to form a colour anti-triplet, which together interact like a 5 quark with the considered t quark. So the total gluon interaction of the considered t quark is the same as it would have with a single f quark. In this case the u-channel gluon contribution should equal that of the t-channel. Thus we should compare the effective gluon coupling strength 2 x e:t N 2 x 2 = 4 with ( 1 6 / 1 1 ) x 2g ,2 /2 N 16 x 1 . 0 / 2 = 8 from the Higgs particle. This leads to an increase of Eknding by a factor of (y)2 = ( 3 / 2 ) 2 , giving our final result:

Ebinding = (s) 999," mt

We are now interested in the condition that this bound state should become tachyonic, miound < 0 , in order that a new vacuum phase could appear due to Bose-Einstein condensation. For this purpose we consider a Taylor expansion in g: for the mass squared of the bound state, crudely estimated from our non-relativistic binding energy formula:

= ( 1 2 m t ) 2 ( 1 -

Assuming that this expansion can, to first approximation, be trusted even for large g t , the condition miound = 0 for the appearance of the above phase transition with degenerate vacua becomes to leading order:

&]phase transition = ( g)1'4 N 1.24

We have of course neglected several effects, such as weak gauge boson exchange, s-channel Higgs exchange and relativistic corrections. In particular quantum fluctuations in the Higgs field could have an important effectlo in reducing gtlphase transition by up to a factor of a. It is therefore quite possible that the value of the top quark running Yukawa coupling constant, predicted from our vacuum degeneracy fine-tuning principle, could be in agreement with the experimental value gt(pweak)ezp M 0.98 f 0.03.

331

M, = 173 GeV -

M, = 135 GeV-

Assuming this to be the case, we can now estimate the fundamental to weak scale ratio by using the leading order RGE (4) for the top quark SM Yukawa coupling gt(p). It should be noticed that, due to the relative smallness of the fine structure constants ai = gp/41r and particularly of Qg(,%fundamental), the beta function Ps, for the top quark Yukawa coupling constant, Eq. (4), is numerically rather small at the fundamental scale. Hence we need many e-foldings between the two scales, where gt(pfundamenta1) 21 0.39 and gt(pweak) EZ 1.24. The predicted scale ratio is quite sensitive to the input value of Q3(,%fundamental). When we input the value of a3 N 1/54 evaluated at the Planck scale, from the phenomenological value of AQCD using the RGE for the SM fine structure constants, we predict the scale ratio to be:

p fundamental/pweak - lo2' (16)

M, = 173 GeV M, = 135 G e V -

0.2

0.0

Figure 1. Plots of gt and X as functions of the scale of the Higgs field q5 for degenerate vacua with the second Higgs VEV at the Planck scale q5vacuum2 = 1019 GeV. The second order SM renormalisation group equations are formally applied up to a scale of loz5 GeV.

- - -

" " I " " I "

The running of the top quark Yukawa coupling is shown in Figure 1 as a function of log,,, 4. We note that, as can be seen from Eq. (4), the rate of logarithmic running of gt(p) increases as the QCD gauge coupling constant g3(p) increases. Hence the value of the weak scale is naturally fine-tuned to be a few orders of magnitude above the QCD scale. Using the RGE for

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the SM Higgs self-coupling

- = - [ 12x2 + 3(4g," - 392" - gpp + -g2 + p g 1 + ,91 - dX 1 9 4 3 2 2 3 4 d lnp 16r2 4

(17) and the boundary value at the fundamental scale, Eq. ( 6 ) , we can calculate the running of X(p). The results are also shown in Figure 1. The value of X(pweak) obtained can be used to predict5 the SM H i s s mass

M H = 135 f 9 GeV (18)

5. Properties of the exotic bound state

Strictly speaking, it is a priori not obvious within our scenario in which of the two degenerate electroweak scale vacua discussed in Section 4 we live. There is however good reason to believe that we live in the usual Higgs phase without a condensate of new bound states rather than in the one with such a condensate. The point is that such a condensate is not invariant under the SU(2) x U(1) electroweak gauge group and would contribute to the squared masses of the W* and Zo gauge bosons. Although these contributions are somewhat difficult to calculate, preliminary calculations indicate that these contributions would make the pparameter deviatell significantly from unity, in contradiction with the precision electroweak data.

We expect the new bound state to be strongly bound and relatively long lived in our vacuum; it could only decay into a channel in which all 12 constituents disappeared together. The production cross-section of such a particle would also be expected to be very low, if it were just crudely related to the cross section for producing 6 t and 6 ? quarks. It would typically decay into 6 or more jets, but it would probably not be possible to reconstruct the multi-jet decay vertex precisely enough to detect its displacement from the bound state production vertex. There would be a better chance of observing an effect, if we optimistically assume that the mass of the bound state is close to zero (i.e. very light compared to 12mt M 2 TeV) even in the phase in which we live. In this case the bound state obtained by removing one of the 12 quarks would also be expected to be light. These bound states with radii of order l/mt might then be smaller than or similar in size to their Compton wavelengths and so be well described by effective scalar and Dirac fields respectively. The 6 t + 6 ? bound state would couple only weakly to gluons whereas the 6t + 5 5 bound state would be a colour triplet. So the 6t + 5 ? bound state would

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be produced like a fourth generation top quarka at the LHC. If these 11 constituent bound states were pair produced, they would presumably decay into the lighter 12 constituent bound states with the emission of a t and a T quark. The 6t + 6T bound states would in turn decay into multi-jets, producing a spectacular event.

6. Summary and Conclusion

In this talk, we have put forward a scenario for how the huge scale ratio between the fundamental scale p fundamental and the electroweak scale ,uw,,k

may come about in the pure SM. We appeal to a fine-tuning postulate - the Multiple Point Principle - according to which there are several different vacua, in each of which the energy density (cosmological constant) is very small. In fact our scenario requires a landscape of 3 degenerate SM vacua, in contrast to the

The existence of an exotic bound state of six top quarks and six anti-top quarks is crucial to our scenario. Furthermore the binding of this dodecaquark state, due mainly to Higgs particle exchange, must be so strong that a condensate of such bound states can form and make up a phase in which essentially tachyonic bound states of this type fill the vacuum. The calculation of the critical top quark Yukawa coupling gtlphase transition

for which such a vacuum should appear involves no fundamentally new physics. It is a very difficult SM calculation, but would provide a clean test of the Multiple Point Principle a~ gtlphase transition is predicted to equal the experimentally measured value gt (hweak)ezp . Within the accuracy of our crude extrapolation (15) of the non-relativistic Bohr formula, our Multiple Point Principle estimate of g t (pweak) is in agreement with experiment.

In addition to the 2 degenerate electroweak scale vacua, we postulate the existence of another degenerate vacuum in which the SM Higgs field has a vacuum expectation value of order the fundamental scale. We thereby obtain a prediction (8) for the value of gt(,ufundamental) in terms of the electroweak gauge coupling constants. The crucial point now is that we need an appreciable running of g t ( p ) , in order to make its fine-tuned values at /J&eak and at flfundamental compatible. That is to say we need a huge scale ratio (16), since the running is rather slow due to the smallness of the SM fine structure constants a i from the renormalisation group point of

or so string vacua1’2.

aHowever there would be very little mixing with the top quark, due to the small overlap of their wave functions.

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view. It also naturally follows that the electroweak scale lies within a few orders of magnitude above RQCD.

Finally we remark that, in our scenario, there are still quadratic divergencies in the radiative corrections to the Higgs mass squared at each order of perturbation theory. However the Multiple Point Principle fine-tunes the bare parameters at each order of perturbation theory, so as to ensure the equality of the energy densities in the three different SM vacua. Indeed we obtain a prediction for the Higgs mass: MH - 135 GeV.

Acknowledgements

I should like to thank Michael Vaughn and the Organising Committee for their warm hospitality at this enjoyable and successful symposium. I should also like to acknowledge helpful comments from Paul Frampton, Peter Hansen and Michael Vaughn. Finally I must mention the many useful discussions with my collaborators Larisa Laperasvili and Holger Bech Nielsen.

References 1. M. Dine, These Proceedings. 2. N. Arkani-Hamed, These Proceedings. 3. D.L. Bennett, C.D. Froggatt and H.B. Nielsen, in Proc. of the 27th Znterna-

tional Conference on High Energy Physics, p. 557, ed. P. Bussey and I. Knowles (IOP Publishing Ltd, 1995); Perspectives in Particle Physics '94, p. 255, ed. D. Klabuear, I. Picek and D. Tadii: (World Scientific, 1995) [arXiv:hep- ph/9504294].

4. C.D. Fkoggatt and H.B. Nielsen, Origin of Symmetries (World Scientific, 1991) 5. C.D. Fkoggatt and H.B. Nielsen, Phys. Lett. B368, 96 (1996) [arXiv:hep

ph/9511371]. 6. C.D. Fkoggatt and H.B. Nielsen, Talk at 5th Hellenic School and Workshop

on Elementary Particle Physics, Corfu, 1995; arXiv:hepph/9607375. 7. S.W. Hawking, Phys. Rev. D37, 904 (1988);

S. Coleman, Nucl. Phys. B307, 867 (1988); ibid B310, 643 (1998). 8. C.D. Froggatt and H.B. Nielsen, Sum. High Energy Phys. 18, 55 (2003)

[arXiv:hepph/0308144]; Pmc. to the Euroconference on Symmetries Beyond the Standard Model, p. 73 (DMFA, Zaloznistvo, 2003) [arXiv:hepph/0312218].

9. C.D. Fkoggatt, H.B. Nielsen and Y. Takanishi, Phys. Rev. D64, 113014 (2001) [arXiv:hepph/0104161].

10. C.D. Froggatt, H.B. Nielsen and L.V. Laperashvili, To be published in the Proceedings of the Coral Gable Conference on High Energy Physics and COS- mology, Florida, 2003; arXiv:hep-ph/0406110.

11. M. Veltman, Phys. Lett. B391, 95 (1980).

CP, CPT and Lorentz Invariance Violation


RECENT RESULTS FROM BELLE

SHIRO SUZUKI Department of Physics, Saga University,

1 Honjo-machi, Saga 840-8502, Japan E-mail: su~uZZlkisi~cc.saga-u.ac.jp

Recent results on CP violation and heavy flavor physics obtained by the Belle collaboration are reported. Time dependent CP violation in various b -P s decay processes are measured with high luminosity (253ft-l) and the average “sin2&” value is found to be 2.40 away from the world average for b 4 c- decays. Evidence for direct CP violation in Bo + K + T - is found with 3.90 significance. B -P

K(*)l+l- FCNC process is measured with high statistics, and a first look at the forward-backward asymmetry is presented.

1. Introduction

In this paper we focus on recent results from Belle/KEKB on the topics of potential new CP phases from penguin dominated decay processes, new evidence for direct CP violation (DCPV), and flavor changing neutral current (FCNC) rare processes, which are expected to be sensitive to physics beyond the standard model.

KEKB is an asymmetric energy collider of 8 GeV electrons and 3.5 GeV positrons. Belle detector consists of a Si vertex detector with 3(4) layers of DSSD, a small cell central drift chamber, aerogel Cherenkov counters for particle identification (PID), time of flight counters for PID and trigger, and an electromagnetic calorimeter with Tallium doped CsI crystals. Out- side of the solenoid magnet, we have a p / K L catcher. The achieved peak luminosity was 1.39 x 1034cm-2s-1, and integrated luminosity reached 253 fb-l on resonance and 28 fb-’ off resonance. In total we have accumulated 274 milion BB events. This report, is based on the full data sample.

2. Time Dependent CP Violation

The main subject of Belle/KEB is the exploration of the mechanism of CP violation. In the Kobayashi-Maskawa scheme, CP violation is attributed to an irreducible complex phase in the quark mixing matrix (KM matrix).

337

338

The unitarity condition on the KM matrix requires combinations of bilinear products of certain matrix elements to form a closed triangle in the complex plane. The inner angles of this triangle are related to CP violating observables (fig.-1).

/7

Iud'

Figure 1. Unitarity triangle in the p / q plane.

A notable feature of the asymmetric energy B-factory is to measure time dependent CP violation. Interference of two amplitudes, Bo + fcp and BO + Bo + fcp via BO/B" mixing, may result in CP asymmetry of time dependent decay rate. The general form of time dependent asymmetry is expressed as:

21mX A = w, and X = [ f g w (& denotes CP where S = ,X,2+1,

eigenvalue of the final state f ) . The sine term S and cosine term A represent the strengths of mixing induced and direct CP violation, respectively. For the case of a single decay diagram such as Bo + J/$Ks (fig.-2), only the S term contributes and the asymmetry depends on the CP violating angle 41 (S = -[f sin241). Similarly sin241 is measured by b - c d and b + sqq decays, and sin242 will be measured by b + uEd decay if only tree level diagrams contribute.

1x1 +1 P (PCPIBJ

Figure 2. Interference of mixed and unmixed decay diagrams.

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2.1. sinwl in the golden mode

b + CCS decay processes have large branching fractions and provide precise sin241 information. With the data sample of 140 fb-l available in the summer of 2003, a total of 5400 events are used to obtain CP parameters’. The fit gives sin241 = 0.728fO.O56(stat.) fO.O23(syst.) and 1x1 = 1.007f 0.041 f 0.033. The parameter 1x1 is consistent with no direct CP violation (1x1 = 1) in these processes as the SM predicted. Together with high precision measurements from BaBar and others, the present world average for sin241 is 0.736 f 0.049; this is used as the SM reference2.

2.2. “sin2q51” in the Penguin dominated modes

For a stringent test of the SM, sin241 measurements are needed with different decay processes like b -+ ccd and b + sqq. Among these, penguin dominated b + sqq decays are extremely interesting. For Bo 4 q5Ks decay, we will obtain the same results as B + J / $ K s if we assume only SM particles in the penguin diagram contribute. However, we might expect new particles in the loop diagram besides the top quark (fig.S), which may have different CP phases and also give rise to direct CP violation. Any difference in CP asymmetry from the SM will be a hint of new physics.

Figure 3. contribution of new physics (right).

Standard electro-weak penguin diagram for Bo 4 4Ks (left) and possible

In the summer of 2003, Belle presented results3 for Bo + ~ K s , qlKs and K+K-Ks. All are consistent with A = 0 (no direct CP violation). However, the sin241 values for Bo + K+K-Ks and Bo -+ v’Ks are in the range of - 0.43 - 0.50, which are smaller than the b + cCs case, and the value for Bo + 4Ks is -0.96 f 0.50?::;;, which is 3.5 (T away from the SM (0.736 f 0.049). This may be a serious difference, and obviously measurement with higher statistics and additional decay modes are required.

Since last autumn we have accumulated 1lOfb-’ of data (data set II), and the total integrated luminosity reached 253%-l. Before analyzing b 4

340

sqg processes, we have carried out a series of consistency checks between the two data sets. Taking B --* J / @ K o data as the SM control sample, the CP asymmetries were measured separately for data set I and 11. We obtained S = 0.696 f 0.061 and A = 0.011 f 0.043 for data set I, and S = 0.629 f 0.069 and A = 0.035 f 0.044 for data set 11. The results are stable for both data sets. Also cross checks are made for the B meson lifetime in the decays to J / @ K o , 4K+ and +KO, and null asymmetry in the D*lv and #K+ final states. From these checks, the resolution function and flavor tagging properties are shown to be consistent for data sets I and 11.

0.0 < r < 0.5 t E -1

-0.5 -1 -7.5 -5 -2.5 0 2.5 5 7.5

0.5 < r 5 1.0

At (PSI ' 0 0.10.20.3OAO.SO.60.7O.BO.9 1

P; (GeV/c)

Figure 4. Left: beam constrained mass k f b c for B 4 q5Ks (top) and center of mass momentum p& for B + q5K~ candidates (bottom). Right: Raw asymmetry A(t)Cp for q5Ks and ~ K L combined. The solid line is the fit. The dashed line is the SM prediction.

With 274 million BB events, we analysed the processes Bo 4 q!~Ks, ~ K L , r]'Ks, K+K-Ks, W K S , fo(980)Ks and r0Ks. As shown in fig.-4 we have obtained 139 f 14 4Ks events and 36 f 15 ~ K L events. Note that for the new analysis we include the ~ K L process and Ks 4 r0ro decays. The time dependent asymmetry A ( t ) ~ p for @KO (4Ks and ~ K L combined) is also shown in fig.-4. The obtained parameters are

-Ej . S = +0.06 f 0.33 f 0.09 and A = +0.08 f 0.22 f 0.09,

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where the first error is statistical and the second error is systematic. The absolute value of -0 . S parameter (= effective value of sin 241) is smaller and more positive than in the previous data sample, but is still 2 . 2 ~ away form the SM expectation. Inclusion of the ~ K L sample pushes up the value slightly to the positive side, but the statistical contribution is small.

Bo + q'Ks and K+K-Ks processes give high statistics samples, 512f 27 and 399 f 28 events,, respectively (fig.-5). A(t)cp for Bo + q'Ks is quite close to the SM (fig.-6), and the obtained parameters are -<f . S = 0.65f0.18f0.04 and A = -0.08f0.12f0.07. The parameters for K+K-Ks are -[j . S = 0.49 f 0.18 f 0.04+:::,7 and A = -0.19 f 0.11 f 0.05, which are not much different from before. However, -0 . S may be larger (closer to the SM) if CP even fraction of K+K-Ks system is less than 100%.

Figure 5. Mbc for ~ ' K s (left) and K+K-Ks (right).

1

0.5

0 2 Zi -0.5 E 5 -1

$ 1 $ 0.5 c11

0

-0.5

-1 -7.5 -5 -2.5 0 2.5 5 7.5 -7.5 -5 -2.5 0 2.5 5 '

At (PSI At (PSI

5

Figure 6. Raw asymmetry A( t )Cp for T ~ K S (left) and K+K-Ks (right).

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3. Evidence for direct CP violation

Another key prediction of KM model Direct CP violation in decay occurs amplitudes that have different CP partial rate asymmetry;

r(B -, r(B+ ACP =

In the case of B + Kn process, the complex CP phase and penguin

Belle Results on Time-deuendent CP Violation in b-+s

in B -+ K n decay

.s the existence of direct CP violation. by the interference of more than two

arid strong phases, and will be seen as

f ) - r (B + f) f ) + r ( B + f ) ‘

interference of the tree amplitude with amp:.itudes with real phase is expected to

-1.5 -0.9 -0.3 b.3 0.9 1.5

Figure 7. Summary of %ir/291” for b + sqq processes.

All the b -, sqq results from average “sin2&” value for all b +

only). This is 2.4 c away from indicates direct CP violation,

are summarized in fig.-7. The

A parameters, which is +0.43?0,::; (statistical error

for all channels.

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I; €3:

d

Figure 8.

n- -d

Tree

Tree and penguin diagrams

% 500 z 400

UJ Q b 300 E 200

too

0

-

0 I

- b

B: d

that

Penguin

contribute to Bo + K +

% 500 3 !I e! 400 'i: 300 C

200

100

c

K+

7t -

?r- decay.

BE (GeV) AE (GeV)

Figure 9. A E distributions for Bo + K-n+ (left) and Bo + K + s - (right) candidates. The histograms represent the data, while the curves represent the various components from the fit: signal (dashed), continuum (dotted), three-body B decays (dash-dotted), background from mis-identification (hatched), and sum of all components (solid).

give a CP asymmetry (fig.-8). Measurement of Bo + K+n- processes requires suppression of contin-

uum background and good K / n identification. Continuum background was efficiently eliminated using modified Fox-Wolfram moments4. K / r identification efficiency and fake rates are calibrated from real K / n samples obtained from D*+ -, Don+ and subsequent Do + K-n+ decay. Signals for both charge states are shown in fig.-9. We have obtained a signal yield of 2139 f 53 events. The asymmetry value is measured to be

Acp(K+n-) = -0.101 f 0.025 f 0.005.

The effect is appreciable and statistical significance is 3.9~7. We also measured B+ + K+no process, and obtained 728 f 34 events

(fig.-10). The measured asymmetry value is Acp(K+n0) = +0.04 f 0.05 f 0.02, which is consistent with zero. Naively, K+no is supposed to be same as K+n-, but Acp(K+n0) is - 2.40 away from that of K+n-. This may suggest the contribution of large EW penguins diagram in the B+ -, K+ro process.

344

80 : 70 t ! K+XO

AE (GeV) AE (GeV)

Figure 10. dates. The curves are described in the caption of fig.-9

AE distributions for B- -+ K-no (left) and B+ -+ K+ao (right) candi-

4. Radiative and electro-weak penguin processes

t

Ci,d u,d

Figure 11. Electrc-weak penguin and box diagram for FCNC b -.) sl+l- decay.

Another place where we can expect contributions from new physics is FCNC processes suppressed at the tree level SM. These are mediated by electro-weak penguins and box diagrams (fig.-11). b 4 sl+Z- processes have an extra suppression factor of aem compared to b 4 sy processes. The branching fraction is less than Hence, new particles in the loop diagram may make appreciable contribution to decay rates and asymmetries, and these processes are a good testing ground for the standard model and new physics.

The first b 4 sl+Z- processes observed in the Belle experiment were K(*)e+e- and K ( ' ) p + p - exclusive processes5. These are updated with the high statistics sample (fig.-12). Assuming equal production rates for charged and neutral B meson pairs from T(4S) and isospin invariance, the following combined branching fractions are obtained:

B ( B + K1+1-) = (5.50?::7,: f 0.27 f 0.02) x and

B ( B 4 K*l+Z-) = (16.5+::; f 0.9 f 0.4) x

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%

2 10

5 15 9 N . J a,

5

0

20

15

10

5

0

30

20

10

9.2 5.225 5.25 5.275 5.2 5.225 5.25 5.275 5.3 GeV

Figure 12. Mbc distributions for K(*) l+l - ,

where the first error is statistical, second is systematic and the third is model dependence. The statistical significances are 11.00 and 10.10, respectively. B ( B + K*l+l-) is slightly higher than our previous result but these are consistent with the SM model.

The ratio of branching fractions of B + K p + p - and B + Ke+e- ( R K ~ ) is sensitive to neutral Higgs emission from internal loop in the two Higgs doublet model. If the Higgs contribution is sizable the ratio is greater than one. The same ratio R~.11 for B + K*l+l- is sensitive to the photon pole, and predicted to be - 0.75 in the SM. These values are measured to

consistent with the SM. The q2 dependence of the forward-backward asymmetry for B +

K*l+l- carries information on the sign of the Wilson coefficient. Hence it is sensitive to new physics, which may not be seen in other modes like b + sy decays. Fig.-13 shows the first result, compared with the SM prediction and that the sign of C, is reversed. For now the asymmery for K*l+l- is consistent with both SM and the wrong sign C7 case.

be R ~ l l = 1.38-0,41-0,07 +0.39+0.06 and R K . ~ = 0.98'0,:3,: k 0.08. So far they are

346

Figure 13.

q2 GeV2/c2

q2 dependence of AFB for K’1+1-. J / $ and +’ regions are removed.

5. Summary

Based on the data sample of 253fk1, time dependent CP violation for b + sqg penguin processes are measured in various decay modes. The effective ‘‘sin2&’’ value for +KO is updated to be +0.06+0.33f0.09. Compared to the data from 2003, the value shifted closer to the SM, but it is still 2 . 2 ~ away. The average value for all b + s processes is +0.43+,0::; and 2 . 4 ~ away from the SM. Evidence of direct CP violation in Bo --t K+T- is found with 3 . 9 ~ significance. We also find a hint that Acp(K+r-) # Acp(K+7r0). Accumulation of FCNC B + K(*)Z+Z- processes made progress, and the first look at forward-backward symmetry is reported.

We need more data to draw strong conclusions on new physics. Further study with high luminosity is important and a super B-factory project may be needed to find new physics.

References 1. K. Abe et al. (Belle collaboration), HEP-EX/U4U8111 (2004). 2. Particle Data Group, Phys. Letters B 592 1 (2004). 3. K. Abe et al. (Belle collaboration), Phys. Rev. Letters 91, 261602 (2003). 4. G. C. Fox and S. Wolfram, Phys. Rev. Letters 41, 1581 (1978). 5. K. Abe et al. (Belle collaboration), Phys. Rev. Letters 88, 021801 (2002);

A. Ishikawa et al. (Belle collaboration), Phys. Rev. Letters 91, 0261601 (2003).

EFFECTS OF CP PHASES ON THE PHENOMENOLOGY OF SUSY PARTICLES

A. BARTL AND S. HESSELBACH Institut fur Theoretische Physik, Universitat Wien, A-1090 Vienna, Austria

We review our recent studies on the effects of CP-violating supersymmetric (SUSY) parameters on the phenomenology of neutralinos, charginos and third generation squarks. The CP-even branching ratios of the squarks show a pronounced dependence on the phases of At, Ab, p and M I in a large region of the supersymmetric parameter space, which can be used to get information on these phases. In addition we have studied CP-odd observables, like asymmetries based on triple product correlations. In neutralino and chargino production with subsequent three-body decays these asymmetries can be as large as 20 %.

1. Introduction

The Lagrangian of the Minimal Supersymmetric Standard Model (MSSM) contains several complex parameters, which are new sources of CP- violation. In the sfermion sector of the MSSM the trilinear scalar couplings A f and the higgsino mass parameter p can be complex. In the chargino and neutralino sector the parameter p and the U( l ) gaugino mass parameter MI can be complex, taking M2 real.

The phases of the complex parameters are constrained or correlated by the experimental upper limits on the electric dipole moments of electron, neutron and the atoms lg9Hg and '''Tl. In a constrained MSSM the restrictions on the phases can be rather severe. However, there may be cancellations between the contributions of different complex parameters, which allow larger values for the phases l.

The study of production and decay of charginos (2;) and neutralinos (24) and a precise determination of the underlying supersymmetric (SUSY) parameters M I , M2, p and tan ,O including the phases ( P M ~ and 'pp will play an important role at the International Linear Collider (ILC) 2 . In methods to determine these parameters based on neutralino and chargino mass and cross section measurements have been presented. In the impact of the SUSY phases on chargino, neutralino and selectron production has been

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analyzed and significances for the existence of non-vanishing phases have been defined. In CP-even azimuthal asymmetries in chargino production at the ILC with transversely polarized beams have been analyzed.

Concerning the determination of the trilinear couplings A f , detailed studies in the real MSSM have been performed in 6 . In the complex MSSM the polarization of final top quarks and tau leptons from the decays of third generation sfermions can be a sensitive probe of the CP-violating phases 7.

In the effects of the CP phases of A,, p and MI on production and decay of tau sleptons (71~) and tau sneutrinos (6,) have been studied. The branching ratios of ?1,2 and 6, can show a strong phase dependence. The expected accuracy in the determination of A, has been estimated to be of the order of 10 % by a global fit of measured masses, branching ratios and production cross sections. The impact of the SUSY phases on the decays of the third generation squarks has been discussed in 9f10i11 and will be reviewed in Sec. 2.

In order to unambiguously establish CP violation in supersymmetry, including the signs of the phases, a measurement of CP-odd observables is inevitable. T-odd triple product correlations between momenta and spins of the involved particles allow the construction of CP-odd asymmetries already at tree level 12J3. Asymmetries of this kind in scalar fermion decays have been discussed in 14. T-odd asymmetries in neutralino and chargino production with subsequent two-body decays have been analyzed in 15. For leptonic two-body decays asymmetries up to 30 % can occur. CP-odd observables involving the polarization of final T leptons from two-body decays of neutralinos have been studied in 16. T-odd asymmetries in neutralino and chargino production with subsequent three-body decays 17*18 will be discussed in Sec. 3.

2. Decays of Third Generation Squarks

In 9i10i11 we have studied the effects of the phases of the parameters A t , Ab, p and M1 on the phenomenology of the third generation squarks, the top squarks ;1,2 and the bottom squarks b l ~ . We have focused especially on the effects of p~~ and PA^ in order to find methods to determine these parameters. The third generation squark sector is particularly interesting because of the effects of the large Yukawa couplings. The phases of A f and p enter directly the squark mass matrices and the squark-Higgs couplings, which can cause a strong phase dependence of suitable observables. In the case of top squarks the p term in the off-diagonal element of the mass

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B 0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.5 1 1.5 2 0 0.5 1 1.5 2

P A h PA, Jn

Figure 1. (a) Partial decay widths r and (b) branching ratios B of the decays + g f b (solid), ti + gyt (dashed), 21 -+ g z b (dashdotted) and i 1 -+ %$t (dotted) for tanp = 50, Mz = 233.2 GeV, IMlI/M2 = 5/3 tan2 Ow, lpl = 377.0 GeV, IAtJ = 498.9 GeV, vr = ( P M ~ = (PA* = 0, mt, = 530.6 GeV, mZz = 695.9 GeV, mi1 = 606.9 GeV, M o > Mij and mH+ = 416.3 GeV. From g.

matrix is suppressed by 1/ tanp, hence the phase dependence of the decay matrix elements is essentially determined by (PA& in a large part of the SUSY parameter space with lAtl >> Ipl/ tan& This can lead to a strong phase dependence of many partial decay widths and branching ratios.

In the case of bottom squarks the mixing is smaller because of the smaller bottom quark mass. It is only important for large tan@, when the p term in the off-diagonal element of the mass matrix is large. However, in the squark-Higgs couplings the phase PA^ appears independent of the bottom squark mixing. This can lead to a strong PA^ dependence of bottom squark and top squark partial decay widths into Higgs bosons.

We first discuss the PA^ and PA^ dependence of top squark and bottom squark partial decay widths and branching ratios. We have analyzed fermionic decays I$ -, gFq', & + i : q and bosonic decays & + i$H*, & + giW*, & + &Hi, & + &Z of i 1 , 2 and &,2. In the complex MSSM the CP-even and CP-odd neutral Higgs bosons mix and form three mass eigenstates H1,2,3 19. This can also cause a phase dependence of the widths of the decays into Higgs bosons.

In Fig. 1 we show the partial decay widths r and branching ratios B of the 51 . All partial decay widths, especially I ' ( i 1 + g t b ) , have a pronounced (PA& dependence, which leads to a strong PA^ dependence of the branching ratios. This (PA& dependence of the partial decay widths is caused by that of the top squark mixing matrix which enters the respective couplings. In the case of the heavy & many decay channels can be open and can show a strong PA^ dependence.

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Figure 2; (a) Partial decay wjdths r and (b) branching r$%x B of the decays 61 -+ gyb (solid), bl +. 28b (dashed), bl +. H - ~ I (dotted) and bl -+ W - & (dashdotted) for tanp = 30, M2 = 200 GeV, IMlIIMz = 513 tan2 Ow, [ P I = 300 GeV, IAbl = IAtl = 600 GeV, p,, = A, p~~ = ~ / 4 , p~~ = 0, m- = 350 GeV, mi2 = 700 GeV, mil = 170 GeV, M a > M f i and mH* = 150 GeV. From '. b l

In Fig. 2 we show 61 decay widths and branching ratios. Only I'(& +

H - i l ) shows a pronounced PA^ dependence, which leads to a strong PA^ dependence of the branching ratios. This is caused by the PA^ dependence of the H * ~ L ~ R coupling. The other partiai decay widths depend only very weakly on PA^. This is typical for the PA^ dependence of the &,2 decays. Only the partial decay widths into Higgs bosons, I?(& -+ H-fl) for 61

and r(b2 -+ H - ~ I , ~ ) , r(i2 -+ H1,2,3&1) for i2, can show a strong phase dependence for large tan@.

In order to estimate the precision which can be expected in the determination of the underlying SUSY parameters we have made a global fit of the top squark and bottom squark decay branching ratios as well as masses and production cross sections in ll. In order to achieve this the following assumptions have been made: (i) At the ILC the masses of the charginos, neutralinos and the lightest Higgs boson can be measured with high precision. If the masses of the squarks and heavier Higgs bosons are below 500 GeV, they can be measured with an error of 1 % and 1.5 GeV, respectively. (ii) The masses of the squarks and heavier Higgs bosons, which are heavier than 500 GeV, can be measured at a 2 TeV e+e- collider like CLIC with an error of 3 % and 1 %, respectively. (iii) The gluino mass can be measured at the LHC with an error of 3%. - (iv) For the production cross sections u(e+e- -+ & r j ) and a(e+e- -+ &&) and the branching ratios of the & and & decays we have taken the statistical errors, which we have doubled to be on the conservative side. We have analyzed two scenarios, one with small tan p = 6 and one with large tan p = 30. In both scenarios

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we have found that Re(At) and IIm(At)l can be determined with relative errors of 2 - 3 %. For Ah the situation is considerably worse because of the weaker dependence of the observables on this parameter. The corresponding errors are of the order of 50%. For the squark mass parameters A40, MG, M E the relative errors are of order of 1 %, for tan0 of order of 3 % and for the other fundamental SUSY parameters of order of 1 - 2%. In this analysis we have used the tree-level formulae for the top squark and bottom squark decay widths l l . One-loop corrections are discussed in 2o

and the references therein.

3. T-odd Asymmetries in Neutralino and Chargino

We have studied T-odd asymmetries in neutralino l7 and chargino l8 production with subsequent three-body decays

Production and Decay

e+e- + zi + z j + r7.i + zyff(’), (1)

where the full spin correlations between production and decay have to be included 21. Then in the amplitude squared ITI2 of the combined process products like z ~ ~ ~ ~ ~ p ~ p ~ p $ ~ of the momenta p’ of the involved particles appear in those terms which depend on the spin of the decaying neutralino or chargino. Together with the complex couplings these terms can give real contributions to suitable observables at tree-level. Examples are the triple products ?; = ge- . (p ) x P)-,I,> of the initial electron momentum Fe- and the two final fermion momenta jif and f i j ( 1 ) or 12 = jie- . (& x p’r) of the initial electron momentum ge-, the momentum of the decaying neutralino or chargino jicj and one final fermion momentum Sf. With these triple products we define the T-odd asymmetries

where j”lT126Lips is proportional to the cross section n of the process (1). AT is odd under naive time-reversal operation and hence CP-odd, if higher order final-state interactions and finite-widths effects can be neglected.

We first consider neutralino production and subsequent leptonic three- body decay e+e- + 2; + 2; + 2: + zyl+l- and define the triple product ?; = &- . (@”+ x &-) and the corresponding asymmetry AT. Then AT can be directly measured without reconstruction of the momentum of the decaying neutralino. We show in Fig. 3 the asymmetry AT and the corresponding cross section for zyzi production and subsequent decay of zi,

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M2/GeV AT in % MZIGeV 500

400

300

200

100

n

0 100 200 300 400 500 0 100 200 300 400 500 IPIlGeV IPlIGev

Figure 3. Contours (a) of the T-odd asymmetry AT in % and (b) of the cross section u(e+e- -+ %:%: -+ g:g:t!+e-), summed over e = e, p , in fb, respectively, for tan p = 10, mi,. = 267.6 GeV, miR = 224.4 GeV, IMlI/Mz = 513 tan2 Ow, ' p ~ ~ = 0 . 5 ~ and 'pp = 0 with f i = 500 GeV and P,- = -0.8, P,+ = +0.6. The dark shaded area marks the parameter space with m-* < 103.5 GeV excluded by LEP. In the light shaded area

XI the analyzed three-body decay is strongly suppressed because m-o > mz + mny or m-o >mi . From 1 7 .

x 2

x2 R

e+e- + 2: + 2; + 2: + jj:l+l-. As can be seen, asymmetries AT of the order of 10% can be reached in the parameter region where the cross section is of the order of 10 fb. Also for the associated production and decay of 2; and g:, e+e- + 2: + 2; -, 2: + j j :C+C-, the asymmetry AT has values O(10 %) in large parameter regions, where the corresponding cross sections u are of the order of 10 fb. For e+e- --$ 2: + 2; + 2: + z:l+l- we have obtained asymmetries AT M 6 %, however the cross section is only

We have also studied chargino production and subsequent hadronic three-body decay e+e- + 2; + 2: + 2; + 2:Sc. As an example we consider the triple product II = fie- . (9s x fit) and the corresponding asymmetry AT. In this case it is important to tag the c jet to discriminate between the two jets and to measure the sign of 71. For the associated production and decay of 2; and X;, e+e- -, 2; + 2: + 2i + ~ : S C , asymmetries AT of the order of 10% are possible (Fig. 4). In the scenario of Fig. 4 the corresponding cross sections are in the range of 1 - 5 fb. In Fig. 4 (b) it is remarkable that large asymmetries AT % 10% are reached

u ~ l f b .

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AT in % AT in %

0 0.5 1 1.5 2 'PM, /.

0 0.5 1 1.5 2 ' p p h

Figure 4. T-odd asymmetry AT for e+e- + 2; + 2; +. 2; + 2ySc in the scenario Mz = 150 GeV, IMlI/M2 = 5/3 tanZ Ow, = 320 GeV, tan0 = 5, mfi = 250 GeV and maL = 500 GeV with f i = 500 GeV (a) for 'pM = 0 and (b) for 9~~ = 0 and beam polarizations Pe- = -0.8, Pe+ = +0.6 (solid), Pe- = +0.8, Pe+ = -0.6 (dashed).

for almost real parameter p around pp = 7r. In the chargino sector even for the pair production and decay process e+e- --+ 2; + 2; --f 2: + 273c asymmetries AT x 5% can appear, which can only originate from the decay process, because all couplings in the production process are real. This means that the contributions from the decay to AT play an important role in chargino production with subsequent hadronic decays. This can also be seen in Fig. 4 (a), where AT can be large for pw = 0 and p~~ # 0. It is furthermore remarkable that o(e+e- -, 2: + 2: + 17.1 + 27Sc) can be rather large, for example 117 fb in the scenario MZ = 350 GeV, lpl = 260 GeV and the other parameters as in Fig. 4 (a), where AT x 4 %.

If the momentum of the decaying chargino 2: can be reconstructed, for example with help of information from the decay of the 2;, the process e+e- + xi+%: + 2;+x -yC+v can be analyzed, where the chargino decays leptonically. Then the triple product 12 = ge- . (p'-+ x &+) can be used to define AT. For the associated production and decay of 2; and 22, e+e- -, 2; + 2; + 2; + 2ye+v, asymmetries AT 2 20 % can occur. But in the region with largest asymmetries around 1p1 = 320 GeV and M2 = 120 GeV the cross section is very small (c x 0.1 fb). However, for decreasing Ipl the cross section increases and reaches ~7 x 2 fb for 1pl = 220 GeV and M2 = 120 GeV.

x1

4. Conclusions

Using the CP-violating MSSM as our framework we have studied the impact of the complex parameters At , Ab, p and M I on the decays of top

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squarks and bottom squarks. In the case of top squark decays all partial decay widths and branching ratios can have a strong PA^ dependence because of the large mixing in the top squark sector. If tanb is large and decay channels into Higgs bosons are open, top squark and bottom squark branching ratios can show also a strong PA^ dependence. This strong phase dependence of CP-even observables like branching ratios has to be taken into account in SUSY particle searches at future colliders. It will affect the determination of the underlying MSSM parameters. In order to estimate the expected accuracy in the determination of the MSSM parameters we have made a global fit of masses, branching ratios and production cross sections in two scenarios with small and large tanP. We have found that At can be determined with an error of 2 - 3%, whereas the error of Ab is likely to be of the order of 50 %. Furthermore tan p can be determined with an error of 3% and the other fundamental MSSM parameters with errors of 1 - 2%.

The measurement of CP-odd observables is inevitable to unambiguously establish CP violation in supersymmetry. This will allow to determine the phases including their signs. We have studied T-odd asymmetries in neutralino and chargino production with subsequent three-body decays. These asymmetries are based on triple product correlations between incoming and outgoing particles. They appear already at tree-level because of spin correlations between production and decay. The T-odd asymmetries can be as large as 20% and will therefore be an important tool for the search for CP violation in supersymmetry and the unambiguous determination of the phases of the SUSY parameters.

Acknowledgments

A.B. is grateful to the organizers of PASCOS’04/NathFest for creating an inspiring atmosphere at this conference. This work has been supported by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00149 “Physics at Colliders” and by the “Fonds zur Forderung der wissenschaftlichen Forschung” of Austria, FWF Project No. P16592-N02.

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PHENOMENOLOGICAL IMPLICATIONS OF RESONANT LEPTOGENESIS

A. PILAFTSIS' School of Physics and Astronomy, University of Manchester,

Manchester M13 9PL, United Kingdom

Recently, we have presented [hepph/0408103] a variant of resonant leptogenesis in which the baryon asymmetry in the Universe is created by resonant lepton- tebaryon conversion of an individual lepton number, for example that of the 7-

lepton. We briefly outline the phenomenological implications of this scenario for the production of sub-TeV heavy Majorana neutrinos at efe- linear colliders and for low-energy processes, such as Oupp decay and p + ey.

One of the central themes in particle cosmology is the origin of our matter-dominated Universe. Recently, the cosmic baryon-to-photon ratio of number densities has been measured with the unprecedented precision of 10% to be: VB x 6.1 x l. Therefore, finding a laboratory testable solution to this problem will set new milestones on our endeavours of understanding fundamental physics. A consensus has now been reached that a possible solution to the cosmological problem of the baryon asymmetry in the Universe (BAU) requires physics beyond the Standard Model (SM). In this context, an interesting suggestion has been that neutrinos, which are strictly massless in the SM, may acquire their observed tiny mass naturally by the presence of superheavy partners through the sclcalled seesaw mechanism '. These superheavy neutrinos have Majorana masses that violate the lepton number ( L ) by two units. Their out-of-equilibrium decay in an expanding Universe may initially produce a leptonic asymmetry, which is then converted into the observed BAU by means of in-thermal equilibrium ( B + L)-violating sphaleron interactions '.

In ordinary seesaw models embedded in grand unified theories (GUTS), the natural mass scale of the heavy Majorana neutrinos is expected to be

*Plenary talk given at PASCOS '04, 16-22 August 2004, Northeastern University, Boston, USA.

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of order the GUT scale 10l6 GeV. However, the reheating temperature Treh

in these theories is of order lo9 GeV, thus requiring for one of the heavy neutrinos, e.g. N1, to be unnaturally light below Treh, so as to be abundantly produced in the early Universe. On the other hand, successful leptogenesis and compatibility with the solar and atmospheric neutrino data put a lower bound on the N1-mass: m N l ,> lo9 GeV 5@. To avoid this narrow window of viability of the model around lo9 GeV, one needs to assume that the second heaviest neutrino N2 is as light as N1 7, an assumption that makes this thermal GUT leptogenesis scenario even more unnatural.

The aforementioned Treh problem can be avoided by appealing on low- scale thermal leptogenesis, by means of resonant leptogenesis 819t12. Res- onant leptogenesis takes place when a pair of heavy Majorana neutrinos has a mass difference comparable to their decay widths, in which case the leptonic asymmetry could reach values of order one. Thus, after the relevant heavy-neutrino self-enegy graphs loyll are resummed ’, the leptonic asymmetry in a model with two heavy Majorana neutrinos is given by ’:

where i, j = 1 ,2 (i # j ) , are the tree-level decay widths of the heavy Majorana neutrinos Ni and h” is the Yukawa coupling matrix (see Eq. (2) below). The corresponding expression for a 3 generation-mixing model is more involved and given in 12. Hence, the scale of leptogenesis can be lowered to the TeV range ‘7’ in complete agreement with the solar and atmospheric neutrino data 12. Although our focus will be on a minimal non-supersymmetric 3-generation model, the present discussion could be extended to unified 13714715916 and supersymmetric theories 17718719.

Recently, we have presented 2o a variant of resonant leptogenesis, where a given individual lepton number is resonantly produced by out- of-equilibrium decays of heavy Majorana neutrinos of a particular family type. For the case of the .r-lepton number, we called this mechanism ws- onant r -1eptogenesis. Since sphalerons preserve the individual quantum numbers B - L,,,,T/3 21 in addition to the B - L number, an excess in LT will be converted into the observed BAU, provided the possible &violating reactions are not strong enough to wash out such an excess. Moreover, a chemical potential analysis 21 shows that the generated baryon asymmetry is B = -$ L, at temperatures T above the electroweak phase transition, i.e. for T 2 T, NN 15Ck200 GeV. Hence, generating the BAU from

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an individual lepton-number excess is very crucial to render the resonant 7-leptogenesis scenario presented below phenomenologically testable.

The model under discussion is the SM symmetrically extended with one singlet neutrino per family. The leptonic flavour structure of the Yukawa and Majorana sectors of such a model may be described by the Lagrangian

j=1 /

where 6 is the isospin conjugate of the Higgs doublet @. We define the individual lepton numbers in (2) in the would-be charged-lepton mass basis, where the charged-lepton Yukawa matrix h1 is diagonala. Likewise, the heavy Majorana-neutrino mass matrix is defined to be positive and diagonal. Our selection of the weak basis could be justified from arguments based on decoherentional in-thermal equilibrium dynamics. Specifically, all SM reactions, including those that involve the e-Yukawa coupling, will be in thermal equilibrium for temperatures T 5 10 TeV of interest 22, thereby favouring the would-be charged-lepton mass basis 23. Likewise, decoherentional effects on heavy Majorana neutrinos, whose decays are thermally blocked already at temperatures T 2 3 m ~ ~ 24, will also favour the heavy Majorana-neutrino mass basis.

We now present a generic scenario for resonant T-leptogenesis. The neutrino Yukawa sector of this scenario has the following maximally CP- violating form:

E, a e-i~I4 a e i ~ I 4 i. ,c, c e- i~/4 c earl4 ) (3) h' = E be-ir/4 .

For nearly degenerate Ni masses in the range m N , M 0.5-1 TeV, the parameters a, b have to be smaller than about for phenomenological reasons to be discussed below. On the other hand, the requirement to protect an excess in L, from wash-out effects leads to the relatively stronger constraint c 5 Furthermore, the parameters q, with 1 = e, p, T , are taken to be small perturbations of the order of the e-Yukawa coupling, i.e. E Z - "An implicit assumption made here is that also sphaleron transitions are flavour diagonal in the same weak basis.

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Here, we should stress that at least 3 singlet heavy neutrinos N1,2,3 are needed to obtain a phenomenologically relevant model.

It is now important to notice that for exactly degenerate heavy Ma- jorana neutrinos m N , = m N and ~1 = 0, the light-neutrino mass matrix vanishes identically, i.e. (mv)llt = - t~~ Ci=1,2,3 mi: &hFi = 0, where v x 175 GeV is the usual SM vacuum expectation value. The resulting vanishing of the light neutrino masses will be an all-orders result, protected by a hidden leptonic symmetry of a linear combination of L, and L, 25.

low7 x m N l b , the entries of m” are within the observed region of less than - 0.1 eV for m N 1 - 1 TeV. At this point, we should emphasize that our scenario is radiatively stable. For a TeV-scale leptogenesis scenario, renormalization-group running effects on m” are very small 28. In addition, there are Higgs- and 2-boson-mediated threshold effects 6m” of the form 29:

In the presence of non-zero &l - lop6 and for A m N , , , = m N , , , - m N , <

Given the constraints on the Yukawa parameters discussed above, one may estimate that the finite radiative effects 6m” stay well below 0.01 eV for A m N i / m N l 5 and a, b 5 Hence, the perturbation parameters €1 and A m N i / m N l provide sufficient freedom to describe the solar and atmospheric neutrino data. For our resonant .r-leptogenesis scenario, the favoured solution is an inverted hierarchical neutrino mass spectrum with nearly bi-maximal u,u, and upuT mixings 30.

It is instructive to present an order-of-magnitude estimate of the BAU generated by resonant r-leptogenesis. In such a model, only the heavy Majorana neutrino Nl will decay relatively out of thermal equilibrium. Instead, N2 and N3 will decay in thermal equilibrium predominantly into e and p leptons. To avoid erasure of a potential L, excess, the decay rates of N2 and N3 to r-leptons should be rather suppressed. To be specific, in this framework the predicted BAU is expected to be

bWe shall not address here in detail the origin of these small breaking parameters, but they could result from different sources, e.g. the Frogatt-Nielsen mechanism 12*26, Plank- or GUT-scale lepton-number breaking 8 ,27 , etc.

361

where 6Lll computed analytically in l2 for a 3-generation model, is the T-lepton asymmetry and K N ~ = rN1/H(z) is an out-of-equilibrium measure of the N1-decay rate rNl with respect to the Hubble rate H ( z ) %

17m~l / ( z2Mplanc~) , with z = mNl/T. For ~l - we find that K N ~ - 10-100. Thus, if 161;JlI is resonantly enhanced to order 0.014.1 and E , / c - lob2, then the generated BAU T/B will be of order namely at the level observed. It is important to remark here that the size of V B is determined by a few key parameters: K N ~ , 6h1 and &,/c. Instead, the parameters a and b could be as large as N potentially giving rise to observable effects of lepton-number violation at colliders and laboratory experiments (see discussion below).

We now perform a simplified numerical analysis of the BAU in this scenario of resonant T-leptogenesis. Since the N2 and N3 heavy Majorana neutrinos will be in thermal equilibrium, their contribution to the L, asymmetry will be vanishingly small. More explicitly, our conservative numerical estimate will be based on the Boltzmann equations:

where ny is the photon number density, and 7 3 ~ ~ and qL, are the NI- number-to-photon and L,-number-to-photon ratios of number densities, respectively. We follow the conventions of l2 for the collision terms y z G related to the decays and inverse decays of the N1,2,3 neutrinos into Le+, and @. The collision term y iT describes the L,-violating 2 - 2 scatterings: (i) L,@ ~1 L&,@t and L,@ H Le,,@; (ii) L,Lz,,, H @@ and L,L,,, H

@at. The collision terms for the reactions (i) can be shown to be always smaller than x:=l yF:o for the temperature domain z 5 10 of interest, while those for the reactions (ii) are suppressed by factors proportional to AmN,/T, (m”):j/v2 and (a + b)2 N and can therefore be neglected. To obtain a conservative estimate, we set yZr = xi”=, y::@.

Figure 1 shows our numerical estimates of q ~ , as functions of z = mN1/T, for a resonant T-leptogenesis scenario, where E ~ , ~ , , = 0.5 x a = b = 3 x ArnN2/mN1 = lo-’, Arnrv,/rnlv, = 3 x and mN1 = 1 TeV. In this scenario, it is 6bl x -0.007 and

c = 0.4 x

362

KNi = 20. We find that qL,(T) at temperatures just below mN1 is independent of the initial values q$, and qp,. Thus, the heavy neutrino mass mNl could in principle be as low as the critical temperature T,, namely at the electroweak scale. Employing the lepton-to-baryon conversion formula for T << T, 12, x - & ? j ~ ~ , we find that our numerical results are compatible with the estimate in (5).

1

in in - q N l = ' 7 qL= = in in - - qN, =o, = o in - in = 1 ............... q N i - 0, ~ L Z

-4: l o

10

r ' : -

-.. -....-. I.._..,

10 **-..* 7 -7 - \

10

10

10 -9

-10 1 10 *

10 -l 1 10

z = m N , / T

Figure 1. Numerical estimate for 7 / ~ , versus z = mNl / T , for different initial conditions for qN1 and 7 / ~ , (see discussion in the text for the choice of the model parameters). The horizontal dotted Line at q ~ , % -3 x indicates the value needed to obtain the observed BAU, while the vertical dotted line corresponds to T = Tc = 200 GeV.

In the following, we will briefly discuss the phenomenological implications of the above model of resonant T-leptogenesis. These could be summarized as follows:

0 The model can lead to a sizeable neutrinoless double beta (Ovpp)

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decay with an effective neutrino mass ( (my)ee( M 0.05-0.4 eV 31

(within the ballpark of 32), as a consequence of the inverted light- neutrino mass spectrum. Also, effective neutrino masses larger than 0.1 eV will strongly point towards a degenerate heavy neutrino spectrum 6J2.

6.10-4 x (u2b2v4)/m& in the heavy-neutrino limit 33. Confronting this prediction with the experimental limit BeXP(p + er) 5 1.2 x 30, the resulting constraint is (ahi2)/m$1 5 1.4 x lo-*. For sub-TeV-scale heavy neutrinos and a,b - 3 x there should be observable effects in foreseeable experiments sensitive to B ( p + ey) - A positive signal of p -+ ey and a simultaneous observation or even non-observation of p --$ eee and p-e conversion in nuclei in future experiments would probe the presence of possible sizeable non-decoupling terms of the form ( ~ ~ b ~ v ~ ) / r n & ~ , which dominate for a, b 2 1 36. The latter would enable us to get some idea about the size of the heavy neutrino mass scale mN1. Assuming that only p + ey was observed, this would favour relatively light heavy Majorana neutrinos of TeV or even sub-TeV mass.

0 The possible existence of sub-TeV heavy neutrinos N2,3 with a p preciable e-Yukawa couplings a - 3 x could then be directly tested by studying their production at e+e- linear colliders 37. Finally, since N2,3 play an important synergetic role in resonantly enhancing 6hl, potentially large CP-violating effects in their decays will determine the theoretical parameters further.

Consequently, a phenomenological answer could be obtained to the question of whether electroweak-scale resonant .r-leptogenesis plays a relevant role in the solution to the cosmological problem of the baryon asymmetry in the Universe.

0 The model predicts p + ey with a branching ratio B ( p + ey)

34,35.

Acknowledgments

This work is supported in part by PPARC, research grant no: PPA/G/0/2001/00461. The author thanks K. Zioutas and K. Tamvakis for discussions and for a critical reading of the manuscript.

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R. Slansky, in Supergravity, eds. D.Z. Freedman and P. van Nieuwenhuizen (North-Holland, Amsterdam, 1979); T. Yanagida, in Proc. of the Workshop on the Unified Theory and the Baryon Number in the Universe, Tsukuba, Japan, 1979, eds. 0. Sawada and A. Sugamoto; R. N. Mohapatra and G. SenjanoviC, Phys. Rev. Lett. 44 (1980) 912.

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T. Hambye, Y. Lin, A. Notari, M. Papucci and A. Strumia, hep-ph/0312203. 8. A. Pilaftsis, Phys. Rev. D56 (1997) 5431; Nucl. Phys. B504 (1997) 61. 9. A. Pilaftsis, Int. J. Mod. Phys. A14 (1999) 1811; L. Boubekeur, hep-

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227; S. Pascoli, S. T. Petcov and W. Rodejohann, hepph/0302054. 14. G.C. Branco, R. Gonzalez Felipe, F.R. Joaquim, I. Masina, M.N. Rebelo and

C.A. Savoy, Phys. Rev. D67 (2003) 073025; R. Gonzalez Felipe, F.R. Joaquim and B.M. Nobre, hep-ph/0311029.

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S. Dar, S. Huber, V.N. Senoguz and Q. Shafl, Phys. Rev. D69 (2004) 077701. 20. A. Pilaftsis, hepph/0408103. 21. J.A. Harvey and M.S. Turner, Phys. Rev. D42 (1990) 3344; H. Dreiner and

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G. Cvetic, C.S. Kim and C.W. Kim, Phys. Rev. Lett. 82 (1999) 4761.

SFERMIONS AND GAUGINOS IN A LORENTZ-VIOLATING THEORY

ROLAND E. ALLEN AND SEIICHIROU YOKOO

In Lorente-violating supergravity, sfermions have spin 1/2 and other unusual properties. If the dark matter consists of such particles, there is a natural explanation for the apparent absence of cusps and other small scale structure: The Lorentz- violating dark matter is cold because of the large particle mass, but still moves at nearly the speed of light. Although the R-parity of a sfermion, gaugino, or gravitino is +1 in the present theory, these particles have an “S-parity” which implies that the LSP is stable and that they are produced in pairs. On the other hand, they can be clearly distinguished from the superpartners of standard supersymmetry by their highly unconventional properties.

In Lorentz-violating supergravity192, sfermions have spin 1/2. For one SO (10) generation, there are 16 fields which are initially massless and right-handed. If half of these are converted to left-handed charge-conjugate field^^.^, one obtains Lagrangian densities of the form

where $R and $L are 2-component left-handed and right-handed fields, with 5 0 - - u 0 = 1 and Ck = -ak as usual, and qPu = diag(-l,1, 1 , l ) . (The spacetime coordinate system in which the initial fields are right-handed is defined above (9) of Ref. 3.) One can change from the dimension 3/2 fields $ to conventional dimension 1 bosonic fields q5 by absorbing a factor of m-1/2, but that would have no effect in the following arguments. The energy scale m is not determined by the present theory, but is assumed to lie well above 1 TeV- perhaps as high as lo9 TeV or even the Planck scale.

Because of the minus sign in (2), one cannot couple left-and right- handed fields with a Dirac mass in the normal way. A Majorana mass is also not appropriate (except possibly in the case of sneutrinos). In the present paper we therefore consider a Lorentz-violating mass which is postulated to arise from supersymmetry breaking, at some high energy scale,

366

367

within the present Lorentz-violating theory. The Lagrangians then become

ck = m-’f”$fidpdu$R + $LiOpdp$R - mR$fi$R

.c; = f i - ’ q p u $ L a p a , , ~ L - +Liapa,$L - mL$L$L

(3) (4)

where the prime indicates that a mass has been added and integrations by parts have been performed in the action S = J d4x L. The mass term is invariant under a rotation, but not under a boost, since the transformation matrix for a boost is not unitary.

The equations of motion are

m-’?ydpd,,$R + iaCldp$R - mR$R = 0 ( 5 )

(6)

+ = C an$n , $n (2) = A, xn ea5“ , a, ( t ) = e-awntun (0) (7)

where n w fl, X and there are two possibilities for the %-component spinor

m-l pu q

As in Ref. 1, let each field be represented as

- iaWp@L - m L $ L = 0.

n

Xn:

If we require only that each an$n satisfy the equation of motion for $, there are 4 solutions:

f 2 w = - V i f 4 E 2 + 4 p 2 f 4 7 E ( p + m ) (9)

where m represents either mR or m L and the three independent f signs have the following meaning: The first f sign, on the left side of the equation, is + for a right-handed field $R and - for a left-handed field $ L . The last f sign, under the radical, is + for a right-handed solution containing X R and - for a left-handed solution containing X L . Finally, the middle f sign, preceding the radical, indicates that there are two solutions for a given $ and x, with the + sign corresponding to the normal solution, for which w + 0 as lpl -+ 0, and the - sign to an extremely high energy solution, for which (w( --+ m/2 as

We will now see that not all of the 4 above solutions are physical, because one of them may correspond to negative-norm states which are inadmissible in a proper positive-norm Hilbert space. After discarding any unphysical solution, however, we are still left with enough basis functions $, (2)to have a complete set of functions for (i) representing an arbitrary classical field

-0 .

368

+ (2)and (ii) satisfying the quantization condition below. The canonical momenta are (in the notation of Ref. 1)

where the upper sign holds for a right-handed field and the lower for a left-handed field. We again quantize by interpreting + and as operators, and requiring that

where a and ,B label the two components of $ and TL. Following the same logic as in Ref. 1, one finds that this requirement can be satisfied if

2772-l f w,l > 0. (13) Depending on the 3-momentum F, there are either three or four solutions which meet this condition (and which are therefore included in the representation of the physical field operator $R or + L ) :

L f2w = -a- J1522+4jd2 - 4m(14 - m) all I P ~

where 2P+ = m f dm2 - 4mm. Here the first column indicates whether the solution involves X R or XL, and again the upper and lower signs hold for +R and $ L respectively. The same calculation as in (4.36) of Ref. 1 shows that the quantization condition above can be satisfied by choosing

A i A n = (Iwnl V)-' 12m-l f u:'l-'

A i A , = 2(Iunl V)-' 12fi-l fui ' l- '

(14)

(15)

in the case when there are 2 solutions for a given x (either X R or X L ) and

when there is only one solution. As in Ref. 1 (and in standard physics), when the original energy w

is negative we reinterpret the destruction operator a for a particle as the

369

R L

creation operator bt for an antiparticle. If we now discard the extremely high energy solutions, and also restrict attention to momenta that are not extremely large, we have the following possibilities for both sfermions and ant isfermions:

w = lpl + m all lfl w = - l f l + m lpl<m *

With a Lorentz-violating mass m, therefore, the group velocity v = B w / a lpl is 1 for right-handed sfermions (or antisfermions), in units with f i = c = 1, and - 1 for left-handed sfermions. I.e., these Lorentz-violating particles have a highly Lorentz-violating energy-momentum relation, which implies that they travel essentially at the speed of light even though their masses are presumably comparable to 1 TeV. Furthermore, the velocity of a left- handed sfermion is antiparallel to its momentum, and there are no left- handed sfermion (or antisfermion) states for momenta with lp’l c > mc2.

The present theory also contains gauginos etc. which can be candidates for the dark matter, but let us suppose here that the lightest supersymmetric partner is a neutral sfermion - i.e., a sneutrinw with the highly unconventional properties discussed immediately above. For simplicity, consider a circular orbit of radius r about a mass M . Let us restore c in the equations for clarity, and write p = lfl. For pc << mc2 the general formula for the centripetal force implies that

pvlr = G M m / r 2 or r = Rs (mc2/2pv) (16)

where Rs = 2GM/c2 is the Schwarzschild radius. For nonrelativistic standard cold dark matter (CDM), the kinetic energy is pv/2, and for Lorentz- violating dark matter (LVDM) with m << m it is pv with v x c . ~ For a given kinetic energy, therefore, LVDM and CDM have orbits of comparable size in this simplistic picture. This can also be seen from the virial theorem5 (pv) = (p’. i7) = - Fa r‘ = (GM ( r ) mlr) which implies that the kinetic energy is equal in magnitude to the gravitational potential energy for LVDM, and to one-half the potential energy for CDM, so dark matter with a given distribution of kinetic energies will have comparable large-scale trajectories in both models. On the other hand, the binding energy is vastly smaller for LVDM, and this leads to the hope that LVDM can provide a natural explanation for the apparent discrepancy between observations and CDM simulations in regard to cusps and other small scale structure. There is some tentative confirmation of this idea in preliminary computer simulations5.

( - )

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In Lorentz-violating supergravity2, at energies low compared to f i , the fermion fields @ j and sfermion fields @ b are coupled in the following way to the gauge fields A;, the gaugino fields a;, the gravitational vierbein e$, and the gravitino field E::

SL = J d4a: e ~t ~E:O"'D, @ , D~ = a, - iALt,. (17)

We have generalized the usual vocabulary in a natural way, so that the superpartner of the graviton is defined to be the gravitino, and the superpartners of gauge bosons to be gauginos, even though these fermions have quite unconventional properties. In particular, each superpartner has both the same quantum numbers and the same spin as the Standard Model particle. This means that the conventional R-parity is +1 for sfermions, gauginos, and gravitinos. However, since each vertex involves an even number of supersymmetric partners we have conservation of an "S-parity", which is -1 for sfermions, gauginos, and gravitinos, and +1 for their Standard Model counterparts. (This is also true when the kinetic terms for the force fields are included, as will be discussed elsewhere.) Then the lightest supersymmetric partner (LSP) cannot spontaneously decay into lighter Standard Model particles. For the same reason, sparticles are always created in pairs. Parton-parton or lepton-lepton interactions will lead to production of an even number of supersymmetric partners, just as in standard supersymmetry with R-parity conservation. However, the sparticles predicted by the present theory are clearly distinguished by their highly unconventional properties.

References 1. R. E. Allen, in Proceedings of Beyond the Desert 2002, edited by H. V.

Klapdor-Kleingrothaus (IOP, London, 2003), hep-th/0008032. 2. R. E. Allen, in Proceedings of Beyond the Desert 2003, edited by H. V.

Klapdor-Kleingrothaus (IOP, London, 2004), hep-th/0310039. 3. R. E. Allen and S. Yokoo, Nuclear Physics B Suppl. (in press), hepth/0402154. 4. R. E. Allen and S. Yokoo, in Proceedings of the Third Meeting on CPT and

Lorentz Symmetry, edited by V. A. Kosteleckf (Singapore, World Scientific, in press).

5. A, R. Mondragon and R. E. Allen, in Proceedings of PASCOS 2001, edited by P. H. Frampton (Rinton Press, Princeton, 2001), astro-ph/0106296; and work in progress.

SUSY Phenomenology


LEPTON MASSES AND MIXINGS IN NEXT-TO-MINIMAL SUSY SO(10) GUT

M. M A L I N S K ~ Scuola Internazionale Superiore d i Studi Avanzati (S. I.S.S.A.)

Via Beirut 2-4, 34014 Trieste, Italy E-mail: malinskyQsissa.it

in collaboration with

s. BERTOLINI~ AND M. FRIGERIO~ Scuola Internazionale Superiore d i Studi Avanzati (S. I.S.S. A . )

Via Beirut 2-4, 34014 Trieste, Italy Department of physics, University of California,

Riverside, C A 92521, USA

We examine a simple extension of the minimal renormalizable supersymmetric SO(10) grand unified theory by adding a 120-dimensional Higgs representation. This brings new antisymmetric contributions to the relevant quark and lepton mass sum rules and leads to a better fit of the measured values of lepton masses and mixings together with a natural completion of the renormalizable Higgs sector within the SUSY SO(10) framework.

1. Introduction

The class of supersymmetric grand unified theories (GUT) based on the SO(10) gauge group seems to be one of the most promissing frameworks to describe the physics beyond the Standard model, including massive neutrinos. Though the scale at which the GUT symmetry should be realized is very large (-J 10l6 GeV) there could be observable consequences of such scenarios at the laboratory energies, be it the tiny neutrino masses, measurable proton decay rate, anomalous electric dipole moments and other phenomena. The high scale symmetry propagates into the low-energy observables by means of effective relations among quantities which are in general uncorrelated within the SM framework. This makes such scenarios quite predictive and thus very attractive from the point of view of the low- energy phenomenology. In this talk we give a short overview of the minimal

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3 74

renormalizable SUSY SO(10)172, in particular the effective mass sum rules for quark and lepton mass matrices arising from the GUT-scale physics and their consequences on the leptonic sector, namely the predictions for the neutrino masses and the PMNS lepton mixing matrix. Then we define a simple renormalizable extension of the minimal model by including one additional (almost decoupled) 120-dimensional Higgs multiplet . We argue that even a tiny admixture of its bidoublet components within the light MSSM Higgs doublets can lead to substantial effects in the predicted values of the neutrino masses and PMNS mixing angles.

2. Minimal renormalizable SUSY SO( 10) model

In the past few years one may notice a 'renaissance' 3t4151677 of the renormalizable SUSY SO(10) model. It was shown that this framework can accommodate good R-parity conserving SO( 10) -+ S M breaking patterns being more constrained than any realistic GUT model based on the SU(5) gauge group3. Moreover, there is an intriguing relationship between the (approximate) maximality of the atmospheric mixing in the leptonic sector and the br unification provided the neutrino mass matrix is dominated by the type-I1 seesaw contribution.

Structure of the minimal renormalizable SUSY SO(10) One of the most appealing features of any SO(10) GUT model is the fact that the SM fermions of each generation (including the right-handed neutrinos) reside in one irreducible 16-dimensional representation of the gauge group, the spinorial 1 6 ~ . Concerning the Yukawa part of the superpotential, the matter bilinear 1 6 ~ x 1 6 ~ can couple at the renormalizable level only to three types of Higgs multiplets, namely the 10-dimensional vector multiplet 1 0 ~ , the 126-dimensional antiselfdual 5-index antisymmetric tensor 1268 and the 120-dimensional three-index antisymmetric tensor 1208. It was ~ h o w n ~ > ~ that in order to obtain a realistic leptonic spectrum it is sufficient to consider only the 108 and 1268 Higgs multiplets. (In addition the 210-dimensional four-index antisymmetric tensor 2 1 0 ~ is needed to break properly the SO(10) group down to the MSSM and mix the 108 and 126~ to generate the left-handed triplet VEV entering the seesaw formula together with the proper mixings among the components entering the two light Higgs doublets of the MSSM).

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Sum rules for quark and lepton mass matrices in MRM Inspecting the SU(3), x s U ( 2 ) ~ x U(1)y structure of these multiplets one sees that the quark and lepton mass matrices obey

provided Y ~ o and Y126 are the (symmetric) Yukawa matrices parametrizing the coupling of the matter bilinear to the relevant Higgs multiplets and

10,126 are the VEVs of bidoublets contained in 1 0 ~ and 126~.

Lepton mixing in MRM with dominant type-I1 seesaw Assuming that the type-I1 contribution dominates the neutrino mass- formula one can obtain the following sum-rules for the charged lepton and neutrino mass matrices6 (in the basis where Md is diagonal = Dd; the tilde denotes the mass matrix normalized to its largest eigenvalue)

Here k and r are functions of the v-parameters in (1). Looking at (2) one can appreciate the predictivity of the model: i) It is very nontrivial to get a good fit of the charged lepton mass ratios at the LHS of (2) by varying only the quark masses and mixings within their experimental ranges and using the freedom in r and the remaining 6 complex phases (in D,) on the RHS. ii) Whenever one finds a region in the parametric space where the charged lepton mass ratios fit well, the neutrino mass matrix is known up to just one phase. Thus the model is very predictive in the neutrino sector. Moreover, if mb approaches mT (and the relative phase of the two terms is adjusted properly), the 33-entry of m, is comparable with the other entries in the 23 sector, what leads to the almost bimaximal structure of the U ~ M N S ' . Coming to the numerical analyses6i7 (with the CP-phases switched off for simplicity; their effects were shown to be subleading in most cases6 usually worsening the fit of the charged lepton formula), the following predictions are obtained at the 1-0 level 6t7: /Ue31 2 0.15, sin2 2813 2 0.85, sin2 2823 5 0.97. The lower bound for ]Uesl turns out to be very rigid and can be a 'smoking gun' of the model. Moreover, the solar mixing tends to be too large, while the atmospheric is never maximal.

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3. Extending the minimal renormalizable model

Though the minimal model predictions are in a reasonably good agreement with the experimental data by extending the analysis to 2-0 level one may ask whether some extensions of MRM may perform better. However, often the price to be paid is the lack of predictivity and one should look for extensions that are constrained enough to remain as predictive as possible.

Adding a 120-dimensional Higgs representation One of (few) such renormalizable generalizations of the MRM is the scenario with one additional quasidecoupled 120-dimensional Higgs representation (to which we refer as the next-to-minimal renormalizable model, NMRM)7. The key observation is that the 1 2 0 ~ multiplet can be naturally heavier than the GUT scale, because it does not participate at the GUT-symmetry breaking. Its scalar mass parameter M120 is not constrained by potential- flattness conditions and can be naturally as large as the cut-off, be it the Planck scale. This means that the weights of the bidoublet components entering the weak-scale MSSM Higgs doublets may be naturally suppressed with respect to those coming from 1 0 ~ and 126~. Therefore the relations (1) are only slightly modified thus preserving most of the good features of the MRM. Let us write down the new quark and lepton mass formulae :

MU = Y~OUA' + Y12621A26 + Y12021A2O M d = Y 1 o V i o f Y12621:26 f Y 1 2 0 2 1 : ~ ~

Mi = y 1 0 ~ ~ ~ - + Y 1 2 0 2 1 ~ ~ ~ M, C( y 1 2 6 ( ( 1 , 3 , + 2 ) m ) (3)

The inequality M 1 2 o >> MGUT translates into w:20 << u c i 2 6 . Equivalently, one can write (diagonalizing the quark mass matrices by means of biunitary transformations M, = V.DxVk , x = u , d and denoting W = V: V f , VCKM f V," Vk and Y{20 = V d R T Y 1 2 0 V k )

T T

T

k v d RT ~ l v k = W T D , V C K M f T B d + Y,',o(k&l - &, - T&d)

with E,,d,l V,,d,l/mt,b,r. Since Y120 is antisymmetric, the M,,d,l axe no longer symmetric and the unknown right-handed quark mixing matrix W appears at the RHS of (4). However, since this setup is a perturbation of the MRM one can expand the W matrix around VCKM by means of the small parameters in the game (neglecting the CP phases): W = VCKM + 2 (-&,Z,VCKM + & d V C K M Z d ) + O(E:) where the Z, matrices axe

given by (z~)ij = ( Y L ) i j / [(b,)ii f (B&)j j ] and yi V C K M Y & ) V ~ K M ,

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Yi = Y:Z0. Therefore, for small E, the predictions of this model are expected to be close to those of MRM. It can be shown that the tiny antisymmetric corrections change the PMNS angles linearly in E'S while the masses are affected at the second order, what makes the perturbative method self- consistent and the fit of the charged lepton mass matrix stable enough to preserve the good features of the MRM. On the other hand, the 1-2 entries of the 2, matrices can be strongly enhanced by the small 'B11,22' terms in the denominator.

Lepton mixing in NMRM with dominant type-I1 seesaw The numerics shows' that even for E N the non-decoupling effects of the 1 2 0 ~ contributions to the P M N S angles can reach several tens of percent. For instance the MRM lower bound for lUe31 can be relaxed to about lUe31 > 0.1, while the atmospheric angle can be maximal and the solar bound is changed to about sin2 2813 > 0.75, all this at 1-0 level even with the CP-phases switched off.

4. Conclusions

We have argued that the lepton sector predictions of the minimal renormalizable SUSY SO(10) model are very sensitive to the magnitude of the antisymmetric Yukawa structure be it an additional small renormalizable coupling of 1 2 0 ~ Higgs multiplet to the matter bilinear (or an effective vertex generated by dynamics beyond the GUT-scale). Therefore such terms should be taken into serious consideration when discussing the phenomenological implications of such class of grand unified theories.

References 1. T. E. Clark, T. K. Kuo and N. Nakagawa, Phys. Lett. B 115 (1982) 26. 2. C. S. Aulakh and R. N. Mohapatra, Phys. Rev. D 28 (1983) 217. 3. C. S. Aulakh, B. Bajc, A. Melfo, G. Senjanovic and F. Vissani, arXiv:hep-

ph/0306242. 4. B. Bajc, A. Melfo, G. Senjanovic and F. Vissani, arXiv:hep-ph/0402122. 5. B. Bajc, G. Senjanovic and F. Vissani, Phys. Rev. Lett. 90, 051802 (2003)

[arXiv:hep-ph/0210207]. 6. H. S. Goh, R. N. Mohapatra and S. P. Ng, Phys. Lett. B 570, 215

(2003) [arXiv:hep-ph/0303055], Phys. Rev. D 68, 115008 (2003) [arXiv:hep- ph/0308197].

7. S. Bertolini, M. F'rigerio and M. Malinsky, arXiv:hep-ph/0406117. 8. B. Brahmachari and R. N. Mohapatra, Phys. Rev. D 58 (1998) 015001

[arXiv: hep-ph/9710371].

HIGH SCALE STUDY OF POSSIBLE B d + ~ K s CP PHYSICS

TING T. WANG*

Michigan Center for Theoretical Physics, Ann Arbor, MI 48109, USA

Some rare decay processes are particularly sensitive to physics beyond the Standard Model (SM) because they have no SM tree contributions. We focus on one of these, B d -) 4K5. Our study is in terms of the high scale effective theory, and high scale models for the underlying theory, while previous studies have been focusing on the low scale effective Lagrangian. We show that models with family dependent Kahler potential can provide large non-SM CP effects to the Bd + 4K5 process.

1. Introduction

CP violation is a fascinating subject in particle physics. Complex parameters in the Lagrangian could give rise to numerous interesting observables in low energy experiments. Many, especially those associated with the third generation of quarks, are either not well measured or controversial. They represent new opportunities to discover new physics and uncover new flavor physics in the near future. If possible FCNC parameters are small, some presently unknown mechanism or symmetry has to be found to explain why they are small. Sizable new CP violations, as well as the KM phase itself, probe fundamental flavor physics, which is closely related to supersymmetry breaking and string theory.

Recently the time dependent CP asymmetry in the B d -+ $Ks process has been measured by both Belle and BARBAR groups. The SM prediction is

S ~ K _N S+K -N sin 2@ = 0.736 (1)

The latest results from BaBarl is S ~ K ~ = 0.29 f 0.31 and from BELLE' is S ~ K ~ = 0.00 f 0.33. The average is S ~ K ~ = 0.15 f 0.23. Because this process is one of a few that are unusually sensitive to CP physics beyond

*email: tingwangOumich.edu

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the SM, it is worthwhile studying it in detail however the experimental situation is finally resolved. And of course if a deviation from the SM prediction is confirmed it is exceptionally important. In this paper, we show that unification scale supersymmetric models with family dependent Kahler potential can give large CP violation contributions to the B d -+ C#JKS process without violating other experimental bounds.

2. Models with family Dependent Kahlar Potential

2.1. Large d;iLR case. In this model, we take the Yukawas as given parameters and they have the following form at the high scale:

Y, = au x diag{m,,m,,rn~}

Yd = ad x VCKM diag{md, m,, mb} . U (2) In these equations, a, and a d are normalization factors depending on tan p. VCKM and the quark masses should take their high scale values. U is a matrix which generates a right handed down-type quark mixing:

1 0 u = (o cosw ei@sinw

o -e-@ sin w cos w O ) (3)

To calculate the soft terms, we assume a mixed dilaton/moduli susy breaking scenario and assume we are in the weakly coupled heterotic orbifold vacuum. The soft terms take the form as

ml/2 = h m 3 p sin ee-'Ys

m: = m&,(l+ 3cos2 edi.02)

~~~k = - h m 3 p ( s i n ee-i7s + cOs e C e- iYa 0,

(4) (5)

(6)

+

3

CY=l

x [I + n: + ny + n: + (T, + T:)d, log(~,jk)])

In these formulas, i , j , k denote different MSSM fields. 6 is the goldstino angle and C 0; = 1 where a = 1,2,3 correspond to diagonal T-moduli fields associated with 3 compactified complex planes. For simplicity we assume only these moduli fields are relevant for soft term calculations. The ys and ya are the phases for Fs and FT, and we set them to be zero, so all the CP violation sources are in the Yukawas. n: are the modular weights of a field i with respect to a-moduli. They are negative fractional numbers.

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Generally, the Yulcawas x j k depend on moduli fields so that the last term in eq.(6) is nonzero. Since we take the Yukawas as input numbers, there derivatives respect to T, can not be determined. Thus we have to assume their contributions to the trilinears are small and neglect them.

In this model, we use the following parameters:

1 7r m3/2 = 420GeV sin0 = - w = - t anP = 40. (7) J5 4

The relevant modular weights and 0, are shown in Table 1.

I TI I T?

I nn, l o I -1 -1 0 I nn, l o I -1 ~ I 0

~

I I I

I nH.. l o I -1 I d --1

We assume the other matter fields have family independent modular weights. Therefore, all other fields have family independent trilinear and scalar masses. Notice no2 and no3 are different which leads to different Kahler potential for D2 and D3 and generates a relative large &tiLR at the high scale.

Taking these parameters as high scale input, we use RGEs to run the soft Lagrangian to the weak scale and calculate observables such as S ~ K ~ , the strange quark CEDM, BR(b -+ sy), the higgs mass, etc. We scan the phase 4 in eq.(3) from 0 to 27~. For this large 6tiLR case, it’s the EDM bound that is most difficult to satisfy. We show the correlation between predictions of S $ K ~ and ed? in the left panel of Figure 1. The upper bound on led:[ is 5.8 x 10-25ecm. The theoretical prediction shown in this figure has a large uncertainty. If we allow factor 3 theoretical uncertainty, from this figure we find the smallest S ~ K ~ is about -0.3. We checked that for this model, other experimental constraints such as BR(b + sy) and the higgs mass bound are satisfied.

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2.2. Large d;iLR case

To make this MI large, we need a large mixing of the 2nd and 3rd generation left-handed quarks. Thus we use the following Yukawas:

Y, = a, x U * diag{m,,m,,mt} Yd = a d x U . VCKM . diag{md, m,, mb}

The parameters are:

(8) 1 7r

m3/2 = 360GeV sin8 = - w = - t a n p = 24

The 0, and modular weights are shown in Table 2. Due to the different fi 7

modular weights for Q2 and Q 3 , 6,d;LR will be large. We assume modular weights for other particles are family independent. We scan the phase q5 in the U matrix. In this large case, the SUSY contribution to S ~ K ~ is mainly constrained by BR(b + sy). We show the correlations between them in the right panel of Figure 1. From this figure, we see that the smallest 5 ’ 4 ~ ~ we can get without violating the BR(b + sy) bound is about 0.13.

3. Conclusions

Fundamental flavor physics is expected to manifest itself in the flavor structure of the soft supersymmetry Lagrangian at the supersymmetry breaking scale. In principle, there could be large CP violating parameters in the soft Lagrangian. They could give rise to deviations from Standard Model prediction of the CP violating observables. One example is the recent PO- tential deviation in SG,K~ of B d + q5Ks . On the other hand, CP violation in the soft Lagrangian at high scale is constrained by low energy observables such as EDM and b + sy. We found several models which could produce

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Large 6irn scenario

20 . . . , I . . . . , . . I . , . . . .

. - Experimental Bound h

B I O -

O- 1 1 I -0.5 0.5

SOKS

Figure 1. Left: CEDM vs. S ~ K ~ in the large 6;iLR scenario. We scan the phase $ in eq.(3) from 0 to 27r. Two horizontal lines are the experimental bounds on the s-quark CEDM. If we use the exact CEDM bound, the smallest S+,K~ we can get in this model is around 0.14. If we allow a factor 3 theoretical uncertainty on the CEDM prediction, the smallest S ~ K ~ in this model is around -0.33. Right: BR(b + sy) vs. S + K ~ in the large scenario. Two horizontal lines are the bounds on BR(b + sy). The smallest S ~ K ~ in this model without violating the 6 -) sy constraint is around 0.13.

large deviation from SM in processes such as Bd -+ ~ K s and satisfy the constraints at the same time. Thus data such as the CP asymmetry in the Bd + q5Ks process could point toward some classes of high scale theories as favored. Further investigation along those directions are clearly very interesting and important.

Acknowledgment

The author would like to thank G. Kane, H.b. Wang and L.T. Wang for fruitful collaboration. The research is supported in part by the U.S. De- partment of Energy.

References

1. Taken from talks by BARBAR and BELLE groups given at this conference. 2. G. L. Kane, H. b. Wang, L. T. Wang and T. T. Wang, arXiv:hep-ph/0407351. 3. A. Brignole, L. E. Ibanez and C. Munoz, arXiv:hep-ph/9707209.

B 3 X, + 7 IN SUPERSYMMETRY WITHOUT R-PARITY

R. D. VAIDYA* AND O.C.W. KONG Department of Physics,

National Central University, Chung- Li 3.2054 TAIWAN

We present a systematic analysis of the decay B + X , + y at the leading log within the framework of Supersymmetry without R-parity. We point out some new contributions in the form of bilinear-trilinear combination of R-parity violating (FWV) couplings that are enhanced by large tan 0. We also improve by a few orders of magnitude, bounds on several combinations of RPV parameters.

1. Introduction

The large number of talks on supersymmetry (SUSY) in this conference provides ample proof of the inadequacy of standard model (SM) as a complete theory, and the appeal of SUSY as a most popular candidate for the physics beyond SM. In our opinion, the minimal supersymmetry standard model with conserved R-parity, lacks the much needed solution to neutrino mass problem which is naturally addressed in models with R-parity violation (RPV). However, the large number of a pri- on' arbitrary RPV couplings must be constrained from phenomenology in all possible ways. In this talk we shall discuss the influence of RPV on the decay channel I? 4 X, + y . Being loop mediated rare decay, it is sensitive to physics beyond SM. It has already been well measured by CLEO, BELLE, ALEPH and BABAR and hence can be used to put upper bounds on RPV couplings. The experimental world average' is Br [B + X, + y (Ey > 1.6GeV)I SM = (3.57 f 0.30) x lo-*. Within la this matches very well with the SM prediction2 of (3.57*0.30) x lo-*. The good agreement between SM prediction and the experimental number at la can be used to constrain the large number of a priori arbitrary parameters of SUSY without R-parity.

'Speaker at the conference.

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There have been few studies on the process within the general framework of R-parity vi0lation~3~. Ref.3, fails to consider the additional 18 four- quark operators which, in fact, give the dominant contribution in most of the cases. Ref.4 has considered a complete operator basis. However, we find their formula for Wilson coefficient (WC) incomplete, and they do not report on the possibility of a few orders of magnitude improvement on the bounds for certain combinations of RPV couplings, as we present here5. In fact, the particular type of contributions - namely, the one from a combination of a bilinear and a trilinear R-parity violating (RPV) parameters, we focused on 6 , has not been studied in any detail before. Here we shall briefly report the results. For the analytical details we refer the readers t 0 ~ 7 ~ .

We adopt an optimal phenomenological parametrization of the full model Lagrangian - the single single-wev parametrization. It is essentially about choosing a basis for Higgs and lepton superfields in which all the “sneutrino” vev vanish. The formalism is discussed at length in7.

2. Formalism

The partonic transition b + s + y is described by the magnetic penguin diagram. Under the effective Hamiltonian approach, the corresponding WC of the standard &7 operator has many RPV contributions at the scale Mw. For example, we separate the contributions from different type of diagrams as C7 = C,W + CF + (2;- + Cj?” + C;- + C:” corresponding to W- boson, gluino, chargino, neutralino, colorless charged-scalar and colorless neutral-scalar loops (for details please see5). Apart from the 8 SM operators with additional contributions, we actually have to consider many more operators with admissible nonzero WC at Mw resulting from the RPV couplings. These are the chirality-flip counterparts a 7 and & of the standard (chromo)magnetic penguins Q 7 and &8, and a whole list of 18 new relevant four-quark operators. For the lack of space, we list 8 important operators below. -

QS-l l=(SLa 7’ b L 4 ) ( 4 R p 7’ qRa) I q = d, s, b; I

&9-13=(sRa 7’ bR4) (QLp 7 p qLa) , q d, s, b, %c; (1) and six more operators from A” couplings5. The interplay among the full set of 28 operators is what makes the a d y s i s complicated. The effect of the QCD corrections proved to be very significant even for the RPV parts.

After the QCD running of WC from scale Mw to m b , dictated by 28 x 28 anomalous dimension matrix, the effective WC are given as (at leading log

385

order)5 :

c,ff (mb) = -0.351 cgff (Mw) + 0.665 cFff (Mw) + 0.093 cEff (&) -0.198 Cgff(Mw) - 0.198 Cl"off(Mw) - 0.178 C7r(Mw) ,

(?;ff (mb) = 0.381 p(&) + 0.665 eFff (Mw) + 0.093 5:' (Mw) -0.198 (?gff (M,) - 0.198 Gf(M, ) - 0.178 Gf(Mw) +0.510 @ezff(Mw) + 0.510 + 0.381

-0.213 @sff(Mw) . (2)

The branching fraction for Br(b + s + y) is expressed through the semileptonic decay b * ulcev (so that the large bottom mass dependence (- m:) and uncertainties in CKM elements cancel out) with Br,,,(b +

ulcefi,) = 10.5% and r(b * sy) c( (lC;""(p~b)(~ + 15;ff(pt,)(2). Note that we have also to include RPV contributions to the semi-leptonic rate for consistency.

3. Results: Impact of bilinear-trilinear combination of parameters

Analytical Appraisal. -We implement our (1-loop) calculations using mass eigenstate expressions, hence free from the commonly adopted mass- insertion approximation. While a trilinear RPV parameter gives a vertex, a bilinear parameter now contributes only through mass mixing matrix elements characterizing the effective couplings of the mass eigenstate running inside the loop. The pi's are involved in fermion, as well as scalar mixings. There are also the corresponding soft bilinear Bi parameters involved only in scalar mixings7. Combinations of pi's and Bi's with the trilinear parameters are our major focus.

There are two kinds of Bi-A' combinations that contribute to b --$ s + y at 1-loop: (a) B:Alj2, and (b) B&. These involve quark-slepton loop diagrams. Case (a) leads to the bL * s, transition (where SM and MSSM contribution is extremely suppressed) whereas case (b) leads to SM-like 4 + sL transition. For the purpose of illustration, we will assume a degenerate slepton spectrum and take the sleptonic index i = 3 as a representative. For the j values, the charged loop contributions are still possible by invoking CKM mixings. Consider the contribution of case (a) with IB3*A$321 to the c7, for instance. Through the extraction of the bilinear mass mixing effect under a perturbative diagonalization of the mass matrices7: we

386

obtain6,

for the charged and neutral colorless scalar loop, respectively. Here zt stands for (m:/M:) with an obvious replacement for 2 6 . Here F12(34) (xt) =

quFl(3) (zt) + F2(4) (zt) where Fi are the well known loop functions5. In the above equations, proportionality to tanP shows the importance of these contributions in the large tan@ limit. The M:, 42, A&?, are all scalar (slepton/Higgs) mass parameters. The term proportional to yt above has chirality flip inside the loop. Thinking in terms of the electroweak states, it is easy to appreciate that the loop diagram giving a corresponding term for 8f'" (cf. involving @m3@i,) requires a Majorana-like scalar mass insertion, which has to arrive from other RPV couplings7. In the limit of perfect mass degeneracy between the scalar and pseudoscalar part (with no mixing) of multiplet, it vanishes. Dropping this smaller contribution, together with the difference among the Inami-Lim loop functions and the fact that the charged loop has more places to attach the photon (with also larger charge values) adding up, we expect the 6;- to be larger than e:'. We corroborate these features in our numerical study.

Numerical Results. - We take non-vanishing values for relevant combinations of a bilinear and a trilinear RPV parameters one at a time, and stick to real values only. Our model choice for parameters is (with all mass dimensions given in GeV): squark masses 300, down-type Higgs mass 300, po = -300 sleptons mass 150 and gaugino mass M2 = 200 (with MI = 0.5Mz and M3 = 3.5M2), tanp = 37 and A parameter 300. We impose the experimental number to obtain bounds for each combination of RPV parameters independently (given in Table I). Consider, for instance, the case (b) combination JB3Al:3). We obtain a bound of 5.0. when normalized by a factor of pt. Since this is a & --+ s, transition, the RPV contribution interferes with the SM as well as the MSSM contribution. Over and above the loop contributions there are contributions coming from four-quark operator with C11 (.: Y b ) which is stronger than the other two four-quark quark coefficients 810,13 0; y s . Since the neutral scalar loop contribution is proportional to the loop function F1 (which is of order .Ol ) , it is suppressed compared to current-current contributions. Also here the charged scalar contribution comes only with chirality flip inside the loop and

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Table 1. Our Bounds for the absolute value of products of FU’V couplings.

Product Bound Product bound Product bound 4.5.10-2 Pi.&$ 2 .2 .

pL- .X! PO

5.0.10-5

7.4.10-3 6.5.10-2 y 2 1.0 .

2 .3 . 10-3 B i $ l ~ $ 8.0 . 10-2 Pi”i3$ 8.0 . n PO

has a CKM suppression. So the current-current is dominant. It has a more subtle role to play when one writes the regularization scheme-independent C+* = C7 - C11 at scale Mw(see5). Due to dominant and negative sign chargino contribution (because At po < 0), the positive sign CII interferes constructively with C7 and enhances the rate.These features can be verified from Fig1 of Ref.‘. We have done the similar analytical and numerical exercise for all possible combinations of bilinear and trilinear couplings and quote the relevant bounds obtained €or the first time in Tablel.

Conclusions. - We have systematically studied the influence of the combination of bilinear-trilinear RPV parameters on the decay b + s + 7 analytically as well as numerically. These contributions are enhanced by large tan@. We also demonstrate the importance of QCD corrections and obtain strong bounds on several combinations of RPV parameters for the first time. Numerical study has also been performed on combinations of trilinear parameters5. We quote here a few exciting bounds under a similar sparticle spectrum. For instance I & . X:;,l for i = 2,3 should be less than 1.6 . to be compared with rescaled existing bound of 2 .

Acknowledgment

We would like to thank National Science Council of Taiwan and Physics department (NCU) for the support under post-doc grant number NSC 92- 281 1-M-008-012.

References

1. C. Jessop, SLAC-PUB-9610. 2. A. Buras et al. Nucl. Phys.B631 219(2002). 3. B. de Carlos and P.L. White Phys. Rew.D55 4222 (1997). 4. Th. Besmer and A. Steffen Phys. Rev. D63 055007 (2001). 5. O.C.W. Kong and R. Vaidya hep-ph/0403148. 6. O.C.W. kong and R. Vaidya hep-ph/0408115. 7. O.C.W. Kong, 1nt.J.Mod.Phy. A19, 1863 (2004).

MSUGRA DARK MATTER AND THE B QUARK MASS

M. E. GOMEZ* Dept. de Fisica Aplicada, Uniuersidad de Huelua, 21071 Huelua.

E-mail: mario.gomezOdfa. uhu. es

T. IBRAHIM Dept. of Physics, Faculty of Science, University of Alexandria,

Alexandria, Egypt' Dept. of Physics, Northeastern University, Boston, M A 02115-5000, USA

E-mail: tarekOneu.edu

P. NATH Dept. of Physics, Northeastern University, Boston, M A 021 15-5000, USA.

E-mail: nath@neu. edu

S. SKADHAUGE Dept. de Fisica, Instituto Superior Tkcnico, 1049-001 Lisboa, Portugal.


We extend the commonly used mSUGFt.4 framework to allow complex soft terms. We show how these phases can induce large changes of the SUSY threshold corrections to the b quark mass and affect the neutralino relic density predictions of the model. We present some specific models with large SUSY phases which can accommodate the fermion electric dipole moment constraints and a neutralino relic density within the WMAP bounds.

1. Introduction

The recent Wilkinson Microwave Anisotropy Probe (WMAP) data allows a determination of cold dark matter (CDM) to lie in the range' = 0.1126~!:~!~. In this analysis we extend the mSUGRA framework to include CP phases in the gaugino sector 2 , which affects the loop corrections to the

*Speaker. tpermanent address of T. I.

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389

b quark mass and also affects the mixing between the neutral Higgs bosons. These corrections then affect relic density computations in important ways.

- 21400 2, E 1200

1000

800

600

400

200

0

M (GeV)

I

....... ....

50 100 150 200 250 300 350 m, (GeV)

Figure 1. Darker areas are favored by the phenomenological constraints described in Ref. [7].

Neutralino relic density (left) and dx-p (right) in mSUGRA at t a n p = 50.

2. mSUGRA Dark Matter

In most of the mSUGRA parameter space the LSP is almost purely a Bin0 B with a large relic density. However, we can classify three regions where R, can reach the WMAP bounds: (i) Coannihilation region 4: Relic abundance decreases due to coannihilations x - f when mi N m,. (ii) Hyperbolic Branch/Focus-point (HB/FP) region5: The p-term is small, such that xo may have large Higgsino fraction which enables a faster annihilation. (iii) Resonances on Higgs mediated channels6: Relic abundance constraints are satisfied by annihilation though resonant s-channel Higgs exchange. In Fig. 1, we present some representative mSUGRA predictions for R,h2 at large tanP7. In the next section we analyze the impact of enlarging this picture including CP phases using the following point in the point in the mSUGRA parameter space

tanp = 50, mo = ml/2 = = 600GeV. (1)

3. Phase Generalized mSUGRA

Within mSUGRA there are only two physical phases, B,,@A which are phases of p and Ao. These phases must be small (5 lo-') to satisfy the

390

60

55

a I B M

45

Phases (rad)

Figure 2. relic density constraints for the parameters oh Eq. (1) of the text.

Amb as a function of the indicated phases (left) and regions allowed by the

$(m3

Figure 3. R,h2 as a function of c 3 and ON for the parameters on Eq. (l), using the theoretically predicted value of Amb (black lines), Amb = 0 (light lines). Solid lines include all contributions, dashed lines (dot-dashed lines) only s-channel HI ( H 3 ) mediated annihilation to b6.

electric dipole moments (EDM) constraints

Id,( < 4 . 2 3 ~ 1 0 - ~ ~ e c r n , ldnl < 6 . 5 ~ 1 0 - ~ ~ e c m , C H ~ < 3 . 0 ~ 1 0 - ~ ~ c m . (2)

Large phases can be accommodated in several scenarios such as models with super heavy sferrnions for the two first generations * or models with a non-trivial soft term flavor structureg. Here, we assume a cancellation mechanism lo which becomes possible if we assume an extended SUGRA parameter space characterized by the parameters

m0, m1/2r tan P7 I A O l , Ofi, aA7 el 7 6 7 6 , (3)

where, ti is the phase of the gaugino mass Mi. The value of 1p1 is determined by imposing electroweak symmetry breaking.

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3.1. Loop Comct ion to the b Quark Mass

At the loop level the effective b quark coupling with the Higgs is given by"

= (hb 4- 6hb)bRbLHf + bhbbRbLHi* 4- H.C. (4) The correction to the b quark mass is then given directly in terms of Ahb and 6hb so that

Ahb bhb hb hb

Amb = [Re(-) tanb + R e ( - ) ] . ( 5 )

A full analysis of Amb is used12. Amb depends strongly on t3 and eP and weakly on CXA, (1, <2 as we see from the left panel of Fig. 2. The consequences for Rxh2 arising from the changes of Amb in this range can be understood from the qualitative analysis on the right panel of Fig. 2, where Am, is used as a free parameter. We observe that for a fix value of t anp the pseudo scalar Higgs mass can reach values in the range mA N mx/2 , allowing predictions for Rxh2 a on the WMAP bounds.

3.2. The Higgs sector CP-even CP-odd Mixing

CP violating phases induce mixing at one loop of the CP-even, H , h, and CP-dd, A , neutral tree level Higgs bosom: (H,h ,A) + ( H I , Hz,H3), where Hi (i=1,2,3) are the mass eigen states. The relevant couplingsb for the s-channel neutralino annihilation become

L = (Si + iS;y5)Hkxx + (C," + iC,Pys)Hkbb. (6)

The resonant annihilation cross-section behaves as

We observe that imaginary couplings (S") will dominate, since the real ones are suppressed by the factor 1 - 4 m i / s . Fig. 3 shows the changes of Rxh2 with & and 6,. The long light lines are obtained by setting Am, = 0, which implies that the Higgs masses remain almost constant along these lines and hence we see the effects of the CP phases on the vertices without the variation due to Am,. Also, mH1 - mH, and FH, - r H 3 , therefore large mixing are possible as we see in the partial contributions of H I (dash) and H3 (dot-dash) mediated s-channels.

use micrOMEGAs l3 for the computations of Rxh2 without phases. bWe use CPsuperH l4 in our computations.

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1

Figure 4. The neutralino relic density as a function of tanP for the three cases (i), (ii), (iii) of the text (left). Lines (I), (11) and (111) correspond to similar set of SUSY parameters for the case of vanishing phases. On the right we present the corresponding values of Am(,.

4. CP-Phases, EDM’s and Neutralino Relic Density

In Fig. 4 the neutralino relic density is displayed as a function of tanp for three cases given by: (i) mo = m1/2=IA~I = 300 GeV, C X A ~ = 1.0, <I = 0.5, 5 2 = 0.66, <3 = 0.62, 6, = 2.5; (ii) mo = m l p = = 555 GeV, ( Y A ~ = 2.0, <1 = 0.6, <2 = 0.65, <3 = 0.65, 6, = 2.5; (iii) rno = ml/2 = lAol = 480 GeV, ( Y A ~ = 0.8, <I = 0.4, <2 = 0.66, 53 = 0.63, 8, = 2.5. In all cases the EDM constraints (2) are satisfied for tan@ = 40 and their values are exhibited in table 1. We also observe that the WMAP bounds are also satisfied in the range of t anp exhibited in Fig.4.

Table 1. The EDMs for tanp = 40 for cases of text.

(ii) I 1.29 x I 1.82 x I 6.02 x lo-”” (iii) I 9.72 x I 4.19 x I 1.41 x lopz7

5. Conclusions

The SUSY threshold correction to mb, Amb, can induce large changes on the two heavier neutral Higgs bosons. For a given m, and certain values of Am, the resonance condition mHi - 2m, can be satisfied. This implies a neutralino relic density inside the WMAP bounds. Amb is strongly dependent on the SUSY phases 53 and d, . Hence, these phases can drive ah2 to the WMAP region leaving the possibility of choosing the other phases such

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that a cancellation mechanism keeps the fermion EDM predictions below the current experimental bounds.

Acknowledgments

MEG acknowledges support from the ’Consejeria de Educaci6n de la Junta de Andalucia’ and the Spanish DGICYT under contract BFM2003-01266. The research of TI and P N was supported in part by NSF grant PHY- 0139967. SS acknowledges support from the European RTN network HPRN-CT-2000-00148.

References 1. C. L. Bennett et al., Astrophys. J. Suppl. 148, 1 (2003); D. N. Spergel et al.,

Astrophys. J. Suppl. 148, 175 (2003). 2. M. E. Gomez, T. Ibrahim, P. Nath and S. Skadhauge, Phys. Rev. D 70,035014

(2004). 3. For a review see, A. H. Chamseddine, R. Arnowitt and P. Nath, Nucl. Phys.

Proc. Suppl. 101, 145 (2001) [arXiv:hep-ph/0102286]. 4. J. R. Ellis, T. Falk, K. A. Olive and M. Srednicki, Astropart. Phys. 13, 181

(2000); M. E. Gomez, G. Lazarides and C. Pallis, Phys. Rev. D 61, 123512 (2000); Phys. Lett. B 487, 313 (2000); Nucl. Phys. B 638, 165 (2002).

5. K.L. Chan, U. Chattopadhyay and P. Nath, Phys. Rev. D 58, 096004 (1998); J. L. Feng, K. T. Matchev and T. Moroi, Phys. Rev. D 61, 075005 (2000).

6. A. B. Lahanas, D. V. Nanopoulos and V. C. Spanos, Phys. Rev. D 62, 023515 (2000); J. R. Ellis, T. Falk, G. Ganis, K. A. Olive and M. Srednicki, Phys. Lett. B 510, 236 (2001); H. Baer and J. O’Farrill, JCAP 0404, 005 (2004).

7. D. G. Cerdefio et al, JHEP 0306, 030 (2003). 8. P. Nath, Phys. Rev. Lett.66, 2565(1991); Y. Kizukuri and N. Oshimo, Phys.

Rev. D 46,3025(1992). 9. S. -4be1, S. Khalil and 0. Lebedev, Nucl. Phys. B 606, 151 (2001);

G. C. Branco et al., Nucl. Phys. B 659, 119 (2003). 10. T. Ibrahim and P. Nath, Phys. Lett. B 418, 98 (1998); Phys. Rev. D57,

478(1998); 11. M. Carena and H. E. Haber, Prog. Part. Nucl. Phys. 50, 63 (2003) 12. T. Ibrahim and P. Nath, Phys. Rev. D 67, 095003 (2003). 13. G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys.

14. J. S. Lee et a1 Comput. Phys. Commun. 156, 283 (2004). Commun. 149, 103 (2002); for an updated version see hep-ph/0405253.

NEUTRINO MASSES, MIXINGS IN A MINIMAL SO(10) MODEL AND LEPTONIC FLAVOR VIOLATION*

B. DUTTA*, Y . MIMURA*, R.N. MOHAPATRA~ *Department of Physics, University of Regina,

Regina, Saskatchewan S4S OA2, Canada Department of Physics, University of Maryland,

College Park, M D 2OY42, USA

The minimal SO(10) model with one 10 and one 126 Higgs can fit the neutrino masses and mixing angles. Recently, a detailed analysis shows that the fit of solar, atmospheric mixing angles and mass ratios in this very predictive minimal model requires the Kobayashi-Maskawa phase to be either 0 (Type I seesaw) or to be much larger than 100°(Type I1 seesaw). In this talk, we propose a parity symmetric SO(10) model with an additional 120 Higgs multiplet. The parity symmetry makes the Yukawa matrices hermitian and all the parameters in Higgs sector are real. The model is more predictive since the number of parameters is less than the original minimal model. I will elaborate the predictions of this more minimal model.

1. Introduction

The simplest grand unified model which explains the small neutrino masses appears to be the SO(10) model due the following reasons: (i) the existence of right handed neutrino, NR, needed to implement the seesawl mechanism since it fits in with other standard model fermions in the 16 dimensional spinor representation (ii) it also contains the B-L symmetry2i3 needed to explain the right handed neutrino masses below the Planck scale and provides a group theoretic explanation of why neutrinos are necessarily Majorana particles.

The SO(10) models are appealing for neutrino mass studies, but a detailed quantitative predictions generally involve too many parameters. The most predictive SO(10) uses only one 10 and one 126 Higgs multiplet to generate fermion masses4. The predictions of this model are in contra-

*This work is supported in part by a National Science Foundation Grant PHY-0354401 and in part by Natural Sciences and Engineering Research Council of Canada.

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diction with the recent experimental data for the neutrinos and requires the CKM phase to arise from new p h y ~ i c s ~ 9 ~ . However the model with 10 + 126 + 120 Higgs fields7*' seems to successfully explain the quark masses, CKM angles, phase and the neutrino data. The model' has less nos. of parameters and will be described here.

2. The Parity Invariant Model for Neutrinos

The Yukawa superpotential of this model involves the couplings of the 16- dimensional matter spinor $i with 10 ( H ) , 126 (A), and 120(A) dimensional Higgs fields:

- -

The Yukawa couplings, h and f, are symmetric matrices, whereas h' is an anti-symmetric matrix due to SO(10) symmetry. They are all complex matrices in general. Under the G422 = SU(4), x s U ( 2 ) ~ x s U ( 2 ) ~ decomposition we have the following representations that contain the Higgs doublets of up and down type: one pair arises from H 3 (1,2,2), one pair comes from 3 3 (15,2,2), and two pairs come form A 3 (1,2,2) + (15,2,2). We assume that one pair of their linear combinations, H, and Hd, remains massless (mass is - O(u,k)) and become the MSSM Higgs doublets.The SO( 10) spinor $ is decomposed as $ = +$: We write SU(4) indices by p, v, s U ( 2 ) ~ indices by a, p and s U ( 2 ) ~ indices by dr, 8. We now define the parity transformation in the G422 basis.

H (qhp":)*, d26' c-) d2e. (2)

In the Higgs sector, the transformations of the (1,2,2) and (15,2,2) sub- multiplets under G422 are:

A'"": H (Aa":)*, A,""": ~f (Apya":)*.

A consequence of the parity symmetry (4), is that the coupling matrices h and f real and symmetric and h' antisymmetric and all the parameters in the Higgs potential are real. This considerably reduces the number of parameters in the theory and further makes the mass matrices for all charged fermions hermitian. The model has 15 parameters to fit the fermion masses and mixing angles and hence we can have three predictions in the neutrino sector.

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Figure 1. Predictions for different set of mass signatures (a) and (b) in Eqs.(4,5) are given.

The atmospheric mixing angle is plotted as a function of mass squared ratio.

Figure 2. tally allowed region in 3u of recent data fitting is 0.3 < tan' Osol < 0.6 and IUe31 < 0.26.

The relation between solar mixing angle and IUe3( is plotted. The experimen-

3. Solution for Neutrino Oscillation

In Fig.1, the prediction of the atmospheric mixing sin2 28A is shown as a function of the mass squared ratio. The lines (a) and (b) are for the mass signatures described in the following equations

(a) D, =diag( f , - ,+) , Dd =diag(-,+,+) D, =diag(f,-,+),(4)

(b) D, = diag(f , -, +), D, = diag(f, +, +).(5)

which give acceptable solutions for neutrinos. The strange quark mass plays an important role and we find solutions for neutrinos for m,(l GeV) in the range of 100-220 MeV. The experimental constraint sin2 28A > 0.9, needs the parameter region where the strange mass has a larger value and a larger

Dd = diag(+, +, +)

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1 0 -

0 5

L .

2 0 0 :

.E

n.5

value of bottom mass is preferred to obtain large atmospheric mixing. Once all quark, lepton data, the mass squared ratio R(=Amg/Ami)

and the solar mixing angle are fitted, we can predict lUe31 and ~ M N S P .

We show the correlation between solar mixing tan2 Osol and lUe31 in Fig.2. The two lines correspond to different mass squared ratios, R = 0.02 and 0.07. From the figure, we can see that Ami/Ami = 0.07 is not favored in the 3a range of the experimental data of solar neutrino and Ue3 for the case (a). In Fig.3, the prediction of the sin6MNSp is plotted in the case where R = 0.02. In Fig.4, we plot lUe31 randomly. Each point is obtained for different quark mass and mixings which are randomly generated in the experimentally allowed region. The model predicts a lower limit for IUe31 of about 0.1. In Fig.5, we show the KM phase dependence on the predictions of the model. The lines in the figure are drawn for the mass squared ration to be 0.02.

Since all the parameters of the model are now determined, it can be used to make other predictions. For example, The BR[p -+ e + 71 can be as large' as

~

: -

Figure 3. The prediction of MNSP phase is plotted as a function of the solar mixing angle. These lines (a) and (b) are plotted in the case A m ~ o , / A m ~ = 0.02 for different mass signature.

References

1. T. Yanagida, in Proceedings of the Workshop o n the Unified Theory and Baryon Number in the Universe, edited by 0. Sawada and A. Sugamoto (KEK, Tsukuba, 1979); S. L. Glashow, Cargese lectures, 1979; M. Gell-Mann, P. Ra- mond, and R. Slansky, in Supergravity, edited by D. Freedman and P. van

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0.05

0.w - 0.30 035 040 045 0 1 055 OM

Figure 4. The prediction of IUe31 and solar mixing angle is plotted as dots for randomly generated quark masses (with signature) and mixings in the experimentally allowed region. The lower bound of IUe31 exists in this model.

Figure 5. The relation between solar mixing angle and [Ue31 is plotted for various KM phases. Each plot is given in the case of mass signature (a) and mass squared ratio is 0.02.

Nieuwenhuizen (North-Holland, Amsterdam, 1979); R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).

2. J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974). 3. R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett. 44, 1316 (1980). 4. K. S. Babu and R. N. Mohapatra, Phys. Rev. 70, 2845 (1993). 5. H. S. Goh, R. N. Mohapatra and S. P. Ng, Phys. Lett. B 570, 215 (2003). 6. B. Dutta, Y. Mimura and R. N. Mohapatra, Phys. Rev. 69, 115014 (2004). 7. S. Bertolini, M. F'rigerio and M. Malinsky, hep-ph/0406117; W. M. Yang and

Z. G. Wang, hepph/0406221. 8. B. Dutta, Y . Mimura and R. N. Mohapatra, hep-ph/0406262.

PROTON DECAY AND THE PLANCK SCALE

DANIEL T. LARSON Theoretical Physics Group, Lawrence Berkeley National Laboratory, University

of Calafornia, Berkeley, CA 94720, USA

Department of Physics, University of California, Berkeley, CA 94720, USA

Even without Grand Unification, proton decay can be a powerful probe of physics at the highest energy scales. Supersymmetric theories with conserved R-parity contain Planck-suppressed dimension 5 operators that give important contributions to nucleon decay. These operators are likely controlled by flavor physics, which means current and near future proton decay experiments might yield clues about the fermion mass spectrum. I present a thorough analysis of nucleon partial lifetimes in supersymmetric one-flavon Froggatt-Nielsen models with a single V( l)x family symmetry which is responsible for the fermionic mass spectrum as well as forbidding R-parity violating interactions. Many of the models naturally lead to nucleon decay near present limits without any reference to grand unification.

1. Two Myths

It is often loosely stated that the observation of proton decay implies the existence of a grand unified theory (GUT). However, it is well known that generic supersymmetric (SUSY) theories possess nonrenormalizable operators that violate baryon- and lepton-number ( B and L, respectively). In an effective field theory these operators are necessarily present, and can be dangerous even when suppressed by the Planck scale, Mpl [l].

It is also often sloppily said that R-parity prohibits proton decay in SUSY theories. Though R-parity prohibits the renomalzzable B- and L-violating operators, it still allows the nonrenormalizable superpotential terms &QQQL and &?ODE, which contain dimension five operators that can lead to rapid proton decay. In fact, with generic O(1) coefficients, weak scale squark masses, and M N Mpl, the proton lifetime comes out to be sixteen orders of magnitude below the current experimental limit! This embarrassment has been called SUSY’s “dirty little secret”.

This “dirty secret” is most likely cleaned up by the physics that generates flavor. If a broken flavor symmetry is the source of the small Yukawa

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couplings, as in a F'roggatt-Nielsen model [2], then that same flavor symmetry will govern the coefficients of the higher dimensional operators mentioned above, allowing the suppression of their coefficients to be predicted.

In what follows we will survey the predictions for nucleon lifetimes as computed in [3] for the class of specific, string-motivated models introduced in [4]. These models are based on a single, anomalous U( l )x Froggatt- Nielsen flavor symmetry but do not require grand unification.

2. Proton Decay Operators

In GUT theories the exchange of X gauge bosons generates B- and L- violating four-fermion operators suppressed by two powers of MGUT, yielding the proton decay rate - *m:. For the standard model, MGUT - 1015 GeV leading to a proton lifetime around 1031 years, well below the current limits which now exceed years for many decay modes [5]. In a SUSY-GUT the unification scale is a factor of 10 higher, suppressing the rate from these dimension six operators by four more orders of magnitude, evading the experimental constraint. However, colored Higgs exchange generates dimension five couplings between fermions and their superpartners which lead to four-fermion operators that are suppressed by one power of MGUT and one power of the scalar soft mass, msoft. The proton decay rate becomes r - A m : . Since we expect msoft << MGUT, proton decay from these operators is relatively enhanced and very dangerous, excluding the minimal SU(5) SUSY-GUT [6] .

Even without grand unification an effective field theory should contain all allowed higher dimensional operators suppressed by Mpl, including the dimension-5 B and L violating operators mentioned above. They lead to proton decay with a rate r - MZ$2 mg. If such operators were present with C3( 1) coefficients it would be disastrous. Therefore we need to consider the degree to which these coefficients are suppressed by flavor physics.

MGUT

M G U T ~ ~ ~ ~ ~

PI soft

3. Flavor Model Framework

In the class of models presented in [4] the MSSM superfields are charged under a horizontal U( l )x symmetry that is spontaneously broken when a flavon field, A, gets a nonzero VEV generated by string dynamics. Both the MSSM Yukawa terms and the higher dimensional operators are then suppressed by the ratio F. = (A)/Mpl raised to the the appropriate power necessary to conserve U(1)x. The string dynamics predicts E - sin Bc. The

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X-charges for the MSSM superfields are restricted by sum rules that ensure anomaly cancellation through the Green-Schwarz mechanism [7], and further constrained by requiring that they lead to the observed fermion mass spectrum and mixings, including neutrinos and the MNS matrix, and by requiring that R-parity be an exact, accidental symmetry of the low energy theory. It is non-trivial that these requirements can be simultaneously fulfilled. There are 24 distinct models with these properties, parametrized by three integers z, y, and z that are related to tanp, the CKM texture, and the MNS texture, respectively. (See [4, 31 for details.)

4. Results

The 24 models each make predictions for the parametric size of the coefficients appearing in front of the dimension-five B and L violating operators, allowing us to compute the lifetime of the proton predicted by each model.

There are two types of uncertainties that enter into our predictions. The first type of uncertainty comes from our ignorance of P p , the overall scale of the matrix elements for proton decay as computed using the chiral Lagrangian technique [8], and our ignorance of the superpartner mass scale msoft. These two uncertainties will hopefully be reduced with time. The second type of uncertainty is inherent in our effective field theory framework and comes from the unknown O( 1) coefficients that appear in front of each higher dimensional operator. We estimate the effect of the unknown phases of these coefficients by adding contributing amplitudes either incoherently, destructively, or constructively.

Figure 1 shows the partial lifetimes for the most constraining mode, p + K+fi, for all 24 models labeled by the parameters 2, y, and z . Already many of the models are disfavored, unless they have significant cancellations between contributing amplitudes. The models that are least constrained are those with lower tanP (higher x). However, the uncertainties in PP and msoft can potentially change the overall scale of the prediction by the factors shown graphically to the right of Figure 1.

For the proton the next modes to appear after p -, K+fi are generally p + Toe+, p + T O P + , and p + KopL+. In Figure 2 we show the expected lifetime for those four modes in the 12 models with tanP 5 10. We see that most models which survive the constraint from p -+ K+fi have a lifetime for p + r0p+ that is within two or three orders of magnitude of the experimental bound, while p + Kop+ is only slightly larger, and p --+ Toe+ can potentially be smaller. This raises the exciting possibility of two or

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a d a=1 x=2 1=3 a d a=1 a=2 a=3 (Anarchy) ,el (Semi-Anarchy)

Models

Figure 1. Plot of proton partial lifetime in years for the mode p -+ K+p. Within each half t an0 decreases from left to right. The error bars are not lo bars, but show the shift from incoherent addition of amplitudes (central value) due to destructive and constructive interference. The horizontal line shows the experimental lower limit of 1.6 x years. The scales on the right show the overall shift caused by changing either msoft or pp away from = 1 and Op = 0.01.

Figure 2. Comparison of proton lifetime in years for four different decay modes. The upper plot shows the computed lifetime for p 4 K f C (x , left axis) and p -+ @p+ (A, right axis). The lower plot shows p + roe+ ( 0 , left axis) and p -+ x0p+ (0, right axis).

three decay modes being detected in the coming round of experiments. Figure 3 shows the partial lifetimes in 11 proton and neutron decay

modes for three models, illustrating how various modes can discriminate between models. For example, any mode involving a muon in the final state can differentiate Model 1 from Models 2 and 3.

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Figure 3. Plot of nucleon lifetime in years for eight proton decay modes (left side) and five neutron decay modes (right side). The different symbols represent different U(l)x charge assignments. The experimental limit for each mode is shown as a vertical column.

5. Conclusion

Focusing on a class of string motivated Froggatt-Nielsen models that explain the masses and mixings of all SM fermions while automatically enforcing R-parity, we have shown that nucleon decay is a powerful probe of Planck scale physics. In these models Planck suppressed operators lead to nucleon lifetimes that are generically right near the current experimental limits, even without grand unification. Since current bounds constrain many of the 24 models of this type, we conclude that proton decay is already probing physics at the Planck scale.

Acknowledgments

DTL thanks R. Harnik, H. Murayama, and M. Thormeier for fruitful collaboration. This work was supported in part by the DOE under contract DEAC03-76SF00098 and in part by NSF grant PHY-0098840.

References 1. H. Murayama and D. B. Kaplan, Phys. Lett. B 336, 221 (1994) [arXiv:hep

ph/9406423]. 2. C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B 147, 277 (1979). 3. R. Harnik, D. T. Larson, H. Murayama and M. Thormeier, arXiv:hep

ph/0404260. 4. H. K. Dreiner, H. Murayama and M. Thormeier, arXiv:hep-ph/0312012. 5. E. Kearns, Talk at Snowmass 2001,

http://hep.bu.edu/-kearns/pub/kearns-pdk-snowmass.pdf. 6. H. Murayama and A. Pierce, Phys. Rev. D 65, 055009 (2002) [arXiv:hep

ph/0108104]. 7. M. B. Green and J. H. Schwarz, Phys. Lett. B 149, 117 (1984). 8. M. Claudson, M. B. Wise and L. J. Hall, Nucl. Phys. B 195, 297 (1982).

SUPERSYMMETRIC CONTRIBUTIONS TO THE CP ASYMMETRY OF THE B + cPK~ AND B v‘Ks

SHAABAN KHALIL Mathematics Department, German University in Cairo- GUC, New Cairo city,

El Tagamoa El Khames, Egypt.

We analyze the CP asymmetry of the B -+ 4Ks and B + 7’Ks processes in general supersymmetric models. We adopt the QCD factorization method for evaluating the corresponding hadronic matrix elements. We show that chromomagnetic type of operator may play an important role in accounting for the deviation of the mixing CP asymmetry between B -+ 4Ks and B -+ J /$Ks processes observed by Belle and BaBar experiments. We aIso show that due to the different parity in the final states of these processes, their supersymmetric contributions from the R- sector have an opposite sign, which naturally explain the large deviation between their asymmetries.

1. Introduction

One of the most important tasks for B factory experiments would be to test the Kobayashi-Maskawa (KM) ansatz for the flavor CP violation. The flavor CP violation has been studied quite a while, however, it is still one of the least tested aspect in the standard model (SM). Although it is unlikely that the SM provides the complete description of CP violation in nature (e.g. Baryon asymmetry in the universe), it is also very difficult to include any additional sources of CP violation beyond the phase in the CKM mixing matrix. Stringent constraints on these phases are usually obtained from the experimental bounds on the electric dipole moment (EDM) of the neutron, electron and mercury atom. Therefore, it remains a challenge for any new physics beyond the SM to give a new source of CP violation that may explain possible deviations from the SM results and also avoid overproduction of the EDMs. In supersymmetric theories, it has been emphasised that there are attractive scenarios where the EDM problem is solved and genuine SUSY CP violating effects are found.

Recently, BABAR and Belle collaborations announced a 2.70 deviation from sin2P in the B -+ +Ks process 2 v 3 . In the SM, the decay process of

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B -+ 4K is dominated by the top quark intermediated penguin diagram, which do not include any CP violating phase. Therefore, the CP asymmetry of B .+ J/$Ks and B -+ 4Ks in SM are caused only by the phase in B" - B mixing diagram and we expect S J / Q K s = S + K ~ where Sf,, represents the mixing CP asymmetry. The B .+ q'Ks process is induced by more diagrams since q' meson contains not only sd state but also ufi and ddstates with the pseudoscalar mixing angle 0,. Nevertheless, under an assumption that its tree diagram contribution is very small, which is indeed the case, one can expect S + K ~ = S,,IK~ 2 f 4 as well. Thus, the series of new experimental data surprised us:

---o

rxp. = 0.734 f 0.054, r x p .

Syks = 0.33 f 0.41

J/dJKs +Ks = -0.39 f 0.41,

It was pointed out that the discrepancy between Eq. (1) and Eq. (2) might be explained by new physics contribution through the penguin diagram to B -+ ~ K s . However, in that case, a simultaneous explanation for the discrepancy between Eq. (2) and Eq. (3) is also necessary. We show our attempts to understand all the above experimental data within the Supersymmetric models.

Including the SUSY contribution, the effective Hamiltonian H,affBB=l for these processes can be expressed via the Operator Product Expansion (OPE) as

where A, = vp,V,*,, with Vpb the unitary CKM matrix elements satisfying + Au + A, = 0, and Ci = c i ( p b ) are the Wilson coefficients at low energy

scale ,Ub 21 mb.

As emphasised in 6 9 7 , the leading contribution of both gluino and chargino to AB = 1 processes come from the chromomagnetic penguin operator Og(Og). The corresponding Wilson coefficient is given by

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and

where x W i = m&/mii, xi = mZi/fi2, Z i = f i 2 / r n i i , and x i j = mzi/mzi. The loop functions P z ( L R ) ( x ) and also the functions M i ( x ) , i = 1 ,3 ,4 can be found in Ref.8. Finally, U and V are the matrices that diagonalize chargino mass matrix.

It is now clear that the part proportional to LR mass insertions in C& which is enhanced by a factor m g / m b would give a dominant contribution. Also the part proportional to the LL mass insertion in C& is enhanced by m x / m b and could also give significant contribution.

2. Can we explain the experimental data of 5 ' 9 ~ ~ in SUSY?

Following the parametrisation of the SM and SUSY amplitudes in Ref.', S ~ K ~ can be written as

where R+ = (ASUSY/ASM(, 06 = arg(ASUSY/ASM), and 612 is the strong phase.

We will discuss in the following whether the SUSY contributions can make 5 ' 4 ~ ~ negative. For me = mg = 500 GeV and adopting the QCD factorization mechanism to evaluate the matrix elements, one obtains

R Q C D F I - 9 - - (-0.14 e - i o . l ( 6 : ~ ) 2 3 - 127 X e - i 0 ' 0 8 ( 6 f ~ ) 2 3 } + { L * R} , (9)

4

while in the case of chargino exchange with gaugino mass M 2 = 200 GeV, p = 300 GeV, and f i f R = 150 GeV, we obtain, for t a n p = 40

RZCDFIX* N 1.89 x e-i0.07 (6; , )32 - 0.11 x e-i0.17 ( 6 g L ) 3 2

+ 0.43 x edio.O7 ( 6 t L ) 3 1 - 0.02 x e-i0.17 (6&)31 . (10)

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Figure 1. gluino and chargino contributions respectively.

S ~ K ~ as a function of arg[(b&&3] (left) and arg[(bFL)32] (right) with

From results in Eqs.(S)-(lo), it is clear that the largest SUSY effect is provided by the gluino and chargino contributions to the chromomagnetic operator which are proportional to ( 6 $ ~ ) 2 3 and (6:~)32 respectively. However, the b --+ sy constraints play a crucial role in this case. For the above SUSY configurations, the b + sy decay set the following constraints on gluino and chargino contributions, respectively 1(6$R)231 < 0.016 and 1(6EL)321 < 0.1. Implementing these bounds in Eqs.(S)-(lo), we see that gluino contribution can achieve larger value for R4 than chargino one.

We present our numerical results for the gluino and chargino contributions to CP asymmetry S Q K ~ in Fig. 1. We plot the CP asymmetry as function of the phase of (6&)23 for gluino dominated scienario and Arg[(bEL)32] for the chargino dominanted model. We have scanned over the relavant SUSY parameter space, assuming SM central values as in table 1. Namely, the average squark mass f i , gluino mass mg. Moreover we require that the SUSY spectra satisfy the present experimental lower mass bounds. In particular, mg > 200 GeV, f i > 300 GeV. In addition, we scan over the real and imaginary part of the corresponding mass insertions, by requiring that the branching ratio (BR) of b + sy and the Bo - Bo mixing constraints are satisfied at 95% C.L.. Also we have scanned over the full range of the QCD factorization parameters P A , H and ~ A , N , We remind here that these parameters are taken into account for the (unknown) infrared contributions in the hard scattering and annihilation diagrams respectively.

As can be seen from this figure, the gluino contributions proportional to (6iR)23 have chances to drive S Q K ~ toward the region of larger and negative

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values. While in the chargino dominated scenario negative values of 5’4 cannot be achieved. The reason why extensive regions of negative values of 5’4 are excluded here, is only due to the b -+ sy constraints. Indeed, the inclusion of (6:,)32 mass insertion can generate large and negative values of 5’4, by means of chargino contributions to chromomagnetic operator &sg which are enhanced by terms of order mX*/mb. However, contrary to the gluino scenario, chargino contributions to c 8 g are not enhanced by colour factors. Therefore, large enhancements of the Wilson coefficient Cs,, leave unavoidablly to the breaking of b 4 sy constraints. As shown in Ref.7, by scanning over two mass insertion but requiring a common SUSY CP violating phase, a sort of fine tuning to escape b -+ sy constraints is always possible, and few points in the negative regions of 5’4 can be approached.

3. What happened to the B -, q’Ks process?

Although B 4Ks and B -+ rfKs are very similar processes, the parity of the final states can deviate the result. In B -+ ~ K s , the contributions from Ci and ci to the decay amplitude are identically the same (with the same sign), while in B -+ $Ks, they have sign difference. This can be simply understood by noticing that

(4KslQi IB) = (4KslQi IB) . (11)

which is due to the invariance of strong interactions under parity transformations, and to the fact that initial and final states have same parity. However, in case of B -+ q’Ks transition, where the initial and final states have opposite parity, we have

( d K s IQi IB)QCDF = - ( d K s IQi WQCDF. (12)

As a result, the sign of the RR and RL in the gluino contributions are different for B -+ 4Ks and B -+ $Ks lo. Using the same SUSY inputs adopted in Eqs. ( lo) , (9). For gluino contributions we have

while for chargino exchanges we obtain

409

-1 ' I x 0 n/Z .id2 n

I QCDF (6uLL)32 1 1 I n .n/Z 0 n12 n

Figure 2. gluino and chargino contributions respectively.

S q f ~ s as a function of arg[(biR)23] (left) and arg[(6xL)32] (right) with

We show our results for gluino and chargino contributions in Fig. 2, where we have just extended the same analysis of B -+ ~ K s . Same conventions as in Fig. 1 for B -, $Ks have been adopted here. As we can see from these results, there is a depletion of the gluino contribution in Sq/ , precisely for the reasons explained above. Negative regions are disfavoured, but a minimum of S,! N 0 can be achieved. Respect the chargino contributions, it is clear that it can imply at most a deviation from SM predictions of about f 2 0 %.

4. Conclusions

We studied the supersymmetric contributions to the CP asymmetry of B + 4Ks and B -+ rfKs in a model independent way. We found that the observed large discrepancy between SJ,+,K~ and S ~ K ~ can be explained within some SUSY models with large (6~j3)23 or ( 6 ~ ~ ) 2 3 mass insertions. We showed that the SUSY contributions of (6RR)23 and (6RL)23 to B ---$ q5Ks and B -+ r]'Ks have different signs. Therefore, the current observation, s + ~ ~ < s , , ! ~ ~ , favours the (6RR,RL)23 dominated models.

References

1. S. Abel, S. Khalil and 0. Lebedev, Nucl. Phys. B 606, 151 (2001). 2. K. Abe et al. [Belle Collaboration], arXiv:hep-ex/0207098. 3. B. Aubert et al. [BABAR Collaboration], arXiv:hep-ex/0403026. 4. B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 91 (2003) 161801.

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5. Y. Nir, Nucl. Phys. Proc. Suppl. 117 (2003) 111 [arXiv:hep-ph/0208080]. 6. S. Khalil and E. Kou, Phys. Rev. D 67 (2003) 055009 [arXiv:hep-

ph/0212023]. 7. D. Chakraverty, E. Gabrielli, K. Huitu and S. Khalil, Phys. Rev. D 68:

095004 (2003) [arXiv:hep-ph/0306076]. 8. E. Gabrielli, K. Huitu and S. Khalil, in preparation. 9. M. Ciuchini, E. F’ranco, A. Masiero and L. Silvestrini, Phys. Rev. D 67 (2003)

0750 16 [arXiv: hep-ph/02 123971. 10. S. Khalil and E. Kou, arXiv:hepph/0303214.

HADRONIC DECAY OF THE GRAVITINO IN THE EARLY UNIVERSE AND ITS IMPLICATIONS TO INFLATION*

MASAHIRO KAWASAKI Institute for Cosmic Ray Research, University of Tokyo

Kashiwa, Chiba 277-8582, Japan

KAZUNORI KOHRI+ Department of Earth and Space Science, Osaka University

Toyonaka, Osaka 560-0043, Japan

TAKE0 MOROI Department of Physics, Tohoku University, Sendai 980-8578, Japan

We discuss the effects of the gravitino on the big-bang nucleosynthesis (BBN), paying particular attention to the hadronic decay mode of the gravitino. We will see that the hadronic decay of the gravitino significantly affect the BBN and, for the case where the hadronic branching ratio is sizable, very stringent upper bound on the reheating temperature after inflation is obtained.

1. Introduction

It has long been regarded that the gravitino, superpartner of the graviton in the supergravity theory, may cause serious problem in cosmology1i2. This is because the primordial gravitinos produced in the very early universe decay with vary long lifetime. In particular, if the gravitino mass m312 is smaller than N O(10 TeV), its lifetime is expected to become longer than 1 sec so the primordial gravitinos decay after the big-bang nucleosynthesis (BBN) starts. Even with the inflation, such a problem may not be solved; even if the gravitinos are diluted by the inflation, gravitinos are produced by the scattering processes of the thermal particles after the reheating.

'This work is supported by the Grants-in Aid of the Ministry of Education, Science, Sports, and Culture of Japan No. 14540245 (MK), 15-03605 (KK) and 15540247 (TM). +Present address: Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cam- bridge, MA 02138, U.S.A.

41 1

412

As we will discuss later, the gravitino abundance becomes larger as the reheating temperature becomes higher. Consequently, in order not to spoil the success of the BBN scenario, upper bound on the reheating temperature is obtained, and the primordial gravitinos provide stringent constraints on cosmological scenarios based on (local) supersymmetry.

Thus, the effects of the gravitinos on the BBN have been intensively studied in many studies. In particular, in the past, the BBN with hadrodis- sociation processes induced by hadronic decays of long-lived particles was studied in several articles3. After those studies, however, there have been significant theoretical, experimental and observational progresses in the study of the BBN. Thus, we performed a new analysis taking account of those progresses, paying a special attention to the hadronic decay of the gravitino4. The most important improvements compared to the old works are as follows. (i) We carefully take into account the energy loss processes for high-energy nuclei through the scattering with background photons or electrons. In particular, dependence on the cosmic temperature, the initial energies of nuclei, and the background 4He abundance are considered. (ii) We adopt all the available data of cross sections and transfered energies of elastic and inelastic hadron-hadron scattering processes. (iii) The time evolution of the energy distribution functions of high-energy nuclei are computed with proper energy resolution. (iv) The JETSET 7.4 Monte Carlo event generator5 is used to obtain the initial spectrum of hadrons produced by the decay of gravitino. (v) The most resent data of observational light element abundances are adopted. (vi) We estimate uncertainties with Monte Carlo simulation which includes the experimental errors of the cross sections and transfered energies, and uncertainty of the baryon to photon ratio.

2. Outline of the Analysis

Let us briefly discuss the outline of our analysis. In our study, we first calculate the gravitino abundances as a function of the reheating temperature,

which is defined as TR sM:r;nf ) , where rinf is the decay rate of the inflaton, and g* = 228.75 is the effective number of the massless degrees of freedom in the MSSM. By numerically solving the Boltzmann equations governing the evolution of the number density of the gravitino, we found that the number density of the gravitino n3/2 normalized by the entropy

114

(

413

DECAY

I Hadronic I Radiative

Figure 1. Outline of our analysis.

density s is well approximated by

Thus, the number density of the primordial gravitino is approximately proportional to TR and hence, with higher reheating temperature, effects on the BBN becomes more significant.

Once the primordial abundance of the gravitino is given, we consider the effects of the decay of the gravitino on the BBN. Importantly, the interaction of the gravitino is well constrained by the local supersymmetry and its

414

lifetime is calculable. (For the detailed values, see the next section.) Here, we assume several reasonable values of the hadronic branching ratio B h of the gravitino, and calculated the abundances of the light elements taking account of the effects of hadronic and electro-magnetic showers induced by the decay products of the unstable gravitino. Outline of our treatments of the hadronic and electro-magnetic processes is schematically shown in Fig. 1, and the details of our study are discussed in the full papers4y6. (Notice that there are other recent studies on the BBN scenario with late-decaying exotic particles7.)

We compare the theoretically predicted values of the light-element abundances with the observations. As observational constraints on the light element abundances, we adopt the following values, D/H = (2.8f0.4) x 4He mass fraction Yp = 0.238f0.002f0.005 by Fields and Olive (FO)9 and Yp = 0.242~0.002(f0.005),y,t by Izotov and Thuan (IT)lO, 10g,,(~Li/H) = -9.66 f 0.056(f0.3),y,t11, 6Li/7Li < 0 .07(20)~~ , and 3Hr/D < 1.13(20)13. The above errors are at lo level unless otherwise stated. Then, we derive upper bound on the reheating temperature requiring that the theoretical predictions be consistent with the observations.

3. Results

Now, we show the results of our analysis. Here, we consider two typical case. The first case is that the gravitino dominantly decays into the gluino pair; in this case, the hadronic branching ratio is expected to be 1 and, in addition, the lifetime of the gravitino is estimated to be

( 10Xd-3 ~ 3 / 2 ( $ ~ , , -+ g + 8) II 6 x lo7 sec x

The second case is that the gravitino decays into the photon and the lightest neutralino (which we assume to be the photino); then, the hadronic branching ratio is expected to be N 0(10-2 - and the lifetime is given by

For these two cases, we have derived the upper bound on the reheating temperature.

In Fig. 2, upper bound on the reheating temperature after the inflation is plotted as a function of the gravitino mass. The results indicates that, as the hadronic branching ratio becomes larger, constraint on the reheating temperature becomes severer. For the case with B h = 1, for

415

log 3/2/sec log,o(.r,/,/sec) 6 4 2 0 - 2 8 6 4 2 0 - 2

10'0 10'0

100 1 OD

2 108 108

107 107

E-p: 108 108

105 105

104 104

0)

102 103 104 1 0 5 102 103 104 105 m 3 / 2 (GeV) m 3 / 2 (GeV)

Figure 2. branching ratio of the gravitino is taken to be 1 (left) and regions are the excluded region for the case with B h = 0.

Upper bound on the reheating temperature after the inflation. The hadronic (right). The shaded

example, behavior of the upper bound can be understood in the following way. For m3/2 5 200 GeV, energetic neutron is likely to decay before scattering off the background nuclei. In this case, effects of the hadronic decay is not so significant while, in this case, effects of the photodissoci- ation becomes comparable to or more significant than the hadrodissocia- tion. Then, the strongest constraint is from the overproduction of 3He; For 200 GeV 5 m3/2 5 7 TeV, energetic hadrons (in particular, neutron) is hardly stopped by the electro-magnetic processes and hence the hadrodis- sociation processes become the most efficient. In particular, in this case, non-thermal productions of D and 6Li provide the most stringent constraint; For 7 TeV 5 m3/2 5 100 TeV, the p H n conversion processes are efficient and significant amount of p may be converted to n resulting in the enhancement of 4He. In this case, the constraint from the overproduction of 4He is the most significant; For m3/2 2 100 TeV, gravitino decays before the BBN starts. In this case, no upper bound is obtained on the reheating temperature.

4. Summary

In our study, we have studied the effects of the unstable gravitino on the BBN, paying particular attention to the hadronic decay modes. As we have emphasized, as the hadronic branching ratio becomes larger, constraints become more stringent.

416

Our results have significant implications. In particular, for the gravity- mediated supersymmetry breaking, gravitino mass is expected to be - O(100) GeV. In this case, even if the hadronic branching ratio is - 0(10-3), the reheating temperature is constrained to be smaller than lo6 - lo8 GeV. If the gravitino mass is much larger than - O(100) GeV, the constraint on TR may be relaxed. With such gravitino, however, the hadronic branching ratio would be close to 1 since, in such a case, all the superpartners of the standard-model particles are expected to be lighter than the gravitino from the naturalness point of view. (Such a mass spectrum may be realized in the anomaly-mediated supersymmetry breaking scenario14 .) For m3/2 - O(10 - 100) TeV with Bh - 1, the upper bound is given by TR 5 lo7 - 1O1O GeV. For the cosmology, since the reheating temperature is required to be very low when the gravitino mass is O(100 GeV - 1 TeV), baryogenesis should occur with very low reheating temperature. This fact imposes significant constraints on some of the scenarios of the baryogenesis, in particular for the leptogenesis scenario with right-handed neutrinos15.

References

1. S. Weinberg, Phys. Rev. Lett. 48, 1303 (1982). 2. For more details of the gravitino problem, see, for example, T. Moroi,

arXiv:hep-ph/9503210, and reference therein. 3. S. Dimopoulos et al., Astrophys. J. 330, 545 (1988); Nucl. Phys. B 311, 699

(1989); M. H. Reno and D. Seckel, Phys. Rev. D 37, 3441 (1988). 4. M. Kawasaki, K. Kohri and T. Moroi, arXiv:astro-ph/0408426. 5. T. Sjostrand, Comput. Phys. Commun. 82, 74 (1994). 6. M. Kawasaki and T. Moroi, Prog. Theor. Phys. 93, 879 (1995); Astrophys. J.

452, 506 (1995); E. Holtmann, M. Kawasaki, K. Kohri and T. Moroi, Phys. Rev. D 60, 023506 (1999); M. Kawasaki, K. Kohri and T. Moroi, Phys. Rev. D 63, 103502 (2001); K. Kohri, Phys. Rev. D 64, 043515 (2001).

7. R. H. Cyburt et al., Phys. Rev. D 67, 103521 (2003); K. Jedamzik, arXiv:astro- ph/0402344; arXiv:astro-ph/0405583.

8. D. Kirkman et al., Astrophys. J. Suppl. 149, 1 (2003). 9. B.D. Fields and K.A. Olive Astrophys. J.506, 177 (1998). 10. Y. I. Izotov and T. X. Thuan, arXiv:astro-ph/0310421. 11. P. Bonifacio et al. Astron. Astrophys. 309, 91 (2002). 12. V.V. Smith, D.L. Lambert, and P.E. Nissen, Astrophys. J. 408 262 (1993). 13. J. Geiss, in Origin and Evolution of the Elements, edited by N. Prantzos et

al., 89 (Cambridge University, 1993). 14. L. Randall and R. Sundrum, Nucl. Phys. B 557, 79 (1999); G. F. Giu-

dice, M. A. Luty, H. Murayama and R. Rattazzi, JHEP 9812, 027 (1998); J. -4. Bagger, T. Moroi and E. Poppitz, JHEP 0004, 009 (2000).

15. M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986).

INFLATION AND LEPTOGENESIS IN SUSY LEFT-RIGHT MODEL AND ITS CONJECTURE ABOUT b - T

UNIFICATION

QAISAR SHAFI AND ARUNANSU SIL Bartol Research Institute,

21 7 Sharp Labomto y, Newark, Delaware, USA

E-mail: shafiObartol.udel.edu; asilQbartol.udel.edu

We present a leptogenesis scenario through the decay of heavy triplets following smooth hybrid inflation within a moderate extension of the left-right supersymmetric gauge group S u ( 2 ) ~ x s U ( 2 ) ~ x U ( ~ ) B - L . Baryon number conservation and the solution of the fi-problem are provided by additional symmetries. All the right handed neutrinos are heavier than inflatons. Then we have studied neutrino mass matrix generated by the triplet VEVs having the parameters constrained by leptogenesis and inflation scenario and found that the production of the required lepton asymmetry of the universe is not only right enough to be consistent with the neutrino mass and mixing angles predicted by the recent data but also can provide an estimate of b - T unification in the context of a minimal susy SO(10) model.

I. Introduction

The recent evidence for neutrino masses and mixings indicates the physics beyond the Standard Model. The possibility that the neutrino masses are associated with the lepton number violation put the mechanism of leptogenesis' on a firm basis as an explanation of the baryon asymmetry of the universe (BAU). Another important aspect of recent cosmology and particle physics is the inflationary scenario2 of the universe. Our aim is to relate the inflation, leptogenesis and neutrino mass in a single framework. Lepton asymmetry can be produced also by the decay of the heavy triplets3 and the ingredients are present in left-right supersymmetric model. We choose the symmetry of our model in such a way that the right handed neutrinos are not playing any role either to the generation of lepton asymmetry or to the neutrino mass as their masses turn out to be - 10l6 GeV, the ( B - L ) breaking scale. Here we consider the smooth hybrid inflation

417

418

model4 consistent with our choice of symmetry group. The left-handed triplets are responsible for the leptogenesis as well as for the production of light neutrino mass. As SUSY LR model can be embedded in a minimal supersymmetric SO(10) model5, we have a chance to relate the neutrino masses and mixings with the inflation and leptogenesis parameters which can also provide some predictions about b - T unification.

11. The Model

Our supersymmetric model is based on the gauge group SU(3), x s U ( 2 ) ~ x su(2)R x U ( ~ ) B - L . The quark and lepton superfields have the transformation properties: Q = (3,2,1,$); Qc = (3*,1,2,-5); L = (1,2,1, -1); L, = (1,112,1) where the numbers in the brackets denote the quantum numbers under SU(3),, s U ( 2 ) ~ and sU(2)R and U ( ~ ) B - L respectively. The higgs sector consists of H = (1 ,2 ,2 ,0); AE = (1,3,1,2); A; = (1 ,3 ,1 , -2) (a = 1,2); A R = (1 ,1 ,3 , -2); AR = (1 ,1 ,3 , 2)a. We have the whole superpotential,

W = Wl + ~ A ; A ; A R A ~ + -HHAZiiR 7" + f;LLA; + MS MS 6

f 2 L c L c A ~ + Y'LLCH + Y'QQCH + T S H H A R i i ~ (1)

where WI = S [ w - p 2 ] is responsible for inflation. Here the SU(2), generation and color indices are suppressed. S is a gauge singlet superfield, p is a superheavy mass scale and M s is a cutoff scale of the theory. The U(1)R and 2 4 charges of the superfields are defined as follows:

MS

Ms

Table 1. perlields.

The u ( 1 ) ~ and 2 4 charges are specified for different su-

Charges S A: A; AR AR H L L, Q Qc

2 4 1 - 1 1 1 - l i - i 1 - i 1 R 1 0 1 0 o o + $ + $

11.1. Inflation and Reheating

Using the D-flatness condition A& = A;, we see that the SUSY vacuum lies at [ ( A R ) ~ = I(AR)I = M = &- and (S) = 0. The Wl leads to a

*CP violation for the production of lepton asymmetry is not possible with a single A L , so we have to consider two pairs of conjugate left handed superfields A L , ~ L .

419

smooth hybrid inflation scenario4. There is no need to consider radiative corrections to get slope of the inflationary path, an in-built inclination is present here in contrast with other hybrid inflation. Now to be consistent with COBE results6, the quadrupole anisotropy the number of e-foldings NQ - 60 and assuming M - 10l6 GeV, we get p N 6.15 x 1014 GeV and MS N 1.63 x 1017 GeV.

After the end of inflation, the inflaton fields enter smoothly into an era of damped oscillation about the SUSY minima and thus decay. The inflaton consisting of two complex scalar fields S and 8 = (& + @)/& (&#I = A; - M and d$ = A; - M ) which can decay into a pair of triplets (A;, A;) and their fermionic partners (A;, A;). The corresponding decay width of both S and 8 turns out to be rinf = &(-&)2Cy~minf where minf = 2&g is the common mass of the inflaton. Another possible decay of inflaton to right-handed neutrinos is not possible as the right handed neutrinos are heavier than inflaton. S + H H is then the dominant decay channel of S. We know that the reheating temperature is defined as the temperature attained at the beginning of the radiation dominated epoch7 and defined as TRH - in MSSM. We have two fields to decay. So even the S field decays earlier (through S + fifi), the 8 field is oscillating after that (because of having longer lifetime) and when it is also converted into radiation at that time the radiation density corresponding to the decay of S becomes dilutedb. Hence reheating temperature, is effectively determined by the decay width of 8 field and thus turns out to be TRH - 0.22Ma. The acceptable value of reheating temperature can be taken as8 TRH 5 10l1 GeV.

- 6 x

11.2. Solution of p-problem and proton stability

Our solution of p-problem is through the presence of the term A S H H A R A ~ in the superpotential. Following Dvali et. a19, the presence Ms of a soft term as 2m3/2p2S in the low energy scalar potential along with the term 8 M 6 ISI2 obtained from the F-term of the superpotential can shift the VEV of S from zero to ( % ) 2 in a gravity mediated supersymmetry breaking scenario. Then for m3/2 N few TeV, the VEVs of S and AR will generate a p term with p = -4 N TeV. We also see from Table 1 that

F$

msl28

bThis is because of the fact that after 71 = rZ1, the corresponding radiation density falls off as R-4 ( R is the cosmic scale factor) and when the point 7 2 = r,' is reached, the radiation density is mainly dominated by that of 9, not of S.

420

the combined R-charges of the baryon number violating product of the superfields QQQ or QcQcQc turns out to be greater than one and we do not have any field in our model carrying negative R-charges. Hence the baryon number violating operators are forbidden and proton stability is achieved.

11.3. Leptogenesis

We now turn to the discussion of the generation of BAU. The fitted value of the observed BAU is according to WMAP resultslO. In our model, inflaton decays into left-handed triplets generating lepton asymmetry. We will follow the analysis by Hambye et al.3. From the superpotential W , the following decays can be estimated:

- 8 x

A;++ + L+L+ ; A;++ +, Z+L+ +H+H+ ; +, H + H + (2)

and similarly for A:++, A:++ also. We assume that we are working in the basis where Ma is real and diagonal without loss of generality. We then calculate the amount of CP violation from their decay leading to a lepton asymmetry ( E ) produced by the interference of the tree level process with the one-loop diagrams and is given by

where ny and s are the photon number density and entropy. Considering maximal CP violating phase we can parametrize E by f and g (ga = y a x ) assuming lgll - 1g21 - g, 1f:I - I f f 1 and taking Cijlf;ij12 = f 2 for further simplification.

M.9

11.4. Neutrino mass: relation with b - r unification

So far we have not considered the neutrino mass term Mu = 2 f f i j v & - rn;M;'mo = MUrl - mu^. A t acquires a small VEV, v8, = &v2 and the neutrino mass matrix M,,II is therefore

where v - 250 GeV. The contribution of the Type I seesaw term (M,I) to the neutrino mass is smaller than mu^^ as in our model all the right handed

421

neutrino masses (- f 2 M ) are expected to be order of the (B - L)-breaking scale - 10l6 GeV.

In view of the dominance of the type I1 term in generating the neutrino mass, we can think of relating this mass matrix with that obtained from the minimal SUSY SO(10) Model5 so that some relations with neutrino mixings and b--7 unification can be established" with the same parameters generating the lepton asymmetriesC. In this susy SO(l0) model, the pure type I1 seesaw mass term, M I I , is proportional to (Md - Ml) , where Md and M, are the down quark and charged lepton mass-matrices respectively. We work in a basis where the up-quark mass matrix Mu and MZ are diagonal. Therefore Md = VCKM Md diag t V,,, at the GUT scale leads to

mb mbX3 - m7 1 md + m,X2 - me m,X ma - m, - m,X2 + mbX4 mbX2 - m,X2 ( :;;3 mbX2 - maX2

MII - a

(5) as a rough estimation without considering CP violation where the cabbibo angle X - 0.22 keeping in mind - X2. Now to get the mixing angles and mass-squared differences, we will diagonalize this matrix step by step. Let us consider, m, = -xmbX2,m, = -ym,X2 and mb - m, =

- zmaX2; 2, y, z > Od.

Then we get,

Thus we see the maximal atmospheric angle demands a cancellation between the terms in the denominator. Then it follows

(x - 1 ) X z + 1 + xx2

tan2ol3 - -J-i - (0.06 - 0.08)

for values of z - (1 - 2) and x - 0.45 which is well within the CHOOZ bound, sin013 5 0.2. Assuming 013 + 0, we get the approximate solar

-0.4515 and AT: M (l;-ldZA7)2 x J1 + tan2 Z+l+ZA mixing as tan200 - z-l-sx

Amatm With the data from solar, atmospheric, reactor and accelerator neutrino experiments13 we studied the variation of Am&, tan2 for fixed Amit,. For example when z - 1.23, we get Am& - 5.1 x eV2, tan2 00 - 0.41 and M I I ~ ~ - 2.73 x

and M I I ~ ~ with z

e eV for Am;,, - 2.5 x

'Susy L-R model can be embedded into the minimal Supersymmetric SO(10) model5. dObserving the extrapolated values of the quark and lepton masses at GUT scale we see z N (0.4 - O.S),y N 1.22 from12.

422

Ml(GeV) TRH (GeV) P I Parameters

V2. This indicates an approximate b - r unification: Imb - mtaU I N 0.06mb. The requirement eatrn + n/4 demands zmb-[ym,-mb(x-xX2-X2)] << mb which in turn indicates a value of y around 1.4 for x - 0.45, z N 1.23. Fixing Am: at 5.1 x eV2, we have also checked that the value of z allowed by the range of Am:,, as above specified, is given by 1.19 5 z 5 1.28. Afterwords we will restrict ourselves to this range of z with above Am& and eo.

E L P

111. Results and Conclusions

Once we know the M I I 3 3 , we can compare it with M v ~ ~ 3 3 obtained in our case from eq.(4) and then it is possible to get the range of parameters f and g for specific choice of Ma using the bounds14 1 . 8 ~ 10-l' 5 < 2 . 3 ~ and 1.6 x eV2 5 Am:,, < 3.9 x eV2. We parametrize M I and M2 by the degeneracy factor p as M I = pM2; p > 1. The value of the heaviest triplet mass M I then in turn determine the reheating temperature. We then scan the f ' - g parameter space (f' = f:',,) to achieve final lepton asymmetry in the correct range corresponding to different degree of degeneracies between Ml and M2.

Table 2. metries are tabulated.

Corresponding to some choices of parameters, final lepton asym-

5 x10'0 1.1 x 1010 2.0 3.5 x 1 0 - ~ 1.27 x 10-3 1.93 x i o - l o 3.0 2.3 x 10-3 1.45 x 10-3 1.87 x in-10

In Table 2, we figure out few possible values of parameters for which right amount of final lepton asymmetry can be obtained along with the appropriate neutrino masses and mixings. Those observations suggest that corresponding to lower reheating temperature or lower mass of MI we need higher degeneracy, i.e. lower p . This is evident from our observation as we have seen the parameter spaces are more constrained for lower MI. So in summary, we consider a model where inflation, leptogenesis and neutrino masses are concerned. We resolve the p-problem and dimension-5 proton decay operator is also forbidden. We have shown a connection between the

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lepton asymmetry production parameters with the reheating temperature, neutrino mass and also with b - T unification.

Acknowledgments This work was supported by DOE under contract number DE-FG02- 91ER40626. A. Sil thanks V.N. Senoguz for discussions.

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PRECISION CALCULATION OF MASS SPECTRA IN THE MSSM

I. JACK, D.R.T. JONES AND A..F. KORD Department of Mathematical Sciences,

University of Liverpool, Liverpool L69 3BX, U.K.

E-mail: dijOIiv.ac.uk

We present the full three-loop ,%functions for the MSSM generalised to include additional matter multiplets in 5, 10 representations of SU(5). We analyse the effect of three-loop running on the sparticle spectrum for the MSSM Snowmass Benchmark Points. We also consider the effect on these spectra of additional matter multiplets (the semi-perturbative unification scenario).

1. Introduction

The LHC will soon resolve the question as to whether low energy supersymmetry is the solution to the hierarchy problem; and if it is, moreover, the LHC and a future e+e- linear collider (LC) will lead to very precise measurements of the sparticle spectrum and couplings. The success of gauge unification in the MSSM suggests a Desert, the existence of which would mean that extrapolation of the MSSM couplings and masses to high scales will lead to immediate information about the underlying theory; for example regarding the commonly assumed universality of soft scalar masses, gaugino masses and cubic scalar interactions.

One component of this analysis is the running of masses and couplings between the weak and gauge unification scales, which is governed by the renormalisation group ,&functions. In this talk we present the state of theart of higher-order ,&function calculations for the MSSM and then we compare the results for the MSSM spectra obtained using one, two and three-loop @-functions. In each case we have generally used the same one- loop corrections for the relationship between running and pole masses for the various particles, with some use of two-loop results such as for the top quark mass. As we shall see, the effect of three-loop running is surprisingly large.

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2. N = 1 Supersymmetry

A general N = 1 theory is described by the superpotential:

where @i are c h i d superfields. The corresponding Lagrangian is:

LSUSY = LG + LM (2)

(3)

where 1 1 4 2

L G = --GP”G ,” + iX“cr:,D,? + -D2

and

LM = i$g.D$+ (D,I$/~ +- FiFi + FiWi + FiWg

- -wij$i$j 1

+ g(Ra) i j [Da4iqP - f id iXa$ j -

- zwij$i$3 1 2

(4)

where @ = (&)* etc, W i = a h etc, and ( R a ) i j are the representation matrices for the representation of the gauge group to which 0 belongs.

The following standard terms are usually added to break supersymmetry:

~ i & ~ = (m2)j i@4j

+ ( $ h i i k & 4 j $ k + $bii4g4j + ;MAX + h.c.) . ( 5 ) but there is no reason not to have the following non-standard terms as well (unless there are gauge singlet fields):

( 2 ) 1 jk i (6) = T T i 4 ( b j 4 k + $mFi’$i$j + mAia$iXa -I- h.c.

None of these terms introduce quadratic divergences so we say they preserve naturalness. But it’s usual to ignore L(2) .

3. The &functions

3.1. The supersymmetric theory

The renormalisation of a supersymmetric theory is governed by the gauge B-function(s) &(g, Y, Y * ) and the matter multiplet anomalous dimension yij(g, Y, Y*); the latter governs both mass and Yukawa &functions.

#k = y P ( i j y k ) ) , = y i j ~ 7 k p + ( l c + + i ) + ( k + , j )

(7) o ; = p “ P(i Y j I p

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In DRED (Dimensional Reduction) Pg has been calculated through four loops1 and 7; through three loops in general2 and through four loops in the ungauged case3. (In QCD four loops marks the first appearance in & of higher order group invariants; these cancel in the supersymmetric case).

There is an exact form for the gauge &function given by

1 pgNsvz - g [ Q - 2r-'tr[yC(R)] 16n2 1 - 2g2C(G)(16n2)-l '

where Q = T(R) - 3C(G). C(R), C(G) are the quadratic Casimirs for the representation (Ra)ij and for the adjoint representation, while tr(R,Rb) = T(R)&b. However, this result is valid in a particular regularisation scheme which is not the one normally employed for explicit calculations, namely Dimensional Reduction (DRED). For instance at three loops we have

= g{3X1 + 6x3 + X4 - 6g6Qtr[C(R)2]

- 4g4C(G)tr[PC(R)]}

+ g7QC(G)[4C(G) - 91

T

Pf)Nsvz = g{2Xl + 4x3 - 4g6Qtr[C(R)2] T

- 4g4C(G)tr[PC(R)]} + 4g7C(G)2Q

where Pij = i Y i k l ~ k l - 2C(R)aj and

X1 = g2Y klmPnl C( R)'mYknp, 4 klm

X2 = 9 Y C(R)nlC(R)PmYknp,

X3 = g4tr[PC(R)2], X4 = g2tr[P2C(R)]. (11)

(12)

These results can be reconciled' with a redefinition of g,

6g = ig3 [r-ltr [PC(R)] - g2QC(G)]

which is essentially unique apart from the overall constant, and leads to

its required. Similar results apply at 4

3.2. The soft @-functions

The non-standard soft P-functions have been calculated through 2 loops5, and the &function of the Fayet-Iliopoulos term through 3 loops6.

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All the standard soft p-functions can be expressed exactly in terms of the &(g, Y, Y') and the y(g, Y, Y * ) of the unbroken theory7-':

p11.l= 2 0 (5 ) where

(r)ij = uyij. The soft scalar mass &function is8

pm.= 2 0 0 * + 2 1 M I g -+ Y-+cc +x- y [ &la (',"Y ) sag]

g 3 r-ltr[rn2C(R)] - M M * C ( G ) XNsvz = -2- 16n2 1 - 2C(G)g'(16~')-~

XDRED' is known through three loopss. This means that we can generate (by computer) the soft p-

functions through three loops starting from the three-loop @-functions and anomalous dimensionslO. For the MSSM the three loop P s have now been calculated11712 and made publicly available at http://www.liv.ac.uk/-dijlbetas.

4. The Minimal Supersymmetric Standard Model

The Minimal Supersymmetric Standard Model (MSSM) is a particular example of an N = 1 supersymmetric theory. The superpotential is :

W = H2QKtC + HIQYbb" + HiLYTTc + pH1Hz (18)

Here tC is a 3-vector in flavour space and correspondingly Yt etc are 3 x 3 matrices of Yukawa couplings.

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The soft breaking terms are:

+ [HzQhtt" + HiQhbb" + HILh,TC + h.~ . ]

= m+$Hl$Hz H;Qitt" (2)

+ H,*Q&,bc + H,*Lh,r" + h.c. (19)

It is usual to take & , , r to be diagonal and to write ht = A t K , hb = AbYb, h, = &Y, etc, with At&,, also diagonal. We assume that at the unification scale the soft masses have a common value mo, the gaugino masses a common value M, and that At = Ab = A, = A l .

4.1. R-parity Violation

Unless we impose "Matter Parity" conservation Q , L , uc, d", e" + -Q, -L , -uc, -dc, -ec we can have in the superpotential:

W2 = i (AE)ecLL + i(hU)ucdcd" + (A,)dcLQ + p'LH2 (20)

and in the soft breaking terms:

L!iLT = miH, 'L + rngLH2 + ihEeCLL +ih&d"d" + hDd"LQ + h.c., (21)

and in the soft breaking NS terms:

LFdkT = m,.+L+H2 + RIL*Quc + R2H1H,*ec +R3u"e"dc* + iR4QQdC* + h.c. (22)

In our current analysis we have omitted R-parity violating and non- standard couplings but they could easily be incorporated in the running analysis.

4.2. Higher-order behaviour of P-finctions

It is manifest from the form of the @-functions that higher-order terms will be significant. For instance, retaining the top Yukawa coupling only, we have:

@Yt = 6y: - 2 2 ~ : + [I02 + 36<(3)]yr - [678 + 696<(3) - 216<(4) + 1440C(5)] y:, (23)

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where we note the increasing coefficients. Again, the three loop result for @,; 11i12 in the approximation that

we retain only yt and 123, in the MSSd with 3 generations and 125,1210

additional 5, lO multiplets is given by

5. Precision QFT and the MSSM

Calculations of sparticle spectrum resulting from given assumptions about the underlying theory have become increasingly refined, with several public programs available (ISAJET, SOFTSUSY, SPHENO, SUSPECT ...) that incorporate two-loop Renormalisation Group Equations (RGEs) and one- loop radiative correction~l~.

We have repeated this analysis for a selection of MSSM “benchmark” points14 and extended it to include three-loop @-function correctionsll 9 12.

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IPoint t a n p M m0 A signp SPSla 10 250GeV lOOGeV -100GeV + PSlb 30 400GeV 200GeV 0 +

10 400GeV 9OGeV 0 + lkgg 5 300GeV 150GeV -lTeV t

For weakly interacting particles the three loop running corrections are small but for the squark masses they tend to be the same or bigger than the two loop ones. We have also extended this analysis to incorporate additional matter representations. As the amount of matter is increased the effect of two and three loop corrections becomes more dramatic”. The Snowmass Benchmark Points are given in Table 1.

Particle 1 loop 2 loops 3 loops AKP 9 743 729 727 718-728 GL 684 677 668 676-684 ii_. 658 656 646 653-660

LSP 128 120 120 119-121 h 115 115 115 112-119

t 2 243 257 240 232-258

Table 1. Some CMSSM benchmark points

SPSla: “Typical” MSUGRA point. SPSlb: “Typical” MSUGRA point with larger tanp. SPS3: Light stau, almost degenerate with the neutralino LSP. SPS5: Large A-parameter, leading to a light stop quark.

In Tables 2-4 we show examples of spectra for SPS5 (for two different values of the top quark mass) and SPSla. We compare our results with the spread of results obtained using earlier programs taken from Ref. 15, denoted AKP (see also Ref. 16). Note how sensitive the light stop mass is to the input top quark mass for SPS5.

5.1. Semi-perturbative Unification

Complete SUS representations do not (at one loop) change the prediction of sin2 Ow (or alternatively of g $ ( M z ) ) that follows from imposing ~ 1 , 2 , 3 gauge unification.

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Table 3. SPS5 benchmark point: (mt = 178.OGeV)

particle 1 loop 2 loops 3 loops AKP

B 743 729 727 719-72 C r. 684 677 668 676-68 I

f i R 658 656 646 655-66 t a 265 278 263 258-28

LSP 128 120 120 119-12 h 117 118 118 116-12

Table 4. SPSla benchmark point

particle 1 loop 2 loops 3 loops AKP

400 399

114 114 LSP 104 95.6-97.

As n5 + 3n10 + 6, a3 increases at high energies. The regime when unification occurs close (but not too close) to

its Landau pole is called “semi-perturbative” unification”. Although at one loop the gauge unification scale, MX is unchanged, at higher loops it increases and can approach the string scale. The resulting sparticle spectrum can differ markedly from that obtained in the MSSM case. For example, the gaugino mass ratio M2/M1 can approach 1, and the squark/slepton spectrum is compressed.

In Fig 1 we plot, for the SPS5 point, the ratio of the i l ~ and gluino masses against nlo for n5 = 0; as already noted in Ref. 17, the mass increases with nlo. It is interesting that the effect of the three-loop correction to this ratio almost precisely cancels the two-loop correction, for all 7110. We contrast this with Fig 2 where we show the behaviour of the light stop mass for the same SPS point; in this case the ratio decreases smoothly, and the three-loop correction only cancels the two-loop one at n10 = 0. For the SPS5 point the electroweak vacuum fails around nl0 = 0.48.

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Figure dotted

1. Plot of the CL/gluino mass ratio against n10 for SPS5. Solid, lines correspond to one, two and three-loop running respectively.

0 0.05 0.1 0.15 0 2 025 0.3 0.35 0.4 0.15

"10

15

dashed and

Figure 2. and dotted lines correspond to one, two and three-loop running respectively.

Plot of the light stop/gluino mass ratio against n10 for SPS5. Solid, dashed

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6. Conclusions

The LHC and an e+e- linear collider will measure sparticle masses with high accuracy. Very precise theoretical calculations will be required to disentangle the parameters of the underlying theory from the observations, and to distinguish, for example, nonuniversal boundary conditions from extra matter in the Desert or R-parity violation. We anticipate that by LHC-time complete two-loop threshold corrections will be available; indeed they are currently being performed". The effect of these we would expect to be of the same order of magnitude as the effect of using the three- loop P-functions, which, as we have seen, is surprisingly large for squarks. Completely consistent calculations at this order will thus be possible.

Acknowledgments

DRTJ was supported by a PPARC Senior Fellowship and a CERN Re- search Associateship, and was visiting CERN and the Aspen Center for Physics while part of this work was done. AK was supported by an Iranian Government Studentship.

References 1. I. Jack, D.R.T. Jones and C.G. North, Nucl. Phys. B486 (1997) 479 2. I. Jack, D.R.T. Jones and C.G. North, Nucl. Phys. B473 (1996) 308 3. P.M. Ferreira, I. Jack and D.R.T. Jones, Phys. Lett. B392 (1997) 376 4. I. Jack, D.R.T. Jones and A. Pickering, Phys. Lett B435 (1998) 61 5. I. Jack and D.R.T. Jones, Phys. Lett. B457 (1999)lOl; Phys. Rev. D61 (2000)

095002 6. I. Jack, D.R.T. Jones and S. Parsons, Phys. Rev. D62 (2000) 125022 7. I. Jack and D.R.T. Jones Phys. Lett. B415 (1997) 383 8. I. Jack, D.R.T. Jones and A. Pickering, Phys. Lett. B432 (1997) 114 9. L.V. Avdeev, D.I. Kazakov and I.N. Kondrashuk, Nucl. Phys. B510 (1998)

10. P.M. Ferreira, I. Jack and D.R.T. Jones, Phys. Lett. B387 (1996) 80 11. I. Jack, D.R.T Jones and A.F. Kord, Phys. Lett. B579 (2004) 180 12. I. Jack, D.R.T Jones and A.F. Kord, Ann. Phys. (to appear) 13. D.M. Pierce, J.A. Bagger, K.T. Matchev and R.J. Zhang, Nucl. Phys. B491

14. B.C. Allanach et al., Eur. Phys. J. C 25, 113 (2002) 15. http://kraml.home.cern.ch/kraml/comparison 16. B.C. Allanach, S. Kraml and W. Porod, JHEP 016 (2003) 0303 17. C.F. Kolda and J. March-Russell, Phys. Rev. D55 (1997) 4252 18. S.P. Martin, contribution to these Proceedings

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(1997) 3


String Theory, String Phenomenology,

Extra Dimensions


HOW STUECKELBERG EXTENDS THE STANDARD MODEL AND THE MSSM

BORIS KORS Center for Theoretical Physics, Laboratory for Nuclear Science, and

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

kors Olns. mit. edu

PRAN NATH Department of Physics, Northeastern University,

Boston, MA 02115, USA nathaneu. edu

Abelian vector bosons can get massive through the Stueckelberg mechanism without spontaneous symmetry breaking via condensation of Higgs scalar fields. This appears very naturally in models derived from string theory and supergravity. The simplest scenarios of this type consist of extensions of the Standard Model (SM) i)r the minimal supersymmetric standard model (MSSM) by an extra U ( l ) x gauge group with Stueckelberg type couplings. For the SM, the physical spectrum is extended by a massive neutral gauge boson Z' only, while the extension of the MSSM contains a CP-even neutral scalar and two extra neutralinos. The new gauge boson Z' can be very light compared to what one has in other models with V(1)' extensions. Among the many new features of the Stueckelberg extension of MSSM, perhaps the most striking one in the possibility of a new lightest supersymmetric particle (LSP) & which is mostly composed of Stueckelberg fermions. In this scenario the LSP of MSSM xy is unstable and decays into x!&. Such decays alter the signatures of supersymmetry and have impact on searches for supersymmetry in accelerator experiments. Further, with R-parity invariance, xgt is the new candidate for dark matter.

1. Stueckelberg mechanism for gauge boson masses

The Stueckelberg mechanism', as an alternative to spontaneous symmetry breaking in the Higgs effect2, describes a way to make the naive Lagrangian of a massive abelian vector boson A,, such as

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gauge invariant. To achieve this, one replaces A, --+ A, + i a ,u , where u is an axionic scalar that takes the role of the longitudinal mode of the massive vector, and defines the gauge transformation

6A, = a,€ , 60 = -me.

The resulting Lagrangian is gauge invariant and ren~rmalizable~ >4. The physical spectrum contains just the massive vector field, and this mass growth occurs without the need for a charged scalar field developing a vacuum expectation value, without spontaneous symmetry breaking and accordingly without the need for a Higgs potential. In the above the mass parameter m is called a topological mass4. One can now go through the procedure of gauge fixing, so that A, and u decouple in the final theory,

1 m2 1 1 m2 4 pv 2 , 2J 2 2 Lst + L,f = - - F FPV - - A A , - -((a,A,)2 - -a,uPa - <-u2 .

A number of properties of this Lagrangian should be stressed. i) The vector A, has absorbed the real scalar o in the process of getting a mass, with nothing left. ii) As the global subgroup of the gauge transformation, one can shift the scalar by a constant, 6a = c. This is a Peccei-Quinn like shift symmetry, and is the reason why we call u an axionic pseudoscalar, which only appears with derivative Couplings.” iii) Currently it appears possible to write such a gauge invariant Stueckelberg Lagrangian only for an abelian gauge symmetry, not for non-abelian gauge transformations3. However, as will become clearer from the string theoretic embedding of the Stueckelberg mechanism into D-brane models, the relevant U(1) gauge group can become a subgroup of some non-abelian and simple grand unified gauge group in higher dimensions.

2. Stueckelberg couplings in string theory and supergravity

One immediate way to see that Stueckelberg couplings appear in dimensional reduction of supergravity from higher dimensions, and in particular string theory, is to consider the reduction of the ten-dimensional N = 1 supergravity coupled to supersymmetric Yang-Mills gauge fields, in the presence of internal gauge fluxes. The ten-dimensional kinetic term for the anti-symmetric 2-tensor Brj involves a coupling to the Yang-Mills Chern- Simons form, schematically a [ r B J ~ ~ + A [ r F j ~ l + . . . , in proper units. Di-

&But u does not necessarily have to couple to the QCD gauge fields in the usual topological term. In fact, we assume such couplings to be absent.

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mensional reduction with a vacuum expectation value for the internal gauge field strength, (Fij) # 0, leads to

8,Bij + A,Fij + . . . - a,(z + mA, , after identifying the internal components Bij with the scalar (z and the value of the gauge field strength with the mass parameter m, which is indeed a topologiucal quantity, related to the Chern numbers of the gauge bundle. Thus A, and (z have a Stueckelberg coupling of the form A,aj'"o. These coufilings play an important role in the Green-Schwarz anomaly cancelation mechanism. In the effective four-dimenional theory, for instance abelian factors in the gauge group can have an anomalous matter spectrum, whose ABJ anomaly is canceled by Green-Schwarz type contributions. These involve the two terms mA,a,u + C ( T F , ~ ~ ~ ~ in the Lagrangian.

* + - = O m c

As can be read from the left Feynman-diagram, the contribution to the anomalous 3-point function is proportional to the product of the two couplings, m . c, while the mass parameter in the Stueckelberg coupling is only m. Therefore, any anomalous U(1) will always get massive through the Stueckelberg mechanism, since m . c # 0, but a non-anomalous U(1) can do so as well, if m # 0, c = 0. Since we do not want to deal with anomalous gauge symmetries here, we shall always assume that m # 0, c = 0. The mass scale that determines m within models that derive from string theory can, at leading order, also be derived from dimensional reduction. It turns out to be proportional to the string or compactification scale in many cases5, but can in principle also be independent6.

The fact that an abelian gauge symmetry, anomalous or non-anomalous, may decouple from the low energy theory via Stueckelberg couplings was actually of great importance in the construction of D-brane models with gauge group and spectrum close to that of the SM7. Roughly speaking, these D-brane constructions start with a number of unitary gauge group factors, which are then usually broken to their special unitary subgroups via Stueckelberg couplings of all abelian factors, except the hypercharge,

Stueckelberg -+ U ( 3 ) x U ( 2 ) x U(1)2 SU(3) x SU(2)L x U ( 1 ) y

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The mass matrix for the abelian gauge bosons is then block-diagonal, and only the SM survives. In order to ensure this pattern, one has to impose an extra condition on the Stueckelberg mass parameters, beyond the usual constraints that follow from the RR charge cancellation constraints, namely that the hypercharge gauge boson does not couple to any axionic scalar and remains massless7. In the language of these D-brane models, we will here relax this extra condition, and allow the hypercharge gauge boson to have Stueckelberg type couplings, and thus mix with other abelian gauge factors beyond the SM gauge group, which seems a very natural extension of the SM in this frame work.

3. Minimal Stueckelberg extension of the Standard Model

To keep things as simple as possible, and study the essence of the Stueck- elberg effect in its minimal v e r s i ~ n ~ > ~ , we consider an extension of the SM with only one extra abelian gauge factor U(1)x. Along with the U( l )x we also allow a hidden sector with charged matter fields. Since the Stueck- elberg mechanism cannot break the non-abelian s U ( 2 ) ~ in the process of electro-weak symmetry breaking, we cannot replace the usual Higgs mechanism, but only add the Stueckelberg masses for the abelian gauge bosons of hypercharge and U( l )x on top of the usual Higgs mechanism. Schemat- ically, the couplings are thus given by

Higgs - SU(3) x SU(2)L x U( l )y xU(1)x

SU(3) x SU(2)L x U(1)y x U( l )x . - Stueckelberg u

The extra degrees of freedom beyond the SM spectrum are then the new gauge boson C, plus the axionic scalar u which combine into the massive neutral gauge field Z’. Explicitily, we add the Lagrangian

to the relevant part of the SM Lagrangian 1 1 4 LSM = --trF,,FP” - qB,,Bp” +g2AiJip +gyB,J:

4,@tD’L@ - V(@t@) + . . . .

The scalars and vectors decouple after adding gauge fixing terms in a standard fashion. To keep the model as simple as possible, we impose the

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following constraints on the charged matter spectrum: We assume that the fermions of the SM are neutral under U ( l ) x , and vice versa we require that all the fields of the hidden sector, which are potentially charged under U ( l ) x , be neutral under the SM gauge group.b Finally, one has to make sure that the hidden sector dynamics can really be ignored, i.e. that there is no spontaneous breaking of the U(l)x in the hidden sector.

4. Stueckelberg effects in the Standard Model

All effects of the minimal Stueckelberg extension on the SM Lagrangian can be summarized by the modified mass matrix of the, now three, neutral gauge bosons. In the basis (C,,B,,A:) for U ( l ) x , hypercharge and the 3-component of iso-spin, The vector boson mass matrix is

The mass matrix above has one massless eigenstate, the photon with My2 = 0, and two massive ones with eigenvalues

V 2 M; = a ( g i + g$) + O(6) , Mi2, = M 2 + O(6) .

In the above we have introduced two parameters M and 6, defined by M 2 = M; + M i and 6 = M2/Ml to parametrize the Stueckelberg extension, which are a mass scale and a small coupling parameter. The bounds for these two, the values for which the Stueckelberg extension is still safely within experimentally allowed ranges, have been given in8 as

M > [150GeV] , 6 < 0.01 . For later use, we also define M i = v2(g; + g$)/4. Roughly speaking, one may notice that all couplings which allow communication of SM fields to the Stueckelberg sector, are suppressed by 6. In models where an extra U(1)' addition to the SM gauge group is broken by a Higgs condensate, the suppression factor comes from the propagator of the massive gauge boson, and is of the order of Mi/MZ, - 0.Ol1OJ1J2. This demands that the mass

bThese conditions are very hard to satisfy in computable D-brane models, such as intersecting D-brane models with tori or toroidal orbifolds as compactification space. As a matter of principle, this should be an artifact of the too simple toroidal geometry and topology, and we expect that one can get around this problem in more general Calabi-Yau compactifications.

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of the new gauge boson has to be in the range of TeV, much larger than the bound on M as above. For the Stueckelberg model, the suppression can easily be achieved by demanding 6 be sufficiently small.c

A regime in the parameter space which would also be very interesting to investigate is the region M I , It12 + 00, 6 = finite. This scaling limit corresponds to the expected behaviour in string theoretic models with a high mass scale, but basically arbitrary coupling 6. One may speculate that even though the Z’ becomes very heavy, and cannot be produced directly, small effects may remain observable. In essence, the Z’ does not decouple from the low energy theory, because the mass matrix is not diagonal, and this may have important consequences.

To proceed further, one can now diagonalize the vector boson matrix by an orthogonal matrix 0 = 0 ( 0 , 4 , $), a function of three variables {0,4, $}, with (see’ for the details of the notation)

mass angle

The bounds on M and 6 translate into 4,$J < lo and 0 - Ow, which is the electro-weak mixing angle. Inserting the mass eigenstates into the interaction Lagrangian

Lint = Q ~ A E Jll” + gYB, JG + gxC, J g

one finds the interactions of the phyical vector fields. As the most striking example one gets for the photonic interactions

which implies that the electric charge unit is now slightly redefined by

and that the charge unit of fields in the hidden sector, with charge under

cTo date a real global fit of the experimental data within the Stueckelberg extended model has not been performed, and thus these statements cannot be made completely quantitative as yet.

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U ( l ) x , is irrational and very small.d Since all exotic couplings are suppressed by powers of 6 , computable finite quantities that lead to definite predictions are best found in terms of ratios of such couplings. Examples are the branching ratios of the Z‘, or the forward-backward asymmetry at the Z’ peak in resonant producation via e+e- collision8. One finds for example that the total width of the Z’ is extremely small, at least as long as the hidden sector charged fields are heavy enough, i.e.,

r(Z‘ + ff) - O(1O)MeV.

Therefore, Z’ would appear as a very sharp peak in e+e- annihilation and in other collider data.

5. Stueckelberg extension of the MSSM: StMSSM

The supersymmetrized version of the Stueckelberg coupling is related to the so-called linear multiplet formalism, see13. For the present minimal Stueckelberg extension of the MSSM, which we call StMSSM, it readsg

Lst = J d20d2e (MlC + M2B + S + S)’

Here S is the Stueckelberg chiral multiplet, and B , C are the abelian vector multiplets of hypercharge and U(1)x. The degrees of freedom in components are given by S = (x, p + io, F ) , B = (B,, X B , D B ) , and C = (C,, XC, Dc), and the Lagrangian becomes

fp(M1Dc + k f 2 0 ~ ) + [X(kfif(c + M 2 x B ) + h . ~ . ] + 21FI2 . One may of course also add Fayet-Iliopoulos terms for each one of the two abelian factors, which would contribute to the scalar potential. As is usually done for the MSSM, we will however assume that their contributions to the breaking of supersymmetry are small compared to other sources, and therefore neglect these terms throughout, leaving a more complete analysis to future work. Eliminating the auxiliary fields F, DB, D c one thus finds corrections to the usual D-term potential of the MSSM through the coupling

dNote that a hidden sector with such an irrational electric charge is a possible consequence of the Stueckelberg scenario, which survives as a potentially observable fact in the stringy limit M --t 00, 6 = finite. While the mass of the gauge boson Z’ becomes very large, the hidden sector charged matter fields can be massless at the string scale, and thus be much lighter and in the end observable in experiment.

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of the D-fields to p. To complete the action, we add the following softsupersymmetry breaking terms for the neutral gauginos, and the scalars{hi,h2,p},

1 -•- -^fhi\B\B - 2

I 2 - -J l ~ 2h.c.

An important difference between the Stueckelberg Lagrangian and the La-grangian that involves the Higgs mechanism is the fact that the chiralfermion x 'IS neutral under the gauge group. A scalar Higgs condensatewould need to be charged under U(l)x to break the gauge symmetry spon-taneously. But then, the standard coupling of the fermionic partner h of theHiggs scalar in the form gyB^ha^h, gxC^ha^h would imply a contribu-tion ot the AB J triangle anomaly, and a second Higgs multiplet of oppositecharges would be needed to cancel the anomaly, just as in the MSSM. Thisis not the case for the Stueckelberg mechanism, where x does not have suchcouplings, and no second chiral multiplet is needed.

Putting things together, the new effects in the StMSSM are in the scalarpotential and the neutralino mass matrix, in addition to the mass matrixof the neutral gauge bosons, which is identical to that of the extended SM.The scalar potential with the three CP-even scalars p, hi , h2 is

M2 •m2)p2

l(ml pgYM2)\h2\2 + ro§(/n • h2 + h.c.

where V$ssu(hi,h2) is the usual D-term potential of the MSSM for theHiggs fields hi,h2 of MSSM. The new scalar p modifies the Higgs massterms through its vacuum expectation value. Shifting p -> vp + p with\gvM2vp\ < 10 4 A/|, one however finds that this induces only very tinyeffects, as for instance in the electroweak symmetry breaking constraint

2 _2 ° ~ tan2(/3)-l ' cos(2^) '

The CP-even scalar mass matrix in the basis (hi,h2,p) reads

-(Mo +TO24)s/3C/3

-(Ml + jU-2.21V10 S0

2 .,2AC0

445

Going through the details, one again finds a very narrow resonance for the third mass eigenstate ps in the J = O+ channel, similar to the Z', i.e.,

r ( p S + tf) - O(10) MeV . Perhaps the most interesting sector of the StMSSM is the fermionic neutralino mass matrix, which now involves the usual two higgsinos and two gauginos of MSSM, plus the new gaugino of U ( l ) x and the chiral fermion x of the Stueckelberg multiplet. In the basis (x, Xc, Xg, X3, h1, h 2 ) it reads

0 Mi Mi *;u

M2 0 0 0 0 0 0 0

with abbreviations c1 = CDse, c2 = spse, c3 = cqce, c4 = soso, further so, so, etc., standing for the sin and cos of Ow and P, where tan@) = ( h z ) / ( h l ) . It is convenient to number the eigenstates of the 4 x 4 MSSM mass matrix according to mny < ml;g < m2g < m%$, and the two new ones by

1 1 m p , m-o = M; + - f ig f -fix + O(6) ,

The parameter space now easily allows the situation that the lighter one among the two new neutralinos becomes the lightest among all six. It therefore is the LSP, and with R-parity conservation, the natural dark matter candidate of the model. In this case, when mriz < m p , we call 2; the Stueckelino 2gt, and important modifications of the usual signatures for searches for supersymmetry in accelerator experiments14 would follow. One would actually observe decay cascades (see alsol5), in which the lightest neutralino of the MSSM would further decay into the true LSP, the Stueckelino, via emitting fermion pairs,

mng 2 mn; . X6 J 4 2

2: .+ 1.i. -0 z ZXSt , e9&t 7 z2;t . This would lead to decays of the form

for sleptons or

446

for charginos, and similarly for squarks and gluinos. The analysis above shows that there will be multilepton final states in collider experiments which would be a characteristic signature for this of scenario.

6. Summary

We summarize now the most important features of the minimal Stueckel- berg extension of the SM and of the MSSM:

(1) The Stueckelberg mechanism provides ti gauge invariant, renormalizable method to generate masses for abelian gauge bosons with minimal extra residual scalar fields in the system. The mass of the massive vector boson is “topological” in nature.

(2) It naturally appears in many models that descend from string theory and higher-dimensional SUGRA.

(3) The Stueckelberg extension is very economical and distinct, even at the level of the degrees of freedom, compared to Higgs models with extra U ( 1)’ gauge factors.

(4) In the Stueckelberg extension of SM, only the vector boson sector is affected, as it introduces an extra Z’ boson which is typically a very narrow resonance. In addition, one has small exotic couplings of the photon and the Z with hidden matter, if present.

( 5 ) In the Stueckelberg extension of the MSSM, the vector boson sector, the Higgs sector and the neutralino sectors are affected. The effect on the vector boson sector is identical to what one has in the Stueckelberg extension of the SM. In the neutral CP even Higgs sector one has mixing among the two CP-even Higgs of MSSM and a new CP-even Stueckelberg scalar field p. In the neutralino sector, one finds two more neutralinos which are mostly mixtures of neutral Stueckelberg fermions, in addition to the four neutralinos of MSSM.

Acknowledgements

The work of B. K. was supported by the German Science Foundation (DFG) and in part by funds provided by the U S . Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818. The work of P. N . was supported in part by the U.S. National Science Foundation under the grant NSF-PHY-0139967.

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References 1. E.C.G. Stueckelberg, Helv. Phys. Acta. 11 (1938) 225; V. I. Ogievetskii and

I.V. Polubarinov, JETP 14 (1962) 179. 2. P. W. Higgs, Phys. Lett. 12 (1964) 132; Phys. Lett. 13 (1964) 508; Phys. Rev.

145 (1966) 1156. See also G. S. Guralnik, C. Hagen, and T. W. B. Kibble, Phys. Rev. Lett. 13 (1964) 585; F. Englert and R. Brout, Phys. Rev. Lett. 13 (1964) 321.

3. J.M. Kunimasa and T. Goto, Prog. Theor. Phys. 37 (1967) 452; A. A. Slavnov, Theor. Math. Phys. 10 (1972) 305; M. Veltman, Nucl. Phys. B7 (1968) 637; L. Faddeev and A. A. Slavnov, Theor. Math. Phys. 3 (1970) 18; A. I. Vain- shtein and I. B. Khriplovich, Sov. J. Nucl. Phys. 13 (1971) 198; K. Shizuya, Nucl. Phys. B121 (1977) 125; Yu. Kafiev, Nucl. Phys. B201 (1982) 341; D.Z. Freedman and P.K. Townsend, Phys. B177 (1980) 282; D.G.C. McKeon, Can. J . Phys. 69 (1991) 1249.

4. T. J. .411en, M. J. Bowick and A. Lahiri, Mod. Phys. Lett. A 6 (1991) 559. 5. D. M. Ghilencea, L. E. Ibanez, N. Irges and F. Quevedo, JHEP 0208 (2002)

016 [hep-ph/0205083]; D. M. Ghilencea, Nucl. Phys. B 648 (2003) 215 [hep- ph/0208205].

6. L. E. Ibanez, R. Rabadan and A. M. Uranga, Nucl. Phys. B 542 (1999) 112 [hep-th/9808139].

7. L. E. Ibanez, F. Marchesano and R. Rabadan, JHEP 0111 (2001) 002 [hep- th/0105155]; see also: R. Blumenhagen, V. Braun, B. Kors and D. Lust, hepth/0210083.

8. B. Kors and P. Nath, Phys. Lett. B 586 (2004) 366 [hep-ph/0402047]. 9. B. Kors and P. Nath, hep-ph/0406167. 10. M. Cvetic and P. Langacker, Phys. Rev. D 54 (1996) 3570 [hep-ph/9511378];

M. Cvetic and P. Langacker, Mod. Phys. Lett. A 11 (1996) 1247 [hep- ph/9602424]; A.E. Farragi and M. Masip, Phys. Lett. B 388 (1996) 524 [hep- ph/9604302].

11. V. Barger, C. W. Chiang, P. Langacker and H. S. Lee, Phys. Lett. B 580 (2004) 186 [hep-ph/0310073]; V. Barger, P. Langacker and H. S. Lee, Phys. Rev. D 67 (2003) 075009 [hep-ph/0302066]; T. Gherghetta, T. A. Kaeding and G. L. Kane, Phys. Rev. D 57 (1998) 3178 [hep-ph/9701343]; E. Ma, Phys. Rev. Lett. 89 (2002) 041801 [hep-ph/0201083]; 0. C. Anoka, K. S. Babu and I. Gogoladze, hep-ph/0401133; F. M. L. Almeida et al., hep-ph/0405020; T. Han, P. Langacker and B. McElrath, hep-ph/0405244.

12. For reviews see: J. L. Hewett and T. G. Rizzo, Phys. Rept. 183 (1989) 193; M. Cvetic and P. Langacker, hep-ph/9707451; A. Leike, Phys. Rept. 317 (1999) 143 [hep-ph/9805494].

13. M. Klein, Nucl. Phys. B 569 (2000) 362 [hep-th/9910143]; S. V. Kuzmin and D. G. C. McKeon, Mod. Phys. Lett. A 17 (2002) 2605 [hep-th/0211166].

14. P. Xath and R. Arnowitt, Mod. Phys. Lett. A 2 (1987) 331. For a more recent update see H. Baer, T. Krupovnickas and X. Tata, JHEP 0307 (2003) 020 [hep-ph/0305325]; and the references therein.

15. U. Ellwanger and C. Hugonie, Eur. Phys. J. C 5 (1998) 723 [hep-ph/9712300].

FERMION MASSES AND PROTON DECAY IN STRING-INSPIRED SU(4) ~ s U ( 2 ) ~ XU(l)x*

T. DENT! G. LEONTARIS AND J. FUZOS Theoretical Physics Division, Dept. of Physics

University of Ioannina, Ioannina 45110, GREECE

We present a supersymmetric S U ( ~ ) X S U ( ~ ) ~ x U ( l ) x model of fermion masses with fundamental and antisymmetric tensor representations only. The up, down, charged lepton and neutrino Yukawa matrices are split by SU(4) and s U ( 2 ) ~ Clebsch-Gordan coefficients. We obtain a hierarchical light neutrino masa spectrum with bi-large mixing. The condition that anomalies be cancelled by a Green- Schwars mechanism consistent with gauge unification leads to fractional U( 1 ) ~ charges, which exclude B violation through dimension-4 and -5 operators.

The values of the adjustable parameters of the Standard Model (SM) La- grangian may be an important clue to physics beyond it (BSM). Exper- imental values of gauge coupling constants, fermion masses and mixing angles (including the neutrino sector) can be compared to the predictions of models with full or partial gauge unification and/or flavour symmetries. Recent data on atmospheric and solar neutrinos', implying large 1-2 and 2-3 mixing angles, present challenges for any unified framework in which neutrinos form part of a multiplet with quarks.2.

We use a variant of the string-inspired 4-2-2 models3 for which fermion masses were previously investigated in Ref. 4. In such models large Higgs representations are not required, the doublet-triplet splitting problem is absent, third generation fermion Yukawa couplings are unified5 up to small Corrections, and unification of gauge couplings is allowed and, if one assumes the model embedded in supersymmetric string or some other higher- dimensional framework, might be predicted.6 The field content is summarized in Table 1, where i ranges from 1 to 3. The singlet q5 obtains a v. e. v. to satisfy the U( l ) ~ D-flat condition including an anomalous Fayet-Iliopoulos

*Work supported by the European Union under RTN contract HPRN-CT-2000-00148. temail: [email protected]

448

449

Table 1. Field content and U ( l ) x charges

term. The sneutrino-like components of H and H also obtain v. e. v.’s as follows:

( H ) = (H,) = MG, ( H ) = (Hv) = MG.

Small effective couplings arise from nonrenormalizable operators involving r$ and the SU(4) XSU(2)~-breaking Higgses. MSSM Yukawa couplings arise from the operators

where M s is the string scale (or scale of quantum gravity) and only the 33 element is nonvanishing at renormalizable level. The constants y? and y’: are nonvanishing and generically of order 1 whenever the U( 1)x charge of the corresponding operator vanishes: there are two a priori independent expansion parameters depending on the sign of charge (since m, n > 0). Right-handed neutrinos (RHNs) are automatically present and obtain Ma- jorana masses from the operators

. .

Again, the U( l)x charges & + ti!j + 22 +pz or &i + fij + 22 + q(z + 2) must be zero if the couplings p z and p l y , respectively, are not to vanish. More details of the model are discussed in Ref. 7.

For nonrenormalizable Dirac mass terms involving n products H H / M ; , the gauge group indices may be contracted in different ways4 leading to Clebsch factors Czu,d,e,u) multiplying the effective Yukawa coupling: these are generically numbers of order 1 and may be zero in some cases. This freedom is exploited to fit the different mass hierarchies in the u, d and e sectors and produce the CKM mixing. Although the Cij for a particular operator On may vanish at order n, the coefficient for the operator O(n+a);b

450

containing a additional factors ( H R ) and b factors of q5 will generically be nonzero. Dirac mass terms at the scale MG are then

where riZ = -(ai - a3 + dj - 6 3 ) and f i = -(ai - a3 + bj - the 3rd generation mass scale is m3 = V u , d y p , with v, and V d being the uptype and down-type Higgs v.e.v.’s respectively, and we define

We suppress higher-order terms involving products E E ’ . Majorana mass terms are MZvfv; with

where MR = M i / M s 5 E‘Ms, p = - (d i+dj+222) and q = - (&+&j+22);

p and q range from zero upwards. Note that the fermion mass terms and H N D couplings are invariant

under the family-independent shifts of U( l ) ~ charge

ai + a2 + c, 6i + di + ( x - 2 - C z+z++. (5)

Thus for the purpose of investigating fermion masses, we are free to assign a 3 = 6 3 = 0. We choose charges 2 + 3 = 1, ,z = -1, a1 = -4, dl = -2 , a2 = -3, d2 = 1, and substitute E and E’ by a single expansion parameter 77 ‘v 0.06 via E = 77, E’ = fi, giving a hierarchical common Dirac mass matrix and (2-dependent) Majorana RHN mass matrix. We then have to specify Clebsch coefficients for operators involving one or more powers of H H / M g . In specific string models these coefficients are calculable; in the absence of a specific string construction we impose that the C? should be either small integers or simple rational numbers. We can take all Cij equal to unity apart from the following: C:’ = Cz2 = 5, C:3 = 3 7 u C1’ = Ci2 = Czl = C22 = Cil = 0, all multiplying the leading (smallest n) entry. The ratio C$/Cj2 = 3 is the usual Georgi-Jarlskog factor. When CAi vanishes for the leading term, the next-to leading term is smaller by a factor v3l2. Dirac mass matrices at the GUT scale take the form

aThe signs of riZ and A must be the same as z and z + a: respectively.

45 1

up to complex phases and factors of order 1 , where mbO = m70 = mto/tanp. The order-1 factors arise from couplings yg, etc., which are SU(4) symmetric. Fermion masses and quark mixings yield acceptable fits to observables: for example, the CKM mixing angles 612 and 613 are Jii 0.24 and q2 N 0.0035 respectively; the ratio (m,/mt)lMz is q9 /2 / (3 N 6 x where ( N 0.83 accounts for the RG evolution of yu,c,t in the large tanP (fixed point) regime.

The light neutrino mass matrix depends on x through the RHN Majo- rana masses: we find two possible values, x = 1 and x = 3/2. We have

up to order 1 factors and corrections due to RG evolution, displaying leading terms in q. The RHN mass hierarchy combines with a hierarchical neutrino Dirac mass matrix to produce a seesaw mass matrix with e l e ments of similar magnitude. One can then obtain a neutrino mass spectrum with normal hierarchy and bi-large mixing. For x = 1, tanP = 40 and MR = _N 1.2 x 10l6 GeV, the largest entries in mu are of order 0.06eV. The RHN's comprise two superheavy states of mass (1 f q/2)kfR and one at q4MR 21 1.5 x 10l1 GeV. Alternatively, with x = 3/2, tanP = 40 and MR set to Jii times the heterotic string scale 5 x 1017 GeV, the scale of m, is of order 0.09eV and the RHN masses are ( J S ~ M R , q 3 l 2 M ~ , q 5 k f ~ N 9 X 10" GeV).

For U ( 1)x mixed anomalies to be cancelled by a Green-Schwarz mechanism, given level 1 string unification, the anomaly coefficients must satisfy A4 = A ~ L = A ~ R . For any initial U(1)x charge assignment, one may satisfy the anomaly conditions, without altering the fermion mass matrices, by using the charge shifts C, c. For example, if x = 1 , we have < = 13/6 and

= -1/6. Such fractional shifts eliminate many operators, for example FFFF which would lead to D = 5 proton decay has charge E a + 26/3, thus cannot be cancelled by any singlet combination of H , B and r$ fields. The only surviving operators coupling matter to coloured H i s s triplets axe FFD1 and FFHH. For D = 5 proton decay, mass terms for intermediate states are also required, either DID1 or DlHH; but the charges of these operators are both shifted by 4c = -213 and such mass terms do not exist.

Mass terms for the H-H-Dl-Dz system follow from the superpotential

452

It may be verified that the colour triplet d-like states arising both from H and and the D fields obtain masses near the GUT scale, satisfying bounds from D = 6 proton decay; there is also no transition between 31 and either 31 or $8, at any order. Other B- and L-violating operators are

Fi Hh, cab F ~ A Fp F ~ B HBb , Fi Fi $‘k H , Fi Fj Fk H h (9)

giving rise to superpotential terms LH,; QDcL and LE“L; UcDcDc; and QQQHd respectively. These may be eliminated by imposing an R-parity acting either on F and F , or on H and H . If we choose not to do so, the operators F 3 8 and F3Hh still vanish due to their fractional U ( l ) x charge, while FHh and FFFH receive integer shifts and so are suppressed by some powers of 7, or disallowed if a: is half-integer. The mu term arises from the product hh whose U(1)x charge is -4, receiving only a mild suppression: hence we would require an additional symmetry to solve the “mu-problem”.

In conclusion, the model is consistent with gauge unification and with all known elementary fermion masses and mixings. The lightest RHN mass is about 10l1 GeV: rather large for standard thermal leptogenesis8 if one takes into account gravitino production, for m3/2 = 1-lOTeV, but possible for nonthermal leptogene~is.~ The lightest neutrino mass eigenstate is of order ~ / 2 II 0.03 times the heaviest and the neutrino mixing angle 013 is of order JiZl2 21 0.12. Issues for further investigation include gauge unification, L violation if R-parity is not imposed, CP violation, supersymmetry-breaking and flavour-changing effects in quark and lepton sectors, and cosmology including inflation, baryogenesis and dark matter.

References

1. For recent reviews, see M. Maltoni, T. Schwetz, M.A. Tortola and J. W. F. Valle, Phys. Rev. D 68,113010 (2003); J. N. Bahcall, M. C. Gonzalez- Garcia and C. Peiia-Garay, hepph/0406294; G. Altarelli, hep-ph/0405182.

2. J. C. Pati, Phys. Rev. D 68, 072002 (2003). 3. I. Antoniadis and G.K. Leontaris, Phys. Lett. B 216, 333 (1989); I. An-

toniadis, G.K. Leontaris and J. Rizos, Phys. Lett. B 245, 161 (1990); G. K. Leontaris and J. Rizos, Nucl. Phys. B 554, 3 (1999).

4. B. C. Allanach and S. F. King, Nucl. Phys. B 459, 75 (1996); B. C. Allanach, S. F. King, G. K. Leontaris and S. Lola, Phys. Rev. D 56, 2632 (1997).

5. B. C. Allanach and S. F. King, Phys. Lett. B 353 477 (1995). 6. G. K. Leontaris and N. D. Tracas, Phys. Lett. B 372 219 (1996). 7. T. Dent, G. Leontaris and J. Rizos, hep-ph/0407151. 8. See e.g. W. Buchmiiller, P. Di Bari and M. Pliimacher, hep-ph/0406014. 9. G. Lazarides and Q. Shafi, Phys. Lett. B 258, 305 (1991); T. Dent,

G. Lazarides and R. Ruiz de Austri, Phys. Rev. D 69 075012 (2004).

CAN OUT-OF-EQUILIBRIUM PLASMA EFFECTS CREATE EMBEDDED STRINGS?

F. FREIRE+*, P. S A L M I ~ , N. D. ANTUNES~, A. ACHUCARRO~ Instituut-Lorentz, Universiteit Leiden, The Netherlands

Centre for Theoretical Physics, University of Sussex, England *E-mail: freireQlorentz.leidenuniv.nl

We show that out-of-equilibrium plasma effects provide the conditions for the formation of embedded vortex-strings. The necessary conditions are an efficient dissipation mechanism for the field modes forming the string and a not too large scalar self-coupling right below the transition scale. We discuss the cosmological implications at the chiral, electroweak and GUT symmetry breaking scales.

1. Introduction

It has been suggested recently by Nagasawa and Brandenberger (NB) that embedded vortex-string configurations can be stabilised, even if only for a finite period of time, due to out-of-equilibrium effects'. This can have observable cosmological implications. Two examples are the electroweak (EW) and the pion string, which are formed after the EW transition and the chiral symmetry breaking transition, respectively. Embedded strings are also conceivable at earlier cosmological transitions, for instance at the GUT scale. However, these strings are non-topological and we have learnt that the stability analysis of these type of configurations reveal some surprises due to competing effects that determine the overall balance2.

Both EW and pion strings are unstable for physical values of the parameters of the SM when all fields are in thermal equilibrium2t3. The instability is well exemplified by the O(4) linear sigma model. In this model we can embed a global string involving only two scalar fields. This configuration is stable if these two were the only fields in the model because the vacuum manifold would be M = S'. However, in the O(4) model M = S3. Therefore the phase of the fields can be unwound around the string core. Embedded strings are also dynamically unstable due to an unfavourable balance between their gradient and potential energies. The string is only stable for parameter values deep into the unphysical region.

453

454

The NB stabilisation mechanism requires that: (a) there are both electrically charged and neutral scalar fields; (b) the embedded string is made up of the neutral scalars; and (c) the charged scalar fields are minimally coupled to an electromagnetic field that is in thermal equilibrium at a temperature T . Under these conditions only the charged scalars receive a thermal mass which effectively reduces M from S3 to S’. The analysis by NB shows that if the embedded string is formed it is stable, but there is no discussion concerning the actual formation of the string.

In our study we analyse the necessary conditions for a favourable formation of embedded strings. We place particular emphasis on the role of dissipation in the dynamics of the fields out of equilibrium and the formation of the background condensate necessary for string formation.

We assume the neutral scalars are not in thermal equilibrium below the critical temperature T,. This is natural if the phase transition were first order but not for a second order or a smooth crossover. On the other hand, the charged scalars remain in thermal equilibrium with a heat bath of photons. Of course, the neutral scalars are not cut-off from the heat bath due to the scalar self-interaction A. How quickly they thermalise depends on the relative magnitudes of A and the coupling to the heat bath. We will show explicit studies of the competing effects between these two scales.

Furthermore, we note that the embedded strings of interest require a condensate of neutral scalars only. Dynamically, the fields that form the condensate need to dissipate the “hot” kinetic energy that they carry from above T,. One can expect a certain degree of competition between the charged and the neutral scalars to select which one will condense. We are interested in this situation because it brings us closer to the desired reduction in the vacuum manifold. The dissipation or viscosity coefficient of the neutral fields is a free parameter in our study.

With these assumptions we simulate the transition dynamics by studying the evolution of field theoretical Langevin equations. This approach has been used in recent years with excellent results that helped to clarify several aspects of vortex string formation and their thermodynamics4. It also has the advantage of freeing us from the notion of a finite temperature effective potential which is not suitable with fields out of equilibrium.

2. The model and Langevin field dynamics

We write the Lagrangean for the O(4) linear sigma model as

455

where we use the standard sigma-pion meson notation, T+ = (n-)* is the charged pion and u and no are neutral mesons. As is well known, the topology of the vacuum manifold S3 does not allow for vortex-string topological defects. However, this type of configuration can be embedded by freezing out some of the fields. We are interested in those embedded strings for which n* = 0, ie. made up of collective neutral scalar modes. As we mentioned, they are neither topological nor dynamically stable.

The equations of motion for any of the scalar fields, say &, read

[(a," - v2> - p2 + x (a2 + + 21n+12) + ~ i a t ] 4i = ti, (2) with p2 = Xu2 and where qi and ti are respectively the viscosity and the Gaussian noise characterising the coupling to the heat bath

(<i(z)tj(2/)) = Ri&jb(x - 9). (3) For a field in thermal equilibrium pRi = 27)i, with p = l / k ~ T . Otherwise, Ri and qi are not necessarily correlated so we can set ti = 0 and qi # 0, a situation we consider for the out-of-equilibrium neutral fields.

0.8 t

0.2 t

P Figure 1.

We run our simulations for successive temperatures of the heat bath determined by R+, the amplitude for the noise of the charged mesons. In Figure 1 we plot the order parameter for the condensate of neutral and charged scalars versus the temperature. These are, respectively, ( Im0 I) and (In+n-l), where (lq5iq5jjl) = Jv q 5 , ( ~ ) ) ~ are averages over the entire lattice. We use X = 1 and a 503 box. We decouple the neutral scalars from the heat bath below T,. The dissipation is kept the same below T, for all the scalars, qa = qo = q+. We show three different types of decoupling: instantaneous, exponential and power law indicating the way R, and Ro are suppressed below T,. In all three cases the neutral scalars

456

2.0

1.5b

condense. The details of the decoupling do not seem to affect the outcome of the condensation. Therefore, as long as the dissipation terms are all comparable the formation of neutral embedded strings is favoured.

The outcome is reversed for q, = qo = 0 below T,, as shown in Figure 2a. The reason why the charged scalar sector condenses in this case is illustrated by Figure 2b, where we plot the kinetic energy for both the charged and neutral scalars. Above T, the curves are indistinguishable. However, below T, the kinetic energy of the charged scalars continues to follow the expected equipartition curve while for the neutral fields it remains almost constant because there is no direct mechanism for dissipation.

, , , , , , , I a : = 0.005 3

equilibrium CUNB

1=16.0

I

0 3 , h ' ' b ' I b ' ; I ' 12 P

' I ' I ' I ' I '

8 . ' I ' I ' I ' I ' I ' I ' I ' I '

-- in thermal bath

1-

0 3 ' h * s ' ; ' ; ' ; ' ; ' & ' ; l ' l z

P

Before we discuss the implications of our analysis we illustrate how the magnitude of the dissipation coefficient and the scalar self-coupling affects the thermalisation of the neutral scalars. In Figure 3a the kinetic energy is plotted for q = .005 (= qo = q,) and P = 6.5 in a loo3 box. The neutral fields start with a kinetic energy below the equilibrium value of the charged fields. The larger X is, the closer the kinetic energy will get to the equipartition value. In Figure 3b we keep the self-coupling the same for

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all the curves, X = 8, and vary q. We observe that the larger qo is, the further away the kinetic energy moves from the equipartition value. Indeed, it is only for no dissipation that the neutral fields do eventually reach the equilibrium curve. This also supports the view that the NB mechanism can only be realised i f there i s a sizeable dissipation in the neutral fields to guarantee that they remain suficiently out-of-equilibrium. For analogous reasons the self-coupling should not be too large either.

3. Discussion

As a preamble to the study of the formation of embedded strings in models with M = S3 we looked at the conditions for the formation of a condensate of neutral mesons in an O(4) linear sigma model. Our study shows that the neutral fields cannot form a collective state unless there is an efficient mechanism for them to lose their “hot” kinetic energy. These fields should also have a relatively weak self-coupling so that the rate of interaction with the particles in the heat bath is small. Further studies will focus on quantitative bounds on the different time scales involved.

A possible dissipation mechanism is provided by the expansion of the universe. In the standard model this dissipation term is not large enough to support alone the formation of either the pion string or the EW string. At the late cosmological time when both these transitions occur, the expansion rate is far too small. However, the situation is more optimistic in extensions of the SM in which the EW transition can be first order. Also at an earlier transition, e.g. at the GUT scale, it is very plausible that the fast decimation of energy strongly favours the formation of embedded strings via the NB mechanism.

References 1. M. Nagasawa and R. H. Brandenberger, Phys. Lett. B 467 (1999) 205 [hep

ph/9904261]; Phys. Rev. D 67 (2003) 043504 [hep-ph/0207246]. 2. A. Achhcarro and T. Vachaspati, Phys. Rept. 327 (2000) 347 [hep-

ph/9904229] and references therein. 3. X. Zhang, T. Huang and R. H. Brandenberger, Phys. Rev. D 58 (1998) 027702

[hepph/9711452]. 4. N. D. Antunes, L. M. A. Bettencourt and M. Hindmarsh, Phys. Rev. Lett. 80

(1998) 908 [hep-ph/9708215].

A SUSY ORIGIN OF GAMMA RAY BURSTS

L. CLAVELLI’ Department of Physics and Astronomy

University of Alabama lhscaloosa AL 35487

Bright bursts of gamma rays from outer space have been puzzling Astronomers for more than thirty years and there is still no conceptually complete model for the phenomenon within the standard model of particle physics. Is it time to consider a supersymmetric (SUSY) origin for these bursts to add to the astronomical indications of supersymmetry from dark matter?

Several times each day, satellites in near earth orbit observe a prodigious burst of gamma rays and the accumulating data has resisted a complete physical explanation for decades. How long should one wait until decid- ing that it is time to consider explanations beyond the standard model? Astronomers, while recognizing a persistent mystery behind the “central engine” of violent astrophysical events, have been notoriously slow to en- courage discussion of new physics solutions.

Ideally, in a situation like that of the gamma ray bursts (grb’s), there should be a broad search for conceptually complete explanations that predict the primary characteristics of the phenomenon in zeroth order. Violent astrophysical events are complicated processes and one must expect difficult numerical computations to be necessary even if the underlying physical model is conceptually complete. By conceptually complete I mean that each step of the process is completely based on known or plausibly proposed physical principles. We suggest that supersymmetry does provide a conceptually complete picture of the burst phenomenon while the standard model of particle and astrophysics does not.

Within the standard approaches to gamma ray bursts there are significant conceptual gaps. Among these are the mechanisms for the enormous energy release required, the mechanism for the transfer of this enormous

* 1clavellQbama.ua.edu

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energy into a narrow range of the gamma ray spectrum, and the mechanism for the strong angular collimation of the bursts if such exists or the extra energy release otherwise required. The places where more physics input is needed are currently marked by the signs “central engine”, “firecones” , “sub-jet s” etc.

In addition, the standard approaches do not predict the primary quantitative characteristics of the gamma ray bursts except as related to free parameters in the theory. Most of the studies of these bursts take as a starting point the unexplained production of relativistic outgoing jets by a dying star and proceed to model the afterglow left in their wake. Several good reviews l2 of the standard approaches are available.

The primary characteristics of the phenomenon are:

(1) Dominant photon energies in the E7 = 0.1 to 1 MeV range. (2) The distribution of burst durations ranges from approximately 0.02

s to 300 s with a pronounced dip in the distribution at 2 s. Bursts of less than 2 s duration are referred to as “short” bursts and those above 2 s are defined to be “long” bursts.

(3) Total burst energy of the order

ET M 3 . 1053-ergs AR 4n

where AR is the opening solid angle of the burst. If AR is 4n (isotropic burst), the energy output is equivalent to or greater than that observed in supernovae The difference, of course, is that supernovae put most of their energy release into neutrinos and into kinetic energy of heavy particles and nuclei, eventually appearing in a broad range of the electromagnetic spectrum. Subject to some possibly strong assumptions, arguments have been given that the jet opening angles are M 5 deg. This would correspond to AR M 7.6 . and ET M 1.8 . 1050 ergs. For comparison with the implied burst energies, the rest mass of the sun is 1.9. ergs and that of the earth is 6 . ergs.

Thus, the shortest bursts can be compared to a hypothetical near solar mass of matter disappearing into MeV scale photons in 20 ms. Relativity would then require that this solar mass should be initially compressed into the radius of the earth ( R E / c = .02s) or less. Such is the typical density of white dwarf stars. However, the sun is prevented from converting its

460

full rest energy to photons by baryon number conservation which is well tested on time scales up to yr. In addition, compact objects such as white dwarfs are Fermi degenerate systems restricted in the energy they can emit by the Pauli principle. The electrons occupy all energy levels up to the Fermi energy. The average and total kinetic energy of a degenerate electron gas in a white dwarf of solar mass and earth radius are, respectively 0.1 MeV and 1.2. 1050 ergs, not far from the observations quoted above assuming a 5" opening angle.

This suggests that one consider the elimination of the Pauli blocking by a transition to a system of Bosons as is possible (perhaps even necessary) given supersymmetry and other physical considerations5. A neutral system of density p, fully relieved of Pauli blocking, will collapse to a black hole in a time

-112 -112 t = ; ( 3 ) 8XGNP ~ ! . 1 . 5 ~ ( & ) .

Thus one might expect that the bursts of duration somewhat above 1.5 s will be depleted and might cause a dip in the duration distribution as is observed at 2 s. Bursts of duration much less or much longer than this time might be relatively unaffected by the cutoff due to gravitational collapse. A full treatment, however, requires studying the collapse during a gradual lifting of the degeneracy pressure.

Although a SUSY phase transition to a system of Bosons can easily explain the primary characteristics of the gamma ray bursts as outlined below, my astronomer friends tell me that substantial progress is also being made in the standard astrophysical approaches to the short duration bursts. Among the most recent ideas that have been proposed is the following '.

ergs of energy could be released into a neutrino-antineutrino cloud which could then be converted on a short time scale into an efe- cloud and made available for the production of a relativistic fireball. Assuming a one percent conversion efficiency, the fireball would have about the right rest energy, lo5' ergs, to account for the gamma ray bursts. It would seem that such standard model explanations for the bursts still do not have the predictive power or the conceptual com- pleteness of the SUSY phase transition model. Similarly, standard model attempts at a unified model of the bursts with a dip in the duration distribution at 2 s often assume the dip is due to a viewing angle dependence l17. This, however, would seem to require a somewhat strange and ad hoc shape of the gamma ray wave packet.

Depending on various parameters, up to

46 1

The SUSY phase transition model for gamma ray bursts is based on the following scenario.

(1) In a region of space with a high level of fermion degeneracy there is a phase transition to an exactly supersymmetric ground state. In the SUSY phase, electrons and their SUSY partners (selectrons) are degenerate in mass as are the nucleons and snucleons, photons and photinos etc. We assume that these masses are at most those of the standard model particles in our normal world of broken supersymmetry. This assumption is, perhaps, supported by the superstring prediction of low mass (massless) ground state supermultiplets.

(2) In the SUSY phase, electron pairs undergo quasi-elastic scattering to selectron pairs

e(p1) + 4P2) + q P 3 ) + e'(P4) (3) which, uninhibited by the Pauli principle, can fall into the lowest energy state via gamma emission. These gamma rays are radiated into the outside (non-SUSY) world. Selectrons and other SUSY particles are confined to the SUSY bubble being too low in energy to penetrate into the broken SUSY world where their mass would be orders of magnitude greater. Since there are about

N = 6 . (4)

electrons in a white dwarf and the average squared momentum in the degenerate sea is

(5) 3 3N 2'3

< p >= - 5 (-) ~ T V . ( 2 ~ h ) ~ x 0.15(MeV/~)~,

the total energy released if all the electrons convert is

ET = N ( < E > -mc2) = 1 .2 . 105'MeV (6) as advertised above.

(3) The highly collimated jet structure might be produced by the stimulated emission of sfermions and photons; in a bath of pre-existing sfermions, new sfermion production goes preferentially into momentum space bins with large occupation numbers.

(4) Simultaneous with electron conversion into selectrons, nucleons within heavy nuclei convert into snucleons. With no further support from the electron degeneracy, the star collapses to nuclear density under gravitational pressure.

462

(5) Remaining nucleon pairs then undergo the analogous conversion to snucleon pairs with the cross section mediated by the strong exchange of supersymmetric pions. This process can be temporarily interrupted by brief periods of fusion energy release but then continues until all kinetic energy is radiated away or until the star falls below the Schwarzschild radius and becomes a black hole.

In this model many of the bursts are due to the decay of isolated white dwarfs, or neutron stars which are absolutely table in standard astrophysics. We therefore predict the existence of low mass black holes below the Chan- drasekhar limit. Other bursts might come from the SUSY transition during a stellar contraction when a stage of fermion degeneracy is reached. If the duration is linearly related to the radius of the star, the longest duration bursts would come from a star of radius lo4 times that of earth (lo2 times that of the sun). However, if the photons undergo a random walk with a mean free path of earth radius before emerging from the dense star, durations would depend quadratically on radius

R2 7- x -.

CRE (7)

In such a case, the longest duration bursts would come from stars of radius 100 times that of earth. Many other effects, however, including fusion reignition and the free collapse time of a star relieved of Pauli blocking, can also influence the duration of the burst so the situation needs more study.

There are now accumulating indications from accelerators and astrophysics that we live in a broken SUSY world with a positive vacuum energy E x 3560 MeV/m3. On the other hand string theory seems to persistently predict that the true vacuum is an exact SUSY world with zero vacuum energy. In this situation it is inevitable that bubbles of true vacuum will spontaneously form in our world and, if of greater than some critical size, R,, will expand to engulf the universe. The potential of some effective scalar field may be as indicated in figure 1 where our local minimum of broken SUSY is labelled “US” and the true vacuum is exactly supersymmetric. In reality, @ may stand for a complicated linear combination of string moduli and there may be a large number of local minima most of which are of a height O(lO1oo) times greater than E .

In the simplified single field situation the critical radius is obtained by an instanton calculation *.

3s R, = 7.

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us SUSY

Figure 1. Effective potential in vacuum. The broken SUSY world is indicated with a postive vacuum energy density while the SUSY phase has zero ground state energy density.

Here S is the surface tension of the bubble of true vacuum. Since it is to be expected that the probability to nucleate a bubble of radius r is a steeply falling function of r, for sufficiently large 9 the broken SUSY phase is metastable in dilute matter. Many authors have discussed a catalysis of the phase transition in the presence of dense matter lo, ll. If the effective field, 9, is a gravitational modulus, it is reasonable to assume that the critical radius is related to the difference in as suggested by figure 2; that is

3s c + A p

R, = -

ground state energy densities

(9)

where Ap is the ground state matter density, p, in the broken SUSY phase minus the corresponding ground state matter density, ps, in the SUSY phase. In a degenerate star Ap is the kinetic energy density of the Fermi gas since, in the SUSY phase, the ground state will consist mainly of Bosons in the lowest energy level. The phase transition to the exact SUSY

464

us SUSY

Figure 2. Effective potential in the presence of matter.

vacuum is then enormously exothermic. For the typical white dwarf

Ap = 6 . MeV/rn3 (10)

Thus we might reasonably expect that the ratio of the critical radius in a white dwarf star to that in vacuum is

Here we have made the simplest assumption, namely that the surface tension is independent of the density. Other possibilities might be explored. If the critical radius m, then the critical radius in vacuum is many times larger than the galactic radius. Thus a supercritical SUSY bubble in a white dwarf would grow only to the edge of the star where it would then be stabilized from further growth.

A transition to a Bosonic final state as proposed in this model not only allows the release of the fermion kinetic energy but also suggests significant jet structure due to the Bose enhancement effect familiar from laser physics.

in a white dwarf is of order

465

In addition to the momentum dependence of the elementary process, the matrix element for the emission of a selectron pair with momenta F3 and $4 in process 3 in the presence of a bath of previously emitted pairs is proportional to

M N< 4 5 3 ) + 1,n@4) + 1 I at@3)at(p’4) I n@3),n@4) > d l 4 m T m . (12)

The cross section is, therefore, proportional to (n(p77) + l)(n(j&) + 1) . The calculation l2 of the process 3 has been recently extended l3 to

include the electron mass effects which are important in the exact SUSY phase.

p3 p1 p4 - . I . - -*--- ’ 1- - +- - - -

p2 p4 p2 p3

Figure 3. jorana) photino exchange.

Feynman graphs for the conversion of an electron pair to selectrons via (Ma-

There are two types of scalar electron (partners of the left and right handed electron) which must be treated separately. The three different final states are found to have the elementary squared matrix elements

I M L R ~ ~ = Ess~ (MaML + MbMJ + MaMJ + MbMi)

4mz (mz - mz) (t - M$)(u - M$) + t * u +

466

where C,,! denotes averaging over the spins of incoming electrons.

For example the conversion rate per unit volume in the degenerate electron gas into left selectrons is given by

(n~(@3) + l ) (n~@4) + 1). (15)

An event generator for this process has been constructed l3 where, in principle, one generates one event at a time continually updating the probability distribution in accordance with the Bose enhancement factors. Initially, all the occupation numbers are zero but, after the first event, the next event is four times as likely to be into the same phase space cell as into any other. Because of the huge number of available states, the second transition is still not likely to be into the first phase space cell. However, as soon as some moderate fluctuation of selectrons has been produced in a single cell, the number in that state escalates rapidly producing a narrow jet of selectrons. As these decay down to the ground state via bremstrahlung, a narrow jet of photons is created around the direction of the initial selectron jet. After a large number of events, a jet structure emerges as the distribution locks in on particular cells in the final state phase space. To speed up the emergence of the jet structure, the occupation numbers are, in some studies, incremented by two units at each event. The distribution in momentum magnitude, polar angle cosine, and azimuthal angle after 6 - lo5 events is shown in table 1. The selectron momentum will essentially be totally converted to gamma rays as the selectrons fall into the ground state although we have not as yet treated this radiative process in detail.

Interesting features of the distribution are the jagged nature of the angular distribution and preference for momenta above the average of the Fermi sea. This latter effect suggests that earlier (more probable) emissions will be of greater energy than later ones in agreement with indications from burst observations 14.

There are obviously many details of the behavior of a dense supersymmetric system remaining to be investigated but the SUSY phase transition picture provides an alternative framework for thinking about gamma ray bursts that predicts some of the basic characteristics and addresses some of the conceptual gaps of the standard approaches.

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Table 1. Selectron momentum and angular distributions showing the effect of boson enhancement.

0.075 76 0.124 229 0.174 553 0.224 1000 0.274 1867 0.324 2831 0.373 9285 0.423 107484 0.473 2886144

case ~ ( c o s ~ ) -0.90 71969 -0.70 111973 -0.50 34399 -0.30 19424 -0.10 632776 0.10 114240 0.30 1272163 0.50 383288 0.70 86555 0.90 282697

4 N(4) 0.314 43430 0.942 70097 1.571 472570 2.199 34781 2.827 117413 3.456 358925 4.084 29715 4.712 1813451 5.341 26526 5.969 42576

Acknowledgements This work was supported in part by the US Department of Energy

under grant DE-FG02-96ER-40967. The work described here was done in collaboration with George Karatheodoris and Irina Perevalova. In addition we gratefully acknowledge discussions with Doug Leonard, Phil Hardee, and Yongjoo Ou.

References 1. Bing Zhang and Peter MbzAros, Astro-ph/0311321 2. S. Rosswog, Astro-ph/0401022 3. C. Kouveliotou et al., Astr0phys.J. 413, LlOl (1993) 4. Frail et al., ApJ 526, 152 (1999) 5. L. Clavelli and G. Karatheodoris, hep-ph/0403227 6. W.H. Lee, E. Ramirez-Ruiz, and D. Page, Astrophys. J. 608 ,L5 (2004) (astro-

ph/0404566) 7. R. Yamazaki, K. Ioka, and T. Nakamura, astro-ph/0401142 8. S. Coleman, Phys. Rev. D15, 2929 (1977); C.G. Callan and S. Coleman, Phys.

Rev. D16, 1762 (1977); S. Coleman and Frank DeLuccia, Phys. Rev. D21, 3305 (1980)

9. Paul H. F’rampton, hep-th/0302007 10. AS. Gorsky and V.G. Kiselev, Phys. Lett. B304,214 (1999) 11. M.B. Voloshin, Phys. Rev. D49, 2014 (1994) 12. W.-Y. Keung and L. Littenberg, Phys. Rev. D28, 1067 (1983) 13. L. Clavelli and I. Perevalova, hep-ph/0409194 14. E. Pian et al., astro-ph/9910235

TADPOLES AND SYMMETRIES IN HIGGS-GAUGE UNIFICATION THEORIES

C. BIGGIO Institut de Fisica d'Altes Energies (IFAE),

Uniuersitat Autbnoma de Barcelona, E-08193 Bellaterra (Barcelona), SPAIN


M. QUIROS Institucio' Catalana de Recerca i Estudis AvanGats (ICREA) and

Institut de Fisica d 'Altes Energies (IFAE), Universitat Autbnoma de Barcelona,

E-08193 Bellaterra (Barcelona), SPAIN E-mail: [email protected]

In theories with extra dimensions the Standard Model Higgs fields can be identified with internal components of bulk gauge fields (Higgs-gauge unification). The bulk gauge symmetry protects the Higgs mass from quadratic divergences, but at the fixed points localized tadpoles can be radiatively generated if V ( 1) subgroups are conserved, making the Higgs mass UV sensitive. We show that a global symmetry, remnant of the internal rotation group after orbifold projection, can prevent the generation of such tadpoles. In particular we consider the classes of orbifold compactifications T d / Z ~ ( d even, N > 2 ) and T d / Z z (arbitrary d) and show that in the first case tadpoles are always allowed, while in the second they can appear only for d = 2 (six dimensions).

1. Introduction

Among the possible motivations for studying theories in extra dimensions with Higgs-gauge ~nificationl-~ there is the so called little hierarchy problem5. The latter consists in the one order of magnitude discrepancy between the upper bound for the Standard Model cutoff AEW coming from the requirement of stability of the Higgs mass under radiative corrections and the lower bound arising from the non-observation of dimension-six four- fermion operators6.

Up to now the best solution to the little (and grand) hierarchy problem

468

469

is supersymmetry. Indeed in this framework the Standard Model cutoff is identified with the mass of supersymmetric particles, while R-parity conservation induces a suppression in the loop corrections to four-fermion operators which solves the little hierarchy problem. However, since the minimal supersymmetric Standard Model extension is becoming very constrained, it is useful to propose possible alternative solutions which may fill the gap between the sub-TeV scale required for the stability of the electroweak symmetry breaking and the multi-TeV scale required by precision tests of the Standard Model.

One possible alternative solution is Higgs-gauge unification. In these theories the internal components of higher dimensional gauge bosons play the role of the Standard Model Higgses and can acquire a non-vanishing vacuum expectation value through the Hosotani mechanism7. The Higgs mass in the bulk is protected from quadratic divergences by the higher- dimensional gauge theory and only finite corrections o( (l/R)' (R is the compactification radius) can appear. The Standard Model cutoff is then identified with 1/R and the little hierarchy between 1/R and A, which is now the cutoff of the higher dimensional theory, is protected by the higher- dimensional gauge symmetry. However at the fixed points the bulk gauge symmetry is broken and localized terms consistent with the residual symmetries can be generated by quantum corrections*. While a direct localized squared mass (- A') for the Higgs-gauge fields is forbidden by a shift symmetry remnant of the original bulk gauge symmetry', if a U(1) symmetry is conserved at a given fixed point then the corresponding field strength can be radiatively generated, giving rise to a quadratic divergent mass for the Higgs2y3. This is a generic feature of orbifold compactifications in dimensions D 2 6 and has been confirmed in six-dimensional orbifold field4 and ten-dimensional stringg theories. One way out4 is that local tadpoles vanish globally, but a more elegant way, explored in Ref. [lo] and discussed in this talk, is to find a symmetry which forbids the generation of these localized terms. This symmetry is precisely the subgroup of the tangent space group SO(D - 4) whose generators commute with the orbifold group elements leaving the considered fixed point invariant. How this symmetry comes out and how it is related to the generation of localized tadpoles will be discussed in the following.

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2. Symmetries at the fixed points and allowed localized

We consider a gauge theory (gauge group G ) coupled to fermions in a D = d+4 > 4 dimensional space-time parametrized by coordinates zM = (zf i , yi) where p = 0 ,1 ,2 ,3 and i = 1,. . . , d. The Lagrangian is

terms

with F i N = d M A i - dNA& - gfABCAEAs , DM = aM - igA&TA and where rg are the I?-matrices corresponding to a D-dimensional space-time. The local symmetry of (1) is the invariance under the (infinitesimal) gauge transformations

(2) 1 9

ScA& = - a ~ < ~ - fABCEBA$, dci3 = i tATAi3.

Now we compactify the extra dimensions on an orbifold. Firstly we construct a d-dimensional torus T d by modding out Rd by a d-dimensional lattice Ad and then we define the orbifold by modding out T d by (6, where (6 is a discrete symmetry group acting non-freely (i.e. with fixed points) on itll. The orbifold group is generated by a discrete subgroup of SO(d) that acts crystallographically on the torus lattice and by discrete shifts that belong to the torus lattice. The action of Ic E (6 on the torus is k. y = Pk y + u, where Pk is a discrete rotation in SO(d) and u E Ad; y and Ic . y are then identified on the orbifold. Since the orbifold group is acting non-freely on the torus there are fixed points characterized by k . yf = yf. Any given fixed point yf remains invariant under the action of a subgroup Gf of the orbifold group.

The orbifold group acts on fields q 5 ~ transforming as an irreducible representation R of the gauge group 9 as

k. ~ R ( Y ) = @ P," q 5 ~ ( k - l . Y) (3)

where A: is acting on gauge and flavor indices and Pt, where u refers to the field spin, on Lorentz indices. In particular one finds for scalar fields Pl = 1 and for gauge fields P," = pk for a discrete rotation (P," = 1 for a lattice shift), while for fermions P$ can be derived requiring the invariance of the lagrangian under the orbifold action. On the other side A& depends on the gauge structure and the gauge breaking of the orbifold action.

In general the orbifold action breaks the gauge group in the bulk 9 = { T A } to a subgroup 'Hf = {T"f} at the fixed point yf. It can be shown that the subgroup Xf left invariant by the orbifold elements k E Gf is

2

471

defined by the generators that commute with Ak, i.e. [Ak,Tz] = 0. The latter condition must be satisfied by any irreducible representation R of 8.

We now consider the effective four-dimensional lagrangian. This can be written as:

Lzff = / ddy [ L D + dd) (9 - 1/ j ) L j ] (4) f

where LD is given by (1) and Lj is the most general lagrangian consistent with the symmetries localized at the fixed point yf. In order to write the most general Cj we need to know which are the symmetries present at each fixed point. First of all the operators must be invariant under the action of the orbifold group [Gf] and the 4D Lorentz group [S0(1,3)]. Then we have to consider the bulk gauge symmetry E: when applied to the orbifold fixed points y j it reduces to the four-dimensional gauge symmetry % j = {Tat} that applies to the four-dimensional gauge fields AEf which are also invariant under the orbifold action. This consists in the usual gauge invariance under %-transformationsa 6tAE = a, ta /g - f abc<bAE. However this is not the only symmetry generated by the original gauge symmetry E. Indeed by localizing the transformations (2) at the orbifold fixed point 9 j and keeping the orbifold invariant terms one can define an infinite set of transformations (remnant of the bulk gauge invariance) induced by derivatives of < A that we can call lc-transformations2. Then only % and lc-invariant quantities are allowed at the orbifold fixed points.

The presence of the remnant gauge symmetry K: is very important in order to prevent the appearance of direct mass terms for gauge fields localized at the orbifold fixed points. Indeed if the gauge field A6 is invariant under the orbifold action, where T k E E/%, the remnant “shift” symmetry 6cAt = &<‘/g - fkbetbA? prevents the corresponding zero mode from acquiring a localized mass.

Now that we know which are the symmetries at the fixed points we can write the most general lagrangian Lf. Invariant operators are (Fiv)2 , which corresponds to a localized kinetic term for A; and F:”ptv which is a localized anomaly. Moreover if for some (i, j) FG is orbifold invariant (this is model-dependent), (Fa.)2 can be non-zero at a fixed point and if also A: is orbifold invariant, ( F i ) 2 can be present too. These last two lagrangian terms give rise, respectively, to localized quartic couplings and localized

&From here on we will remove for simplicity the subscript “f“ from the gauge group and the corresponding generators.

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kinetic terms for A:. All these operators are dimension four, that is they renormalize logarithmically. However if 3t contains a U(1) factor

FG = &A: - ajAY - g f ageA!Af z j , ( 5 ) where Q is the U ( 1 ) quantum number, is 31 and Ginvariant as well as orbifold and Lorentz invariant. If this operator is allowed, we expect that both a tadpole for the derivative of odd fields, aiAja, and a mass term for the even fields, faieAfA!, will be generated on the brane by bulk radiative corrections. Moreover, since these operators have dimension two, we expect that their respective renormalizations will lead to quadratic divergences, making the theory ultraviolet sensitive.

Apart from the case of D = 5 where the term Fij does not exist, for D 2 6 it does and we expect the corresponding mass terms to be generated on the brane by radiative corrections. This has been confirmed by direct computation in six-dimensional orbifold field2-4 and ten-dimensional stringg theories. Of course if these divergent localized mass terms were always present, Higgs-gauge unification theories would not be useful in order to solve the little hierarchy problem. One way out can be that local tadpoles vanish globally, but this requires a strong restriction on the bulk fermion content4. A more elegant and efficient solution would be finding another symmetry forbidding the generation of localized tadpoles: this symmetry exists and has been studied in Ref. [lo].

When compactifing a d-dimensional space to a smooth Riemannian manifold (with positive signature), a tangent space can be defined at each point and the orthogonal transformations acting on it form the group SO(d)12. When an orbifold group acts on the manifold it also breaks the internal rotation group SO(d) into a subgroup O f at the orbifold fixed point yf. Indeed here a further compatibility condition between the orbifold action and the internal rotations is required. In particular, if the given fixed point yf is left invariant by the orbifold subgroup Gf , only Gf -invariant operators @ R , ~ couple to yf, i.e.

l c . @R,o(Yf) = @R,o(Yf) * (6) Acting on @ R , ~ with an internal rotation we get a transformed operator that should also be Gf-invariant. This means, using Eq. (3), that the subgroup Of is spanned by the generators of SO(d) that commute with P,", i.e. they satisfy the condition

[Of , 721 = 0 (7)

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for k E Gf and arbitrary values of o. In particular in the presence of gauge fields AM = ( A p , Ai) an invariant operator can be Fij with R = Adj and o = 2. The internal components Ai transform under the action of the orbifold element k E Gf as the discrete rotation 4. At the orbifold fixed point yf only the subgroup O f E SO(d) survives and the vector representation Ai of SO(d) breaks into irreducible representations of Of.

We have then identified an additional symmetry that the lagrangian Cf at the fixed point yf must conserve. Summarizing, the invariances that we have to take into account are the following: four-dimensional Lorentz invariance [SO(1,3)], invariance under the action of the orbifold group [GI, usual four-dimensional gauge invariance [%I, remnant of the bulk gauge invariance [K] and invariance under rotations of the tangent space [Of].

Now we consider the tadpole term of Eq. (5) in order to see if and when it is invariant under the last discussed symmetry. We have just seen that the vector representation Ai of SO(d) breaks into irreducible repre sentations of the internal rotation group Of SO(d). In particular if the rotation subgroup acting on the (i, j)-indices is SO(2) then eijF$, where cij is the Levi-Civita tensor, is invariant under Of and so it can be radiatively generated. On the other hand if the rotation subgroup acting on the (i, j)-indices is SO(p) ( p > 2) then the Levi-Civita tensor would be ci1i2-.ip

and only invariants constructed using p-forms would be allowed. In other words a sufficient condition for the absence of localized tadpoles is that the smallest internal subgroup factor be ,SO@) ( p > 2).

3. Tadpoles for T D / Z ~ orbifolds

To show explicitly how the above discussed symmetry arguments apply, we consider the class of orbifolds (6 = ZN for even d. The generator PN of the orbifold group is defined by

where ki are integer numbers (0 < ki < N ) and J2i-1,2i is the generator of a rotation with angle 27r% in the plane (y2+1,y2i). All orbifold elements are defined by Pk = Ph (k = 1, ..., N - 1) and satisfy the condition P# = 1. The generator of rotations in the (y2i-1, yZi)-plane can be written as J2iW-1,2i = diag(0,. . . , 02,. . . ,0) where the Pauli matrix o2 is in the i-th two-by-two block. Therefore the generator PN can be written as PN = diag(R1, . . . , Rd/2) where the discrete rotation in the (yzi-1, yzi)-plane is

474

defined as

with ci = cos(2~ki/N), si = sin(2nki/N). Let yf be a fixed point that is left invariant under the orbifold subgroup

Gf = Z N ~ where Nf 5 N. We now define the internal rotation group Of as the subgroup of SO(d) that commutes with the generator of the orbifold Z N ~ , P N ~ as given by Eq. (8) with N replaced by Nf. In general, if Nf > 2 O f is trivially provided by the tensor product:

d/2 Of = g) SO(2)i (10)

i= 1

where SO(2)i is the SO(2) E SO(d) that acts on the (~2i-1,~2i)-~ubspace. In every such subspace the metric is 61 J and the Levi-Civita (antisymmetric) tensor €IJ ( I , J = 2i - 1,2i , i = 1,. . . , d/2) such that we expect the tadpoles appearance at the fixed points yf as

2i

i=l I , J=2 i - l

If Nf = 2 then the generator of the orbifold subgroup Gf = Z2 is the inversion P = -1 that obviously commutes with all generators of SO(d) and Of = SO(d). In this case the Levi-Civita tensor is and only a d-form can be generated linearly in the localized lagrangian. Therefore tadpoles are only expected in the case of d = 2 ( D = 6).

The last comments also apply to the case of Z2 orbifolds of arbitrary dimensions (even or odd) since in that case the orbifold generator is always P = -1 and the internal rotation group that commutes with P is Of = SO(d) for all the fixed points. Again tadpoles are only expected for D = 6 dimensions while they should not appear for D > 6.

Since every operator in Cf that is consistent with all the symmetries should be radiatively generated by loop effects from matter in the bulk (unless it is protected by some other -accidental- symmetry) explicit calculations of the tadpole in orbifold gauge theories should confirm the appearance (or absence) of them in agreement with the above symmetry arguments .

In Ref. [lo] we have considered the class of orbifold compactifications Td/Z2 and we have explicitly calculated the tadpole at one- and two-loop level. The result we found, in agreement with our general conclusions,

4 75

is that the tadpole is zero for D > 6 , while it can be non-zero only for D = 6. In particular we found that gauge and ghost contributions are always zero, even in six dimensions, due to an extra parity symmetry that inverts separately the internal coordinates. As for the fermion contribution, it is always zero except for D = 6 , where it is proportional to e i j , in agreement with Eq. (11). Since in six dimensions there are two possibilities for Pi, we observed that in one case the result is chiral-independent, while in the other the sign depends on the six dimensional chirality of fermions. In the latter case this means that starting with Dirac fermions can imply a vanishing tadpole. Unfortunately, due to the existing relation between the tadpole and the mixed U ( 1)-gravitational anomaly, a vanishing tadpole corresponds to a non-vanishing anomaly. The condition for a vanishing tadpole coincides with the one for a vanishing anomaly only when dealing with chiral fermions of equal six-dimensional chirality, where this reduces to

4. Conclusions and outlook

In orbifold field theories Standard Model Higgs fields can be identified with the internal components Ai of bulk gauge fields. Higher-dimensional gauge invariance prevents the Higgs from acquiring a quadratically divergent mass in the bulk while a shift symmetry, remnant of the bulk gauge symmetry at the fixed points, forbids it on the branes. Nevertheless, if a U(1) subgroup is conserved at the fixed points, the corresponding field strength is gauge invariant and can be radiatively generated giving rise, through its non-abelian part, to a quadratically divergent mass for the Higgs. However an additional symmetry must be taken into account. Indeed, being the Higgs fields identified with the internal components of bulk gauge fields, they transform in the vector representation of the tangent space rotation group SO(d). This is valid in general on a d-dimensional compact manifold, but at the orbifold fixed points the orbifold projection must be taken into account. This induces a reduction of the tangent space group to the subgroup Of whose generators commute with the orbifold subgroup leaving the corresponding fixed points invariant. If Of = SO(2), then the Levi-Civita tensor with two indices cij exists and the corresponding field strength can be generated on the brane as &Fij. On the contrary if Of 2 SO(p) with p > 2 then only invariant terms built with the pform eg1.,.'p are allowed and so the tadpole cannot be generated. We have shown that in the class

. .

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of orbifolds T d / Z ~ (d even, N > 2) Of = SO(2) 8 SO(2) 8 ... 8 SO(2), i.e. tadpoles are always allowed. On the contrary on orbifolds T d / Z 2 (arbitrary d ) Of = SO(d) that means that tadpoles can be generated only in six dimensions (d = 2).

However the absence of tadpoles seems to be a necessary but not sufficient condition in order to build a realistic theory of electroweak symmetry breaking without supersymmetry. First of all it seems compulsory considering models with more than six dimensions, since in five dimensions the absence of quartic coupling leads to too low Higgs masses, while in six dimensions electroweak symmetry breaking is spoiled by localized tadpoles. We have seen that for T d / Z 2 orbifolds with d > 2 tadpoles cannot be generated, but in general these models predict the existence of d Higgs fields, leading to non-minimal models. Of course the conditions that preclude the existence of quadratic divergences for Higgs fields do not forbid the radiative generation of finite N (1/R)2 masses, that vanish in the R + 00 limit. Some of the above Higgs fields can acquire different masses and even not participate in the electroweak symmetry breaking phenomenon, depending on the models. Moreover, even if we were able to build a model with only one Standard Model Higgs field, its mass should be in agreement with LEP bounds and precision measurements.

Another issue that must be addressed in order to construct a realistic model is the flavour problem. One possibility should be putting matter fermions in the bulk and coupling them to an odd mass that localizes them at different locations13. This seems nice since, for instance, it can also be used to explain fermion replica14, but problems can arise with CP violation and flavour changing neutral currents15. Another possibility is to consider localized matter fermions which can develop Yukawa couplings through Wilson line interactions after the heavy bulk fermions have been integrated out 3~16.

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491; I. Antoniadis and K. Benakli, Phys. Lett. B 326 (1994) 69 [arXiv:hep- th/9310151]; H. Hatanaka, T. Inami and C. S. Lim, Mod. Phys. Lett. A 13 (1998) 2601 [arXiv:hep-th/9805067]; H. Hatanaka, Prog. Theor. Phys. 102 (1999) 407 [arXiv:hep-th/9905100]; M. Kubo, C. S. Lim and H. Yamashita, arXiv:hep-ph/0111327; G. R. Dvali, S . Randjbar-Daemi and R. Tabbash, Phys. Rev. D 65 (2002) 064021 [arXiv:hep-ph/0102307]; I. Antoniadis, K. Be- nakli and M. Quiros, Nucl. Phys. B 583 (2000) 35 [arXiv:hep-ph/0004091];

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I. Antoniadis, K. Benakli and M. Quiros, New Jour. Phys. 3 (2001) 20 [arXiv:hep-th/0108005].

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5. R. Barbieri and A. Strumia, arXiv:hep-ph/0007265; G. F. Giudice, Int. J. Mod. Phys. A 19 (2004) 835 [arXiv:hep-ph/0311344].

6. S. Eidelman et al., “Review of Particle Physics”, Phys. Lett. B 592 (2004) 1. 7. Y. Hosotani, Phys. Lett. B 126 (1983) 309; Y. Hosotani, Annals Phys. 190

(1989) 233; N. Haba, Y. Hosotani and Y. Kawamura, Prog. Theor. Phys. 111 (2004) 265 [arXiv:hep-ph/0309088]; N. Haba, M. Harada, Y. Hosotani and Y. Kawamura, Nucl. Phys. B 657 (2003) 169 [Erratum-ibid. B 669 (2003) 3811 [arXiv:hep-ph/0212035]; Y. Hosotani, S. Noda and K. Takenaga, Phys. Rev. D 69 (2004) 125014 [arXiv:hep-ph/0403106]; Y. Hosotani, S. Noda and K. Takenaga, arXiv:hep-ph/0410193.

8. H. Georgi, -4. K. Grant and G. Hailu, Phys. Lett. B 506 (2001) 207 [arXiv:hep- ph/0012379]; H. Georgi, A. K. Grant and G. Hailu, Phys. Rev. D 63 (2001) 064027 [arXiv:hep-ph/0007350]; R. Contino, L. Pilo, R. Rattazzi and E. Trincherini, Nucl. Phys. B 622 (2002) 227 [arXiv:hep-ph/0108102]; G. von Gersdorff, N. Irges and M. Quiros, Nucl. Phys. B 635 (2002) 127 [arXiv:hep- th/0204223].

9. S. Groot Nibbelink, H. P. Nilles, M. Olechowski and M. G. A. Walter, Nucl. Phys. B 665 (2003) 236 [arXiv:hep-th/0303101]; S. Groot Nibbelink and M. Laidlaw, JHEP 0401, 004 (2004) [arXiv:hep-th/O311013]; S. Groot Nibbe- link and M. Laidlaw, JHEP 0401 (2004) 036 [arXiv:hep-th/0311015].

10. C. Biggio and M. Quiros, to appear in Nucl. Phys. B [arXiv:hep-ph/0407348]. 11. L. J. Dixon, J. -4. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 (1985)

678; L. J. Dixon, J. A. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 274 (1986) 285.

12. See e.g. M. B. Green, J. H. Schwarz and E. Witten, “Superstring The- ory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology,” Cambridge, Uk: Univ. Pr (1987) (Cambridge Monographs On Mathematical Physics).

13. G. Burdman and Y. Nomura, Nucl. Phys. B 656 (2003) 3 [arXiv:hep- ph/0210257].

14. C. Biggio, F. Feruglio, I. Masina and M. Perez-Victoria, Nucl. Phys. B 677 (2004) 451 [arXiv:hep-ph/0305129].

15. A. Delgado, A. Pomarol and M. Quiros, JHEP 0001 (2000) 030 [arXiv:hep- ph/9911252].

16. C. A. Scrucca, M. Serone and L. Silvestrini, Nucl. Phys. B 669 (2003) 128 [arXiv:hep-ph/0304220].

VECTOR/TENSOR DUALITY IN FIVE DIMENSIONS AND APPLICATIONS

EMILIAN D U D A S ~ J tLPT Univ. de Paris-Sud, Bi t . 210, F-91405 Orsay Cedex, France

CPhT, UMR du CNRS 7644, Ecole Polytechnique, F-91128 Palaiseau, France

The five dimensional version of the Green-Schwarz mechanism can be invoked to cancel abelian anomalies on the boundaries of brane world models. In five dimensions there are two dual descriptions that employ either a two-form tensor field or a vector field. We present the supersymmetric extensions of these dual theories using four-dimensional superspace. The vector formulation always contains singular boundary mass terms which are absent in the tensor formulation. This apparent inconsistency is resolved by showing that in either formulation the propagator of the anomalous U( 1) gauge (super)field exhibits finite (non-local) boundary mass terms, which generically lead to a finite nonzero mass for the lowest lying Kaluza- Klein mode. Talk based on the work done in collaboration with T. Gherghetta and S. Groot-Nibbelink.

1. Introduction

The quantum consistency of gauge theories coupled to matter, specifically the absence of anomalies, has proven to be one of the most important guid- ing principles for model building beyond the Standard Model. Therefore not surprisingly there have been many investigations' of anomalies in brane world models, in particular orbifold theories. The general outcome of these analyses is that the anomalies localize on even dimensional hypersurfaces and are determined by the local fermionic spectrum of theory there. (This conclusion also remains true in the context of warped compactifi~ations~). In odd dimensions anomalies do not need to cancel locally, since as long as they cancel globally, a bulk Chern-Simons term can be used to ensure that the theory is gauge invariant at the quantum level. Therefore, an interesting question is whether it is possible to have even more general localized anomalies that can still lead to consistent theories.

*unit6 mixte de recherche du cnrs (umr 8627).

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One of the major breakthroughs in the development of string theory was the realization of Green and Schwarz that so-called factorisable anomalies can be compensated by anomalous variations of anti-symmetric tensor fields. Studies of heterotic string compactifications to four dimensions has revealed that a four dimensional version of the Green-Schwarz mechanism is relevant for phenomenological string model building. This mechanism simultaneously cancels both pure and mixed U ( 1) anomalies provided that the spectrum satisfies a specific universality condition. In particular, there will be a mixed gravitational-gauge anomaly which in the N = 1 supersymmetric context is directly related to one-loop induced Fayet-Iliopoulos terms4 that has been confirmed in the string theory context5.

This provides motivation to investigate how the local Green-Schwarz mechanism can be implemented in a five dimensional setting. In odd dimensions the prime example of the Green-Schwarz-like mechanism is given by eleven dimensional M-theory. An interesting aspect of the Green-Schwarz theory is that it allows for two dual descriptions. In five dimensions the Green-Schwarz interaction can be described by a rank two tensor or a vector. This raises the intriguing possibility that the model can be described by two equivalent descriptions. As we will show this indeed occurs in our five dimensional theory even though the two descriptions are not always manifestly equivalent', For example, the vector formulation contains singular boundary mass terms for the anomalous U(1) gauge field which are absent when the two-form formulation is used. However, the proper inclusion of mixing terms in the vector formulation resolves this apparent inconsistency. In fact, the consistency of this duality provides confirmation of the S(0) regularization proposed in Ref. [6] .

An elegant and simple way to have explicit supersymmetric theories in higher dimensions is to employ four dimensional N = 1 superspace techniques in five dimensions'. In particular, for the tensor formulation we use the five dimensional tensor multiplet and find that contrary to the conventional four dimensional case the N = 1 linear supermultiplet is in- sufficient to contain all the bosonic components resulting from the dimensional reduction. In addition the Green-Schwarz theory contains Chern- Simons interactions. The five dimensional completion of the four dimensional Chern-Simons (threeform) superfield will require an additional chiral Chern-Simons superfield.

Although we are mainly concerned about a single U ( ~ ) A anomalous gauge group, it is also possible to consider the generalization to multiple U(1) gauge groups. These results can also be conveniently written in terms

480

of a prepotential. We expect that the five dimensional formalism developed here in the

context of models with large extra dimensions and a low fundamental scale could be useful for the fermion mass hierarchy and supersymmetry breaking.

2. Bosonic Green-Schwarz mechanism in five dimensions

In this section we will review the basics of the Green-Schwarz anomaly cancellation mechanism in the context of a five dimensional gauge theory on the interval S1/Zz. The spacetime coordinates are denoted by x M = ( x o , . . . x3 , x5 = y), with 0 5 y 5 T (where we have set the radius R of the extra dimension to unity). For simplicity we will consider the gauge potential A M , A4 = 0 , . . . 3 , 5 , that corresponds to a single U ( ~ ) A gauge group with field strength F M N . As is well-known bulk fermions and brane chiral fermions give rise to gauge anomalies localized on the boundaries. When the anomalies on the two boundaries are equal and opposite, they can be cancelled by a five dimensional Chern-Simons interaction. However this is a very special form for the anomaly and more general boundary anomalies can instead be cancelled by invoking the Green-Schwarz mechanism. In five dimensions the Green-Schwarz mechanism can be described by either a rank two tensor B M N or a vector AM that have field strengths HMNP and F M N , respectively.

2.1. Anomalous U ( ~ ) A gauge field in the bulk

The gauge field AM in the bulk has the standard Yang-Mills action

where we have introduced form notation for brevity and compliance with the standard literature on anomalies.

By assumption the effective action r , f f (A ) , obtained by integrating out all chiral brane and bulk fermions, is not invariant under the gauge transformation 6A1 = 6ao. However, because of the Wess-Zumino consistency conditions and the fact that gauge anomalies do not exist in five dimensions, we infer that the variation of the effective action takes the form

with

48 1

where t rq; denotes the cubic sum of charges of the chiral fermions at the boundary I = O , T , and trqi that of the bulk fermions. Note that the bulk fermions give equal anomalies on both boundaries. With the notation 11 we emphasize that all fields are to be evaluated at the boundary y = I .

In the Green-Schwarz anomaly cancellation mechanism a crucial role is played by the Chern-Simons three-form

R3(A) = AlF’ , 603(A) = F: , (4)

for the anomalous U ( ~ ) A gauge potential one-form A l . Under a gauge transformation, the gauge variation of the Chern-Simons three form is given by

6,,Qj(A) = 6aoF2 . (5)

In five dimensions a vector AM is dual to a two-form tensor B M N . (Aspects of the vector-tensor duality in five dimensions were discussed in the past, see for example [12], in the context of the Type I1 - heterotic duality.) A convenient starting point to describe this duality is to use the action

which contains the non-dynamical two forms F 2 and B2.

extended to the Green-Schwarz action The vector/tensor action 6, described in the previous subsection, is

with the Chern-Simons three-form 03(A) given in (4). We have also included the standard kinetic term (1) of the gauge field one-form A,; it will be a “spectator” as far as the duality is concerned. The two-form X2(A) only has support at the end points of the interval

This action has been chosen such that it cancels the anomalous variation of the effective action reff obtained by integrating out the brane and bulk fermions:

60, (reff -k SGS) = 0 . (9)

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Since the anomalous variation 2 of the effective action reff does not contain the Hodge *-dualization, we infer that the Green-Schwarz tensors BMN and FMN transform (up to exact terms) as A

A

6,,B2 = -aoF2 , 6,,F2 = - *6(6,,B2) = * ( 6 a o F ~ ) . (10)

The gauge variation of the Green-Schwarz action is given by

This variation indeed cancels the anomalous contribution of r,.f given in 2. It should be stressed that this anomalous variation is independent of which dual formulation of the theory is being used, since we did not yet choose to eliminate either p2 or B2 from the theory.

As in the previous subsection we can go to the one- or two-form formulation of the Green-Schwarz mechanism by integrating out B2 or p2. The equation of motion for B2 is 6 ( p 2 - *%(A) + X2(A) ) = 0 which is solved by

F 2 = 6A1+ *!23(A) - X2(A) , (12) h

using the one-form A1. In order for F2 to have the required gauge transformation it follows that A1 transforms under the anomalous U ( ~ ) A as

6,,Al = 1 CIQO6(Y - I)SY . (13)

Note that this anomalous gauge transformation only affects 6,,A5 = CIaOb(y - I ) , while the four dimensional components are invariant &,A, = 0. After substituting (12) into the action (7) the oneform formulation of the Green-Schwarz mechanism becomes

1 SGsl = / [ T *6AiSA1+

1 1 -*(dAi+*fk(A)-X2(A)) 2 ( ~ A I + * % ( A ) - - X ~ ( A ) ) 5 * Rs(A)%(A)]

= 1 [z *6AlbA1+ 1 *6A16A1- 03(A)6A1 - ( * 6A1 - 5 *X2(A))X2(A)] .

Notice that the term &(A)X, (A) is zero. If there are multiple non- anomalous U(l)’s then this term does not cancel and leads to the generalized Chern-Simons terms with two gauge field one-forms.

The term -RS(A)SA1 is not gauge invariant, and therefore the gauge variation of 14 does not equal ( l l ) , unless the four dimensional components

(14)

1 1 2

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of the Green-Schwarz vector vanishes at the boundaries A,II = 0. This action is only defined formally since it involves squares of delta functions:

However, since it has been obtained from the regular action 7, there is a unique and well-defined way to deal with the singular contribution S(0) in this case.

The dual formulation using the two-form B2 with F 2 = - * 6B2 is manifestly free of any singular looking terms

SGS2 = / [i * 6A16A1+ - *(dB2 +%(A) ) (SB2 + % ( A ) ) - SB2X,(A)] .

One therefore finds that the two-form description is free of singular terms, while in the dual formulation involving a six-form the singular S(0) term arises as well.

Furthermore, by introducing the gauge invariant three-form field strength 2 = SBZ + u3(A) one obtains the anomalous Bianchi identity SH = F:. Finally, the duality relation between the two- and one-form formulation is modified to

1 2

(16)

h

2.2. The Anomalous U1 photon mass

We have shown that there are two equivalent dual formulations of the Green-Schwarz mechanism in five dimensions. Let us now consider the mass of the anomalous U ( ~ ) A photon which is determined by the boundary mass terms. In the vector formulation (15) there are divergent boundary mass terms which appear to be absent in the dual tensor formulation (16). This would seem to contradict the equivalence of the two dual formulations. However we will see that both formulations give rise to the same (non-local) boundary mass term for the photon. The key observation is that both the vector and tensor formulations contain mixing terms between the vector or tensor and the anomalous U ( ~ ) A gauge field that need to be correctly taken into account.

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Consider first the tensor formulation which contains the mixing term

/ -6B2X2(A) = -: ~ E I /d5x PnPqd,BnpAq S(y - I ) . (18) I

Using the two-form propagator

- a (BmnBpq) = pq$ICgZmp)7nq - ICgZmqICgZnp) ’ (19)

where k is the four-dimensional momentum, oe gets the boundary photon mass term’

where we have have explicitly shown the R dependence. Interestingly, in the Kaluza-Klein reduction only the B,, zero mode contributes to the boundary photon mass, while the nonzero Kaluza-Klein modes give no mass contribution. In fact this is the reason for why there is no singular b(0) term in (20) which would correspond to an infinite divergent sum over the nonzero Kaluza-Klein modes. A similar mass contribution was also calculated in four dimensions in [lo].

To calculate the corresponding boundary mass term in the vector formulation we need to include the mixing between A1 and A1 given in (15). In the Landau gauge the field A5 decouples and only A , gives a mass contribution. Using the gauge field propagator

-i k2 + k:

this gives rise to the mass contribution

Contrary to the tensor field B2, where only the zero mode gave a mass contribution, it is only the nonzero Kaluza-Klein modes of A , which give a mass contribution for the (odd) vector field A1. The corresponding infinite divergent sums can be regularised in the following way6

1 d(0) = c 1, x ( - 1 y = o ,

,=--00 n=--00

(23)

and leads to the 6(0) term in ( 2 2 ) . However we now see that the first term in ( 2 2 ) precisely cancels the 6(0) term in (15). The remaining boundary

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mass term is finite and agrees with the tensor formulation result (20). Thus, as expected, both dual formulations give the same result for the boundary photon mass term. Notice also that this boundary mass term contains brane-to-brane contributions which are non-local in five dimensions.

The boundary mass term (20) can now be used to calculate the anomalous photon mass spectrum. Together with the Kaluza-Klein mass contributions arising from the bulk photon kinetic energy an infinite dimensional mass matrix (see also [17]) can be written whose mass eigenvaIues are determined by the equation

Notice that only if t o +tiT = 0 then m = 0 is a solution of (24). In this case the zero mode photon is massless and the anomaly is cancelled by the bulk Chern-Simons term. However when t o + tiT # 0 the zero mode solution of this equation becomes approximately

for t o + CiT << 1. The anomalous photon mass is finite at tree level, and only depends on the orbifold radius R. We will see later that in the supersymmetrized version the photino masses will be similarly obtained.

3. Phenomenological aspects

In this section we investigate some simple consequences of supersymmetric five dimensional theories with an anomalous matter spectrum. As before we consider the theory on an interval or equivalently the orbifold S1/Zz. The matter may be either bulk hypermultiplets or chiral multiplets on the boundaries. All matter is only charged under the anomalous U ( ~ ) A , but does not couple directly to the Green-Schwarz multiplet, irrespectively of whether it is described by the vector or tensor multiplet formulation.

Suppose that on the y = 0 boundary there are two charged fields Q 0 , @+, of charges -1 and 2, respectively. Supersymmetry must also be spontaneously broken and in four dimensions this occurs by adding the gauge invariant superpotential term X@o@o@+. Etom a five dimensional perspective, supersymmetry can be broken by adding the gauge invariant superpotential term X@o@o@+(y = 0) on the y = 0 boundary. It is indeed straightforward to check that the five dimensional F and D flatness conditions FO = F+ = F- = D = D = 0 have no solution in this model. However, compared to four dimensions, the supersymmetry breaking scale

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will be suppressed by the volume of the compact dimension via the wave function of the bulk field a+, using the mechanism proposed in a different context in [18]. Thus, unlike in four dimensions, the FI term in this five dimensional model can induce supersymmetry breaking at the TeV scale.

If instead the charged matter fields live on different boundaries then fermion mass hierarchies can be generated in a different way. For example, if (aT) are charged matter boundary fields at y = 0 (y = T ) with U ( ~ ) A charges -1 (+2), then the gauge invariant superpotential term that breaks supersymmetry will involve the Wilson line operator

Wsusy-breaking = @o~o,zfiso" dy S b I Y ) ( P n . (26)

The Wilson line operator can induce a large hierarchy between the fundamental scale and the supersymmetry breaking scaleg. All quark and charged lepton fields should live on the y = 0 boundary in order to avoid a (too large) suppression in the fermion masses arising from the Wilson line operator. On the other hand, if neutrino singlets Ni live on the y = T boundary, then the extra suppression from the Wilson line operator allows a nice way of getting very small Dirac neutrino masses. The relevant neutrino mass operator can be written as

Together with (26) the superpotential term (27) predicts a relation between the neutrino mass scale and the supersymmetry breaking scale.

Acknowledgments

I would like to thank my collaborators T. Gherghetta and S. Groot- Nibbelink for an enjoyable collaboration and the organizers of PASCOSO4 and Nathfest for providing a stimulating atmosphere. Work supported in part by the RTN European Program HPRN-CT-2000-00148 and the CNRS PICS no. 2530.

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ANOMALY-INDUCED INFLATION

ILYA L. SHAPIRO Departamento de Fisica, ICE,

Universidade Federal de Juiz de Fom, MG, Brazil E-mail: [email protected]’. br

The anomaly-induced inflation (modified Starobinsky model) is based on the a p plication of effective quantum field theory (QFT) approach to the Early Universe. We present a brief review of this model in relation to recent results concerning quantum field theory in curved space-time.

The original Starobinsky model 1 7 2 * 3 takes into account the vacuum quantum effects of the free, massless, conformal matter fields on classical curved background in the matter-free universe. Quantum corrections to the vacuum Einstein equations

(1) 1

R,, - 5 Rg,, = 8rG < Tpu >

produce a non-trivial effect because the anomalous trace of the stress tensor

<TL>= - (wC2+bE+cV2R) (2)

is non-zero. Here w, b, c are nothing but the @-functions for the corresponding higher derivative terms in the classical action of vacuum (see 4 ~ 5

for the introduction). Taking trace of (1) we arrive (in the k = 0 case) at the following equation for the conformal factor a( t ) of the metric:

M2 a a 2 ( 4b)$ ( ”) = 0 . (3) .... ..2 a 3 a a a - + - + - - 5 + - - - 2 - + - - - a a2 a2 arc a a2 3

A remarkable particular solution a( t ) = exp (Ho . t ) with

indicates the possibility of inflation generated by vacuum quantum effects, without any special entity like, e.g., inflaton. It is easy to see that the

488

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inflationary solution is stable under perturbations of a = lna in case c > 0 and unstable for c < 0 2 ~ 6 .

is based on the unstable case. Heavy fine-tuning of initial data is required for giving sufficient time to inflation and also in order to avoid the universe falling into uncontrollable run-away type solutions. There is, however, another approach using both stable and unstable cases ’. The p-functions 20, b, c in (2) depend on the number of scalars No, fermions N1/2 and vectors N1 in the underlying QFT. b < 0 for any particle content while the sign of c varies and the condition of stable inflation c > 0 can be cast into the form

The original version of the Starobinsky model

The Eq. (5) admits a simple physical interpretation in terms of effective QFT. Let us notice that (5) holds for MSSM with N1,1/2,0 = (12,32,104) and also for any realistic SUSY model. Hence, if we assume that the high energy particle spectrum is supersymmetric, we arrive at the stable inflation which does not depend on the choice of initial data for a(t) . At the same time, the condition (5) does not hold for MSM with Nl , l / z , o = (12,24,4). Remember that SUSY (if it really takes place) should be broken at law energies and that the most probable reason is that all s-particles are very heavy. Hence we can link the transition from stable to unstable inflation to the Appelquist-Carazzone like decoupling of s-particles at lower energies. This simple observation enable one to formulate the modified Starobinsky model 7 9 9 3 1 0 .

One of important problems is to explain why the energy scale should decrease during inflation. In order to address this issue we developed a new semi-phenomenological method of evaluating vacuum effective action for the massive fields ’. This method is appropriate when the masses of quantum matter fields are smaller than the typical energy of the external graviton tail in the Feynman diagram. It turns out that taking the masses into account slows down inflation. The behaviour of u can be approximated by the simple expression

a(t) = Hot - H i - p , 4

where the parameter f depends on the masses of the quantum fields. The total amount of the e-folds depends on the point when the Hubble parameter H = c j becomes equal to M,, that is the scale of transition from stable to unstable inflation. For the TeV-scale SUSY breaking we have

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a huge lo3' e-folds of inflation, of course only the last 60-70 of them are physically relevant. The expected temperature of the universe after the end of inflation can be evaluated from Einstein equation in a usual way T N d m x 10l1 GeV, which is a standard estimate in the framework of inflaton-based models.

Beyond the inflationary region, the effective QFT - based models look as a perfect tool in explaining the stability of the low-energy universe in the presence of higher derivative quantum corrections. It is interesting that the stability under the perturbations of c is conditioned by the presence of the positive cosmological constant lo. In order to see this, we notice the existence of a second particular solution of Eq. (3): a(t) = exp ( H , . t ) , with

Direct calculation shows that this solution is stable in the present-day universe with A > 0, where all fields except the photon decoupled.

Another nice property of the anomaly-induced inflation is related to the non-amplification of the tensor perturbations 6,10J1. The equation for these perturbations has been derived in 12$ (see also 13). The small difference between the two calculations is exclusively due to the different choice of the conformal functional which emerges when we integrate the conformal anomaly. On the top of that there is another ambiguity related to the arbitrary choice of the coefficient of the conformal higher derivative term in the classical action of vacuum 6. However, all these divergences disappear at the last stage of the anomaly-induced inflation lo, where the equation is well defined, with all coefficients depending on the universal p-functions w and b. The analysis of this equation shows that tensor perturbations can be perfectly kept under control in case M, 5 10-5Mp that is indeed the case for the low energy SUSY. Moreover, after crossing horizon gravitational waves with the wavenumbers below the Planck scale do not amplify. This property emerges automatically, without fine-tuning of any sort and represents an advantage of the anomaly-induced inflation with compared to the conventional inflaton models.

The QFT foundation of the modified Starobinsky model includes, along with the mentioned method of evaluating the high-energy effective a c t i ~ n ~ ~ ' ~ , the detailed investigation of the decoupling of quantized matter field in an external gravitational background14, including the complicated case of a theories with Spontaneous Symmetry Breaking15. The compli-

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cated calculation of the decoupling l4 has been performed in the linearized gravity framework and also by using the prominent solution for the heat- kernel in the second order in curvature approximation16. Both methods are equivalent and enable one to use the mass-dependent physical scheme of renormalization in curved space. Another interesting application has been presented in 17, where the long-standing contradiction between the expressions for the conformal anomaly derived in dimensional regularization l8 and other regularization schemes (e.g. point-splitting 19) has been successfully resolved. The most important aspect is that the existing ambiguity does not threat the anomaly-induced inflation scheme.

Unfortunately, the power of the existing calculational methods compatible with the mass-dependent renormalization scheme is restricted to the higher derivative terms in the effective action and are not efficient for the renormalization of the Einstein-Hilbert and cosmological terms. At the same time the conventional MS (minimal subtraction) scheme enables one to address all sectors of the effective action, but it is not suitable for observing decoupling. Indeed, the limits of existing QFT methods constitute the main difficulty of the anomaly-induced inflation. The point is that the approach of is valid in the high-energy limit, when the masses of the particles may be treated as small perturbations to the exact solution for the anomaly-induced effective action 20. On the other hand, the most developed calculational methods such as Schwinger-DeWitt technique or the heat-kernel solution l6 deal with the low-energy approximation when the expansion is performed in the series in curvature tensor. There are no methods, so far, to deal with the intermediate regime, when the masses of the fields have the same order of magnitude as Hubble parameter or energy of metric perturbations.

The lack of the appropriate QFT method is the main difficulty of the modified Starobinsky model. At the present stage it represents an interface between two asymptotic regimes: the high energy one with unbroken SUSY and stable inflation and the low-energy limit with broken SUSY and unstable inflation (or, looking further into IR, with the late universe stable to higher derivative quantum corrections). There is no consistent theoretical method to deal with the most interesting intermediate regime. Therefore, our knowledge is not sufficient to describe, e.g. reheating or making definite predictions about the density perturbations in the last few e-folds of the stable inflation or in the transition period (indeed, 22 can be considered as an example of such investigation, despite it concerns a different version of the anomaly-induced inflation).

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After all, despite the anomaly-induced inflation is not as developed as inflaton models (see, e.g. 21), it represents an attractive alternative to them. In particular, it enables one to avoid the standard fine-tuning in the choice of initial data, gives a good chance to have a natural graceful exit and also control the amplitude of the gravitational perturbations. Only further theoretical and phenomenological study of this and other models and their comparison with existing and future experimental/observational data may eventually show which of the models is closer to the realm of our Universe in the first instants of its history.

References 1. M.V. Fischetti, J.B. Hartle and B.L. Hu, Phys. Rev. D20 1757 (1979). 2. A.A. Starobinsky, Phys. Lett. 91B (1980) 99. 3. A. Vilenkin, Phys. Rev. D32 2511 (1985). 4. N.D. Birell and P.C.W. Davies, Quantum Fields in Curved Space (Cambridge

Univ. Press, Cambridge, 1982). 5. I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro, Eflective Action in Quantum

Gravity (IOP Publishing, Bristol, 1992). 6. J.C. Fabris, A.M. Pelinson, I.L. Shapiro, Grav. Cosmol. 6 (2000) 59 [gr-

qc/9810032]; Nucl. Phys. B597 539 (2001). 7. I.L. Shapiro, Int. Journ. Mod. Phys. 11D 1159 (2002). 8. T. Appelquist and J. Carazzone, Phys. Rev. 11 2856 (1975). 9. I.L. Shapiro, J. Sol&, Phys. Lett. 530B 10 (2002). 10. A.M. Pelinson, I.L. Shapiro and F.I. Takakura, Nucl. Phys. 648B 417 (2003). 11. J.C. Fabris, A.M. Pelinson, I.L. Shapiro and F.I. Takakura, Nucl. Phys. B

12. A.A. Starobinsky, JETP Lett. 30 (1979) 719; 34 (1981) 460. 13. S.W. Hawking, T. Hertog and H.S. Real, Phys. Rev. D63 083504 (2001). 14. E.V. Gorbar and I.L. Shapiro, JHEP 02 021 (2003); 06 004 (2003). 15. E.V. Gorbar and I.L. Shapiro, JHEP 02 060 (2004). 16. I. G. Avramidi, Sow. Journ. Nucl. Phys. 49 1185 (1989);

17. M. Asorey, E.V. Gorbar and I.L. Shapiro, Class. Quant. Grav. 21 163 (2003). 18. M.J. Duff, Nucl. Phys. 125B (1977) 334;

19. S.M. Christensen, Phys. Rev. 17D (1978) 948. 20. R.J. Reigert, Phys.Lett. 134B 56 (1980);

E.S. Fradkin and A.A. Tseytlin, ibid, 187. 21. A. H. Guth, Phys.Rept. 333 555 (2000). 22. V.F. Mukhanov and G.V. Chibisov, JETP Lett. 33 (1981) 532; JETP (1982)

(P.S.) 127 159 (2004).

A.O. Barvinsky and G.A. Vilkovisky, Nucl. Phys. 333B 471 (1990).

D.M. Capper and M.J. Duff, Nuovo Cim. 23A (1974) 173.

258.

STABILIZING MODULI WITH STRING COSMOLOGY

S. WATSON Department of Physics,

Brown University, Providence, RI 02912, USA

E-mail: watsonohet. brom.edu

In this talk I will discuss the role of finite temperature quantum corrections in string cosmology and show that they can lead to a stabilization mechanism for the volume moduli. I will show that from the higher dimensional perspective this results from the effect of states of enhanced symmetry on the one-loop free energy. These states lead not only to stabilization, but also suggest an alternative model for ACDM. At late times, when the low energy effective field theory gives the appropriate description of the dynamics, the moduli will begin to slow-roll and stabilization will generically fail. However, stabilization can be recovered by considering cosmological particle production near the points of enhanced symmetry leading to the process known as moduli trapping.

1. String Gases in 1OD

1.1. Initial Conditions

One problem in string cosmology is the issue of initial conditions. Not only must models of string cosmology address the standard initial condition problems in cosmology, but string theory also predicts the existence of extra dimensions. The usual prescription for dealing with the extra dimensions is to take them small, stable, and unobservable. However a complete model of string cosmology should explain how this came about and why the explicit breaking of Lorentz invariance should be allowed. A step in this direction was first proposed by Brandenberger and Vafa'. They argued that, by considering the dynamics of a string gas in nine, compact spatial dimensions initially taken at the string scale, one could explain why three dimensions grow large while six stay compact. The crux of their argument is based on the fact that in addition to the usual Kaluza Klein modes of a particle on a compact space, strings also possess winding modes. These extra degrees of freedom will generically halt cosmological expansion; however, if these modes could annihilate with their anti-partners this would

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allow the dimension they occupy to expand. Then, the fact that strings generically intersect (interact) in at most three spatial dimensions means that in the remaining six dimensions thermal equilibrium cannot be main- tained. Thus, the winding modes will drop out of equilibrium and the six spatial dimensions will be frozen near the string scale. Furthermore, once all the winding modes in the three large dimensions annihilate, the universe emerges filled with a gas of momentum modes, which evolves as a radiation dominated universe.

1.2. String Gases at Finite Temperature

The usual starting point of string cosmology is the action,

1 S = - d d + 1 x f i e - 2 v R + ~ ( V ( P ) ~ - - H 2 + O(cr') 2K2 'J 12

For simplicity we will ignore the Ramond sector and set H = 0. Here we have in mind the heterotic string on a toroidal background. Motivated by the Brandenberger-Vafa scenario we will take the background to be R4 x T6, where we assume that the three spatial dimensions have grown large enough to be approximated by an FRW universe and the six small dimensions are toroidal and near the string scale. To include time dependence we make use of the adiabatic approximation, which implies that we can replace static quantities by slow varying functions of time.

The action (1) represents a double expansion in both the string coupling gs - e(9) and the string tension, T = A. We now want to include terms coming from gs corrections at finite t e m p e r a t ~ r e ~ ? ~ . Let us consider the 1-loop free energy

where 21 is the one loop partition function, M ( n , w , N , N) is the string mass, and p = 1/T is the inverse temperature. In the early universe we are interested in temperatures near or below the string scale (@ 2 a) where the major contribution to the one-loop free energy can be seen to come from the massless modes of the string.

In the case of the heterotic string there are additional massless states that occur at the special radius R = a. These extra states include winding and momentum modes of the string. To understand the dynamics that result by including these states we can find the energy density and

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1.2 '1

Figure 1. The evolution of the volume modulus of the extra dimensions for different initial values near the self-dual radius. We see that the moduli are eventually stabilized at the self-dual radius where the pressure is found to vanish.

pressure, which follow from the free energy as

where V is the spatial volume and Ri is the scale factor in the ith direction. As discussed above, we take initial conditions where three dimensions have grown large and six remain near the string scale. For such initial data, the above pressure in three dimensions corresponds to the equation of state of a radiation dominated universe p3 = i p , whereas the pressure in the small dimensions gives the behavior,

pfj < o for > and pfj > o for RG < a. (4)

In dilaton gravity negative pressure implies a contracting universe, whereas positive pressure leads to expansion. Thus, as can be seen in Figure 1, pressure leads to a stabilizing effect for the scale factor of the extra dimensions driving the radius toward the enhanced symmetry point R(6) = where the pressure vanishes. At this location the gauge symmetry of the heterotic string is enhanced, ES x E8 + U(1)6 - ES x E8 + SU(2)6.

In addition to stabilizing the volume moduli, it has been shown that the remnant string modes, if taken in the dark sector, can also lead to an

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interesting cold dark matter andi id ate^?^ (see also6).

2. Moduli Trapping and Stabilization in 4 0

If one attempts to extend the arguments above to the 4 0 effective field theory, one finds that the stabilization mechanism no longer holds. This is not surprising since the pressure in the extra dimensions has no analog from the 4 0 perspective. However, one thing that should remain is the idea of enhanced symmetry.

Recall that it was the contribution of the enhanced symmetry states near R = &? that led to the pressure terms in (4) stabilizing the extra dimensions. We can account for these enhanced states from the effective field theory (EFT) perspective by considering the effects of particle production near the enhanced symmetry point, R = a.

To understand how this mechanism works let us consider the simplest case of heterotic strings on the background R4 x S1. The low energy effective action comes from the compactification of the action (1). The dynamics are then given by dilaton gravity coupled to a chiral U(1) gauge theory,

1 1 - L, = - -(F,u)2 - -(F,u)2,

4g2 4g2 ( 5 )

where F = d A ( E = d A ) is the left (right) gauge theory resulting from the compactification of the higher dimensional metric and flux and g is the gauge coupling. The scalar u gives the radius of the compactification and can be scaled to measure the departure from the self-dual radius, i.e. u = 0 at R = &?.

We see that u has only a kinetic term and the lack of a potential implies the radius is free to take any value. However, as the modulus passes near the self-dual radius we have noted that there are additional massless degrees of freedom. If our theory is to be complete these extra degrees of freedom must be included in the low energy effective action. This is accomplished by lifting the effective lagrangian in (5) to a non-abelian gauge theory, in this case chiral SU(2) . We introduce the covariant derivative,

This leads to a time dependent mass for the new vector A, 1 -g2u( t)2AE. 2 (7)

This time dependent mass implies particle production in our cosmological space-time. The stabilization of the radius u can now be realized as follows;

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initially u is dominated strictly by the kinetic term, however once it passes near the enhanced symmetry point u = 0, A, particles will be produced. Then, as u continues its trajectory the mass of the A,’s will increase and this leads to backreaction on u. This force, along with friction from the COS-

mological expansion, will eventually stabilize o at the enhanced symmetry point u = 0.

This is a simple example of moduli trapping7>’?’. Although we have considered here a simple toy model of a string on a circle, points of enhanced symmetry are present in nearly all string and M-theory compactifications. Moreover, it is worth mentioning that this mechanism need not apply only to volume moduli. In fact, in Kofman, et. al? the modulus of interest was the distance between two branes which is of course related by T-duality to the case we have considered here.

3. Conclusions

We have seen that from the 1OD perspective it is possible to stabilize the volume modulus of a heterotic string compactification on a T6. The stabilization was found to be the result of the pressure exerted on the compact space due to the presence of enhanced symmetry states contributing to the one-loop free energy of the strings at finite temperature. In the 4 0 effective field theory such effects can be understood by considering particle production near the enhanced symmetry point. Near this point additional massless states are allowed which can be particle produced in the cosmological space-time. These states then get masses via the string Higgs Effect and backreact on the modulus stabilizing the extra dimensions at the self-dual radius.

References 1. R. H. Brandenberger and C. Vafa, Nucl. Phys. B 316, 391 (1989). 2. J. Kripfganz and H. Perlt, Class. Quant. Grav. 5, 453 (1988). 3. A. A. Tseytlin and C. Vafa, Nucl. Phys. B 372, 443 (1992) [arXiv:hep

th/9109048]. 4. S. S. Gubser and P. J. E. Peebles, arXiv:hep-th/0407097. 5. S. S. Gubser and P. J. E. Peebles, arXiv:hepth/0402225. 6. T. Battefeld and S. Watson, JCAP 0406, 001 (2004) [arXiv:hep-th/0403075]. 7. L. Kofman, A. Linde, X. Liu, A. Maloney, L. McAllister and E. Silverstein,

JHEP 0405, 030 (2004) [arXiv:hepth/0403001]. 8. S. Watson, arXiv:hep-th/0404177. 9. L. Jarv, T. Mohaupt and F. Saueressig, arXiv:hepth/0311016.

PROBING ORIENTIFOLD BEHAVIOR NEAR NS BRANES

DMITRI BURSHTYN, SHMUEL ELITZUR AND YAAKOV MANDELBAUM Racah Institute of Physics

Hebrew University Givat Ram, Jerusalem, Ismel E-mail: dima@phys. hujji. ac.il

The effect of NS 5 branes on an orientifold is studied. The orientifold is allowed to pass through a pile of k NS branes forming a regularized CHS geometry. Its effect on open strings in its vicinity is used to study the change in the orientifold charge induced by the NS branes.

1. Introduction

We will use the regularized background of k parallel N S branes to study the effect of such branes on the RR charge of an orientifold. An 0 6 will be put into this background 2 , 3. The world volume of this orientifold intersects the S U ( 2 ) manifold at two points. A probe of N 0 4 branes is further connected to the pile of k N S branes. The strings connecting these D branes to their 0 6 images will be studied. As this probe of 0 4 branes is rotated with respect to the orientifold, passing the equator of the S3 sphere, a phase transition is encountered in the gauge theory on these D branes. For odd k it will be a transition from an S O ( N ) to an S p ( $ ) gauge group. This talk is based on l .

2. Open Strings near NS Branes

Consider a stack of k > 1 NS-5 branes spanning the hyper-plane (012345) in 10 dimensional type IIA model, at x6 = x7 = x8 = x9 = 0. The near horizon geometry formed by these branes is

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where I, J, K , M run from 6 to 10, p, v run fiom 0 to 5 and 1zI2 = ( x ~ ) ~ + (z7)2 + ( z ~ ) ~ + (z9)’. This background is described by an exact CFT with the target space R5?l x R4 x sU(2)k. The su(2)k factor consists of the bosonic angular coordinates parameterizing S3 in the 4 dimensional space transverse to the branes, and the three corresponding fermions X I , x 2 , x3.

A group element g E SU(2) can be parameterized as g = [z71 + i(z8,1 + z902 + x603)]. As it stands the background is singular. Regularization of this background is (see and references therein)

R195 X SL(2)k/U(1) X su(2)k/u(1) (2)

Now, let a stack of ND4 branes end on the NS 5 branes. These are stretched along the (0123) hyper-plane with their fifth coordinate in the (6,7) plane forming an angle f with the z6 axis. Let another identical stack end on the NS branes forming an angle 7r - f with the z6 axis in the (6,7) plane. A NS vertex operator for emitting a light open string connecting the ith 0 4 brane of the first stack to the j th brane of the second is

where cp is the bosonized susy ghost,the factor expi(Cik,zp) describes the 4-dimensional motion of the emitted open string,the factor [%!,:I belongs to the SL(2)/U(l) , and [Vms,(i”’V&] belongs to the SU(2)/U(1).

For a = 0, calculating the mass of the string emitted by (3) one can find that only the two values (m1,mz) = (y,l) and (m1,mz) = ( - y , - l ) correspond to massless particles. Turning on Q moves their mass off zero. For the state with ml + m2 = we have k

a M: = ; k and for the state with m1+ m2 = -h we have

a 7r

= --

(4)

( 5 )

3. Orientifold Action on Symmetry Generators

The su(2)k/u( l ) part of this background can be analyzed using Su(2)k WZW model with systematically controllable constraints on coset 4.

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The conserved currents are defined by J = dgg-'; = -g-'ag. These currents with their equations of motion S J = aJ = 0 generate an SU(2), x SU(2), symmetry of the WZW model.

We choose the orientifold position such that it identifies configuration related by g ( z , Z) --+ g- l ( -Z , - z ) , which implies transformation J ( z , Z) --+

-J(-Z, - z ) ; J ( z , 2) --f - J(-Z, -2) of the currents. In the geometry described above the 0 4 brane intersect S3 in one point,

thus it is described by a pointlike D brane on the group manifold. Choose its location at g = h such that its mirror image under the orientifold transformation is at the antipodal point, namely, h-' = -h. And consider open string connecting the D brane at h to its mirror image at -h. Varying the action with the boundary condition induced by D brane we get for the conserved modes

J , --t (-l)"hJ,h-l (7) Denote J, = tE2=lJ:cra. We have for the orientifold action J," -+ (-l),J; and J;f2 --t -(-l)nJ$2. In particular for the global SU(2) generators

(8 ) f

5: --+ J i J$ --+ -J,

The two points g = f h correspond to the two trivial conjugacy classes of SU(2)preserved by the branes boundary conditions, the points f l , shifted by the element h . According to Alekseev and Schomerus 5, there are altogether k + 1 (shifted) conjugacy classes which are allowed to inhabit branes which preserve this SU(2) symmetry. Each of these classes corresponds to one of the k + 1 primary fields of SU(2)k. In particular the class at h corresponds to the primary field of spin 0, while that at -h corresponds to the primary field of spin $.This correspondence, due to Cardy 6 , implies that the open strings stretched between two branes, belong to the representations of the chiral algebra which appear in the fusion of the primary fields corresponding to these branes. In our case then, the strings stretched between the brane at h and that at -h belong to the representation of the SU(2) group generated by J,-s with spin 4. The lightest of those strings form a degenerate multiplet of k + 1 members transforming

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in the spin $ representation under the global S U ( 2 ) generated by Jo. Let Im > be the state of such a light string which satisfies Jilm >= mlm >. Here, $ >_ m 2 -$ and t - m is an integer. Let V, be the vertex operator to emit an open string in the state Im >.

If the sign of the orientifold is such that it preserves the state I$ >, then, since J:changes sign under the orientifold action, all the open strings of types 1: -2n > will be preserved as well, while the string states I$ -2n+l > will be projected out by the orientifold. In particular, in such a case, the open string generated by V-g is preserved for k even but projected out for k odd.

When there are N D branes, rather than one, each open string state connecting the ith brane to the mirror j th brane, carries a pair of Chan- Paton indices ( i , j ) . The orientifold identification for l ( i , j ) ; t > should read now

For the plus sign in (9), the arguments above imply that l(z,j); $ - 2n > survives the orientifold projection only in symmetric combinations of the Chan-Paton indices , while for \ ( i , j ) ; $ - 2 n + 1 > only antisymmetric Chan-Paton combinations survive. The opposite assignments occur for the minus sign in (9) .

Consider moving the D brane away from the equator by angle a12 with mirror brane at appropriate position, i.e. at g = h-l = -heicro3. Varying the action with these boundary conditions we get spectrally flowed version of SU(2) , ’. Redefining -

the modified generators j:, j: and j; generate a standard affine SU(2)k algebra. For the case a = 0 we had the k + 1 vertex operators V,, all of dimension f , creating open strings connecting the brane at h to its mirror image at h-l. Turning on a! continuously, these operators remain in spin 4 representation of the modified S U ( 2 ) generated by the j,. The modes 3, transform under orientifold identification for general a! in a same manner as before, namely j: + (-1)”j:; j$ + -(-1)”j$. In particular for global modes ji + $; jof + -jof. The same behavior of the vertex operators as for a = 0 holds for general value of a, i.e. for N branes the operators of the form VLi’ji, -- should be symmetric in the Chen-Paton indices while

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those of the form V:’in+l are antisymmetric.

4. Phase Transition in Presence of an Orientifold

Now let us come back to our set up using these results. Let an 0 6 orientifold be stretched along the (0123457) plane, at the location z6 = z8 = z9 = 0, in the system described in previous section. This amounts to gauging the world sheet symmetry

zP(2, z ) + z’l(-E, -2) IzI(2, 5) --f 121(-z, - z ) g ( 2 , E ) + g-l(-z, -2) (11)

The action of the orientifold on the current modes is

with same type of action on fermionic modes and, thus for the modes of the total current J = J B + JF.

Let us now using the results of the previous discussion follow the low energy behavior of the theory on the stack of N D 4 branes as a function of the parameter a. To have a 4 dimensional physics assume that these branes connect our pile of NS5 branes to another NS brane , call it the NS’ brane, sitting in the (6,7) plane at a finite distance from the origin . The mirror images of these D branes should of course end at the mirror image of this NS’ brane under the orientifold projection. We change the value of a by moving the NS’ brane around in the (6,7) plane. Start with the supersymmetric case Q: = 0. In the absence of the orientifold the two stacks of 0 4 branes are not identified with each other. We expect at low energy a U(N) x U ( N ) gauge group. We found two types of massless open strings connecting one stack to the other, corresponding to ml + m2 = fh. The vertex operator with ml -I- m2 = $ gives rise to a complex massless chiral superfield Q, in the bi-fundamental representation (N, N ) . The operator with ml + m2 = -$ gives rise to an anti-chiral superfield denoted by Q*, in the same representation. Q in the (N, N ) and Q in the ( N , N) representation are two chiral fields with J: charge f . The orientifold identifies the two sets of branes, turning the gauge group into a single U ( N ) . For even k, we found that both Q and Q* have the same orientifold behavior. If the sign of the orientifold is chosen to take the vertex operators corresponding to Q and Q* to themselves, their Chan-Paton symmetric part survives the projection and the resulting U ( N ) gauge theory contains two

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chiral matter superfields, one in the symmetric representation and the other in its conjugate. For the other choice of the sign of the orientifold only the antisymmetric part survives and we get a theory with two fields, one in the antisymmetric representation of U ( N ) and the other in its conjugate. For an odd k we find that the two vertex operators corresponding to Q and Q* have opposite behavior under the orientifold transformation. In that case then the low energy gauge theory will contain two massless matter fields, Q in the symmetric representation and Q in the conjugate antisymmetric one (or vice versa).

Now rotate the 0 4 branes to get a > 0. Notice that the preceding analysis of the orientifold action on Q and Q* applies for any value of a. By eq.(4) the mass squared ME becomes negative. The scalar component of the field Q becomes tachyonic. Suppose the sign of the orientifold is such that this field is in the symmetric representation of U ( N ) . From field theory point of view this means that Q gets a non zero vacuum expectation value. An expectation value for a symmetric field breaks the gauge group U ( N ) down to SO(N) . From a geometrical point of view the appearance of a tachyon is a sign of an instability of the system caused by the rotation of the branes into a non zero a position. Presumably the system decays into a more stable position in which the 0 4 branes detach from the pile of the N S branes connecting the NS' brane directly to its image, meeting the orientifold at right angle at a point with x7 > 0.

The latter configuration is definitely more stable than the original one, since it preserves space-time supersymmetry. Combining together field theory and geometry we conclude that when the 0 4 branes meet the orientifold at positive x7, the resulting low energy gauge group on them is SO(N). Now let a be negative. According to ( 5 ) the scalar component of Q becomes tachyonic and gets non zero expectation value. For even k, we have a similar phenomenon. Q is also a symmetric field and the resulting gauge symmetry is again SO(N). Geometrically this SO(N) occurs when the D branes detach from the N S pile meeting the orientifold at x7 < 0.

For generic values of a the theory on the 0 4 branes is S O ( N ) , at the critical point a = 0 there is an enhancement of the gauge symmetry to

If k is odd the picture is different. Unlike Q, the field Q is antisymmetric. Its expectation value breaks the U ( N ) group down to Sp($) . Now the enhanced symmetry point a = 0 is a phase transition from an S O ( N ) phase for positive a to a different, S p ( g ) phase for negative a. Geometrically it means that 0 4 branes meeting the orientifold at x7 > 0 have S O ( N ) gauge

U ( N ) .

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group at low energy, while those that meet the orientifold at x7 < 0 have Sp( $) as their gauge group.

So,we can conclude that in the system of 2 stacks of 04 branes ending on a pile of k NS5 branes, in which instabilities occur for generic angle between them 4, these instabilities, in the presence of orientifold, could be shown to break the gauge symmetry on the D branes either to an orthogonal or to a symplectic group. The choice between these two possibilities characterizes the sign of orientifold. A change in the sign between a positive and negative angle was identified for odd k. In this way we could follow the change in the nature of the orientifold as a phase transition in the gauge theory on the D branes which end on the pile of N S branes.

References 1. D.Burshtyn, S.Elitzur, Y.Mandelbaum, JHEP 03 022 (2004) 2. C.V.Johnson, Phys.Rev. D56, 5160 (1997). 3. S.Forste, D.Ghoshal, S.Panda, Phys.Lett. B 411, 46 (1997). 4. S.Elitzur, A.Giveon, D.Kutasov, E.Rabinovici,G.Sarkissian, JHEP 0008 046

5. A.Y.Alekseev, VShomerus, Phys. Rev. D 60, 061901 (1999). 6. J.L.Cardy, NucLPhys B 324, 581 (1989). 7. C .Callan, J .Harvey, A. St rominger, hep-t h/9 1 12030

(2000).

Recent Theoretical Developments


A PURE COTTON KINK IN A FUNNY PLACE

R. JACKIW Center for Theoretical Physics,

Massachusetts Institute of Technology, Cambridge, MA 02139-4307


When the %dimensional gravitational Chern-Simons term is reduced to two dimensions a dilaton-like gravity theory emerges. Its solutions involve kinks, which therefore describe 3-dimensional, conformally flat spaces.

Among the mathematical physics topics that have interested me over the years two are relevent to the meeting at which I am now speaking: these are first, gravity theory and second, toplogical entities both in mathematical settings like characteristic classes and in physical realizations like kink profiles on a line. So for this event I shall present the results of an investigation that unites these diverse elements.

Let me begin with lineal kinks. Consider the field equation

up - cp + p3 = 0, (1)

where C is a positive constant. When we look for a lineal kink solution, we take cp to depend on a single spatial variable. Then (1) reduces to

-p” - ccp + 9 3 = 0, (2)

and has the well-known kink solution,

cpk(z) = d??tanh/$, (3)

which interpolates between the “vacuum” solutions cpo = *@. The kink has interesting roles in condensed matter physics where it triggers fermion fractionization. Other kinks in other models give rise to completely solvable field theories, both in classical and quanta1 frameworks. These stories do not belong here. But I shall return to the above kink later.

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Next let me consider the Chern-Simons characteristic class. In non- Abelian gauge theory it is constructed from a matrix gauge connection (Aa)Pv as

W ( A ) = - 1 / d 3 x ~ a o T ‘ t r ( i A a 8 p A 7 + 4n2

This gauge theoretic entity finds physical application in the quantum Hall regime, perhaps also in high T superconductivity. When added with strength m to the usual Yang-Mills action, the Chern-Simons term gives rise to massive, yet gauge invariant excitations in (2 + 1) - dimensional space-time. Also for consistency in the quantized version of a non-Abelian theory m must be an integer multiple of 2n. This is a precise field theoretic analog of Dirac’s celebrated quantization of magnetic monopole strength. Finally in an important mathematical application, the Chern-Simons term gives a functional integral formula for knot invariants.

The gauge theoretic Chern-Simons term (4) can be translated into a 3- dimensional geometric quantity by replacing the matrix gauge connection ( A a ) p v with the Christoffel connection rgv.

But it is to be remembered that is constructed in the usual way from the metric tensor, gpv, which is taken as the fundamental, independent variable. When W ( r ) is varied with respect to gpv there emerges the Cotton tensor, which has an important role in 3-dimensional geometry.

6W(F) = -- d3x6gpV h C P v (6) 4nJ ‘ S

[In (7) one may freely replace the Ricci tensor RE by the Einstein tensor Gf = Rf - i&Ra,.] The Cotton tensor is like the covariant curl of the Ricci or Einstein tensor. It is manifestly symmetric; it is covariantly conserved and traceless because it is the variation of the diffeomorphism and conformally invariant W ( r ) . Furthermore, the Cotton tensor replaces the Weyl tensor, which is absent in three dimensions, as a template for conformal flatness: C p v vanishes if and only if the space is conformally flat.

Cpv = 0 confornially flat space (8)

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Absence of the 3-dimensional Weyl tensor has the consequence that 3- dimensional geometries satisfying Einstein’s equation carry non-vanishing curvature only in regions where there are sources. Therefore, there are no propagating excitations. However, upon extending Einstein’s gravity equation by adding CP’ to the Einstein tensor [equivalently, adding $ W ( r ) to the Einstein-Hilbert action] the theory acquires a propagating niode with mass rn, all the while preserving diffeomorphism invariance! Here we have another perspective on the absence of propagating modes in 3-dimensional Einstein theory: to regain the Einstein equations from the modified equations, we must pass m to infinity, whereupon the super-massive propagating mode decouples.

This is a well known story, with which I do not concern myself now. Rather, I consider the opposite limit of the extended theory, where only the Cotton tensor survives, and the equation that I shall examine demands its vanishing, i .e . eq (8). But as indicated previously that equation is not sufficiently restrictive to be interesting: any conformally flat space-time [coordinates ( t , z, y) ] is a solution. So I shall place a further restriction: the solution that I seek should be independent of the y coordinate in a Kaluza-Klein dimensional reduction from ( 2 f l ) to (1+1) dimensions of the gravitational Chern-Simons term W(F) (5) and of the Cotton tensor CP”

To effect the dimensional reduction, We begin by making a Kaluza-Klein (7).

Ansatz for the 3-dimensional metric tensor. It is taken in the form

3-d metric tensor = cp (9)

where the 2-dimensional metric tensor gap, vector a,, and scalar cp depend only on t and z. [Henceforth, Greek letters from the beginning of the alphabet (a, p, y, ...) index 2-dimensional (t, z)-dependent geometric entities, which are written with lower case letters; in three space-time dimensions geometric entities are capitalized (save the metric tensor) and are indexed by middle Greek alphabet letters ( p , v, p . . . ) . ]

It is easy to show that under infinitesimal diffeomorphism, which leave the y-coordinate unchanged, gap, a, and cp transform as 2-dimensional coordinate tensor, vector and scalar respectively, and moreover a, undergoes a gauge transformation.

With the above Ansatz for the 3-dimensional metric, the Chern-Simons action becomes

d ‘ x f i ( . f r + f 3 ) . 8n2

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Here g = detg,p, &a, - dpa, = f a p E f i E a p f , and r is the 2- dimensional scalar curvature. The absence of a cp - dependence is a consequence of the conformal invariance of the gravitational Chern-Simons term, and this also ensures that the Cotton tensor is traceless. Henceforth we set cp to 1.

The above expressions look like they are describing 2-dimensional dil* ton gravity, with f taking the role of a dilaton field? However, in fact f is not a fundamental field, rather it is the curl of the vector potential a,. Alternative expressions for the action (10) are Jda( r + f 2 ) , (where da is a 2-form; this exposes the topological character of our theory), and J d2z8 f ,p, 0 = r + f ’, (which highlights an axion-like interaction in 2-dimensional space-time).

Variation of a, and goo produces the equations

The first is solved by

(13) r + 3 f 2 = constant C.

Eliminating r in the second equation, and decomposing it into the trace and trace-free parts leaves

O = D 2 f -Cf + f3, (14)

(15) 1 2 0 = D,Dpf - -gapD2f.

Note that the equations are invariant against changing the sign of f (the action then also changes sign).

A homogenous solution that respects the f - f symmetry is

However, there also is a “symmetry breaking” solution.

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Forms for and a, that lead to the above results are

( a ) f = 0, r = C > O : g a p = - - a, = (O,O),

( b ) f = 0, r = C < 0 : g a p = - a, = (O,O),

In the first case, the 2-dimensional space-time is desitter; in the last

The 3-dimensional scalar curvature R, with metric tensor as in OW two, it is anti-deSitter.

Ansatz (9) at cp = 1, is related to the 2-dimensional curvature r by

(21) 1 2

R = r + - f 2 .

Hence for the three cases, the 3-dimensional curvature and line element read

2 (b) R = C < 0, ( d ~ ) ~ = - [ ( $)2 - ( $)2] - ( d ~ ) ~ , (23) PI

2 3 2 (c) R = --C < 0, ( d ~ ) ~ = *-dtdy - (*) - ( d ~ ) ~ . (24)

2 f i x f i x Although all three solutions carry constant 3-dimensional curvature,

the “symmetry breaking” solution, (c) above, possesses greater geometrical symmetry in 3-dimensions: it supports 6 Killing vectors that span SO(2.1) x SO(2.1) = S0(2.2), the isometry of 3-dimensional anti-deSitter space. Moreover one verifies that, as expected, Rf = BSI R = -1 2 v P C - the 3-dimensional space-time is maximally symmetric. The “symmetry preserving” solutions, (a) and (b) above, admit only 4 Killing vectors that span SO(2.1) x SO(2).

Since the Cotton tensor vanishes, we expect that the above spacetimes, are (locally) conformally flat. This can be seen explicitly for the “symmetry preserving” solutions. In (22) set T = t cosh f iy , Y =

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t sinh f i y , and X = x to find

(dS)2 = C(T2 - Y2)

while in (23) the coordinate transformation X = x c o s d y y , Y =

xsin f l y , T = t gives the line element

The relevant transformation, which takes the “symmetry breaking” solution into a conformally flat coordinate has been found by two graduate students.

It is given by

6

Then the line element (24) becomes

and puts into evidence the anti-deSitter geometry.

between the “symmetry breaking” solutions (17). One can verify that The equations (14), (15) also posses a kink solution, which interpolates

fl f(z) = fi tanh - x, 2

with

) , Sap = ( l/cos;Qx 0

-1

satisfies the relevant equations. That the solution depends only on one variable (only x, not both t , x) is a general property (provided coordinates are selected properly). Thus in (14), (15) one is dealing with a system of second-order ordinary (not partial) differential equations, whose solution involves two integration constants. One integration constant is the trivial origin of the 2 coordinate [taken to be x = 0 in(29), (30)]. The other involves choosing an integration constant in a first integral, so that one

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achieves a kink: a profile that interpolates between f f i as x + f m . (Other choices for this second constant lead to the same local geometry, but to different global properties. This has been thoroughly explained by Grumiller and Kummer. 7,

The 2-dimensional curvature corresponding to (28) is 3 c

cash (Tx) r=-2C+ 2~ .

Also the 3-dimensional line element, for (27), (28) reads 2

cosh2 $ x (ds)2 = -(dx)2 - dtdY - (dYI2,

and the 3-dimensional scalar curvature is according to (21), (27) and (28) 3c 5 c R = - - +

2 c o s h 2 q x’ (33)

The 3-dimensional scalar curvature (33) clearly tends at large x to its “symmetry breaking” value (24), but the relation of the line element (32) to the one on (24) is not evident. To expose it, we change coordinates in (32) by

1 2 d E - t = t + y / 2 , 3 = -sinh -x, y = y JE 2 (34)

Then the ”kink” line element (32) becomes conformal to the “symmetry breaking” line element (24).

It is now clear that at large 3 the “kink” and the‘kymmetry breaking” line elements coincide, while at finite 3, the kink provides a deformation of anti-deSitter space. Expression (35) for the “kink” line element, together with the previously given transformation (27) [replace (t, x, y) by (t, 2, jj)] determines a conformally flat “kink” line element.

Once again we see that (36) tends to (28) at large X . This then is the kink in a “funny place” - in conformally flat (2+1)-

dimensional space-time. A question remains: can one understand a prioo71 that such a kink should exist in that geometry? This question may be posed in a more general setting.

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Observe that the flat space kink in Eqs. (1)-(3), possesses the same profile as (29), except for a change in scale. In fact this is a general feature. The following can be proven. If the non-linear equation in flat space-time,

0 cp + v’ (cp) = 0, (37)

possesses a kink solution v k (x) = Ic (x), then the curved (l+l)-dimensional space-time equations

D2 f + V ’ ( f ) = 0, (38)

are solved by

f = Ic(s/Jz), (40)

(41)

T = -V”(F) . (42)

with 2-dimensional line element

(ds)2 = V(f ) (d t )2 - (dx)2,

leading to a 2-dimensional curvature

In another generalization one can consider the supersymmetric extension of the Chern-Simons section and Cotton tensor. [8] In a dimensional reduction, similar to the one discussed previously, one is led to a supersymmetric generalization of (10). Then the profiles that we have found, together with vanishing values for the fermionic fields, continue to be solutions of the supersymmetric equations. One may also determine the supersynime- try properties of our solutions. One finds that the “symmetry preserving” solutions (16), (18) and (19) are not preserved by any of the supersymmetric transformations. The “symmetry breaking” solution (17), (20), as well as the kink solution (29), (30) preserve half the of supersymmetries. Furthermore, it is interesting to notice that our model has also emerged in a recent study of BPS solutions to N = 2, D = 4 gauged supergravity. lo

References 1. See for example, R. Jackiw “Quantum Meaning of Classical Field Theory”

Rev. Mod. Phys. 49, 681 (1977). 2. R. Jackiw and J. Schrieffer, “Solitons with Fermion Number 1/2 in Con-

densed Matter and Relativistic Field Theories” Nucl. Phys. B190, 253 (1981).

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3. S. Deser, R. Jackiw, and S. Templeton, “Three Dimensional Massive Gauge Theories” Phys. Rev. Lett. 48, 975 (1982); “Topologically Massive Gauge Theories” Ann. Phys. 140, 372 (1982), (E) 185, 406 (1988).

4. G. Guralnik, A. Iorio, R. Jackiw and S.-Y. Pi, “Dimensionally Reduced Grav- itational Chern-Simora Term and its Kink” Ann. Phys. 308, 222 (2003).

5. D. Grumiller, W. Kurnmer and D. Vassilevich, “Dilaton Gravity in Two Dimensions” Phys. Ftept. 369, 377 (2002).

6. M. Brigante and R. Sharma (unpublished). 7. D. Grumiller and W. Kummer, “The Classical Solutions of the Dimensionally

Reduced Gravitational Chern-Simons Theory” Ann. Phys. 308, 211 (2003). 8. S. Deser and J. Kay, “Topologically Massive Supergravity” Phys. Lett. B

120, 97 (1983). 9. L. Bergarnin, D. Grumiller, A. Iorio and C. Nuiiez (in preparation).

10. S. Cacciatori, M. Caldarelli, D. Klemm and D. Mansi “More on BPS Solutions of N = 2, D = 4 Gauged Supergravity” hepth/0406238.

PLANAR EQUIVALENCE: FROM TYPE 0 STRINGS TO QCD

A. ARMONI Department of Physics, Theory Division CERN, CH-1211 Geneva 23, Switzerland

M. SHIFMAN * William I. Fine Theoretical Physics Institute,

University of Minnesota, Minneapolis, MN 55455, USA

G. VENEZIANO Department of Physics, Theory Division CERN, CH-1211 Geneva 23, Switzerland

This talk is about the planar equivalence between N = 1 gluodynamics (super-Yang-Mills theory) and a non-supersymmetric “orientifold field theory.” We outline an “orientifold” large N expansion, analyze its possible phenomenological consequences in one-flavor massless QCD, and make a first attempt at extending the correspondence to three mas5 less flavors. An analytic calculation of the quark condensate in oneflavor QCD starting from the gluino condensate in N=1 gluodynamics is thoroughly discussed.

1. Genesis of the idea

Kachru and Silverstein studied [l] various orbifolds of R6 within the framework of the AdS/CFT correspondence. Starting from N = 4, they obtained distinct four-dimensional (daughter) gauge field theories with matter, with varying degree of supersymmetry, = 2,1,0, all with vanishing p functions.

Shortly after, Bershadsky and Johansen abandoned the string theory set- up altogether. They proved [2] that non-supersymmetric large-N orbifold field theories admit a zero beta function in the framework of field theory per se.

The first attempt to apply the idea of orbifoldization to non-conformal field theories was carried out by Schmaltz who suggested [3] a version of

*conference speaker

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Seiberg duality between a pair of non-supersymmetric large-N orbifold field theories .

After a few years of a relative oblivion, the interest in the issue of planar equivalence was revived by Strassler [4]. In the inspiring paper en- titled “On methods for extracting exact non-perturbative results in non- supersymmetric gauge theories” he shifted the emphasis away from the search of the conformal daughters, towards engineering QCD-like daughters.

Unfortunately, it turned out [5,6] that Strassler conjecture could not be valid. The orbifold daughter theories “remember” that they have fewer vacua than the parent one, which results in a mismatch [5] in low-energy theorems. In string theory language, the killing factor is the presence of tachyons in the twisted sector. This is clearly seen in light of the calculation presented in Ref. [6].

2. Orientifold field theory and h/= 1 gluodynamics

Having concluded that, regretfully, the planar equivalence of the orbifold daughters does not extend to the non-perturbative level, we move on to another class of theories, called orientifolds, which lately gave rise to great expectations. We will argue that in the N 4 00 limit there is a sector in the orientifold theory exactly identical to N= 1 SYM theory, and, therefore, exact results on the IR behavior of this theory can be obtained. This sector is referred to as the common sector.

The parent theory is N = 1 SUSY gluodynamics with gauge group SU(N). The daughter theory has the same gauge group and the same gauge coupling. The gluino field A$ is replaced by two Weyl spinors q[ijl and @I, with two antisymmetrized indices. We can combine the Weyl spinors into one Dirac spinor, either iPpj1 or iPLijl. Note that the number of fermion degrees of freedom in is N 2 - N , as in the parent theory in the large-N limit. We call this daughter theory orientafold A .

There is another version of the orientifold daughter - orientzfold 5’. In- stead of the antisymmetrization of the two-index spinors, we can perform symmetrization, so that A$ --+ (qij} , t{Zj}). The number of degrees of freedom in is N2 + N . The field contents of the orientifold theories is shown in Table 1. We will mostly focus on the antisymmetric daughter since it is of more physical interest; see Sect. 4.

The hadronic (color-singlet) sectors of the parent and daughter theories are different. In the parent theory composite ferrnions with mass scaling as N o exist, and, moreover, they are degenerate with their bosonic SUSY counterparts. In the daughter theory any interpolating color-singlet current

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Table 1. The field content of the orientifold theories Here, 7 and < are two Weyl fermions, while A,, stands for the gauge bosom. In the left (right) parts of the table the fermions are in the twdndex symmetric (antisymmetric) representation of the gauge group SU(N).

11 SU(N) UV(1) uA(1) 11 SU(N) UV(1) UA(1)

-1 -1

0 0 0 0

with the fermion quantum numbers (if it exists at all) contains a number of constituents growing with N . Hence, at N = 00 the spectrum contains only bosons.

2.1. Perturbative equivalence Let us start from perturbative considerations. The Feynman rules of the planar theory are shown in Fig. 1.

The difference between the orientifold theory and N = 1 gluodynamics is that the arrows on the fermionic lines point in the same direction, since the fermion is in the antisymmetric representation, in contrast to the supersymmetric theory where the gaugino is in the adjoint representation and the arrows point in opposite directions. This difference between the two theories does not affect planar graphs, provided that each gaugino line is replaced by the sum of q.1 and $..I.

Figure 1. theory. (c) Orientifold daughter.

(a) The fermion propagator and the fermion-fermion-gluon vertex. (b) N = 1 SYM

There is a one-bone correspondence between the planar graphs of the two theories. Diagrammatically this works as follows (see, for example, Fig. 2). Consider any planar diagram of the daughter theory: by defini-

519

tion of planarity, it can be drawn on a sphere. The fermionic propagators form closed, non-intersecting loops that divide the sphere into regions. Each time we cross a fermionic line the orientation of color-index loops (each one producing a factor N) changes from clock to counter-clockwise, and vice- versa, as is graphically demonstrated in Fig. 2c. Thus, the fermionic loops allow one to attribute to each of the above regions a binary label (say hl), according to whether the color loops go clock or counter-clockwise in the given region. Imagine now that one cuts out all the regions with a -1 label and glues them again on the sphere after having flipped them upside down. We will get a planar diagram of the SYM theory in which all color loops go, by convention, clockwise. The number associated with both diagrams will be the same since the diagrams inside each region always contain an even number of powers of g, so that the relative minus signs of Fig. 1 do not matter.

\ , ....

Figure 2. Hooft notation for (b) N=1 SYM theory; (c) orientifold daughter.

(a) A typical planar contribution to the vacuum energy. (b and c) The same in the 't

Thus, all perturbative results that we are aware of in Af= 1 SYM theory apply in the orientifold model as well. For example, the ,d function of the orientifold field theory is

{ 1 + 0 (i) } . 1 3Na2 p=-- 2~ 1 - ( N Q ) / ( ~ T )

In the large-N limit it coincides with the N= 1 SYM theory result [7]. Note that the corrections are 1/N rather than 1/N2. For instance, the exact first coefficient of the ,d function is -3N - 413 versus -3N in the parent theory.

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2.2. Non-perturbative equivalence proof

Now we will argue that the perturbative argument can be elevated to the nonperturbative level in the case at hand. It is essential that the fermion fields enter bilinearly in the action, and that for any given gauge-field configuration in the parent theory there is exactly the same configuration in the daughter one. Our idea is to integrate out fermion fields for any j xed gluon-field configuration, which yields respective determinants, and then compare them.

Consider the partition function of N = 1 SYM theory,

z0 = J V A V A exp ( ~ s [ A , A, 4) ,

where J is any source coupled to color-singlet gluon operators. (Appropriate color-singlet fermion bilinears can be considered too.)

For any given gluon field, upon integrating out the gaugino field, we obtain

(2.3)

where Tidj is a generator in the adjoint representation. If one integrates out the fermion fields in the non-supersymmetric orien-

tifold theory, at fixed A, one arrives at a similar expression, but with the generators of the antisymmetric (or symmetric) representation instead of the adjoint, Tidj +. T&,, or TCmm.

Since the theory, being vector-like, is anomaly free, the determinant in Eq. (2.3) is a gauge-invariant object and, thus, can be expanded in Wilson- loops operators

Wc[A~dj] = P exp (2 1 AETZdj d.") . (2.4) C

For other representations, the Wilson-loop operator is defined in a similar manner. Thus, one can write

D = det (i P+ &"Tidj - m) = ac Wc[A~dj] . (2 .5) C

Using

= Ta @ 1 + 1 @ T a ,

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imply that

D = X a c T r P exp (il A: (Ta €3 1 + 1 @5?)dx” C

Moreover, since the commutator is such that

[pa €3 l), (1 €3 Fa)] = 0 ,

the determinant (2.7) can be rewritten as

D = c ac Tr P exp (i J, A;Ta dx”) Tr P exp (i 1 A:Ta dd‘) . (2.8) C

As a result, the partition function takes the form

2 0 = C a c W C [ A O l Wl:[Aol) . (2-9) C

One of the two most crucial points of the proof is the applicability of factorization in the large-N limit,

2 0 = c ac (We(&) ) ( Wl:(AO) ) = X ac ( We(&) )2 * (2.10) C C

In the second equality in Eq. (2.10) we used the second most crucial point, the reality of the Wilson loop,

( W e ) = ( W E ) . (2.11)

The partition function (2.10) is exactly the same as that obtained in the (extended) orientifold theory upon exploiting factorization,

Zorientifold = c aC ( WC(&) ) ( WC(&) ) (2.12) C

The third ingredient is independence of the expansion coefficients ac of the fermion representation.

At this point we can take the square root of the determinants of the two theories. Owing to the non-vanishing mass, the determinants do not vanish and no sign ambiguities arise. In the parent theory we recover the (softly broken) super-Yang-Mills determinant, while in the daughter theory, given the planar equivalence of the symmetric and antisymmetric representations, we recover either one of them. We finally take the (supposedly) smooth massless limit and, thus, prove our central result.

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3. Orientifold large-N expansion

3.1. General features (Nf > 1 jixed, N large)

We will now abandon for a while the topic of planar equivalence, and look at the SU(N) orientifold theories with Nf flavors from a more general perspective. We will focus on the antisymmetric orientifold theories, assuming that Nf does not scale with N at large N, say, Nf = 1, 2 or 3. We will return to the issue of planar equivalence later.

If N = 3 (i.e. if the gauge group is SU(3)) the two-index antisymmetric quark is identical to the standard quark in the fundamental representation. Therefore, it is quite obvious that extrapolation to large N, with the subsequent 1/N expansion, can have distinct starting points: (i) quarks in the fundamental representation; (ii) quarks in the two-index antisymmetric representation; (iii) a combination thereof. The first option gives rise to the standard ’t Hooft 1/N expansion [8,9], while the second and third lead to a new expansion, to which we will refer as the orientifold large-N expansion.a

It is clear that the ’t Hooft expansion underestimates the role of quarks. This was noted long ago, and a remedy was suggested [ll], a topological expansion. The topological expansion (TE) assumes that the number of flavors Nf scales as N in the large-N limit, so that the ratio Nf/N is kept fixed.

The graphs that survive in the leading order of TE are all planar diagrams, including those with the quark loops. This is easily seen by slightly modifying [ll] the ’t Hooft double-line notation - adding a flavor line to the single color line for quarks. In the leading (planar) diagrams the quark loops are “empty” inside, since gluons do not attach to the flavor line. Needless to say, obtaining analytic results in TE is even harder than in the ’t Hooft case.

The orientifold large-N expansion opens the way for a novel and potentially rich large-N phenomenology in which the quark loops (i.e. dynamical quarks) do play a non-negligible role. An additional bonus is that in the orientifold large-N expansion, one-flavor QCD gets connected to supersymmetric gluodynamics, potentially paving the way to a wealth of predictions.

In order to demonstrate the difference between the standard large-N expansion and the orientifold large N expansion we exhibit a planar contribution to the vacuum energy in two ways in Fig. 3. Moreover, we will illustrate the usefulness of the orientifold large-N expansion at the qualitative, semi-

a It is curious that Corrigan and b o n d suggested [lo] to replace the ’t Hooft model by a model with one two-index antisymmetric quark Qtijl and two fundamental ones qi,z, as early as 1979.

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quantitative and quantitative levels by a few examples given below.

.............. .... 0 a ..:..

...................

.............. ..... b .......

.... ..................

.... ................ c ( ...... 0 ”’\ +>- .> - : ;-<. - -9

..... ................

Figure 3. (a,) A typical contribution to the vacuum energy. (b.) The planar contribution in ’t Hooft large-N expansion. (c.) The orientifold large-N expansion. The dotted circle represents a sphere so that every line hitting the dotted circle gets connected “on the other side.”

Probably, the most notable distinctions from the ’t Hooft expansion are as follows: (i) the decay widths of both glueballs and quarkonia scale with N in a similar manner, as 1/N2; this can easily be deduced by analyzing appropriate diagrams with the quark loops of the type displayed in Fig. 3c; (ii) unquenching quarks in the vacuum produces an effect that is not suppressed by 1/N; in particular, the vacuum energy density does depend on the quark masses in the leading order in 1/N.

3.2. Qualitative results for one-flavor QCD from the orientifold expansion

In this subsection we list some predictions for one-flavor QCD, keeping in mind that they are expected to be valid up to corrections of the order of 1/N = 1/3 (barring large numerical coefficients):

(i) Confinement with a mass gap is a common feature of oneflavor QCD and SUSY gluodynamics.

(ii) Degeneracy in the color-singlet bosonic spectrum. Even/odd parity mesons (typically mixtures of fermionic and gluonic color-singlet states) are expected to be degenerate. In particular,

where 7’ and CJ stand for the lightest 0- and O+ mesons, respectively. This follows from the exact degeneracy in N = 1 SYM theory. Note that the u meson is stable in this theory, as there are no light pions for it to decay into.

The prediction (3.1) should be taken with care: a rather large numerical coefficient in front of 1/N is not at all ruled out, since the q‘ mass is anomaly- driven (the Witten-Veneziano (WV) formula [12,13]), whereas the cr mass is more “dynamical.”

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In more general terms the mass degeneracy is inherited by all those “daughter” mesons that fall into one and the same supermultiplet in the parent theory. The accuracy of the spectral degeneracy is expected to improve at higher levels of the Regge trajectories, as 1/N corrections that induce splittings are expected to fall off (see e.g. [ll]).

(iii) Bifermion condensate. N = 1 SYM theory has a bifermion condensate. Similarly we predict a condensate in one-flavor QCD. A detailed calculation is discussed in Sect. 4.

(iv) One can try to get an idea of the size of 1/N corrections from perturbative arguments. In one-flavor QCD the first coefficient of the function is b = 31/3, while in adjoint QCD with N j = 1 it becomes b = 27/3. One can go beyond one loop too. As was mentioned, in the very same approximation the p function of the one-flavor QCD coincides with the exact NSVZ p function,

1 9a2 p=- - 27r 1 - 3 a / 2 ~ *

Thus, for the (relative) value of the tweloop &function coefficient, we predict +3cr/27r, to be compared with the exact value in one-flavor QCD,

134 a a +-- - - 4.32 - . 31 27r 27r

We see that the orientifold large-N expansion somewhat overemphasizes the quark-loop contributions, and, thus, makes the theory less asymptotically free than in reality- the opposite of what happens in ’t Hooft’s expansion. Parametrically, the error is 1/N rather than 1/N2. This is because there are N 2 - 1 gluons and N 2 - N fermions in the orientifold field theory.

4. Calculating the quark condensate in one-flavor QCD from supersymmetric gluodynamics

The gluino condensate in SU(N) SYM theory is

(A:Aa,a) = -6Nh3 . (4.1)

It is not difficult to show that the correspondence between the bifermion operators is as follows:

(A:Aaq H (G 9). (4-2)

The left-hand side is in the parent theory, the right-hand side is in the orientifold theory A, and they project onto each other with the unit coefficient. It is worth emphasizing that Eq. (4.2) assumes that (AzAa@) is real and

525

negative in the vacuum under consideration (which amounts to a particular choice of vacuum). It also assumes that the gluino kinetic term is normalized non-canonically, as in Eq. (4.4).

Thus, the planar equivalence gives us a prediction for the quark condensate in the one-flavor orientifold theory at N = co, for free. Our purpose here is to go further, and to estimate the quark condensate at N = 3, i.e. in one-flavor QCD.

Table 2. ficients of the 8 function.

Comparison of the anomalous dimensions and the first two coef-

The anomalous dimension y of the fermion bilinear operators is normalized in such a way that

and p and Q denote the normalization points. For our present purposes we can limit ourselves to the two-loop p functions and the one-loop anomalous dimensions. We can easily check that the various coefficients of the orientifold theory A go smoothly from those of YM ( N = 2) through those of If-QCD ( N = 3) to those of SYM theory at N + 00. Note, however, that some corrections are as large as N 2 / N . This is an alarming signal.

is renormalization-group-invariant (RGI) at any N , and so is A. This is not the case for the quark condensate in the non-SUSY daughter (i.e. the orientifold theory) at finite N . Unlike SUSY theories, where it is customary to normalize the gluino kinetic term as

The gluino condensate (X$Xa

1 - -AXig)X, g2 (4.4)

the standard normalization of the fermion kinetic term in non-SUSY theories

526

is canonic,

Hence, in fact, the correspondence between operators is

In the canonic normalization (4.5), the RGI combination is

where

and Y 19 6 ( N ) 3 1 - - = O(l /N) , 6(3) = - .

P O 31 (4-9)

Combining Eqs. (4.1), (4.6) and (4.7) we conclude that

= -6(N - 2) A& K ( p , N ) , (4.10)

where p is some @ed normalization point; the correction factor

K:(p,N=CO)=l, (4.11)

simultaneously with 6(N = 00) = 0. At finite N the correction factor W) K(p, N ) - 1 = O(l/N) and K depends on p in the same way as [g2(p)] .

The combination (g2) on the left-hand side is singled out because of its RG invariance. Equation (4.10) is our master formula.

The factor N - 2 on the right-hand side of Eq. (4.10), a descendant of N in Eq. (4.1), makes ( (g2)1-6(N) $!I! vanish at N = 2. This requirement is obvious, given that at N = 2 the antisymmetric fermion loses color. In fact, we replaced N in Eq. (4.1) by N - 2 by hand, assembling all other 1/N corrections in K:, in the hope that all other 1/N corrections collected in K: are not so large. There is no obvious reason for them to be large. Moreover, we can try to further minimize them by a judicious choice of p.

It is intuitively clear that 1/N corrections in K: will be minimal, provided that p presents a scale “appropriate to the process”, which, in the case at hand, is the formation of the quark condensate. Thus, p must be chosen

l - b ( N ) -

)

527

as low as possible, but still in the interval where the notion of g 2 ( p ) makes sense. Our educated guess is

[ g y p ) ] 6(3) x 4.9.

As a result, we arrive at the conclusion that in one-flavor QCD

= - 1 . 2 A k K ( N ) .

(4.12)

(4.13)

Empiric determinations of the quark condensate with which we will confront our theoretical prediction are usually quoted for the normalization point 2 GeV. To convert the RGI combination on the left-hand side of Eq. (4.13) to the quark condensate at 2 GeV, we must divide by [g2(2 GeV)]1-6(3) M 1.4. Moreover, as has already been mentioned, we expect non-planar corrections in K: to be in the ballpark f l / N . If so, three values for K:,

K: = {2/3, 1, 4/3} 1 (4.14)

give a representative set. Assembling dl these factors together we end up with the following prediction for one-flavor QCD:

( p [ijl 9 [ijl = - (0.6 to 1.1) A&. (4.15)

Next, our task is to compare it with empiric determinations, which, unfortunately, are not very precise. The problem is that one-flavor QCD is different both from actual QCD, with three massless quarks, and from quenched QCD, in which lattice measurements have recently been carried out 1141. In quenched QCD there are no quark loops in the running of a,; thus, it runs steeper than in one-flavor QCD. On the other hand, in three-flavor QCD the running of a, is milder than in one-flavor QCD.

(the 't Hooft coupling) we resort to the following procedure. First, starting from a , ( M T ) = 0.31 (which is close to the world average) we determine A g . Then, with this A used as an input, we evolve the coupling constant back to 2 GeV according to the one-flavor formula. In this way we obtain

To estimate the input value of

X(2 GeV) = 0.115. (4.16)

A check exhibiting the scatter of the value of X(2 GeV) is provided by lattice measurements. Using the results of Ref. [15] referring to pure Yang-Mills theory one can extract a,(2 GeV) = 0.189. Then, as previously, we find A E , and evolve back to 2 GeV according to the one-flavor formula. The

528

result is

A(2 GeV) = 0.097. (4.17)

The estimate (4.17) is smaller than (4.16) by approximately one standard deviation u. This is natural, since the lattice determinations of a, lie on the low side, within one u of the world average. In passing from Eq. (4.13) to Eq. (4.15) we used the average value X m ( 2 GeV) = 0.1.

One can summarize the lattice (quenched) determinations of the quark condensate, and the chiral theory determinations extrapolated to one flavor, available in the literature, as follows:

= - (0.4 to 0.9) A&. (4.18) 2 GeV, “empiric”

We put empiric in quotation marks, given all the uncertainties discussed above.

Even keeping in mind all the uncertainties involved in our numerical estimates, both from the side of supersymmetry/planar equivalence 1/N corrections, and from the “empiric” side, a comparison of Eqs. (4.15) and (4.18) reveals an encouraging overlap.

References 1. S. Kachru and E. Silverstein, Phys. Rev. Lett. 80, 4855 (1998) [hep-th/9802183]. 2. M. Bershadsky and A. Johansen, Nucl. Phys. B 536, 141 (1998) [hepth/9803249]. 3. M. Schmaltz, Phys. Rev. D 59, 105018 (1999) [hep-th/9805218]. 4. M. J. Strassler, On methods for extracting exact non-perturbative results in non-

supersymmetric gauge theories, hepth/0104032, unpublished. 5. A. Gorsky and M. Shifman, Phys. Rev. D 67, 022003 (2003) [hepth/0208073]. 6. D. Tong, JHEP 0303, 022 (2003) [hepth/0212235]. 7. V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B

229, 381 (1983); Phys. Lett. B 166, 329 (1986). 8. G. ’t Hooft, Nucl. Phys. B 72, 461 (1974). 9. E. Witten, Nucl. Phys. B 160, 57 (1979).

10. E. Corrigan and P. Ramond, Phys. Lett. B 87, 73 (1979). 11. G. Veneziano, Nucl. Phys. B 117, 519 (1976). 12. E. Witten, Nucl. Phys. B 156, 269 (1979). 13. G. Veneziano, Nucl. Phys. B 159, 213 (1979). 14. L. Giusti, C. Hoelbling and C. Rebbi, Phys. Rev. D 64, 114508 (2001), (E) D65,

079903 (2002) [heplat/0108007]. 15. M. Liischer, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys. B 413, 481 (1994) [hep

lat/9309005]; M. Guagnelli, R. Sommer and H. Wittig [ALPHA collaboration], Nucl. Phys. B 535, 389 (1998) [heplat/9806005].

KNOT ENERGIES AND THE GLUEBALL SPECTRUM*

THOMAS w. KEPHART~ AND ROMAN v. BUNKS Department of Physzcs and Astronomy

Vanderbilt University Nashville, T N 37235

E-mail: thomas.w.kephartBvanderbi1t. edu E-mail: roman@uoregon. edu

Systems of tightly knotted, linked, or braided quantized flux tubes have a universal mass-energy spectrum, since the length of fixed radius flux tubes depend only on the topology of the configuration. We concentrate on the model of glueballs as knotted QCD flux tubes, but other applications are also briefly discussed.

1. Review of previous physical results on tight knots and links

It is known from plasma physics that linked magnetic flux tubes are much more stable than an unknotted single loop [l], [2]. Linked flux tubes carry topological charges, and these can be thought of as a conserved (at least to lowest order) physical quantum numbers. Similarly, knotted flux tubes carry topological quantum numbers, and one can think of a knot as a self- linked loop.

If tubes have fixed uniform thickness and circular cross-sections (we could relax this condition), then each knot and link has a completely specified length if the configuration is tight, i.e., is of the shortest length with the cross section of the tubes undistorted and non-overlapping. This provides a geometric invariant length for each topology. For tubes with uniform cross sections, as can be approximately the case with magnetic or electric flux tubes carrying quantized flux, the length of the tight knot is proportional to the mass (or energy) of the knot. This, we claim, generates a universal mass

Work supported by DOE grant no. DE-FG05-85ER40226. tTalk at PASCOS 2004. SPresent address: Institute of Theoretical Science, University of Oregon, Eugene, OR 97403.

529

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(energy) spectrum for knotted/linked configurations of objects of this type. The lengths of tight knots were not studied until the mid-1990s [3], and only recently have accurate calculations of large numbers of tight knots [4] and links [5] become available. These results now make it possible to examine physical systems and compare them with the knot spectrum. The first physical example studied was tightly knotted DNA [3]. More recently, we have examined the glueball spectrum in QCD [6]. These particles [7] are likely to be solitonic states [8] that are solutions to the QCD field equations. While QCD will be our main focus, there are many more cases where tight knots may play a role.

Two time scales are of key importance in determining if a system of topologically nontrivial flux tubes falls into the universal class of having a tight knot energy spectrum. These are the lifetime of the soliton T~ and the relaxation time 7,. necessary to reach the ground state of a tight knot configuration. So a tight knot/link requires r3 > r,.. The soliton lifetime depends on the rates of flux tube breaking, rearrangement, and reconnection. All these processes change topological charge, and their partial widths compete more or less favorably with each other depending on the parameters that describe the system.

While no knot lengths have been calculated exactly, it is possible to calculate the exact lengths of an infinite number of links and braids [5]. For links these calculations are possible in the case where each individual elements of the link lies in a plane. For braids, exact calculations are possible when the elements of the braid are either straight sections or when they follow uniform paths 191. The shortest of all links, the double donut, is exactly calculable. The two elements lie in perpendicular planes and are tori of equal length. The shortest non-trivial braid is a helically twisted pair. “Weyl’s tube formula” is the ideal tool for calculating the volume of a flux tube [lo, 111. For the case at hand, the formula states that for a tube of constant cross-section 0 normal to a path of length 1 , the volume of the tube is VT = la in flat 3D.

2. Plasma physics

The example of linked flux in electromagnetic plasma is conceptually simple and provides motivation for what is to come. Movement of fluids often exhibits topological properties (for a mathematical review see e.g. Ref. [12]). For conductive fluids, interrelation between hydro- and magnetic dynamics may cause magnetic fields, in their turn, to exhibit topological properties

531

as well. For example, for a perfectly conducting fluid, the (abelian) magnetic helicity K = Jd3a:eijkAidjAk is an invariant of the motion [13], and this quantity can be interpreted in terms of knottedness of magnetic flux lines [14]. In MHD systems, the change in magnetic helicity K in a volume V with boundary S , is due to: (i) flow of helicity through S, and (ii) a resis- tive dissipation term proportional to JQJ . BdV. But, in an ideal plasma the resistivity vanishes, so in an isolated region, K is conserved. Here we are assuming a classical system, so the analysis does not include quantum effects like rearrangement, etc. that can change K .

Consider an abelian gauge potential 1-form A = Aidxi and the corresponding field-strength 2-form F = iFijdxedxj. The helicity for the field inside volume V is defined by

, = L A . (1)

Under a gauge transformation, A + A + d$ and F + F , and so using the Bianchi identity dF = 0 and the Stokes theorem we find

K + K + J ~ , $ F .

The helicity is thus gauge invariant if the normal component of the field F vanishes on the surface BV.

It is easy to calculate the helicity for two linked flux tubes with fluxes and @2. Considering first infinitely thin tubes (centered around the

curves C1 and C2) and integrating over their cross sections we find

Stokes’ theorem now leads to

K = 2n@1@2, (4)

where n is the Gauss linking number of the two tubes (the algebraic number of times that one tube crosses the surface spanned by the other tube). It is straigtforward to generalize this to the case of linked and/or self-linked thick flux tubes.

Now we begin to step toward QCD by considering a non-abelian plasma. Since both A and F are conserved during the plasma motion, any combination of A and F is a candidate for a conserved quantum number K . The choice of expression is narrowed by requiring that K is a topological quantity. In particular, this requires that K be a surface integral. By analogy

532

with the abelian case, for a conserved non-abelian helicity, we choose the corresponding expression with topological properties,

K = ( A d A + $ A 3 ) (5)

A (time-independent) gauge transformation A --+ g - l A g + g - l d g leads to

We consider only a limited set of gauge transformations such that do not change the condition vPAP = 0; this implies the constraint vi&g = 0. Be- cause of this constraint only two of the components g-'&g are independent and so ( g - ' ~ i g ) ~ = 0. The helicity is invariant if the normal component of A vanishes on the surface d V . This condition is more restrictive than the one needed in the abelian case and implies the latter.

3. QCD and Glueballs

What is the ideal physical system in which to discover and study tight knots and links? We will argue that it is Quantum Chromodynamics (QCD). We support this claim with the following facts:

(1) QCD is a solidly based part of the standard model of particle physics, and much about color confinement and the quark model is already well understood in this context, making much of the previous work transferable to the problem of tightly knotted flux tubes.

(2) Unlike plasmas, fluids or other condensed matter systems where flux tubes are excitations of some media with many parameters that could hide universal behavior, flux knots in QCD can exist in the vacuum. Thus continuum states are absent and there are no media parameters to vary and obscure the universality. Hence, the results in QCD can be far less ambiguous.

(3) The hadronic energy spectrum has been measured over a large range of energies (140 MeV to 10 GeV) and already many hundreds of states are known. We expect that among these, a few dozen can be classified as tightly knotted/linked flux tubes states. These states must have no valance quarks (i.e., no flavor quantum numbers) in order to be classified as glueballs.

(4) Knotted solitons in QFT are already known to exist.

533

( 5 ) One can efficiently search for new glueball states at accelerators. (Also, data from older experiments still exist and can be reanalyzed to check the predictions of new states described below.)

Consider a hadronic collision that produces some number of baryons and mesons plus a gluonic state in the form of a closed QCD flux tube (or a set of tubes). From an initial state, the fields in the flux tubes quickly relax to an equilibrium configuration, which is topologically equivalent to the initial state. (We assume topological quantum numbers are conserved during this rapid process.) The relaxation proceeds through minimization of the field energy. This process occurs via shrinking the tube length, and halts to form a “tight” knot or link. Flux conservation and energy minimization force the fields to be homogeneous across the tube cross sections. The radial scale will be set by The energy of the final state depends only on the topology of the initial state. For a tube of radius a and length 1, the energy is proportional to the length. The dimensionless ratio K. = 2/(2a) is a geometric invariant and the simplest definition of the “knot energy.” Many knot energies have been calculated by Monte Carlo methods [3] and more are being calculated [5 ] .

Most of the knot energies shown in Table 1 have been taken from Ref. [3], but we have independently calculated the energy of 2:, 4; and 6: exactly and the energy for several other knots and links approximately. We find ~ ( 2 : ) = 47r M 12.57, to be compared with the Monte Carlo value 12.6. We also find 44;) = 67r + 2 and 46:) = 87r + 3. Other exactly calculable links can be found in Ref. [5] and an example of a link with energy 1 0 ~ + 5 is shown in Fig. 1.

In our model, the chromoelectric fields Foi are confined to knotted/linked tubes. After an initial time evolution, the system reaches a static equilibrium state which is described by the energy density

& = itrFoiFoi - V. (7) Similar to the MIT bag model, we have included a constant potential energy V needed to keep the tubes at a fixed cross section. The chrornoelectric flux @E is conserved and we assume flux tubes carry one flux quantum. The energy is proportional to 1 and thus the minimum of the energy is achieved by shortening 1 , i.e., tightening the knot.

Lattice calculations, QCD sum rules, electric flux tube models, and constituent glue models agree that the lightest non-qq states are glueballs with quantum numbers J++ = O++ and 2++. Therefore, we will model all J++ states i.e., all f~ and f$ states listed by the PDG [7], some of which will

534

Figure 1. An example of an exactly calculable link.

be identified with rotational excitations, as knotted/linked chromoelectric QCD flux tubes. The lightest candidate is the f0(600), which we identify with the shortest knot/link, i.e., the 2y link; the fo(980) is identified with the next shortest knot, the 31 trefoil knot, and so forth. All knot and link energies have been calculated for states with energies less then 1680 MeV. Above 1680 MeV the number of knots and links grows rapidly, and their energies are just now being calculated [5]. However, we do find knot energies corresponding to all known fJ and f(l states, and so can make preliminary identifications in this region. (We focus on f~ and f(l states from the PDG summary tables. The experimental errors are also quoted from the PDG. There are a nuniber of additional states reported in the extended tables, but some of this data is either conflicting or inconclusive.)

Our detailed results are collected in Table 1, where we list f~ and fi masses, our identifications of these states with knots and the corresponding knot energies. In Fig. 2 we compare the mass spectrum of f~ states with the identified knot and link energies. Since errors for the knot energies in Ref. [3] were not reported, we conservatively assumed the error to be 1%.

i.e., glueballs.

535

10 15 20 25 30 35 40

4 K ) Figure 2. Relationship between the glueball spectrum E ( G ) and knot energies E ( K ) . Each point in this figure represents a glueball identified with a knot or link. The straight line is our model and is drawn for the fit (8).

A least squares fit to the most reliable data (below 1680MeV) gives

E(G) = (23.4 f 46.1) + (59.1 f 2.1)~(K) [MeV], (8)

with x2 = 9.1. The data used in this fit is the first seven f~ states (filled circles in Fig. 2) in the PDG summary tables. Inclusion of the remaining seven (non-excitation) states (unfilled circles in Fig. 2) in Table 1, where either the glueball or knot energies are less reliable, does not significantly

536

Table 1. Comparison between the glueball mass spectrum and knot energies.

400 - 1200 980 f 10

1275.4 f 1.2 1281.9 f 0.6

1426.3 f 1.1 M 1430)

1200 - 1500 1507 f 5

1518 f 5) 1525 f 5

1546 f 12) 1638 f 6)

......................

1713 f 6

2011t,6; 2025 f 8

2297 f 28 2339 f 60

12.6 [4n] 16.4

[6n + 21 21.2

(21.4) 24.2 24.2

(24.7) 24.9 24.9 24.9

(25.9) ((27.3))

[8n + 31 ((28.6))

28.9 (30.5) 29.1 (30.5) [8n + 41

29.2 29.3 30.5 30.9 31.0 32.7 33.2 33.9 37.0 37.6 38.3 40.0 44.8 47.0

........... . . . . .

768 [766] 993

[1256] 1277

(1289) 1454

1454 + 6’ (1484) 1496

1496 + 6 1496 + 36

(1555) ((1638))

((1714))

, . . . . . . . . . . . . [1686] f

1732 (1827) 1744 (1827)

(17451 1750 1756 1827 1850 1856 1957 1986 2028 221 1 2247 2288 2389 2672 2802

~

a: Notation nk means a link of 1 components with n crossings, and occurring in the standard table of links on the kth place. K#K’ stands for the knot product (connected sum) of knots K and K’ and K * K’ is the link of the knots K and K’. b : Values are from Ref. [3] except for our exact calculations of 2:, 2: * 01, and (2: * 01) *Oi in square brackets, our analytic estimates given in parentheses, and our rough estimates given in double parentheses. c : E ( G ) is obtained from K. using the fit (8). d : States in braces are not in the Particle Data Group (PDG) summary tables. e: This is the link product that is not 2: * 2:. f: Resonances have been seen in this region, but are unconfirmed 171.

537

alter the fit and leads to

E(G) = (26.9 f 24.9) + (58.9 f l.O)e(K) [MeV], (9)

with x2 = 10.1. The fit (8) is in good agreement with our model, where E(G) is proportional to E ( K ) . Better HEP data and the calculation of more knot energies will provide further tests of the model and improve the high mass identification.

For comparison, we have fitted the same data except that the first glueball is missed out; this results in x2 = 231. Similarly, we have also fitted the same data when the first link is missed out; this results in x2 = 355. This is strong evidence that our identification is appropriate.

In terms of the bag model, the interior of tight knots corresponds to the interior of the bag. The flux through the knot is supported by current sheets on the bag boundary (surface of the tube). Knot complexity can be reduced (or increased) by unknotting (knotting) operations [21, 221. In terms of flux tubes, these moves are equivalent to reconnection events. Hence, a metastable glueball may decay via reconnection. Once all topological charge is lost, metastability is lost, and the decay proceeds to completion. Two other glueball decay processes are: flux tube (string) breaking; this favors large decay widths for configurations with long flux tube components; and quantum fluctuations that unlink flux tubes; this would tend to broaden states with short flux tube components. We have assumed one fluxoid per tube. There may be states with more than one fluxoid, but these states would presumably be above the mass range of known glueball candidates.

4. Discussion and conclusions

Knotted/linked solitonic physical systems can have universal mass-energy spectra and we have demonstrated this with a detailed example from QCD. Other candidates include superfluid 4He, type 11 superconductors, Bose- Einstein condensates such as dilute s7Rb, cosmic strings etc. To summarize, a system is a candidate if it contains line solitons, or defects that can (somewhat paradoxically) relax to their tight knot/link ground state in a time shorter than their decay time. We hope future work will tie knot theory to many subfields of physics or to other sciences.

References 1. H. K. Moffatt, J. Fluid Mech. 159, 359 (1985). 2. P. M. Bellan, Spheromaks, Imperial College Press, 2000.

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3. V. Katritch, et al., Nature 384, 142 (1996); V. Katritch, et al., Nature 388, 148 (1997).

4. K. Millett and E. Rawdon, J: of Comp. Phys. 186, 426 (2003). 5. R. V. Buniy, T. W. Kephart, M. Piatek and E. Rawdon, in preparation. 6. R. V. Bmiiy and T. W. Kephart, Phys. Lett. B 576, 127 (2003). 7. K. Hagiwara et al., Phys. Rev. D 66, 010001 (2002). 8. H. B. Nielsen and P. Olesen, Nucl. Phys. B 61, 45 (1973). 9. R. V. Buniy and T. W. Kephart in Numerical methods, simulations, and

computations in knot theory and its applications, eds. J. Calvo, K. Millet, and E. Rawdon, World Scientific, Singapore, 2005 (in press).

10. H. Weyl, Am. J. Math. 61, 461 (1939). 11. A. Gray, Tubes, Addison-Wesley, 1990. 12. V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics,

13. L. Woltier, Proc. Nat. Acad. Sci. 44, 489 (1958). 14. H. K. Moffatt, J. Fluid Mech. 35, 117 (1969). 15. M. Barranco and J. R. Buchler, Phys. Rev. C34, 1729 (1980). 16. G. Pang and H. Zhm, J. Phys. A : Math. Gen. 25, L527 (1992). 17. H. Muller and B. D. Serot, Phys. Rev. C52, 2072 (1995). 18. V. Baran, M. Colonna, M. Di Tor0 and A. B. Larionov, Nucl. Phys. A632,

19. V. Baran, M. Colonna, M. Di Tor0 and V. Greco, Phys. Rev. Lett. 86, 4492

20. X. Yan, Phys. Rev. E61, 4745 (2000). 21. D. Rolfsen, Knots and Links, Publish or Perish, 1990. 22. L. H. Kauffman, Knots and Physics, World Scientific, 2001.

Springer, 1998.

287 (1998).

(2001).

ON THE EXPERIMENTAL TESTING OF DIRAC EQUATION*

ALFRED0 MACiAS AND ABEL CAMACHO Departamento de Fa'sica

[Jniversidad Auto'noma Metropolitana-Iztapalapa Apartado Postal 55-534, C.P. 09340, Mhico , D.F., M6xico.


The advent of phenomenological quantum gravity has ushered us in the search for experimental tests of the deviations from general relativity predicted by quantum gravity or by string theories, and as a by-product of this quest the possible modifications that some field equations, for instance, the motion equation of spin- 1/2-particles, have already been considered. In the present work a modified Dirac equation, whose extra term embraces a second-rder time derivative, is taken as mainstay, and two different experimental proposals to detect it are put forward. The novelty in these ideas is that two of them do not fall within the extant a p proaches in this context, to wit, redshift, atomic interferometry, or Hughes-Drever type-like experiments.

1. Introduction

One of the bedrocks beneath our present description of the fundamental laws of physics is embodied by Lorentz symmetry. The significance of this symmetry in the theoretical realm clearly justifies the long-lasting interest in testing it 1,21334. One of the profits in this context, the one can be readily seen with a fleeting glimpse to the corresponding experimental constructions, is the fact that the involved precisions have undergone a remarkable improvement.

*This work was supported by conacyt grant 42191-F.

539

540

2. A modified Dirac equation

Our mainstay is the introduction of a second-rder time derivative in Dirac equation. To wit, from square one we assume the following motion equation

A factor l/mc2 in the term containing the second-rder time derivative has been introduced, in order to have a dimensionless parameter E . For = 0, the introduced equation reduces to the usual Dirac situation. Additionally, a fleeting glimpse to ( 1 ) shows us that Lorentz-covariance is violated.

In the usual Dirac equation the non-relativistic limit is deduced by splitting up the energy into two parts, namely, (i) the rest energy, and (ii) additional contributions to the energy. This is attained introducing $ = 4 exp (- imc2t) . The non-relativistic limit is obtained assuming that the rest energy is much larger than any other kind of energy involved.

Proceeding as usual 5, which means that here 4 = (:), we arrive at the

following expression

Here X denotes the Compton wavelength of the particle. Even if e is very small, the term mc2e2(2 + E ) could have an order of magnitude similar to the remaining energies present in ( 2 ) .

The introduction of the coupling with an electromagnetic field is achieved resorting to the minimal coupling procedure. Therefore the resulting equation reads

In (3) we have introduced the vector potential, A, and the scalar one, @. The non-relativistic limit of this last expression renders the generalized Pauli equation

541

Two new terms have been introduced in (4), to wit, the magnetic field, B, and the spin operator, S , respectively.

3. Experimental proposals.

3.1. Spreading t ime of a wave packet

Let us now consider a solution to (2) in the form

( 5 )

This Ansatz allows us to cast (2) in the following form h2k2 h

[I - 2e(1+ c)]tw = - + E - X W ~ + mc2e2(2 + 6 ) . 2m c

It is readily seen that this last expression defines w as a function of k. Indeed,

1 2€X

w ( k ) = -{ [l - 241 + E)]C f

€A h2k2 Ch 2m

~ / [ l - 2 4 1 + € ) I 2 - 4-[mc2e2(2 + E ) + - 1 ) . (7)

The group and phase velocity are defined by, us = and vp = f, respectively. Eq. (7) defines a cutoff in the permitted wave number. Forsooth, the square-root, in (7), entails, in order to have real-valued frequency, the following condition

Assuming 161 << 1, the cutoff, in terms of the momentum becomes, approximately p 5 z. A condition always fulfilled within the non- relativistic realm. Consider now a oneedimensional wave packet constructed as a superposition of plane waves, in such a way that this packet is sharply peaked around k = ko, with a width given by Ak

{ 1 l W $(o, t ) = - 1 A(k - ko) exp iko - iwt dk.

J21; -W (9)

The condition upon the manner in which this wave packet has been constructed implies that A(k - k ~ ) M 0 if Ik - kol > Ak. Expanding ko - wt around k = ko allows us to cast (9) in the following form

$(o, t ) = exp ikoo - iw(ko ) t } { d2w dk2 lko

5 - [us - q- ] t ) } d q .

542

Here we have defined q = k - ko. Since it has been assumed from the very beginning that A(k - ko) x 0 if Ik - kol > Ak, then (10) will be dominated by values of q in the range [-Ak, Ak]. Hence, we are allowed to put forward the following relation q$$lko = *Avg.

Knowing that the Fourier transform is dominated by those parts satisfying the condition z - vgt M 0 (as long as (Ak)2$lkot << l ) , then it is reasonable to define the spreading time of the wave packet. To first order in E this spreading time reads

t , = -{1- 2 4 1 - ,,I}. Am; h(Ak)2

3.2. Probability density and its dependence upon E

Let us now hark back to (4), with the initial assumption of vanishing magnetic field, namely, B = 0. Proceeding in the usual manner it is possible to deduce a probability conservation law associated to (4). Indeed, under these circumstances $f + V . J = 0, with

x c[l - 241 + E ) ]

p = 1 - € }44* - i E { c[l - 241 + €)]

and

[4w* - 4*v4 h

x J = i

2 4 1 - 241 + E ) ]

A44*. - %[l - 241 + E ) ]

If E = 0 is implemented, then everything reduces to the usual conservation law '. The probability density not only hinges upon first-order time derivatives, it also displays a dependence on the charge of the involved particle. Both characteristics are absent in the usual model.

4. Discussion

The possibility of resorting to the spreading time of a wave packet in order to detect an extra term, like the one encompassed by ( l ) , is cognate with the fact that the experimental resolution, At, has to be smaller than the difference between the spreading times in our proposal, ( l l ) , and the spreading time in the usual model, henceforth denoted by is. It will be

543

possible to detect, within the realm of the first proposal, an extra term like the one here considered if At < h&y,. { 1 - ~ } I E I .

Consider a particle at rest, whose wave func6on embodies a linear superposition of plane waves, its initial form is Gaussian. Two screens will be located at two different points, if their positions are denoted by 0 < S1 < SZ and Az(t = 0) is the root-mean-squaredeviation at time t = 0, then Az(t = 0) < S1. As time goes by the packet spreads, a time tl will come in which Az(t = t l ) = S1. Screen 1, at this moment, emits a photon. The same situation will be associated to the second screen, at time t 2 , the root-mean-squaredeviation fulfills Az(t = t 2 ) = S2. The time interval between these two photons will be related to the spreading velocity of the packet, and since we know the distance between the two screens, S2 - S1, the knowledge of these two factors would allow us to set a bound to the E parameter. The possibility of measuring time intervals down to 50 fs is already within the technological developments ’.

The possibility of employing the probability density to detect the extra term is related to the fact that, the probability density hinges upon the charge of the corresponding particle, whereas in the usual theory it does not. Therefore, the change q -+ -9, leads us to conclude that there must be a change in the probability density, associated to the modification of the charge of the involved particle is not present in the usual situation, and defines a trait that could lead to the detection of the new term.

References 1. H. P. Robertson, Rev. Mod. Phys. 21, 378 (1949). 2. R. M. Mansouri and R. U. Sexl, Gen. Rel. Grav. 8, 809 (1977). 3. C. Braxmaier et al, Phys. Rev. Lett. 88, 010401 (2002). 4. P. Wolf et al, Phys. Rev. Lett. 90, 060402 (2003). 5. A. Camacho and A. Macias, Phys. Lett. B582 229 (2004).

PERTURBATIVE CALCULATION OF QUASI-NORMAL MODES*

GEORGE SIOPSIS Department of Physics and Astronomy, The University of Tennessee, Knoxville,

E-mail: [email protected] TN 37996 - 1200, USA.

I discuss a systematic method of analytically calculating the asymptotic form of quasi-normal frequencies. In the case of a four-dimensional Schwarzschild black hole, I expand around the zeroth-order approximation to the wave equation proposed by Mot1 and Neitzke. In the case of a five-dimensional Ads black hole, I discuss a perturbative solution of the Heun equation. The analytical results are in agreement with the results from numerical analysis.

1. Introduction

This report summarizes recent work I did largely in collaboration with S. Musiri 172,3.

Quasi-normal modes (QNMs) describe small perturbations of a black hole. They are obtained by solving a wave equation for small fluctuations subject to the conditions that the flu be ingoing at the horizon and outgoing at asymptotic infinity. In general, one obtains a discrete spectrum of complex frequencies. The imaginary part determines the decay time of the small fluctuations (Sw = +).

2. Schwarzschild black holes

For a Schwarzschild black hole,

w,/TH = (2n + 1)ri + In3 (1) This was first derived numerically 4,5,67778 and subsequently confirmed analytically '. Sw, is large and easy to understand because the spacing of

'This work is supported in part by the US Department of Energy under grant DEFGOS- 91ER40627.

544

545

frequencies is 27riT~, same as the spacing of poles of a thermal Green function. On the other hand, 8wn is small. Its analytical value was conjectured by Hod lo and is related to the Barbero-Immird parameter. It is intriguing from the loop quantum gravity point of view suggesting that the gauge group should be SO(3) rather than SU(2). Thus the study of QNMs may lead to a deeper understanding of black holes and quantum gravity.

The analytical derivation of the asymptotic form of QNMs offered a new surprise as it heavily relied on the black hole singularity. It is intriguing that the unobservable region beyond the horizon influences the behavior of physical quantities. In the Schwarzschild background metric the wave equation for a spin-j perturbation of frequency w is

lPQ 1 dz2 w -- + 1 V[r(z)]Q = Q , V(T) = (1 - y ) (V + 9) (2)

where z = w ( r + ro ln(r/ro - 1) - i m o ) and TO is the radius of the horizon. For QNMs we demand (assuming 8w > 0) ei"3 N 1 as z + +cm and near the horizon ( z -+ -m), eizQ N e2ir . The latter boundary condition is implemented by demanding that the monodromy around the singular point r = TO in the complex r-plane be M(r0) = e-4awro along a contour running counterclockwise. We may deform the contour so that the only contribution to the monodromy comes from the black hole singularity (r = 0) '. The potential can be written as a series in ,/5, which is also a formal expansion in l/fi. We may then solve the wave equation perturbatively. To zeroth order, we obtain

Q; + { -t l}Qo = 0 (3)

whose solutions can be written in terms of Bessel functions. They lead to a zeroth-order monodromy M(r0) = -(1+ 2cos(7rj)) yielding a discrete spectrum of complex frequencies (QNMs)

w,/TH = (272 + 1)ri + In( 1 + 2 cos(nj)) + o(l / f i ) (4)

Expanding the wavefunction, XP = QO -t -& * I + o(l/w), the first-order correction obeys

( 5 ) 3t(t+l)+l-j2 2-3/2

3;+{$&+1}Q1=dvQo, d V = - 6 4

By solving eq. (5), one obtains an o(l / f i ) correction to the monodromy which yields the QNM frequencies

e * 3 * / 2 Wn/TH = (272 + 1)ni + h(l+ 2cOs(rj)) + m d f o(l/n) ,

546

This analytical result is in agreement with numerical results for scalar (j = 0) and gravitational waves ( j = 2). 6~11 In the latter case, it also agrees with the result from a WKB analysis. l2

3. Kerr black holes

Extending the above discussion to rotating (Kerr) black holes is not straightforward. Numerical results paint a complicated picture l3 and an+ lytical calculations have yet to produce results. We have obtained explicit results in the case a = J / M << 1, where J is the angular momentum of the black hole and M is its mass. This regime includes the Schwarzschild black hole (a = 0). For the asymptotic range of frequencies

working as in the Schwarzschild case, we obtain

% = & ln( l+ 2cos.rrj) + ma + .(a2) (8) where m is the azimuthal eigenvalue of the wave and R % a is the angular velocity of the horizon. In the Schwarzschild limit, the range of frequencies (7) extends to infinity (l/a .--) co) and we reproduce our earlier result (4). It would be interesting to extend this result to asymptotic frequencies for arbitrary values of a (0 < a < 1).

4. Ads Black Holes

According to the AdS/CFT correspondence, QNMs for an Ads black hole are expected to correspond to perturbations of the dual CFT. The estab- lishment of such a correspondence is hindered by difficulties in solving the wave equation. In three dimensions, it reduces to a hypergeometric equation which is analytically solvable 14. Numerical results have been obtained in four, five and seven dimensions 15.

In five dimensions, the wave equation reduces to a Heun equation which cannot be solved analytically. We shall build a perturbative expansion in the case of a large black hole based on an approximation to the wave equation which is valid in the high frequency regime ’. The singularities are located at r2 = *T$, where ro is the radius of the horizon. In higher dimensions, there are more singularities, all lying on the circle Irl = TO in the complex r-plane. Even though these are unphysical singularities (with

547

the exception of T = T O ) , they seem to play an important role in determining the QNMs, as in the Schwarzschild case '.

Setting the Ads radius R = 1, the wave equation for a massive scalar of mass rn, frequency w and transverse momentum p'reads

(y(y2 - 1)Q')' + { $ & - $ - rn2y)B = 0 (9)

where y = <, 2 = &, p = &. After isolating the behavior at the two singularities, y = fl ,

T O

iqy) = (y - 1)-iG/4(y + 1 ) - 2 / 4 ~ ( y ) (10) it reduces to the Hem equation

y(y2 - 1)F" + { (3 - ~ 2 ) y2 + 9 Ljy - 1) F'

-2 } (11) + w T - l - i ) y - m y + ( ~ - z ) $ - % 2 F = O G (iG

For large 2 and in the physical regime T > TO, this may be approximated by the hypergeometric equation

(y2 - 1)F: + ((3 - Y L j ) y + y2) Fi + ( 4 ($ - 1 - i) - m2} F' = 0 (12)

Two independent solutions are

IC* = (X + l)-"*F(aj-, c - U,;Q - ay + 1; l / ( ~ + 1)) (13) whereak = h* - 2, c = 3 2 2 - 2: 2, h* = I f d m a n d Z = 9. The solution which is well-behaved at the boundary (T + 00) is FO = K+. Near the horizon (x + 0) , we have FO N do + where

(14) A - W-cF(l(-a-+a+; ,

= r(;-;F{;;,a_+,;p-l r(1-a-p- I - ~ + ~ +

For a QNM, we demand regularity at the horizon, so 230 = 0. This yields the spectrum

Ljn = -2(1 + i ) ( n + h+ - 4) , n = 1,2, ... (15) which agrees with numerical results 16.

The first-order correction may be written as 00 x X l F o - K+(~.J~~ K - Z F o Fib) = n-(4sz +w

where W is the Wronskian and

x1= &{i& + (i - l)? + $}

1 - ( r(a':r(c-a+) +) so

At the horizon, Fl(x) N dl + B1zl-', where B - r c-1 r l+a - -a 00 K X l K +

+ w

548

The QNM frequencies to first order are solutions of Bo + f31 = 0. To solve this equation, we need to calculate the integral in (18). A somewhat tedious calculation leads to corrections to the zeroth-order expression for QNM frequencies (15) which are of order l /h+ - l/m.

5. Conclusions

Even though QNMs have long been known numerically, we have only recently made progress toward calculating them analytically. Their behavior appears to rely on unobservable, or even unphysical singularities. It is desirable to fully investigate the behavior of QNMs in general space-time backgrounds. In asymptotically AdS spaces, we would like to understand their relevance to the AdS/CFT correspondence. In asymptotically flat spacetimes, they should lead to the equivalent of Bohr’s correspondence principle for the gravitational force which seems to exist in the case of Schwarzschild black holes but remains elusive for Kerr black holes. This may shed some light on the quantum theory of gravity.

References 1. S. Musiri and G. Siopsis, Class. Quant. Grav. 20 (2003) L285. 2. S. Musiri and G. Siopsis, Phys. Lett. B579 (2004) 25. 3. S. Musiri and G. Siopsis, Phys. Lett. B563 (2003) 102; B576 (2003) 309.

G. Siopsis, Phys. Lett. B590 (2004) 105. 4. S. Chandrasekhar and S. Detweiler, Proc. R. SOC. London, Ser. A 344 (1975)

441. 5. E. W. Leaver, Proc. R. SOC. London, Ser. A 402 (1985) 285. 6. H. P. Nollert, Phys. Rev. D47 (1993) 5253. 7. N. Andersson, Class. Quant. Grav. 10 (1993) L61. 8. A. Bachelot and A. Motet-Bachelot, Annales Poincarb Phys. Theor. 59

(1993) 3. 9. L. Motl, Adv. Theor. Math. Phys. 6 (2003) 1135.

L. Motl and A. Neitzke, Adv. Theor. Math. Phys. 7 (2003) 2. 10. S. Hod, Phys. Rev. Lett. 81 (1998) 4293. 11. E. Berti and K. D. Kokkotas, Phys. Rev. D68 (2003) 044027. 12. A. Maassen van den Brink, J. Math. Phys. 45 (2004) 327. 13. E. Berti, et al., Phys. Rev. D68 (2003) 124018. 14. V. Cardoso and J . P. S. Lemos, Phys. Rev. D63 (2001) 124015.

D. Birmingham, et al., Phys. Rev. Lett. 88 (2002) 151301. 15. G. T. Horowitz and V. E. Hubeny, Phys. Rev. D62 (2000) 024027.

A. 0. Starinets, Phys. Rev. D66 (2002) 124013. R. A. Konoplya, Phys. Rev. D66 (2002) 044009; ibid. 084007.

16. A. Nliiiez and A. 0. Starinets, Phys. Rev. D67 (2003) 124013.

RESOLUTIONS OF THE INFRARED DIVERGENCE IN QFT

YEONG-SHYEONG TSAI Dept. of Applied Math., National Chung Hsing University, Taichung, Taiwan

ystsai@nchu. edu. tw

HUNG-MING TSAI Dept. of Physics, National Taiwan University, Taipei, Taiwan

[email protected]

PO-YU TSAI Dept. of Chemistry, National Taiwan University, Taipei, Taiwan

r93223016@ntu. edu. tw

LU-HSING TSAI Dept. of Physics, National Taiwan Normal University, Taipei, Taiwan

fsors777@yahoo. com. tw

Though QFT is well developed and successfd in theoretical aspects and applications, there are some defects in QFT [l]. Ultraviolet divergences and infrared divergences are serious defects of QFT. The ultraviolet divergence will not be discussed in this article since it was discussed by Hung-Ming Tsai et al. in 2003 [2]. There are two approaches to remove the infrared singularities. The first part of this article is the mathematical proof of the existence of a positive lower bound of the energy of emitted photons or gluons. Therefore, the soft divergence is removed. The second part of this article is to show the existence of a regular basis of the solution space of the field equation. The second quantization is based on this regular basis. With this regular basis, the infixed divergences, both the soft divergence and the collinear divergence, are expected to be removed.

1. Introduction

Almost one year ago, Hung-Ming Tsai el al. suggested, in the 3rd International Symposium on Quantum Theory and Symmetries (QTS3), Cincinnati, Ohio, 10- 14 September 2003, that the thermal factors,

were introduced into QFT [2]. With these thermal factors, the ultraviolet divergence is removed automatically. Therefore, the only divergence left is the infrared divergence. In speaking of the infrared divergence, there are two kinds of divergences, the soft divergence and the collinear divergence [3]. First, it will be

549

550

shown that there exists a positive lower bound of the energy of emitted photons or gluons. Therefore, the soft divergence is removed. Then, some regular basis of the solution space of the field equation is introduced. The second quantization is based on this regular basis. With this quantization, both kinds of the infrared divergences are expected to be removed.

2.

Consider some physical event. An electron emits a photon. Let At denote the time interval for an electron to emit this photon. Hypothetically, provided that the interval At is fixed, we imagine that this particular event repeats 2n times. The energies of this photon in these 2n-time emissions might not be the same because of the uncertainty principle. Assume that the energies of this photon are el, E ~ , . . . , E ~ ~ for these 2n-time emissions respectively. From the uncertainty principle, we have AEAt 2 h/2 , where AE is the uncertainty of energy. For simplicity, we assume that At = 1 unit. Then AE 2 h/2 , in the numerical value. Though it is assumed to be a hypothetical phenomenon, it is not excluded in real cases. Let b=AE.Let

A Positive Lower Bound of the Energy of Massless Particles

Clearly, E , which is the mean value of E, 's , is more meaninghl than any E,

when AE is the same order as B . By the definition of AE ,

= b2.

Now, we are going to find the minimum of B subjected to constraint (2) and the constraints E, > 0 , i = 1,2, ..., 2n. With a large number of variables and constraints, the nonlinear mathematical programming is troublesome. In order to avoid the mathematical difficulty mentioned above, we assume that E, 2 0. Let si = x,' . The constraints, E, 2 0 , are satisfied automatically. Therefore, (1) becomes

and (2) becomes

551

Consequently, equation (4) is the only constraint left. The work is to minimize E subjected to the constraint (All)' = b2 . By the method of Lagrange multipli- ers, let

E = E - R ( ( A E ) 2 4 2 ) .

Taking the partial derivative of are obtained.

with respect to x, , the following equations

aE/&, = 2xk/ (2n)-4 i l x i / ( 2 n ) - c x f X, 4n') = 0 , ( ( : ) I ( 1 that is,

From the natural phenomenon, it is reasonable to assume that the distribution of E, 's is statistically symmetric, that is, the number of E, 's, E, > , is the same as that of E, 's, E, < E . Therefore, the values of R and are independent of n , which must go to infinity in the sense of statistics. One of the solutions of (5) and (4) is

0, k = l , 2 )..., n.

xk = {@, k = n + l , n + 2, ..., 2n.

The other solutions are obtained by reordering of x, 's. The minimum of E is b , and b is the lower bound of all E 's.

3.

Without the second quantization, QFT will lead to some serious difficulties, such as the negative energy problem. The second quantization is based on the Fourier expansion of the field variables. In speaking of the Fourier expansion, it is nothing but finding an orthogonal basis or an orthonormal basis, and expanding the field variables in terms of this basis [ l ] . In the continuous type, the orthonormal- ity is replaced by independency. It is well known that there are infinitely many

Regular Bases of the Solution Space of the Field Equation

552

admissible bases in a vector space. Therefore, it is possible to find a desired basis, a regular basis, in QFT. For simplicity, we take the Maxwell field with the Coulomb gauge as an example. Usually, the second quantization of the Maxwell field is based on the basis

(s) i(k.x-mf) (s) -i(k.x-of)

{ q k "' l k , s '

with some appropriate coefficients in order to fit some requirements, such as

and

where qt) is the linear polarization vector, and N,? = uL((")a;) is the occupation number operator [ 11. From the viewpoint of mathematics, the set

(s) i,&(k.x-ar) (s) -i&(k.x-or)

3akqk 1k.s A = { akt)k

is a basis of the solution space of the field equation if w = lklc, where the constants ak and PI, are independent of space and time. Obviously, there are many ak and pk which can meet the requirements of (7) and (8). One pair of them is

pk =(A~/(mc~))"~ and ak =(l/&)(h'o/(mc2)Y2)c,

where m is the mass of the Dirac field which associates with the Maxwell field. mc2 can be replaced by k,T , where k, is the Boltzmann constant and T is the temperature. This replacement might make the basis more meaningful.

It is obvious that each member of A is a solution of the Maxwell equation. It will be shown that A is a set of independent vectors. Therefore, combined with the annihilator .;I"' and the creator u:('), the Maxwell field is quantized. Clearly, the infrared singularities are removed.

4. Conclusion and Discussion

Physics is one of the basic sciences. Since it is so basic and fundamental, most people expect that it must be very simple. Yes, the equations of Dirac, Maxwell, etc., are very simple in the sense of mathematics. Compared to the field equation or its related Lagrangian density, the renormalization theory is too complicated. It is the renormalization theory that makes QFT bulky.

553

Now, by introducing the thermal factor and the regular basis, the bulky QFT might be simplified since all divergences are removed without applying the renormalization theory. We hope the final version of QFT might be more successful and simpler. Furthermore, there might be a new approach to design some experiments to test whether the new QFT is wrong or right since the new QFT predicts that most phenomena of QFT depend on the temperature T .

Appendix

If we can prove that the set A is an independent set, then it is a candidate for the basis of the solution space of the Maxwell equation. Now, we are going to prove that the set A is an independent set. It is well known that the set

A, = (eAf,eh',...,e4']

is a set of independent vectors if all A, are different. With the generalization of this result, it is easy to prove that A is an independent set. Consider any finite subset of A ,

A, = ( alerhki.X ,aze18iki'x Y..., a n e'8nk'x)

where all k, are different, and for simplicity, the linear polarization vectors qt) have been omitted. Let all the components of the momentum in A, be mapped on the real line Iwl . Let s' and M' be the minimum and the maximum, respectively, of the set whose members are the distances between all pairs of distinct points mapped on W' . Let S = min (8 , l ) and M = M' + 1 . Let u = M / S . Let

Clearly, all members of A, are eigenvectors of D and their eigenvalues are distinct. Therefore, A, is an independent set, and so is A . Consequently, A can span the solution space of the Maxwell equation.

References

1. D. Lurie, Particles andfields, Wiley, New York (1968). 2. Yeong-Shyeong Tsai, Hung-Ming Tsai, Po-Yu Tsai and Lu-Hsing Tsai,

"The solution of the Dirac equation with the interaction term," in Proceed- ings of the 3rd International Symposium on Quantum Theory and Symme- tries (QTS3), Cincinnati, Ohio, 10-14 September 2003, World Scientific, Singapore (2004), pp. 43 1-436.

3. T. Muta, Foundations of Quantum Chromodynamics, 2nd ed., World Scien- tific, Singapore (1998).

ELECTROMAGNETIC FIELDS I N STATIONARY CYCLIC SYMMETRIC 2+1 GRAVITY

ELOY AYON-BEATO Centro de Estudios CientGcos (CECS)

Casilla 1469, Valdivia, Chile. E-mail: [email protected]

MAURICIO CATALDO Departamento de Fisica, Facultad de Ciencias, Universidad del Bio-Bio,

Avenida Collao 1202, Casilla 5-C, Concepcidn, Chile. E-mail: mcataldoQubiobio. cl

ALBERT0 A. GARCiA Departamento de Fisica.

Centro de Investigacidn y de Estudaos Avantados del IPN. Apdo. Postal 14-740, 07000 Mkxico DF, MEXICO.

E-mail: aagarciaaj7.s. cinvestav.mx

It is established the general structure of the Maxwell electromagnetic field for a 2+1 spacetime endowed with stationary and cyclic symmetries. The family of possible fields splits in two disjoint classes. The general solution for c # 0 and its limiting cases are explicitly exhibited.

1. Introduction

During the last two decades three-dimensional gravity has received much attention, in particular, in topics such as: search of exact solutions, black hole physics, quantization of fields coupled to gravity, cosmology, topological aspects, and so on. The list of references on this respect is extremely vast. This interest in part has been motivated by the discovery , in 1992, of the (2+l ) stationary circularly symmetric black hole solution which possesses certain features inherent to (3+1) black holes. On the other hand, it is believed that (2+1) gravity may provide new insights towards a better understanding of the physics of (3+1) gravity.

The purpose of this contribution is to provide a new approach on the

554

555

search of electromagnetic -gravitational solutions, allowing for stationary and cyclic symmetries, to the Einstein-Maxwell fields of the (2+1) gravity in the presence of a cosmological constant.

In Sec. 2, a theorem for stationary cyclic metrics coupled to the electromagnetic field *F = adt + bd+ + c g , , d r / f i is stated. In Sec.3, the general solution for F = c g , , d r / f i is exhibited; its sub- branches are presented.

The outline of the present paper is as follows:

2. Electromagnetic field for stationary and cyclic symmetric 2+1 spacetime; a theorem

In this section the demonstration of a theorem for (2+1) solutions is oulined. To start with, we consider a stationary cyclic symmetric spacetime with

signature (-,+,+), i.e., a space endowed stationary symmetry k = &,k .k < 0, such that x k g = 0, and cyclic symmetry m = 84, m . m > 0, such that L‘,g = 0, with closed integral curves from 0 to 27r, which in turn commute [k,rn] = 0. Hence the Killing vector fields k and m generate the group SO(2) x R. The electromagnetic field, described by the antisymmetric tensor field F = $F,,dx@ Adz” , is assumed to be stationary cyclic symmetric, i.e., LkF = 0 = €,F. It should be pointed out that, in contrast to the general 3+1 stationary cyclic symmetric spacetime case, any 2+1 stationary cyclic symmetric spacetime is necessarily circular, i.e., the circularity conditions

k A m A dk = 0 = k A m A d m (1)

are identically fulfilled because of their 4-form character. There exit the discrete symmetry when simultaneously t 4 -t and C#J -+ -6. One may find a coordinate system such that the metric tensor components g(k dr ) = 0 and g ( m d r ) = 0, where the coordinate direction d r is orthogonal to the surface spanned by k A m. Commonly one introduces in 2+1 gravity the coordinate system {t , 4, r } .

Theorem A: The general form of stationary cyclic symmetric electromagnetic fields in 2+1 dimensions is given by

(2) *F = adt + bd4 + c z d r , g T T

where the constants a, b and c are subjected, by virtue of the Ricci circu-

556

larity conditions, to the equations

(3) a c = 0 = bc

which give rise to two disjoint branches

c # 0 , *F = c z d r , Srr

and

c = 0, *F = adt + bdq5,

(4)

(5)

with its own subclasses a = 0 or b = 0.

uses the source-free Maxwell equations To establish that the field * F possesses the form given by Eq. (2) one

d F = 0 = d * F (6)

where * denotes the Hodge star operation.

one uses the Ricci circularity conditions To derive the vanishing conditions fulfilled by the constants a, b and c

1 m A k A R ( k ) = - d t ( m A k A d k ) z O + m A k A T ( k ) = 0 ,

2 1

k A m A R(m) = - d t ( k A m A d m ) = 0 + k A m A T(m) = 0 , (7) 2

where dt := (-1)’ * d*. The implied results arise from the use of the Einstein equations R,, - +Rg,, = KT,,, where the electromagnetic energy momentum tensor components are defined through the electromagnetic field F,, = -F,, as 4nT,, = F,,FVu - ~g,,FaoFau, arriving in this way at the vanishing conditions for the constants, which on their turn, give rise to the sub-branches of possible solutions.

3. General solution for electromagnetic field c # 0 = a = b.

For the stationary cyclic metric

with an electromagnetic field *F = c%dr, the general solution is given by the structural functions:

(9) 4 12

F ( r ) = - ( r - q ) ( r - T Z ) ,

557

r - r1

r - 1-2 f ( r ) := -,

with the constant c related with r1 and 7-2 through

(1 - a). (13) 2 (r2 -

14 c =

3.1. Uncharged limit; the BTZ solution

For vanishing electromagnetic field, i. e., for constant c = 0 4 a = 1 the metric (2) can be brought to the BTZ solution by accomplishing the coordinate transformations

( r l - r 2 K : J 2 + 2 - ( r 2 - r I ) d 2 ) , Ki (14c) 1

r = 1-K;J2

where with prime, ', we denote the corresponding BTZ coordinates, which ought to be accompanied with the identification

~ + K ; J ~ M = 21K1 '

3.2. Static limit; Cataldo electromagnetic solution

A second possible limit of the studied solution arises in the static case ,i.e.) for vanishing parameter J, which gives rise to the branch of electromagnetic static (2+1) solutions. The static form of the studied metric is given by

F 1 g = - y d t 2 + Ha2d4l2 + -dr2,

Hs F

558

with 4 1 F(T) = T ( r - T l ) ( T - T Z ) ,

H, 2 = H ( r , J = 0) 2 = 1

(T - TI) ( 1 + f i ) / 2 ( T - r2)(1-fi)/2

(17) 2Kl (T2 - T1) fi

which, via the coordinate transformations, can be brought to the form

It corresponds to the static charged solution reported in Ref. 2.

Acknowledgments

This work has been partially supported by: FONDECYT Grant 1040921 and Fundaci6n Andes Grant D-13775 (EAB), CONICYT grant No 7010485 and FONDECYT No 1010485 (MC), CONACyT grant 38495-E (AAG), and CONICYT/CONACyT Grant 2001-5-02-159 .

References 1. M. Baiiados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849. 2. M. Cataldo, Phys. Lett. €3529, 143 (2002).

Photographs and List of Participants


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ATTENDEES

Julio Abad (Universidad de Zaragoaa) Carl Albright (Northern Illinois U. and Fermilab) Roland Allen (Texas A&M University) George Alverson (Northeastern University) Luis Anchordoqui (Northeastern University) Alexey Anisimov (Max Planck Institute for Physics) Stefan Antusch (University of Southampton) Nima Arkani-Hamed (Barvard University) Richard Arnowitt (Texas A&M) Howard Baer (Florida State University) Alexander Barabanschikov (Northeastern University) Emanuela Barberis (Northeastern University) Vernon Barger (University of Wisconsin) Alfred Bart1 (University of Vienna) Sergei Bashinsky (Princeton University) Antonio Bassetto (Dept. of Physics PADUA) Rachel Bean (Princeton/WMAP) Katherine Benson (Emory University) Rita Bernabei (Univ. Roma “Tor Vergata”/INFN-&ma2) Pushpalatha Bhat (Fermilab) Tirthabir Biswas (McGill University) Matt Bouchard (Northeastern University) Gustavo C. Branco (Instituto Sup. Tecnico) Robert Brandenberger (Brown University) Dmitri Burshtyn (&ah Institute, Hebrew University of Jerusalem) Freddy Cacahazo (Institute for Advanced Study) Kanokkuan Chaicherd- sakul (UT Austin) Ali Chamseddine (American University of Beirut/CAMS) Max Chaves (Universidad de Costa Rica) Wenfeng Chen (NCTS, Taiwan) Roberto Chierici (CERN) Marco Cirelli (Yale University) Louis Clavelli (University of Alabama) David Cline (UCLA) Hael Collins (Carnegie Mellon University) Fred Cooper (National Science Foundation) Ruben Corderc+Elizalde (ESFM-Instituto Politecnico Nacional, MEXICO)

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Mirjam Cvetic (University of Pennsylvania) Thomas Dent (University of Ioannina) Michael Dine (UC Santa Cruz) Emilian Dudas (Orsay) Leanne Duffy (University of Florida) Bhaskar Dutta (University of Regina) Damien Easson (Syracuse University) Marco Fabbrichesi (INFN/SISSA) Malcolm Fairbairn (Universite' Libre de Bruxelles) Cory Fantasia (Northeastern University) Daniel Feldman (Northeastern University) Sergio Ferrara (CERN) Roberto Floreanini (INFN - Sezione di Trieste) Kristen Flowers (Northeastern University) Paul Frampton (UNC-Chapel Hill) Filipe Freire (Leiden University) Colin Froggatt (Glasgow) Mary K. Gaillard (U C Berkeley) A. A. Garcia (CINVESTAV-IPN) James Gates (University of Maryland) Margaret Geller (Smithmnian Astrophysical Observatory) Chiara Genta (INFN and University of Florence) Chamkaur Ghag (University of Edinburgh) Bairn Goldberg (Northeastern University) Mario E. Gomez (Universidad de Huelva) Javier Gonzalez (Northeastern University) Rajan Gupta (Los Alamos National Lab) Gerald Guralnik (Brown University) Alan Guth (MIT) Nicholas Hadley (The University of Maryland) Benjamin Harms (University of AIabama) Takayuki Hirayama (University of Toronto) William Hogan (Northeastern University Physics alumnus) Deog Ki Hong (Pusan National University) Greg Huey (UIUC) Tarek Ibrahim (Northeastern University) Ian Jack (University of Liverpool) Roman Jackiw (MIT) Kenji Kadota (Fermilab)

571

Kyungsik Kang (Brown University) Alexander Kaganovich (Ben Gurion University) Thomas Kephart (Vanderbilt University) Shaaban Khalil (Durham University) Peyman Khorsand (Northeastern University) Steve King (University of Southampton) Hans Volker Klapdor-Kleingrothaus (MPI Fiir Kernphysik) Boris Kors (MIT) Alan Kostelecky (Indiana University) Takahiro Kubota (Department of Physics, Osaka University) Stefan0 Lacaprara (INFN & Padova University) Daniel Larson (UC Berkeley & LBNL) Albion Lawrence (Brandeis University) Alfredo Macias (Universidad Autonoma Metropolitana-Iztapalapa) K.T. Mahanthappa (University of Colorado) Rupak Mahapatra (Univ. of California, Santa Barbara) Rahul Malhotra (University of Texas at Austin) Michal Malinsky (SISSA, Trieste, Italy) J& Maneira (Queen’s University) William Marciano (Brookhaven) Antonio L. Maroto (Universidad Complutense de Madrid) Stephen Martin (Northern Illinois University and Fermilab) Philip Maurone (Northeastern University Physics alumnus) Thomas McCauley (Northeastern University) Bob McElrath (UC Davis) Maggie McKee (New Scientist magazine) Merlin Miller (Northeastern University Physics alumnus) Abdulaziz Mohammad (Tufts University) b b i n d r a Mohapatra (University of Maryland) Take0 Moroi (Tohoku) Stephen Morris (Perimeter Institute) Satya Nandi (Oklahoma State University) Pran Nath (Northeastern University) Hans Peter Nilles (Universitat Bonn) Carlos Nunez (MIT) Megha Padi (Harvard University) Sandip Pakvasa (University of Hawaii) Eleftherios Papantonopoulos (National Technical University of Athens) W a n 4 Park (Korea Advanced Institute of Science and Technology)

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Stefan0 Passaggio (INFN - Sezione di Genova) Jogesh Pati (University of Maryland) Thomas Paul (Northeastern University) Apostolos Pilaftsis (University of Manchester) Peter Polito (Springfield College) Paul Pond (Northeastern University Physics alumnus) Anastasios Psinas (Northeastern University) Mariano Quiros (IFAE) Erik Ramberg (Fermilab) Rudnei Ramos (Rio de Janeiro State University) Eugenie Reich (New Scientist) Efrain Rojas-Marcial (Universidad Veracruzana, MEXICO) Durga Prasad Roy (Tata Institute for Fundamental Research) Gerard Sajot (LPSC-Universite Grenoble 1) Norisuke Sakai (Tokyo Instute of Technology) Stefan Schlenstedt (DESY Zeuthen) Markus Schumacher (Physikalisches Institut , Universitat Bonn) Delia Schwartz Perlov (Tufts University) Jihye Seo (Harvard University) Ilya Shapiro (Universidade Federal de Juiz de Fora) M. Shifman (University of Minnesota) Masatoshi Shoji (University of Nebraska-Lincoln) Ernst Sichtermann (LBNL) Pierre Sikivie (University of Florida) Arunansu Sil (Bartol Research Institute, University of Delaware) Joseph Silk (University of Oxford) Eva Silverstein (SLAC, Stanford) George Siopsis (University of Tennessee) Constantinos Skordis (University of Oxford) Lee Smolin (Perimeter Institute) Yogi Srivastava (Northeastern University) Ion Stancu (University of Alabama) Shankaranarayanan Subramaniam (Abdus Salam ICTP, Trieste) Richard Suitor (Northeastern University Physics alumnus) Shiro Suzuki (Saga University) John Swain (Northeastern University) Raza Syed (Notheastern University) Torno Takahashi (University of Tokyo) Gregory Tar16 (University of Michigan)

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Xerxes Tata (University of Hawaii) Tomasz Taylor (Northeastern University) Nikolai Tolich (Stanford University) Hung-Ming Tsai (National Taiwan University) Dmitri Tsybychev (SUNY at Stony Brook) Rishikesh Vaidya (National Central University Taiwan) Michael Vaughn (Northeastern University) Alex Vilenkin (Tufts University) Bryant Villeponteau (Arogen Sciences) Carlos Wagner (Argonne) Ting Wang (University of Michigan) Stephanie Ward (Northeastern University) Scott Watson (Brown University) Tom Weiler (Vanderbilt Uaiversity) Steven Weinberg ( U T Austin) Darien Wood (,dortheastern University) Masahiro Yamagucbi (Sendai) Masahide Yamaguchi (Aoyama Gakuin University) Un-ki Yang (University of Chicago) Bruno Zumino (U C Berkeley) Barton Zwiebach (MIT)

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Particles, Strings and Cosmology PASCOS 2004 (589 pages)

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