Introduction —The Past and Future of Elementary Particle Physics— Sachio KOMAMIYA and Yoshitaka KUNO 1 Department of Physics, The University of Tokyo, Tokyo 113-0033 1 Department of Physics, Osaka University, Osaka 560-0043 (Received May 12, 2007; accepted May 29, 2007; published November 12, 2007) Near the end of the last century, remarkable progress was made in elementary particle physics both experimentally and theoretically, which led to the development of the Standard Model. However, we consider the model as a realization of a more fundamental theory at low energies, since there are so many unexplained physics in the model. Now, elementary particle physics is entering an exciting period in which a new paradigm of the field will be opened by the new discoveries expected in experiments at high-energy frontier colliders. The Higgs boson and supersymmetry are the main targets in these experiments. In parallel to energy frontier physics, experimental and theoretical studies attack the mystery of quark and lepton flavors. The combined efforts in cosmology and particle physics give synergy effects in understanding the history and the current state of the Universe. KEYWORDS: the Standard Model, Higgs boson, supersymmetry, grand unification, LHC, ILC, flavor physics DOI: 10.1143/JPSJ.76.111001 1. A Brief History and Present Status of Elementary Particle Physics We, particle physicists, are entering an exciting period in which a new paradigm of the field will be opened on the TeV energy scale by the new discoveries expected in experiments at high-energy frontier colliders. Namely, the long-awaited discoveries of the Higgs boson and supersymmetric particles, or some new particles or phenomena alternative to these, have been strongly anticipated in the experiments at the Large Hadron Collider (LHC). 1,2) The LHC, the highest- energy proton–proton colliding accelerator at CERN, will start taking data in 2008 at the full center-of-mass energy of 14 TeV. The underlying principles of the new physics are expected to be uncovered by precision measurements at the International Linear Collider (ILC), which is the highest- energy electron–positron (e þ e ) colliding accelerator sched- uled to be constructed via international efforts in the 2010s. 3,4) In the beginning of the last century, quantum mechanics and the relativity were established. These became the bases of elementary particle physics. In the 1920s, only the electron, proton, and photon were the elementary particles. In the 1930s and 40s, the energy frontier experiments used cosmic rays. The antiparticle was predicted by Dirac, 5) and the positron was discovered by Anderson in the cloud chamber exposed to cosmic rays. 6) In 1935, Yukawa predicted existence and the mass of %-mesons as intermedi- ate bosons for the short-range nuclear force. 7) Efforts to discover Yukawa’s %-mesons was made in cosmic ray experiments, and a new lepton, the muon, was accidentally discovered in 1937. Eventually Yukawa’s pion was discov- ered in 1947 also in cosmic rays. Strongly interacting particles, mesons and baryons and their antiparticles, are generally called hadrons. New type of hadrons with the strangeness quantum number were also discovered in 1947 in cosmic ray. After World War II, high-energy accelerators with beam energies over 10 GeV were constructed, and dozens of strongly interacting hadrons were discovered. To classify these hadrons, the quark model was proposed by Gell-Mann and Zweig in 1964. 8) There were only three quarks (up, down and strange) in the initial quark model. The mixing of (quark) flavors was proposed by Cabbibo before the quark model was proposed. 9) The experimental evidence of the quarks was discovered as constituents of nucleons by electron deep-inelastic scattering with nucleons in the early 1970s. At the time, four types of leptons, the electron, muon, electron–neutrino and muon-neutrino, were known to exist. If the quarks are considered to be elementary particles that are on the same level as leptons, the fourth quark, the charm, must have been missing. Near the end of the twentieth century, remarkable progress was made in particle physics both experimentally and theoretically, which led to the development of the Standard Model. Rapid progress was triggered by the so- called November Revolution in 1974, which marked the discovery of the J = particle or the charm quark. 10) This discovery clarified the direction of particle physics towards the establishment of the Standard Model. The time was ripe for the revolutionary era, since the theoretical framework of the electroweak theory had already been established by then. 11) The strong interaction was formulated as Quantum Chromo Dynamics (QCD), which is a non-Abelian gauge theory based on the SU(3) gauge group. 12) In these theories interactions are mediated by spin 1 gauge bosons. The mediator of the strong interaction, the gluon, was directly discovered in 1979 at the PETRA e þ e collider. The weak bosons, W and Z 0 , were discovered in 1982 at the CERN p p collider. The existence of the third-generation quarks was predicted by Kobayashi and Maskawa in 1973, 13) before the discovery of the charm quark. Their prediction is based on the argument that the CP violation observed in neutral kaon decay 14) is due to the complex phase in the quark mixing matrix, and three quark generations are needed to have such a phase. The third-generation lepton ( was discovered in SPECIAL TOPICS Journal of the Physical Society of Japan Vol. 76, No. 11, November, 2007, 111001 #2007 The Physical Society of Japan Frontiers of Elementary Particle Physics, the Standard Model and Beyond 111001-1
Transcript
Introduction
—The Past and Future of Elementary Particle Physics—
Sachio KOMAMIYA and Yoshitaka KUNO1
Department of Physics, The University of Tokyo, Tokyo 113-00331Department of Physics, Osaka University, Osaka 560-0043
(Received May 12, 2007; accepted May 29, 2007; published November 12, 2007)
Near the end of the last century, remarkable progress was made in elementary particle physics bothexperimentally and theoretically, which led to the development of the Standard Model. However, weconsider the model as a realization of a more fundamental theory at low energies, since there are so manyunexplained physics in the model. Now, elementary particle physics is entering an exciting period inwhich a new paradigm of the field will be opened by the new discoveries expected in experiments athigh-energy frontier colliders. The Higgs boson and supersymmetry are the main targets in theseexperiments. In parallel to energy frontier physics, experimental and theoretical studies attack themystery of quark and lepton flavors. The combined efforts in cosmology and particle physics givesynergy effects in understanding the history and the current state of the Universe.
KEYWORDS: the Standard Model, Higgs boson, supersymmetry, grand unification, LHC, ILC, flavor physicsDOI: 10.1143/JPSJ.76.111001
1. A Brief History and Present Status of ElementaryParticle Physics
We, particle physicists, are entering an exciting period inwhich a new paradigm of the field will be opened on the TeVenergy scale by the new discoveries expected in experimentsat high-energy frontier colliders. Namely, the long-awaiteddiscoveries of the Higgs boson and supersymmetric particles,or some new particles or phenomena alternative to these,have been strongly anticipated in the experiments at theLarge Hadron Collider (LHC).1,2) The LHC, the highest-energy proton–proton colliding accelerator at CERN, willstart taking data in 2008 at the full center-of-mass energy of14 TeV. The underlying principles of the new physics areexpected to be uncovered by precision measurements at theInternational Linear Collider (ILC), which is the highest-energy electron–positron (eþe�) colliding accelerator sched-uled to be constructed via international efforts in the 2010s.3,4)
In the beginning of the last century, quantum mechanicsand the relativity were established. These became the basesof elementary particle physics. In the 1920s, only theelectron, proton, and photon were the elementary particles.In the 1930s and 40s, the energy frontier experiments usedcosmic rays. The antiparticle was predicted by Dirac,5) andthe positron was discovered by Anderson in the cloudchamber exposed to cosmic rays.6) In 1935, Yukawapredicted existence and the mass of �-mesons as intermedi-ate bosons for the short-range nuclear force.7) Efforts todiscover Yukawa’s �-mesons was made in cosmic rayexperiments, and a new lepton, the muon, was accidentallydiscovered in 1937. Eventually Yukawa’s pion was discov-ered in 1947 also in cosmic rays. Strongly interactingparticles, mesons and baryons and their antiparticles, aregenerally called hadrons. New type of hadrons with thestrangeness quantum number were also discovered in 1947in cosmic ray.
After World War II, high-energy accelerators with beamenergies over 10 GeV were constructed, and dozens of
strongly interacting hadrons were discovered. To classifythese hadrons, the quark model was proposed by Gell-Mannand Zweig in 1964.8) There were only three quarks (up,down and strange) in the initial quark model. The mixing of(quark) flavors was proposed by Cabbibo before the quarkmodel was proposed.9) The experimental evidence of thequarks was discovered as constituents of nucleons byelectron deep-inelastic scattering with nucleons in the early1970s. At the time, four types of leptons, the electron, muon,electron–neutrino and muon-neutrino, were known to exist.If the quarks are considered to be elementary particles thatare on the same level as leptons, the fourth quark, the charm,must have been missing.
Near the end of the twentieth century, remarkableprogress was made in particle physics both experimentallyand theoretically, which led to the development of theStandard Model. Rapid progress was triggered by the so-called November Revolution in 1974, which marked thediscovery of the J= particle or the charm quark.10) Thisdiscovery clarified the direction of particle physics towardsthe establishment of the Standard Model. The time was ripefor the revolutionary era, since the theoretical framework ofthe electroweak theory had already been established bythen.11) The strong interaction was formulated as QuantumChromo Dynamics (QCD), which is a non-Abelian gaugetheory based on the SU(3) gauge group.12) In these theoriesinteractions are mediated by spin 1 gauge bosons. Themediator of the strong interaction, the gluon, was directlydiscovered in 1979 at the PETRA eþe� collider. The weakbosons, W� and Z0, were discovered in 1982 at the CERNp �pp collider.
The existence of the third-generation quarks was predictedby Kobayashi and Maskawa in 1973,13) before the discoveryof the charm quark. Their prediction is based on theargument that the CP violation observed in neutral kaondecay14) is due to the complex phase in the quark mixingmatrix, and three quark generations are needed to have sucha phase. The third-generation lepton � was discovered in
SPECIAL TOPICSJournal of the Physical Society of Japan
Vol. 76, No. 11, November, 2007, 111001
#2007 The Physical Society of Japan
Frontiers of Elementary Particle Physics, the Standard Model and Beyond
1975 at the Stanford Linear Accelerator Center (SLAC) justafter the discovery of the charm quark.15) For the third-generation quarks, the bottom quark was discovered in 1977in the fixed target experiment at Fermilab,16) and finally thetop quark was discovered in 1994 at the Tevatron p �pp
collider.17) The number of the quark and lepton generationswas determined to be three from the precise measurementsof the cross section on and near the Z0 peak at the LEP andSLC eþe� colliders.18,19) The origin of the three generationsis not yet theoretically understood.
In 1998, neutrino oscillation was discovered in theanalysis of atmospheric neutrinos at SuperKamiokande.20)
This is the first evidence that the neutrinos have tiny butnonzero masses. The neutrino oscillation was also observedin solar neutrinos and reactor neutrinos. The mixing patternof neutrino flavors is measured to be very different from thatof quark flavors. The smallness of neutrino masses is the hintfor a new area of physics on the large-energy scale of 1011–1016 GeV.
Recently, CP violation, which have been observed only inthe neutral K meson decays, was discovered in the bottomhadron sector at the B factories in KEK and SLAC.21)
The Standard Model is made up of three main pillars: (1)quarks and leptons and their antiparticles are the fundamen-tal constituents of matter, (2) their interactions (electro-magnetic, weak and strong) are governed by the gaugeprinciple, and (3) the origin of the mechanism for generatingthe masses of elementary particles is the Higgs boson (seeTable I).
The Standard Model has been tested through manyexperiments with high precision, and its success has becomeincreasingly firm. Thus far, all the quarks and leptons of 12species in three generations have been discovered. If quarksand leptons were not point like elementary particles and theywere made up of much smaller constituents, the effectswould be observed at high-energy colliders. At present, theexperimental upper limits of the size of quarks and leptonsare on the order of 10�19 m, and they are still considered tobe elementary.19)
Nevertheless, the Higgs boson has not been discoveredyet. Thus, one of the pillars of the Standard Model is stillmissing and awaits experimental confirmation. In addition,the Standard Model does not include gravity in its frame-work. Since the common origin of these interactions is notknown, their unification is out of the question within theStandard Model. Furthermore, it cannot explain why thereare three generations and twelve kinds of quarks and leptons,
why they have different masses, and why mixing betweendifferent generations occurs. Hence, there are so manyunexplained free parameters, that should be determinedexperimentally in the model. It is obvious that the StandardModel is not the ultimate theory of particle physics;therefore we consider the model as a realization of morefundamental theory at low energies.
2. Particle Physics at Highest Energy Frontier
2.1 Electron–positron colliders and hadron collidersThe evolution of elementary particle physics is stimulated
by the development of accelerator technologies. Theconventional high-energy physics experiments are fixed-target experiments, in which accelerated beams are smashedon to a stationary target. The reaction energy of the collision,which can be used for new particle production, is given byffiffiffiffiffiffiffiffiffiffi
2EMp
at high energies, where M is the target particle massand E is the beam particle energy. The advent of collidingbeam accelerators (colliders) brought a drastic increase inreaction energy. For collider experiments the two beams areaccelerated in opposite directions and the beams collide witheach other. A particle detector surrounds the beam collisionpoint with an almost 4� solid angle coverage. In this casethe reaction energy is 2E for the head-on collision of twoparticles with the same mass m and the same energy E,hence the energy efficiency is significantly higher than thatfor fixed target experiments. The initial colliders wereconstructed in the late 1960s. After the 1970s, most of thenew particles, including J= , the �-lepton, gluons, W- andZ-bosons and the top quark, were discovered in colliderexperiments at the highest energy frontier. There are twotypes of colliders: eþe� colliders and hadron colliders(proton–proton or proton–antiproton).
Experiments at eþe� colliders surpass those at hadroncolliders in terms of simplicity of the processes andcleanness of the experimental environment. An electronand a positron annihilate each other, so that all the reactionenergy can be efficiently used for new particle production.The event rate as well as background level are relatively lowin eþe� collider experiments. Since electrons and positronsdo not interact strongly with matter, the experimentalenvironment is clean.
Meanwhile, protons and antiprotons are composite par-ticles made up of three quarks bounded by gluons; hencetheir interactions are complicated. The fundamental inter-actions take place as a collision of partons (quarks orgluons), so that only a fraction of the total energy 2E can beused for new particle production and the reaction energyvaries for each event. Also experiments at proton–proton(proton–antiproton) colliders suffer from a high backgroundrate and a high radiation level. In order to select a fewinteresting events in the high background, a sophisticatedtriggering system must work in a short decision time. Newparticles with distinctive features can only be searched for athadron colliders.
However, in bending the electron or positron beams ina circular collider with a dipole magnetic field, the beamparticles loose their energy by emitting synchrotronradiation. At high energies the energy loss by the synchro-tron radiation per one turn in a circular accelerator (�E) isgiven by the formula �E ¼ ð4�=3Þe2ðE=mÞ4=R, where E is
Table I. List of all the elementary particles in the Standard Model. Each
quark has three color degree-of-freedoms, as a gluon has eight color
combinations.
SpinElementary particles
Leptons Quarks
Generation 1 2 3 1 2 3
Matter 1/2 �e �� �� u c t
fermions e� �� �� d s b
Interactions EM Weak Strong
Gauge bosons 1 � W� Z0 gi (i ¼ 1{8)
Higgs boson 0 H0
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the beam particle energy, m is the beam particle mass, and R
is the bending radius of the particle in the magnetic field.The electric power used for the acceleration is proportionalto ðE=mÞ4=R2. In any case, emission of synchrotron radiationis a serious drawback for realizing higher-energy eþe�
circular colliders beyond the LEP energies.There are two remedies for this problem. One of them is
to give up eþe� collision and to use a proton as the beamparticle. Protons are about 1800 times heavier than electronsor positrons, so that the energy loss per turn caused bysynchrotron radiation can be reduced by a factor ofðme=mpÞ4. Since higher energy is essential to creating newheavy particles, it is worth sacrificing the cleanness of theexperiment. In fact, the proton collider LHC uses the sametunnel that was previously used for the eþe� collider LEP.The center-of-mass energy is increased by a factor of 70from that for LEP. The layout of the LHC acceleratorcomplex is shown in Fig. 1, and the inside of the LHCaccelerator tunnel is shown in Fig. 2.
Since eþe� collisions have so many advantages asdescribed above, we should not simply give it up. Therefore,the other remedy is to set the bending radius R to infinity, sothat no synchrotron radiation can be emitted. The conceptof the eþe� linear collider was proposed along this line.In the linear collider, electron and positron beams areaccelerated face-to-face in two separate linear accelerators,
and they collide with each other in the experimental halllocated at the center of the accelerator system. Circularcolliders have an advantage that particles can be acceleratedgradually when they circulate and pass the same acceleratorcavities. For linear colliders the acceleration is one-shot,which is a serious disadvantage. Therefore, the acceleratinggradient in the linear accelerator must be very high,otherwise the length of the collider becomes too long andhence the cost becomes too high. The other technical issue isto focus the beams into a very small size at the collisionpoint in order to achieve large luminosity. Technicaldevelopment of eþe� linear colliders began started in the1980s. Now, the linear collider has become an internationalproject. In 2004, ICFA (International Committee for FutureAccelerators) decided that the superconducting acceleratorcavities be used for the linear collider, and named the projectILC (International Linear Collider). The layout of the ILCaccelerator tunnels is shown in Fig. 3.
In 2008, the LHC experiments will start taking data witha center-of-mass energy of 14 TeV and will survey newphysics on the TeV scale. The ILC experiments will thenperform precision measurements in the clean environmentof eþe� collisions to reveal physics principles behind thenew phenomena observed at the LHC. Experiments at theLHC and ILC are expected to establish a new paradigm ofparticle physics, which will be as important as formerbreakthroughs such as the discovery of antiparticles or theestablishment of the gauge principle. The most importantphysics expected to be discovered in the near future aregiven below.
2.2 Higgs bosonsAll elementary particles in the Standard Model should be
massless if gauge symmetries are not broken. The Higgsboson breaks the gauge symmetries and give mass toelementary particles.22) The weak gauge bosons (W� and Z0)
Fig. 1. Layout of the LHC accelerator at CERN.
Fig. 2. Inside of the LHC tunnel, formerly used for the LEP eþe� collider.
Superconducting dipole-magnets for the proton beams are installed in the
tunnel.
Fig. 3. Layout of the ILC tunnel. In the left tunnel, the RF system such as
klystrons and its power supplies are installed. This tunnel can be accessed
for maintenance even during machine operation. In the right tunnel,
superconducting accelerator cavities are aligned. The beam is accelerated
in these cavities.
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obtain masses through the Higgs mechanism, which is thedirect consequence of the breakdown of the gauge symme-tries.23) The quarks and leptons obtain masses from theYukawa coupling to the Higgs boson. Past measurementsconstrain the mass of the Higgs boson in the Standard Modelto be between 114 GeV and about 200 GeV. The upper limitof 114 GeV is from the direct Higgs boson searches at LEP,and the lower limit of about 200 GeV is from preciseelectroweak measurements including those from the experi-ments from LEP and Tevatron.19,24) Thus the Higgs bosonpostulated in the Standard Model will be discovered at theLHC.2) It will be the discovery of the elementary scalarparticle. Elucidation of the properties of the Higgs bosonwill be the first step in understanding the structure ofvacuum, the inflation of the Universe and dark energy, forwhich some kind of scalar particle may be responsible. TheILC will be a Higgs boson factory and scrutinize theproduction and decay of the Higgs boson, preciselydetermine its mass, spin, coupling constants for couplingto other elementary particles, and its self-coupling. Throughthese measurements, the ILC will verify that gauge symme-try breaking is the origin of elementary particle masses.4)
Furthermore, there is a possibility of finding the direction inwhich to extend the Standard Model after uncovering thephysics behind the electroweak symmetry breaking.
2.3 SupersymmetrySupersymmetry25) is regarded as the most promising
paradigm beyond the Standard Model. Supersymmetry isan extension of symmetry in the Poincare group, and is asymmetry between Fermion and Boson fields. If super-symmetry exists, every elementary particle has its super-symmetric partner, whose spin differs by 1/2 unit from thatof the corresponding Standard Model particle. The groupof supersymmetric partners can be discovered through theexperiments in the TeV energy scale. There exist thefollowing pieces of indirect evidence for TeV scale super-symmetry: (1) the Higgs boson mass, which is not protectedfor divergence by any symmetry, is kept sufficiently lightwith the advent of supersymmetry, (2) three types ofinteractions, strong, electromagnetic, and weak interactions,are unified at a very high energy scale of 1016 GeV with thesupersymmetry, (3) the lightest supersymmetric particle isa leading candidate for dark matter. Supersymmetry, ifdiscovered, will give an important clue to superstring theory,which is a candidate for the ultimate unified theory thatincludes gravity.
Experiments at the LHC are expected to discoverevidence for supersymmetry at an early stage in datataking.2) If supersymmetric particles are within the reach ofthe ILC, their properties will be understood from precisemeasurements of their production and decays.4) Throughthese measurements, we will clearly know if dark matter canbe identified as the lightest supersymmetric particle, and thisinfluences the strategy for terrestrial dark matter searchexperiments. No supersymmetric partners of the StandardModel particles have far been discovered; hence the super-symmetry must be broken. By combining the informationfrom the LHC and ILC experiments on the mass spectrum ofthe supersymmetric partners, the mechanism of supersym-metry breaking is expected to be studied.
2.4 Something unexpectedAccording to the present understanding of elementary
particle physics, a light Higgs boson should exist. Manyresearchers also expect that supersymmetry will be discov-ered at an energy below 1 TeV. However, future experimentsmay encounter something unexpected. In particular, if wefind no Higgs boson but some new particle or some newphenomenon that is responsible for the electroweak sym-metry breaking or the mass generation, it will open a newfrontier of physics in both theoretical and experimentalstudies.
One such scenario can be realized in models with extradimensions. In superstring theories,26) there must be 6-dimensional space beyond our 4-dimensional space–timerealm. Since effects of the extra dimensions are not observedanywhere, these extra dimensions can be either compactifiedinto a very small size or the Standard Model particles canonly live in our space–time realm of 4 dimensions. Sometheorists are building models with extra dimensions withoutthe ordinary Higgs boson or without TeV-scale supersym-metry. The effects of the extra dimensions might be seenin the high-energy experiments.2,4)
3. Particle Physics at Precision Frontier
Even if the light Higgs boson is discovered andsupersymmetry is experimentally proven to be a newframework of physics, there still remain the followingfundamental questions: Why do quarks and leptonshave three generations and why do they have differentmasses? Why do they mix among different generations(flavor mixing)? Why do they violate the symmetrybetween particles and anti-particles (CP violation)? Whatis the origin of the neutrino masses? Did the neutrinosplay an important role in creating the matter-dominantUniverse? To address these fundamental questions, anexperimental approach at the high-precision frontier can beexplored.
3.1 Quark flavor physicsThe word of ‘‘flavor’’ implies a quantum number to
distinguish different kinds of elementary particles. The 12different kinds of elementary particles have 12 differentflavors. Flavor physics searches for a breakthrough to answersome of the questions mentioned above, in particular onflavor mixing. Here, the flavor mixing causes the transitionof one type of quark to another. Parameters of the flavormixing originate from the Yukawa couplings, which are thestrength of the interactions between the Higgs particle andquarks or leptons. The Yukawa couplings in the StandardModel include many unknown parameters that need to bedetermined experimentally. Precision measurements of themixing parameters are a necessary step towards under-standing of flavor mixing. For quarks,27) these parametersare presented in a 3� 3 unitary matrix, which is called theCabbibo–Kobayashi–Maskawa (CKM) matrix, VCKM.9,13)
It is given by
d0
s0
b0
0B@
1CA ¼
Vud Vus Vub
Vcd Vcs Vcb
Vtd Vts Vtb
0B@
1CA
d
s
b
0B@
1CA; ð1Þ
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where (d0; s0; b0) and (d; s; b) are the eigenstates of down-typequark fields in the weak interaction (flavor eigenstate) andthe mass eigenstate fields, respectively. It implies that onequark state, when the weak interaction takes effects, can berepresented by a mixture of different quarks in their masseigenstate. As a consequence, the square of the correspond-ing matrix element, jVqq0 j2, is proportional to the transitionprobability from one quark q to another q0.
The CKM matrix, since it is a 3� 3 unitary matrix, hasthree angles as well as one imaginary phase, �CKM. Instead ofusing the three angles and one imaginary phase, the CKMmatrix is usually represented in an expansion of � ¼ cos C,where C is the Cabibbo angle and � ¼ 0:22. This is calledthe Wolfenstein parameterization, which is shown in thefollowing.
VCKM ¼
1��2
2� A�3ð� i�Þ
�� 1��2
2A�2
A�3ð1� � i�Þ �A�2 1
0BBBBB@
1CCCCCA
þ Oð�4Þ;
ð2Þ
where � , A, , and � are parameters. The parameters ofVCKM, in particular, and �, which were not preciselydetermined before, have been extensively studied anddetermined experimentally using K mesons28) and B mesons.Now, one of the long-standing questions in quark flavorphysics is CP violation, which was observed in the K0
L
decays in 1964.14) One potential explanation is that thisviolation is due to an imaginary phase, �CKM or equivalently�, in the CKM matrix. To resolve this question, B factories,both at KEK in Japan and at SLAC in the US, were built forthese studies. The B factories are electron–positron colliderswith high beam currents at a center-of-mass energy of10.58 GeV. In 2001, large CP violation in the B mesonsystem was discovered at both KEK and SLAC.21) Inaddition, the parameters of and � have been preciselydetermined. Figure 4 shows experimental constraints on and � from various B decays and K decays. In Fig. 4, goodconsistencies among various measurements can be seen. It
has been proven with good precision that the CKM theorycan account for the flavor mixing and CP violation in thequark sector.
Upon the upgrading of the KEK B factory, researcherswill search for deviations from the Standard Model in somekey observables of flavor physics in the clean environmentof a lepton collider. Deviations from the Standard Model, ifdiscovered, will quantify the effect of physics beyond theStandard Model. Furthermore, such discoveries may uncovernew physics that cannot be detected at the energy frontier.In particular, new sources of CP violation and new right-handed currents, which may arise from supersymmetry orother physics beyond the Standard Model, can be searchedfor from precision measurements of the b! s transitionsthat are not measured precisely at the moment.
3.2 Lepton flavor physicsTurning to neutrino physics,29) it was originally assumed
in the Standard Model that the neutrinos are massless andtherefore no neutrino mixing occurs. However, the discoveryof neutrino oscillation phenomena at Super-Kamiokande,20)
and later at SNO,30) and KamLAND,31) verified thatneutrinos have masses and they are mixed. The neutrinomixing matrix, which is known as the Maki–Nakagawa–Sakata–Pontecorvo (MNSP) matrix,32,33) is given by
�e
��
��
0B@
1CA ¼
Ue1 Ue2 Ue3
U�1 U�2 U�3
U�1 U�2 U�3
0B@
1CA
�1
�2
�3
0B@
1CA; ð3Þ
where �l (l ¼ e; �; �) are the flavor eigenstates of neutrinofields and �i (i ¼ 1; 2; 3) are the mass eigenstates. Since theMNSP matrix is a 3� 3 unitary matrix, it can also bepresented by three angles, such as 12, 23, and 13, and oneimaginary phase �MNSP. If neutrinos are Majorana-typeparticles, the neutrino mixing matrix would include twoadditional imaginary phases (called the Majorana phases).Here, the Majorana-type particle is a particle that is identicalto its antiparticle, in contrast to the Dirac-type particle.The two large angles of 12 and 23 are determined frommeasurements of the oscillation of neutrinos from the sun(solar neutrinos) and those from the earth’s atmosphere(atmospheric neutrinos), respectively (sin2 23 � 0:5 andsin2 12 � 0:3), and the 3rd angle 13, which is the smallest,is yet to be determined. The Double Chooz experiment (inFrance) using reactor neutrinos, as well as the T2K experi-ment (in Japan) and the No�A experiment (in the US) usingaccelerator-based neutrinos, are under preparation to meas-ure the 13 angle. The studies of neutrino oscillation not onlyverified that neutrinos are mixed but also showed that thedifferences of the neutrino masses are quite small; �m2
21 ¼m2
2 � m21 ¼ 8:0þ0:6
�0:4 � 10�5 eV2, and j�m232j ¼ jm2
3 � m22j ¼
ð2:5� 0:5Þ � 10�3 eV2, where mi is the mass of neutrino�i.
34) This indicates the existence of physics at a very highenergy scale if we assume the seesaw mechanism, whichwas proposed to explain the smallness of the neutrinomasses.35,36) Also, since the MNSP matrix includes animaginary phase �MNSP, CP violation in the neutrinooscillation, which is determined from a difference in theoscillation probabilities of neutrinos and antineutrinos, canbe expected to be observed, in particular, in the case when
γ
β
α
γ+β2
smΔdmΔ dmΔ
Kε
cbVubV
ρ-1 -0.5 0 0.5 1
η
-1
-0.5
0
0.5
1
Fig. 4. Experimental constraints on the and � parameters in the
CKM matrix from various B and K decays. This result presents good
consistencies among the various measurements.
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13 is large. It is noted that the information of the twoadditional CP-violating Majorana phases will not be ob-tained from the neutrino oscillation.
We have also understood that the flavor mixing ofneutrinos shows a different pattern from that of quarks.The determination of the neutrino mixing parameters withas good precision as in the quark mixing parameterswould be desirable. By combining the mixing parametersof quarks and leptons, it might be possible to gain deepinsight into the flavor structures of quarks and leptons, by forinstance scrutinizing the quark–lepton complementarity,12 þ C � �=4, where C is the Cabbibo angle in the quarkmixing matrix.37,38)
Furthermore, there are the following important issues inneutrino physics ahead of us:(1) Determination of whether neutrinos are a Dirac-type
or Majorana-type particleSearches for neutrinoless double beta (0���) decays,which can occur only when the neutrinos are Major-ana-type, are carried out extensively for various ��decay nuclei. The rates of the 0��� decays areproportional to the square of the effective electronneutrino mass, hm�ei ¼
Pj jUejj2mj expði jÞ, where j
is a Majorana phase of the neutrino �j. The presentsensitivity to hm�ei is on the order of 0.1 eV and futureaimed sensitivities will be on the order of 0.01 eV. Ifthe neutrinoless double beta decays are observed, theMajorana phases, which might be indirectly related tothe required CP violation in leptogenesis (mentionedlater), would be estimated in conjunction with theneutrino mixing parameters.
(2) Determination of the absolute masses (not the massdifference) of neutrinosThe direct measurement of the electron–neutrino (�e)from tritium decays is planned with a sensitivity of0.1 eV. Also, the 0��� decays enable us to determinehm�ei which can be used to evaluate the absoluteneutrino masses.
(3) Determining the neutrino mass hierarchyTwo types of neutrino mass spectra can be considered;one is the case in which two neutrinos are light and oneneutrino is heavy (normal hierarchy), and the other isthe case in which two neutrinos are heavy and oneneutrino is light (inverted hierarchy). The neutrinomass hierarchy can be determined by planned long-baseline neutrino oscillation experiments using mattereffects.
Turning to charged leptons, in contrast to neutrinos, thetransitions from one type to another have never beenobserved yet. They are lepton-flavor-violating (LFV) proc-esses of charged leptons.39) Examples are �! e� decay,coherent �� e conversion in nuclei (�� þ N ! e� þ N),�! �� decay, and �! e� decay. They violate theconversation of the lepton flavor quantum numbers betweenthe initial and final states by one unit. In the Standard Model,the LFV processes of charged leptons are heavily suppressedby small neutrino masses against the W boson mass. If theyare discovered, we will be able to develop an entirely newphysics of charged-lepton mixing. This will open a new wayto obtain indirect evidence for supersymmetry and theseesaw mechanism and grand unification.
In order to ascertain the origin of the flavor structure, itseems likely that our understanding of physics at the energyscale of grand unification will play a key role. In the future,with all of the basic precision measurements mentionedabove (for quarks, neutrinos and charged leptons) becomingavailable, the origin of supersymmetry breaking beingunderstood, and proton decay modes being measured, thereis a possibility of making a theoretical breakthrough on theorigin of the flavor structure.
3.3 Precision measurements of particle propertiesThe highest-precision frontier in particle physics exists in
the measurements of the properties of elementary particles.The objective of this kind of study is to search for anydeviations from the expectations of the Standard Model,where potential deviation can be caused by new physics ornew unseen heavy particles through quantum effects.Examples are the electric dipole moments (EDMs) ofelementary particles,40) such as an electron, a neutron, adeuteron, and diamagnetic atoms, and the magnetic dipolemoments (MDMs) such as those of an electron and a muon.EDM is known to violate the invariance of time reversal (T-odd and CP-odd) as well as that of space inversion (P-odd),and it has attracted much attention in terms of potentials ofsearching for unseen sources of CP violation to explain thebaryon asymmetry in the Universe (as discussed later). In theStandard Model, EDM is estimated to be vanishingly small,and therefore, clean searches for nonzero EDMs, withoutthe Standard Model contributions, can be realized. The mostsensitive measurement of the EDM search was carried outon the 199Hg diamagnetic atom, whose upper limit of about10�28 e�cm was achieved. Although nonzero EDMs have notbeen observed, the next-generation experiments are planned.Regarding MDM, the precise measurement of the anomalousmagnetic moment [ðg� 2Þ=2] of the muon at the Brookhav-en National Laboratory (BNL) in the US was carried out. Asensitivity of less than 1 ppm has been achieved, and it hasrevealed a 3.4 � discrepancy from the value predicted by theStandard Model.39)
The search for decays of protons (the lightest baryon),such as p! eþ�0 and p! Kþ�, which violates theconservation of the baryon numbers (B violation), is oneof the most important issues in particle physics, becauseit would have a potential to reveal hints for the GrandUnification Theory (GUT), which is a theory that unifies thefundamental gauge forces, as well as the two differentsectors of quarks and leptons, and the GUT allows thebaryon-number-violating interaction. The unification of thefundamental forces is Einstein’s dream, which he was unableto achieve in his late research life. Such studies need manyprotons, and therefore, water has been used for most casesas a target material and at the same time, as a detector bydetecting the Cherenkov lights of the decay products ofprotons. The Super-Kamiokande is such a water cherenkovdetector of large size (50 ktons). However, the proton decaysare yet to be detected, and only the upper limit of the protonhalf life time of 1033 years has been obtained.34) Next-generation experiments are under consideration worldwide.
4. Particle Physics in the Intersection with Cosmology
Turning to observations in the intersection of particle
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS S. KOMAMIYA and Y. KUNO
111001-6
physics and cosmology, we now confront the astonishingfact that known particles, which form ordinary matter,accounts for only 4% of the total energy density of theUniverse. All the rest consists of dark matter41) (20%) anddark energy42) (76%) which are not identified in terms ofparticle physics. It is likely that dark matter is made up ofweakly interacting particles that were created in the earlyUniverse and survived until now.43) These particles weregathered by the attractive gravitational force and becomeseed galaxies. Dark energy is carried by vacuum and isresponsible for the accelerating expansion of the currentUniverse. Dark energy is consistent with the cosmologicalconstant that was introduced by Einstein into the fundamen-tal equation of general relativity. Both dark matter and darkenergy should be explained in terms of particle physics inthe future.
Another important issue in the intersection betweenparticle physics and cosmology is how the matter-dominantUniverse was created in the very early Universe.44) At thebeginning of the Big Bang, it is thought that equal numbersof matter and antimatter were created. Since then, as theUniverse expended and cooled, matter and antimatterannihilated each other. If the amounts of matter andantimatter were kept the same in the early Universe, therewould be no matter in the present Universe and we wouldnot exist. There must be some mechanism of producing anexcess of matter over antmatter. This mechanism is called‘‘baryogenesis’’ (implying baryon genesis). If baryogenesisdid not occur, we humans would not exist. It is now knownthat baryogenesis needs Sakharov’s three conditions;45)
namely (1) violation of the baryon numbers (B violation),(2) asymmetry between matter and antimatter (CP violation),and (3) interactions out of the thermal equilibrium. Theoriginal baryogenesis scenario based on the simple grandunification theory without neutrino masses turned out notto work and needs to be modified because of anomalousbaryon-number-violating processes (sphaleron processes) inthe Standard Model at high temperature. Instead, baryo-genesis from massive neutrinos (leptogenesis46)) has nowattracted attention. All these phenomena should be explainedby particle physics.
5. Summary
In the last century, remarkable progress was made inelementary particle physics both experimentally and theo-retically, which led to the establishment of the StandardModel. It is indeed a great triumph of particle physics,achieved jointly by theoretical and experimental physicists.However, we consider the Standard Model as an approx-imation of more fundamental theory at low energies, sincethere are so many unexplained issues in the Standard Model.
In this special issue on elementary particle physics, thesuccess of the Standard Model of particle physics isdescribed from various aspects. At the same time, futureprospects of finding physics phenomena beyond the Stand-ard Model are also emphasized. There are still manyfundamental questions that are yet to be answered andshould be addressed by elementary particle physics in thefuture. Some of the fundamental questions are listed below.(1) Are there new symmetries or new physical laws of the
Universe? (supersymmetry)
(2) Do all the forces become unified? (grand unification,superstring)
(3) Are there extra dimensions of space? (superstring)(4) Why are there so many kinds of particles? (flavor
physics)(5) What are neutrinos telling us? (the origin of neutrino
mass, Dirac or Majorana?)(6) What is dark energy? What is dark matter?(7) How did the Universe begin? (Big Bang)(8) What happened to antimatter? (baryogenesis)
By addressing these questions, we, particle physicists, areabout to take the critical steps towards a revolutionaryunderstanding of the Universe. We believe that the oppor-tunity for new discoveries about the fundamental nature ofthe Universe will be enormous in the near future.
1) http://lhc.web.cern.ch/lhc/
2) S. Asai: J. Phys. Soc. Jpn. 76 (2007) 111013.
3) http://linearcollider.org/cms/
4) H. Yamamoto: J. Phys. Soc. Jpn. 76 (2007) 111014.
5) P. A. M. Dirac: Proc. R. Soc., Ser. A 126 (1930) 360.
6) C. D. Anderson: Phys. Rev. 43 (1932) 491.
7) H. Yukawa: Proc. Phys.-Math. Soc. Jpn. 17 (1935) 48.
8) M. Gell-Mann: Phys. Lett. 8 (1964) 214; G. Zweig: CERN Rep. 8419/
Th (1964) 412.
9) N. Cabbibo: Phys. Rev. Lett. 10 (1963) 531.
10) J. J. Aubert et al.: Phys. Rev. Lett. 33 (1974) 1404; J. E. Augustin
et al.: Phys. Rev. Lett. 33 (1974) 1406.
11) S. L. Glashow: Nucl. Phys. 22 (1961) 579; S. Weinberg: Phys. Rev.
Lett. 19 (1967) 1264; A. Salam: in Elementary Particle Theory, Proc.
8th Nobel Symp., ed. N. Svartholm (Wiley-Interscience, New York,
1968).
12) D. Gross and F. Wilczek: Phys. Rev. Lett. 30 (1973) 1343;
H. D. Politzer: Phys. Rev. Lett. 30 (1973) 1346.
13) M. Kobayashi and T. Maskawa: Prog. Theor. Phys. 49 (1973) 652.
14) H. J. Christenson, J. Cronin, V. Fitch, and R. Turlay: Phys. Rev. Lett.
13 (1964) 138.
15) M. L. Perl et al.: Phys. Rev. Lett. 35 (1975) 1489.
16) W. Innes et al.: Phys. Rev. Lett. 39 (1977) 1240.
17) F. Abe et al. (CDF Collaboration): Phys. Rev. Lett. 73 (1994) 225.
18) ALEPH, DELPHI, L3, OPAL, and SLD Collaborations: Phys. Rep.
427 (2006) 257.
19) T. Kawamoto: J. Phys. Soc. Jpn. 76 (2007) 111003.
20) Y. Fukuda et al. (SuperKamiokande Collaboration): Phys. Lett. B 436
(1998) 33; Y. Fukuda et al. (SuperKamiokande Collaboration): Phys.
Rev. Lett. 81 (1998) 1562.
21) B. Aubert et al. (BaBar Collaboration): Phys. Rev. Lett. 87 (2001)
091801; K. Abe et al. (Belle Collaboration): Phys. Rev. Lett. 87
(2001) 091802.
22) Y. Okada: J. Phys. Soc. Jpn. 76 (2007) 111011.
23) Y. Nambu: J. Phys. Soc. Jpn. 76 (2007) 111002.
24) B. Heinemann: J. Phys. Soc. Jpn. 76 (2007) 111004.
25) M. Yamaguchi: J. Phys. Soc. Jpn. 76 (2007) 111012.
26) T. Yoneya: J. Phys. Soc. Jpn. 76 (2007) 111020.
27) A. I. Sanda: J. Phys. Soc. Jpn. 76 (2007) 111005.
28) L. S. Littenberg: J. Phys. Soc. Jpn. 76 (2007) 111006.
29) K. Nakamura: J. Phys. Soc. Jpn. 76 (2007) 111008.
30) Q. R. Ahmad et al. (SNO Collaboration): Phys. Rev. Lett. 89 (2002)
011301.
31) K. Eguchi et al. (KamLAND Collaboration): Phys. Rev. Lett. 90
(2003) 021802.
32) Z. Maki, M. Nakagawa, and S. Sakata: Prog. Theor. Phys. 28 (1962)
From Yukawa’s Pion to Spontaneous Symmetry Breaking
Yoichiro NAMBU
Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, U.S.A.
(Received February 6, 2007; accepted April 9, 2007; published November 12, 2007)
An overview is given of the development of particle physics since the 1930s to the 1960s when theconcept of spontaneous symmetry breaking (SSB) was established as an essential component of particletheory and eventually led to the Standard Model. A brief account of SSB as a general phenomenon inphysics is also added.
What we now call particle physics started first as nuclearphysics in the 1930s, and I have once proposed to call ErnestLawrence and Hideki Yukawa the two founders of particlephysics:1) The first invented the cyclotron, the second‘‘invented’’ the meson theory, which together opened thedoor to the exploration of ever higher energy phenomenaand to the discovery of particles of ever increasing masses.The problems of partcles physics have consisted of threequestions:
a) What are the elementary constituents of matter?b) What are the interactions, i.e., the forces or fields
operating among them, and their basic symmetryproperties?
c) What is the mathematical formalism to deal with thesequestions?
The present standard model has answered most of thesequestions successfully up to the energy scales currentlyavailable: it contains the fundamental fermions for a), thegauge fields for b), and the quantum field theory withrenormalization and spontaneous symmetry breaking for c),but with the question of the auxiliary field (Higgs field)sector still awaiting verification.
Up until the 1930s, physicists had known only twoelementary matter particles, the electron and the proton, plusthe photon, the quantum of the only force beside gravity.The nature of atomic nuclei and that of the weak processesbelonged to an unknown territory. But they thought theyhad already found the fundamental constituents of matter.It was a surprise and embarrassment when the neutron wasdiscovered. On the other hand, quantum mechanics hadsuccessfully solved the mysteries of atomic phenomenawhich involved energies of the order of electron volts, butit could not resolve the difficulties of infinite self-energiesinherited from classical theory. When dealing withnuclear phenomena involving energies a million timeshigher than the atomic counterparts, they inclined to believethat quantum mechanics had to be replaced with a newmechanics but did not conceive the possibility of newelementary particles. It was Yukawa2) who showed that onecan keep quantum field theory but admit new particlesin understanding the problem of nuclear forces. Hisprediction was confirmed later by the discovery of the
pion.3) The difficulty of infinities was circumvented later bythe idea of renormalization by Tomonaga, Schwinger, andFeynman.4)
To have a general perspective of what has happened sincethe 1930s, it is useful to recall the so-called three-stagestheory5) by Taketani, who was a disciple and collaboratorof Yukawa. He argued that the advance of physics goesthrough cycles of three stages: Faced with a set of newphenomena, one has first to find some regular patterns inthem; then one tries to understand or explain the regularitiesqualitatively in terms of concrete models; in the last stagea more precise and quantitative theory is developed. Butsooner or later new phenomena or discrepancies with theestablished theory will show up, and one starts the threestages over again. In actuality, these stages do notnecessarily proceed in sequence. The three kinds of attemptsmay go on simultaneously. The path may take a zigzagpattern.
2. Search for Symmetries
Yukawa’s theory originally envisioned only a uniquevector charged boson, but soon they found it necessary toexamine various spin and neutral versions, simultaneouslydeveloping theoretical formulations as they went along. Theapparent equality of forces among protons and neutronsled Kemmer6) to introduce the concept of isospin symmetry.But problems related to the divergent properties ofderived nuclear forces made clearcut conclusions difficult.There was a confusion when the cosmic ray muon wasdiscovered7) and incorrectly but understandably identifiedwith Yukawa’s meson. As more and more new particles,now called hadrons, showed up in cosmic ray events aswell as in reactions in the more powerful accelerators,there followed attempts to make sense of the rich spectrumof particles. In this period they started to search forsymmetry properties of reactions even though they wereonly approximate symmetries. The isospin symmetry forproton and neutron was extended to include the strangehadrons,8) then to the flavor SU(3) symmetry.9) In parallelthere also were attempts to understand these regularitiesin terms of constituent particles: the nucleon pair modelof the pion,10) generalized to include Lambda baryons,11)
and finally the logically simpler quark model,12) whichintroduced a new set of fundamental constituents below the
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level of hadrons. But the problem of statistics, when thehadrons are regarded as composites of quarks, led furtherto the concept of quarks endowed with an SU(3) ofcolor.13) The origin of strong interaction were then to betransferred to that among the quarks, and the hadrons wouldbecome dynamical composites of them.
3. Search for Dynamical Principles
The problems c) of the mathematical formalism, may bedivided into two different kinds:
1) Search for general and rigorous properties of localquantum field theory independent of the particularLagrangian or perturbation theory, which led to resultslike the spin-statistics and CPT theorems. There wasalso a period of groping for alternatives to quantumfield theory which persisted even after the successes ofrenormalization. These attempts include Heisenberg’sS-matrix theory, its descendants in the form ofdispersion theory and the Regge trajectory theory. Itis important to emphasize that all these efforts haveplayed an important, if not direct, role in arriving at thestandard model.
2) Establishment of the gauge principle for the funda-mental interactions among particles. Yukawa’s mesontheory was a phenomenological one from today’sviewpoint. The non-Abelian gauge theory of Yang andMills14) was a purely theoretical construct inspired bythe isospin symmetry. The crucial problem for prac-tical applications was its masslessness. Another wasthat the flavor symmetries are only approximatewhereas gauge symmetries are exact. As for the weakinteraction, its true origins were only a subject ofspeculations, but when its V-A nature was establish-ed,15) there arose the possibility that it might also beattributed to a gauge field. As it has turned out, thegauge principle was then not to be applied to the flavorsymmetries of the hadrons, but to the strong (color),electromagnetic and weak interactions at the level ofquarks and leptons.
Regarding the problem of mass, actually there was thelong known plasma mode, i.e., the massive Coulombfield, as well as its generalization to the transverse counter-parts16) But these happened in ionized media, not in therelativistic ‘‘vacuum’’. A crucial hint to the understandingand resolution of these problems came from the theoriesof superconductivity. The BCS theory17) assumed a con-densate of charged pairs of electrons or holes, hencethe medium was not gauge invariant. There were foundintrinsically massless collective excitations of pairs (NGmodes) that restored broken symmetries, and they turnedinto the plasmons by mixing with the Coulomb field. TheFermi sea of electrons in the BCS theory developedfermionic excitations with an energy gap, remindingone of the mass gap of the nucleons (and the quarks andleptons). Transporting these results to relativistic theoriesrequired one to abandon the concept of the relativistic‘‘void’’ as Dirac once did in the interpretation of hisequation. The spontaneous symmetry breaking thus emergedas a universal phenomenon in physics, and was firstapplied to the chiral symmetry breaking18) and (the con-stituent) mass generation for nucleons. Yukawa’s pion plays
the role of the NG boson. The fact that the chiral symmetryis only approximate, and not gauged, makes the pionmassive for reasons different from the case of the weakbosons.
An alternative and in fact older description of super-conductivity was one by Ginzburg and Landau,19) whichturned out to be a phenomenological (effective) representa-tion of the BCS theory. In this transcription the complexscalar ‘‘Ginzburg–Landau’’ field is equivalent to the Higgsfield, representing a bound pair of electrons or holes, and itsphase and amplitude components correspond respectively tothe massless NG and the massive Higgs type excitations. Asit happens, the collective excitations of both the NG andHiggs types do exist in all phenomena of the superfluiditytype. The precise transcription and utilization (or sometimesindependent derivations) of these condensed matter exam-ples to relativistic theories were carried out in due course bya number of people.20) The essential point of the problemmay be explained as follows. In London’s phenomenologicaldescription, the static induced current ji (@i ji ¼ 0) in asuperconductive medium is of the form (in arbitrary staticgauge)
ji ¼ KikAk;
KikðqÞ ¼ �ik �qiqk
q2
� �Kðq2Þ:
ð1Þ
The Meissner effect implies that Kð0Þ 6¼ 0: There are twopossibilities as to the origin of the second term: Either it isof dynamic origin so that in the nonstatic case q2 would bereplaced by q2 � v2!2, implying some accoustic-type(massless) collective excitations, or else the pole remainsstatic. But the latter could not happen unless the 1=q2
singularity was built into the Hamiltonian from the begin-ning. In fact it is the Coulomb interaction (of the electroncomponent, the compensating background charge remaininginert) that actually produces the pole In a relativisticallyinvariant medium, the above eq. (1) will be replaced by itsrelativistic version:
j� ¼ K�� j�;
K��ðqÞ ¼ ��� �q�q�
q2
� �Kðq2Þ
ð2Þ
The two alternatives still exist. It is due to the lack ofmanifest Lorentz invariance of gauge potentials that thesecond alternative escapes the NG massless boson theorem,and KðqÞ takes the form K ¼ q2=ðq2 � m2
V Þ, mV being thegauge boson mass. It is now history that this mechanismof mass generation of gauge bosons was most successfullyapplied to the weak interaction sector, rather than the stronginteraction sector, in a form of Ginzburg–Landau–Higgs(G–L–H) description of spontaneous symmetry breaking(SSB) in the electroweak unification of Glashow, Salam, andWeinberg (GSW) for the hypothetical W and Z vectorbosons, which were to be confirmed experimentally later.The gauge theory of strong interactions sector also joined theGSW theory later to complete the full Standard Model. Itcontains two SSB phenomena: the gauged one in the weakinteraction sector explicitly displayed in the G–L–H form tomake the weak bosons massive, whereas the other ungaugedchiral symmetry in the strong interaction sector for lightquarks is only implicit and approximate.
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4. General Remarks on SSB
The general properties of SSB may be characterized asfollows.
1) Degeneracy of the ground state2) The degrees of freedom N !1 (the thermodynamic
limit). Finite systems do also exhibit similar dynamicalproperties, but the SSB description becomes an exactone only in infinite systems.
3) Superselection rule (contraction of the Hilbert space)This means that the Hilbert space of the system is builtup from one of the ground states, and other Hilbertspaces built on other ground states become inaccessiblebecause there are no local observables that can connectthem.
4) The Nambu–Goldstone (NG) modesThe lost symmetry is nevertheless visible in the formof the NG modes. which are massless since in thelong wave limit it reduces to the global symmetryoperation.
The conscious use of the symmetry principle in physicsdates back to the 19th century. Curie21) used symmetryconsiderations to derive a kind of selection rules for physicaleffects, an example of which is the Wiedemann effect: Aconducting cylinder suffers a twist t when a current J ispassed and simultaneously a magnetic field B is appliedparallel to it. The behavior of t under various spacereflection operations matches that of either B or J. So heargued that an effect is possible only if its symmetriesSeff are compatible with those of the environment Senv.The SSB, on the other hand, may be characterized by aself-diminution of Senv from SL, the symmetries of theLagrangian L of the system. It was not envisioned by Curie,but such phenomena were already known in classicalsystems. The spontaneous deformation of a rotating bary-tropic body from a sphere to a Jacobi ellipsoid is anexample, which clearly shows that symmetry breakingis a dynamical problem. More relevant examples for us,however, came after Curie. The ferromagnetism is theprototype of today’s SSB, as was explained by the works ofWeiss,22) Heisenberg,23) and others. Ferromagnetism havesince served us as a standard mathematical model of SSB. Itis no coincidence that Heisenberg made use of it later in hisattempt at a unified theory. Examples of BCS-type SSB aresuperconductivity and the superfluidity of He3. They havequasifermions, NG and Higgs bosons satisfying simple massrelations among them.24)
Acknowledgement
This work was supported by the University of Chicago.
1) Y. Nambu: Particle Accelerators (Gordon & Breach, Boston, 1990)
Vol. 26, p. 1.
2) H. Yukawa: Proc. Phys.-Math. Soc. Jpn. 17 (1935) 48.
3) C. M. G. Lattes, H. Muirhead, G. P. S. Occhialini, and C. F. Powell:
Nature 159 (1947) 694.
4) S. Tomonaga: Prog. Theor. Phys. 1 (1946) 27; Z. Koba, T. Tati, and S.
Tomonaga: Prog. Theor. Phys. 2 (1947) 101; Z. Koba, T. Tati, and S.
Experimental Electroweak Physics at Lepton Colliders
Tatsuo KAWAMOTO
ICEPP, the University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033
(Received April 29, 2007; accepted July 3, 2007; published November 12, 2007)
Progress in the electroweak physics at the eþe� colliders are reviewed, focusing mainly on the latestand highest energy colliders LEP and SLC. The results on the Z and W boson properties are discussed,and the implications of these precision results in the standard model are shown.
It would be fair to say that electron–positron (eþe�)colliders have made vital contributions to the progresson particle physics. The spectacular success in the 1970sincludes the discovery at SPEAR of c�cc resonancesðJ=�;�0; . . .Þ, charmed mesons and � leptons, the study of� resonances at DORIS and CESR, and discovery of Bmesons, the observation of quark and gluon jets at PEP andPETRA, followed by TRISTAN in 1980s towards highercollision energies. A great advantage of eþe� colliders is thewell defined initial state of the point-like electrons, leadingto very clean final states, allowing both precise measure-ments and clear signatures of new phenomena. LEP1) andSLC2) are on the wave front (Fig. 1), conceived as themachines to explore physics at the electroweak scale, byproducing a large number of Z bosons, and at a later stageW pairs at LEP-2. This report gives a brief summary ofelectroweak physics at eþe� colliders, focusing mainly onthe results from LEP and SLC.
2. Colliders and Experiments
The SLC collider at SLAC was based on the existing 3 kmlinear electron accelerator originally built in the 1960s. Itwas upgraded and modified in the 1980s to accelerate bothelectrons and positrons to the beam energies sufficient toproduce the Z boson. The beams were brought into collisionat a single interaction point after guided along two arcs.In June 1989 the first eþe� collisions at the Z energy wererecorded by the Mark II detector. By August 1989, the firstresults of the Z resonance parameters were produced.3)
Since 1992, the new SLD detector4) took data at the SLC.The machine was also upgraded to provide longitudinallypolarised electron beam,5) the feature unique to the SLCwhich is absent in LEP. By 1998 when the SLC was finallystopped, SLD detector collected 0.6 million Z decays. Fora large fraction of the dataset, the level of polarisation wasover 70%.
The LEP collider was built at CERN in the 1980s. It was acircular accelerator of its 27 km circumference, the largestaccelerator in the world. The large size was dictated by therequirement of limiting the energy loss due to synchrotronradiation, which goes as E4
beam, to a manageable level.Existing CERN accelerators such as PS and SPS were usedto ramp the energy of electrons and positrons to the LEPinjection energy at 20 – 22 GeV. The eþe� beams were
further accelerated in the LEP ring and brought intocollisions at four interaction points where the four LEPdetectors ALEPH,6) DELPHI,7) L3,8) and OPAL,9) werelocated.
LEP started its operation in 1989 at the centre-of-massenergy near to the Z mass. Until the end of 1995, four LEPexperiments each collected about 4.5 million Z decay events.
In the second phase of LEP operation (LEP-2) from 1996until 2000, centre-of-mass energy was increased progres-sively. Pair production of the W boson became possiblefor the first time in the eþe� collision, allowing precisemeasurements of W boson mass and detailed studies ofW pair production mechanism. Another important topicthroughout the LEP programme was searches for the Higgsboson and other possible new physics signatures whichcould appear only at high energies. In the last year of LEPoperation, the centre-of-mass energy of LEP reached to itshighest energy of 209 GeV.
3. The Standard Model in Brief
At the tree level in the standard model of electroweakinteraction, properties of Z and W bosons are determinedpractically by three independent parameters correspondingto the two gauge couplings of SUð2ÞL � Uð1ÞY and thevacuum expectation value of the Higgs field. A basicparameter set suited for the Z studies is GF, mZ and �, sincethey are measured precisely. Other parameters like W mass
SPECIAL TOPICS
10
10 2
10 3
10 4
10 5
0 20 40 60 80 100 120 140 160 180 200 220
Centre-of-mass energy (GeV)
Cro
ss-s
ecti
on (
pb)
CESRDORIS
PEP
PETRATRISTAN
KEKBPEP-II
SLC
LEP I LEP II
Z
W+W-
e+e−→hadrons
Fig. 1. (Color online) The hadronic total cross-section in eþe� collision
as a function of centre-of-mass energy. The solid curve is the prediction
of the standard model, and the points are the experimental measurements.
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Frontiers of Elementary Particle Physics, the Standard Model and Beyond
mW, the electroweak mixing angle sin2 �W and the Zcouplings to fermions are all derived using the standardmodel relations. For example, mW is related to Fermiconstant GF by:
GF ¼��ffiffiffi
2p
m2W sin2 �W
; ð1Þ
and
m2W ¼ �m
2Z cos2 �W; ð2Þ
with � ¼ 1 in the minimal model with one Higgs doublet.The Z couplings to left- and right-handed fermions are:
gL ¼ffiffiffi�pðT3 � Q sin2 �WÞ;
gR ¼ �ffiffiffi�p
Q sin2 �W: ð3ÞHere T3 is the third component of the weak iso-spin, and Q isthe fermion charge. Alternatively vector and axial-vectorcouplings are defined as:
gV ¼ gL þ gR; gA ¼ gL � gR ð4Þ
These simple relations are modified by radiative correc-tions, which depend, among others, on the top mass mt andthe Higgs mass mH. The parameters GF, mZ, and mW can bedefined as physically measured quantities and eq. (2) holdswith � ¼ 1, introducing a corrected sin2 �W. The electro-magnetic coupling � is replaced by the running value �ðmZÞ.Equation (1) is modified accordingly. For the Z resonancein eþe� collisions, effects on parameters � and sin2 �W ineq. (3) can be largely absorbed by defining correspondingeffective complex quantities.13) It is possible to split thesecorrected quantities into two parts; real and constantparameters (the effective parameters) and in general verysmall complex remnants. Furthermore, the constant effectiveparameters can be treated in a very good approximation14)
as if they are independent parameters, allowing a (nearly)‘‘model independent’’ parametrisation with the small rem-nants calculated using the standard model.
The basic strategy of LEP and SLC electroweak measure-ments is to summarise the measured observables in terms ofthese ‘‘model independent’’ pseudo-observables, and com-pare them to the standard model predictions to test the modeland determine or constrain the standard model parameterssuch as mt and mH.
4. Z Parameters
The process eþe� ! f�ff is mediated in the s-channel bytwo spin-1 neutral gauge bosons, a massless photon and amassive Z boson. The cross-section can be parametrised by aBreit–Wigner resonance for the Z contribution and remain-ing contributions from photon exchange and �–Z interfer-ence:
�fðsÞ ¼ �0f
s�2Z
ðs� m2ZÞ
2 þ s2�2Z=m
2Z
þ �ð�Þ þ �ð� � ZÞ; ð5Þ
where s is the centre-of-mass energy squared, mZ and �Z aremass and total width of the Z boson, and �0
f is the peakcross-section of the resonance. Here the s-dependent width isused in the definition. Measurement of the resonance curve,the Z lineshape, allows precise determination of the threeparameters mZ, �Z, and �0
f . The peak cross-section is relatedto the partial widths for initial and final state fermions, �e
and �f , with
�0f ¼
12�
m2Z
�e�f
�2Z
: ð6Þ
By combining lineshape measurements for all visiblefermion final states f = e, , �, and hadrons (q), partialwidths �f can be extracted.
Due to the parity violating couplings of the Z to fermions,various asymmetries arise in the process eþe� ! Z! f�ff.This can be seen by writing down the cross-sections fordifferent helicity combinations �ij (i = L, R, j = L, R):
�LLðcos �Þ ¼ CðsÞ � g2Leg
2Lfð1þ cos �Þ2 ð7Þ
�LRðcos �Þ ¼ CðsÞ � g2Leg
2Rfð1� cos �Þ2 ð8Þ
�RLðcos �Þ ¼ CðsÞ � g2Reg
2Lfð1� cos �Þ2 ð9Þ
�RRðcos �Þ ¼ CðsÞ � g2Reg
2Rfð1þ cos �Þ2 ð10Þ
where CðsÞ is the s-dependent common coefficient, and i
indicates left(L)- or right(R)-handed initial-state electronand j for L or R final-state fermion f. Left- and right-handedcouplings are each denoted by gLf and gRf .Angular distributions for unpolarised beams,ignoring the final-state helicityCross-section is obtained by averaging over initial-statehelicities and summing over final-state helicities.
�ðcos �Þ ¼1
2ð�LL þ �LR þ �RL þ �RRÞ
/ ðg2Ve þ g2
AeÞðg2Vf þ g2
AfÞ� ð1þ cos2 � þ 2AeAf cos �Þ; ð11Þ
where the asymmetry parameter Af is defined by:
Af ¼2gVfgAf
g2Vf þ g2
Af
: ð12Þ
Forward–backward asymmetry AFB ¼ ð�F � �BÞ=ð�F þ �BÞarise from the term proportional to cos �, and is given by:
AFB ¼3
4AeAf ð13Þ
Polarisation of final-state fermion, unpolarised beamsIn the �-pair final-state, polarisation of taus can be measuredbased on the V–A charged current decays of taus. The taupolarisation P�ðcos �Þ ¼ ð�R � �LÞ=ð�R þ �LÞ is given by:
The forward–backward symmetric term is proportional toA� , while the anti-symmetric term is proportional to Ae.From the measurement of �-polarisation, Ae and A� areextracted simultaneously and almost independently.Polarised beam, cross-section asymmetriesMeasured by the SLD experiment at SLC using the polarisedelectron beam. Left–right cross-section asymmetry directlydetermines Ae.
Here �ij is the cross-section integrated over cos � withinforward–backward symmetric acceptance. Alternatively,
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS T. KAWAMOTO
111003-2
forward–backward asymmetry of the left–right asymmetry,AFBLR, gives the asymmetry parameter of the final-statefermion:
AFBLR ¼3
4Af ð16Þ
Forward–backward asymmetries are measured at LEP forcharged leptons e, and �, and for b and c quarks. Using themeasurements of Ae (from electron AFB and � polarisation),the parameters A, A�, Ab, and Ac can be extracted usingeq. (13). At SLC ALR and AFBLR measurements determinedthe parameters Ae, A, A� , Ab, and Ac. The LEP and SLCresults form a complementary and practically complete setof the Af measurements.
If the Z couplings follow the standard model structure, theratio of vector and axial-vector couplings can be representedby the effective mixing angle sin2 �feff
gVf
gAf
¼ 1�2Q
T3
sin2 �feff : ð17Þ
Figure 2 shows the dependence of Af , which is a functionof gVf=gAf , on sin2 �feff for leptons, b and c quark. For theactual value of sin2 �feff � 0:23, the leptonic A‘ is themost sensitive to the variation of sin2 �feff , while Ab is veryinsensitive. Therefore sin2 �feff is best determined frommeasurements of the leptonic asymmetry parameter A‘.
4.1 Lineshape and leptonic AFB
The Z mass mZ, the total width �Z, and pole cross-section�0
f are determined from the lineshape measurement (Fig. 3).How well these parameters be determined depends on howaccurately the ‘‘lineshape’’ is determined. Since the hadronicfinal-state is the dominant decay mode of the Z (�87%of visible decays), mZ and �Z are mainly determined fromthe hadronic lineshape. The data sample from 1993 –1995is particularly important. In this running period, precisionenergy scans have been performed at three centre-of-massenergy points on the peak and two off-peak pointsapproximately 1.8 GeV above and below the peak. About36 pb�1 of integrated luminosity were collected by eachof the LEP experiments at the off-peak points. In 1994 alldata were collected on peak and about 55 pb�1 werecollected. Measurements of �0
f and the pole asymmetryA0;f
FB profit from the high statistics measurements at thepeak.
Systematic uncertainties must also be under control inorder to make full use of the statistical precision. Key issuesare:
. Event selection efficiency and background.
. Determination of luminosity.
. Precise calibration of LEP beam energy.
. Precision calculation of radiative corrections.All of these were studied carefully. Systematic uncertaintieson the cross-section and asymmetry measurements werecontrolled to better than 0.1%. Luminosity determinationwas based on the measurement of the rate of small angleBhabha scattering (eþe� ! eþe�). The experimental un-certainty was well below 0.1% while the commmon theoryuncertainty was 0.06%. Successful calibration of LEPenergy achieved a relative accuracy of 10�5 level.10)
Uncertainties on mZ and �Z arising from LEP energyuncertainty are �1:7 and �1:2 MeV, respectively. Thecross-section formula eq. (5) is largely modified due toradiative corrections, mainly by initial-state radiation. Inorder to interpret the precisely measured quantities, qualityof theoretical calculations must match the experimentalprecision. Achieved theoretical precision of the calcula-tions as implemented in the programs ZFITTER11) andTOPAZ0,12) is typically to the 10�4 level.13)
Using these data, lineshape and lepton AFB are analysedby fitting theoretical parameterisations to the data. Thestandard set of the parameters are:
. mZ and �Z, defined in eq. (5).
. The hadronic pole cross-section
�0had ¼
12�
m2Z
�e�had
�2Z
ð18Þ
. Partial width ratios:
R0‘ ¼ �had=�‘; ‘ ¼ e; ; � ð19Þ
-1
-0.5
0
0.5
1
0 0.2 0.4 0.6 0.8 1
Af
sin2θefff
b
ce
Fig. 2. (Color online) Dependence of Af on sin2 �feff for lepton, b and c
quark. The vertical line indicates the measured value of sin2 �lepteff .
0
5
10
15
20
25
30
35
88 89 90 91 92 93 94 95√s(GeV)
Cro
ss-s
ecti
on (
nb)
199019911992199319941995
hadrons
OPAL
-0.01
0
0.01
89.4 89.5
data
/fit
- 1
91.2 91.3 92.9 9.3 93.1
Fig. 3. (Color online) An example of hadronic lineshape. In the top plot,
the points are data and the curve is the result of lineshape fit. The bottom
plots show deviation of data from the fit.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS T. KAWAMOTO
111003-3
. Leptonic pole asymmetries.
A0;‘FB ¼
3
4AeA‘ ð20Þ
Results from the four LEP experiments are combined14,15)
to obtain the best results from LEP. The Z mass hasbeen determined to a relative precision of 10�5, mZ ¼91:1875� 0:0021 GeV. The ratio of invisible width, �inv ¼�Z � �e � � � �� � �had, to the leptonic width is:
�inv=�‘ ¼ 5:943� 0:016: ð21Þ
The uncertainty is largely due to the luminosity uncertain-ties. Assuming the invisible width is due to neutrinos,and using the standard model expectation for this ratio for asingle neutrino species ð�=�‘ÞSM ¼ 1:99125� 0:00083, thenumber of light neutrino species is obtained as:
N ¼ ð�inv=�‘Þ=ð�=�‘ÞSM ¼ 2:9840� 0:0082 ð22Þ
which is consistent with three. Alternatively, assuming N ¼3 additional contribution to the invisible width is
��inv ¼ �2:7� 1:6 MeV; ð23Þ
and the 95% confidence level upper limit on the additionalwidth is
��inv < 2:0MeV ð24Þ
4.2 � polarisationThe fermion pairs produced via the Z resonance is
polarised according to eq. (14). In the �-pair production,polarisation can be measured from the kinematic distribu-tions of � decay products. Observed distributions are ingeneral distorted from the original distribution due todetector effects, biases arising from event selection andradiative effects, and backgrounds. Monte Carlo simulationis used to take into account such effects. Expecteddistribution is parametrised as a linear combination of twodistributions, one corresponding to helicity +1 and the otherto �1, and the relative fraction of the two contributions,giving the best fit to the data, corresponds to the �polarisation.
The LEP experiments have analysed all relevant decaymodes: �! �� , ��, a1�, e�e, �. Depending on thenature of each decay mode, an optimal kinematic observableis constructed and used for fitting the polarisation. Combined� polarisation measurements15) are shown as a function ofcos � in Fig. 4. The data behave as expected from eq. (14).Separately extracted Ae and A� from the � polarisationmeasurements are Ae ¼ 0:1498� 0:0049 and A� ¼0:1439� 0:0043, consistent with equality as expected fromthe lepton universality. The combined leptonic asymmetryparameter A‘ from the � polarisation measurements isA‘ ¼ 0:1465� 0:0033.
4.3 b and c widths and asymmetriesZ decays into b and c quarks can be identified with good
efficiency and purity. Using the tagged b and c samples, theZ partial widths into b and c quarks, and asymmetries weremeasured.
. Ratios of b and c partial width to the hadronicwidth:
R0b ¼ �b=�had; R0
c ¼ �c=�had ð25Þ
. Forward–backward pole asymmetries:
A0;bFB ¼
3
4AeAb; A0;c
FB ¼3
4AeAc ð26Þ
. Asymmetry parameters for b and c from AFBLR:
Ab; Ac ð27Þ
Various tagging techniques were developed. The leptontag is based on the weak semi-leptonic decays of b/cflavoured hadrons. Due to the large mass of b/c hadrons,these leptons tend to have high momentum (p), and hightransverse momentum (pt) with respect to the jet axis.Separation between b and c quarks is made based ondifferent ðp; ptÞ spectra for b! ‘, b! c! ‘, and c! ‘.For the asymmetry measurements, charge of the leptonprovide information to distinguish q from �qq. A powerfulmethod of b tagging is based on the long life time (typically1.5 ps) and large mean charged multiplicity of B hadrondecays. The LEP/SLD detectors were equipped withprecision vertex detectors which allow detecting signaturesof displaced decay vertex, either by directly reconstructingthe secondary vertex, or based on a large number of trackswith significant impact parameter. Large invariant mass ofthe particles from the secondary vertex is also an indicationof B hadron decays. These sensitive variables are oftencombined using artificial neural network or likelihoodtechnique. Figure 5 shows an example of such b-taggingvariables. The SLD experiment profited from the small beamsize and small beam pipe of the SLC collider which enabledtagging b/c quarks with higher efficiency than LEP experi-ments. This allowed SLD to perform competitive measure-ments of R0
b and R0c despite much lower statistics of the Z
sample (�1=10 of LEP).Results from the LEP experiments and SLD using different
techniques are found to be consistent. A combination15) hasbeen made using these measurements to produce a set of band c quark measurements on R0
b, R0c , A0;b
FB, A0;cFB, Ab, and Ac.
Measured Pτ vs cosθτ
-0.4
-0.3
-0.2
-0.1
0
0.1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Pτ
cosθτ
ALEPH
DELPHI
L3
OPAL
no universalityuniversality
-
-
Fig. 4. P� as a function of cos �. The points are data, and the curves
are expectation from eq. (14) with and without assuming equality of Ae
and A� .
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS T. KAWAMOTO
111003-4
4.4 Z couplingsResults of the asymmetry parameter Af are summarised15)
graphically in Fig. 6. Leptonic asymmetry A‘ is obtainedfrom a combination of leptonic AFB, � polarisation and theALR measurements assuming lepton universality. For bquark, Ab is directly from the SLD AFBLR measurement.These two measurements are presented in the figure byvertical and horizontal bands. Forward–backward asymme-try for b quark, A0;b
FB, is a product of A‘ and Ab, thereforerepresented by the diagonal band in the A‘–Ab plane. Thethree bands have a common region in the plane as indicatedby the confidence level contours. However the combinedvalue15) of Ab ¼ 0:899� 0:013 is 2.7 standard deviationaway from the standard model expectation of 0.934. This ismainly because the intersect of A0;b
FB and A‘ bands deviateslargely from the standard model value. In the standardmodel, as seen in the figure, Ab is rather stable quantityinsensitive to variations of any of the standard model
parameters. The large deviation of the combined contourmay be simply due to fluctuation of one or both of the A0;b
FB
and A‘ measurements, or could be arising from some newphysics effects which involve non-standard b couplings.Direct determination of Ab is not precise enough todistinguish the two possibilities.
From the asymmetries Af and partial decay widths �f ,vector and axial-vector couplings can be extracted; Af
determines (gVf=gAf), while �f determines (g2Vf þ g2
Af).Results for lepton, b and c couplings obtained from LEPand SLD partial widths and asymmetry measurements areshown Fig. 7. Results for the three lepton species agreewell. The combined result assuming lepton universality isalso shown in the plot. The vertical arrow (��) shows theexpectation where only the running of � is considered in theradiative correction (gA‘ ¼ �1=2). The observed value ofgA‘ is significantly different from this expectation, whichdemonstrate the effect of electroweak correction �� on� ¼ 1. Also shown is the expectation of the standard model.The result of lepton couplings prefers low Higgs mass. The bcoupling is not quite consistent with the standard model, asalready seen in the A‘–Ab plot.
Assuming the standard model structure, the ratio gVf=gAf
can be represented by the effective mixing angle sin2 �feff[eq. (17)]. As seen above in Fig. 2, Ab (and Ac) areinsensitive to sin2 �feff . Therefore A0;b
FB and A0;cFB can be used to
determine A‘ via A‘ ¼ ð4=3ÞA0;bFB=Ab with Ab constrained
according to the standard model relation. The effectivemixing angle, sin2 �lept
eff , determined from A‘ measurementsfor different processes are summarised in Fig. 8. Two mostprecise determinations, one from ALR by SLD and the otherfrom A0;b
FB at LEP, differ by 3.2 standard deviation. Thisdifference is reflected in the overall �2 of the average; �2/d.o.f = 11.8/5, corresponding to a probability of 3.7%. Theorigin of this difference is as already seen in the comparisonof Af in the discussion of asymmetries. If there were somenew physics contribution that affects the b coupling tolargely deviate from the standard model, the extraction ofsin2 �lept
eff from the A0;bFB does not work.
5. Measurements above the Z
By the end of LEP-2 operation in the year 2000, theintegrated luminosity collected by each of the LEP experi-ments at the centre-of-mass energies above the Z reached700 pb�1. The range of centre-of-mass energy is 130 –209 GeV.
W physics at LEP-2Each LEP experiments collected about 12,000 W pair
events eþe� !WW from the full LEP-2 data sample. Thisallows precise measurements of W properties (mass, widthand decay branching ratios), and direct tests of the uniquestructure of three gauge boson couplings through systematicstudies of production cross-section, angular distribution andW helicity.
A W boson decays either hadronically (W! q�qq) orleptonically (W! ‘ �). In the eþe� collisions at LEP-2,all decay final states from W pair production are identifiedwith good efficiency and purity. W pair cross-sectionand W decay branching ratios are determined from datawithout relying much on the standard model. Figure 9 shows
0.8
0.9
1
0.14 0.145 0.15 0.155
Al
Ab
68.3 95.5 99.5 % CL
SM
Fig. 6. (Color online) The measurements of combined LEP+SLD A‘,SLD Ab, and LEP A0;b
FB (bands), compared to the standard model
expectations (arrows). Also shown are probability contours from a joint
analysis.
DataMC
Mass (GeV/c2)
No
. of
Hem
isp
her
es
bc
0
500
1000
1500
2000
2500
3000
3500
0 1 2 3 4 5 6
cuds
Fig. 5. (Color online) An example b-tagging variable, the vertex mass of
SLD.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS T. KAWAMOTO
111003-5
the measured W pair cross-section as a function of centre-of-mass energy. The measured data are compared to thestandard model expectation [solid (green) curve] obtainedusing the latest precision calculations.16) Also shown in thefigure are two dashed curves which correspond to twoscenarios where one or both of the gauge boson self-couplings were absent. It clearly indicates that there are infact self-couplings of gauge bosons and the structure of thecouplings are as given in the standard model. Furtherdetailed studies have been performed which can be foundelsewhere.17)
The W boson mass contains, through the standard modelrelations and radiative corrections, information on the otherstandard model parameters. In particular, given the preciselyknown values of �ðmZÞ, mZ, and GF, radiative correctionsare sensitive to the values of top mass and yet unknownHiggs mass.
W mass were measured at LEP-2 by reconstructingthe invariant mass of the W decay products with aconstraint of the LEP beam energy. Precise calibrationof LEP energy is therefore vital for the W mass measure-ment. Two decay modes were used. In the WW! q�qqq�qq
channel, both W bosons decay into quark pairs, leading to4-jet final states. Another channel is the WW! ‘q�qq
final state where one W decays into ‘ and the otherinto q�qq. Though the neutrinos are invisible, due to the beamenergy constraint, complete kinematic reconstruction ispossible.
Combined mW from the results of the four LEP experi-ments17) is shown in Fig. 9. The statistical uncertainty of theLEP mW is 25 MeV. Without systematic effects, it would beabout 20 MeV. However, due to uncertainties on the effectsof final state hadronic interactions,17) the weight of theWW! q�qqq�qq channel has been reduced in the combination.Also shown is the measurement from the p�pp colliderincluding the new preliminary result from Tevatron RUN-II.18,19) These two measurements are based on quite differenttechniques, and they agree well, yielding an average value of80:398� 0:025 GeV.19)
6. Higgs Boson Searches
Higgs boson was searched at LEP and LEP-2. In eþe�
collisions at the LEP/LEP-2 energies, Higgs boson isproduced mainly via Higgsstrahlung process,
eþe� ! ZH: ð28Þ
Since the cross-section decreases as the Z goes off-shell,the LEP-2 Higgs searches are limited to the Higgs mass
10 2
10 3
0.23 0.232 0.234
sin2θlept
eff
mH
[GeV
]
χ2/d.o.f.: 11.8 / 5
A0,l
fb 0.23099 ± 0.00053
Al(Pτ) 0.23159 ± 0.00041
Al(SLD) 0.23098 ± 0.00026
A0,b
fb 0.23221 ± 0.00029
A0,c
fb 0.23220 ± 0.00081
Qhad
fb 0.2324 ± 0.0012
Average 0.23153 ± 0.00016
Δαhad= 0.02758 ± 0.00035(5)
mt= 178.0 ± 4.3 GeV
Fig. 8. (Color online) Comparison of determinations of sin2 �lepteff from
asymmetry measurements. Also shown is the prediction of the standard
model as a function of Higgs mass.
-0.041
-0.038
-0.035
-0.032
-0.503 -0.502 -0.501 -0.5
gAl
g Vl
68% CL
l+l−
e+e−
μ+μ−
τ+τ−
mt
mH
mt= 178.0 ± 4.3 GeVmH= 114...1000 GeV
Δα
-0.36
-0.34
-0.32
-0.3
-0.54 -0.52 -0.5 -0.48
gAb
g Vb
68.3 95.5 99.5 % CL
SM
0.16
0.18
0.2
0.22
0.47 0.5 0.53
gAc
g Vc
68.3 95.5 99.5 % CL
SM
Fig. 7. (Color online) Contours in the gVf–gAf plane for leptons, b and c quark. Also shown are the predictions of the standard model.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS T. KAWAMOTO
111003-6
up to approximatelyffiffisp� mZ. The negative results of the
searches yielded a LEP combined lower bound on thestandard model Higgs boson of 114.4 GeV at the 95%confidence level.20)
7. Global Analysis in the Standard Model
The precision electroweak measurements at LEP/SLCand other experiments can be used in the global analysis inthe framework of the standard model to see whether all dataare compatible with the standard model expectations for acommon set of the standard model parameters, and to deriveor place constraint on the parameters. Particularly interestingof these parameters is the Higgs boson mass which is the lastunknown standard model parameter.
The following data are used in the analysis discussedhere.15,19)
. The Z parameters– lineshape and lepton asymmetry at LEP: mZ, �Z,�0
had, R0‘ , and A0;‘
FB.– A‘ from � polarisation at LEP– A‘ from polarised left–right asymmetry from SLD– Heavy quark measurements at LEP and SLD: R0
b, R0c ,
A0;bFB, A0;c
FB, Ab, and Ac
– sin2 �lepteff from inclusive quark forward–backward
asymmetry at LEP. W mass mW and width �W from LEP and Tevatron. Top mass mt at Tevatron21)
. Light quark contribution to the running of �:��ð5Þh ðmZÞ22)
In the determination of ��ð5Þh ðmZÞ, data of cross-section foreþe� ! hadrons at low energies are particularly important.
The set of the standard model parameters used are: �ðmZÞ,mZ, and GF as the three basic parameters of the electroweakinteraction, and �sðmZÞ for QCD corrections. In addition mt
and mH are needed to calculate radiative corrections. Thelatest version of ZFITTER and TOPAZ0 programs are used.Some of the measurements in the list are used to directlyconstrain these parameters ðmZ;��
ð5Þh ðmZÞ;mtÞ, while other
observables are calculated in the standard model andcompared to the experimental results.
As seen in §3 effects of electroweak radiative correctionappear in �� and sin2 �feff . They control the size of Z partialwidths �f and asymmetry parameters Af . In case of thehadronic width �had, QCD final state correction is alsoincluded, therefore it is sensitive to the strong couplingconstant �sðmZÞ. There are additional corrections to the bquark couplings arising from the vertex correction, leadingto the unique dependence of �b on mt. Finally the W massinvolves the correction �r to eq. (1).
The Z observables are functions of quantities such asmZ, �f , sin2 �feff . Depending on how the observables areconstructed from these quantities, the sensitivity to thestandard model parameters are different. For example, theasymmetry A‘ is a function of sin2 �lept
eff , which is sensitive tomt and mH. On the other hand partial width ratio R0
‘ ¼�had=�‘ is insensitive to the values of mt and mH due toa large cancellation of universal corrections included inboth of �had and �‘, but it is sensitive to the strongcoupling constant �sðmZÞ which is involved in the QCDcorrection on �had. The hadronic pole cross-section �0
had ¼ð12�=m2
Z�e�had=�2Z has only weak sensitivity to any of the
standard model parameters. This is because in addition to thecancellation of universal corrections in the Z widths, theQCD correction, which is included both in �had and �Z, alsocancels largely. However, �0
had is sensitive to the invisiblewidth, or, number of neutrino species N, which directlydetermines the total width �Z.
To see how these observables place constraints on mt
and mH through radiative corrections, Fig. 10 shows in themH–mt plane the constraints set by the measurements of thefour sensitive observables:23) �‘, sin2 �lept
eff , R0b ¼ �b=�had,
and mW. Note that R0b is insensitive to mH and provide
constraint on mt almost independently of mH. The contourshows the 68% confidence level region determined fromthe Z data only, without using mW and mt. This can becompared to the constraint band from mW. They are inagreement.
As an another way of presentation, Fig. 11 shows in themt–mW plane the constraint using indirect measurementsonly, and comparison with the direct measurements. Direct
Fig. 9. (Color online) Left: Cross-section for W pair production as a function of centre-of-mass energy. The points are data. See text about the curves.
Right: LEP and Tevatron W mass measurements and the average. LEP/SLD and LEP/SLD/mt points are indirect determinations from the standard
model fits (§7).
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS T. KAWAMOTO
111003-7
and indirect determinations are consistent, an important testof the standard model at the level of radiative corrections.Also shown in Fig. 11 are trajectories corresponding togiven values of Higgs boson mass mH. Comparison of theselines and confidence contours indicates that the data aresensitive to the Higgs boson mass, and they prefer low mH.
Using all data including mW and mt a constraint on theHiggs boson mass is obtained. Figure 12 shows ��2 ¼�2 � �2
min of the fit as a function of mH. The minimum �2/d.o.f is 18.2/13 which corresponds to the probabilityof 15.1%. Higgs mass at the minimum �2 is 76 GeV. The95% upper bound on mH, corresponding to the value which
gives ��2 ¼ 2:7, is 144 GeV taking into account theoreticaluncertainties, while the lower bound on mH obtainedfrom the direct searches at LEP is 114.4 GeV at the 95%confidence level.
8. Conclusion
Twelve years of running of the two eþe� colliders, LEPand SLC, was very fruitful. Determination of the number oflight neutrino species is one of the most important results.Direct tests of the gauge boson self-couplings should also benoted. Precision results from LEP/SLC allowed tests of thestandard model to the level of radiative correction, and itseems the standard model, based on the gauge symmetryand the spontaneous symmetry breaking due to the Higgsmechanism, is well established. Agreement between indirectand direct determinations of mW and mt is a demonstration ofsuch tests. Similar analysis has been repeated to predict theHiggs mass. Current knowledge on the probable Higgs massis in the range of 114.4 –144 GeV as inferred from the resultsof direct searches at LEP and indirect bound through theanalysis of precision measurements in the framework ofthe standard model.
Further improvements of experimental data will comein future; new information from the current and futureexperiments include a new set of mW and mt measurementsfrom Tevatron. LHC will further contribute to theseprecision measurements, and more importantly discoveryand studies of the nature of the Higgs boson, and probablysomething new beyond the standard model. Future eþe�
linear collider will follow to make a new round of precisionand systematic survey of physics, hopefully be includingnew physics.
Acknowledgements
The author wishes to thank the colleagues of the LEPelectroweak working group.
0
1
2
3
4
5
6
10030 300
mH [GeV]
Δχ2
Excluded Preliminary
Δαhad =Δα(5)
0.02758±0.00035
0.02749±0.00012
Theory uncertainty
mLimit = 144 GeV
Fig. 12. (Color online) ��2 as a function of mH (the curves). The shaded
band indicates the size of theoretical uncertainties. The dashed curve
correspond to an alternative choice of the determination of ��ð5Þhad. The
shaded area is the region of mH excluded by the direct searches.
80.3
80.4
80.5
160 180 200
mt(GeV)
mw(G
eV)
114300
1000mH
LEP+SLD
LEP+Tevatron (prelim.)
68% CL
Fig. 11. (Color online) A comparison in the mt–mW plane of the indirect
determinations from LEP and SLD data and the direct measurements at
LEP and Tevatron. In both cases the 68% probability contours are shown.
Also shown is the standard model relation between two masses for given
values of mH.
140
160
180
200
10 102
103
mH(GeV)
mt(G
eV)
(1)
(1) Z data
Γl sin2θefflept
Rb mW
Rb
sin2θefflept
mW
Γl
Fig. 10. (Color online) The bands show the regions corresponding to
the one standard deviation ranges of the measurements of mW, �‘, R0b, and
sin2 �lepteff . The contour shows the 68% probability area determined from a
global fit to the Z data.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS T. KAWAMOTO
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1) LEP Design Report, CERN-LEP/84-01, CERN, 1984; S. Myers and E.
Picasso: Contemp. Phys. 31 (1990) 387; D. Brandt, H. Burkhardt, M.
Lamont, S. Myers, and J. Wenninger: Rep. Prog. Phys. 63 (2000) 939.
2) SLAC Linear Collider Conceptual Design Report, SLAC-R-229,
SLAC, 1980.
3) G. S. Abrams et al. (Mark-II Collaborations): Phys. Rev. Lett. 63
(1989) 724.
4) K. Abe et al. (SLD Collaboration): Phys. Rev. D 53 (1996) 1023.
5) K. Abe et al. (SLD Collaboration): Phys. Rev. Lett. 78 (1997) 2075.
6) http://aleph.web.cern.ch/aleph/; D. Decamp et al. (ALEPH Collabo-
ration): Nucl. Instrum. Methods Phys. Res., Sect. A 294 (1990) 121; D.
Buskulic et al. (ALEPH Collaboration): Nucl. Instrum. Methods Phys.
Res., Sect. A 360 (1995) 481.
7) http://delphi.web.cern.ch/Delphi/Welcome.html; P. Aarnio et al.
Experimental Highlights of Electroweak Physics at Hadron Colliders
Beate HEINEMANN�
University of California, Berkeley, 366 LeConte #7300, Berkeley, CA 94720-7300, U.S.A.
(Received January 18, 2007; accepted April 24, 2007; published November 12, 2007)
Hadron colliders have played a pivotal role in establishing the electroweak sector of the standardmodel at high precision. Highlights were the discovery of the W and Z bosons in 1983, the discovery ofthe top quark in 1995 and the high precision measurements of the mass of the W boson and the top quark.The Higgs boson is the final missing piece in the theory of electroweak interactions. In this article thediscoveries and precision measurements of the W boson and top quark are reviewed, and the status ofsearches for the Higgs boson at hadron colliders is presented.
KEYWORDS: electroweak, hadron collider, W boson, top quark, Higgs bosonDOI: 10.1143/JPSJ.76.111004
1. Introduction
There have been two hadron colliders, the Sp �ppS and theTevatron, that were pivotal to establishing and to testing theelectroweak sector of the standard model. In 1982 theproton–antiproton collider Sp �ppS started operation at theEuropean Center of Nuclear Physics (CERN) in Geneva/Switzerland. It was the first collider ever built whereantiprotons were accelerated and collided. The proton andantiproton beams were collided at two experiments, calledUA1 and UA2, with a center of mass energy of
ffiffisp¼
540 GeV initially and 630 GeV later. In 1985 first collisionstook place at the Tevatron proton–antiproton collider inthe United States near Chicago. Here the beam energywas 900 GeV, yielding a center-of-mass energy of 1.8 TeV.However, at that time the luminosity was much smaller thanat the Sp �ppS, and it was only in 1989 that the Tevatronphysicists had sufficient data to surpass the measurements bythe UA1 and UA2 collaborations. At the Tevatron there aretwo experiments, called CDF and D0. About 100 pb�1 ofintegrated luminosity was collected between 1989 and 1996in so-called Run I. Then both the accelerator and theexperiments were upgraded, the energy was increased to1.96 TeV, and about 1.5 fb�1 of data have been collected in‘‘Run II’’ between March 2002 and December 2006.
The electroweak force is mediated by the gauge bosons,namely the W� and the Z boson.1) The electroweaksymmetry is broken via a Higgs field that has a vacuumexpectation value of 246 GeV. This mechanism of electro-weak symmetry breaking causes the W� and the Z boson tobe massive, and also the fermions, e.g., the electron or thetop quark. The stronger any particle interacts with the Higgsboson the more massive it is. Within the standard modelthere are precise predictions for the relationships between,e.g., the mass of the W� and Z bosons, the Higgs boson, andthe top quark, and the couplings of the gauge and Higgsbosons to any fermions and bosons. Thus measuring anyof the masses or couplings of these particles is critical tofurther constrain and test the electroweak theory, and hasbeen a major focus of hadron collider experiments.
In the following firstly hadron colliders are introduced
and the experimental challenges are described. Then thediscovery and the precision measurements of the W� and Z
boson quantities are highlighted. A review of the discoveryand precision measurements of the top quark is presented.Finally the current knowledge of the Higgs boson isreviewed and perspectives for a discovery are given.
2. The Challenge of a Hadron Collider
Analyzing data at a hadron collider is significantly morechallenging than at a lepton collider. The reason is thathadrons are not point like fundamental objects, but ratherthey are composed of quarks and gluons. A schematicdiagram of a proton–antiproton collision is shown in Fig. 1for W� production.
This causes several experimental challenges:. The composition of a hadron is known on average but
unknown for any given interaction. Thus even thoughthe p �pp center-of-mass energy is well known the actualenergy of the collision is not. The outgoing non-interacting partons are not measured in the detectorsas they travel along their original direction and thusescape through the beam pipe. Therefore energyconservation which is a powerful constraint at leptoncolliders can not be exploited at hadron colliders.However, since the outgoing partons carry negligibletransverse momentum the constraint that is used athadron colliders is the conservation of the momentumin the transverse plane.
. When one of the partons in the hadron undergoes a hardinteraction the hadron looses its color neutralness, and
SPECIAL TOPICS
p
p
q
q ±W±μ
ν,ν
X
X
Fig. 1. Schematic diagrams for p �pp! W� ! ��� production. The
systems X indicate the particles due to the proton remnant that are
produced in p �pp collision in addition to the hard scattering process.
the remaining partons interact strongly with each otheror the partons from the other hadron. These rather softinteractions cause energy spray into the detectors that isunrelated to the hard scattering process and must beaccounted for experimentally.
. Quarks and gluons are much more likely to interactstrongly than electromagnetically or weakly. In partic-ular at low energy the coupling constant of the stronginteraction, �s, is much larger than the electromagneticcoupling constant, �em, or the weak constant. E.g., atMZ ¼ 91 GeV �s � 0:118 and thus more that 10 timeshigher than �em � 1=128. Therefore there is a largebackground from strong parton interactions to W�
bosons in the quark decay modes. This large back-ground makes it very difficult to observe the W� or Zboson in its decay to quarks. Thus most measurementsat hadron colliders are made using the lepton decaymodes.
3. The Electroweak Gauge Bosons
The dominant processes for W� production at hadroncolliders are �uud! W� and u �dd! Wþ and �uuu! Z and�ddd! Z for Z production. The processes are illustrated inFig. 2.
The best modes for observing W� and Z bosons are thelepton decay modes. The standard model predicts about 10%of all W� bosons decay into an electron (or positron) and anelectron-neutrino, and 10% into a muon and a muon-neutrino. Similarly, about 3% of the Z boson are expected todecay into eþe� and 3% decay into �þ��. These leptonicdecay modes were used for the observation and for thesubsequent precision measurements at hadron colliders.
3.1 Discovery of the W� and Z bosonsThe primary goal of the Sp �ppS collider and its experiments
was the observation of the W� and Z bosons.2) Thepredictions of these bosons had been made as a consequenceof the electroweak theory:1) W bosons were thought to beresponsible for the � decay (q! q0eþ�e) and Z bosons werethe prime candidate for explaining the observation of neutralcurrents (�q! �0q0) at the Gargamelle bubble-chamberexperiment in 1973.3) While in these processes the inter-mediate vector bosons were exchanged in the t channel, aresonance production in the s-channel could occur in quark–antiquark interactions: q �qq! Wþ and q �qq! Z.
For W� bosons the experimental strategy was to searchfor high energy electrons or muons and an imbalance in thetotal transverse energy, called ‘‘missing ET’’ (E=T), measuredin the event. The measurement of E=T was a novel idea at thattime, and the UA1 detector was designed to be as hermeticas possible to allow this quantity to be measured as this
technique only works if all the transverse energy produced inthe collision is detected with high efficiency. In January1983 the UA1 collaboration published the observation of sixW� ! e�� candidate events,4) and measured the W� bosonmass to be 81� 5 GeV/c2 in excellent agreement with theprediction5–7) of 82:0� 2:4 GeV/c2. Only a few monthslater, in June 1983 the UA1 collaboration published theobservation of four Z ! eþe� þ X candidates and one Z !�þ�� þ X event,8) and measured the Z boson mass as95:2� 2:5 GeV/c2. For these discoveries Carlo Rubbia andSimon van der Meer were awarded the Nobel Prize in 1984.
3.2 Measurement of the W boson massSince the neutrino cannot be measured directly and only
its transverse momentum can be inferred from the measure-ment of E=T the W boson mass can not be measured directlyas the invariant mass of the lepton and the neutrino. Insteadthe ‘‘transverse’’ mass is used, which is defined as
where pTð‘Þ and pTð�‘Þ are the transverse momenta of thecharged lepton and the neutrino respectively and �‘� is thedifference in azimuthal angle between the charged leptonand the neutrino.
In the publication of the observation a first massmeasurement was presented at MW ¼ 81� 5 GeV based onthe six candidates. At the end of the Sp �ppS run in 1990, UA2presented a measurement9) of MW ¼ 80:49� 0:43(stat.)�0:21(syst.) GeV/c2. This measurement used an integratedluminosity of 13 pb�1. For this precise measurement theratio of the W boson to the Z boson mass was determined toreduce systematic uncertainties, e.g., on the precision of thelepton energy determination. The W boson mass was thenextracted using the precise determination of the Z bosonmass at the LEP and SLC eþe� colliders.10–14)
In August 1990 the CDF experiment presented the firstW� mass measurement15) at the Tevatron of MW ¼ 79:91�0:39 GeV/c2 based on an integrated luminosity of about4 pb�1, for the first time improving on the precision of theUA2 experiment. At the end of the Tevatron Run I period,in 1996, a luminosity of about 100 pb�1 was available andCDF and D0 presented measurements with substantiallyimproved precision. The CDF and D0 measurements werecombined16–18) to yield a Tevatron average of 80:456�0:059 GeV/c2.
Recently the CDF collaboration has presented a newmeasurement based on a luminosity of 200 pb�1 of Run IIdata. This yields the world’s single most precise measure-ment of 80:413� 0:048 GeV/c2. It is based on a combinedfit to the transverse mass and lepton pT distributions in boththe electron and the muon decay modes. In the electrondecay mode it is based on 63,964 candidate events, and inthe muon decay mode on 51,128 events. The transverse massdistributions and the corresponding mass fits are shown inFig. 3 for electrons and muons. The measurement is mostsensitive to the falling edge of the spectrum 76 < mT <86 GeV/c2. This is by far the highest precision measurementmade at a hadron collider with the remarkable precision ofonly 0.059%. A summary of the W mass measurements athadron and lepton19) colliders is presented in Fig. 4. Theworld average is at present 80:398� 0:025 GeV/c2.
u
d–
W+
u / d
u / d– –
Z
Fig. 2. Lowest order diagrams for W and Z boson production.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS B. HEINEMANN
111004-2
It is interesting to review the uncertainties of the Tevatronexperiments to judge whether further improvements can beexpected in the future. A summary of the dominant sys-tematic uncertainties is given in Table I for the recent CDFmeasurement for the fit of the transverse mass distribution.
The dominant uncertainty is the statistical precision whichwill be improved naturally with more data. The second
largest uncertainty is the lepton energy scale. This isdetermined using data from the J= ! �þ��, �!�þ��, and Z ! �þ�� decays for the track momentumscale, and by using the ratio of the calorimetric energymeasurement to the track momentum measurement forelectrons in W ! e� decays. They will be improved as moredata are collected as their precision is mostly limited by thestatistical precision of these samples. Most other uncertain-ties are derived using the Z ! ‘þ‘� samples, in particularthe recoil and the pTðWÞ model will improve with increasingZ sample size. The only systematic uncertainties that cannot be improved with increasing statistical precision directlyare the parton distribution functions and QED radiationwhere the analysis relies on external input. With 2 fb�1 ofluminosity it is expected that the Tevatron experimentsachieves a precision of 20 – 30 MeV, depending on whetherthe uncertainties on the parton distribution functions and theQED radiation can be further reduced.
3.3 Other measurements of W and Z boson propertiesIn addition to the W boson mass a large number of other
interesting properties has been measured at hadron colliders.They are shortly summarized here.
The width of the W boson is measured directly andindirectly. It is sensitive to possible decays of the W to othernew particles due to contributions from particles beyond theSM. The direct measurement uses also the transverse massdistribution but the width sensitivity arises mostly from thetail of the distribution, 100 < mT < 200 GeV/c2. Combiningall the results from the Tevatron20) it is found to be2078� 87 MeV/c2 in good agreement with the LEP value21)
of 2128� 88 MeV/c2 and the SM prediction22) of 2092� 3
MeV/c2. Indirect determinations are also in good agreementwith the SM value.23)
In addition the leptonic branching ratios of the W ,23)
the coupling of the W and Z bosons to photons,24–26)
the production of two W or Z bosons,27–30) the forward–backward asymmetry of the Z boson,31) and severalmeasurements related to the production of W or Z bosonswere made.
The quark decay modes are very difficult to detect athadron colliders since there is a huge background from QCDdijet production that obscures the signal. The quark decay
(eν) (GeV)Tm60 70 80 90 100
even
ts /
0.5
GeV
0
500
1000
1500
= (80493 ± 48stat) MeVWM
/dof = 86 / 482
-1L dt ≈ 200 pbCDF II preliminary
(μν) (GeV)Tm60 70 80 90 100
even
ts /
0.5
GeV
0
500
1000
= (80349 ± 54stat) MeVWM
/dof = 59 / 482
-1L dt ≈ 200 pbCDF II preliminary
χ χ
Fig. 3. (Color online) Transverse mass distribution for W ! e� (left) and W ! �� (right) events. The data (points) are compared to the simulation
(histogram) using the best fit MW value. Also shown are the fit values for the W mass with the statistical uncertainty. The histograms near the bottom
show the background from non-W production.
)2W Boson Mass (MeV/c80100 80200 80300 80400 80500 80600
World Average 200780398 ± 25
CDF Run II (prel.)80413 ± 48
ALEPH80440 ± 51
OPAL80416 ± 53
L380270 ± 55
DELPHI80336 ± 67
DO Run I80483 ± 84
CDF Run I80433 ± 79
Fig. 4. (Color online) W boson mass measurements from CDF and D0
Run I, from the LEP experiments and from CDF Run II. Also shown is
the world average.
Table I. Systematic uncertainties for Run II CDF W� mass measurement.
Shown are the uncertainties for the fit to the transverse mass distribution,
separately for the electron and muon channel. Also given are the
uncertainties common to both measurements. All values are given in
MeV.
Source W ! e�e W ! ��� Common
Statistical 48 54 0
Lepton energy 31 17 17
Recoil and pT (W) 11 11 11
Selection and backgrounds 12 10 5
Parton density functions 11 11 11
QED radiation 11 12 11
Total 62 60 26
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111004-3
modes were, however, established successfully32) but are notused to make precision measurements of the W� and Z
boson properties.In all of these measurement the electroweak theory was
confirmed and in many of them the precision is similar tothat of lepton colliders.
4. The Top Quark
In the standard model top quarks are mostly produced inpairs and at the Tevatron q �qq annihilation contributes about85% and the remaining 15% arise from gg interactions. Theleading order diagrams are shown in Fig. 5.
The top quark was discovered by the CDF and D0experiments in 1995.33–35) It has a surprisingly large mass ofabout 175 GeV/c2, 40 times larger than the second heaviestquark. Its large mass makes the top quark a critical particlefor the electroweak theory as the Higgs boson coupling toparticles is stronger the larger the mass of that particle is.
For this mass, the top quark nearly always decays into a b
quark and a W boson. The reaction chain is
p �pp! t�tt þ X! WþbW�bþ X!‘þ�b‘��bþ X (1)
q �qqb‘��bþ X (2)
q �qqbq �qqbþ X (3)
8<:
and in the following we call (1) the ‘‘dilepton’’ mode, (2) the‘‘lepton+jets’’ mode, and (3) the ‘‘all-jets’’ mode.
4.1 Observation of the top quarkIn 1995 the CDF and D0 collaborations claimed obser-
vation of the top quark. The observation was based on thedilepton and lepton+jets modes. In the lepton+jets modeCDF required an electron or muon from the W decay, largeE=T and more than two jets in the final state, and at least oneof them should be identified as a b-quark. The b-quark wasidentified by detecting a secondary vertex due to the b
hadron decay since the b hadron has a lifetime of about1.5 ps�1 and flies about 460 mm before it decays. This decaycould be resolved by the new CDF silicon microstripdetector. CDF observed 27 b-tags based on the secondaryvertex algorithm for events with three or more jets comparedto a background of 6:7� 2:1 tags. In addition CDF usedanother algorithm to tag b-quarks exploiting the semi-leptonic decay rates of b hadrons: in this analysis 23 tagswere found compared to a background of 15:4� 2:0. In thedilepton mode 6 events were observed with a backgroundof 1:3� 0:3 events. Combining the excess in each of theanalyses an excess of 4:8� compared to the SM backgroundprediction was found.
The D0 experiment was not equipped with a siliconmicrostrip detector: the observation at D0 was based on thedilepton mode, on the semi-leptonic b-hadron decay tagging,and on the sum of the transverse energies of all jets inlepton+jets events. In total 17 events were observed
compared to a background estimate of 3:8� 0:6 events,corresponding to an excess with a significance of 4:6�.
Both experiment found the events to be consistent withcoming from a top quark, and presented measurements of thetop quark mass of 176� 13 GeV/c2 (CDF) and 199� 30
GeV/c2 (D0). These direct measurements were in excellentagreement with the prediction using electroweak precisionobservables at lepton colliders36) of 150 – 210 GeV/c2
depending on the mass of the Higgs boson.The analysis techniques used for the observation of the
top quark are still used in similar form today by the Tevatronexperiments. After an upgrade between 1996 and 2001 theD0 collaboration also inserted a silicon microstrip detectorand uses this now for identifying top quarks in Run II.
4.2 Precision measurements of the top quark massThe top quark mass has been measured in each of the
signatures, i.e., the dilepton, the lepton+jets, and the all-jetsmode. The main challenges of the measurement are assign-ing the final state particles that originate from the same topquark correctly, understanding the background and control-ling the systematic uncertainties, in particular on the jetenergy measurement.
The highest precision measurements are achieved in thelepton+jets mode as in this sample the background isrelatively small (about 20% when requiring a b-quark tag),the samples are relatively large and nearly all final stateparticles can be measured. In the dilepton mode there are twoneutrinos escaping which complicates the analysis, and thesample is relatively smaller due to the small branching ratioof theW� to leptons. The all-jets samples is large but the largebackground due to QCD multi-jet production complicatesmeasurements in this decay mode. Therefore I will focus ondescribing the lepton+jets measurement in this article.
In the lepton+jets mode the four-momenta of the twob-jets, the two other jets and the lepton and px and pycomponent of the neutrino are measured. The two unknownvariables thus are the longitudinal momentum of theneutrino and the mass of the top quark. In addition thereare three constraints in each event:
. the lepton-neutrino invariant mass must equal the W�
boson mass. the invariant mass of the non-b-quark jets must equal
the W� boson mass. the invariant mass of the decay products of the top must
be the same as that of the anti-topSince there are three constraints but only two unknowns
the system is constrained well enough to measure themass of the top quark. In fact since it is over-constrained it iseven possible to fit for the jet energy scale as an additionalfree parameter. This has an advantage since the jet energyscale is difficult to constrain with high precision a priorito a precision better than 3%. The reason is that jets arecomplicated objects, and in particular the hadronization of aparton to a jet is only understood by phenomenologicalmodels based on data but there are no strict theoreticalcalculations one could rely on. By fitting for the jet energyscale inside the top sample itself the uncertainty becomesstatistical and can improve with increasing data samplesnaturally. It results therefore in a higher precision of themeasurement.
q
q
g
t
t
g
g
t
t
g
g
g
t
t
Fig. 5. Feynman diagrams for top quark pair production.
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The current single best measurement was made by theCDF collaboration38) and used a technique called ‘‘matrix-element’’ method. For each event a probability distributionas function of top quark mass and jet energy scale iscalculated, and the final measurement of the top mass ismade by forming a likelihood as a product of the singleevent probabilities.39) The likelihood is shown in Fig. 6 asfunction of the top quark mass and the jet energy scale. Themeasurement yields mtop ¼ 170:9� 2:2(stat.)� 1:4(syst.)
GeV/c2 where the jet energy scale uncertainty is part ofthe statistical error and the dominant systematic uncertain-ties arise from the modeling of QCD radiation, and residualuncertainties on the jet energy scale.
The top quark mass was additionally measured in theother decay channels38,40–42) by both CDF and D0 experi-ments using Run I and Run II data, and all measurementsare found to agree well with each other as is seen in Fig. 7.They are thus combined43) to give the world average topmass of mtop ¼ 171:4� 2:1 GeV/c2, corresponding to aprecision of 1.2%.
4.3 Other top quark measurementsA large number of other measurements related to the top
quark have also been made and found to all be consistentwith the SM predictions, e.g., measurements of the topquark production cross section,44–48) of the charge of the topquark,49) the helicity of the W boson in top quarkdecays50–53) and the branching ratio54) t! Wb have beenmade. All of them confirmed the SM predictions and nodeviation from the expectations have yet been seen.
Recently a first sign of the single top quark productionprocess that proceeds via the electroweak interaction hasbeen claimed.55,57) The results are consistent with the SMexpectation56) but the precision is rather limited, and moredata are required to claim conclusive observation of singletop production and to test its agreement with the SMprediction with better precision.
5. The Higgs Boson
The Higgs Boson is often called the ‘‘holy grail’’ of theelectroweak sector of the standard model as it plays thecritical role of generating the masses of the other SMparticles through the mechanism known as electroweaksymmetry breaking. It is the only SM particle that has notyet been observed experimentally, and finding it is a majorgoal of the Tevatron and of the large hadron collider (LHC)experiments at CERN that will start operating end of 2007.The best experimental bounds so far have been set by theLEP experiments that exclude masses below 114.4 GeV/c2
at 95% confidence level (CL).58)
5.1 Indirect constraints on the Higgs boson massWithin the context of the standard model the precision
measurements of the W bosons and top quark mass canbe used to constrain the Higgs boson mass via radiativecorrections. The W boson mass is given by
M2W ¼
��ffiffisp
GF sin2 W
1
1��r;
where � is the fine structure constant, GF the Fermi constant,W the Weinberg angle, and �r is a term that contains theradiative corrections. This terms �r is proportional to m2
top
and to lnðMHÞ. Example diagrams that introduce thisdependence are shown in Fig. 8.
The W boson mass is shown versus the top quark mass inFig. 9. Shown are the direct measurements from theTevatron and LEP2 colliders, the indirect constraints fromLEP1 experiments and SLD and lines of constant Higgsboson mass. The direct and indirect measurements arein good agreement. The direct measurements, however,indicate a rather low Higgs boson mass which is alreadyexperimentally excluded by LEP2. The most probable valueis determined as MH ¼ 80þ36
�26 GeV/c2, and at 95% CL Higgsmass valuers above 153 GeV/c2 are excluded. Higher
)2
(GeV/ctm165 170 175
JES
0.95
1
1.05Δln L=8.0
Δln L=4.5
Δln L=2.0
Δln L=0.5
Fig. 6. Two-dimensional likelihood of the jet energy scale versus the top
quark mass. Shown is the value of minimal likelihood (cross) and
contours of equal likelihood corresponding to 1�, 2�, 3�, and 4�. The jet
energy scale is defined as the ratio of the jet energy scale in data and
Fig. 7. (Color online) Top mass measurements by the CDF and D0
collaborations in the individual analysis modes. Also shown is the world
average value.
+W +W
t
b H+W
Fig. 8. Feynman diagrams that for radiative corrections to the W boson
mass due to loops involving the top quark and the Higgs boson.
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precision is required to see whether this slight tensionbetween the direct limit on the Higgs boson mass and theindirect prediction will become more significant or whetherit is just a statistical fluctuation.
The current top quark mass precision of ðmtopÞ ¼ 2:1GeV/c2 corresponds to a Higgs mass uncertainty of ðMHÞ=MH � 18% while the current world W mass precision of 25MeV/c2 corresponds to a Higgs mass precision of ðMHÞ=MH � 35%. Thus currently the W mass precision limits theindirect constraints on the Higgs boson mass, and theprecision on the Higgs boson mass would benefit most fromimprovements in the W mass precision.
5.2 Direct searches for the Higgs bosonIn p �pp collisions at the Tevatron Higgs bosons are
predominantly produced via gg fusion: this cross section isabout 1 pb at MH ¼ 120 GeV/c2, and falls to 0.4 pb at160 GeV/c2. The subdominant production process is Higgs-radiation off a W or Z boson, and the cross sections are 0.15and 0.09 pb for WH and ZH production respectively forMH ¼ 120 GeV. The Feynman diagrams for these processesare shown in Fig. 10. This can be compared to, e.g., the t�tt
production cross section of 7 pb or the W production crosssection of about 2.7 nb. The main difficulty in the Higgsboson search arises from the very small signal yield,compared to the significant backgrounds that are present.
At low mass, MH < 135 GeV/c2, the dominant decaymode of the Higgs is to a b �bb pair. This decay mode cannot bedetected in the gg! H channel due to a large backgroundfrom QCD b �bb production. Thus at low mass the most
sensitive analyses use the WH and ZH production modes inthe leptonic decay channels of the W and the Z. The threemost sensitive analyses are WH ! ‘�‘b �bb, ZH ! ‘þ‘�b �bb,and ZH ! � ���b �bb. The invariant mass of the two b-jets isexpected to peak at the H boson mass, and this is exploitedto discriminate WH production from the dominant back-grounds, Wb �bb=c �cc, and t�tt production. The invariant massdistribution of the two b-jets is shown for the WH analysisin Fig. 11. A review of all low mass Higgs searches is givenin ref. 59.
At larger masses, MH > 135 GeV/c2, the Higgs bosondecays predominantly to WþW�. The standard modelbackground arises mostly from the relatively low crosssection SM WW production process and this analysis canmake use of the large gg! H production cross section. Theanalysis has been performed requiring both W’s to decay toleptons: the analysis channels are WW ! e�ee�e, WW !���e�, WW ! ������. Large E=T is required to indicatethe presence of neutrinos. To discriminate the signal fromthe SM background a large number of selection cuts ismade.60,61) The final discriminant is shown in Fig. 12 forthe D0 analysis of p �pp! H ! WþW� ! e���� ���: it isthe difference in azimuthal angle between the two leptons.Since the Higgs is a scalar particle and due to helicityconservation the two leptons are typically relatively close toeach other, while for the backgrounds the lepton angles areless correlated.
Presently the Tevatron Higgs analyses are not sensitive tothe predicted SM rate of Higgs production. Even whencombining60) all the analyses from both CDF and D0 there isno sensitivity yet to SM Higgs production. This is seen inFig. 13 where the ratio of the experimental cross sectionlimit at 95% CL divided by the SM cross section is shown.At low MH ¼ 115 GeV/c2 the observed limit is a factor of10 larger than the SM prediction, and at MH ¼ 160 GeV/c2
it is a factor 4 above the SM prediction.The Tevatron will increase the integrated luminosity by
about a factor 10 compared to this result. That will improve
80.3
80.4
80.5
150 175 200
mH [GeV]114 300 1000
Δα
mt [GeV]
mW
[GeV
]
68% CL
LEP1 and SLD
LEP2 and Tevatron (prel.)
Fig. 9. (Color online) W boson mass versus top quark mass: shown are
the direct measurements fro LEP2 and the Tevatron, the indirect
constraints from LEP1 and SLD and lines of constant Higgs mass at
114, 300, and 1000 GeV/c2.
g
g
tH
q
q*
WW
H q
q*
ZZ
H
Fig. 10. Feynman diagrams for gg! H (left), q �qq! WH (middle), and
q �qq! ZH (right) production.
)2
Dijet Mass (GeV/c
0 50 100 150 200 250 300 350 400 450 500
)2
Eve
nts
/ 20
(GeV
/c
0
2
4
6
8
10
12
)-1
Data(955pb
W+Heavy Flavor
Mistag
Non-W QCD0Diboson/Z
(6.7pb)+Single Toptt
Background Error
)2=115GeV/cH 10 (m×WH
CDF Run II Preliminary
ττ
Fig. 11. (Color online) Dijet invariant mass distribution of two identified
b jets for events with a W boson. The data (points) are compared to the
SM prediction: the dominant background contributions are W+Heavy
Flavor and t�tt production. Shown as open histogram is also the SM Higgs
contribution, increased by a factor 10 compared to the SM cross section.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS B. HEINEMANN
111004-6
the limits by about a factor 3 if the analyses are keptunchanged which is still not sufficient to be sensitive to SMHiggs production. Thus the experiments are working onfurther improving the analyses and hope to gain anotherfactor of 3 by making experimental improvements, e.g.,increasing the b-jet identification efficiency, improving theacceptance for leptons, improving the Higgs mass resolutionand using advanced analyses techniques (e.g., neural net-works) to gain additional discrimination power. If this isachieved it could be possible for the Tevatron to see afirst glimpse of the Higgs boson. However, ultimately itis expected that the experiments at the LHC will for sureeither detect a Higgs boson or exclude its existence at allmasses. It is expected that by 2010 a Higgs boson is eitherfound or excluded. The discovery of the Higgs boson wouldbe yet another amazing triumph of the electroweak theory.
6. Conclusions and Outlook
The Sp �ppS and Tevatron p �pp colliders have been criticalin establishing the theory for electroweak interactions, andin testing it at high precision. Highlights are the observationof W and Z bosons at the Sp �ppS, the observation of thetop quark at the Tevatron, and the subsequent precisionmeasurements of 0.06% accuracy for the W boson and 1.2%accuracy for the top quark. So far the electroweak theory haswithstood all these experimental tests but just now we maystart to face an interesting situation where there is a slighttension between the indirect determination of the Higgsboson mass and the direct limits.
The Tevatron collider experiments are still in progress andare expected to improve the top quark and W mass precisionby about a factor of two in accuracy. Further improvementsof these precision measurements are expected by theexperiments at the Large Hadron Collider starting in 2008.It is possible that the Tevatron will see a first hint for a Higgsboson in the coming years, but ultimately the LHC colliderwill be required to observe the Higgs boson conclusively andto measure its properties. These measurements will furtherchallenge the electroweak sector of the standard model, andit remains to be seen whether the electroweak theory willcontinue to be so successful or whether it will break down,indicating presence of new particles and new laws of naturethat yet need to be discovered.
1) S. L. Glashow: Nucl. Phys. 22 (1961) 579; A. Salam: Proc. 8th Nobel
Symp., Stockholm, 1968, p. 367; S. Weinberg: Phys. Rev. Lett. 19
(1967) 1264.
2) C. Rubbia, P. McIntire, and D. Cline: Proc. Int. Neutrino Conf.,
Aachen, 1976 (Vieweg, Braunscheig, 1977) p. 683; Study group,
Design study of a proton–antiproton colliding beam facility, CERN/
PS/AA 78-3 (1978).
3) F. J. Hasert et al. (Gargamelle Collaboration): Phys. Lett. B 46 (1973)
138.
4) G. Arnison et al. (UA1 Collaboration): Phys. Lett. B 122 (1983) 103.
5) A. Sirlin: Phys. Rev. D 22 (1980) 971.
6) W. J. Marciano and A. Sirlin: Phys. Rev. D 22 (1980) 2695.
7) C. H. Llewellyn Smith and J. A. Wheater: Phys. Lett. B 105 (1981)
486.
8) G. Arnison et al. (UA1 Collaboration): Phys. Lett. B 126 (1983) 398.
9) J. Alitti et al. (UA2 Collaboration): Phys. Lett. B 241 (1990) 150.
10) G. Abrams et al. (Mark II Collaboration): Phys. Rev. Lett. 63 (1989)
2173.
11) D. Decamp et al. (ALEPH Collaboration): Phys. Lett. B 231 (1989)
519; D. Decamp et al. (ALEPH Collaboration): Phys. Lett. B 235
(1990) 399.
12) P. Aarnio et al. (DELPHI Collaboration): Phys. Lett. B 231 (1989)
539.
13) B. Adeva et al. (L3 Collaboration): Phys. Lett. B 231 (1989) 509; B.
Adeva et al. (L3 Collaboration): Phys. Lett. B 237 (1990) 136.
14) M. Z. Akrawy et al. (OPAL Collaboration): Phys. Lett. B 231 (1989)
530.
15) F. Abe et al. (CDF Collaboration): Phys. Rev. Lett. 65 (1990) 2243.
16) V. M. Abazov et al. (CDF and D0 Collaborations): Phys. Rev. D 70
(2004) 092008.
17) A. Affolder et al. (CDF Collaboration): Phys. Rev. D 64 (2001)
052001.
18) V. M. Abazov et al. (D0 Collaboration): Phys. Rev. D 66 (2002)
012001.
19) T. Kawamoto: J. Phys. Soc. Jpn. 76 (2007) 111003.
20) Tevatron electroweak working group (for the CDF and D0 Collabo-
rations): hep-ex/0510077.
21) The LEP electroweak working group (for the ALEPH, DELPHI, L3
Department of Physics, Kanagawa University, Yokohama 221-8686
(Received February 23, 2007; accepted May 7, 2007; published November 12, 2007)
Without flavor physics, we would not have the standard model of elementary particles today. The goalof flavor physics is to get at the most fundamental theory of nature. This is done by looking forphenomena which can not be explained by the standard model. To do this, we need to study flavorphysics from every angle possible. In support of this, we review how flavor physics has contributed to ourunderstanding of nature over the past 60 years.
In 1940’s and 50’s, particles with very strange behaviorhave been found, for example, K meson, �0, ��, and ��
baryons. When they are produced in collisions:
�þ P! �Kþ; �0Kþ; ð1Þ
their production rates are as large as those of typical stronginteraction. But when they decay, for example, as in:
�! Pþ ��;K� ! �� þ �0;
ð2Þ
their decay rates are too slow compared to a typical strongdecay like N� ! Pþ �. These considerations lead Gell-Mann,1) Nakano and Nishijima2) to postulate that theseparticles possess an internal quantum number which theycalled strangeness. Strangeness assignments are: 0 forproton, neutron, and � meson, �1 for �, �0, ��, and K�,þ1 for Kþ, �2 for ��, etc.
The strangeness is conserved in strong and electromag-netic interactions. The interactions shown in eq. (1) containparticles with opposite strangeness so that the final states asa whole are zero strangeness states — thus these productionprocesses conserve strangeness and they get producedthrough strong interaction.
Particles with strangeness cannot decay to the final statewith 0 strangeness through strong and electromagneticinteractions. Thus they decay through weak interaction,and they have relatively long life time. For example,particles decaying through strong interaction may havetypical lifetime of order 10�23 s, while those decayingthrough weak interaction have typical life time of order10�10–10�15 s.
Flavor physics is a systematic study of particles with thesenew quantum numbers. These particles can live for long time,and they have time to expose interesting phenomena. Thusthe purpose of this study is to search for yet unprobed physicshidden behind flavor changing weak decays, and to learn thefundamental laws which govern interactions of matter.
This discovery of hadrons with the internal quantumnumber ‘‘strangeness’’ marks the beginning of a mostexciting epoch in particle physics that even now, sixty yearslater, has not yet found its conclusion.3)
2. First Suggestion of Parity Violation: The �–� Puzzle
Flavor physics immediately lead to something which wasquite unexpected! Two very strange decays have been foundfor charged strange mesons, namely:
�þ ! �þ�0;
�þ ! �þ�þ��:ð3Þ
The problem arose when ever more precise measurementsfailed to find any significant difference in either the massor the lifetime of the � and � mesons. This constituted the�–� puzzle: how could nature assign the same mass to twodistinct particles? Or even more baffling: how could naturecontrive to generate the same lifetime to two distinctparticles, the major decay channels of which possess totallydifferent phase space?
To explain it further, let us review the parity symmetry.Parity transformation is almost like transforming the worldto its mirror image. If ðr1; r2; . . .Þ denote the coordinatesof everything in this world, the parity symmetry P invertsthe sign of all the coordinates: ð�r1;�r2; . . .Þ.
The �þ was found to be a spinless state, and therefore,the 2� final state is in a positive parity state. (2� in a Swave state is symmetric under the interchange of theseparticles.) The angular distributions of the three pions fromthe �þ decay revealed the final state to carry zero totalangular momentum as well, but with negative parity! Itwas assumed that parity, like angular momentum, wasconserved by the relevant forces. The parity of the initialstate then coincides with that of the final state. With � and �exhibiting different parity, they had to be distinct objectsand thus indeed deserved different names.
It is difficult it to give up what we are use to. This puzzlecould be understood trivially if parity were not absolutelyconserved. For then � and � could represent merely twodecay modes of the same particle. Parity conservation hadbeen tested extensively in strong and electromagnetictransitions. Yet the breakthrough came when Lee and Yangpointed out in 19564) that this symmetry had not been probedyet in weak transitions.
This was how new important feature of weak interaction,the parity violation, was revealed through flavor physics.But this was only a beginning!
Today, we know that we can describe all hadrons interms of the quark model.5) There are six quarks and sixleptons as listed in Table I. The strange particles are thoseparticles whose constituents include a quark which carriesstrangeness �1, namely the stange quark s. For example,particles we mentioned in the introduction have quarkassignments: P ¼ ðuudÞ, N ¼ ðdduÞ, �þ ¼ ðu �ddÞ, Kþ ¼ ðu�ssÞ,�0 ¼ ðsduÞ, �þ ¼ ðuusÞ, and �� ¼ ðssdÞ. (Here �qq denotesthe antiparticle of quark q.) By studying � decays ofnuclei, neutron and muon, it was found that weakinteraction proceeds through ðV � AÞ � ðV � AÞ four-fermiinteraction:
L ¼ 4GFffiffiffi
2p Jy�J
�; ð4Þ
where
J� ¼ �dd����uþ ��������� þ �ee�����e: ð5Þ
and �� ¼ ð1� �5Þ=2. (In the definition of the current J�, qdenotes the quark field operator, and �qq denotes qy�0.) Matrixelements which cause beta decays are, for example:
The Lagrangian given in eq. (4) does not allow starangeparticles to decay. The strangeness changing interaction wasintroduced in eq. (4) in an ingenious way by Gell-Mann andLevy,6) which was verified by Cabibbo.7) The idea was toreplace d which appears in �dd����u by a rotated state:d0 ¼ d cos �c þ s sin �c, where �c is the Cabbibo angle. Intoday’s language, this replacement means that there is adistinction between mass eigenstate of quarks, d and s anda state which participates in weak interaction d0. This state iscalled a weak eigenstate.
The current can now be written as
J� ¼ J�S¼0� þ J�S¼1
� þ ��������� þ �ee�����e: ð7Þ
Here J�S¼0� ¼ cos �c �dd����u is a strangeness conserving
current, J�S¼1� ¼ sin �c �ss����u is a strangeness changing
current. Strangeness changing K decays are, for example:
The weak interaction Lagrangian given in eq. (4) violatesP maximally, as the current consists of both vector andaxial vector components ðV � AÞ. It can shown that thecharge conjugation symmetry, which transforms a particlestate to the corresponding antiparticle state C is alsoviolated in such a way that the combined symmetry CP isconserved.
4. The Standard Model
This theory of weak interaction, which we havebeen discussing, involves local four-fermi interaction,and it is unrenormalizable. We can imagine that the four-fermi interaction is generated by an exchange of a heavycharged spin 1 boson. But a field theory which containsmassive spin one charged boson is known to be alsounrenormalizable. The break-through came when Wein-berg8) wrote both weak and electromagnetic interaction interms of SUð2Þ � Uð1Þ gauge theory. The Lagrangian of thistheory contains no mass terms for the gauge bosons. Theirmasses are generated when the neutral Higg’s boson breaksthe gauge symmetry spontaneously. In this process gaugebosons become massive. This dynamics also generates allquark masses. Since field theory with massless spin oneparticles is renormalizable, this theory could have a chanceof being renornmalizable and contain massive spin onebosons at the same time. Indeed, this has been shown to bethe case.
Notice that in Table I, there is a family of leptonsassociated with each family of quarks. In this gauge theory,we need to consider leptons and quarks together in thefamily structure, otherwise infinities from anomalies arise.So, flavor theory predicts one-to-one correspondence be-tween quarks and leptons.
Because all experimental facts are consistent withthis theory, it is called the standard model (SM). Nomatter how difficult it is to find deviation from thistheory, we know that it is not the most fundamentaltheory. There must be a theory to which the SM is thelow energy effective theory. This is because we expectthe ultimate theory to give answers to all fundamentalquestions. Some of the questions are as follows. Why arethere three generations? Why mt=me � 3� 105? Whyneutrinos are almost massless? Why do quarks have theobserved mass structure, for example, mu ¼ 5 MeV andmt ¼ 178 GeV?
Flavor physics tries to find answers to these questions bysearching for deviations from the SM which will eventuallylead us to the more fundamental theory.
5. Quantum Mechanics of Particle–Antiparticle Mixing
K0, and �KK0 mesons are bound states of s �dd, and d �ss,respectively. Experimentally, they both decay to �þ��
states. Therefore we can imagine an interaction K0 $�þ�� $ �KK0. If they can turn into each other through weakinteraction, they are not eigenstates of the Hamiltonian.
There are many decay channels for the K meson and theHamiltonian is an infinite dimensional matrix. Diagonalizingit is almost impossible. However, Weisskopf–Wigner ap-proximation9) comes to our rescue.
Let us assume that we have an initial state, which is madeout of some linear combination of only K0 and �KK0.
Table I. Three families of elementary particles, quarks and leptons. These
quarks are called u: up; d: down; c: charm; s: strange; b: beauty; and t:
top. Leptons are called e: electron; �: muon; �: tau, and their
corresponding neutrinos �e: electron neutrino; ��: muon neutrino; and
�� : tau neutrino.
Generation First Second Third Electric charge
Quark u c t þð2=3ÞeQuark d s b �ð1=3ÞeLepton e � � �eNeutrino �e �� �� 0
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS I. SANDA
111005-2
�ð0Þ ¼ að0ÞjK0i þ bð0Þj �KK0i: ð9Þ
This state will evolve into some admixture of K0, �KK0, andother hadrons as time goes by. If we are interested only in K0
and �KK0 components at some later time, define
�ðtÞ ¼ aðtÞjK0i þ bðtÞj �KK0i �aðtÞbðtÞ
� �: ð10Þ
The result of Weisskopf–Wigner approximation states that,at some time t which is large compared to the stronginteraction time scale, �ðtÞ obays the following Schrodingerequation:
ih�@
@t� ¼ H�; ð11Þ
with
H ¼hK0 j H j K0i hK0 j H j �KK0i
h �KK0 j H j K0i h �KK0 j H j �KK0i
!: ð12Þ
Let us first show that CP symmetry implies hK0 j H j �KK0i ¼h �KK0 j H j K0i ¼ �. Using the unitarity of CP operator, wehave CPyCP ¼ CP�1CP ¼ 1. Then inserting these unitoperators:
h �KK0 j H j K0i ¼ h �KK0 j CPyCPHCP�1CP j K0i¼ hK0 j H j �KK0i;
ð13Þ
where we have used
CPjK0i ¼ j �KK0i; ð14Þ
and the statement of CP conservation: ½H;CP� ¼ 0.Let us now show that � is complex and thus H is not
hermitian. Note that � contains an absorptive part. That is,for example, � gets contribution from the intermediate stateK0! �þ�� ! �KK0 � ��þ�� . This contribution is complexdue to the scattering of �þ�� intermediate state. It can beshown that ��þ�� ¼ e2i0 j��þ��j, where 0 is the isospin 0,S wave �� phase shift. So, H is not hermitian and it leads tothe decay of the initial state �ð0Þ.
As mentioned above, a typical transition which contrib-utes to � is K0! ��! �KK0 and it is second order in weakinteraction. Therefore, � is truly infinitesimal compared tothe diagonal component which is of order MK . Neverthelessit dictates the eigenstates to be
jK1i �1ffiffiffi2p jK0i þ j �KK0i
� �;
jK2i �1ffiffiffi2p jK0i � j �KK0i
� �:
ð15Þ
as long as j�j jhK0 j H j K0i � h �KK0 j H j �KK0ij.Using eq. (14), we have CPjK1
2i ¼ �jK1
2i. Now, consider
a j�þ��i state. Pj�þ��i ¼ j���þi, and Cj�þ��i ¼j���þi. So, we have
CPj��i ¼ þj��i: ð16Þ
Therefore K1 has the same CP quantum number and it candecay to 2�, i.e.,
K1 ! 2�;
K2 9 2�:ð17Þ
The leading nonleptonic channel for K2 is then
K2! 3�: ð18Þ
The phase space for eq. (18) is very restricted compared tothe 2� decay — 3�M� ’ 420 MeV vs MðK2Þ ’ 500 MeV.Thus we expect the lifetime for the CP odd state K2 to bemuch longer than for the CP even state K1. Since K1 and K2
possess quite different lifetimes, it is customary to refer tothem as KS and KL, respectively, with S (L) referring toshort-lived (long-lived). Likewise we use MðK1Þ ¼ MðKSÞ ¼MS and MðK2Þ ¼ MðKLÞ ¼ ML. State-of-the-art measure-ments gives:10)
�S � �ðKSÞ ¼ ð0:8953� 0:0006Þ � 10�10 s;
�L � �ðKLÞ ¼ ð5:098� 0:021Þ � 10�8 s:ð19Þ
It should be kept in mind, though, that this huge difference inlifetimes, namely �ðKLÞ=�ðKSÞ � 600, reflects a dynamicalaccident. If pions were massless, �L � �S, and as we shallsee below, we might not know about CP violation eventoday!
So, we have seen the existence of two vastly differentlifetimes for neutral kaons. Fun is not over yet!
For an initially pure K0 beam the time evolution for ajK0ðtÞi is
jK0ðtÞi ¼1ffiffiffi2p jKSðtÞi þ jKLðtÞið Þ
¼1ffiffiffi2p
�exp �i MS �
i
2�S
� �t
� �jKSð0Þi
þ exp �i ML �i
2�L
� �t
� �jKLð0Þi
�
¼ fþðtÞjK0i þ f�ðtÞj �KK0i; ð20Þ
where
f�ðtÞ ¼1
2e�iM1te�ð1=2Þ�1t 1� e�i�Mteð1=2Þ��t
; ð21Þ
where �M ¼ M2 �M1, and �� ¼ �2 � �1, and likewisefor j �KK0ðtÞi.
From this expression for the time dependent admixtures,we can compute the probability of finding a K0 and �KK0 inan initially pure K0 beam. We have shown these proba-bilities in Fig. 1.
Fig. 1. The probability of finding K0 in an initial K0 beam as a function of
time, and the probability of finding �KK0 in the same beam.
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111005-3
Clearly what is shown here is an interference phenomen-on. As it appears in many places in physics, we can makevery precise measurements by making use of interferenceeffects. Here it is not an exception. The mass differencebetween KL and KS states is measured to be:
�MK � ML �MS
¼ ð3:4836� 0:0066Þ � 10�12 MeV:ð22Þ
This is one of the smallest mass differences ever measuredby man.
7. KL Was Caught Violating CP Symmetry
KL lives 600 times longer than KS. Indeed, Cronin, Fitch,and collaborators caught KL in the act of CP violation.11)
Their apparatus is shown in Fig. 2. K0 and �KK0 mesons areproduced in a target 57 feet in front of the spectrometer.Over that distance the KS component has decayed awayand a pure KL beam remains. To their astonishment, theyhave found the decay KL! 2�— they have reported theresult:
þ� ¼�ðKL ! �þ��Þ�ðKS! �þ��Þ
¼ ½ð2:0� 0:4Þ � 10�3�2; ð23Þ
which is completely consistent with what is known today.The observations can be summarized by saying that thequantum mechanical KL state contains a small admixture ofa CP even component in addition to its dominant CP oddpart:
If there is a CP violating interaction in the Hamiltonian H,the matrix element of KL! �� can be written as
h�� j H j KLi ¼ ���h�� j H j K1i þ h�� j H j K2i; ð25Þ
where we have assumed that ���2 is small and set thenormalization factor to be 1. The first term is called indirect
CP violation and it is caused by the deviation from CPsymmetry requirement hK0 j H j �KK0i ¼ h �KK0 j H j K0i, andthe second term is called the direct CP violation. If ��� is theonly source of CP violation, i.e., without direct CPviolation, we expect þ� ¼ 00 where
00 ¼�ðKL! �0�0Þ�ðKS! �0�0Þ
: ð26Þ
Since the discovery of CP violation in K decay,many experimentalists have searched for the possiblepresence of the second term in eq. (25). It will causeþ� 6¼ 00. The term commonly used by physicists inthis field is
�0
�¼
1
6
jþ�j2 � j00j2
jþ�j2: ð27Þ
The first measurement for this quantity was made ny theNA31 collaboration12) 29 years after the measurement ofþ�. Recent values obtained by NA4813) and KTeV14)
collaborations averaged by PDG10) is
�0
�¼ ð16:7� 2:3Þ � 10�4: ð28Þ
8. Symmetries Gone Away Side
So, the parity symmetry was the first to go. Now CPsymmetry is gone. Through detailed studies of K mesondecays, experiments have shown that the rate for K0 ! �KK0
is different from the rate for �KK0! K0. Thus T symmetryviolation is also established.15)
Thus we have ended up with a curious situation. The sacredsymmetries of classical mechanics: parity and time reversal,
Fig. 2. A schematic drawing of an apparatus used by Cronin, Fitch, and collaborators. K0 and �KK0 are created on the internal target. They travel through
the collimator so that only the ones traveling parallel to the axis of the apparatus are accepted. Also, most of the KS component has decayed away and
we have almost a pure KL beam. Two detectors which consist of magnets and spark chambers measure three-momenta of the two body decay
product KL ! �þ��.
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111005-4
as well as particle antiparticle symmetry which is seen to bepreserved in strong and electromagnetic interaction areviolated in weak interaction.
In looking for physics beyond the SM, we should alwayskeep in mind that the ultimate symmetry which is valid inany field theory which satisfies locality, CPT symmetry maybe violated at some level. So far, CPLEAR result is:15)
In the previous section, we have considered chargedcurrents in the form of �dd0����u. While the charged currentswere necessary to describe � decays, there is no reason whyneutral current should not play a role. In fact in the standardmodel, neutral current which couples to the Z0 bosonnaturally exists. So, let us consider a neutral current of theform �uu����uþ �dd0����d
0. The second term must be d0 ratherthan d, because d0 is the weak eigenstate. This leads to acurrent–current interaction of the form
L0 ¼GFffiffiffi
2p sin �c cos �cð �dd����sÞ ��������þ h:c: ð31Þ
Now let us look at the data. The first two decay modesshown in Table II are typical strangeness changing weakdecays. But, look at the last decay, eq. (31) causes atransition
h�þ�� j ð �dd����sÞð ��������Þ j ðs �ddÞi ð32Þ
and this decay should have a branching ratio of Oð1%Þ instead of 10�9. Why is it so suppressed? It implies that theneutral current interaction of the form eq. (31) can not exist.
This puzzle has an elegant solution proposed by Glashow,Illiopolous and Maiani.16) They assumed that there wasa new quark named charm. Just as ðu; d0Þ contributed inweak interactions, ðc; s0Þ with s0 ¼ �d sin �c þ s cos �c,which corresponds to the orthogonal state to d0, also shouldcontribute. The neutral component of the hadronic currentnow looks like:
J0� ¼ �uu����uþ �cc����cþ �dd0����d
0 þ �ss0����s0
¼ �uu����uþ �cc����cþ �dd����d þ �ss����s: ð33ÞThe unwanted strangeness changing neutral current has
been canceled out. A simple way to see this is to say that
d0
s0
� �
L
¼ VL
d
s
� �
L
¼cos �c sin �c
� sin �c cos �c
� �d
s
� �
L
ð34Þ
d0
s0
� �
L
����d0
s0
� �
L
¼d
s
� �
L
����d
s
� �
L
ð35Þ
where we have used the fact that VL is unitary.So, by insisting on the absence of neutral strangeness
changing current, again, flavor physics managed to predictthe existence of a new quark, charm. This is the power offlavor phyiscs. But it is not over yet!
10. KM Mechanism
The fact that CP violation must be due to the presence ofa phase in the Hamiltonian can be easily seen as follows.Write the Hamiltonian as
H ¼ chþ c�hy; ð36Þ
where h is some operator which causes a transition,and c is a complex coefficient. Note that the secondterm must be present due to the hermiticity of theHamiltonian H. It can be shown that if h describesinteraction of particles, hy describes interaction of antipar-ticles, and
CPhCP�1 ¼ hy: ð37Þ
So, H in invariant under the CP transformation ð½H;CP� ¼0Þ if and only if c is real.
Since its experimental discovery, many theoretical ideashave been put forward to explain CP violation. Amongthem, multi-Higgs models, superweak theory, etc. Thesimplest version of the multi-Higgs model predicted �0=� 0:0517) which is excluded by experiments. Superweak modelpredicted �0=� ¼ 0, which is again excluded by experiments.The Kobayashi–Maskawa (KM) ansatz escaped scrutinyby these measurements. In fact its prediction of �0=� isconsistent with experiment, although theoretical predictionis plagued with uncertainties coming from the fact that weare unable to reliably compute strong interaction effects.In this section we describe the KM ansatz.18)
We stated that CP violation originates from the phasein the interaction Lagrangian. Could CP violating phaseappear naturally in the SM? The answer is yes!
Let us examine how quarks couple to W bosons, andhow quark masses arise. The Higg’s boson interacts withquarks through Yukawa interaction:
LYukawa ¼Xi; j
ðGUÞijð �UUi;L; �DDi;LÞ��0�
���
�Uj;R
þXi; j
ðGDÞijð �UUi;L; �DDi;LÞ��þ
�0
�Dj;R þ h:c:;
ð38Þ
where UTðL;RÞ ¼ ðu; c; tÞ�T
ð�þÞ and DTðL;RÞ ¼ ðd0; s0; b0Þ�T
ð�þÞ,where q0 indicates the rotated weak eigenstates. The indicesi and j run over 1 to 3, the number of families. Once theneutral Higgs field acquires a vacuum expectation value,h�0i ¼ v, The mass matrices for the up- and down-typequarks are then proportional to the corresponding Yukawacouplings with the scale set by v:
MU ¼ vGU and MD ¼ vGD: ð39Þ
Since the Yukawa couplings are quite arbitrary, so are themass matrices, and in general they will contain complexelements.
To describe CP violation, there must be complex phasesin the Hamiltonian. So, can the phases in these massmatrices be the origin of CP violation?
Table II. Rare decay modes of KL meson.
Decay modeBranching ratio
(%)
KL ! �þe ���e 40:52� 0:15
KL ! �þ���0 12:56� 0:05
KL ! �þ �� ð6:87� 0:11Þ � 10�9
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To, investigate this, let us diagonalize the mass matrix:
VuLMUVyuR ¼ MdiagU
VdLMDVydR ¼ MdiagD ;
ð40Þ
where UmTðL;RÞ ¼ ðu; c; tÞ�T
ð�þÞ and DmTðL;RÞ ¼ ðd; s;bÞ�T
ð�þÞ, arethe mass eigenstates, and M
diagD;U is the diagonalized down-
type mass matrix, and up-type quark mass matrix, respec-tively. The mass eigenstates can be written in terms of weakeigenstates:
UmL ¼ VL
uUL; DmL ¼ VL
dDL: ð41Þ
Now we write the Lagrangian in terms of particles thatparticipate in interactions, i.e., mass eigenstates of quarks.Then the Lagrangian becomes
L ¼ �UUmTL VL
uVLyd �
�DmLW� þ �UUm
L MdiagU Um
R
þ down quark mass termsþ h:c::ð42Þ
There are plenty of phases in V � VLuVLy
d , which we callthe KM matrix. Do they cause CP violating interaction? Theproblem is, as stated before, that experiments can only countnumber of particles. In quantities that can be measured, mostof the phase information is lost. In particular, the phase ofexternal states are not observable. So, we can adjust them tomake the constants which appear in the Lagrangian real.Make the following transformation which does not changeresult of any experiment.
UmL ! diagðei�u ; ei�c ; ei�tÞUm
L ;
DmL ! diagðei�d ; ei�s ; ei�b ÞDm
L ;ð43Þ
where diagða; b; cÞ denotes a diagonal matrix with diagonalelements a; b; c. Of course, the phases of the right handedquarks should be adjusted accordingly so that the massesremain real.
Can we tune quark phases to make all the elements ofV real? Note that the number of parameters in V risesquadratically and the number of adjustable phases increaseslinearly with number of generations. For two generations,there are enough phases we can adjust to make V real. But,for 3 generations, there are not enough free phases to makeall Vij real. For a system of three generation of quarks, therewill be one unremovable phase. This phase may appear inphysical observables.
The existence of CP violation is a natural consequence offact that: (1) Yukawa couplings are in general complex, andthat (2) there exist three generations of quarks which mixthrough weak interaction to all other quarks of the samecharge. For three generations, there is one phase and threemixing angles in the KM matrix. All we have to do is tofigure out how to measure it.
11. Why is K Meson CP Violation Small?
We have argued in the previous section that if there existonly two generations, there is no CP violation within thecontext of the KM ansatz. This means that for CP violatingK meson decays, the third generation must play a crucialrole. We know from quantum mechanics, the second orderperturbative transition
K0! t�tt! �KK0 ð44Þ
can occur, if there is a coupling K0 ! t�tt, and t�tt! �KK0.
Remember that, in the second order purtabation theory, theintermediate state may have a different energy comparedto the initial and final state energy. Indeed for K mesonsystem, t�tt state is the major contribution to the phasenecessary for the CP violating decay KL! �þ��.
We thus understand why CP violation is small in K
decays. The mass of the K meson is only 500 MeV while theintermediate state of t�tt is 350 GeV. The amplitude is highlysuppressed due to the fact that it is proportional to theinverse of the energy difference between initial and theintermediate state.
12. How Do We Extract Phases by Counting Number ofParticles?
In physics experiments, we shoot in particles and theyscatter, then we just count number of final state particles.In other words, output of experiments are just integers. Wehave seen that CP violation effects are controlled by phasesof amplitudes. How could information about phases beobtained from integers?
Measuring intensities of light passing through two slits,Young’s double slit experiment allows us to measure phasedifference of two waves passing through these slits.Intensities is proportional to the number of photons hittingthe screen, i.e., integers.
Now, consider a decay in which there are two differentdecay paths. Experimenters can not tell which decay pathhas been taken, as long as they measure just the finalstate decay products. Having two different decay paths isanologous to having two different slits. The amplitudes forthese two decay paths will interfere. Let us assume that theB0– �BB0 mixing is big enough so that their effect can be seen.Then there are two paths for a B0 meson to decay to f ,as shown in Fig. 3. Since we can not tell which path aparticular decay took, we have to coherently add twoamplitudes. These two amplitudes interfere with each otherwhen we square the total amplitude to obtain the proba-bilities for the B0! f and �BB! f decays. For a specificdecay mode f ¼ KS, we obtain:19,20)
There is a nice way to summarize CP violation param-eters. Note that the CKM matrix is a unitary matrix. So,we have
Fig. 3. With B0– �BB0 mixing and non vanishing amplitudes for both B! f
and �BB! f decays there are two decay chains B! f and B! �BB! f .
Just as in the two-slit interference pattern observed in Young’s experi-
ment in optics, these two decay chains interfere according to the principle
of quantum mechanics.
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111005-6
VduV�ub þ VdcV
�cb þ VdtV
�tb ¼ 0: ð46Þ
If we represent these three complex numbers as vectors on acomplex plane, the unitarity relation is represented by atriangle shown in Fig. 4. One of the angles �1 is given interms of the asymmetry given in eq. (45):
sinð2�1Þ ¼ ImV�tbVtd
VtbV�td
VcbVcs
V�cbVcs
� �: ð47Þ
Also it can be shown that �2 is obtained from the CPasymmetry in the B! �� channel, and �3 can also beobtained, for example, from the asymmetry in B! KD
channel. Three quantities: like three sides of the triangle;two angles and one side; two sides and one angle; fix thetriangle uniquely, and there are six experimentally meas-urable quantities: 3 angles and 3 sides of the triangle. TheKM theory can therefore be tested by performing threeconsistency checks.
To arrive at our result eq. (45), we assumed that: (1) Bmeson exists; (2) there exists B0– �BB0 transition; (3) the Bmeson life time is long enough so that the time dependenceof the asymmetry can be seen; (4) the phase of the KMmatrix elements must be sizable for the asymmetry to beobserved. With all these ‘‘if’s’’, there is no wonder thatnobody paid much attention to this prediction for about 7years.
13. Discovery of Beauty that Mixes
When Kobayashi and Maskawa proposed the KM ansatzfor CP violation, only two families where known to exist.But, as we have seen in §10, with only two families, allphases which appear in the mass matrix due to Yukawainteractions of Higgs’ boson can be removed by redefiningthe invisible quark phases. Their bold prediction was that thethird family must exist.
Theorists predicted the existence of charm to complete thesecond family, and J= which is a bound state of charmquark and anti-charm quark was discovered in 1976.Theorists once again predicted the existence of the thirdfamily in order to explain CP violation. Sure enough theisodoublet of mesons ðBþ;B0Þ, which are bound states �bb
quark and ðu; dÞ, have been found in 1981.21)
It was the MAC collaboration22) working at the PEPring of SLAC that found the first evidence for a longbeauty life time. The discovery was quickly followed byMARKII collaboration, also at PEPII.23) They have observedthat B decay products came not from where B0 and �BB0
where produced, but from some distance away. It showedthat B mesons traveled some observable distance before itdecayed, i.e., their life times were long enough to leavea gap. To me, this was the crucial discovery which leadto many successes at the B factories 20 years later. Then
came another crucial discovery, B0– �BB0 mixing. Let ustake a brief trip into a memory lane. After the discoveryof the B mesons, we knew that the second member of thethird family had to exist. It took more than a decade ofsearch, but sure enough, the top quark was discovered in1995. For a long time there was a rumor that the topquark was about to be found, and its mass was at about50 GeV. This value is huge for those who thought that Bmeson mass of 5 GeV was extraordinary. Theoreticalcalculation showed that the probability for observingmixing is proportional to ðmt=mWÞ4, and we thought itwas too small to be observed if mt ¼ 50 GeV. If we putthe correct value of mt � 174 GeV into our old computa-tions, it is exactly what had been observed. If we hadbeen bold enough, we could have predicted B0– �BB0 mixingbefore its discovery. In spite of the fact that theoristslacked courage to bring out such a bold prediction, theARGUS collaboration24) announced that they have seen it!The role has been reversed. Having seen B0– �BB0 mixing,we knew that the top quark should be found aroundmt � 170 GeV. This again illustrates the power of flavorphysics as long as those who work in it are courageousenough.
So, everything that was assumed in predicting theCP asymmetry in B! J= KS have been verified byexperiments. The next step in understanding the CPviolating phenomena is to experimentally check this pre-diction.
14. How Do We Actually Measure CP Violation?
First, we have to have beams of B0’s and �BB0’s. Theproblem is that B0’s live only for about 1.5 ps. This is soshort that even light travels only 0.45 mm during this time.Before B’s decay, we have to check if we are looking at a B0
decay or �BB0 decay, identify the decay product, and measurethe time when it decayed. The way to produce B’s is to pairproduce it:
eþ þ e� ! B0 þ �BB0: ð48Þ
To identify B0 and �BB0, we take advantage of the fact thatleptonic decay of b ( �bb) quark always contains �� (�þ):b! cþ �� þ ���� and �bb! �ccþ �þ þ �. Since B0 ( �BB0) is abound state of �bbd (b �dd), leptonic decay of B0 always contains�þ and that of �BB0 always contains ��.
B0! �þ þ �� þ hadrons
�BB0! �� þ ���� þ hadrons:ð49Þ
Now, note that pair production eq. (48) always produceB0 �BB0 pair in the relative angular momentum 1 state, i.e., thewave function is antisymmetric in the interchange ofB0 $ �BB0. So, if we identify, for example, the leptonic B
decay which contains �þ at time t, then we know that, at thattime t, the other member of the pair production was a �BB0.This way we managed to produce the �BB0 beam. The ð4SÞ ismost suited for studying CP violation in B decays. At the topof the resonance, the signal to noise ratio for B �BB productionis increased by 1 : 2:5.
We have estimated, back in 1981, that the asymmetryshown in eq. (45) is:
sinð2�1Þ > 0:15: ð50Þ
*dc cbV
*dt tbV*
du ubV
1φ
2φ
3φ
V V
V
Fig. 4. The unitarity triangle representing the unitarity relation eq. (46).
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS I. SANDA
111005-7
Note that this is a huge asymmetry compared to � �2� 10�3. Based on this value we estimated that we needed100 million B0– �BB0 pairs to be produced in one year if wewere to discover large CP violation during that time. Thismeant that we have to have a eþ � e� collider with theluminosity of 1034 cm�2 s�1. This luminosity was 1000 timesmore than that of the existing state of the art machine,CESR. Not only that, the machine had to collide electrons at9 GeV and positron at 3 GeV — it had to be an asymmetricmachine. This was to boost ð4SÞ so that B– �BB pair is movingfast enough to leave a track of about 200 mm before itdecayed.
15. Recent Results
Both KEK and SLAC have accepted this challenge inbuilding the B factory. Sure enough they have both reachedthe design goal in a reasonable time period, and the CPasymmetry was discovered simultaneously at both laborato-ries in the year 2000. The latest number is:25)
sinð2�1Þ ¼ 0:674� 0:026: ð51Þ
This is large compared to the lower limit we used, 15%.Since there is only one phase in the KM matrix, allmeasurements associated with CP violation can be repre-sented by a point on the complex – plane, where , and are real, and imaginary part of the complex number
V�ubV�cb¼ þ i; ð52Þ
respectively. For example, experimental measurement ofjVub=Vcbj gives a region which is a circle, with the centerat the origin of the – plane. B0– �BB0 mass mixing gives aregion which is also a circle centered at ð1; 0Þ on the –plane. The measurement of �1 defines a unitarity trianglewhere the base of the triangle is 1. This is because all threesides of the triangle is divided by jVcbj. The most recentpublished result which summarizes various experimentalresults is shown in Fig. 5.26) In addition to this result, thereare many more interesting results are listed in the abovementioned reference.
16. Future
B physics is still in a very early stage. Remember Kphysics has been an active area of research for already 60years. This implies that B physics, if it started in 1981 withits discovery, should continue to yield interesting results atleast until 2046. The reason why B factories have been sucha success is that the luminosity was set at 1034 cm�2 s�1
which is 10 times more than we needed to measure theasymmetry. Thus it is yielding much more interestingresults than just the value of CP violation in B! KS,sinð2�1Þ. This should be a lesson for future accelerators.The luminosity should be much as we can possibly obtain.
Physics of high enery scale should affect the way Bdecays through quantum corrections. Such new physicseffects should be small but it may be the only way to getat the new physics if its scale is beyond the existingaccelerators.
1) M. Gell-Mann: Phys. Rev. 92 (1953) 833.
2) T. Nakano and K. Nishijima: Prog. Theor. Phys. 10 (1953) 581.
3) I. I. Bigi and A. I. Sanda: CP Violation (Cambridge University Press,
2000). For details and for other very important topics which were
ommitted for lack of space, we refer the reader to this text book.
4) T. D. Lee and C. N. Yang: Phys. Rev. 104 (1956) 254.
5) M. Gell-Mann and Y. Ne’eman: The Eightfold Way (W. A. Benjamin,
New York, 1964).
6) M. Gell-Mann and M. Levy: Nuovo Cimento 10 (1960) 705.
7) N. Cabibbo: Phys. Rev. Lett. 10 (1963) 531.
8) S. Weinberg: Phys. Rev. Lett. 40 (1978) 223.
9) V. Weisskopf and E. Wigner: Z. Phys. A 63 (1930) 54; V. Weisskopf
and E. Wigner: Z. Phys. A 65 (1930) 18.
10) S. Eidelman et al. (Particle Data Group): Phys. Lett. B 592 (2004) 1.
11) J. H. Christensen et al.: Phys. Rev. Lett. 13 (1964) 138.
12) G. D. Barr et al.: Phys. Lett. B 317 (1993) 233.
13) J. R. Batley et al.: Phys. Lett. B 544 (2002) 97.
14) A.-H. Harati et al.: Phys. Rev. D 67 (2003) 012003.
The history of the kaon system in the genesis of theStandard Model (SM) is well-known and for many yearsthere was intense experimental activity in this area. But aftera series of attempts to uncover a non-Superweak1) origin forthe observed CP-violation failed to do so,2,3) experimentalefforts in this field diminished to a much lower level for adecade. Interest revived only gradually, primarily driven bythree developments:(1) The realization that the Kobayashi–Maskawa model4)
of CP-violation implied a potentially observable devia-tion from the Superweak prediction5) for K ! 2�6)
(2) A developing appreciation for the fact that GIMsuppressed7) kaon decays were potentially cleanlysensitive to SM parameters, including those character-izing CP-violation.8–14)
(3) As theories were developed to address the perceivedshortcomings of the SM, it turned out that many ofthese naturally predicted violations of lepton flavor (asin KL! ��e�) or other conservation laws in rarekaon decays. Prominent among these were models ofdynamic symmetry breaking, particularly those underthe rubric of extended technicolor.15)
2. CP-Violation in K! 2�
Reference 6, which predicted a ratio Reð"0="Þ as large as10�1–10�2, stimulated experiments at Brookhaven16) andFermilab17) that could have observed effects of this size.18)
Later predictions were more conservative and more sensitiveexperiments at Fermilab (E731) and CERN (NA31) werelaunched to pursue this measurement. Each exploited thefact that
Reð"0="Þ �1
61�
�ðKL! �0�0Þ=�ðKS! �0�0Þ�ðKL! �þ��Þ=�ðKS! �þ��Þ
� �ð1Þ
This implies that the precision on Reð"0="Þ will be 6times better than that on the ratio of rates. The practicalapplication of this formula was quite different in the twoexperiments. After a series of runs, they eventually reportedmarginally conflicting results: Reð"0="Þ ¼ ð23� 6:5Þ � 10�4
for NA3119) and Reð"0="Þ ¼ ð7:4� 5:2stat � 2:9sysÞ � 10�4
for E731.20) These results are less than 2� apart, but they hadqualitatively different implications for the Superweak Model.This led to another set of experiment at both FNAL andCERN, led by veterans of E731 and NA31. These experi-ments represented large advances on the previous round.
Figures 1 and 2 show the KTeV and NA48 experimentrespectively. Both experiments featured simultaneous KS
and KL beams and detectors capable of measuring all fourK ! 2� reactions at once. In case of KTeV the averagekaon momentum was about 70 GeV/c and the two beamsdiverged by 1.6 mrad in the horizontal so that their centerswhich were 14 cm apart at the beam-defining collimatorbecame separated by 30 cm at the detector. A regenerator inone beam provided the KS’s and was switched between thetwo beams once per minute. For NA48, the KS’s werecreated in a special target partway down the beam line. Asmall fraction of the uninteracted portion of the proton beamdownstream of the KL target was diverted via a siliconmono-crystal, collimated, tagged and transmitted to the KS
target. The two beams converged (vertically at an angle of0.6 mrad) at the detector. In both experiments the decayregion and associated veto counters were situated in largeevacuated tanks. Downstream of the KTeV tank, heliumbags separated the detector elements while in NA48,the detector elements to the end of the spectrometer weresituated in a helium-filled volume. Running conditionsvaried over the life of each experiment, but typically theysaw a instantaneous fluxes of a few MHz of KL. Both hadfour-station drift chamber spectrometers followed by high-quality electromagnetic calorimeters. KTeV used an array ofpure CsI crystals and NA48 a liquid Krypton accordion-typecalorimeter. NA48 had a hadron calorimeter and bothexperiments had muon-identification systems. The highquality of the instrumentation needed for this demandingapplication allowed many other interesting measurements,some of which are discussed below.
In 1996 and 1997 KTeV collected 11.1M KL! �þ��
and 3.3M KL! �0�0 decays. The corresponding numbersof KS decays were 19.2M and 5.6M. Their overall result21)
was Reð"0="Þ ¼ ð20:7� 2:8Þ � 10�4. There was a subse-quent run in 1999 that doubled the statistics previouslycollected, but the result from this data is not yet available.
In 1998 and 1999 NA48 collected 14.5M KL! �þ��
and 3.3M KL! �0�0 decays. The corresponding numbersof KS decays were 22.2M and 5.2M. Their overall result,22)
when combined with that from a lower statistics 1997 run23)
was Reð"0="Þ ¼ ð15:3� 2:6Þ � 10�4. In 2001 NA48 had anadditional run under somewhat different conditions in whichthey collected about half as many events as in 1998 – 99.They obtained a result24) Reð"0="Þ ¼ ð13:7� 3:1Þ � 10�4,which combined with previous running gave Reð"0="Þ ¼ð14:7� 2:2Þ � 10�4.
SPECIAL TOPICSJournal of the Physical Society of Japan
Vol. 76, No. 11, November, 2007, 111006
#2007 The Physical Society of Japan
Frontiers of Elementary Particle Physics, the Standard Model and Beyond
At this point the Particle Data Group25) fits for kaondata result in Reð"0="Þ ¼ ð16:6� 2:6Þ � 10�4, many � from0, so that it is clear that the Superweak Model5) cannot becorrect, and a quest lasting more than 30 years has beensuccessfully concluded. Unfortunately the detailed implica-tions of this result for the origin of CP violation are notclear. At the moment one cannot rule out a StandardModel explanation for this CP-violation, but that is aboutall one can say. There is an enormous literature attemptingto calculate Reð"0="Þ in the Standard Model, but littleagreement on the resulting accuracy. At the point at whichref. 24 was released, the range of up-to-date predictionsvaried from �10� 10�4 to þ30� 10�4. References may befound in ref. 24.
A number of experiments including KTeV and NA48have seen �-level CP-violating effects in other kaon decaymodes including KL‘3,26,27) KL ! �þ���,28) and KL!�þ��eþe�.29)
3. CP-Violating Amplitudes in Rare Kaon Decays
A number of years before the denouement of K ! 2�, itwas appreciated that CP-violation could manifest itself in acleaner way in certain rare kaon decays. These decays hadcontributions from G.I.M.-suppressed7) one-loop amplitudessensitive to fundamental SM parameters such as mt and Vtd.However it was not until it became clear that mt was verylarge that it was realized that these processes might beaccessible to measurement. Even with the large mt thesedecays are also suppressed enough to be potentially verysensitive to BSM physics. These processes include Kþ !�þ� ���, KL! �þ��, KL! �0� ���, KL ! �0eþe�, and KL!�0�þ��. In the latter three cases the one-loop contributionsviolate CP. In KL! �0� ��� this contribution completelydominates the decay.13) Diagrams for such loops are shownin Fig. 3. Since the GIM-mechanism enhances the contri-bution of heavy quarks, in the SM these decays are sensitive
Fig. 2. (Color online) Layout of the NA48 experiment.
120 140 160 180z (m) = distance from kaon production target
-1
0
1x
(m)
Beams
Regenerator BeamVacuum Beam
AnalysisMagnet
Regenerator
DriftChambers(DC1-DC4)
TriggerHodoscope
MuonSystem
Steel
CsI
MaskVeto
VacuumVeto
SpectroVeto
CsIVeto
Fig. 1. Layout of the KTeV experiment.
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to the product of couplings V�tsVtd � �t. Although it isperhaps most natural to write the branching ratio for thesedecays in terms of the real and imaginary parts of �t,
30,31) forcomparison with results from the B system it is convenientto express them in terms of the Wolfenstein parameters,A, , and . Figure 4 shows the relation of these rare kaondecays to the unitarity triangle. The dashed triangle isthe usual one derived from V�ubVud þ V�cbVcd þ V�tbVtd ¼ 0
(� �u þ �c þ �t), the solid one indicates the informationavailable from rare kaon decays. The apex, ð; Þ, can bedetermined from either triangle, and disagreement betweenthe K and B determinations implies physics beyond the SM.In Fig. 4 the branching ratio closest to each side of the solidtriangle can be used to determine the length of that side.KL ! �þ��, which can in principle determine the bottomof the triangle (), is the decay for which the experimentaldata is the best but for which the theory is most problem-atical. KL ! �0� ���, which determines the height of thetriangle () is the cleanest theoretically, but for this modeexperiment falls far short of the SM-predicted level.Kþ ! �þ� ���, which determines the hypotenuse, is nearlyas clean as KL! �0� ��� and has been observed (albeit withonly three events). Prospects for the latter are probably thebest of all since it is both clean and already within reachexperimentally.
3.1 KL! �0� ���KL! �0� ��� is considered the most attractive target in the
kaon system, since(1) it is direct CP-violating to a very good approxima-
tion13,32) (in the SM BðKL ! �0� ���Þ / 2) and(2) the rate can be rather precisely calculated in the SM or
almost any alternative.33)
The hadronic matrix element can be obtained from Ke3 viaan isospin transformation.34) Unlike the case of its chargedanalog, Kþ ! �þ� ���, discussed below, it has no significantcontribution from charm. Consequently, the intrinsic theo-
retical uncertainty connecting BðKL ! �0� ���Þ to the funda-mental short-distance parameters is less than 1%.35) In theSM BðKL ! �0� ���Þ is directly proportional to ðIm �tÞ2 andIm �t ¼ �J =f�½1� ð�2=2Þg where J is the Jarlskoginvariant.36) Thus a measurement of BðKL ! �0� ���Þ deter-mines the area of the unitarity triangles with a precisiontwice as good as that on BðKL! �0� ���Þ itself.
BðKL! �0� ���Þ can be bounded indirectly by measure-ments of BðKþ ! �þ� ���Þ through a nearly model-independ-ent relationship pointed out by Grossman and Nir.37) Theapplication of this to the E787/949 results discussed belowyields BðKL! �0� ���Þ < 1:4� 10�9 at 90% CL. This is fartighter than any extant direct experimental limit. To actuallyobserve KL ! �0� ��� at the SM level (3� 10�11), one willneed to improve on the current state of the art by some 4orders of magnitude.
The first dedicated KL ! �0� ��� experiment, KEKE391a,38) mounted at the KEK 12 GeV proton synchrotron,aimed to achieve sensitivity comparable to the indirectlimit. It was meant to serve as a test for a more sensitiveexperiment to be performed in the future at J-PARC.39)
E391a features a carefully designed ‘‘pencil’’ beam40) withaverage KL momentum 2 GeV/c. Figure 5 shows a layoutof the detector.
A particular challenge of the E391a approach is to achieveextremely low photon veto inefficiency in order to reject themuch more copious background decay modes with addi-tional photons, such as KL! �0�0. The photon veto systemconsisted of two cylinders. The inner, more upstream barrelwas intended to suppress beam halo and reduce confusionfrom upstream KL decays. Roughly 4% of the KL’s decayedin the 2.4 m fiducial region between the end of the innercylinder and the charged particle veto upstream of thephoton detector. Signal photons were detected in a multi-element CsI-pure crystal calorimeter.41) The entire apparatusoperated in vacuum. Physics running began in February2004 and two more runs occurred in 2005.
–
– –
Fig. 3. (Color online) Loop contributions to K decays.
Fig. 4. (Color online) K decays and the unitarity plane. The usual unitarity triangle is dashed. The triangle that can be constructed from rare K decays is
solid. See text for further details.
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In this experiment events with two showers in thecalorimeter and no additional activity were examined todetermine whether the assumption of a decay vertex at anypoint along the fiducial section of the beamline results in areconstructed mass consistent with a �0. If so, the pT couldthen be determined. Cuts were imposed on the showerpatterns and energies, the ZV and the pT. In addition, eventsconsistent with ! �� were discarded.
Figure 6 shows the distribution of residual candidates inpT vs ZV, when all other cuts are applied for the first 10% ofthe 2004 run. The observed versus predicted events areshown in several test regions where background is expectedto dominate. The agreement is statistically acceptable. Noevents were observed in the two divisions of the signalregion, with expected backgrounds of 0:4þ0:7
�0:2 and 1:5� 0:7events. With ð1:67� 0:04Þ � 109 KL decays (determinedfrom observed KL! �0�0 events) and an acceptance of(0:657� 0:016)%, they obtained a limit of BðKL! �0� ���Þ <2:1� 10�7 at 90% CL,38) an improvement by a factor 2.8over the previous limit42) on this decay.
The proposed future program begins with moving theE391a detector to a 16� neutral beam at J-PARC. The beam
is slightly lower in energy, but much more intense with 40
times more useful KL per hour than at KEK. The CsIcalorimeter will be replaced by one with superior resolutionand granularity, the vetoes thickened, and the electronicsupgraded. A single event sensitivity of 8� 10�12 per eventand a signal : background of 1:4 : 1 (assuming the SMbranching ratio) are the goals. A later stage experimentwith optimized beam and detector aims at a >100 eventmeasurement with a signal : background of 5 : 1.
3.2 KL! �0‘þ‘�
Like KL! �0� ���, KL! �0�þ��, and KL! �0eþe� areGIM-suppressed neutral current reactions sensitive to short-distance SM and BSM effects, but whose experimentalconsiderations are qualitatively different. In the SM, likeKL ! �0� ���, they are sensitive to Im �t, but in general theyhave different sensitivity to BSM effects43) and the combi-nation of the measurements of both can be quite informativein BSM scenarios.44)
Although they are more tractable experimentally thanKL ! �0� ���, they are subject to a background that has noanalogue in KL ! �0� ���: KL! ��‘þ‘�. This process, aradiative correction to KL! �‘þ‘�, is 103–104 timesmore copious than KL! �0‘þ‘�. Kinematic cuts areeffective, but it is quite difficult to improve the signal :
background beyond about 1 : 1:545) and still maintain apractical acceptance. Both KL! ��eþe�46) [BR�>5 MeV ¼ð5:84� 0:15stat � 0:32systÞ � 10�7] and KL! ���þ��47)
[BRm��>1 MeV/c2 ¼ ð10:4þ7:5�5:9stat
� 0:7systÞ � 10�9] have beenobserved at rates that agree reasonably well with theoreticalprediction. These branching ratios are to be compared withthe SM prediction48) of the direct CP-violating part ofBðKL ! �0eþe�Þ, ð4:4� 0:9Þ � 10�12 and that of BðKL!�0�þ��Þ, ð1:8� 0:4Þ � 10�12 (these numbers are updatedbelow).
In addition to this background, there are two othercontributions that complicate extracting short-distanceinformation from KL! �0‘þ‘�. Recent experimental andtheoretical advances have mitigated their effects but theystill have substantial impact:(1) an indirect CP-violating amplitude from the K1
component of KL that is proportional to �AðKS!�0‘þ‘�Þ.
(2) a contribution of similar order, but CP-conserving, ismediated by KL ! �0��.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
200 250 300 350 400 450 500 550 600Zvtx (cm)
PT (
GeV
/c)
(a) 0 (0.9 ± 0.2) (b) 3 (4.0 ± 1.6)
(c) 0 (0.4 ± 0.70.2 ) (d) 0 (1.5 ± 0.7) (e)
(f)
(g) 0 (0.1 ± 0.70.1 ) (h)
(e) 1 (3.0 ± 1.4)
(h) 9 (5.6 ± 1.1)
(f) 4 (3.2 ± 1.0)
signal region
Fig. 6. (Color online) pT vs ZV E391a events passing all other cuts (from
first 10% of Run 1). Signal region is the 5-sided region at the center. The
observed (predicted) background events in each subregion are indicated.
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The first contribution is of the same order of magnitude asthe direct CP-violating amplitude and can interfere with it.In the case of KL! �0eþe�, it yields:48)
BðKL ! �0eeÞCPV
��15:7a2
S � 6:2aS
Im �t
10�4þ 2:4
Im �t
10�4
� �2�� 10�12
ð2Þ
where
BðKS! �0eeÞ � 5:2a2S � 10�9 ð3Þ
For KL! �0�þ��, the corresponding expressions are:48,49)
BðKL ! �0��ÞCPV
��3:7a2
S � 1:6aS
Im �t
10�4þ 1:0
Im �t
10�4
� �2�� 10�12
ð4Þ
and
BðKS! �0��Þ � 1:2a2S � 10�9 ð5Þ
There are now measurements for both these processesfrom NA48, based on 7 and 6 events for the electronic andmuonic cases respectively. They find BðKS! �0eþe�Þ ¼ð5:8þ2:8
�2:3 � 0:8Þ � 10�9 (ref. 50) from which eq. (3) thenyields jaSj ¼ 1:06þ0:26
�0:21 � 0:07 and BðKS! �0�þ��Þ ¼ð2:9þ1:5
�1:2 � 0:2Þ � 10�9 (ref. 51) from which eq. (5) thengives jaSj ¼ 1:54þ0:40
�0:32 � 0:06. Averaging the electron andmuon results yields a best estimate of jaSj ¼ 1:2� 0:2.48)
Another contribution of potentially similar size, butCP-conserving, is mediated by KL ! �0��. In principlethis contribution should be predictable from measurementsof KL! �0��, of which thousands of events have beenobserved. The matrix element for this decay is given by:52)
MðKL ! �0��Þ
¼G8�
4���ðk1Þ��ðk2Þ Aðk�2 k
�1 � k1 � k2g
��Þ þ B2
m2K
ðpk � k1k�2 p
�K þ pK � k2k
�1p�K � k1 � k2p
�K p
�K � g��pK � k1pK � k2Þ
� �;ð6Þ
where k1 and k2 refer to the photons. The A amplitudecorresponds to J�� ¼ 0; B is a mixture of J�� ¼ 0 andJ�� ¼ 2. G8 is the octet coupling constant in �PT.Equation (6) leads to
@2�ðKL ! �0��Þ@y@z
¼mK
29�3z2jAþ Bj2 þ y2 �
1
4�ð1; r2�; zÞ
� �2
jBj2" #
;
ð7Þ
where z � ðk1 þ k2Þ2=m2K , y � pK � ðk1 � k2Þ=m2
K , r� �m�=mK , and �ða; b; cÞ � a2 þ b2 þ c2 � 2ðabþ acþ bcÞ.Since in KL! �0eþe� the effect of A is greatly suppressedby helicity conservation and B ¼ 0 at leading order in �PT,it was initially thought that the CP-conserving contributionto BðKL! �0eþe�Þ would be very small. However thepossibility of a substantial vector meson dominance (VDM)contribution to B was pointed out by Sehgal.53) Such acontribution can arise at Oðp6Þ in �PT. The most recentmeasurements of the branching ratio for this decay, byNA48:54) 1:36� 0:03stat � 0:03syst � 0:03norm and and byKTeV:55) 1:30� 0:03stat � 0:04sys agree well. The m��spectra also appear to agree, although the KTeV result isstill preliminary, and they are consistent with a very smallCP-violating contribution to KL! �0eþe�.56,57)
The muonic case is quite different. There is very littlehelicity suppression of the J�� ¼ 0 part of the CP-conservingcontribution to BðKL ! �0�þ��Þ and this is in factexpected to dominate. Reference 48 shows that althoughthere are considerable uncertainties in the absolutecalculation of the CP-conserving contribution, the ratioBCP-consðKL ! �0�þ��Þ=BðKL! �0��Þ should be calcu-lable to 30%. Using their value for this ratio the average ofthe KTeV and NA48 results for BðKL! �0��Þ givesBCP-consðKL ! �0�þ��Þ ¼ ð4:9� 1:5Þ � 10�12.
Plugging aS ¼ 1:2� 0:2 and Im �t ¼ ð1:31� 0:11Þ �10�4 into eqs. (2) and (4).
BðKL! �0eeÞCPV � ½ð22:6� 7:5Þmix � ð9:75� 1:8Þint
þ ð4:12� 0:69Þdir � 10�12 ð8Þ
and
BðKL! �0��ÞCPV � ½ð5:3� 1:8Þmix � ð2:5� 0:5Þint
þ ð1:7� 0:3Þdir � 10�12 ð9ÞTo obtain a prediction for the complete branching ratios,
one needs to fix the sign of the interference between directand indirect CP-violation, and decide what to take for theCP-conserving part. Positive interference has long beenpreferred by theorists.57) If we accept this choice, eqs. (8)and (9) give BðKL ! �0eeÞCPV � ð3:65� 0:77Þ � 10�11 andBðKL ! �0��ÞCPV � ð0:95� 0:19Þ � 10�11, respectively.The CP-conserving contribution is not significant for theelectron mode so that:
BðKL ! �0eþe�Þ � ð3:7� 0:8Þ � 10�11 ð10Þ
In the muonic mode, using the value discussed above, onegets:
BðKL ! �0�þ��Þ � ð1:4� 0:2Þ � 10�11 ð11Þ
The current experimental status of KL ! �0‘þ‘� issummarized in Table I and Fig. 7. About 30% more KL!�0�þ�� data is expected from the KTeV 1999 run.
As can be seen from Table I and Fig. 7, background inboth modes is observed at a sensitivity an order ofmagnitude short of what is needed to observe the directCP-violating signal. The problems of extracting a value ofIm �t from these modes have been discussed in ref. 61among other places, and summaries of various schemesto deal with these problems are given in refs. 57 and 48, butrecent developments make it worthwhile to take anotherlook:
Table I. Results on KL ! �0‘þ‘�.
Mode90% CL
upper limit
Estimated
background
Observed
eventsReference
KL ! �0eþe� 2:8� 10�10 2:05� 0:54 3 58
KL ! �0�þ�� 3:8� 10�10 0:87� 0:15 2 59
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111006-5
(1) The observation of KS! �0‘þ‘� with higher thanexpected rates imply that the SM branching ratios ofthe KL modes are also higher than previously believed.
(2) If we compare the single event sensitivity of the mostrecent data with that of the residual KL! ��‘þ‘� (theleast tractable background), the ratio is much less thanan order of magnitude. The 1999 KTeV KL ! �0ee
data, had a single event sensitivity of 1:04� 10�10 andan estimated residual KL! ��ee background of0:99� 0:35 events.58) This implies a S : B (signal tobackground) of 1 : 2:5. In their 1997 KL ! �0�� data,KTeV had a single event sensitivity of 0:75� 10�11
and a calculated residual KL! ���� background of0:37� 0:03 events59) which implies62) S : B ¼ 1 : 1:9.
(3) The interest in KL! �0‘þ‘� has evolved from its roleas a possible source of information on Im �t, to that ofan arena for probing BSM effects.
With this in mind one can ask, for example, what singleevent sensitivity would be needed to establish a factor twoeffect at 3� in these modes, given the background levelsmentioned above. For the electronic case, the answer is10�12; for the muonic case, 0:4� 10�12.
Judging by the KaMI proposal at Fermilab,63) a next-generation experiment could reach this level in about 3 yearsof running. If instead, the SM were correct, a 30%measurement of the BR would result. Thus it seems ashame that no such experiment is on the near horizon.However such an experiment has been mentioned as apossible follow-up to the CERN Kþ ! �þ� ��� proposaldiscussed below.
3.3 Kþ ! �þ� ���Recent developments in theory have rendered Kþ !
�þ� ��� nearly as clean as KL! �0� ���. As in the neutral case,the often problematical hadronic matrix element can becalculated to <1% via an isospin transformation from that ofKþ ! �0eþ�e.
34,35) The hard GIM suppression minimizesQCD corrections and the long-distance contributions tothis decay are very small. A recent discussion of the latterwith references to previous work can be found in ref. 64.
Kþ ! �þ� ��� is directly sensitive to the quantity �t as canbe seen in eq. (12):65)
BðKþ ! �þ� ���Þ
¼ þ
"Im �t
�5XðxtÞ
� �2
þRe �c
�PcðXÞ þ
Re �t
�5XðxtÞ
� �2#;
ð12Þ
where � � sin �Cabibbo, xt � ðmt=mWÞ2, XðxtÞ, and Pc containthe top and charm contributions respectively and will bediscussed below, and þ ¼ ð5:17� 0:03Þ � 10�11.
The Inami–Lim function,11) XðxtÞ characterizing the GIMsuppression of the top contribution is also given in ref. 66.For the current measured value of mt, XðxtÞ ¼ 1:464�0:041. The QCD correction to this function is 1%. Thecharm contribution is given by:
�4PcðXÞ ¼2
3Xe
NL þ1
3X�NL; ð13Þ
where the functions XlNL are those arising from the NNLO
calculation.67,68) The QCD correction leads to a 30%reduction of the charm Inami–Lim function which is nowknown to about 10%. From eq. (12)
where the last component contains uncertainties due tothe CKM matrix elements and to mt, which will naturally bereduced as data is improved. This would make a highprecision measurement of BðKþ ! �þ� ���Þ very interestingfrom a BSM point of view.
As mentioned above the branching ratio can also bewritten in terms of the improved Wolfenstein variables69)
and one finds it is proportional to
A2X2ðxtÞ1
�½ð� �Þ2 þ ð0 � �Þ2;
where
� � ð1� �=2Þ�2
and
0.00
0.05
0.10
0.15
0.20
0.25x 10
-2
0.46 0.48 0.50 0.52 0.54m (GeV/c2)
P⊥2 (
GeV
/c)2
Fig. 7. (Color online) Signal planes showing candidates for KL ! �0eþe� (left from refs. 60 and 58) and KL ! �0�þ�� (right from ref. 59).
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS L. S. LITTENBERG
111006-6
0 � 1þPcðXÞA2XðxtÞ
:
To a good approximation the amplitude is proportional tothe hypotenuse of the solid triangle in Fig. 4. This is equalto the vector sum of the line proportional to Vtd=A�
3 andthat from ð1; 0Þ to the point marked 0. The length 0 � 1
along the real axis is proportional to the amplitude forthe charm contribution to Kþ ! �þ� ���. More precisely,BðKþ ! �þ� ���Þ determines an ellipse of small eccentricityin the ð �; �Þ plane centered at ð0; 0Þ with axes r0 and r0=�where
Kþ ! �þ� ��� has been observed in the E787/949 seriesof experiments at the BNL AGS. These experiments usedstopped Kþ which gives direct access to the Kþ center ofmass, and is conducive to hermetic vetoing. The cylindri-cally symmetric detector, mounted inside a 1 Tesla sol-enoid,70) is shown in Fig. 8. An 80% pure beam of >107Kþ
per AGS cycle.71) traversed a Cerenkov counter foridentifying Kþ and �þ and was tracked by MWPC’s. Itwas then slowed by a BeO degrader followed by a showercounter and beam hodoscope. About a quarter of the Kþ
survived to enter a scintillating fiber stopping target. Ahodoscope surrounding the stopping target demanded asingle charged particle leave the target after a delay of2 ns. The particle was tracked by a cylindrical driftchamber giving momentum resolution 1%. Additionaltrigger counters required it to exit the chamber radiallyoutward and enter a cylindrical array of scintillators andstraw chambers, the ‘‘Range Stack’’ (RS), in which it wasrequired to stop in order for the event to be considered aKþ ! �þ� ��� candidate. In this configuration, the range andkinetic energy of the particle could be measured to 3%.Comparison of range, energy and momentum is a powerfuldiscriminator of low energy particle identity. In addition,transient recorder readout of the RS photomultipliersallowed the �þ ! �þ ! eþ decay chain to be used to
identify �þ’s. The combination of kinematic and life-cycletechniques can distinguish pions from muons with amisidentification rate of Oð10�8Þ. Surrounding the RS wasa cylindrical lead-scintillator veto counter array and adjacentto the ends of the drift chamber were endcap photon vetoarrays of undoped CsI modules.72) There were also a numberof auxillary veto counters near the beamline as well as aveto in the beamline downstream of the detector.
Monte Carlo estimation of backgrounds was in general notreliable since it was necessary to estimate rejection factors ashigh as 1011 for decays occurring in the stopping target.Instead, methods to measure the background from thedata itself were developed, using the primary data stream aswell as data from special triggers taken simultaneously.The principles adhered to included:
. The signal acceptance region was kept hidden whilecuts are developed.
. Cuts were developed on 1/3 of the data (evenlydistributed throughout the run) but residual backgroundlevels determined only from the remaining 2/3 after thecuts were frozen.
. Background sources were identified a priori and twoindependent high-rejection cuts were developed foreach background. Each cut was reversed in turn as theother was studied. After optimization, the combinedeffect of the cuts could then be calculated as a product.
. Background calculations were verified though compar-ison with data near the signal region.
In this way backgrounds could be reliably calculated atthe 10�3 to 10�2 event level.
All factors in the acceptance besides those of solid angle,trigger and momentum interval were determined from data.
Three Kþ ! �þ� ��� events were observed, two byE78773,74) and one by E949.75) The range versus kineticenergy distribution of these events is shown in Fig. 9. Thecombined result was a branching ratio BðKþ ! �þ� ���Þ ¼ð1:47þ1:30
�0:89Þ � 10�10. This is about twice as high as theprediction of eq. (14), but statistically compatible with it.
The total background to the two E787 events wasmeasured to be 0.15 of an event and that of the E949 event
Fig. 8. (Color online) Elevation view of the E949 detector.
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111006-7
0.3 of an event. Thus E787/949 has developed methods toreduce the backgrounds to a level sufficient to make aprecise measurement of Kþ ! �þ� ���.
From the first observation published in 1997, E787’sresults for BðKþ ! �þ� ���Þ have been rather high withrespect to the SM prediction. Although there has neverbeen a statistically significant disagreement with the latter,it has stimulated a large number of predictions in BSMtheories43,76–94)
The Kþ ! �þ� ��� data also yield an upper limit on theprocess Kþ ! �þX0 where X0 is a massless weaklyinteracting particle such as a familon.95) For E787 this wasBðKþ ! �þX0Þ < 5:9� 10�11 at 90% CL.
E949,96) which ran in 2002 was based on an upgrade ofthe E787 detector. It was improved in a number of ways withrespect to E787: thicker and more complete veto coverage,augmented beam instrumentation, higher capacity DAQ,more efficient trigger counters, upgraded chamber electron-
ics, auxiliary gain monitoring systems, etc. Using the entireflux of the AGS for 6000 hours, E949 was designed to reacha sensitivity of 10�11/event. In 2002 the detector operatedwell at fluxes twice those typical of E787, but unfortunatelyDOE support of the experiment was terminated after thatfirst run. Further progress in Kþ ! �þ� ��� will have to comefrom experiments yet to be mounted.
There are currently two initiatives for future Kþ ! �þ� ���experiments. One is a J-PARC LOI97) for a higher-sensitivitystopping experiment. This is very like E787/949 inconception, but with many improvements in detail. Theseinclude a lower incident beam momentum (leading to ahigher stopping efficiency and a better signal/random rateratio), higher granularity (leading to greater rate capabilityand muon rejection power), brighter scintillators, and a morecapable DAQ. The goal of this experiment is to observe50 events at the SM-predicted level. The schedule forrunning at J-PARC would be some time after 2012.
Although the stopped-Kþ technique is now well-under-stood, and one could be reasonably sure of the outcome ofany new experiment of this type, to get really large samplesof Kþ ! �þ� ��� (�100 events), it will almost certainly benecessary to go to an in-flight configuration. There havebeen a series of attempts to initiate such an experiment,most recently the P326 proposal to CERN.98) This experi-ment exploits newly developed tracking technology to allowthe use of an extremely intense (1 GHz) unseparated75 GeV/c beam. Charged beams of this intensity have beenused to search for good-signature kaon decays such asKþ ! �þ�þe�,99) but P326 is a departure for a poor-signature decay for which high-efficiency vetoing is re-quired.
Figure 10 shows the layout of the proposed experiment.Protons from the 400 GeV/c SPS will impinge on a 40 cmBe target. Positive secondaries with momenta within �1%of 75 GeV/c will be taken off in the forward direction. The6% of Kþ in the beam will be tagged by a differentialCerenkov counter (CEDAR). The 3-momenta of all tracks
E787/E949
Energy (MeV)
Ran
ge
(cm
)
28
30
32
34
36
38
40
42
90 100 110 120 130 140 150
Fig. 9. (Color online) Range vs energy of �þ for the E787 (circles) and
E949 (triangles) samples. Events near E ¼ 108 MeV are Kþ ! �þ�0
background. The expected distribution of signal events is indicated
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111006-8
will be measured in a beam spectrometer with three setsof ‘‘GIGATRACKER’’ detectors (fast Si micro-pixels andmicro-mega TPCs). The expected performance is �p ¼0:5%, �� ¼ 16 mr, and �t ¼ 150 ps. The beamline has beencarefully designed to hold the muon halo that contributes todetector random rates to the order of 10 MHz. The beam willcontinue through the apparatus in vacuum. About 10% of theKþ will decay in a 60-m-long fiducial region. The remainderof the beam will be conducted in vacuum out of the detectorregion. Photons from Kþ decays will be detected in a seriesof ring vetoes (at wide angles), by an upgraded version of theNA48 liquid Krypton calorimeter (at intermediate angles)and by two dedicated inner veto systems. Charged decaytracks will be momentum-analyzed in a two-dipole straw-tube spectrometer (<1% resolution on pion momentum and50 – 60 mr resolution on �K� are necessary). Downstream ofthe spectrometer a RICH filled with Ne at 1 atm will helpdistinguish signal pions from background muons. This is tobe followed by a charged particle hodoscope of multigapglass RPC design (100 ps resolution is required). Behind thehodoscope is the ‘‘MAMUD’’ muon veto, a magnetized iron-scintillator sandwich device to complete the pion/muondistinction. Its 5T-m bending power serves to kick the beamout of the way of the small angle photon veto at the back ofthe detector.
The collaboration proposes to build this experiment intime to begin taking data in 2011. A two-year wouldaccumulate 100 events with a 8 : 1 signal to background.
4. Beyond the Standard Model
Although in principle LFV processes like KL! �e andKþ ! �þ�þe� can proceed through neutrino mixing, theknown neutrino mixing parameters limit the rate through thismechanism to a completely negligible level.100) Thus theobservation of LFV in kaon decay would require a newmechanism. Figure 11 shows KL! �e mediated by ahypothetical horizontal gauge boson X, compared with thekinematically very similar process Kþ ! �þ� mediated bya W boson. Using measured values for MW, the KL and Kþ
decay rates and BðKþ ! �þ�Þ, and assuming a V � A formfor the new interaction, one can show:101)
MX � 220 TeV/c2gX
g
� �1=4 10�12
BðKL! �eÞ
� �1=4
ð16Þ
so that truly formidable scales can be probed if gX g
(see also ref. 102). In addition to this generic picture,there are specific models, such as extended technicolor inwhich LFV at observable levels in kaon decays is quitenatural.15)
There were a number of K decay experiments primarilydedicated to lepton flavor violation at the Brookhaven AGSduring the 1990’s. These advanced the sensitivity to suchprocesses by many orders of magnitude. In addition, several‘‘by-product’’ results on LFV and other BSM topics haveemerged from the other kaon decay experiments of thisperiod and from more recent ones. Table II summarizes thestatus of work on BSM probes in kaon decay. The relativereach of these processes is best assessed by comparing thepartial rates rather than the branching ratios.
This table makes it clear that any deviation from the SMmust be highly suppressed. In a real sense the kaon LFVprobes have become the victims of their own success. Byand large the particular theories they were designed to probehave been forced to retreat to the point where meaningfultests in the kaon system would be very difficult (althoughthere are exceptions110)). The currently more populartheoretical approaches tend to predict a rather small degreeof LFV in kaon decays. For example, although these decaysdo provide the most stringent limits on strangeness-changingR-violating couplings, the minimal supersymmetric exten-sion of the Standard Model (MSSM) predicts LFV in kaondecay at levels far below the current experimental sensitiv-ity.111) Decays such as Kþ ! ���þ�þ, that violate leptonnumber as well as flavor, are also allowed in the MSSM,but are much more strongly suppressed. However there aremodels involving sterile neutrinos in which such processesare conceivably observable.112)
s
d
μ+
e-gx gx
Xs
u
μ+
νμ
gsinΘc
gW+
Fig. 11. Horizontal gauge boson mediating KL ! �e, compared with W
mediating Kþ ! �þ�.
Table II. Current 90% CL limits on K decay modes violating the SM. The violation codes are ‘‘LF’’ for lepton flavor, ‘‘LN’’ for lepton number, ‘‘G’’ for
generation number,101Þ ‘‘H’’ for helicity, ‘‘N’’ requires new particle.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS L. S. LITTENBERG
111006-9
There have been some recent exceptions to the waningof theoretical interest in kaon LFV, including models withextra dimensions,113) but even in an improved motivationalclimate, there would be barriers to rapid future progress.Although K fluxes significantly greater than those used in thelast round of LFV experiments are currently available,commensurate rejection of background is a significantchallenge.
At the moment no new kaon experiments focussed onLFV are being planned. Interest in probing LFV has largelymigrated to the muon sector. Note that a new experiment toprobe KL ! �0‘þ‘� would tend to be sensitive to severalLFV modes as well.
One exception to the poor prospects for dedicated BSMsearches in kaon decay is the search for T-violating (out-of-plane) �þ polarization in Kþ ! �0�þ�. There’s a propos-al114) to continue the work of the current experiment, KEKE246, at the J-PARC facility currently under construction.The proponents seek to make an advance in precision of afactor 20 on the measurement of the polarization. Since thisis an interference effect the advance is roughly equivalent tothat of 400 in BR sensitivity. This measurement is quitesensitive to BSM physics, particularly multi-Higgs modelsincluding certain varieties of supersymmetry.115–119)
5. Further Study
The subject of kaon decay is much richer than can bepresented in a short review such as this. Fuller accounts canbe found in refs. 120–125.
Acknowledgment
This work was supported by the U.S. Department ofEnergy under Contract No. DE-AC02-98CH10886.
1) The Superweak hypothesis postulates that CP violation is solely due
to an extremely weak �S ¼ 2 interaction.
2) M. Banner, J. W. Cronin, C. M. Hoffman, B. C. Knapp, and
M. J. Shochet: Phys. Rev. Lett. 28 (1972) 1597.
3) M. Holder et al.: Phys. Lett. B 40 (1972) 141.
4) M. Kobayashi and T. Maskawa: Prog. Theor. Phys. 49 (1973) 652.
5) L. Wolfenstein: Phys. Rev. Lett. 13 (1964) 562.
6) F. J. Gilman and M. B. Wise: Phys. Lett. B 83 (1979) 83.
7) S. L. Glashow, J. Iliopoulos, and L. Maiani: Phys. Rev. D 2 (1970)
1285.
8) M. K. Gaillard and B. W. Lee: Phys. Rev. D 10 (1974) 897.
9) J. R. Ellis, M. K. Gaillard, and D. V. Nanopoulos: Nucl. Phys. B 109
(1976) 213.
10) F. J. Gilman and M. B. Wise: Phys. Rev. D 21 (1980) 3150.
11) T. Inami and C. S. Lim: Prog. Theor. Phys. 65 (1981) 297.
12) C. Dib, I. Dunietz, and F. J. Gilman: Phys. Rev. D 39 (1989) 2639.
13) L. S. Littenberg: Phys. Rev. D 39 (1989) 3322.
14) G. Buchalla, A. J. Buras, and M. K. Harlander: Nucl. Phys. B 349
(1991) 1.
15) E. Eichten, I. Hinchliffe, K. D. Lane, and C. Quigg: Phys. Rev. D 34
(1986) 1547.
16) J. K. Black et al.: Phys. Rev. Lett. 54 (1985) 1628.
17) R. H. Bernstein et al.: Phys. Rev. Lett. 54 (1985) 1631.
18) "0 is the parameter governing �S ¼ 1-mediated CP violation in
the decay K0 ! 2� whereas " is the CP-violating state-mixing
parameter.
19) G. D. Barr et al.: Phys. Lett. B 317 (1993) 233.
20) L. K. Gibbons et al.: Phys. Rev. Lett. 70 (1993) 1203.
21) A. Alavi-Harati et al.: Phys. Rev. D 67 (2003) 012005.
22) A. Lai et al.: Eur. Phys. J. C 22 (2001) 231.
23) V. Fanti et al.: Phys. Lett. B 465 (1999) 335.
24) J. R. Batley et al.: Phys. Lett. B 544 (2002) 97.
25) W. M. Yao et al.: J. Phys. G 33 (2006) 1.
26) C. Geweniger et al.: Phys. Lett. B 48 (1974) 483.
27) A. Alavi-Harati et al.: Phys. Rev. Lett. 88 (2002) 181601.
28) E. Abouzaid et al.: Phys. Rev. D 74 (2006) 032004.
29) E. Abouzaid et al.: Phys. Rev. Lett. 96 (2006) 101801.
30) L. Littenberg: talk given at 35th Rencontres de Moriond, 2000;
hep-ex/0010048.
31) S. H. Kettell, L. G. Landsberg, and H. H. Nguyen: Phys. Atom. Nucl.
67 (2004) 1398.
32) G. Buchalla and G. Isidori: Phys. Lett. B 440 (1998) 170.
33) D. Bryman, A. J. Buras, G. Isidori, and L. Littenberg: Int. J. Mod.
Phys. A 21 (2006) 487.
34) W. J. Marciano and Z. Parsa: Phys. Rev. D 53 (1996) 1.
35) F. Mescia and C. Smith: Phys. Rev. D 76 (2007) 034017.
36) C. Jarlskog: Phys. Rev. Lett. 55 (1985) 1039.
37) Y. Grossman and Y. Nir: Phys. Lett. B 398 (1997) 163.
38) J. K. Ahn et al.: Phys. Rev. D 74 (2006) 051105.
39) J. Comfort et al.: Proposal for k0L ! �0� ��� experiment at j-parc
(2006), j-PARC Proposal 14.
40) H. Watanabe et al.: Nucl. Instrum. Methods Phys. Res., Sect. A 545
(2005) 542.
41) M. Doroshenko et al.: Nucl. Instrum. Methods Phys. Res., Sect. A
545 (2005) 278.
42) A. Alavi-Harati et al.: Phys. Rev. D 61 (2000) 072006.
43) A. J. Buras, G. Colangelo, G. Isidori, A. Romanino, and L.
Silvestrini: Nucl. Phys. B 566 (2000) 3.
44) G. Isidori, F. Mescia, P. Paradisi, C. Smith, and S. Trine: J. High
Energy Phys. JHEP08 (2006) 064.
45) H. B. Greenlee: Phys. Rev. D 42 (1990) 3724.
46) A. Alavi-Harati et al.: Phys. Rev. D 64 (2001) 012003.
47) A. Alavi-Harati et al.: Phys. Rev. D 62 (2000) 112001.
48) G. Isidori, C. Smith, and R. Unterdorfer: Eur. Phys. J. C 36
(2004) 57.
49) G. D’Ambrosio, G. Ecker, G. Isidori, and J. Portoles: J. High Energy
Phys. JHEP08 (1998) 004.
50) J. R. Batley et al.: Phys. Lett. B 576 (2003) 43.
51) J. R. Batley et al.: Phys. Lett. B 599 (2004) 197.
52) G. Ecker, A. Pich, and E. de Rafael: Nucl. Phys. B 303 (1988) 665.
53) L. M. Sehgal: Phys. Rev. D 38 (1988) 808.
54) A. Lai et al.: Phys. Lett. B 536 (2002) 229.
55) E. Cheu: talk given at Heavy Quarks and Leptons, 2006.
56) F. Gabbiani and G. Valencia: Phys. Rev. D 66 (2002) 074006.
57) G. Buchalla, G. D’Ambrosio, and G. Isidori: Nucl. Phys. B 672
(2003) 387.
58) A. Alavi-Harati et al.: Phys. Rev. Lett. 93 (2004) 021805.
59) A. Alavi-Harati et al.: Phys. Rev. Lett. 84 (2000) 5279.
60) A. Alavi-Harati et al.: Phys. Rev. Lett. 86 (2001) 397.
61) L. Littenberg: hep-ex/0212005.
62) They observed a total of two (presumably background) events. I
assume backgrounds other than KL ! ���� can be suppressed in a
future experiment.
63) T. Alexopoulos et al.: ‘‘A Proposal for a Precision Measurement of
the Decay KL ! �0� ��� and Other Rare Processes at Fermilab Using
The Main Injector—KAMI’’ (2001), proposal.
64) G. Isidori, F. Mescia, and C. Smith: Nucl. Phys. B 718 (2005) 319.
65) G. Buchalla and A. J. Buras: Nucl. Phys. B 548 (1999) 309.
66) A. J. Buras, F. Schwab, and S. Uhlig: hep-ph/0405132.
67) A. J. Buras, M. Gorbahn, U. Haisch, and U. Nierste: Phys. Rev. Lett.
95 (2005) 261805.
68) A. J. Buras, M. Gorbahn, U. Haisch, and U. Nierste: J. High Energy
Phys. JHEP11 (2006) 002.
69) A. J. Buras, M. E. Lautenbacher, and G. Ostermaier: Phys. Rev. D 50
(1994) 3433.
70) M. S. Atiya et al.: Nucl. Instrum. Methods Phys. Res., Sect. A 321
(1992) 129.
71) J. Doornbos et al.: Nucl. Instrum. Methods Phys. Res., Sect. A 444
(2000) 546.
72) I. H. Chiang et al.: IEEE Trans. Nucl. Sci. 42 (1995) 394.
73) S. C. Adler et al.: Phys. Rev. Lett. 79 (1997) 2204.
74) S. Adler et al.: Phys. Rev. Lett. 88 (2002) 041803.
75) V. V. Anisimovsky et al.: Phys. Rev. Lett. 93 (2004) 031801.
76) A. J. Buras, P. Gambino, M. Gorbahn, S. Jager, and L. Silvestrini:
Phys. Lett. B 500 (2001) 161.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS L. S. LITTENBERG
Department of Physics, University of Oxford, Keble Rd, Oxford OX1 3RH, U.K.
(Received January 24, 2007; accepted February 6, 2007; published November 12, 2007)
A brief introduction is given to high energy electron–proton interactions at the HERA collider. Thearticle focuses on deep-inelastic scattering at high momentum transfers, electroweak interactions ofneutral and charged current processes with polarised e� beams and one example of a search for signalsbeyond the standard model of particle physics.
KEYWORDS: high energy physics, deep-inelastic scattering, QCD, quarks, gluons, electroweak, neutral current,charged current, isolated leptons, BSM
DOI: 10.1143/JPSJ.76.111007
1. Introduction
HERA is a unique facility providing collisions ofelectrons (or positrons) with beam energy of 27.5 GeV withprotons of beam energy 920 GeV. It is situated at the DESYlaboratory in Hamburg. During 2000 – 2003 the collider wasupgraded to provide higher luminosity and polarised e�
beams. This article will concentrate on the physics studiedby the two general purpose detectors, ZEUS and H1. Twoother projects have used HERA beams in fixed targetmode — HERA-B studied charm and beauty (two of the so-called ‘‘heavy flavour quarks’’) production in hadron-hadroninteractions using the proton beam on a wire target andHERMES continues to study fully polarised electron-hadronscattering using the polarised e� beams on a polarisedhadronic target.
HERA has operated in two phases, HERA-I during theperiod 1992 – 2000 for much of which the proton beamenergy was slightly lower at 820 GeV and the e� beams wereunpolarised and HERA-II as described above. The HERAcollider will cease operations at the end of June 2007. Asthis review is intentionally brief, only the highlights ofHERA physics can be covered (the topics selected inevitablyreflecting the interests of the author).
2. Deep-Inelastic Scattering
Deep-inelastic scattering at HERA is the process ep!eX in which there is a large transfer of energy andmomentum from the electron to the target through theexchanged force quantum, here a virtual photon or Z0,causing the target proton to break up. As the exchangedquanta have zero electric charge this type of scattering isknown as ‘‘neutral current’’ or NC. At very large momentumtransfers the process ep! �X may also occur in which theexchanged quantum is now a charged W�— ‘‘chargedcurrent’’ or CC. In this part of the HERA phase space onemay study the electroweak (EW) properties of the proton’sconstituents. Many of the most important results fromHERA concern the strong interaction, quantum chromo-dynamics (QCD). By measuring the final state, bothinclusively and in more detail, one can learn somethingdirectly about the constituents of the proton (quarks) that
couple to EW quanta and infer quite a bit about otherconstituents (gluons, the quanta of the strong force field). Agood analogy is to imagine the combined HERA+detectorsetup as an enormous electron microscope that allows oneto ‘‘see’’ inside the proton with a resolution given roughlyby the uncertainty relation �x � h�c=jqcj, where jqcj is themagnitude of the momentum transfered from electron toproton. At the largest values of momentum transfer one isprobing a distance scale of 10�18 m, or something like athousandth of the size of a proton.
The main components of a general purpose HERAcollider detector are shown in Fig. 1. This is an eventdisplay from the H1 detector of a neutral current deep-inelastic scattering event. The beam line runs horizontallythrough the middle of the detector, which is cylindrical.The 27.5 GeV electron beam enters from the left and the820 GeV proton beam from the right. The beam electron hasbeen scattered through a large angle and emerges in theproton (forward) direction as the single charged particlein the upper half. In the lower half there is a nice example of
SPECIAL TOPICS
Q = 25030 GeV , y = 0.56, M = 211 GeV22
E /GeV
jete+
φη
t
Fig. 1. (Color online) A deep-inelastic ep! eX event in the H1 detector.
The inner open ‘‘boxes’’ represent the tracking detectors, the green (light
grey) shows the electromagnetic calorimeter and the orange (grey) the
hadronic calorimeter. All of these components are inside a superconduct-
ing solenoid of diameter 6 m providing a magnetic field of 1.15 T.�E-mail: [email protected]
Journal of the Physical Society of Japan
Vol. 76, No. 11, November, 2007, 111007
#2007 The Physical Society of Japan
Frontiers of Elementary Particle Physics, the Standard Model and Beyond
a hadronic ‘‘jet’’ of charged particles that has emergedfrom the break up of the proton in the interaction. There isalso a small amount of debris recorded around the forwardbeam pipe. The panels on the right of the figure show thatin a NC event the momentum of the scattered electrontransverse to the beam line is balanced by that of thehadronic jet — a very distinctive feature. A CC event wouldhave a hadronic jet balanced by an undetected neutrino, thusgiving events with large missing ‘‘transverse energy’’, ET.The main features of the detector are a charged particletracking system within a solenoidal magnetic field, followedby a calorimeter to measure both charged and neutralparticle energies. The calorimeter is usually divided into twosections, often with different technologies, the first [electro-magnetic (EM)] with a depth of around 25 radiation lengthsto measure the energies of electrons and photons followedby the hadronic section with a depth of up to six or seveninteraction lengths to measure hadronic energy. Muons donot interact strongly and penetrate to chambers (not shown)outside the calorimeter and coil. Final state neutrinos cannotbe detected. The H1 calorimeter uses liquid argon with leadabsorber plates for the EM section, followed by steel platesfor hadron absorption and the whole detector is optimisedfor electron identification and measurement. The ZEUSdetector has a similar structure, but with a uranium andscintillator sampling calorimeter for both EM and hadronicsections surrounding a coil of diameter 2 m providing a fieldof 1.43 T and the whole detector is optimised for hadronic jetmeasurements.
Deep-inelastic scattering has been intimately connectedwith the development of QCD and has a history as old asnuclear physics — going back through the results from muonand neutrino beams at CERN and Fermilab, the seminalresults from SLAC, the earlier work by Hofstader atStanford on nuclei to Rutherford Scattering and the discov-ery of the nucleus. The connection between the elasticscattering of the earliest experiments and deep-inelasticscattering is made clear in Fig. 2, the virtual photon isscattering elastically off one of the charged constituents inthe proton. The inelastic ep! eX cross-section is then theincoherent sum of the elastic cross-sections over all chargedpartons (parton is the generic name for all constituents inthe proton, charged or neutral, now known to be quarks,antiquarks and gluons). At a fixed ep centre-of-mass energytwo variables characterise an inelastic interaction. Thesecould be the energy and scattering angle of the scatteredbeam electron, but it is better to use Lorentz invariantquantities. Using the notation in the figure to define the 4-momenta of the particles concerned, the two most commonly
used variables are Q2 ¼ �ðk � k0Þ2, the negative of thesquared momentum transfer from the electron, and x ¼ Q2=ð2Pp � qÞ, which may be interpreted as the fraction of theincoming proton’s momentum carried by the quark involvedin the so-called ‘‘hard-scatter’’. A third variable y ¼ Pp �q=Pp � k, which is related to the scattering angle in theelectron-quark frame, is also useful. The three variables arerelated by Q2 ¼ sxy, where
ffiffisp
is the ep centre-of-massenergy and particle masses have been ignored. The doubledifferential cross-section for ep! eX takes the form
d2�
dx dQ2¼
2��2
xQ4f½1þ ð1� yÞ2�F2ðx;Q2Þ � y2FLðx;Q2Þg; ð1Þ
where F2 and FL are the structure functions of the proton —that there are two follows from the restrictions of a parityconserving interaction (electromagnetism) between twospin-1/2 particles — at large Q2 the NC cross-sectionrequires a third structure function xF3 which contains theparity violating terms from Z0 exchange. Equation (1) showsthat for Q2 M2
Z , the cross-section factorises into a piececontaining ð�=Q2Þ2 coming from the virtual photon ex-change with a coupling strength given by the fine-structureconstant � and the structure functions containing theinformation on the dynamical structure of the proton. F2
may be written as F2ðx;Q2Þ ¼P
i e2i x fiðx;Q2Þ, where ei is
the electric charge (relative to the proton’s charge) of the i-thquark and fiðx;Q2Þ is an unknown function (the partonmomentum density function or pdf) giving the probabilitythat i-th quark has a momentum xPp. FL has a morecomplicated relationship to the proton pdfs, including adependence on the gluon momentum density, xgðx;Q2Þ.However, as the contribution of FL to the cross-section issmall, except at large values of y, no details are given here.More introductory information about deep-inelastic scatter-ing in general and at HERA in particular may be found in thebook by the author and Cooper-Sarkar.1)
The pre-QCD quark-parton model predicted the aboveform for F2, that the fi would be independent of Q2 and thatFL ¼ 0. Both results were supported approximately by thefirst data on deep-inelastic scattering. The more general Q2
form has since been proved by the ‘‘factorisation’’ theoremsof QCD. Although the pdfs cannot be calculated completelyfrom QCD, the rate of change with Q2 is calculable. Thecrucial ingredients are the ‘‘splitting’’ functions which arederived from the basic processes: q! gq; g! q �qq andg! gg. The first two are analogous to bremsstrahlung andpair production in quantum electrodynamics (QED), but thethird is different and follows from the fact the gluons carrythe colour charge. Very roughly @ f =@ lnQ2 � �sðQ2Þ xgþOð�2
s Þ, where �sðQ2Þ is the QCD analogue of �. Thedependence of �s on Q2 is a consequence of quantumfluctuations and, unlike QED, �s decreases as Q2 increases— the phenomenon known as ‘‘asymptotic freedom’’ —making perturbative calculations possible for processes witha ‘‘hard’’ scale (for example Q2 larger than a few GeV2). Theimportance of the pdfs is that they are intrinsic to the proton,so if determined from deep-inelastic scattering3) then otherhard scattering processes for example p �pp! W�X can bepredicted. The accurate determination of parton densities iscrucial for all hadron collider physics and in particular forthe the large hadron collider (LHC) at CERN. The pdfs are
e+k k' e+
q = k − k'P'q
q pP Px=
Ppp
Zγ , 0
Fig. 2. Deep-inelastic scattering in the quark-parton model.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS R. DEVENISH
111007-2
also determined by ‘‘global’’ fits to hard scattering data fromhigh energy ep, �p, and p �pp experiments.4)
3. Proton Structure
The left plot of Fig. 3 is a summary of the data on F2 fromep! eX deep-inelastic scattering experiments from SLAC,CERN, Fermilab and HERA. The data are plotted at fixedvalues of x as a function of Q2, HERA results populate theband from top left (small x and Q2) to bottom right (large x
and Q2) with the lower energy fixed target data in the regionto the left at smaller Q2 values. It is clear that the predictionof the early quark-parton model is not supported by the data,for small values of x F2 rises with Q2 and for large values ofx the opposite occurs. This pattern of x and Q2 dependenceis just what is predicted by QCD. The splitting functionsderived from q! gq and gluon splitting g! gg behavelike 1=x for small x. The gluon density rises and, throughg! q �qq, so does the density of q �qq pairs that contributedirectly to F2. As Q2 increases at small x the momentumof the proton is shared amongst an increasing number ofparticles and thus the momentum of any one of themdecreases. The right-hand plot of Fig. 3 shows F2 data atQ2 ¼ 15 GeV2 as a function of x in more detail. The qualityof the agreement between the QCD calculations and the datacan be seen from this plot. The rapid decrease F2! 0 asx! 1 is simply a statement that it is very unlikely for asingle quark to carry almost all the proton’s momentum.In the region of the pre-HERA fixed target data, 3� 10�2 <x < 0:2 (triangles), F2 increases slowly by about 20%. Incontrast in the HERA region, 3� 10�4 < x < 3� 10�2, F2
increases by a factor of 3. This is a dramatic increase indeedand one of the most important results from HERA and itraises an interesting question — will the rising behaviourpersist to yet smaller x-values, as the calculated curvessuggest?
That the rise of F2 cannot continue indefinitely followsfrom a very general consideration. For values of Q2 M2
Z
deep-inelastic scattering is equivalent to virtual-photonproton scattering at a centre-of-mass energy, W , given byW2 ¼ Q2ð1=x� 1Þ. So small x corresponds to large W . Nowunitarity (conservation of probability) limits how rapidly anytotal cross-section can increase with energy. Given that�ð��pÞ ¼ ð4�2�=Q2ÞF2, a limit on the large W behaviour of�ð��pÞ is equivalent to a limit on the low x behaviour ofF2. Another, more physical approach, is to realise that as thegluon density increases at low x, other terms will start tobecome important, particularly the possibility of gluonrecombination gg! g. The recombination cross-section isof order �s=Q
2, thus one might expect to see ‘‘saturation’’effects when ð�s=Q2Þxgðx;Q2Þ � �R2, where R is the radiusof the proton. Taking R � 0:8 fm and Q2 � 10 GeV2 givesxg � 2000, quite a bit above the largest projections at thesmallest values of x reachable at HERA.5,6) Much effort hasbeen expended to find other signs in the HERA data forsaturation effects and, although there are some tantalisinghints, there are competing explanations. What is undoubt-edly true is that the rising gluon density has sparkedenormous interest in trying to understand high-density gluondynamics theoretically and to extrapolate from the HERAmeasurements to heavy-ion collisions (in which these effectsare enhanced) at the Brookhaven relativistic heavy-ioncollider (RHIC) and the LHC.
The gluon density is only measured indirectly from the Q2
dependence of the inclusive structure function. The excellenttracking of the HERA detectors enables measurement ofprocesses involving the heavy charm and beauty quarks(masses around 1.3 and 4.5 GeV/c2 respectively). This canbe done by reconstruction of a D� meson7) (c �qq combination)or by identifying events with secondary vertices or trackswith large impact parameters (both indicative of long livedheavy meson decay). For beauty measurements in particular,the identification of the electron or muon from the weaksemi-leptonic decay of b-quark containing hadrons or jetswith large impact parameter tracks is crucial.8) The dominant
Q2 (GeV2)
F2(
x,Q
2 ) +
c(x)
H1
ZEUS
BCDMS
E665
NMC
SLAC
0
1
2
3
4
5
6
7
8
9
10
10-1
1 10 102
103
104
105
106
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
10-4
10-3
10-2
10-1
1
Q2 = 15 GeV2
x
F2em
H1 96/97
ZEUS 96/97
NMC, BCDMS, E665
CTEQ6D
MRST (2001)
Fig. 3. Left: Fproton2 at fixed x values as a function of Q2.2) Right: The rise of F2 at small x.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS R. DEVENISH
111007-3
process for the production of a heavy q �qq pair in deep-inelastic scattering is so-called photon–gluon fusion (shownin the left-hand plot of Fig. 4). The rate for this processdepends directly on the gluon pdf. The crucial point is that aheavy q �qq pair is extremely unlikely to be produced in theprocess by which a quark ‘‘fragments’’ into the jet of hadronsseen in the detector. The right-hand plot of Fig. 4 shows theQ2 dependence of the cross-section for ep! eD�X. Themeasured data are compared with a next-to-leading order(i.e., diagrams to order �2
s ) QCD calculation using a gluondensity derived from fitting the ZEUS structure functiondata. The good agreement over five orders of magnitudegives one confidence in both QCD calculations of hardprocesses and the consistency of the HERA measurements.
4. Electroweak Standard Model Measurements
The ability of HERA to allow the measurement of bothNC (ep! eX) and CC (ep! �X) total cross-sections tovalues of Q2 well above the mass scale of the W and Z weakbosons offers a nice ‘‘textbook’’ picture of the unity of theelectromagnetic and weak interactions at large Q2. This isshown in the left-hand plot of Fig. 5 which summariesthe Q2 dependence of the four cross-sections for NC and CCscattering with incident electron and positron beams. As
discussed after eq. (1), the dominant diagram for NCscattering behaves as �=Q2. For the CC process thebehaviour is M2
WGF=ðM2W þ Q2Þ, where GF is the Fermi
constant as measured in low energy beta decay. For Q2 M2
W (MW ¼ 80:4 GeV/c2), the CC cross-section is tending toa constant value much smaller than that of the NC cross-section which is enhanced by 1=Q4 as Q2 decreases. At largevalues of Q2, near and above M2
W, the NC and CC cross-sections are of comparable magnitude and both eventuallydecreasing as 1=Q4. This is EW unification in action — thesimplest explanation is that � � M2
WGF — which, with a fewextra constants, is what is predicted by the Weinberg-Salamtheory. The small but clear dependence of the NC cross-section on the charge of the beam e� follows from the parityviolating Z0 exchange. For the CC cross-sections parity isalso violated, but the charge dependence is larger because ofthe quark charge (and flavour) selection by W� exchange asdiscussed in the next section.
Another textbook plot — exploiting the polarised e�
beams at HERA-II — is shown in the right-hand plot ofFig. 5. This shows the dependence of the CC cross-sectionson the beam polarisation. In the standard model, neutrinoshave only left-handed helicity (spin projected along themomentum direction — analogous to circular polarisation of
)2
(GeV2Q
-110 1 10 210 310
)2
(nb
/GeV
2/d
Qσ
d
-410
-310
-210
-110
1
10
ZEUS DIS BPC D* (prel.) 98-00
ZEUS DIS D* 98-00
=1.35 GeV, ZEUS NLO pdf fitcHVQDIS, M
ZEUS
Fig. 4. Left: photon–gluon fusion to c �cc. Right: Q2 dependence of the cross-section for ep! eD�X.
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1
10
103
104
HERA
H1 e+p CC 94-00
H1 e-p CC
ZEUS e+p CC 99-00
ZEUS e-p CC 98-99
SM e+p CC (CTEQ6D)
SM e-p CC (CTEQ6D)
H1 e+p NC 94-00
H1 e-p NC
ZEUS e+p NC 99-00
ZEUS e-p NC 98-99
SM e+p NC (CTEQ6D)
SM e-p NC (CTEQ6D)
y < 0.9
Q2 (GeV2)
dσ/
dQ
2 (p
b/G
eV2)
eP-1 -0.5 0 0.5 1
(p
b)
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σ
0
10
20
30
40
50
60
70
80
90
100
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y < 0.9
Xν →p +e
H1
ZEUS
SM (MRST)
Xν →p e
H1 (prel.)
H1
ZEUS (prel.)
ZEUS
SM (MRST)
Charged Current ep Scattering (HERA II)
Fig. 5. (Color online) Left: NC and CC cross-sections at large Q2. Right: Polarisation dependence of the CC cross-section for eþ and e� beams.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS R. DEVENISH
111007-4
light) and anti-neutrinos only right-handed helicity. Theexpectation is then that the e�p! �X cross-section willshow a ð1� PÞ dependence (where P is the beam polar-isation) and the eþp! ���X a ð1þ PÞ dependence. This iswhat the measurements in the figure show.9,10) The differ-ence in magnitude of the two cross-sections follows from thecharge of the W: for e�p the dominant contribution comesfrom uW� ! d; for eþp it is dWþ ! u but in this case thecontribution is suppressed by an angular factor ð1� yÞ2. Thecloseness of the data to the expected polarisation depend-ence can be used to extract limits on possible ‘‘right-handed’’weak exchanges, beyond the standard model. Accuratemeasurements of both NC and CC cross-sections at large x
and Q2 are also an important input to the fits that are usedto extract proton pdfs as they depend on the u and d
quarks — the so-called valence quarks which give the protonits charge and other quantum numbers. Through theinterference of the virtual photon and the Z0, the NCcross-section also depends on the axial and vector EWcouplings of the valence quarks, allowing these to bemeasured as well.11) All these interesting measurementswill be completed when the full HERA-II data set isavailable.
5. High pT Leptons
Beyond the standard model (BSM) searches at HERAcover a vast range of topics. No definitive signals have yetbeen seen, but lower limits on the masses of states in manymodels have been set. For example, leptoquarks (particlesformed by fusing a quark and lepton) which occur in manyBSM models could be seen at HERA as a resonant excitationin either NC or CC deep-inelastic scattering. Most searchesproduce a limit on a combination of coupling and mass.Typical lower limits on the masses of such particles are inthe range 200 – 300 GeV/c2 depending on the model as-sumption for the coupling. For such direct searches the massreach is limited by the HERA centre-of-mass energy,318 GeV.
Isolated leptons ðe; �; �Þ with large ET in events withmissing energy are a good signature for BSM physics. Suchevents can also be produced by the standard model process
ep! eWX followed by the leptonic decay of the W , see theleft-hand diagram of Fig. 6 in which the f�ff from the W decayis here ‘ ���‘. Isolated lepton events have generated a lot ofinterest in recent years and are an unsolved problem for theHERA community. H1 see an excess of events above theexpectation from W production, particularly in eþp scatter-ing. ZEUS, on the other hand, find agreement with thestandard model expectation in all channels. The discrepancyis most clear for large values of PX
T the total transversemomentum of the hadronic system in the event, as thestandard model W production mechanism tends to produceisolated lepton events with low values of this quantity. Theright-hand plot of Fig. 6 shows the PX
T spectrum from H1 forisolated e and � events in eþp scattering. A clear excess ofevents above the standard model expectation is observed.Table I gives a summary of the observed and expected eventnumbers from both experiments, for the highest PX
T bins.13)
Taking the e and � results together the H1 eþp data give anexcess of around 3:4�. For both ZEUS channels and for H1e�p data there is no excess. Both collaborations have triedhard to identify any differences in detector performance oranalysis technique that could explain the different results,without success. An example of a possible BSM mechanismthat could match the pattern of the H1 data comes from R-parity violating supersymmetry using 3rd-generation fields,eþ þ d! t ���� via ~bb (SUSY partner of the b-quark) ex-change. The top quark then decays by t! ‘�‘ (‘ ¼ e or �)with large missing energy. The equivalent process in e�p
scattering could not produce the positive charge-2/3 topquark. For a more general discussion of these events in thecontext of R=p SUSY, see ref. 14.
channelsμ (GeV) e and XTP
0
Eve
nts
-110
1
10
210 H1 Data (prelim.)All SMSignal
= 28DataN 2.6±= 18.5 SM N
)-1p, 158 pb+ events at HERA 1994-2004 (emissTl+P
10 20 30 40 50 60 70 80
Fig. 6. Left: diagram for W production at HERA; Right: H1 PXT spectrum for events with a high pT electron or muon and missing energy in eþp
interactions.12)
Table I. Summary of isolated lepton results from H1 and ZEUS.13Þ
Events with PXT > 25 GeV
Electrons
data/SM
Muons
data/SM
eþp H1 158 pb�1 9=3:3� 0:4 6=2:3� 0:4
ZEUS 106 pb�1 1=1:5� 0:1 1=1:5� 0:2
e�p H1 184 pb�1 3=3:8� 0:6 0=3:1� 0:5
ZEUS 143 pb�1 3=2:9� 0:5 2=1:6� 0:2
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS R. DEVENISH
111007-5
6. Conclusions
This brief survey has given a very partial overview ofsome of the many physics topics that have been studied inhigh-energy ep scattering at the HERA collider. There aremany other areas that have not been covered. These include‘‘diffractive scattering’’ in which the proton remains intact oremerges in a low lying excited state with the same quantumnumbers (ep! epX) and the many interesting results onhadronic final states, QCD jet physics and the hadronicproperties of the photon from photoproduction reactions.Photoproduction events, �p! X, are those in which anenergetic and nearly real photon, radiated from the beam e�
interacts with the proton. The scattered e� remains un-detected in the beam pipe. Many more details about the twoexperiments, the published results and conference reportsmay be found at the H1 and ZEUS websites.15)
Acknowledgements
I would like to thank my colleagues on the ZEUS and H1experiments for many fruitful years of physics at HERA andMark Bell for a careful reading of the manuscript.
1) R. Devenish and A. Cooper-Sarkar: Deep Inelastic Scattering (Oxford
University Press, Oxford, U.K., 2004).
2) W.-M. Yao et al. (Particle Data Goup): J. Phys. G 33 (2006) 1.
3) For example, S. Chekanov et al. (ZEUS Collaboration): Eur. Phys. J.
C 42 (2005) 1.
4) Full details on global pdf fits may be found at the Durham University
HEPDATA website: http://durpdg.dur.ac.uk/HEPDATA/
5) J. Breitweg et al. (ZEUS Collaboration): Eur. Phys. J. C 7 (1999) 609.
6) C. Adloff et al. (H1 Collaboration): Phys. Lett. B 520 (2001) 183.
7) S. Chekanov et al. (ZEUS Collaboration): Phys. Rev. D 69 (2004)
012004.
8) A. Aktas et al. (H1 Collaboration): Eur. Phys. J. C 45 (2006) 23.
9) S. Chekanov et al. (ZEUS Collaboration): Phys. Lett. B 637 (2006)
210.
10) A. Aktas et al. (H1 Collaboration): Phys. Lett. B 634 (2006) 173.
11) A. Aktas et al. (H1 Collaboration): Phys. Lett. B 632 (2006) 35.
12) H1 Collaboration, Prelim-06-62: submitted to ICHEP06, Moscow,
July 2006; See also, V. Andreev et al. (H1 Collaboration): Phys. Lett.
B 561 (2003) 241.
13) C. Diaconu: talk in the BSM session ICHEP06, Moscow, July 2006;
hep-ex/0610041; to be published in the proceedings.
14) S. Y. Choi et al.: hep-ph/0612302.
15) H1: www-h1.desy.de; ZEUS: www-zeus.desy.de
Robin Devenish was born in Trinidad, West
Indies in 1942, but has lived in the UK since
1944. He obtained BA (1964), MA, and Ph. D.
(1969) from Cambridge University. He held re-
search associate positions at Lancaster University
and UCL and was on the staff of the DESY
Laboratory in Hamburg for six years. In 1979 he
moved to Oxford where he has been ever since. He
has worked on the photo- and electro-production of
High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801
(Received April 28, 2007; accepted September 7, 2007; published November 12, 2007)
This paper reviews the current understanding of neutrino mass and mixing parameters in the frame-work of three active neutrino species and experimental results which established neutrino oscillations.Future prospects are also discussed.
KEYWORDS: neutrinoDOI: 10.1143/JPSJ.76.111008
1. Introduction
In 1998, long sought neutrino oscillation was discoveredby the Super-Kamiokande group in atmospheric neutrinoobservations.1) Since then, neutrino oscillation has also beenestablished in solar neutrino observations and long-baselineexperiments utilizing reactor- and accelerator-producedneutrinos.
Neutrino oscillations imply finite neutrino mass as wellas flavor mixing in the neutrino sector. In the StandardModel, neutrinos are massless because no right-handedneutrinos exist in the theory. Therefore, neutrino oscilla-tion is the first evidence of physics beyond the StandardModel.
An unexpectedly large mixing observed in atmosphericneutrino oscillations indicated that the neutrino mixingmatrix is quite different from the well-known quark mix-ing matrix. Comparison of the mass and mixing in thequark sector and those in the lepton sector would be animportant clue to investigate physics beyond the StandardModel. Therefore, obtaining complete knowledge ofneutrino mass and mixing is the goals of future neutrinophysics.
From the cosmic microwave background (CMB) dataalone, neutrino mass is constrained as2–4)
Xi
�i < 2 eV: ð1Þ
A stronger constraintXi
�i < 0:7 eV ð2Þ
is obtained using the combination of CMB and othercosmological data.2,5) Therefore, the mass of the heaviestneutrino is less than 0.7 – 2 eV. Compared with the massesof quarks and charged leptons,6) the neutrino masses areextremely small. If neutrinos are Dirac particles, a minimalextension of the Standard Model may allow a Dirac neutrinomass term LD ¼ mDð�L�R þ �R�L) by introducing the right-handed neutrino field, but the tiny neutrino mass means atiny Yukawa coupling compared to Yukawa couplings ofquarks and charged leptons. There is no explanation for thismysterious relation between the neutrino mass and other
fermion masses. However, because the neutrino is a neutralparticle, a Majorana mass term LM ¼ Mð�cR�R þ �R�
cR) may
be introduced, where �cR is the charge conjugate of �R. The
Majorana mass term violates the lepton number by 2, andthe neutrino and antineutrino states cannot be distinguished.Then, neutrinos are Majorana particles. The existence ofboth Dirac and Majorana mass terms leads to a naturalexplanation of the tiny left-handed neutrino mass, m �m2
D=M, by introducing heavy right-handed neutrinos withmass M > 1012 GeV, where mD is a typical quark orcharged-lepton mass. Conversely, the tiny neutrino massindicates the existence of a very high mass scale. This iscalled the seesaw mechanism.7–9)
Whether neutrinos are Dirac or Majorana particles isof fundamental importantance. If neutrinos are Majoranaparticles, ‘‘leptogenesis,’’ proposed by Fukugita andYanagida,10) is an attractive mechanism for the origin ofbaryon asymmetry in the universe.
In the following, we will first discuss in §2 the presentknowledge of neutrino mass and mixing. In §3, we willbriefly survey the history of neutrino oscillation experi-ments. Then, we will review experiments which establishedneutrino oscillations. Atmospheric neutrino experiments andaccelerator long baseline neutrino oscillation experimentswhich measured �m2
32 and �23 are reviewed in §4, and solarneutrino experiments and the KamLAND reactor neutrinooscillation experiment which measured �m2
12 and �12 in §5.In §6, future prospects of neutrino experiments are dis-cussed. Conclusions are given in §7.
2. Neutrino Mass and Mixing
2.1 Neutrino oscillation parametersWithin the framework of three active neutrino species,11)
the flavor eigen states �� (� ¼ 1; 2; 3) and the mass eigenstates �i (i ¼ 1; 2; 3) with eigen mass mi are related by thefollowing equation.
�e
��
��
0B@
1CA ¼ ½U�i�
�1
�2
�3
0B@
1CA; ð3Þ
where U is a unitary 3� 3 mixing matrix. It is defined as aproduct of three rotation matrices: the parameters involvedare three mixing angles �12, �23, and �13, and a CP-violatingphase �:
where Sij and Cij stands for sin �ij and cos �ij, respectively.Historically, Maki, Nakagawa, and Sakata first formulated
two-flavor neutrino oscillations in 1962.12) Therefore, themixing matrix U is called Maki–Nakagawa–Sakata or MNSmatrix after them. Prior to them, Pontecorvo predictedneutrino-antineutrino oscillations in 1957.13) Therefore, U isalso called Maki–Nakagawa–Sakata–Pontecorvo or MNSPmatrix.
If neutrinos are Majorana particles, there are additionaltwo CP-violating phases called Majorana phases, but theyhave no effects on neutrino oscillations. The generalexpression for the probability of neutrino oscillation ��!�� in vacuum at distance L is given by
Pð��! ��Þ ¼Xi
U��iU�ie�iðm2
i =2EÞL
�����
�����2
; ð5Þ
where E is the neutrino energy. It is apparent that theneutrino mass enters into the neutrino oscillation probabilityas mass square differences, �m2
ij ¼ m2i � m2
j . For threeactive neutrinos there are only two independent values of�m2, because
�m212 þ�m2
23 þ�m231 ¼ 0: ð6Þ
In the analyses of neutrino oscillation experiments, thetwo-neutrino oscillation framework is often used. In vacu-um, the neutrino oscillation probability Pð2�Þð��! ��Þ for� 6¼ � is given by
Pð2�Þð�� ! ��Þ ¼ sin2 2� sin2ð1:27�m2L=EÞ ð7Þ
Here, the units of �m2, E, and L are eV2, GeV, and km,respectively. In fact, the atmospheric neutrino oscillationindicated that it is almost pure two-neutrino oscillation invacuum, �� $ �� ,
with �m2atm � ð2{3Þ � 10�3 eV2, and a nearly maximal
mixing angle, sin2 �atm � 1.The solar neutrino measurements by SNO (Sudbury
Neutrino Observatory) and Super-Kamiokande showedevidence for neutrino flavor conversion. A plausible explan-ation is neutrino oscillation dictated by matter effect[Mikheyev–Smirnov–Wolfenstein (MSW) effect14)] with�m2
sol � ð5{10Þ � 10�5 eV2 and a large mixing angle�sol � 30� in the two-neutrino framework. This parameterregion is called the MSW large mixing angle (LMA)solution. Later, the KamLAND reactor ���e disappearanceexperiment determined �m2
sol with much better accuracy.Now, the values of �m2
atm and �m2sol are clearly different,
they are identified with the two independent �m2 in eq. (6).At this point, it is worth noting that the LSND group claimeda positive signal15) observed in a short baseline neutrinooscillation experiment, ����! ���e. If confirmed, the corre-sponding �m2 should lie in the range of 0.1–10 eV2. Thismeans that there are at least three independent values of�m2, indicating the existence of at least four neutrino
species. Since the number of light active neutrino species isconstrained to three,11) an additional one should be a sterileneutrino species. However, this LSND ‘‘anomaly,’’ has notbeen confirmed by the recent short baseline ��! �eoscillation experiment MiniBooNE.16) If the oscillations ofneutrinos and antineutrinos are the same, the MiniBooNEresult excludes neutrino oscillations claimed by LSND at98% CL (confidence level). Therefore, the three-neutrinoframework now stands on a solider basis.
By convention, we define
�m2sol � �m2
21 ¼ m22 � m2
1 > 0: ð9Þ
With this definition, the largest component of �1 is �e. Forlarger mass splitting, we define
�m2atm � j�m2
32j ¼ jm23 � m2
2j: ð10Þ
Note that two-flavor �� $ �� oscillation is vacuum oscil-lation which is symmetric under the change of sign of �m2
32.In order to determine signð�m2
32Þ, matter effects are needed.If neutrino mass spectrum is hierarchical, there are two
possibilities; one is ‘‘normal mass hierarchy’’ m3 > m2 > m1
and the other is ‘‘inverted mass hierarchy,’’ m2 > m1 > m3.It is one of important tasks of future neutrino oscillationexperiment to resolve this ambiguity. There is anotherpossibility of ‘‘quasi-degenerate’’ mass spectrum, m1 ’ m2 ’m3 > 0:1 eV. The lower mass limit is obtained from theobserved values of j�m2
32j and �m221. In this case, neutrino-
less double beta decay experiments will have sensitivities toexplore the relevant mass region in near future.
In the framework of three-neutrino oscillation, to theextent that �m2
12 is neglected, the oscillation probabilities invacuum can be approximately expressed with the two mix-ing angles, �23 and �13, and �m2
32. The �� $ �� oscillationprobability is given as
Note that eq. (12) has an identical form with the two-neutrino disappearance probability in vacuum, Pð2�Þð ���e!���eÞ with a mixing angle �13. From a short baseline reactorneutrino oscillation experiment, CHOOZ,17) �13 is known tobe small, sin2 2�13 < 0:19. Then, eq. (11) reduces to thetwo-neutrino oscillation in vacuum. From eqs. (8) and (11),
�23 �atm: ð13Þ
Writing the survival probability of solar �e in terms of theMSW effects with two-neutrino mixing as
Psolarð2�Þ ð�e! �eÞ � Pð2�Þð�e! �e; �m2
sol; sin2 2�solÞ; ð14Þ
if �13 is small the survival probability of solar �e with three-neutrino mixing is approximately written18) as
Psolarð�e! �eÞ¼ cos4 �13Pð2�Þð�e! �e; �m2
21; sin2 2�12Þ þ sin4 �13: ð15Þ
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS K. NAKAMURA
111008-2
For �13 � 0, therefore,
�12 �sol: ð16Þ
Measurement of the small mixing angle �13 and the CPviolating phase � in the MNS matrix and measurement of thesign of �m2
32 are important problems left for future neutrinooscillation experiments.
2.2 Absolute neutrino mass scaleThe measurement of the beta decay spectrum of tritium,
3H! 3Heþ e� þ ���e, has been the most sensitive method ofdirect neutrino mass measurement. Because of the neutrinomixing, what is measured is the quantity6)
m2(eff)�e�Xi
jUeij2m2i : ð17Þ
The best upper limit (95% CL) so far was obtained by the
Mainz tritium beta decay experiment,19)
ffiffiffiffiffiffiffiffiffiffiffiffim2(eff)�e
q< 2:3 eV.
If neutrinos are Majorana particles, neutrinoless doublebeta decay ð0���Þ, ðZ;AÞ ! ðZ þ 2;AÞ þ 2e�, occurs. Thedecay rate is proportional to the square of the effectiveMajorana mass hmi,
where �21 and �31 are the Majorana CP-violating phases. Ingeneral, cancellations may occur for hmi.
So far, the best limit of hmi < 0:2{1:1 eV has been ob-tained with an isotope of 130Te using a cryogenic detector.20)
The wide range of upper limit reflects uncertainties in thetheoretical nuclear matrix elements. Although there is aclaim of positive observation of neutrinoless double betadecay using 76Ge, corresponding to hmi ¼ 0:1{0:9 eV,21) theresult has not been confirmed.
From the present knowledge of the neutrino oscillationparameters, the relation between the effective Majoranamass and the lightest neutrino mass can be evaluated foreach possibility of the neutrino mass spectrum, as shown inFig. 1. If the neutrino mass spectrum is quasi-degenerate,ð0���Þ will be observed with hmi 50 meV. If ð0���Þ isobserved with hmi � 20{50 meV, the inverted mass hier-archy is the likely neutrino mass spectrum.
3. History of Neutrino Oscillation Experiments
Searches for neutrino oscillations have been made using avariety of neutrino sources; reactors, accelerators, atmos-pheric neutrinos and solar neutrinos. Before the discovery ofatmospheric neutrino oscillations, accelerator and reactorneutrino oscillation experiments have been conducted with abaseline length of typically hundreds of meters and tens ofmeters, respectively. Using the present terminology, thesewere short baseline experiments.
The sensitivity of a neutrino oscillation experiment to�m2 is given by �E=L for vacuum oscillation. For shortbaseline reactor experiments, typical parameters are E ¼1{10 MeV, L ¼ 10{50 m, and therefore the sensitivity to�m2 is 10�2–1 eV2. This is compared with the sensitivity ofshort baseline accelerator experiments, E ¼ 0:1{100 GeV,L ¼ 0:1{1 km, �m2 10�1 eV2. For atmospheric neutrinos,however, the baseline is as long as the Earth’s diameter,E ¼ 1{10 GeV, L � 107 m, �m2 ¼ 10�4{10�3 eV2 and forsolar neutrinos E ¼ 1{10 MeV, L � 1011 m, �m2 ¼ 10�11{
10�10 eV2 (note, however, that if matter effect is at work, itcan significantly change the sensitivity to �m2 from thatcorresponding to the vacuum oscillation).
Soon after the initial result23) of the chlorine solar neutrinoexperiment indicated significantly smaller flux than thetheoretical predictions in 1968, Gribov and Pontecorvopointed out the possibility of solar neutrino oscillation.24)
However, due to various astrophysical uncertainties itwas not possible to unambiguously verify neutrino flavorconversions until 2001 (see §5).
Searches for neutrino oscillations using the data fromreactor and accelerator neutrino experiments started around1976.25) Thereafter, extensive efforts have been made, but noevidence was found because the exlored �m2 range was notrelevant. Though in a few cases oscillations were report-ed from reactor experiments,26,27) they were not confirmed.
It was fortuitous that the value of �m232 was well-matched
with the sensitivity of the atmospheric neutrino oscillationexperiment and the mixing angle was almost maximal.Otherwise, it would not have been so obvious whether theatmospheric neutrino experiment could discover the neutrinooscillation first. It should be noted, however, that therewere already hints for neutrino oscillations before Super-Kamiokande’s discovery.1) Since 1988, Kamiokande Collab-oration reported28–30) that the observed �=e ratio (the ratio ofthe muon events to electron events, both induced by atmos-pheric neutrinos) was significantly smaller than the theoret-ically expected �=e ratio. More interesting results presentedby the Kamiokande Collaboration was zenith-angle de-pendence of R = (observed �=e)/(expected �=e) for eventshaving a visible energy > 1.33 GeV.30) Kamiokande Collab-oration suggested neutrino oscillations to explain theseobservations.28–30) Some of the other atmospheric neutrinoexperiments31,32) supported Kamiokande’s observation ofR < 1, but other experiments33,34) reported R to be consistentwith unity. The high statistics data from an order-of-magnitude bigger detector, Super-Kamiokande, was neededto disentangle this controversy.
0.001 0.010 0.100 1.000minimum neutrino mass mνmin
(eV)
0.001
0.010
0.100
1.000
effe
ctiv
e m
ass
<m
ββ>
(eV
)
Degenerate
Inverse
Normal
Fig. 1. The relation between the effective Majorana mass and the lightest
neutrino mass for different neutrino mass spectrum. the shaded band
coresponds to the known best values of the oscillation parameters and
�13 ¼ 0. The width of the band reflects unknown Majorana phases. The
region enclosed by the dashed lines further reflects the errors of the
oscillation parameters. This figure is taken from ref. 22.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS K. NAKAMURA
111008-3
4. Measurements of j�m232j and �23
4.1 Atmospheric neutrino resultsThe first evidence for the neutrino oscillation was
presented by the Super-Kamiokande Collaboration in1998.1) The zenith-angle distributions of the �-like events(mostly muon-neutrino initiated charged-current interac-tions) showed clear deficit compared to the no-oscillationexpectation.
Super-Kamiokande is a 50-kton water Cherenkov detectorlocated in the Mozumi mine in Gifu Prefecture, about250 km west of Tokyo, at a depth of 2,700 m of waterequivalent. The inside of the water tank is opticallyseparated into the cylindrical inner detector and a 2.75 mthick outer shield region (anti-counter) by opaque sheets.This experiment started observation in April 1996. InNovember 2001, Super-Kamiokande suffered from anaccident in which substantial number of photomultipliertubes were lost. The detector was rebuilt within a yearwith about half of the original number of photomultipliertubes. The experiment with the detector before the accidentis called Super-Kamiokande-I (SK-I), and that after theaccident is called Super-Kamiokande-II (SK-II). Later in2005 – 2006, the detector was rebuilt again to recover theoriginal number of photomultiplier tubes, and the experi-ment entered a new phase of Super-Kmaiokande-III (SK-III). The complete SK-I atmospheric-neutrino results with a92 kton�yr (1489 live-day) exposure to atmospheric neutri-nos are reported in ref. 35.
Neutrino events in Super-Kamiokande are classified intofully contained (FC) events and partially contained (PC)events. The FC events are required to have their vertexposition inside the 22.5 kton fiducial volume, defined to be>2 m from the PMT wall of the inner detector, and to haveno visible energy deposited in the anti-counter. The PCevents are those events which have at least one prong thatpenetrates the inner detector. Obviously, the total visibleenergy can be measured for the FC events, but not for the PCevents.
Neutrino events are also classified according to thenumber of Cherenkov rings. A Monte Carlo study showsthat more than 90% of the FC single-ring events haveleptons which remember the flavor of the parent neutrinos;the contribution from neutral-current (NC) reactions is lessthan 10%. For the FC multi-ring events, the contaminationof NC interactions is significant (�30%).
FC events are subjected to particle identification of thefinal-state particles. On the other hand, all the PC eventswere assumed to be �-like since the PC events comprise a98% pure charged-current �� sample. The method adoptedfor the FC events identifies the particle types as e-like or�-like based on the pattern of each Cherenkov ring. A ringproduced by an e-like ðe�; Þ particle exhibits more diffusepattern than that produced by a �-like ð��; �Þ particle,since an e-like particle produces an electromagnetic showerand low-energy electrons suffer considerable multipleCoulomb scattering in water.
For FC events, another classification of neutrino events isdefined in terms of the visible energy, Evis. The sub-GeVevents are defined as those events with Evis < 1:33 GeVand a lower momentum cut of 100 (200) MeV/c for e-like
(�-like) events. The multi-GeV events are defined as thoseevents with Evis > 1:33 GeV.
The zenith-angle dependence of each category of eventsobserved in SK-I is shown in Fig. 2. Here, the FC sub-GeVevents are separately shown in two parts (P 400 MeV/cand P > 400 MeV/c, where P denotes the lepton momen-tum). The box histograms show the Monte Carlo predictionfor the hypothesis of no neutrino oscillations. One noticesthat the zenith-angle distributions of the FC �-like eventsand PC events show a strong deviation from the Monte Carloexpectations. The zenith-angle distributions of the upward-going muons also show deviations from the Monte Carloexpectations. On the other hand, the zenith-angle distribu-tion of the FC e-like events is consistent with the MonteCarlo expectation.
The observed zenith-angle dependence of atmosphericneutrino events suggests �� $ �� oscillations. A two-neutrino oscillation analysis with the hypothesis of �� $�� was made with use of all categories of events shown inFig. 2 as well as upward-going muon events.35) The best fitwas obtained at a slightly unphysical value of sin2 2� ¼ 1:02
and �m2 ¼ 2:1� 10�3 eV2 with �2min/DOF (degrees of
freedom) = 174.5/177. By limiting the parameter space tothe physical region, the minimum �2 value, �2
min=DOF ¼174:8=177, was obtained at sin2 2� ¼ 1:00 and �m2 ¼ 2:1�10�3 eV2. On the other hand, assuming no oscillation(sin2 2� ¼ 0 and �m2 ¼ 0), a �2 value was found to be478.7 for 179 DOF. It is very striking that the observed�� $ �� mixing is consistent with maximal mixing.Figure 3 shows the 68%, 90%, and 99% CL allowed regionsin the ðsin2 2�;�m2Þ plane. In Fig. 2, the expectation for�� $ �� oscillations with the oscillation parameters corre-sponding to the �2 minimum in the physical region is shownby the solid lines.
Though the SK-I atmospheric neutrino observations gavecompelling evidence for neutrino flavor conversion �� $ ��which is consistent with vacuum neutrino oscillations,35)
0
100
200
300
-1 -0.5 0 0.5 1
Sub-GeV e-likeP < 400 MeV/c
0
100
200
300
-1 -0.5 0 0.5 1
Num
ber
of E
vent
s Sub-GeV e-likeP > 400 MeV/c
0
50
100
150
-1 -0.5 0 0.5 1cosθ
Multi-GeV e-like
0
100
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300
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0
20
40
60
-1 -0.5 0 0.5 1
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0
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-1 -0.5 0 0.5 1
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150
-1 -0.5 0 0.5 1cosθ
Multi-GeV μ-like
0
50
100
150
200
-1 -0.5 0 0.5 1cosθ
PC
Fig. 2. The zenith angle distributions for fully contained 1-ring events,
multi-ring events, and partially-contained events. The points show the
data, boxes show the non-oscillated Monte Carlo events, and the solid
histograms show the best-fit expectations for �� $ �� oscillations. The
height of the boxes show the statistical error of the Monte Carlo.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS K. NAKAMURA
111008-4
other exotic explanations such as neutrino decay36,37) andquantum decoherence38) cannot be completely ruled outfrom the zenith-angle distributions alone. The firm evidencefor neutrino oscillation is to confirm characteristic sinusoidalbehavior of the conversion probability as a function ofneutrino energy E for a fixed distance L in the case of longbaseline neutrino oscillation experiments, or as a function ofL=E in the case of atmospheric neutrino experiments.
By selecting events with high L=E resolution, evidence forthe dip in the L=E distribution was observed at the rightplace expected from the interpretation of the SK-I data interms of �� $ �� oscillations39) (see Fig. 4). This dip cannotbe explained by alternative hypotheses of neutrino decay andneutrino decoherence, and they are excluded at more than 3�in comparison with the neutrino oscillation interpretation. At90% CL, the constraints obtained from the L/E analysis are1:9� 10�3 < �m2 < 3:0� 10�3 eV2 and sin2 2� > 0:90.These results are consistent with the SK-I final resultsof the combined zenith-angle analysis of fully-contained,
partially-contained, and upward-going muon events (seeFig. 3).
When the Super-Kamiokande Collaboration announcedthe compelling evidence for atmospheric �� $ �� oscil-lations in 1998,1) two other underground experiments,MACRO and Soudan 2, reported supporting evidencefrom respective atmospheric neutrino observations. Thefinal results from these experiments also indicate �� $ ��oscillations consistent with the SK-I atmospheric neutrinoresults.
The Soudan 2 detector was a 963 ton fine-segmented iron-tracking calorimeter with a 770 ton fiducial volume, locatedat a depth of 2070 m of water equivalent underground ofthe Soudan Mine in Minnesota. The Soudan 2 group hasanalyzed the neutrino L=E distributions with the hypothesesof neutrino oscillations and no neutrino oscillations.40) Theprobability of the no oscillation hypothesis is 5:8� 10�4,and the 90% CL allowed region in the ðsin2 2�;�m2Þ plane,consistent with that from the SK-I atomospheric neutrinoobservations, is obtained.
MACRO was a large-area multipurpose undergrounddetector located at the Gran Sasso Laboratory at an averageoverburden of 3700 m water equivalent. The active detectorelements were limited streamer tube planes for tracking andliquid scintillation counters. As a neutrino detector, MACROmeasured the angular distribution of the upward-going muonflux.41) The oscillation analysis of the MACRO data41,42)
supports �� $ �� oscillations with parameters �m2 ¼ 2:5�10�3 eV2 and sin2 2� � 1. This result is also consistent withthe SK-I allowed region.
4.2 Results from accelerator experimentsThe �m2 2� 10�3 eV2 region can be explored by
accelerator-based long baseline experiments with typicallyE � 1 GeV and L � several hundred km. With a fixedbaseline distance and narrower neutrino energy spectrum,the value of �m2, and also with higher statistics, the mixingangle, are potentially better constrained in acceleratorexperiments than from atmospheric neutrino observations.
4.2.1 K2KThe K2K (KEK-to-Kamioka) long baseline neutrino
oscillation experiment43) is the first accelerator-based experi-ment with a neutrino path length extending hundreds ofkilometers. A horn-focused wide-band muon neutrino beamhaving an average L=E� � 200 (L ¼ 250 km, hE�i � 1:3GeV), was produced by 12-GeV protons from the KEK-PSand directed to the Super-Kamiokande detector. K2K aimedat confirmation of the neutrino oscillation in �� disappear-ance in the �m2 2� 10�3 eV2 region with well under-stood flux and composition of the neutrino beam. For thispurpose, the energy spectrum and profile of the neutrinobeam were measured by a near neutrino detector systemlocated 300 m downstream from the production target.
The K2K experiment started stable data-taking in June1999. Super-Kamiokande events caused by accelerator-produced neutrinos were selected based on the timinginformation from the global positioning system. Data wereintermittently taken until November 2004. The total numberof protons on target for physics analysis amounted to0:92� 1020. The observed number of beam-originated
10-3
10-2
0.7 0.75 0.8 0.85 0.9 0.95 1
sin22θ
Δ m2 (
eV2 )
68% C.L.90% C.L.99% C.L.
Fig. 3. Allowed region for the �� $ �� oscillation parameters. Three
contours correspond to the 68% (dotted line), 90% (solid line), and 99%
(dashed line) CL allowed regions. This figure is taken from ref. 35.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1 10 102
103
104
L/E (km/GeV)
Dat
a/P
redi
ctio
n (n
ull o
sc.)
Fig. 4. Results of the L=E analysis of SK-I atmospheric neutrino data. The
points show the ratio of the data to Monte Carlo without neutrino
oscillation, as a function of the reconstructed L=E. The error bars are
statistical only. The solid line shows the best fit with 2-flavor �� $ ��oscillation. The dashed and dotted lines show the best fit expectations for
neutrino decay and neutrino decoherence hypotheses, respectively. This
figure is taken from ref. 39.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS K. NAKAMURA
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FC events in the 22.5 kton fiducial volume of Super-Kamiokande is 112. It is compared with an expectation of158:1þ9:2
�8:6 events without oscillation. The background com-ing from atmospheric neutrinos is negligible. There are 58events in the 1-ring �-like subset of the data. For theseevents, the neutrino energy can be reconstructed from themuon momentum and angle, assuming charged-currentquasielestic kinematics. Figure 5 shows the reconstructedE� distribution for these events. This distribution is com-pared with the expected spectrum at Super-Kamiokande forno neutrino oscillation. The measured energy spectrumshows distortion which is characteristic of neutrino oscil-lations.
A two-flavor neutrino oscillation analysis was performedusing a maximum-likelihood method. The oscillation pa-rameters ðsin2 2�;�m2Þ are estimated by maximizing theproduct of the likelihood for the observed number of FCevents and that for the shape of the reconstructed E�spectrum. The best fit point lies in the unphysical region,ðsin2 2�;�m2Þ = (1.2, 2:6� 10�3 eV2). The allowed �m2
region at sin2 2� ¼ 1:0 is between 1.9 and 3:5� 10�3 eV2 atthe 90% CL with the best-fit value of 2:8� 10�3 eV2. Thisregion is consistent with the allowed region from the SK-Iatmospheric neutrino observations. The E� distributioncalculated with the best-fit parameters in the physical regionis shown in Fig. 5. The observed number of events andenergy spectrum of neutrinos at Super-Kamiokande areconsistent with muon neutrino oscillations. The probabilitythat the observations are due to a statistical fluctuationinstead of neutrino oscillation is estimated by computing thelikelihood ratio of the no-oscillation case to the best fit pointin the physical region. It is 0.0015% or 4:3�.
4.2.2 MINOSMINOS is a long baseline neutrino oscillation experiment
with near and far detectors. Neutrinos are produced by using120 GeV protons from the Fermilab Main Injector. Thefar detector is a 5.4 kton (total mass) iron-scintillatortracking calorimeter with toroidal magnetic field, located
underground in the Soudan mine. The baseline distance is735 km. The near detector is also an iron-scintillator trackingcalorimeter with toroidal magnetic field, with a total mass of0.98 kton. The neutrino beam is a horn-focused wide-bandbeam. Its energy spectrum can be varied by moving thetarget position relative to the first horn and changing thehorn current.
MINOS started the neutrino-beam run in 2005. Initialresults were published44) using tha data taken between May2005 and February 2006. During this period, a ‘‘low-energy’’option was chosen for the spectrum of the neutrino beamso that the flux was maximized in the 1– 3 GeV energyrange. With 1:27� 1020 protons on the production target,215 events with reconstructed neutrino energy E� < 30 GeVwere observed in the far detector. This number is contrastedwith an expectation of 336� 14 events for no neutrinooscillations. Figure 6 compares the reconstructed E� spec-trum at the far detector with the predicted spectrum withand without oscillations. Figure 7 shows the 68 and 90%CL allowed regions, compared with the 90% CL allowed
Eνrec GeV
even
ts/0
.2G
eV
0
2
4
6
8
10
12
14
16
18
0 2 5431
Fig. 5. Reconstructed E� distribution for 1-ring �-like sample. The
solid histogram shows the best-fit spectrum. The dashed histogram
shows the expected spectrum without oscillation. These histograms are
normalized by the number of events observed (58). This figure is taken
from ref. 43.
ν
Fig. 6. Comparison of the far detector spectrum with predictions for no
oscillations for two different analysis methods (shown by the gray lines)
and for �� ! �� oscillations with the best-fit parameters in one of the
analysis methods (shown by the black lines). The last energy bin contains
events between 18– 30 GeV. This figure is taken from ref. 44.
1.5
2.0
2.5
3.0
3.5
4.0
-310×MINOS Best Fit
MINOS 90% C.L.
MINOS 68% C.L.
K2K 90% C.L.
SK 90% C.L.
SK (L/E) 90% C.L.
)23θ(22sin0.2 0.4 0.6 0.8 1.0
)4/c2
| (eV
322mΔ|
1.5
Fig. 7. Allowed region for the �� $ �� oscillation parameters from the
initial MINOS results. The 68 and 90% CL allowed regions are shown
together with the SK-I and K2K 90% CL allowed regions. This figure is
taken from ref. 44.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS K. NAKAMURA
111008-6
regions obtained from the SK-I zenith-angle dependence, theSK-I L=E analysis, and the K2K results. The best fit point isin the unphysical region, with parameters �m2 ¼ 2:72�10�3 eV2 and sin2 2� ¼ 1:01. The initial MINOS results areconsistent with the SK-I and K2K results.
5. Measurements of �m212 and �12
5.1 Solar neutrino observationsObservation of solar neutrinos directly addresses the theo-
ry of stellar structure and evolution, which is the basis of thestandard solar model (SSM). The Sun as a well-defined neu-trino source also provides extremely important opportunitiesto investigate nontrivial neutrino properties such as nonzeromass and mixing, because of the wide range of matter densityand the great distance from the Sun to the Earth.
The solar neutrinos are produced by some of the reactionsin the pp chain or CNO cycle. There have been efforts tocalculate solar neutrino fluxes from these reactions on thebasis of SSM. A variety of input information is needed in theevolutionary calculations. The most elaborate SSM calcu-lations have been developped by Bahcall and his collabo-rators, who define their SSM as the solar model which isconstructed with the best available physics and input data.The currently preferred SSM is BS05(OP) developed byBahcall and Serenelli.45,46) Its prediction for the fluxes fromneutrino-producing reactions is given in Table I. The solar-neutrino spectra calculated with this model45) is shown inFig. 8.
So far, solar neutrinos have been observed by chlorine(Homestake) and gallium (SAGE, GALLEX, and GNO)radiochemical detectors and water Cherenkov detectorsusing light water (Kamiokande and Super-Kamiokande)and heavy water (SNO).
A pioneering solar neutrino experiment by Davis andcollaborators at Homestake using 37Cl started in the late1960’s. This experiment exploits electron neutrino absorp-tion on chlorine nuclei followed by their decay throughorbital electron capture,
37Clþ �e! 37Arþ e� (threshold 814 keV): ð19Þ
The 37Ar atoms produced are radioactive, with a half life(�1=2) of 34.8 days. After an exposure of the detector for twoto three times �1=2, the reaction products are chemicallyextracted and introduced into a low-background proportionalcounter, where they are counted for a sufficiently longperiod to determine the exponentially decaying signal and a
constant background. Solar-model calculations predict thatthe dominant contribution in the chlorine experiment comesfrom 8B neutrinos, but 7Be, pep, 13N, and 15O neutrinos alsocontribute (for notations, refer to Table I).
From the very beginning of the solar-neutrino observa-tion,23) it was recognized that the observed flux was sig-nificantly smaller than the SSM prediction, provided nothinghappens to the electron neutrinos after they are created in thesolar interior. This deficit has been called ‘‘the solar-neutrinoproblem’’.
Gallium experiments (GALLEX and GNO at Gran Sassoin Italy and SAGE at Baksan in Russia) utilize the reaction
71Gaþ �e! 71Geþ e� (threshold 233 keV): ð20Þ
They are sensitive to the most abundant pp solar neutrinos.However, the solar-model calculations predict almost halfof the capture rate in gallium is due to other solar neutrinos.GALLEX presented the first evidence of pp solar-neutrinoobservation in 1992.47) The GALLEX Collaboration for-mally finished observations in early 1997. Since April, 1998,a newly defined collaboration, GNO (Gallium NeutrinoObservatory) continued the observations until April 2003.The complete GNO results are published in ref. 48. TheGNO + GALLEX joint analysis results are also present-ed.48) SAGE initially reported very low flux,49) but laterobserved similar flux to that of GALLEX. The completeSAGE results are published in ref. 50.
In 1987, the Kamiokande experiment succeeded in real-time solar neutrino observation, utilizing �e scattering
�x þ e� ! �x þ e� ð21Þ
in a large water-Cherenkov detector. These experiment takeadvantage of the directional correlation between the incom-ing neutrino and the recoil electron. This feature greatlyhelps the clear separation of the solar-neutrino signal fromthe background. The Kamiokande result gave the first directevidence that the Sun emits neutrinos.51) Later, the high-statistics Super-Kamiokande experiment52) took over theKamiokande experiment. Due to the high thresholds (7 MeV
Table I. Neutrino-producing reactions in the Sun (first column) and their
abbreviations (second column). The neutrino fluxes predicted by the
BS05(OP) model45Þ are listed in the third column.
Reaction Abbr. Flux (cm�2 s�1)
pp! d eþ � pp 5.99 ð1:00� 0:01Þ � 1010
pe�p! d � pep 1.42 ð1:00� 0:02Þ � 108
3He p! 4He eþ � hep 7.93 ð1:00� 0:16Þ � 103
7Be e� ! 7Li �þ ðÞ 7Be 4.84 ð1:00� 0:11Þ � 109
8B! 8Be�eþ � 8B 5.69 ð1:00� 0:16Þ � 106
13N! 13C eþ � 13N 3.07 ð1:00þ0:31�0:28Þ � 108
15O! 15N eþ � 15O 2.33 ð1:00þ0:33�0:29Þ � 108
17F! 17O eþ � 17F 5.84 ð1:00� 0:52Þ � 106
Fig. 8. The solar neutrino spectrum predicted by the BS05(OP) standard
solar model.45) The neutrino fluxes are given in units of cm�2 s�1 MeV�1
for continuous spectra and cm�2 s�1 for line spectra. The numbers
associated with the neutrino sources show theoretical errors of the fluxes.
This figure is taken from the late John Bahcall’s web site, http://
www.sns.ias.edu/jnb/.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS K. NAKAMURA
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in Kamiokande and 5 MeV at present in Super-Kamiokande)the experiments observe pure 8B solar neutrinos becausehep neutrinos contribute negligibly according to the SSM.It should be noted that the reaction (21) is sensitive to allactive neutrinos, x ¼ e, �, and �. However, the sensitivityto �� and �� is much smaller than the sensitivity to �e,�ð��;�eÞ 0:16 �ð�eeÞ.
In 1999, a new real time solar-neutrino experiment,SNO, in Canada started observation. This experimentuses 1000 tons of ultra-pure heavy water (D2O) containedin a spherical acrylic vessel, surrounded by an ultra-pureH2O shield. SNO measures 8B solar neutrinos via thereactions
�e þ d! e� þ pþ p ð22Þ
and
�x þ d! �x þ pþ n; ð23Þ
as well as �e scattering, (21). The charged-current (CC)reaction, (22), is sensitive only to electron neutrinos,while the NC reaction, (23), is sensitive to all activeneutrinos.
The Q-value of the CC reaction is �1:4 MeV and theelectron energy is strongly correlated with the neutrinoenergy. Thus, the CC reaction provides an accurate measureof the shape of the 8B solar-neutrino spectrum. The con-tributions from the CC reaction and �e scattering can bedistinguished by using different cos �� distributions where�� is the angle of the electron momentum with respect to thedirection from the Sun to the Earth. While the �e scatteringevents have a strong forward peak, CC events have anapproximate angular distribution of 1� 1=3 cos ��.
The threshold of the NC reaction is 2.2 MeV. In the pureD2O, the signal of the NC reaction is neutron capture indeuterium, producing a 6.25-MeV -ray. In this case, thecapture efficiency is low and the deposited energy is close tothe detection threshold of 5 MeV. In order to enhance boththe capture efficiency and the total -ray energy (8.6 MeV),2 tons of NaCl were added to the heavy water in the secondphase of the experiment. In addition, discrete 3He neutroncounters were installed and the NC measurement with themis made as the third phase of the SNO experiment.
Figure 9 compares the predictions of the BP05(OP) SSMwith the results of solar neutrino experiments. It is clearlyseen from Fig. 9 that the results from all the solar-neutrinoexperiments, except the SNO’s NC result, indicate signifi-cantly less flux than expected from the solar-modelpredictions.45)
5.2 Evidence for solar neutrino flavor conversionThe solar-neutrino problem had remained unsolved for
more than 30 years. However, there have been remarkabledevelopments in the past six years and now the solar-neutrino problem has been finally solved. In 2001, the initialSNO CC result combined with the Super-Kamiokande’shigh-statistics �e elastic scattering (ES) result53) provideddirect evidence for flavor conversion of solar neutrinos.54)
Later, SNO’s NC measurements further strengthened thisconclusion.55,56) From the recent salt phase measurement,56)
the fluxes measured with CC, ES, and NC events wereobtained as
where the first errors are statistical and the second errorsare systematic. Equation (26) is a mixing-independent resultand therefore tests solar models. It shows good agreementwith the 8B solar-neutrino flux predicted by the solarmodel.45) Figure 10 shows the salt phase result of ð�� or �Þversus the flux of electron neutrinos ð�eÞ with the 68,95, and 99% joint probability contours. The flux of non-�eactive neutrinos, ð�� or �Þ, can be deduced from theseresults. It is
The non-zero ð�� or �Þ is strong evidence for neutrino flavorconversion. A natural and most probable explanation ofneutrino flavor conversion is neutrino oscillation. At thisstage, the LMA (large mixing angle) solution of solarneutrino oscillation in matter14) with �m2 � 5� 10�5 eV2
and a large mixing angle was the most promising. However,with the SNO data alone, other solutions cannot be excludedeven at the 68% CL.
5.3 KamLAND experimentThe solar neutrino problem has been finally solved by the
KamLAND is a 1-kton ultra-pure liquid scintillatordetector located at the old Kamiokande’s site in Japan.Although the ultimate goal of KamLAND is observation of7Be solar neutrinos with much lower energy threshold, theinitial phase of the experiment is a long baseline (flux-weighted average distance of �180 km) neutrino oscillationexperiment using ���e’s emitted from power reactors. Thereaction ���e þ p! eþ þ n is used to detect reactor ���e’sand delayed coincidence with 2.2 MeV -ray from neutroncapture on a proton is used to reduce the backgrounds.
SAGE GALLEX+
GNOSuperK
Kamiokande
SNO SNO
Fig. 9. The predictions of the BP2005 standard solar model45) with the
results of solar neutrino experiments. This figure is taken from the late
John Bahcall’s web site, http://www.sns.ias.edu/jnb/. Note that some of
the experimental data need updating, but the general features remain
unchanged.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS K. NAKAMURA
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With the reactor ���e’s energy spectrum (< 8 MeV) and aprompt-energy analysis threshold of 2.6 MeV, this experi-ment has a sensitive �m2 range down to �10�5 eV2.Therefore, if the LMA solution is the real solution of thesolar neutrino problem, KamLAND should observe reactor���e disappearance, assuming CPT invariance.
The first KamLAND results58) with 162 ton�yr exposurewere reported in December 2002. The ratio of observedto expected (assuming no neutrino oscillation) number ofevents was
Nobs � NBG
NNoOsc
¼ 0:611� 0:085� 0:041 ð28Þ
with obvious notation. This result showed clear evidence ofevent deficit expected from neutrino oscillation.
In June 2004, KamLAND released the results from 766ton�yr exposure.59) In addition to the deficit of events, theobserved positron spectrum showed the distortion expectedfrom neutrino oscillation as can be seen in Fig. 11. Here,the ratio of the observed ���e spectrum to the expectationwithout oscillation is plotted as a function of L0=E withL0 ¼ 180 km. It is a sort of average distance of nuclearreactors contributing to the ���e flux detected in KamLAND,determined as if all anti-neutrinos detected in KamLANDwere due to a single reactor at this distance. Figure 12 showsthe allowed regions in the neutrino-oscillation parameterspace. The best-fit point lies in the region called LMA I.
5.4 Global neutrino oscillation analysisThe SNO Collaboration updated56) a global two-neutrino
oscillation analysis of the solar-neutrino data includingthe SNO’s complete salt phase data, and global solar +KamLAND 766 ton�yr data.59) The best fit parameters for theglaobal solar analysis are �m2 ¼ 6:5þ4:4
�2:3 � 10�5 eV2 andtan2 � ¼ 0:45þ0:09
�0:08. The inclusion of the KamLAND datasignificantly constrains the allowed �m2 region, but shiftsthe best-fit �m2 value. The best-fit parameters for the globalsolar + KamLAND analysis are �m2 ¼ 8:0þ0:6
�0:4 � 10�5 eV2
and tan2 � ¼ 0:45þ0:09�0:07 or � ¼ ð33:9þ2:4
�2:2Þ�.
6. Future Prospects
Some of the important problems left for the futureneutrino experiments are: (1) Absolute neutrino mass; (2)Determining whether neutrinos are Majorana particles orDirac particles; (3) Measurement of the small mixing angle�13 and the CP violating phase � in the MNS matrix; (4)Measurement of the sign of �m2
32.Measurement of absolute neutrino mass will be chal-
lenged by a new tritium beta-decay experiment KATRIN61)
with a sensitivity down to �0:2 eV. As noted in §1, givingan answer to the question of whether neutrinos are Majoranaor Dirac particles is of fundamental importance. Detection ofneutrinoless double beta decay is the only realistic means toshow the Majorana nature of neutrinos. There are a numberof proposed neutrinoless double beta decay experimentsaiming at reaching sensitivities to the effective Majoranamass of 10 – 50 meV. For these experiments, readers arereferred to ref. 22, and references therein.
20 30 40 50 60 70 800
0.2
0.4
0.6
0.8
1
1.2
1.4
(km/MeV)eν/E0L
Rat
io
2.6 MeV promptanalysis threshold
KamLAND databest-fit oscillationbest-fit decay
best-fit decoherence
Fig. 11. Ratio of the observed ���e spectrum to the expectation without
oscillation is plotted as a function of L0=E, where L0 ¼ 180 km. The
curves show the best-fit expectations for oscillation, neutrino decay,36)
and neutrino decoherence38) models. This figure is taken from ref. 59.
Fig. 12. Allowed regions of neutrino-oscillation parameters from the
KamLAND 766 ton�yr exposure ���e data.59) The LMA region from solar-
neutrino experiments60) is also shown. This figure is taken from ref. 59.
)-1 s-2 cm6 10× (eφ0 0.5 1 1.5 2 2.5 3 3.5
)-1
s-2
cm
6 1
0×
(μτφ
0
1
2
3
4
5
6
68% C.L.CCSNOφ
68% C.L.NCSNOφ
68% C.L.ESSNOφ
68% C.L.ESSKφ
68% C.L.SSMBS05φ
68%, 95%, 99% C.L.μτNCφ
Fig. 10. Fluxes of 8B solar neutrinos, ð�eÞ, and ð�� or �Þ, deduced from
the SNO’s charged-current (CC), �e elastic scattering (ES), and neutral-
current (NC) results for the salt phase measurement.56) The Super-
Kamiokande ES flux is from ref. 57. The BS05(OP) standard solar model
prediction45) is also shown. The bands represent the 1� error. The
contours show the 68, 95, and 99% joint probability for ð�eÞ and
ð�� or �Þ. This figure is taken from ref. 56.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS K. NAKAMURA
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Items (3) and (4) are the main goals of the future neutrinooscillation experiments. If �13 is not much less than theCHOOZ limit, sin2 2�13 < 0:19, the ongoing experimentsMINOS and OPERA62) will have a chance to establish non-zero �13. A new experiment under construction in Japan,T2K63) (Phase-I), will have a better sensitivity, sin2 2�13 <0:008 (for the case of � ¼ 0 or ; see the discussion below).T2K will use Super-Kamiokande as a far detector. Anexperiment proposed in US, NO�A,64) will have a similarsensitivity. Both T2K and NO�A plan to use an ‘‘off-axis’’beam. In this scheme, the axis of the beamline componentsis directed a few degrees away from the direction of thefar detector. With this trick, a high-intensity, low-energy,narrow-band neutrino beam can be obtained. The peak ofthe neutrino energy spectrum can be adjusted by varying theoff-axis angle. It will be adjusted close to the oscillationmaximum in order to maximize the sensitivity to theoscillation measurement.
Short baseline reactor neutrino oscillation experimentsalso aim at the measurement of �13. Double-CHOOZ65)
experiment in France will have a sensitivity of sin2 2�13 <0:06 with the far detector only, and < 0:03 when the neardetector is completed. Daya Bay66) experiment in Chinaaims at a sensitivity of sin2 2�13 < 0:01.
In the accelerator neutrino oscillation experiments withconventional neutrino beams, �13 is measured using �� ! �eappearance. With the same approximation used for eq. (11),the dominant term in the probability of ��! �e appearanceis
However, by examining the exact expression for theoscillation probability, it is understood that some of theneglected terms have rather large effects and the unknownCP-violating phase � causes uncertainties in determining thevalue of �13. Actually, from the measurement of �� ! �eappearance, �13 is given as a function of � for a given sign of�m2
32. Therefore, a single experiment with a neutrino beamcannot determine the value of �13 though it is possibleto establish non-zero �13. On the other hand, since thedisappearance probability does not depend on the CP-violating phase, reactor ���e disappearance experiments allowto measure �13 with less ambiguities in contrast to accel-erator ��! �e appearance experiments.67)
In order to determine the value of �13 in the acceleratorneutrino oscillation experiments, simultaneous measurementof the CP-violating phase � is necessary. This will beachieved by measuring the appearance probabilities of ��!�e and ���� ! ���e. To measure CP violation in this way as wellas to determine whether the mass hierarchy is normal orinverted one, a very high intensity neutrino beam and a Mtonclass detector will be required. A ‘‘super-beam’’ means avery high intensity, conventional neutrino beam line (�� or���� beam obtained from decays of horn-focused secondarypions). According to the T2K Phase II (T2K-II) plan, thebeam power should be increased to 4 MW and a 1 Mtonwater Cherenkov detector called Hyper-Kamiokande shouldbe constructed. T2K-II is an example of a super-beamexperiment.
It should be noted there are famous degeneracies ofparameters which give the same appearance probabilities.
There are three types of degeneracies. One is a two-folddegeneracy of (�13, �).68) Another two-fold degeneracy iscaused by the sign of �m2
32.69) If �23 6¼ =4, yet another two-fold degeneracy arises by the interchange of �23 $ =2��23.70) For a single experiment with a given neutrino beamspectrum and a given detector at a fixed baseline distance,it is impossible to resolve these up to eight parameterdegeneracies. Multiple measurements, or synergy of differ-ent experiments, are needed to resolve the degeneracies. Anidea of the T2KK experiment is to use the same off-axisbeam with two 0.5 Mton class water Cherenkov detectorsat different baseline distances. One detector will be locatedat Kamioka with L � 300 km. The other detector will belocated in Korea with L � 1000 km. Since the T2KKexperiment will provide measurements with two differentbaseline distances, it is expected to have a capability ofresolving some of the parameter degeneracies. In fact, ithas been shown that T2KK can resolve mass hierarchyif sin2 2�13 � 0:01.71) For this measurement, the baselinedistance as long as 1000 km is essential because mattereffects are needed. (Note that the T2K experiment with abaseline distance of � 300 km is not sensitive to masshierarchy.) Furthermore, T2KK has high sensitivity to CP-violation measurement and also to resolving �23 $ =2��23 degeneracy72) if sin2 2�13 is within the reach of the Phase-I T2K experiment.
If sin2 2�13 � 0:01, its measurement would be beyond thereach of the on-going projects (Phase-I T2K experiment andreactor experiments, etc.) Even super-beam experimentshave a sensitivity down to sin2 2�13 � a few �10�3. Toenhance the discovery reach, advanced concepts of produc-ing neutrino beams, ‘‘neutrino factory’’ and ‘‘�-beam’’facility have been proposed.
A neutrino factory73) is an intense neutrino source basedon a 20 – 50 GeV muon storage ring. If 1020 { 21 muons peryear decay in the straight section of the ring, a very intenseand highly collimated neutrino beam with well understoodproperties will be produced. The main modes of neutrinooscillations investigated with beams from the neutrinofactory are �e! �� and ���e! ����. If positive muons areaccumulated, the poduced neutrino beam consists of ���� and�e components. In the detector, the CC interactions of thebeam ����’s produce positive muons. On the other hand,negative muons are produced by the CC interactions of ��’sappeared in the beam due to �e! �� oscillations. Thus, thesignal of neutrino oscillations is wrong-sign muons and veryclean measurement is possible.
The concept of a �-beam was proposed by Zuccelli.74)
In this scheme, ���e or �e beams can be produced from thebeta decay of boosted ions; 6
2He! 63Li e� ���e or 18
10Ne!189 Fe eþ�e. High-intensity radioactive ions can be produced,for example, at CERN ISOLDE.75) After acceleration, theyare stored in a storage ring having a long straight section.The resulting neutrino beam is strongly collimated, has well-known energy spectrum and intensity, and contains a purelysingle neutrino flavor. For a recent review, see ref. 76.
Experiments using the neutrino factory beam and the �-beam will have remarkably good physics potential.77) Thesensitivities to the sin2 2�13 measurement reach �10�4. Forthe CP-violation and mass hierarchy measurements, theyhave sensitivities down to sin2 2�13 � a few �10�4.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS K. NAKAMURA
111008-10
7. Conclusions
In the last 10 years, our understanding of neutrino massand mixing has advanced remarkably. Neutrino oscillationshave been firmly established. Finite neutrino mass is thefirst evidence of physics beyond the Standard Model. Theremarkably large mixing observed in the neutrino sector isstrikingly different from the mixing in the quark sector.
In order to gain complete knowledge of the MNS matrix,future neutrino oscillation experiments should measure thelast mixing angle �13 and the CP-violating phase �, anddetermine the sign of �m2
32. Measurement of the absoluteneutrino mass scale and determining whether neutrinos areMajorana particles or Dirac particles are also very importantgoals of future neutrino experiments.
1) Y. Fukuda et al.: Phys. Rev. Lett. 81 (1998) 1562.
2) D. N. Spergel et al.: Astrophys. J. Suppl. Ser. 170 (2007) 377.
3) K. Ichikawa, M. Fukugita, and M. Kawasaki: Phys. Rev. D 71 (2005)
043001.
4) M. Fukugita et al.: Phys. Rev. D 74 (2006) 027302.
5) D. N. Spergel et al.: Astrophys. J. Suppl. Ser. 148 (2003) 175.
6) For a recent summary, see Particle Data Group, W.-M. Yao et al.:
J. Phys. G 33 (2006).
7) P. Minkowski: Phys. Lett. B 67 (1977) 421.
8) M. Gell-Mann, P. Ramond, and R. Slansky: in Supergravity, ed. F. van
Nieuwenhuizen and D. Z. Freedman (North-Holland, Amsterdam,
1979) p. 315.
9) T. Yanagida: in Proc. Workshop on the Unified Theory and the Baryon
Number in the Universe, KEK, 1979, ed. O. Sawada and A. Sugamoto,
1979, p. 95.
10) M. Fukugita and T. Yanagida: Phys. Lett. B 174 (1986) 45.
11) The LEP Collaborations, ALEPH, DELPHI, L3, and OPAL: Phys.
Lett. B 276 (1992) 247.
12) Z. Maki, M. Nakagawa, and S. Sakata: Prog. Theor. Phys. 28 (1962)
Department of Physics, Boston University, Boston, MA 02215, U.S.A.
(Received April 8, 2007; accepted April 18, 2007; published November 12, 2007)
Since its discovery in the 1930s, the muon has played an important role in our quest to understand thesub-atomic theory of matter. The muon was the first second-generation standard-model particle to bediscovered, and its decay has provided information on the (Vector–Axial Vector) structure of the weakinteraction, the strength of the weak interaction, GF , and the conservation of lepton number (flavor) inmuon decay. The muon’s anomalous magnetic moment has played an important role in restrictingtheories of physics beyond the standard standard model, where at present there is a 3:4� differencebetween the experiment and standard-model theory. Its capture on the atomic nucleus has providedvaluable information on the modification of the weak current by the strong interaction which iscomplementary to that obtained from nuclear � decay.
The muon was first observed in a Wilson cloud chamberby Kunze1) in 1933, where it was reported to be ‘‘a particleof uncertain nature’’. In 1936 Anderson and Neddermeyer2)
reported the presence of ‘‘particles less massive than protonsbut more penetrating than electrons’’ in cosmic rays, whichwas confirmed in 1937 by Street and Stevenson,3) Nishina,Tekeuchi, and Ichimiya,4) and by Crussard and Leprince-Ringuet.5) The Yukawa theory of the nuclear force hadpredicted such a particle, but this ‘‘mesotron’’ as it wascalled, interacted too weakly with matter to be the carrierof the strong force. Today we understand that the muon isa second generation lepton, with a mass about 207 timesthe electron’s. Like the electron, the muon obeys quantumelectrodynamics (QED), and can interact with other particlesthrough the electromagnetic and weak forces. Unlike theelectron which appears to be stable, the muon decaysthrough the weak force.
The muon lifetime of 2.2 ms permits one to make precisionmeasurements of its properties, and to use it as a tool tostudy the semileptonic weak interaction, nuclear properties,as well as magnetic properties of condensed matter systems.The high precision to which the muonium (�þe� atom)hyperfine structure can be measured and calculated makesit a significant input parameter in the determination offundamental constants.6) In this review, I will focus on therole of the muon in particle physics.
A beam of negative muons can be brought to rest inmatter, where hydrogen-like atoms are formed, with anuclear charge of Z. The Bohr radius for a hydrogen-likeatom is inversely proportional to the orbiting particle’s mass(rn ¼ ½n2h�c�=½mc2Z��), so that for the lowest quantumnumbers of high-Z muonic atoms, the muon is well insideof the atomic electron cloud, with the Bohr radius of the 1S
atomic state well inside the nucleus. The 2P! 1S x-rayenergies are shifted because of the modification of theCoulomb potential inside the nucleus, and these x rays have
provided information on nuclear root-mean-square chargeradii. The Lamb shift in muonic hydrogen, �E2P�2S, whichis being measured at at the Paul Scherrer Insitut (PSI), isgiven by7) f209:974ð6Þ � 5:226R2
p þ 0:036R3pgmeV, where
Rp is the proton rms charge radius. This experiment shouldprovide a precise measurement of Rp. The weak nuclearcapture, called ordinary muon capture (OMC), of the muonon the atomic nucleus following the cascade to the 1S
ground state, �� þ ZN ! Z�1N þ ��, is the analog tothe weak capture of a K-shell electron by the nucleus, andprovides information on the modification of the weakinteraction by hadronic matter.
The muon mass of �106 MeV restricts the muon to decayinto the electron, neutrinos, and photons. Thus muon decayis a purely leptonic process, and the dominant decay mode is�� ! e� þ �� þ ���e. This three-body decay tells us that theindividual lepton number, electron and muon, is conservedseparately, and that the two flavors (kinds) of neutrinos aredistinct particles.10) Here the �� and e� are ‘‘particles’’ andthe �þ and eþ are the antiparticles. In the 1950s, it becamepossible to make pions, and thus muons, in the laboratory.The energetically favorable decay �þ ! e� was searchedfor and not found8) to a relative branching ratio of<2� 10�5. Also searched for was the neutrinoless captureof a �� on an atomic nucleus,9) �� þN ! e� þN , whichwas not found at the level of �5� 10�4. Such processes aresaid to ‘‘violate lepton flavor’’, and continue to be the objectof present and planned studies reaching to sensitivities of10�14 and 10�16, respectively.
The muon, like the electron, is a spin 1=2 lepton, with amagnetic moment given by
�s ¼ gsq
2m
� �s; � ¼ ð1þ aÞ
qh�
2m; a �
gs � 2
2; ð1Þ
where the muon charge q ¼ �e, and gs, the spin g-factor isslightly greater than the Dirac value of 2. The middleequation above is useful from a theoretical point of view, asit separates the magnetic moment into two pieces: the Diracmoment which is unity in units of the appropriate magneton,eh�=2m, and is predicted by the Dirac equation; and the
anomalous (Pauli) moment, where the dimensionless quan-tity a is referred to as the anomaly. The muon anomaly, likethe electron’s, arises from radiative corrections that arediscussed below.
When the muon was discovered, it was an unexpectedsurprise. Looking at this from our 21st century perspective,it is easy to forget how we reached what is now called the‘‘standard model’’ of subatomic physics, which incorporatesthree generations of leptons, e, �, and � and their neutrinos;three generations of quarks; the electro-weak gauge bosons,�, W , and Z; and the gluons that carry the strong force.When this author joined the field as a graduate student in themid 1960s, none of this was clear. Quarks were viewed bymany as a mathematical device, not as constituent particles.Even after quarks were inferred from deep inelastic electronscattering off the proton, we only knew of the existenceof three of them. While the V–A structure of the weakinteraction was first inferred from nuclear � decay, the studyof muon decay has provided a useful laboratory in which tostudy the purely leptonic weak interaction, to search forphysics beyond the standard model, such as additional termsin the interaction besides the standard-model V–A structure,as well as looking for standard model forbidden decays like�! e�. For many years, the experimental value of themuon’s anomalous magnetic moment has served to constrainphysics beyond the standard model, and continues thatrole today.
2. Muon Decay and GF
The muon decay �� ! e��� ���e is purely leptonic. Sincem� MW , muon decay can be described by a local four-fermion (contact) interaction. While nonrenormalizeable,at low energies it provides an excellent approximation tothe full electroweak theory. The weak Lagrangian is writtenas a current–current interaction, where the leptonic current isof the (V–A) form, �uu� ð1� �5Þu.
Michel11) first wrote down a parameterization of muondecay, defining five parameters, , �, �, , and h, which arecombinations of the different possible couplings allowed byLorentz invariance in muon decay. The standard model hasclear predictions for these parameters and they have beenmeasured repeatedly over the intervening years to search forphysics beyond the standard model. This tradition continuestoday, with the TWIST experiment at TRIUMF, which ismid-way through a program to improve on the precision ofthe Michel parameters by an order of magnitude.12) Whilethere are some scenarios in which new physics wouldconspire to leave the Michel parameters at their standardmodel value,13) a variance from the standard-model valueswould be a clear sign of new physics at work.
The muon lifetime, see Fig. 1 is directly related to thestrength of the weak interaction, which in Fermi theory isdescribed by the constant GF . The standard-model electro-weak gauge coupling g is related to GF by14)
GFffiffiffi2p ¼
g2
8M2W
ð1þ�rÞ; ð2Þ
where �r represents the weak boson mediated tree-levelprocess and its radiative corrections.15) In the standardmodel, the Fermi constant is related to the vacuum expecta-tion value of the Higgs field by GF ¼ 1=ð
ffiffiffi2p
v2Þ.
While the Fermi theory is nonrenormalizeable, the QEDradiative corrections are finite to first order in GF , and toall orders in the fine-structure constant �. This gives therelationship14) between GF and the muon lifetime, ��,
1
��¼
G2Fm�
5
192�3ð1þ�qÞ; ð3Þ
where �q is the sum of phase space, and QED and hadronicradiative corrections. More properly one should write G�since new physics contributions could make G different forthe three leptons.16)
The MuLan experiment at PSI has recently reported anew measurement of the muon lifetime 2.197 013(21)(11) ms(�11 ppm),17) to be compared with the previous worldaverage 2.197 03(4) ms (19 ppm).18) The new worldaverage muon lifetime of 2.197 019(21) ms gives, assumingonly standard-model physics in muon decay, GF ¼1:166 371ð6Þ � 10�5 GeV2 (5 ppm). This experiment shouldeventually reach a precision of 1 ppm on ��.
3. Nuclear Muon Capture
The weak capture of a muon on a proton has much incommon with nuclear � decay. As for other low-energyweak processes, the interaction can be described as acurrent–current interaction with the (V–A) leptonic currentgiven by �uu���
ð1� �5Þu�. Because the strong interactioncan induce additional couplings,19) the hadronic current ismore complicated. The most general form of the vectorcurrent allowed by Lorentz invariance is20)
�uunðp0Þ gV ðq2Þ� þ igMðq2Þ2mN
��q� þgSðq2Þm�
q� �
upðpÞ: ð4Þ
The corresponding form of the axial-vector current is
�uunðp0Þ��gAðq2Þ��5
�gPðq2Þm�
�5q � i
gT ðq2Þ2mN
��q��5
�upðpÞ;
ð5Þ
where m� and mN are the muon and nucleon massesrespectively; the gðq2Þ are the induced form factors: vector,weak magnetism, scalar, axial-vector, pseudoscalar andtensor. The scalar and tensor terms are called ‘‘second classcurrents’’ because of their transformation properties underG-parity, and in the standard model are expected to be quitesmall.19) It is traditional to set these second-class currentsequal to zero.
Nuclear � decay is sensitive to the vector, axial-vectorand weak-magnetism form factors, but in muon capture the
e
e
μ+ W
νμ
(a)
ν
+
GF ν
e
μ+
+
νμ
(b)
e e
FG
μ+
νμ
ν
e(c)
γ+
Fig. 1. (Color online) Muon decay in (a) the standard model; (b) the
Fermi theory; (c) radiative corrections to the Fermi theory.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS B. L. ROBERTS
111009-2
capture rate has a measurable contribution from the inducedpseudoscalar interaction, the least well known of the weaknucleon form factors. Radiative muon capture (RMC),�� þ p! nþ � þ ��, should in principle be more sensi-tive to the induced pseudoscalar coupling than OMC, sincewith the three-body final state, q2 can get closer to the pionpole than is possible in ordinary muon capture, which waspointed out many years ago.21,22) The interested reader isreferred to the review by Gorringe and Fearing for furtherdiscussion.19)
In the past, current algebra and the Goldberger–Triemanrelation expressed gP in terms of gA. With the developmentof quantum chromodynamics (QCD), and chiral perturbationtheory, new interest has developed in the value of gP.19) Thepresently accepted theory value is gPðq2 ¼ �0:88m2
�Þ ¼8:26� 0:23.19)
The experimental history is rather interesting. For manyyears the radiative capture reaction �� þ p! nþ � þ ��was thought to be the ‘‘golden’’ channel to study. However,this experiment is extremely difficult, and it was firstobserved experimentally in the 1990s.23,24) To achieve anadequate muon stopping rate, it was necessary to stop themuons a liquid hydrogen target. The value obtained for gp bythis experiment was in disagreement with both the ordinarymuon capture experiment in liquid hydrogen, and with thetheoretical expectation.
The muon chemistry in hydrogen, especially liquid, israther complicated, and may be the source of thesediscrepancies. The incident �� first forms an atom with aproton in the hydrogen target, but then quickly picks up asecond proton to form a p�p molecule with the protons inthe ortho (J ¼ 1) state. The ortho state of this molecule candecay to the J ¼ 0 (para) state. The ortho and para protonstates have different admixtures of the muon–proton spin:ortho has 3/4 singlet and 1/4 triplet �� p and para has 1/4singlet and 3/4 triplet. Since the muon capture rate is 40times greater from the singlet ��p state than from the triplet,it is essential that �OP, the ortho to para transition rate inthe p�p molecule, is known in order to extract gp fromthe measured capture rate in liquid hydrogen.19) Even aftera recent measurement25) of the transition rate �OP, thedifficulty in accommodating previous results on ordinaryand radiative muon capture results in hydrogen continues.The complications of muon chemistry in liquid hydrogencan be avoided by using a 10 bar ultra-pure hydrogen target,which has a density 1.16% that of liquid hydrogen. Atthis lower density, the sensitivity to �OP is greatly reduced.It is this approach that the recent MuCap experiment atPSI has used.
The MuCap experiment stops �� in a gaseous hydrogentarget that functions as a time projection chamber (TPC),making it possible to determine where the muon stops in thetarget. A comparison of the �� lifetime in this protoniumtarget to the the free �þ lifetime, gives the capture rate anddetermines gp. The MuCap experiment has recently reporteda first result,26) gpðq2 ¼ �0:88m2
�Þ ¼ 7:3� 1:1, consistentwith the expectation from chiral perturbation theory. Theyhave a factor of four more data which are being analyzed.While it is not clear what is wrong with the previousordinary muon capture and radiative capture experiments,the MuCap result seems to indicate that a modern experi-
ment, with a gaseous target and information from the TPC,has settled the long-standing discrepancy.
4. The Magnetic and Electric Dipole Moments
The electric and magnetic dipole moments have been anintegral part of relativistic electron (lepton) theory sinceDirac’s famous 1928 paper, where he pointed out that anelectron in external electric and magnetic fields has ‘‘the twoextra terms
eh�
cð�;HÞ þ i
eh�
c1ð�;EÞ; ð6Þ
. . . when divided by the factor 2m can be regarded as theadditional potential energy of the electron due to its newdegree of freedom27)’’. These terms represent the magneticdipole (Dirac) moment and electric dipole moment inter-actions with the external magnetic and electric fields.
In modern notation, the magnetic dipole moment (MDM)interaction becomes
�uu� eF1ðq2Þ�� þie
2m�F2ðq2Þ�� q
� �u�; ð7Þ
where F1ð0Þ ¼ 1 and F2ð0Þ ¼ a�. The electric dipolemoment (EDM) interaction is
�uu�ie
2m�F2ðq2Þ � F3ðq2Þ�5
� ��� q
�u�; ð8Þ
where F2ð0Þ ¼ a�, F3ð0Þ ¼ d�, with
d� ¼�
2
� �eh�
2mc
� �’ �� 4:7� 10�14 e cm: ð9Þ
(This �, which is the EDM analogy to g for the MDM,should not be confused with the Michel parameter �.)
The existence of an EDM implies that both P and T areviolated.28,29) This can be seen by considering the non-relativistic Hamiltonian for a spin one-half particle in thepresence of both an electric and magnetic field:H ¼ �� B� d E. The transformation properties of E,B, �, and d are given in Table I, and we see that while � Bis even under all three, d E is odd under both P and T .While parity violation has been observed in many weakprocesses, direct T violation has only been observed in theneutral kaon system.30) In the context of CPT symmetry, anEDM implies CP violation, which is allowed by the standardmodel for decays in the neutral kaon and B-meson sectors.
Observation of a non-zero electron or muon EDM wouldbe a clear signal for new physics. To date no permanentEDM has been observed for the electron, the neutron, or anatomic nucleus, with the experimental limits given inTable II. It is interesting to note that in his original paper27)
Dirac stated ‘‘The electric moment, being a pure imaginary,we should not expect to appear in the model. It is doubtful
Table I. Transformation properties of the magnetic and electric fields
and dipole moments.
E B � or d
P � + +
C � � �T + � �
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS B. L. ROBERTS
111009-3
whether the electric moment has any physical meaning,since the Hamiltonian . . . that we started from is real, and theimaginary part only appeared when we multiplied it up in anartificial way in order to make it resemble the Hamiltonianof previous theories’’. Even in the 4th edition of his quantummechanics book from 1958, well after the suggestion ofPurcell and Ramsey31) that one should search for apermanent EDM, Dirac held fast to this point of view.
While CP violation is widely invoked to explain thebaryon–antibaryon asymmetry of the universe, the CPviolation observed to date in the neutral kaon, and in the B
meson sectors is too small to explain it. This CP deficit hasmotived a broad program of searches for EDMs in a range ofsystems. Many extensions to the standard model, such assupersymmetry, do not forbid new sources of CP-violation,and the failure to observe it has placed severe restrictions onmany models.
The magnetic dipole moment can differ from its Diracvalue (g ¼ 2) for several reasons. Recall that the proton’s g-value is 5.6 (ap ¼ 1:79), a manifestation of its quark–gluoninternal structure. On the other hand, the leptons appear tohave no internal structure, and the MDMs are thought toarise from radiative corrections, i.e., from virtual particlesthat couple to the lepton. We would emphasize that theseradiative corrections need not be limited to the standard-model particles, but rather the physical values of the leptonanomalies represent a sum-rule over all virtual particles innature that can couple to the lepton, or to the photon throughvacuum polarization loops.
The standard model value of a lepton’s anomaly, a‘, hascontributions from three different sets of radiative processes:QED—with loops containing leptons (e; �; �) and photons;hadronic—with hadrons in vacuum polarization loops; andweak—with loops involving the bosons W , Z, and Higgs.Examples are shown in Fig. 2. Thus
a(SM)e;� ¼ a(QED)
e;� þ a(hadronic)e;� þ a(weak)
e;� : ð10Þ
The dominant contribution from QED, called theSchwinger term,36) að2Þ ¼ �=2�, and is shown diagrammati-cally in Fig. 2(a). The QED contributions have been
calculated through four loops, with the leading five-loopcontributions calculated.37) Examples of the hadronic andweak contributions are given in Figs. 2(b)–2(e).
The hadronic contribution cannot be calculated directlyfrom QCD, since the energy scale is very low (m�c
2),although Blum has performed a proof of principle calcu-lation on the lattice.44) Fortunately dispersion theory38) givesa relationship between the vacuum polarization loop and thecross section for eþe� ! hadrons,
a�ðHad;1Þ ¼�m�
3�
� �2Z 14m2
�
ds
s2KðsÞRðsÞ; ð11Þ
R � f�totðeþe� ! hadronsÞg=f�totðeþe� ! �þ��Þg, andexperimental data are used as input38,39)
The muon anomaly is sensitive to a number of potentialcandidates for physics beyond the standard model:41)
(1) muon substructure, where the contribution depends onthe substructure scale � as
a�ð��Þ ’m2�
�2�
; ð12Þ
(2) W-boson substructure.(3) new particles that couple to the muon, such as the
supersymmetric partners of the weak gauge bosons,(4) extra dimensions
The potential contribution from supersymmetry hasgenerated a lot of attention,42,43) the relevant diagrams areshown in Fig. 3 below. A simple model with equal masses41)
gives
a(SUSY)� ’
�ðMZÞ8� sin2 �W
m2�
~mm2tan� 1�
4�
�ln
~mm
m�
� �ð13Þ
’ ðsgn�Þ 13� 10�10 tan�100 GeV
~mm
� �2
; ð14Þ
where tan � is the ratio of the two vacuum expectationvalues of the two Higgs fields. If the SUSY mass scalewere known, then a(SUSY)
� would provide a clean way todetermine tan �.
Table II. Measured limits on electric dipole moments, and their standard
tion; (d)–(e) the lowest order electroweak W and Z
contributions. With the present limits on mh, the
contribution from the single Higgs loop is negli-
gible.
+
χ
γ
μ μ0
μ∼ μ∼
γμ μ
ν
χ− −
∼
χ
Fig. 3. The lowest-order supersymmetric contributions to the muon
anomaly. The � are the super-partners of the standard-model gauge
bosons.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS B. L. ROBERTS
111009-4
4.1 Measurement of the anomalous magnetic dipolemoment
Measurement of the magnetic anomaly uses the spinmotion in a magnetic field. For a muon moving in amagnetic field, the spin and momentum rotate with thefrequencies:
!S ¼ �qgB
2m�
qB
�mð1� �Þ; !C ¼ �
qB
m�: ð15Þ
The spin precession relative to the momentum occurs at thedifference frequency, !a, between the spin and cyclotronfrequencies, eq. (15),
!a ¼ !S � !C ¼ �g� 2
2
� �qB
m¼ �a�
qB
m: ð16Þ
The magnetic field in eq. (16) is the average field seen by theensemble of muons. This technique has been used in all butthe first experiments by Garwin et al.,45) which usedstopping muons, to measure the anomaly. After Garwinet al., made a 12% measurement of the anomaly, a series ofthree beautiful experiments at CERN culminated with a7.3 ppm measure of a�.34)
In the third CERN experiment, a new technique wasdeveloped based on the observation that electrostatic quad-rupoles could be used for vertical focusing. With the velocitytransverse to the magnetic field (� B ¼ 0), the spinprecession formula becomes
!a ¼ �q
ma�B� a� �
1
�2 � 1
� ��� E
c
� �: ð17Þ
For �magic ¼ 29:3 (pmagic ¼ 3:09 GeV/c), the second termvanishes; one is left with the simpler result of eq. (16), hencethe name ‘‘magic’’, and the electric field does not contributeto the spin precession relative to the momentum. There aretwo major advantages of using the magic � and a uniformmagnetic field: (i) the knowledge needed on the muontrajectories to determine the average magnetic field is muchless than when gradients are present, and the more uniformfield permits NMR techniques to realize their full accuracy,thus increasing the knowledge of the B-field. The spinprecession is determined almost completely by eq. (16),which is independent of muon momentum; all muonsprecess at the same rate. This technique was used also inexperiment E82140) at the Brookhaven National LaboratoryAlternating Gradient Synchrotron (AGS). The reader isreferred to ref. 38 for a discussion of muon decay relevant toE821, and to ref. 40 for details of E821.
Muons are stored in a storage ring,40) and the arrival timeand energy of the decay electrons is measured. When asingle energy threshold is placed on the decay electrons, thenumber of high-energy electrons is modulated by the spinprecession frequency, eq. (17), producing the time distribu-tion
Nðt;EthÞ ¼ N0ðEthÞe�t=ð��Þ
� ½1þ AðEthÞ cosð!at þ �ðEthÞÞ�:ð18Þ
as shown in Fig. 4.40) The value of !a is obtained from aleast-squares fit to these data. The five-parameter function[eq. (18)] is used as a starting point, but many additionalsmall effects must be taken into account.40)
In E821, both �þ and �� were measured, and assuming
CPT invariance, the final result obtained by E821,40)
aexp� = 116 592 080 (63) � 10�11, is shown in Fig. 5, along
with the individual measurements and the standard-modelvalue. The present standard-model value is38) aSM(06)
� =116 591 785 (61) � 10�11, and one finds �a� ¼ 295ð88Þ �10�11, a 3:4� difference.
One candidate for the cosmic dark matter is the lightestsupersymmetric partner, the neutralino, �0 in Fig. 3. In thecontext of a constrained minimal supersymmetric model(CMSSM), ðg� 2Þ� provides an orthogonal constraint ondark matter46) from that provided by the WMAP survey,as can be seen in Fig. 6.
With the apparent 3:4� difference between theory andexperiment, a new experiment to improve the error by afactor of 2 to 2.5 has been proposed to BrookhavenLaboratory, but at present it is not funded. The theoreticalvalue will continue to be improved, both with the expectedavailability of additional data on eþe� annihilation tohadrons, and with additional work on the hadronic light-by-light contribution.38)
4.2 The search for a muon electric dipole momentWith an EDM present, the spin precession frequency
relative to the momentum must be modified. The totalfrequency becomes ! ¼ !a þ !�, where
s]μs [μTime modulo 1000 20 40 60 80 100
Mill
ion
Eve
nts
per
149
.2n
s
10-3
10-2
10-1
1
10
Fig. 4. The time spectrum of 3:6� 109 electrons with energy greater than
1.8 GeV from the 2001 data set. The diagonal ‘‘wiggles’’ are displayed
Fig. 5. Measurements of the muon anomaly, indicating the value, as well
as the muon’s sign. As indicated in the text, to obtain the value of a�� and
the world average CPT invariance is assumed. The theory value is taken
from ref. 38, which uses electron–positron annihilation to determine the
hadronic contribution.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS B. L. ROBERTS
111009-5
!� ¼ �q
m
�
2
E
cþ �� B
� �� �; ð19Þ
with � defined by eq. (9), and !a by eq. (17). The spinmotion resulting from the motional electric field, �� B isthe dominant effect, so !� is transverse to B. An EDMwould have two effects on the precession, there would be aslight tipping of the precession plane, which would cause avertical oscillation of the centroid of the decay electronsthat out of phase with the !a precession; and the observedfrequency ! would be larger,
The muon limit in Table II, placed by the non-observationof the vertical oscillation, is dominated by systematiceffects. The limit obtained by this method in the CERNexperiment,34) and likely to be obtained by E821, cannotdirectly exclude the possibility that the entire differencebetween the measured and standard-model values of a�could be caused by a muon EDM. Such a scenario wouldimply that the EDM would be d� ¼ 2:4 ð0:4Þ � 10�19 e cm,a factor �108 larger than the current limit on the electronEDM. While this would be a very exciting result, it isorders of magnitude larger than that expected from even themost speculative models47–50)
To reduce systematic errors in the muon EDM measure-ment, a ‘‘frozen spin’’ technique has been proposed51) whichuses a radial electric field in a muon storage ring, operatingat � �magic to cancel the ðg� 2Þ precession. The EDMterm, eq. (19), would then cause the spin to steadily moveout of the plane of the storage ring. Electron detectors aboveand below the storage region would detect a time-dependentup–down asymmetry that increased with time. As in the
ðg� 2Þ experiments, detectors placed in the plane of thebeam would be used, in this case to make sure that theradial-E-field cancels the normal spin precession exactly.Adelmann and Kirsh52) have proposed that one could reacha sensitivity of 5� 10�23 e cm with a small storage ring atPSI. A letter of intent at J-PARC53) suggested that one couldreach <10�24 e cm there. The ultimate sensitivity wouldneed an even more intense muon source, such as a neutrinofactory.
5. The Search for Lepton Flavor Violation
The standard-model gauge bosons do not permit leptons tomix with each other, unlike the quark sector where mixinghas been known for many years. Quark mixing was firstproposed by Cabibbo,54) and extended to three generationsby Kobayashi and Maskawa,55) which is described by a3� 3 mixing matrix now universally called the CKMmatrix. With the discovery of neutrino mass, we know thatlepton flavor violation (LFV) certainly exists in the neutrallepton sector, with the determination of the mixing matrixfor the three neutrino flavors having become a world-wideeffort.
While the mixing observed in neutrinos does predictsome level of charged lepton mixing, it is many orders ofmagnitude below present experimental limits.13) New dy-namics,56–66) e.g., supersymmetry, do permit leptons to mix,and the observation of standard-model forbidden processessuch as
would clearly signify the presence of new physics. Thepresent limits on lepton flavor violation are shown in Fig. 7.
If lepton mixing occurs via supersymmetry, there will be amixing between the supersymmetric leptons (sleptons)which would also be described by a 3� 3 mixing matrix.The schematic connection between lepton flavor violationsand the dipole moments is shown in Fig. 8, and there aremodels that try to connect these processes.67)
In a large class of models, if the �‘ ¼ 1 LFV decay goesthrough the transition magnetic moment, one finds13)
100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800m
0 (G
eV)
m1/2 (GeV)
tan β = 10 , μ > 0
mh = 114 GeV
mχ± = 104 GeV
scal
ar m
ass
gaugino mass
WMAPrestrictionsg-2
± 1σ
± 2σ
Fig. 6. (Color online) Limits on dark matter placed by various inputs in
CMSSM, with tan� ¼ 10. The � between experiment and standard-
model theory is from ref. 38, see text. The brown wedge on the lower
right is excluded by the requirement the dark matter be neutral. Direct
limits on the Higgs and chargino �� masses are indicated by vertical
lines. Restrictions from the WMAP satellite data are shown as a light-blue
line. The ðg� 2Þ 1- and 2-standard deviation boundaries are shown in
purple. The region ‘‘allowed’’ by WMAP and ðg� 2Þ is indicated by the
ellipse, which is further restricted by the limit on Mh. (Figure courtesy of
K. Olive.)
Searches for Lepton Number Violation
10-19
10-17
10-15
10-13
10-11
10-9
10-7
10-5
10-3
10-1
1950 1960 1970 1980 1990 2000 2010
Year
UL
Bra
nch
ing
Rat
io (
Co
nve
rsio
n P
rob
abili
ty)
μ → e γμ- N→ e- Nμ+e-→ μ-e+
μ → e e eKL → π+ μ eKL → μ eKL → π0 μ e
+Projected
Proposed, phase 1
Fig. 7. (Color online) Historical development of the 90% C.L. upper
limits on branching ratios respectively conversion probabilities of muon-
number violating processes which involve muons and kaons. Also shown
is the projected goal of the MEG (�þ ! eþ�) experiment which is
underway at PSI, and the projected sensitivity of the recent letter of intent
to J-PARC for muon-electron conversion. (Figure from ref. 13.)
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS B. L. ROBERTS
111009-6
Bð�N ! eNÞBð�! e�Þ
¼ 2� 10�3BðA; ZÞ; ð23Þ
where BðA;ZÞ is a coefficient of order 1 for nuclei heavierthan aluminum.68) For other models, these two rates can bethe same,13) so in the design of new experiments, the reach insingle event sensitivity for the coherent muon conversionexperiments needs to be several orders of magnitude smallerthan for �! e� to probe the former class of models withequal sensitivity. Detailed calculations of �� e conversionrates as a function of atomic number have also been carriedout,69) and if observed, measurements should be carried outin several nuclei.
From the experimental side, the next generation �! e�experiment, MEG, is now under way at PSI,70) with asensitivity goal of 10�13–10�14. Since the decay occurs atrest, the photon and positron are back-to-back, and shareequally the energy m�c
2. This experiment makes use ofa unique ‘‘COBRA’’ magnet which produces a constantbending radius for the mono-energetic eþ, independent of itsangle. The photon is detected by a large liquid Xescintillation detector as shown in Fig. 9.
Of the various lepton-flavor violating reactions, onlycoherent muon conversion does not require coincidencemeasurements. The decay �! 3e, while theoreticallyappealing, requires a triple coincidence and sensitivity tothe whole phase space of the decay, and thereby isexperimentally more challenging. It is the coherent muonto electron conversion, where with adequate energy reso-lution, the conversion electron can be resolved from back-
ground, that with adequate muon flux can be pushed tothe 10�18 or 10�19 sensitivity. Such a program has beenproposed for J-PARC.71)
The muonium to antimuonium conversion [left-handprocess in eq. (22)] represents a change of two units oflepton number, analogous to K0 �KK0 oscillations. This processwas originally proposed by Pontecorvo.72) An experiment atPSI73) obtained a single event sensitivity of PM �MM ¼ 8:2�10�11 which implies a coupling GM �MM � 3� 10�3GF at 90%C.L., where GF is the Fermi coupling constant. A broadrange of speculative theories such as left–right symmetry,R-parity violating supersymmetry, etc.,74) could permit suchan oscillation.
6. Summary and Conclusions
Since its discovery, the muon has provided an importanttool to study the standard model, and to constrain itsextensions. Experiments in the planning stage for ðg� 2Þ,the search for an electric dipole moment and lepton flavorviolation in muon decay or conversion will continue thistradition. Research and development for new more intensemuon sources, such as the muon ionization cooling experi-ment (MICE),75) will further propel increases in sensitivity.Muon experiments form an important part of the precisionfrontier in particle physics, which will continue to providevital information complementary to that from the highestenergy colliders.
Acknowledgments
I wish to acknowledge A. Czarnecki, T. Gorringe, D.Hertzog, P. Kammel, K. Jungmann, Y. Kuno, W. Marciano,J. P. Miller, Y. Okada, and E. de Rafael, for helpfulconversations, and TG, DH, KJ, PK, and JM for theirexcellent suggestions on this manuscript. This work wassupported in part by the U.S. National Science Foundationand U.S. Department of Energy.
1) See Fig. 5 in P. Kunze: Z. Phys. 83 (1933) 1.
2) C. D. Anderson and S. H. Neddermeyer: Phys. Rev. 50 (1936) 263;
S. H. Neddermeyer and C. D. Anderson: Phys. Rev. 51 (1937) 884.
3) J. C. Street and E. C. Stevenson: Phys. Rev. 52 (1937) 1003.
4) Y. Nishina, M. Tekeuchi, and T. Ichimiya: Phys. Rev. 52 (1937) 1198.
5) M. M. Jean Crussard and L. Leprince-Ringuet: C. R. Acad. Sci. 204
(1937) 240.
6) P. J. Mohr and B. H. Taylor: Rev. Mod. Phys. 77 (2005) 1.
7) A. Antognini et al.: AIP Conf. Proc. 796 (2005) 253.
8) S. Lokanathan and J. Steinberger: Phys. Rev. 98 (1955) A240.
9) J. Steinberger and H. B. Wolfe: Phys. Rev. 100 (1955) 1490.
10) G. Danby et al.: Phys. Rev. Lett. 9 (1962) 36.
11) L. Michel: Proc. Phys. Soc. London 63 (1950) 514; C. Bouchiat and
L. Michel: Phys. Rev. 106 (1957) 170.
12) B. Jamieson et al.: Phys. Rev. D 74 (2006) 072007; A. Gaponenko
et al.: Phys. Rev. D 71 (2005) 071101; J. R. Musser et al.: Phys. Rev.
Lett. 94 (2005) 101805; See C. A. Gagliardi, R. E. Tribble, and N. J.
Williams: Phys. Rev. D 72 (2005) 073002 for a general discussion
of muon decay experiments.
13) J. Aysto et al.: hep-ph/0109217. This report of a CERN working
group lays out a nice case for muon physics at a neutrino factory.
14) T. van Ritbergen and R. G. Stuart: Nucl. Phys. B 564 (2000) 343;
T. van Ritbergen and R. G. Stuart: Phys. Lett. B 437 (1998) 201;
T. van Ritbergen and R. G. Stuart: Phys. Rev. Lett. 82 (1999) 488.
15) These radiative corrections are computed to second order, see M.
Awramik, M. Czakon, A. Freitas, and G. Weiglein: Phys. Rev. D 69
(2004) 053006.
16) W. J. Marciano: Phys. Rev. D 60 (1999) 093006.
Fig. 8. The supersymmetric contributions to the anomaly, and to �! e
conversion, showing the relevant slepton mixing matrix elements. The
MDM and EDM give the real and imaginary parts of the matrix element
respectively.
1m
e+
Liq. Xe ScintillationDetector
γ
Drift Chamber
Liq. Xe ScintillationDetector
e+
γ
Timing Counter
Stopping TargetThin Superconducting Coil
Muon Beam
Drift Chamber
ee++ μμ++ γγEEee == EEγγ == 52.8 MeV52.8 MeV
θθeeγγ == 180°180°
Fig. 9. (Color online) The side and end views of the MEG experiment.
Since the muon is at rest, the photon and positron are at a relative angle of
180 . The positron is tracked in a magnetic field which produces a
constant bending radius, independent of angle.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS B. L. ROBERTS
Electric Dipole Moments of Elementary Particles, Nuclei, Atoms, and Molecules
Eugene D. COMMINS�
Physics Department, University of California, Berkeley, CA 94720, U.S.A.
(Received December 13, 2006; accepted May 17, 2007; published November 12, 2007)
The significance of particle and nuclear electric dipole moments is explained in the broader context ofelementary particle physics and the charge–parity (CP) violation problem. The present status and futureprospects of various experimental searches for electric dipole moments are surveyed.
KEYWORDS: electric dipole moment, CP violation, parity and time reversalDOI: 10.1143/JPSJ.76.111010
1. Introduction
Every physicist knows that the electron, proton, andneutron have spin magnetic dipole moments, as do manyother particles and nuclei with non-zero spin. Can anelementary particle or nucleus also have a spin electricdipole moment (EDM)? In this review we try to explain whythis question has become very important for elementaryparticle physics, and we briefly survey the present status andfuture prospects of various experimental searches for EDMs.
An EDM cannot exist unless both parity (P) and timereversal (T) invariance are violated. This can be seen fromthe non-relativistic Hamiltonian for the interaction of anEDM d with an electric field E, which is HNR
EDM ¼ �d � E.For an elementary particle or nucleus in a non-degeneratestate, the spin angular momentum J is the only vectoravailable to define a direction. Thus d must be collinear withJ, and:
HNREDM ¼ �d � E ¼ �d
J
J� E: ð1Þ
However E is a T-even polar vector while J is a T-odd axialvector. Thus HNR
EDM is odd under P and T transformations.The same conclusion is of course true for the relativisticgeneralization of HNR
EDM.Until now no EDM has been observed, and it is obvious
from the present experimental upper limits, summarized inTable I, that EDMs must be extremely small. For example,the upper limit on the electron EDM de is 8:3� 10�17 �B,where �B is the Bohr magneton. Nevertheless, EDMs maybe non-zero, because P and T are in fact violated in nature.Parity nonconservation [as well as the violation of chargeconjugation (C) invariance] occurs in the weak interaction.Furthermore, combined charge–parity (CP) violation isobserved in neutral K meson and B meson decays.1) If weassume CPT invariance, for which we have very strongconfidence, then this CP violation is equivalent to T
violation. Thus the weak interaction and the mechanismor mechanisms causing CP violation could act jointly togenerate EDMs by P,T-odd radiative corrections to the P, C,T conserving electromagnetic interaction.
Unfortunately, given the present state of our knowledge,such radiative corrections cannot be calculated with con-fidence. Instead, they depend on uncertain theoretical
models of CP violation. According to the standard model,while CP violation in quantum chromodynamics (QCD) canin principle be large, CP violation in the electroweak sectoris described phenomenologically by a single phase thatappears in the Cabibbo–Kobayashi–Maskawa (CKM) quarkmixing matrix.2) This gives a satisfactory account of K- andB-meson CP violation data.3) It can be shown that in thisdescription the neutron EDM dn appears only at the three-loop level of perturbation theory4) while the electron EDMde appears only at the four-loop level5) (and there areadditional suppressions). Thus the standard model electro-weak predictions: dn � 10�32 e cm, de � 10�38 e cm (wheree ¼ 4:8� 10�10 esu is the unit of electronic charge) aremany orders of magnitude smaller than the present exper-imental limits. Indeed, if the standard model mechanismof CP violation is the only one, then given present orforeseeable experimental capabilities, future observationof any EDM is very unlikely.
However, there are good reasons to think that additionalmechanisms exist for CP violation. It is generally acceptedthat if the universe initially was symmetric in baryon–antibaryon number, the presently observed baryon–anti-baryon asymmetry could not have developed without a muchlarger CP violation than is predicted by the standard model.6)
Furthermore, in many theories that attempt to go beyondthe standard model, predicted EDMs are relatively large.For example, in various supersymmetric theories, manynew hypothetical particles and couplings appear, and alongwith them exist new CP violating phases. Thus in many suchmodels the electron and neutron EDMs already appear at the
SPECIAL TOPICS
Table I. Experimental limits on EDMs and the � weak dipole moment
one-loop level, and as a result predictions of de and dn areclose to present experimental limits.7,8) Thus discovery of anEDM by practical experimental methods is a real possibilitywithin the foreseeable future, and such a discovery wouldprovide definite evidence for physics beyond the standardmodel.
The search for EDMs of the neutron and of nuclei isimportant for a related issue of fundamental significance inQCD: the ‘‘strong CP problem’’. A CP-odd term exists inthe effective Lagrangian density for QCD, characterized bythe ‘‘QCD CP-violating parameter’’ ���.9) It can be shown thatthis contributes to dn as: dnð ���Þ � 3� 10�16 ��� e cm. Thus thepresent experimental limit on dn implies ��� � 1� 10�10.Why is ��� so small? A satisfactory answer to this questionis not yet known.
2. Proper Lorentz-Invariant, Gauge-InvariantLagrangian Density
In order to describe the interaction of the EDM of a spin-1/2 fermion with an electromagnetic field we need a gauge-invariant, proper-Lorentz-invariant effective Lagrangiandensity. First let us recall the analogous Lagrangian densityfor an anomalous magnetic moment (‘‘Pauli moment’’) in theDirac theory.10) It is given by the well-known expression:
LPauli ¼ ���B
2�������F��: ð2Þ
Here � is the Dirac field for the fermion, ��� ¼ �y�0 is theDirac conjugate field, ��� ¼ ði=2Þð���� � ����Þ where ��;�
are the usual 4� 4 Dirac matrices,
F�� ¼ @�A� � @�A� ¼
0 Ex Ey Ez�Ex 0 �Bz By
�Ey Bz 0 �Bx
�Ez �By Bx 0
0BBB@
1CCCA ð3Þ
is the electromagnetic field tensor, and � is a suitableconstant. Rewriting eq. (2) in terms of E and B fields, weobtain:
LPauli ¼ ��B���½� � B� i� � E� ð4Þ
where
� ¼� 0
0 �
� �; � ¼
0 �
� 0
� �:
This Lagrangian density results in the Hamiltonian density
HPauli ¼ ���B���½� � B� i� � E� ð5Þ
and in the single-particle Hamiltonian:
HPauli ¼ ���Bð�0� � B� i� � EÞ: ð6Þ
Of course, LPauli of eq. (2) or (4) and HPauli of eq. (6) areeach P- and T-invariant. We can render them P,T-odd byreplacing E by �B and B by E, which is equivalent to thereplacement of F�� by the tensor �F���, where:
F��� ¼1
2"���F
� ¼
0 Bx By Bz
�Bx 0 Ez �Ey�By �Ez 0 Ex�Bz Ey �Ex 0
0BBB@
1CCCA ð7Þ
and "��� is the completely antisymmetric unit 4-tensor.Alternatively we obtain the same Lagrangian density byreplacing ��� in eq. (2) with i����5 (where, as usual, �5 ¼i�0�1�2�3), with no change in F��. Making this lattertransformation and replacing ��B by d we obtain the EDMLagrangian density:
LEDM ¼ �id
2�������5�F�� ¼ d ���½� � E þ i� � B� ð8Þ
which was first described by Salpeter.11) This in turn yieldsthe single-particle Hamiltonian:
HEDM ¼ �dð�0� � E þ i� � BÞ: ð9Þ
In the non-relativistic limit the first term on the right-handside (r.h.s.) of eq. (9) reduces to the r.h.s. of eq. (1).However when the particle of interest is relativistic, the fullexpression on the r.h.s. of eq. (9) must be used, and this hassignificant consequences, as we shall see.
3. The Neutron EDM
In all neutron EDM experiments use is made of the factthat non-relativistic polarized neutrons in collinear E andB fields undergo Larmor precession with frequency � ¼½2�nB� 2dnE=h, where the � sign corresponds to parallel(antiparallel) E and B fields. Thus the presence of an EDM isrevealed by an electric field-dependent shift in � proportionalto the T-odd pseudoscalar E � B. The earliest experimentsemployed neutron beams and the Ramsey method ofmagnetic resonance with spatially separated oscillatingfields and an intense electric field between them.12) Morerecent experiments utilize ultra-cold neutrons that typicallyhave kinetic energies of �10�7 eV or less and undergo totalinternal reflection at any angle of incidence on suitablematerials. These neutrons can be stored without substantialloss in closed vessels permeated by collinear E and B fields,where the oscillating fields for magnetic resonance areseparated in time rather than in space. The most recent andprecise of such experiments was performed at the InstitutLaue–Langevin (ILL) in Grenoble, where the followingresult was obtained:13)
The statistical uncertainty on the r.h.s. of eq. (10) ismainly due to the limited number of ultra-cold neutronsthat could be generated and stored, while the systematicuncertainty arises for the most part from a geometric phaseeffect caused by unintended gradients in B.
New methods are needed if dn is to be determined to muchbetter precision. Several novel projects are under develop-ment, including one at ILL and another at the Los AlamosNational Laboratory (LANL). In the latter experiment,14)
ultra-cold polarized neutrons are produced and stored ina bath of superfluid 4He containing a dilute solution ofnuclear–spin-polarized 3He. A neutron can only exchangemomentum with 4He when the free neutron dispersion curveand the phonon–roton dispersion curve of superfluid 4Heintersect (see Fig. 1), since energy and momentum mustboth be conserved in the collision. Intersections occur at E ¼k ¼ 0 and at k ¼ k�, E ¼ E� (corresponding to the temper-ature T� ¼ E�=kB � 12 K). Polarized neutrons enter thesuperfluid bath, and those with wavelength ¼ 2�=k� �
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS E. D. COMMINS
111010-2
0:89 nm are downscattered to form a polarized ultra-coldneutron sample. The probability of subsequent up-scatteringby absorption of a 4He excitation is very small at the bathoperating temperature T � 500 mK, since the Boltzmannfactor for these excitations is expð�T�=TÞ 1. Thus arelatively large density of ultra cold polarized neutronscan be accumulated in the bath. The polarized 3He actssimultaneously as a neutron spin analyzer and as a co-magnetometer. The cross section for the reaction3He(n,p)3H + 764 keV is very large, but only in the n-3HeJ ¼ 0 state (where the spins of these two species areopposed). Thus observation of this spin-dependent reactionby means of the resulting scintillations in 4He provides away to detect the precession of the neutron spins in the Eand B fields. A number of subtle problems are associatedwith this ambitious experiment. If they can be overcome,an improvement in the present limit on dn of a factor of�100 seems possible.
The neutron and proton EDMs are expected to be roughlycomparable in magnitude. However, as we discuss in thenext section, the proton presents a completely differentchallenge to the experimenter because it is charged.
4. Atomic and Molecular EDMs
4.1 Schiff’s theoremIt has long been considered impractical to search for
an EDM by placing a charged particle (e, p, bare nucleus,. . .) in an electrostatic field, since the particle would quicklybe accelerated out of the region of observation. (Recentproposals for storage ring searches for EDMs of chargedparticles are discussed in §5 and §6). What can we learn byapplying an external electrostatic field to a neutral atomor molecule that contains a nucleus or unpaired electronwith an EDM d? At first sight this approach appears useless,because in the limits where all atomic or molecularconstituents are treated as point charges, and where non-relativistic quantum mechanics applies, the atom or mole-cule cannot possess an EDM da (cannot exhibit a linear Starkeffect) to first order in d. This is Schiff’s theorem15) whichcan be understood intuitively as follows: A neutral atom or
molecule is not accelerated in a uniform external electricfield. Thus the average force on each of the atomic ormolecular constituents must be zero. In the non-relativistic,point charge limits, the only forces are electrostatic; hencethe average electric field at each point charge must be zero.This happens because the external field is cancelled, onaverage, by the internal polarizing field.
We note in passing that Schiff’s theorem is not in conflictwith existence of the so called ‘‘permanent’’ electric dipolemoments of many polar molecules, familiar in chemistry andmolecular spectroscopy. These moments do not violate P orT , nor do they result in a linear Stark effect for sufficientlysmall applied electric fields, in the absence of degeneracy.They have entirely different observational signatures thanexist for the EDMs of interest to us.
4.2 Nuclear EDMsSchiff’s theorem is evaded for a nucleus if one takes into
account magnetic hyperfine structure, and more importantly,for a nucleus of finite size if the nuclear EDM distribution isnot the same as the charge distribution.15) In the latter case,a small residual EDM effect remains, which is expressedin terms of an additional P,T-odd electronic potential VS ¼�e�iS � r�3ðriÞ that must be included in the atomic ormolecular Hamiltonian. Here ri refers to the position of thei-th electron relative to the nuclear center-of-mass, and the‘‘Schiff moment’’ S is a vector proportional to the nuclearEDM depending on the difference between the normalizedcharge and EDM distributions:
S ¼ I; Iz ¼ Ie
10
XZp¼1
r2p �5
3hr2iCh
� �rp
�����
�����I; Iz ¼ I
* +; ð11Þ
where the sum is over all nuclear protons, and I is thenuclear spin. S can be generated by an intrinsic EDM of anunpaired nucleon, and/or by P,T-odd nucleon–nucleon (NN)interactions. Generally speaking, S is largest for the heaviestnuclei, and in particular it is enhanced in octupole-deformednuclei such as 225Ra by roughly a factor of 10 to 100compared to its value for a more symmetric heavy nucleussuch as 199Hg. In addition to the Schiff moment, a nucleuswith nuclear spin I � 1 can possess a magnetic quadrupolemoment M originating from nucleonic EDMs and/orP,T-odd NN interactions. In a paramagnetic atom ormolecule this would couple to the magnetic field resultingfrom the spin and spatial distribution of the unpairedelectron. Because this interaction is magnetic, it would notbe constrained by Schiff’s theorem.16)
The most sensitive nuclear EDM search to date17) was anoptical pumping experiment carried out on the diamagneticatom 199Hg (I ¼ 1=2). The result is:
dð199HgÞ¼ �½1:06� 0:49ðstatÞ � 0:40ðsystÞ � 10�28 e cm
ð12Þ
which gives the limit: jdð199HgÞj � 2� 10�28 e cm. Calcu-lations relating d(199Hg) to S yield the result dð199HgÞ ¼�2:8� 10�17[S=ðe fm3Þ]e cm.18) From this and eq. (12)one obtains Sð199HgÞ ¼ ð3:8� 1:8� 1:4Þ � 10�12 e fm3.The largest contribution to S(199Hg) is estimated to arisefrom a P,T-odd nucleon–nucleon interaction of the form GFð �pppÞð �nni�5nÞ=
ffiffiffi2p
, where GF is Fermi’s coupling constantand is a dimensionless constant. However calculations
E
kk*
E*
n
4He
Fig. 1. Schematic diagram of dispersion curves for the free neutron (n)
and superfluid 4He.
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111010-3
show that there is also a contribution from the intrinsic EDMof the proton; indeed the best current limit on the protonEDM: jdpj � 5:4� 10�24 e cm, is inferred from the exper-imental result (12).19) Very significant improvements in ourknowledge of dp may come from future storage-ringexperiments; see §6. We note in passing that a relativelylarge upper limit also exists for the �0 hyperon;20) seeTable I.
4.3 The electron EDMWe next consider the unpaired atomic electron(s) in a
paramagnetic atom or molecule. Sandars21,22) has shown thatSchiff’s theorem is also evaded here when relativistic effectsare taken into account. The result of Sandars’ analysis[which is based on the first term on the r.h.s. of eq. (9)including the factor �0] may be expressed in terms ofthe ratio da=de or equivalently in terms of the effectiveelectric field Eeff experienced by de. It is convenient to writeEeff ¼ Q� where Q is a factor that includes relativisticeffects as well as details of atomic (or molecular) structure,while � is the degree of polarization of the atom ormolecule by the external electric field Eext. For paramagneticatoms with valence electrons in s1=2 or p1=2 orbitals, suchas Cs and Tl in their ground states,
Q � 4� 1010 V/cm� ðZ=80Þ3; ð13Þ
where Z is the atomic number. Also, for such atoms � �10�3(Eext=100 kV/cm), which is only �10�3 for the max-imum attainable laboratory fields Eext � 100 kV/cm. Sincefor paramagnetic atoms in all practical situations, � isproportional to Eext, the ratio Eeff=Eext is a constant, and isusually called the enhancement factor R ¼ da=de. For theground states of alkali atoms and for thallium, one findsjRj � 10Z3�2 where � is the fine structure constant.Although in these cases � 1, jRj can greatly exceedunity for sufficiently large Z. For example, for the thalliumatom (Z ¼ 81), one calculates R ¼ �585.
Equation (13) also applies for a wide range of heavy polardiatomic paramagnetic molecules with valence electrons in� or � orbitals, such as YbF in the ground 2�1=2 state, orPbO in the metastable a(1) 3�1 state. (In these cases Z is theatomic number of the heavy nucleus.) The main differencebetween atoms and molecules occurs in the factor �. In atypical polar diatomic molecule, nearly complete polar-ization (� � 1) can be achieved with relatively modestexternal fields: (Eext � 102{104 V/cm) because of the veryclose spacing between adjacent spin-rotational levels ofopposite parity. When � � 1, Eeff for a paramagneticmolecule such as YbF or PbO� is approximately 3 orders ofmagnitude larger than the maximum attainable with atoms.
P,T-odd electron–nucleon (eN) interactions can alsocontribute to da in diamagnetic or paramagnetic atoms ormolecules. These, as well as the P,T-odd NN interactions,can appear in one or several non-derivative coupling forms:‘‘scalar’’, ‘‘tensor’’, and ‘‘pseudoscalar’’. (P,T-odd electron-electron interactions are also possible but these only yieldan extremely small contribution.) Finally, C,T-odd (P-even)eN and NN interactions, and possible T-odd beta decaycouplings could cause a P,T-odd atomic or molecular EDMthrough radiative corrections involving the usual weakinteractions of the standard model. For a paramagnetic atom
or molecule the most important contribution to da, inaddition to de itself, is the scalar P,T-odd eN interaction.16)
The most sensitive search for de to date employed 205Tlin an atomic beam magnetic resonance experiment withseparated oscillating fields.23) The result is:
de ¼ ð6:9� 7:4Þ � 10�28 e cm; ð14Þ
assuming Rð205TlÞ ¼ �585 and no contribution from thescalar P,T-odd eN interaction. Equation (14) yields thelimit: jdej � 1:6� 10�27 e cm. At present, many new search-es for de are in progress, using cesium, francium, YbF,PbO�, and the molecular ion HfFþ.24) These experimentswith free atoms or molecules employ various standardmethods of atomic, molecular, and optical physics: laser andrf spectroscopy, optical pumping, atomic and molecularbeams, ion trapping, atom trapping and cooling, etc. Inanother search for de, one applies a large electric field to theparamagnetic solid gadolinium gallium garnet (GGG).25) Inprinciple, the interaction of the EDMs of the unpairedelectrons with the electric field at sufficiently low temper-ature can yield a net magnetization of GGG which can bedetected by a superconducting quantum interference device(SQUID) magnetometer. (It has also been proposed sepa-rately that application of a sufficiently large electric field toa gaseous sample of diamagnetic diatomic moleculescould generate an observable P,T-odd magnetization.26)) Ina complementary experiment, application of an externalmagnetic field to the ferrimagnetic solid gadolinium irongarnet (GdIG) can yield an EDM-induced electric polar-ization of the sample, which is detectable in principle byultra-sensitive charge measurement techniques.27)
The chances are good that at least one of the many newexperimental searches for de will improve the existing limitby at least a factor of 10 in the relatively near future. Theexperimental searches employing paramagnetic molecules(YbF, PbO�, . . .) are of particular interest because thesemolecules have very large Eeff values.
5. The Muon EDM
In most theoretical models, including the standard model,the electron, muon and tau lepton EDMs are proportional orapproximately proportional to their masses. Assuming thisand given the present limit on de, one predicts d� < 3:3�10�25 e cm. However in some theoretical models d� could belarger than this by an order of magnitude or more.28) Thisprovides motivation for d� searches at the 10�24 e cm level.The best current limit on the muon EDM: jd�j � 7� 10�19 e
cm, was obtained in an experiment at the CERN muonstorage ring in the 1970’s, the primary purpose of whichwas a precise measurement of the muon g-factor anomalyað�Þ ¼ ðg� 2Þ=2.29) Since then, storage ring technology hasadvanced considerably, resulting in a much more precisemeasurement of að�Þ in recent years at Brookhaven.30)
This has led to new proposals, not only to improve the limiton d� but also for storage ring searches for the proton,deuteron, and 3He EDMs. The deuteron EDM dD appearsespecially promising (see §6).
In order to understand the main features of these experi-ments, we consider a relativistic particle moving withvelocity � in a horizontal plane, in electric and magneticfields E and B, where B is in the vertical direction (hence
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� � B ¼ 0) and � � E ¼ 0 also. It can be shown31) that theangular velocity ! of spin precession with respect to theparticle momentum is:
! ¼ �e
maBþ
1
2�2� a
� ��� E þ
1
2�ðE þ �� BÞ
� �
ð15Þ
Here, � ¼ ð1� 2Þ�1=2, � ¼ 2dð2mc=eh� Þ�1 where d isthe EDM, and in eq. (15) we employ units where h� ¼c ¼ 1. In the CERN and Brookhaven muon g� 2
experiments the muon energy was chosen so that 2�2 ¼a�1, and also E was negligible. In this situation eq. (15)reduces to:
! ¼ !a þ !e; ð16Þ
where
!a ¼ �e
maB and !e ¼ �
e
2m��� B:
In this case the spin precesses about ! with frequency ! ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2
a þ !2e
p� !a and has a small oscillatory vertical compo-
nent (see Fig. 2). In the CERN experiment this was searchedfor by observing the angular distribution of electrons emittedabove and below the horizontal orbit plane in polarizedmuon decay. However the precision was limited becausethe vertical spin component was so very small, as well asoscillatory, owing to the presence of !a. In a newly proposedmuon EDM search,32) one chooses ð2�2Þ�1 � a and anelectric field is applied of magnitude jEj ¼ �2ajBj and inthe direction �� B. In other words E is radial and in theorbit plane; see Fig. 3. In this case !a is eliminated and thus! is directed along E with magnitude:
! ¼e
2m�ð1þ �2aÞB ð17Þ
Consequently starting from a horizontal orientation the spinprecesses very slowly in the vertical plane and the verticalspin component increases approximately linearly with time,
becoming much larger than in earlier experiments. With thisnew scheme it may be possible to extend the limit on d� to�10�24 e cm.
6. The Deuteron EDM
In a recently proposed storage ring deuteron EDMsearch,33) polarized deuterons with momentum p ¼ 1:5GeV/c are to circulate in a specially designed ring with amagnetic field B ¼ 2 T normal to the orbit plane, and withno applied electric field. In the instantaneous rest frameof the particle, the magnetic and electric fields are:
B0 ¼ �B ð18Þ
and
E0 ¼ �ð�� BÞ ð19Þ
For p ¼ 1:5 GeV/c and B ¼ 2 T, the rest frame electric fieldis E0 5� 108 V/m. In contrast to nuclear EDM searchesemploying neutral atoms, this very large electric field isapplied directly to the deuteron without any ‘‘Schiff’’screening. As usual, the component of precession angularvelocity due to the magnetic moment is directed along B; inthe laboratory frame this is described by the formula !a ¼�ðe=mÞaB. However, in this experiment a novel feature isintroduced: the beam velocity is modulated at frequency !a,(with �= � �1%). As is evident from eq. (19) thisproduces a component of E0 that oscillates in the planeperpendicular to B at the same frequency as the magneticspin precession. The interaction of dd with this oscillatingcomponent of E0 causes a spin reorientation analogous tothat which occurs in conventional magnetic resonance.The net result in the laboratory frame is the developmentof a vertical component of spin polarization proportionalto dD that is approximately linear in the time. Given theparameters we have mentioned, the rate of vertical polar-ization accumulation is �10�9 rad/s for dD ¼ 10�29 e cm.
Detection of the deuteron spin polarization could beachieved as follows. A thin gas jet causes Coulombscattering of a small fraction of beam deuterons on eachturn around the ring. The scattered deuterons strike a thickcarbon target in the shape of an annulus surrounding thebeam. Elastic scattering of D on carbon is spin-dependent
Fig. 2. Schematic diagram (not to scale) showing orientation of vectors in
original storage ring search for d�.29) Angular velocity !a is collinear
with B, while !e is normal to �B plane. Unit vector ss in direction of muon
spin precesses in plane perpendicular to ! ¼ !a þ !e, and thus has an
oscillating vertical component.
Fig. 3. Schematic diagram (not to scale) of vectors in proposed storage
ring search for d�.32) Electric field E in direction of �� B causes
cancellation of !a. Thus ss precesses in �B plane with very small angular
frequency !e.
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due to the spin–orbit interaction, and at 1.5 GeV/c, theanalyzing power is known to be better than 30%. Down-stream from the carbon target is an array of scintillationdetectors, also in the form of an annulus and segmented intofour quadrants (left, right, up, down). The left–rightasymmetry provides the EDM signal, while the up-downasymmetry gives information on the g� 2 precession.Assuming an initial deuteron polarization of 95%, 1012
deuterons in the ring, and a polarization coherence timeof 1000 s, as well as other parameters previously mentioned,it appears possible to achieve a statistical uncertainty indD of 10�29 e cm in several years of running. While such asmall uncertainty is very impressive, it is important to notethat systematic errors in this experiment could be significantand difficult to control, and a thorough analysis of suchsystematics has not yet been completed.
To a good approximation the deuteron EDM may beexpressed as follows:
dD ¼ dn þ dp þ dNucD ; ð20Þ
where dNucD is due to the P,T-odd nucleon–nucleon inter-
action. Since dn � �dp is expected, one has dD � dNucD .
Because of the simplicity of the deuteron, dNucD can be
estimated far more precisely than the corresponding P,T-oddNN interactions in complex nuclei such as 199Hg.
It is expected that the ��� contributions to dD; dn; dp; d3He
Finally, dn and dD can be expressed in terms of the electricdipole moments dq and chromo-electric dipole moments dC
q
of up and down quarks. It can be shown that:
dn � 1:4ðdd � 0:25duÞþ 0:27eðdC
d � dCu Þ þ 0:83eðdC
d þ dCu Þ
ð22Þ
and:
dD � ðdd þ duÞ þ 6eðdCd � dC
u Þ � 0:2eðdCd þ dC
u Þ ð23Þ
We note that the second term on the r.h.s. of eq. (23) ismore than 20 times larger than the corresponding term ineq. (22). These various expressions show that comparison ofthe EDMs of n, p, D, and 3He could yield much valuableinformation that would almost certainly be unobtainablefrom observation of EDMs of heavy nuclei in conventionalatomic or molecular experiments.
7. The � EDM and Weak Dipole Moment
Tau leptons have often been produced in eþe� collisionsat colliding beam accelerators:
eþ þ e� ! �þ þ �� ð24Þ
In lowest order of perturbation theory two distinct ampli-tudes contribute to this reaction: single photon exchange(electromagnetic interaction) and single Z0 exchange (neu-tral weak interaction). P,T-odd radiative corrections to thephoton exchange amplitude introduce the possibility of atau EDM d�, while similar corrections to the Z0 exchangeamplitude involve an analogous weak dipole moment(WDM) ~dd� . Although the EDM and WDM are independentquantities, in most theoretical models they have comparablemagnitudes. When the center-of-mass energy for reaction
(24) is in the vicinity of the Z rest energy (‘‘Z pole’’), as itwas in experiments carried out at the Large Electron–Positron Collider (LEP) at CERN, Z0 exchange greatlydominates in significance over photon exchange. By search-ing for certain P,T-odd correlations at the Z pole betweenthe momenta of the initial electron and the decay productsof �þ and �� in experiments at LEP, it was possible to placethe following upper limit on ~dd�:
34)
j ~dd�j � 5:8� 10�18 e cm: ð25Þ
This result together with plausible theoretical assumptionsleads to the following limit on the EDM d�:
jd�j � 1:1� 10�17 e cm: ð26Þ
8. Can a Neutrino Possess an EDM?
It is not yet known whether a neutrino and an antineutrinoof the same mass eigenstate are distinct particles (‘‘Dirac’’neutrino and antineutrino) or whether a neutrino of givenmass is self-conjugate (‘‘Majorana’’ neutrino). Neutrinomagnetic and electric dipole moments are described by 3�3 matrices �ij, dij respectively, where the diagonal elements�ii, dii refer to the static dipole moments of the i’th masseigenstate. If neutrinos are of the Majorana type, thediagonal elements �ii, dii must be zero because undercharge conjugation, the magnetic dipole and electric dipoleoperators change sign. Of course, no such restriction appliesto Dirac neutrinos.
A neutrino EDM could cause anomalous ionization in adetector because of its interaction with atomic electrons.Making use of this, analysis35) of an experiment carried outby Cowan and Reines in 1957 to detect ���e radiated froma reactor yielded the result jd�Fj � 2� 10�20 e cm, where F
is a form factor.
9. Conclusion
After a half century of search, there is still no exper-imental evidence for an EDM of an elementary particle,nucleus, atom, or molecule. However, widespread appreci-ation of the significance of EDM searches for the generalproblem of CP violation and the development of new andrefined experimental techniques now generate more intenseinterest in EDM searches than ever before. The presentexperimental upper limits for dn, de, and d(199Hg) alreadyprovide serious constraints on various supersymmetricmodels of CP violation. Improvements of factors of 10 –100 in the limits on dn and de, which may come in therelatively near future, should thus be very significant.Finally, success of the proposed deuteron storage ringexperiment would bring the field of nuclear EDMs into anentirely new era.
1) D. Kirkby and Y. Nir: J. Phys. G 33 (2006) 1.
2) M. Kobayashi and T. Maskawa: Prog. Theor. Phys. 49 (1973) 652.
3) F. J. Gilman, K. Kleinknecht, and B. Renk: Phys. Lett. B 592 (2004) 1.
4) A. Czarnecki and B. Krause: Phys. Rev. Lett. 78 (1997) 4339.
5) M. Pospelov and I. B. Khriplovich: Sov. J. Nucl. Phys. 53 (1991) 638.
6) See, for example, W. Bernreuther: in CP Violation in Particle,
Nuclear and Astrophysics, ed. M. Beyer (Springer, Berlin, 2002)
p. 237.
7) A. Faessler, T. Gutsche, S. Kovalenko, and V. Lyubovitskij: Phys.
Rev. D 74 (2006) 074013.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS E. D. COMMINS
1Theory Group, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-08012The Graduate University for Advanced Studies (Sokendai), 1-1 Oho, Tsukuba, Ibaraki 305-0801
(Received June 18, 2007; accepted August 1, 2007; published November 12, 2007)
The Higgs particle is a new elementary particle predicted in the Standard Model of the elementaryparticle physics. It plays a special role in the theory of mass generation of quarks, leptons, and gaugebosons. In this article, theoretical issues on the Higgs mechanism are first discussed, and thenexperimental prospects on the Higgs particle study at the future collider experiments, LHC and ILC, arereviewed. The Higgs coupling determination is an essential step to establish the mass generationmechanism, which could lead to a deeper understanding of particle physics.
KEYWORDS: particle physics, Higgs mechanism, mass generationDOI: 10.1143/JPSJ.76.111011
1. Introduction
Current understanding of the elementary particle physicsis based on two important concepts, gauge invariance andspontaneous symmetry breaking. Out of four fundamentalinteractions of Nature, namely strong, weak and electro-magnetic and gravity interactions, three of them exceptfor gravity are described on the same footing in terms ofgauge theory. The gauge group corresponding to the stronginteraction is SUð3Þ, and the weak and the electromagneticinteractions arise from SUð2Þ and Uð1Þ groups and are calledthe electroweak interaction. Once quarks and leptons areassigned in proper representations of the three gauge groups,all properties of the three fundamental interactions aredetermined from the requirement of gauge invariance.
For more than thirty years, high energy experiments havebeen testing various aspects of gauge symmetry and haveestablished gauge invariance as a fundamental principle ofNature. We have discovered gauge bosons mediating thethree interactions, namely gluon for the strong interaction, Wand Z bosons for the electroweak interaction. The couplingsbetween quarks/leptons and gauge bosons have beenprecisely measured at the CERN LEP and SLAC SLCexperiments, and we have confirmed the assignments of thegauge representations for quarks and leptons.
The gauge principle alone, however, cannot describethe known structure of the elementary particle physics. Inthe Standard Model of the elementary particle physics, allquarks, leptons and gauge bosons are first introduced asmassless fields. In order to generate masses for theseparticles, the SUð2Þ � Uð1Þ symmetries have to be brokenspontaneously.
Spontaneous symmetry breaking itself is not new forparticle physics.1,2) The theory of the strong interaction,QCD, possesses an approximate symmetry among three lightquarks called chiral symmetry. The vacuum of QCDcorresponds to a state where quark and anti-quark pair iscondensed, and the chiral symmetry is broken spontaneous-ly. As a consequence, pseudo scalar mesons such as pionsand kaons are light compared to the typical energy scale of
the strong interaction since they behave approximately asNambu–Goldstone bosons, a characteristic signature ofspontaneous symmetry breaking.
In the case of the electroweak symmetry, it is showntheoretically that the Nambu–Goldstone bosons associatedwith spontaneous breakdown are absorbed by gauge bosons,providing the mass generation mechanism for gauge bosons(Higgs mechanism).3) Although we are now quite sure thatthis is the mechanism for gauge boson mass generation, weknow little about how the symmetry breaking occurs, orwhat is dynamics behind the Higgs mechanism. Clearly, weneed a new interaction other than four known fundamentalforces, but we do not know what it is. The goal of the Higgsphysics is to answer this question.
In this article, I would like to explain what are theoreticalissues of the Higgs sector, what is expected at the futurecollider experiments, LHC and ILC, and what would beimpacts of the Higgs physics on a deeper understanding ofthe particle physics.
2. Higgs Boson in the Standard Model
In the Standard Model, a single Higgs doublet field isincluded for the symmetry breaking of the SUð2Þ � Uð1Þgauge groups. This was introduced in S. Weinberg’s 1967paper ‘‘A Model of Leptons’’,4) and is the simplestpossibility for generating the gauge boson masses.
The Higgs potential is given by
Vð�Þ ¼ ��2j�j2 þ �j�j4; ð1Þ
where the two component complex field is defined as
�ðxÞ ¼�ðxÞþ
�ðxÞ0
!: ð2Þ
In order for the stability of the vacuum the parameter � mustbe positive. The coefficient of the quadratic term, on theother hand, can be either sign. In fact, if the sign is negative,namely �2 > 0, the origin of the potential is unstable, andthe vacuum state corresponds to a non-zero value of the �
field. The states satisfying j�þj2 þ j�0j2 ¼ �2=ð2�Þ � v2=2are degenerate minimum of the potential. We can choose thevacuum expectation value in the �0 direction, h�0i ¼ v=
ffiffiffi2p
,and then there are three massless modes corresponding to the
flat directions of the potential (Nambu–Goldstone modes).When the symmetry is a gauge symmetry, these masslessparticles disappear from the physical spectrum, and becomelongitudinal components of massive gauge bosons. This isseen most clearly if we take the ‘‘Unitary gauge’’ where theNambu–Goldstone modes are removed by an appropriategauge transformation. The kinetic term of the scalar field isdefined as
jD��j2 ¼ @� þ g�a
2Wa� þ
g0
2B�
� ��
��������2
;
and the gauge boson mass terms are obtained by substitutingthe vacuum expectation value into �ðxÞ.
Mass terms of quarks and leptons are also generatedthrough interactions with the Higgs field. This follows fromthe chiral structure of quarks and leptons. Since thediscovery of the parity violation in the weak interaction,5)
chiral projected fermions (Weyl fermions) instead of Diracfermions have been considered as building blocks of aparticle physics model. In particular, only left-handed quarksand leptons are assigned as SUð2Þ doublets because the weakinteraction has a V(vector)–A(axial vector) current structure.Right-handed counterparts are singlet under the SUð2Þgauge group. This unbalance in the SUð2Þ quantum numberassignment forbids us to write direct mass terms for quarksand leptons: the only possible way to generate mass terms isto introduce Yukawa couplings with help of the � field suchas yd�
y �ddRqL where qL ¼ ðuL; dLÞT. After replacing �ðxÞby its vacuum expectation value, this term generates a massof ydv=
ffiffiffi2p
for down-type quarks. Similar mechanism worksfor up-type quarks and charged leptons.
There is one important prediction of this model. Since weintroduce a two-component complex field and three realdegrees of freedom are absorbed by gauge bosons, one scalarparticle appears in the physical spectrum, which is called theHiggs particle (� Higgs boson). In the Unitary gauge,interactions related to the Higgs boson can be obtained byreplacing v with vþ HðxÞ in the Lagrangian where HðxÞrepresents the Higgs boson. The mass of the Higgs boson isgiven by mH ¼
ffiffiffiffiffiffi2�p
v, which means that the Higgs bosonbecomes heavier if the Higgs self-coupling gets larger. Infact, this is a general property of the particle mass generationmechanism due to the Higgs field: A stronger interactionleads to a heaver particle. The mass formula for the W , Zbosons, quarks, leptons, and the Higgs boson at the lowestorder approximation with respect to coupling perturbation(i.e., tree-revel) are summarized in Table I.
3. Naturalness and Physics beyond the Standard Model
Although the Higgs potential in eq. (1) is very simple andsufficient to describe a realistic model of mass generation,we think that this is not the final form of the theory but ratheran effective description of a more fundamental theory. It is
therefore important to know what is limitation of thisdescription of the Higgs sector.
In renormalizable quantum field theories, the form ofLagrangian is specified by requirement for renormalizability.In the case of the Higgs potential, quadratic and quarticterms are only renormalizable interactions. We can thenconsider two kinds of corrections to the potential. One is acalculable higher order correction within the StandardModel. For instance the correction from the top Yukawacoupling constant can be evaluated up to a desired accuracyapplying renormalization procedure of field theory. Anothertype of corrections comes from outside of the present model,presumably from physics at some high energy scale. Wecannot really compute these corrections until we know themore fundamental theory. In this sense, the present theory isconsidered as an effective theory below some cutoff energyscale �.
Although the effective theory cannot include all physicaleffects, it is still useful because unknown correction isexpected to be suppressed by ðE=�Þ2 where E is a typicalenergy scale under consideration. Therefore, as long as thecutoff scale is somewhat larger than E, the theory can makefairly accurate predictions. For example, the correction is0ð10�4Þ when the cutoff scale is around 10 TeV for physicalprocesses in the 100 GeV range. If the theory is valid up tothe Planck scale (�1019 GeV) where the gravity interactionbecomes as strong as the other gauge interactions, thecorrection becomes extremely small. In this way, aneffective theory is useful description as long as we restrictourself to the energy regime below the cutoff scale.
Once we take a point of view that the Higgs sector ofthe Standard Model is an effective description of a morecomplete theory below �, naturalness with regard toparameter fine-tuning becomes a serious problem. Inparticular, the quadratic divergence of the Higgs massradiative correction is problematic, and this has been one ofmain motivations to introduce various models beyond theStandard Model.
In the Higgs potential in eq. (1) the only mass parameteris �2. At the tree level, this parameter is related to thevacuum expectation value v by �2 ¼ �v2 where v is knownto be about 246 GeV. (v ¼ ð
ffiffiffi2p
GF�1=2, where GF is theFermi constant representing the coupling constant of theweak interaction.) If we include the radiative correction, �2
becomes a sum of two contributions �20 þ ��2 where �2
0 is abare mass term and ��2 is the radiative correction. In theStandard Model, the top quark and gauge boson loopcorrections are important and ��2 from these sources arerepresented by a sum of terms of a form Ci½g2
i =ð4�Þ2��2
where gi is the top Yukawa coupling constant, or Uð1Þ orSUð2Þ gauge coupling constant and Ci are Oð1Þ coefficients.Since the radiative correction depends on the cutoff scalequadratically, the fine-tuning between the bare mass termand the radiative correction is necessary if the cutoff scale ismuch larger than 1 TeV. Roughly speaking, the fine-tuningat 1% level is necessary for � ¼ 10 TeV. If the cutoff scaleis close to the Planck scale, the degree of the fine-tuning isenormous: A tuning of one out of 1032 is required. This is thenaturalness problem of the Standard Model, and sometimealso called the hierarchy problem. This problem suggeststhat the description of the Higgs sector by the simple
Table I. Mass formula for elementary particles. g, g0, yf , and � are the
SUð2Þ and the Uð1Þ gauge coupling constants, the Yukawa coupling
constant for a fermion f , and the Higgs self-coupling constant.
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111011-2
potential in eq. (1) is not very satisfactory, and probablywill be replaced by a more fundamental form at a higherenergy scale.
Since the problem arises from the quadratic divergence inthe renormalization of the Higgs mass terms, proposedsolutions involve cancellation mechanism of such diver-gence. Supersymmetry6) is a unique symmetry that guaran-tees complete cancellation of the quadratic divergence inscalar field mass terms. This is a new symmetry betweenbosons and fermions and the cancellation occurs betweenloop diagrams of bosons and fermions. Particle physicsmodels based on supersymmetry such as supersymmetricgrand unified theory (SUSY GUT) have been proposed andstudied since early 1980’s as a possible way out of thehierarchy problem.7) In a realistic model of a supersymmet-ric extension of the Standard Model, we need to introducenew particles connected by supersymmetry to ordinaryquarks, leptons, gauge bosons, and Higgs particles. In1990’s, precision studies on the Z boson were preformed atLEP and SLC experiments, and it was pointed out that threeprecisely measured coupling constants are consistent withthe prediction of SUSY GUT, although the gauge couplingunification fails badly without supersymmetric partnerparticles.8) Supersymmetry is also an essential ingredientof the superstring theory, a potential unified theory includinggravity and gauge interactions. In this way, the super-symmetric model has become a promising candidate beyondthe Standard Model. If supersymmetry realizes at or justabove the TeV scale, it can provide a consistent and unifiedpicture of the particle physics from the weak scale to thePlanck scale.
An opposite idea for solution of the naturalness problem isconsidering that the cutoff scale is close to the electroweakscale. In particular, the Higgs field is considered to be acomposite state of more fundamental objects at a relativelylow energy scale. The simplest form of this model iscalled the technicolor model9) proposed in late 1970’s, inwhich the cutoff scale is about 1 TeV. The technicolor modelis however strongly constrained from precision tests ofelectroweak theory later at the LEP and SLC experiments,10)
but there have been continuous attempts to construct aphenomenologically viable model of a composite Higgsfield. Little Higgs models11) are a recent proposal on thisline, where the physical Higgs boson is dynamically formedby a new strong interaction around 10 TeV. An interestingfeature of this model is that the quadratic divergence ofthe Higgs boson mass term is canceled by loop correctionsdue to new gauge bosons and a heavy partner of thetop quark at one loop level. In this way the hierarchyproblem between the electroweak scale and 10 TeV is nicelysolved.
In addition to supersymmetry and little Higgs models,there have been many proposals for TeV scale physics.Motivations for many of them are solving the naturalnessproblem of the Standard Model or explaining the large(apparent) hierarchy between the weak scale and the gravityscale. Examples are models with large extra dimensions,12)
models with warped extra-dimensions,13) the Higgslessmodel,14) the twin Higgs model,15) and the inert Higgsmodel,16) the split-supersymmetry model,17) etc. All of theseproposals involve some characteristic signals around a TeV
region. These signals are important to choose a correctmodel at the TeV scale and clarify the mechanism of theelectroweak symmetry breaking.
4. Experimental Prospects of Higgs Physics
Higgs physics is expected to be the center of the particlephysics in coming years starting from the commissioningof the CERN LHC experiment. The first step will be adiscovery of a new particle which is a candidate of the Higgsboson. We then study its properties in detail and comparethem with the prediction of the Standard Model Higgsboson. We may be able to confirm that the discoveredparticle is the Higgs boson responsible for the massgeneration for elementary particles. Another possibilitywould be to find some deviation from the Standard ModelHiggs boson. Deviation could be something like smalldifference of production cross section and decay branchingratios from the Standard Model predictions, or more drasticnew signals such as discovery of several Higgs states. At thesame time, we may also find other new particles predicted inextensions of the Standard Model, for example supersym-metric particles in the supersymmetric model or the heavygauge bosons and the top partner in the little Higgs model.In order to accomplish these goals we probably need severalsteps in collider experiments including LHC and ILCexperiments and possible upgrades for these facilities.
If we restrict ourselves to the Higgs boson in the StandardModel, all physical properties are determined by oneparameter, the Higgs boson mass. Present experimentallower bound for the mass of the Standard Model Higgsboson is 114.4 GeV at the 95% confidence level, set by thedirect Higgs boson search at LEP.18) It is remarkable thatwe can also draw an upper bound from a global fit ofelectroweak precision data. Although a heavy Higgs bosonmeans a large self-coupling � , we have not seen anyevidence of such a large coupling in physical observablesrelated to Z and W gauge boson processes. The upper limitof the Standard Model Higgs boson is 166 GeV at the 95%confidence level.19) This implies that a relatively light Higgsboson is favored. If the Higgs boson turns out to be heaverthan 200 GeV, we would expect some additional newparticles that have significant couplings to gauge bosons.
The decay branching ratios of the Higgs boson dependsstrongly on the Higgs boson mass, and therefore thediscovery strategy for the Higgs boson at LHC differs forlight and heavy Higgs bosons. The branching ratios for theStandard Model Higgs boson is shown in Fig. 1. Since theHiggs boson couples more strongly to a heaver particle, ittends to decay to heaver particles as long as kinematicallyallowed. For instance, the Higgs boson mostly decays intotwo gauge bosons if the Higgs boson mass is larger than200 GeV, whereas the bottom and anti-bottom pair is themain decay mode for its mass less than 140 GeV. For thismass range, the Higgs boson search at LHC relies on otherdecay modes such as the loop-induced two photon decaymode, because two bottom modes are hidden by over-whelming QCD background processes. Detail simulationstudies on the Higgs discovery at LHC have been performed,and it is shown that the Higgs boson can be found at LHCexperiments within a few years for the entire mass region aslong as the production and decay properties are similar to the
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Standard Model Higgs boson.20,21) Furthermore, informationon the Higgs couplings is obtained with a higher luminosity.Estimated precision for coupling ratios are typically0(10)%.22)
ILC is a future electron–positron linear collider projectproposed in the international framework.23) One aspect ofthis facility is a Higgs factory. For instance, the number ofproduced Higgs bosons can be 0ð105Þ in the first stage ofexperiments with the center-of-mass collider energy of500 GeV. Under clean environment of the eþe� collider,precise determinations on the mass, quantum numbers, andcoupling constants of the Higgs boson are possible. Typicalproduction and decay processes are shown Fig. 2. Precisionof the coupling constant determination reaches a few % levelfor Higgs-WW , Higgs-ZZ, and Higgs-b �bb couplings for thecase of a relatively light Higgs boson. We can also measurethe Higgs self-coupling from the double Higgs bosonproduction process and the top Yukawa coupling from theHiggs-t�tt production. Figure 3 shows precision of the Higgscoupling constant determination for various particles at ILC.The proportionality between coupling constants and particlemasses is a characteristic feature of the one Higgs doubletmodel where the particle mass formulas involve only onevacuum expectation value. An important feature of ILCexperiments is that absolute values of these couplingconstants can be determined in a model-independent way.This is crucial in establishing the mass generation mecha-nism for elementary particles.
The precise determination of the Higgs coupling constantsis also useful to explore physics beyond the Standard Model.
In some case, the Higgs boson coupling is modified from theStandard Model.
. The Higgs sector of supersymmetric models is differentfrom the Standard Model. In any realistic supersym-metric model, the Higgs sector contains at lease twosets of doublet fields. In the minimal supersymmetricstandard model (MSSM), in particular, the Higgs sectoris a two Higgs doublet model. Furthermore, there is arather strict theoretical upper bound for the the lightestneutral Higgs boson,25) which is about 130 GeV. Sincethis light boson plays a role of the usual Higgs particle,this particle may be the only Higgs particle discoveredat LHC. In such case, the branching ratio measurementfor the lightest neutral Higgs boson is useful to obtaininformation on the masses of heavy Higgs bosons.26–28)
In particular, the tau and bottom coupling constantsshow sizable enhancement if the heavy Higgs bosonexists below 600 GeV. The ratio like BðH! WWÞ=BðH! ��Þ is useful to determine the heavy Higgsmass scale indirectly.
. In models with extra dimensions, there appears a scalarfield called Radion, corresponding to the size of theextra space dimension. Since Radion is a neutral scalarfield, it can mix with the Higgs field. It is pointed outthat Radion-Higgs mixing in the warped extra dimen-sion model could reduce the magnitude of Yukawacoupling constants and WWH and ZZH constant in auniversal way.29) In order to observe such effects,absolute coupling measurements at ILC are necessary.
. The two-gluon width of the Higgs boson is generatedby loop diagrams, so that it can be a probe to virtualeffects of new particles. The same is true for the two-photon width, the measurement of which is improved atthe photon–photon collider option of ILC.30) There aremany new physics models where such loop effects aresizable.
Fig. 1. Decay branching ratio of the Standard Model Higgs boson as a
function of its mass. c, b, t, and � represent charm, bottom, top quarks,
and tau lepton. �, g, W , and Z are photon, gluon, W , and Z bosons.
e
e
Z
Z
H
H
+
−
Fig. 2. Production process of the Higgs boson at ILC (left) and Higgs
boson decay to fermions (right).
1 100Mass (GeV)
0.01
0.1
1
Cou
plin
g co
nsta
nt to
Hig
gs b
oson
(κ ι)
CouplingMass Relation
c τ
b
W Z
Ht
10
Fig. 3. Precision of the coupling-constant determination for various
particles at ILC with the integrated luminosity of 500 fb�1. The Higgs
boson mass is taken to be 120 GeV. For charm, tau, bottom, W , and Z
coupling measurement,ffiffisp¼ 300 GeV is assumed.
ffiffisp¼ 500 GeV
(700 GeV) is taken for the triple Higgs ðt�ttHÞ coupling measurement.24)
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. Explaining the baryon number of the Universe is oneof most outstanding questions for particle physics inconnection with cosmology.31) One possibility is theelectroweak baryogenesis scenario, in which the baryonnumber was generated at the electroweak phasetransition. For a successful electroweak baryogenesis,the Higgs sector has to be extended from that of theminimal Standard Model to realize a strong first-orderphase transition. The change of the Higgs potential canlead to observable effects in the triple Higgs couplingmeasurement.32,33)
As we can see above examples, observations of newphysics effects require precise determination of couplingconstants. This will be an important goal of the future ILCexperiment.
5. Conclusions
The Higgs sector is an unknown part of the particlephysics model. Although a simple potential is assumed in theStandard Model, this description is supposed to be validbelow some cutoff scale, beyond which the theory of theelectroweak symmetry breaking takes a more fundamentalform. If the cutoff scale is as low as 1 TeV, some directsignals on new physics are likely to appear at LHC. If thecutoff scale is much larger, the fine-tuning of the Higgsboson mass term becomes a serious problem. Proposedsolutions to this problem such as supersymmetry or littleHiggs models also predict new physics signals at the TeVscale. These signals are targets of future collider experimentsstarting from LHC.
Experimental prospects for the Higgs physics are quitebright. The Higgs particle can be found and studied at LHC.At the proposed ILC, precise information on couplingconstants between the Higgs boson and other particleswill be obtained. These measurements are an essential stepto establish the mass generation mechanism. At the sametime, the precision measurement may reveal evidence ofnew force and/or new symmetry because these new physicsis most probably related to the physics of electroweaksymmetry breaking, i.e., the Higgs sector. In this way, theHiggs particle will play a special role in determining thefuture direction of the particle physics.
Acknowledgment
This work was supported in part by Grants-in-Aid forScience Research, Ministry of Education, Culture, Sports,Science and Technology Nos. 16081211 and 17540286.
1) Y. Nambu and G. Jona-Lasinio: Phys. Rev. 122 (1961) 345.
2) J. Goldstone: Nuovo Cimento 19 (1961) 154.
3) P. W. Higgs: Phys. Lett. 12 (1964) 132; P. W. Higgs: Phys. Rev. Lett.
13 (1964) 508; P. W. Higgs: Phys. Rev. 145 (1966) 1156.
4) S. Weinberg: Phys. Rev. Lett. 19 (1967) 1264.
5) T. D. Lee and C. N. Yang: Phys. Rev. 104 (1956) 254.
6) J. Wess and B. Zumino: Nucl. Phys. B 70 (1974) 39.
7) N. Sakai: Z. Phys. C 11 (1981) 153; S. Dimopoulos and H. Georgi:
Nucl. Phys. B 193 (1981) 150.
8) P. Langacker and M. X. Luo: Phys. Rev. D 44 (1991) 817; U. Amaldi,
W. de Boer, and H. Furstenau: Phys. Lett. B 260 (1991) 447.
9) L. Susskind: Phys. Rev. D 20 (1979) 2619.
10) M. E. Peskin and T. Takeuchi: Phys. Rev. Lett. 65 (1990) 964;
M. Golden and L. Randall: Nucl. Phys. B 361 (1991) 3; B. Holdom
and J. Terning: Phys. Lett. B 247 (1990) 88.
11) N. Arkani-Hamed, A. G. Cohen, E. Katz, A. E. Nelson, T. Gregoire,
and J. G. Wacker: J. High Energy Phys. JHEP08 (2002) 021;
N. Arkani-Hamed, A. G. Cohen, E. Katz, and A. E. Nelson: J. High
Energy Phys. JHEP07 (2002) 034.
12) N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali: Phys. Lett. B 429
(1998) 263; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and
G. R. Dvali: Phys. Lett. B 436 (1998) 257; N. Arkani-Hamed,
S. Dimopoulos, and G. R. Dvali: Phys. Rev. D 59 (1999) 086004.
13) L. Randall and R. Sundrum: Phys. Rev. Lett. 83 (1999) 3370.
14) C. Csaki, C. Grojean, H. Murayama, L. Pilo, and J. Terning: Phys.
Rev. D 69 (2004) 055006; C. Csaki, C. Grojean, L. Pilo, and
J. Terning: Phys. Rev. Lett. 92 (2004) 101802.
15) Z. Chacko, H. S. Goh, and R. Harnik: Phys. Rev. Lett. 96 (2006)
231802.
16) R. Barbieri, L. J. Hall, and V. S. Rychkov: Phys. Rev. D 74 (2006)
015007.
17) N. Arkani-Hamed and S. Dimopoulos: J. High Energy Phys. JHEP06
(2005) 073.
18) R. Barate et al. (LEP Working Group for Higgs Boson Searches):
Phys. Lett. B 565 (2003) 61.
19) J. Alcaraz et al. (LEP Collaborations and the LEP Electroweak
Prospects of Physics Beyond the Standard Model: Supersymmetry
Masahiro YAMAGUCHI�
Department of Physics, Tohoku University, Sendai 980-8578
(Received July 9, 2007; accepted July 24, 2007; published November 12, 2007)
Despite the success of the standard model of particle physics, there are reasons to expect new physicsbeyond it. Supersymmetric extension is a promising possibility, which solves the fine-tuning probleminherent to the origin of the electroweak scale in the standard model. It also provides a promisingcandidate for dark matter. The verification of low-energy supersymmetry in forthcoming experimentswill lead us to a new paradigm of particle physics, in particular, it may open up the road to the unificationof forces.
KEYWORDS: particle physics, standard model, supersymmetryDOI: 10.1143/JPSJ.76.111012
1. Introduction
Although the standard model (SM) of particle physicssuccessfully describes particle interactions at an energy scalebelow 100 GeV, there are some reasons to anticipate theexistence of new physics beyond it. First, there is the fine-tuning problem (or naturalness problem) associated withthe Higgs field. Second, the SM does not provide anycandidate for the dark matter of the universe. Third, from theviewpoint of attempting to unify all forces and matter, theSM appears to be very far from its goal; the gauge structureas well as the matter representation are complicated, andthere are many parameters, i.e. the gauge coupling constantsand Yukawa coupling constants, in the Lagrangian, whichare obtained experimentally. Related to this, the SM does notincorporate gravitational interaction.
Supersymmetry (SUSY) is a proposed symmetry betweenbosons and fermions, which solves the aforementionedfine-tuning problem of the Higgs boson mass. For everyelementary particle in the SM, there is a correspondingsuperpartner, which differs in spin by 1/2 and obeysopposite statistics. The interactions of particles and theirsuperpartners are related to each other. In the supersym-metric extension of the SM, quadratic divergence, whichwould destabilize the electroweak scale and thus requirefine-tuning, is absent due to the cancellation betweenbosonic contributions and fermionic contributions. Thus,it is a promising, and probably the most widely studied,candidate for physics beyond the SM. It is one of themain targets of the forthcoming experiment, using the LargeHadron Collider (LHC), and one expects its discovery in thenear future. Furthermore supersymmetry will shed light onthe other shortcomings of the SM mentioned above. We willdiscuss these issues in this article.
2. The Fine-Tuning Problem
Strong, weak, and electromagnetic interactions are de-scribed by gauge symmetry in the SM. In fact, electro-magnetism is a typical example of gauge theory, where themassless gauge boson, the photon, mediates the interaction.Weak gauge bosons, W� and Z, become massive as a
consequence of the spontaneous symmetry breakdown ofthis gauge symmetry, i.e., SUð3ÞC � SUð2ÞL � Uð1ÞY downto SUð3ÞC � Uð1Þem. In the SM, this is caused by thepostulated Higgs scalar field.
The scalar potential of the Higgs field, H, is given as
VðHÞ ¼ m2HHyH þ �ðHyHÞ2; ð1Þ
where � is the self-coupling of the Higgs field and m2H is a
mass parameter. The Higgs field develops a non vanishingvacuum expectation value (VEV) and the symmetry isspontaneously broken when m2
H < 0. It is easy to see that theVEV of the Higgs field is hHi ¼
and its mass isffiffiffiffiffiffiffiffiffiffiffi�m2
H
p. Thus, to obtain the correct weak scale of the order
of 100 GeV, this massffiffiffiffiffiffiffiffiffijm2
H jp
should also have the sameorder of magnitude.
The couplings of the Z boson and those of the W boson toquarks and leptons have been precisely measured in the LEPand Tevatron experiments, and are in perfect agreement withthe SM prediction. Furthermore, the SM with a Higgs massof less than roughly 200 GeV is consistent with the electro-weak precision data.
Despite the success of the SM, it suffers from a fine-tuningproblem. This is attributed to the fact that the SM Higgs fieldis an elementary scalar field. The radiative correction to thesquare of the mass of such a scalar field is known to sufferfrom quadratic divergence, and thus it is proportional to thesquare of the UV cut off �. Schematically, it is written
m2H;phys ¼ m2
H;bare þc
16�2�2; ð2Þ
where mH;phys is the physical mass of order 100 GeV, whilemH;bare is the mass parameter given in the bare Lagrangian.The second term of the right-hand side is the divergentquantum correction with c being a number related tocoupling constants. It is plausible that the UV cut off � isaround the Planck scale of 1018 GeV, the energy scale atwhich the quantum effects of the gravitational interactionsbecome important, and presumably the description based onquantum field theory is no longer valid. In this case, a hugeradiative correction proportional to a power of � has to becanceled by the bare mass parameter with a precision of10�18 to obtain the correct electroweak scale of 100 GeV.It is argued that such a fine tuning is unnatural, and thereshould be some way of solving this difficulty.
Supersymmetry is a symmetry that relates bosons withfermions. For the supersymmetry charges Q� (� ¼ 1, 2) and�QQ _�� ( _�� ¼ 1, 2), in terms of two-component spinor notation,
the supersymmetry algebra is represented as
fQ�; �QQ _��g ¼ 2��� _��P�: ð3Þ
Here P� (� ¼ 0, 1, 2, 3) on the right-hand side generatesthe four-dimensional space–time translation, and �� is thePauli matrix generalized to four dimensions, �� ¼ ð�1; �Þ.Intuitively, the supersymmetry transformation is a squareroot of the translation; by applying the supersymmetrytransformation twice, one obtains the space–time translation.A closer look at the Lorentz structure suggests that thesupersymmetry charge Q� should behave as a spinor, andthus should obey the Fermi–Dirac statistics. This is thereason why the above is written in the form of an anti-commutation relation, instead of a commutation relation.When Q� is applied, bosons are transformed into fermionsand vice versa.
Supersymmetry was discovered in the two-dimensionalworld sheet formulation of fermionic string theory.1) Thefour-dimensional Lagrangian formalism was invented byWess and Zumino.2) One of the novel properties of super-symmetry is that it is the maximum symmetry that containsthe space–time Poincare symmetry (i.e., the symmetry gen-erated by translations and Lorentz transformations), and isconsistent with the desired properties of the S-matrix.
It is instructive to illustrate the simplest four-dimensionalsupersymmetric model given by Wess and Zumino. It has acomplex scalar field �, a two component Weyl spinor , anda complex auxiliary field F. Here we introduce the auxiliaryfield F simply to make the supersymmetry transparent. Aswe will see, it does not have a physical degree of freedom,but can be eliminated, using the equation of motion. Thesupersymmetry transformation acts on these fields as
� ¼ffiffiffi2p ;
¼ iffiffiffi2p�� �@��þ
ffiffiffi2pF; ð4Þ
F ¼ iffiffiffi2p
� ����@� :
As anticipated, under the supersymmetry, a boson and afermion form a pair and transform into each other, exactly asunder the isospin symmetry, a proton and a neutron are inthe same multiplet and transform into each other. Further-more, one can see that the number of bosonic degrees offreedom is the same as that of fermions.
The interaction of the Wess–Zumino model is determinedby a single function called superpotential
Wð�Þ ¼1
2m�2 þ
1
3��3; ð5Þ
where m represents the mass and � is the coupling constant,as we will see shortly. � is the superfield which contains �, , and F in a compact form. The Lagrangian of the Wess–Zumino model is written
L ¼ �@���@��þ i � ����@� þ F�F
� F@Wð�Þ@��
1
2
@2Wð�Þ@�2
þ h.c. ð6Þ
After eliminating the auxiliary field with the help of itsequation of motion, one obtains
L ¼ �@���@��þ i � ��@�
� m2���� m�ð��2�þ ���2Þ � �2ð���Þ2
�1
2m � �� þ h.c. ð7Þ
It is straightforward to see that the Lagrangian in eq. (6) isinvariant under the supersymmetry transformation up to thetotal derivative, which is irrelevant when it is integrated overthe whole four-dimensional space–time to obtain the action.
There are two important features that one can immediatelysee from the Lagrangian in eq. (7). First, the mass of theboson, m, is the same as that of its fermionic superpartner.Second, the coupling constant of the scalar quartic inter-action, �2, is related to the Yukawa coupling, � , of scalar-spinor-spinor interaction. These two novel properties arecrucial in supersymmetry, and apply when more than onematter field is introduced and when gauge symmetry isconsidered.
The possible connection of supersymmetry with the realworld was recognized in the early 80 s,3) as a solution to thefine-tuning problem discussed above. Here we illustrate howthe supersymmetry cancels the quadratic divergence of thescalar mass correction in the Wess–Zumino model.
In quantum field theory, all quantum fluctuations con-tribute to physical quantities. The quantum corrections tothe square of the scalar mass in the Wess–Zumino modelinclude the contribution where virtual states are bosons(boson loop) and also the contribution where virtual statesconsist of fermions (fermion loop). Each of the contributionsis quadratically divergent, as we discussed in the previoussection. In fact, the two contributions are the same inmagnitude. However, because of the difference in statistics,the sign of the fermion loop contribution is opposite tothat of the boson loop contribution, and thus the quadraticdivergences of the two contributions cancel with each other,resulting in at most rather harmless logarithmic divergence.This intuitive argument is further supported by the morerigorous non-renormalization theorem, which states that thesuperpotential is not renormalized at all orders in perturba-tive expansions. The radiative corrections to the scalar masssolely originate from the wave function renormalization,which has at most logarithmic divergence. The non-renormalization theorem is valid for more complicatedmodels with more than one matter field and/or with gaugesymmetry. It is thus the basis of the argument that thesupersymmetric SM solves the fine-tuning problem associ-ated with the Higgs boson mass.
4. Supersymmetry Phenomenology
As was discussed above, supersymmetry is a veryattractive candidate for physics beyond the SM, since itsolves the fine-tuning problem of the Higgs boson mass. Thesimplest supersymmetric extension is the minimal super-symmetric standard model (MSSM). Quarks and leptons inthe SM are accompanied by their bosonic superpartnerscalled squarks and sleptons. Associated with the gaugebosons corresponding to the SM gauge group SUð3ÞC �SUð2ÞL � Uð1ÞY , there exist their fermionic superpartners,
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gauginos. For the Higgs sector, the MSSM has two Higgsdoublets with opposite hypercharge, and their superpartners,higgsinos. The extension of the Higgs sector is required tocancel the gauge anomaly and also to describe the Yukawainteractions of the quarks and leptons in the supersymmetricway. Particles in the MSSM are summarized in Table I.
One of the factors that favors low-energy supersymmetrywhen supersymmetry manifests itself down to the energyscale close to the electroweak scale is the argument ofcoupling unification. The gauge coupling constants run,namely, they change when the energy scale changes. Howthey run depends on the particle contents. Using theprecisely measured values for the gauge coupling constantsas the initial conditions and the renormalization groupequations for the particle contents of the MSSM, the threegauge coupling constants meet at one energy scale of1016 GeV.4) This remarkable result may be interpreted as oneof the first pieces of evidence for supersymmetry grandunification, in which the three gauge groups are unified intoa single group such as SUð5Þ or SOð10Þ.5) This fact waswidely recognized in the early 90 s. Together with the factthat most other proposals beyond the SM fail to reconcile theprecision electroweak data,6) the low-energy supersymmetryhas been enthusiastically anticipated as a prime candidate forphysics beyond the SM.
With unbroken supersymmetry, a boson and a fermionin the same supersymmetry multiplet have the same mass.This is in a direct contradiction with experimental facts.For instance, the existence of a superpartner of the electronwith the same mass as the electron is clearly excluded. Thisimplies that the supersymmetry is broken in some way toallow sufficiently heavy superparticle masses.
Supersymmetry breaking and its mediation is one of thecentral issues of supersymmetry phenomenology, i.e., effortsto connect the supersymmetry to the real world. Introducingsupersymmetry-breaking masses does not generically affectits high-energy behavior; thus the absence of the quadraticdivergence is maintained, whereas it would be spoiled ifsupersymmetry was violated in the coupling relations. Suchdesirable masses that do not induce quadratic divergenceare classified as soft supersymmetry-breaking masses. It isinteresting to mention that all superpartners that have notyet been discovered can naturally become heavy withoutgenerating the unwanted quadratic divergence.
Supersymmetry is, on the other hand, probably a funda-mental symmetry of nature. It is thus natural to consider thatit is a local symmetry, in the same spirit as gauge symmetry.Local supersymmetry is called supergravity; it inevitably
includes gravity because the gauge field associated with thelocal supersymmetry is a spin 3/2 field and its superpartneris the graviton with spin 2. The gauge field for the localsupersymmetry is a fermion, called a gravitino.
In supergravity, the aforementioned soft supersymmetry-breaking masses ought not to be given by hand, but shouldarise as a consequence of the spontaneous breaking. In factwhen supersymmetry is spontaneously broken in a sectordifferent from the MSSM sector, it can be mediated to theMSSM sector, resulting in the soft supersymmetry-breakingmasses.7) Different mediation mechanisms of supersymme-try breaking yield different superparticle mass spectra.Therefore, we can aim to discriminate between these spectrain future experiments when supersymmetry is discovered.There are some requirements, in general, that a successfulmediation mechanism should satisfy. Firstly, the massesshould not be very far from the electroweak scale, i.e., of theorder 100 GeV. Second, the soft masses are chosen such thatthey do not cause too large flavor changing neutral current(FCNC). Third, the lightest superparticle (LSP) should beneutral when the R-parity is conserved and the LSP is stable.Furthermore the LSP can constitute (at least partly) the darkmatter of the universe so that its expected abundance shouldnot exceed the observed value of dark matter abundance.
The first of the three requirements is obvious becauseotherwise we would lose the very motivation to induce theweak scale supersymmetry in order to solve the fine-tuningproblem of the Higgs mass. This results in our expectationthat forthcoming energy frontier experiments such as theLHC8) will reveal this secret of nature by discovering thesuperparticles directly.
The second requirement, concerning the suppression ofFCNC, requires further explanation. Here, let us firstconsider how the FCNC is suppressed in the SM. In theSM, the flavor (or generation) mixing processes occur due tothe mismatch between the mass eigenstates and the weak-interaction eigenstates in the quarks. In fact, the FCNCcontribution is absent at the lowest order in the perturbationexpansion, and furthermore, it is suppressed by small quarkmasses and/or small generation mixings of quark massmatrices. This suppression mechanism is known as theGlashow–Illiopoulos–Maiani (GIM) mechanism. In a sense,it suppresses FCNC much more strongly than would benaively expected. When we consider, on the other hand,some extension of the SM, the GIM suppression mechanismwould no longer be operative so that the resulting FCNCwould exceed the experimental bounds by a few orders ofmagnitude. The supersymmetric extension of the SM suffersfrom this generic difficulty. Many mediation mechanismshave been proposed to solve this problem. The first andprobably the most popular approach is gravity mediation,9)
where non-renormalizable interaction suppressed by Planckmass of 1018 GeV mediates the supersymmetry breaking thathas taken in a hidden sector. This mediation mechanism iseconomical because such a non-renormalizable interactionis inevitable in supergravity. The suppression of the FCNCis not automatic in this case. For instance, the minimalsupergravity assumes the universal scalar masses for thesquarks and sleptons (at a high-energy scale where the softterms are given), although the justification of this assump-tion is somewhat unclear. In gauge mediation,10) super-
Table I. Particles in the MSSM.
Fermions Bosons
q quark ~qq squark
l lepton ~ll slepton
~HH1;2 higgsino H1;2 Higgs boson
~gg gluino g gluon
~WW� wino W� W boson
~ZZ zino Z Z boson
~�� photino � photon
~GG gravitino G�� graviton
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symmetry breaking is mediated to messenger (s)quarks and(s)leptons, and then transmitted to the MSSM sector via theSM gauge interaction. In this case, the squarks and sleptonswith the same gauge quantum numbers have generation-independent masses, guaranteeing the suppression of theFCNC contribution from supersymmetric particles. Anothercompelling mechanism is anomaly mediation11) where thesupersymmetry breaking is transfered to superconformalanomaly. In this case, the soft masses are proportional to thebeta functions. This mechanism is elegant, but suffers fromtachyonic masses for the sleptons because both SUð2ÞL andUð1ÞY are asymptotic non-free with positive beta functions,yielding negative contributions to the squares of the sleptonmasses. To circumvent this problem, one has to add anothersource of supersymmetry mediation. Recently, a string-inspired model has been considered, which exhibits anadmixture of the anomaly mediation and the gravitymediation (or the moduli mediation) of supersymmetrybreaking.12) It turns out that this solves the tachyonic sleptonmass problem.
Given the soft supersymmetry-breaking mass parametersat the high-energy scale, the physical masses carry a lotinformation at the mediation scale down to the electroweakscale. Thus the measurements of the superparticle masses areexpected to provide a lot of information on high-energyphysics, including the gauge structure at the high-energyscale as well as the flavor structure.
It is interesting to mention that the FCNC contributionfrom supersymmetry may be marginally suppressed tosatisfy the present experimental constraints but may reap-pear in future experiments. In particular, lepton flavorviolation processes are promising since the SM prediction isnegligibly small, even if small neutrino masses and mixingare taken into account. Here, we would like to point out thata forthcoming experiment called MeG experiment13) tosearch for �! e� will start soon, and it has the potential toreveal the nature of new physics within a few years.
Searches for superparticles in energy frontier experimentshave been carried out for a long time. Null results placelower bounds on the masses of superparticles.14) Gluino andsquark searches at Tevatron constrain their masses to beingheavier than about 300 GeV, but much of the interestingparameter region has not been explored yet. The forth-coming LHC experiment has much greater discoverypotential; its reach of the gluino mass is far above 1 TeV(¼ 103 GeV). One should also emphasize that the detailedstudy of the properties of the superparticles including theprecise determination of their masses and couplings isequally important as the discovery of the superparticles. Alepton collider with, e.g., eþe� collisions is suitable forthis purpose and the International Linear Collider (ILC) hasbeen proposed. More details on the search for and studyof superparticles at the LHC as well as at the ILC will bediscussed elsewhere in this volume.
Another important prediction based on supersymmetry,in particular that of the MSSM, is that there exists at leastone light Higgs boson whose mass is not far from the Z
boson mass. This is because the quartic couplings of theHiggs potential are related to the gauge coupling constantsin the MSSM. In fact, it was shown that, at the tree level,the lightest Higgs boson is lighter than the Z boson.15)
It was recognized some time ago that a large quantumcorrection comes from top-stop contributions, which thisbound.16) Approximately, the lightest Higgs boson mass, mh
is bounded as
m2h . m2
Z þ3GFm
4tffiffiffi
2p�2
lnm2
~tt
m2t
: ð8Þ
The first term on the right-hand side is the classicalcontribution and the second term represents the quantumcorrection. mt is the top mass, m~tt is the stop mass, and GF
denotes the Fermi constant. As a result of this large quantumcorrection, the Higgs mass can easily exceed the tree-levelbound of mZ ’ 91 GeV and it can be as large as about130 GeV. Although this upper bound for the Higgs bosonmass in the MSSM is beyond the capability of the Higgssearch at LEP, it is still well within the reach of the LHCexperiment. Thus, the Higgs boson search at the LHC willprovide us with a clue to low-energy supersymmetry.
5. Connection with Cosmology
At this point, we briefly mention a possible connection ofthe supersymmetric models with cosmology.
In supersymmetry, R-parity is assigned to be even forordinary particles that already exist in the SM and oddfor their superpartners. The conservation of the R-parity isusually assumed since it forbids very rapid proton decaycaused by dimension four operators. In this case, the LSP isstable and thus it should be colorless and neutral (withrespect to electric charge). Among the superparticles in theMSSM, the lightest of the neutralinos often becomes theLSP in many supersymmetric models. Here a neutralino is alinear combination of neutral gauginos, ~�� ~ZZ, and neutralhiggsinos, ~HH0
1 and ~HH02 . The lightest neutralino, if it is the
LSP, is in the category of weakly interacting massiveparticles (WIMP) and is indeed a good candidate for the darkmatter of the universe.17) What we know about dark matter ismerely the necessity of non-baryonic dark matter, whichconstitutes about 1/4 of the total energy of the universe. Thedirect study of the superparticles at the LHC and ILC andvarious searches for dark matter will enable us to identify thenature of dark matter and hopefully understand the earlyhistory of the universe such as when and how the dark matterwas produced and remains until today.
Another cosmological implication of low-energy super-symmetry is related to gravitino production in the earlyuniverse.18) As was mentioned earlier, the gravitino is asuperpartner of the graviton, and thus its interaction is veryweak. If, for instance, the gravitino is not the LSP, then it isinevitably unstable but its lifetime is relatively long. For agravitino mass of the order 1 TeV, the lifetime exceedsone second, and the decay takes place after the big-bangnucleosynthesis (BBN) commences. The gravitino abun-dance at its decay must be severely constrained so as not tospoil the success of the BBN, one of the triumphs of standardcosmology. It is known that the gravitinos are produced byscattering in the thermal bath immediately after the reheatof primordial inflation. Because the yield of the gravitinoproduced this way is proportional to the reheat temperature,the BBN constraint places a severe upper bound on thereheat temperature.19) Furthermore, it was recently pointedout that scalar fields, including inflaton, generically decay
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS M. YAMAGUCHI
111012-4
into gravitino pairs with significant branching ratios.20) Thisplaces yet another severe constraint on the scenarios ofinflation, baryogenesis, and also the mechanism of super-symmetry breaking and mediation.
6. Road to Unification
If low-energy supersymmetry is realized in nature, we willhave a chance to approach the physics of the unificationof forces and matter. In fact, as was mentioned earlier, thethree running coupling constants of the SM gauge inter-actions, SUð3ÞC � SUð2ÞL � Uð1ÞY , meet at one energy scaleat around 1016 GeV if they follow the renormalizationgroup flow of the MSSM. This strongly implies that thegrand unification of forces is achieved at this scale. This is atypical example how to probe physics at the high-energyscale by studying the relations among the parameters inthe Lagrangian. Another example is given by the relationbetween the bottom quark mass and the tau lepton mass.The observed value of this ratio is consistent with thatimplied by supersymmetric grand unification where quarksand leptons are also unified. In supersymmetry, there are newsupersymmetry-breaking mass parameters, such as gauginomasses and squark/slepton masses. They will provide us withcrucial hints on physics at the very high energy scale, andhopefully on grand unification. The flavor violation proc-esses such as �! e� also provide important information onthe structure of generation mixing in high-energy physics.
Another way to probe physics around the grand unifiedscale is to seek rare processes. Proton decay is a typicalexample of this approach. Proton decay such as p! eþ�does not occur within the SM because the baryon number isaccidentally conserved in the SM at the level of renorma-lizable interaction. This is no longer the case when weextend the SM to grand unified theories, where the baryonnumber is not conserved and the exchange of particles withGUT scale mass causes the proton decay. According to theuncertainty principle of quantum mechanics, the transitionprobability of the process with such an extremely energeticvirtual state is highly suppressed, which results in theextreme longevity of the proton. Searches for proton decayhave been made, which have already excluded many grandunified models. The search for this rare process in experi-ments at present or in the next generation experiments mayreveal evidence of grand unification.
As we mentioned in the introduction, the SM does notinclude gravity. At present, superstring theory is a promisingcandidate for ultimate unified theory including gravity.Supergravity is then recognized as a low-energy effectivetheory of the superstring. The experimental verification ofsupersymmetry as well as theoretical developments of thesuperstring theory will shed new light on the understandingthe physics around or beyond the Planckian scale.
1) P. Ramond: Phys. Rev. D 3 (1971) 2415; A. Neveu and J. H. Schwarz:
Nucl. Phys. B 31 (1971) 86; J.-L. Gervais and B. Sakita: Nucl. Phys. B
34 (1971) 632.
2) J. Wess and B. Zumino: Nucl. Phys. B 70 (1974) 39.
3) L. Maiani: in Proc. Gif-sur-Yvette Summer School (Paris, 1980); E.
Witten: Nucl. Phys. B 188 (1981) 513; M. Veltman: Acta. Phys. Pol. B
12 (1981) 437.
4) P. Langacker and M. X. Luo: Phys. Rev. D 44 (1991) 817; U.
Amaldi, W. de Boer, and H. Furstenau: Phys. Lett. B 260 (1991)
447; J. R. Ellis, S. Kelley, and D. V. Nanopoulos: Nucl. Phys. B 373
(1992) 55.
5) S. Dimopoulos and H. Georgi: Nucl. Phys. B 193 (1981) 150; N.
Sakai: Z. Phys. C 11 (1981) 153.
6) M. E. Peskin and T. Takeuchi: Phys. Rev. Lett. 65 (1990) 964.
7) J. Polchinski and L. Susskind: Phys. Rev. D 26 (1982) 3661; L. Hall, J.
Lykken, and S. Weinberg: Phys. Rev. D 27 (1983) 2359.
8) LHC home page (http://lhc.web.cern.ch/lhc/).
9) A. Chamseddine, R. Arnowitt, and P. Nath: Phys. Rev. Lett. 49 (1982)
970; R. Barbieri, S. Ferrara, and C. A. Savoy: Phys. Lett. B 119 (1982)
343; N. Ohta: Prog. Theor. Phys. 70 (1983) 542; L. Hall, J. Lykken,
and S. Weinberg: Phys. Rev. D 27 (1983) 2359.
10) M. Dine and A. E. Nelson: Phys. Rev. D 48 (1993) 1277; M. Dine,
A. E. Nelson, and Y. Shirman: Phys. Rev. D 51 (1995) 1362; M. Dine,
A. E. Nelson, Y. Nir, and Y. Shirman: Phys. Rev. D 53 (1996) 2658.
11) L. Randall and R. Sundrum: Nucl. Phys. B 557 (1999) 79; G. F.
Giudice, R. Rattazzi, M. A. Luty, and H. Murayama: J. High Energy
Phys. JHEP12 (1998) 027.
12) K. Choi, A. Falkowski, H. P. Nilles, M. Olechowski, and S. Pokorski:
J. High Energy Phys. JHEP11 (2004) 076; M. Endo, M. Yamaguchi,
and K. Yoshioka: Phys. Rev. D 72 (2005) 015004; K. Choi, K. S.
Jeong, and K. i. Okumura: J. High Energy Phys. JHEP09 (2005) 039.
13) MeG home page (http://meg.web.psi.ch/).
14) W. M. Yao et al. (Particle Data Group): J. Phys. G 33 (2006) 1.
15) K. Inoue, A. Kakuto, H. Komatsu, and S. Takeshita: Prog. Theor.
Phys. 67 (1982) 1889.
16) Y. Okada, M. Yamaguchi, and T. Yanagida: Prog. Theor. Phys. 85
(1991) 1; J. R. Ellis, G. Ridolfi, and F. Zwirner: Phys. Lett. B 257
(1991) 83; H. E. Haber and R. Hempfling: Phys. Rev. Lett. 66 (1991)
1815.
17) H. Goldberg: Phys. Rev. Lett. 50 (1983) 1419; J. R. Ellis, J. S.
Hagelin, D. V. Nanopoulos, K. A. Olive, and M. Srednicki: Nucl.
Phys. B 238 (1984) 453.
18) S. Weinberg: Phys. Rev. Lett. 48 (1982) 1303.
19) For a recent analysis, see, e.g., M. Kawasaki, K. Kohri, and T. Moroi:
Phys. Rev. D 71 (2005) 083502.
20) M. Endo, K. Hamaguchi, and F. Takahashi: Phys. Rev. Lett. 96 (2006)
211301; S. Nakamura and M. Yamaguchi: Phys. Lett. B 638 (2006)
389; M. Kawasaki, F. Takahashi, and T. T. Yanagida: Phys. Lett. B
638 (2006) 8; T. Asaka, S. Nakamura, and M. Yamaguchi: Phys. Rev.
D 74 (2006) 023520.
Masahiro Yamaguchi was born in Tsuruoka,
Japan, in 1963. He obtained his B. Sc. (1985),
M. Sc. (1987), and D. Sc. (1990) from the Univer-
sity of Tokyo. He was a research associate (1992–
1995), an associate professor (1995– 2003), and is
currently a professor (2003–) at Department of
Physics, Tohoku University. He has worked on
elementary particle theory, in particular physics
beyond the standard model and connection with
cosmology.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS M. YAMAGUCHI
Department of Physics, University of Tokyo, Tokyo 113-0033
(Received March 30, 2007; accepted September 5, 2007; published November 12, 2007)
Physics potential of the LHC (Large Hadron Collider) is summarized in this report, focusing especiallyon two major topics, the Higgs boson and Supersymmetry. ATLAS and CMS collaborations haveexcellent potential to discover them, if they exist at the mass scale less than about 1 and 3 TeV,respectively. Methods and their expected performances, to determine the properties of these newparticles, are summarized.
The most urgent and important topics of contemporaryparticle physics are (1) to understand the origin of ‘‘Mass’’(the Electroweak symmetry braking) and (2) to discover ofthe physics beyond the Standard model (SM). These are themain purpose of the Large Hadron Collider (LHC),1) inwhich two protons collide with the center-of-mass energy of14 TeV. The first physics collision is expected in the summerof 2008 with the lower luminosities of about 1032 cm�2 s�1.The design luminosity of 1034 cm�2 s�1, which correspondsto 100 fb�1 per year, will be achieved within several years.
The production cross-sections of the various high pT andhigh mass elementary processes are expected to be large atLHC, since gluon inside protons can contribute remarkably.Furthermore, because the LHC provides the high luminosityof 10 –100 fb�1 per year, large numbers of the interestingevents will be observed as summarized in Table I. LHChas an excellent potential to produce high mass particles,for example, the top quark, the Higgs boson and SUSYparticles. I focus on the Higgs and Supersymmetry in thisnote, and refer to new gauge symmetry and extra space-dimensions only briefly as alternatives.
2. Detectors
Two general-purpose experiments have been constructed,ATLAS2) and CMS,3) at the LHC. The ATLAS (A ToroidalLHC Apparatus) detector is illustrated in Fig. 1, and itmeasures 22 m high, 44 m long, and weights 7,000 tons. Thecharacteristics of the ATLAS detector are summarized asfollows:4)
. Precision inner tracking system is made with pixel,strip of silicon and TRT (Transition Radiation Tracker)with 2 T solenoid magnet. Good performance isexpected on the B-tagging and the �-conversiontagging.
. Liquid argon electromagnetic calorimeter has finegranularity for space resolution, and longitudinalsegmentation to obtain fine angular resolution andexcellent particle identifications. It has also goodenergy resolution of about 1.3% for 100 GeV e� and �.
. Large muon spectrometer with air core toroidal magnetwill provide a precise measurement on muon momenta(about 2% for 100 GeV-��) even in the forward region.
The CMS (Compact Muon Solenoid) detector (Fig. 2)measures 15 m high, 21 m long, and weights 12,500 tons,with the following features:5)
. Precise measurement of the high pT track is performedwith 4 T solenoid magnet.
. PbW04 crystal electromagnetic calorimeter has anexcellent energy resolution of 0.9% for 100 GeV e�
and �.
3. Higgs Physics
A discovery of one or several Higgs bosons will give adefinite experimental proof of the breaking mechanism ofthe Electroweak gauge symmetry, and detail studies of theYukawa couplings between the Higgs boson and various
SPECIAL TOPICS
Table I. Production cross-section and event numbers for major high pT
and high mass processes with an integrated luminosity of 10 fb�1.
� Event number at LHC
(pb) (L ¼ 10 fb�1)
W� ! ‘�� 6:0� 104 �109
Z0 ! ‘þ‘� 5:7� 103 �108
t�tt 830 �107
jj pT > 200 GeV 105 �109
SM Higgs (M ¼ 115 GeV) 35 �105
~gg~gg (M ¼ 500 GeV) �100 �106
(M ¼ 1 TeV) �3 �104
Fig. 1. Overall layout of the ATLAS detector. Outside blue parts are the
muon system. Red and green parts are hadron and EM calorimeters,
respectively. Inner tracking system is installed inside the EM calorimeter.
Journal of the Physical Society of Japan
Vol. 76, No. 11, November, 2007, 111013
#2007 The Physical Society of Japan
Frontiers of Elementary Particle Physics, the Standard Model and Beyond
fermions will give insights on the origin of lepton and quarkmasses. The mass of the SM Higgs boson itself is nottheoretically predicted, but it’s upper limit is considered tobe about 1 TeV from the unitary bound of the WþW�
scattering amplitudes, or even 200 GeV (95% C.L.) from theElectroweak precision measurements.6) The lower limit ofthe Higgs boson mass is set at 114 GeV (95% C.L.) by thedirect searches at LEP. The Higgs boson should exist in thenarrow mass range of 114 – 200 GeV, and lighter than 130 –150 GeV if the Supersymmetry exists.
The SM Higgs boson, H0SM, is produced at the LHC
predominantly via gluon–gluon fusion (GF)7) and the seconddominant process is vector boson fusion (VBF).8) Theproduction cross-sections are summarized in the report.9)
H0SM decays10) mainly into b�bb and �þ�� for the lighter case
(.130 GeV). On the other hand, it decays into WþW� andZZ with a large branching fraction for the heavier case(&140 GeV). Although its decay into ��, via the one-loopprocess including top quark or W boson, is suppressed(�2� 10�3), this decay mode is very important at the LHC.
3.1 H0SM ! �� in GF and VBF
Although the branching fraction of this decay mode issmall and there is a large background processes viaq�qq! ��, the distinctive features of the signal, high pT
isolated two photons with a mass peak, allows us to separatethe signal from the large irreducible background. Both theATLAS4) and CMS5) detectors have excellent energy andposition resolutions for photon, and the mass resolutionof the H0
SM ! �� process is expected to be 1.3 GeV(ATLAS11)) and 0.9 GeV (CMS12)). Sharp peak appears atHiggs boson mass over the smooth distribution of back-ground events as shown in Fig. 3. This channel is promisingfor the light Higgs boson, whose mass is lighter than140 GeV, and this mode indicates the spin of the Higgsboson candidate.
VBF provides additional signatures in which two high pT
jets are observed in the forward regions, and only the twophotons from the decay of H0
SM will be observed in the widerapidity gap between these jets. The rapidity gap (no jetactivity in the central region) is expected because there is no
color-connection between two out-going quarks. Thesesignatures suppress the background contributions signifi-cantly and improve the signal-to-noise ratio as shown inFig. 4. Although the signal statistics is limited, the back-ground contributions are dramatically suppressed and thedistribution of the background becomes flat. A significanceof about 4.5 � is obtained from VBF processes with aintegrated of 30 fb�1 it gains a significance of the inclusiveH0
SM! �� analysis.
3.2 H0SM ! �þ�� in VBF
H0SM ! �þ�� provides high pT ‘�, when � decays
leptonically, and it can makes a clear trigger. Momentacarried by �’s emitted from � decays can be solvedapproximately by using the 6ET information,13) and the Higgsmass can be reconstructed.14) Figure 5 shows the massdistributions of the reconstructed tau-pair. The mass reso-lutions of about 10%15) can be achieved, and the signal canbe separated from the Z-boson production background. Theperformance of the 6ET measurement is crucial in thisanalysis as the same as in the SUSY search. We can obtain asignificance of about 4 � for this channel15) for mH0
SM< 130
GeV with a luminosity of 10 fb�1, and also this channelprovides a direct information on the coupling between theHiggs boson and a fermion, the tau lepton.
110 120 130 140
7000
6000
5000
4000
3000
mγγ (GeV)110 120 130 140
mγγ (GeV)
100 fb-1 100 fb-1
H γγH γγ
Eve
nts
/ 500
MeV
Eve
nts
/ 500
MeV
600
400
200
0irreducible γγ and "jet" background
Fig. 3. The invariant mass distribution of �� (L ¼ 100 fb�1 at CMS). H0SM
mass is assumed to be 130 GeV. The left figure shows signal plus
background events, and the right one shows the subtracted spectrum. In
addition to the irreducible �� background, there are jet–� and jet–jet
background events, in which a jet is misidentified as a photon.
is assumed, and fake background processes, in which a jet is misidentified
as a photon, are also shown in the green and yellow histograms.
Fig. 2. Overall layout of the CMS detector. Outside white and magenta
parts are the magnetic return yoke (iron) and the muon detectors. Light
blue and blue show the hadron and EM calorimeters, respectively. Inner
tracking system is installed inside the EM calorimeter and 4 T solenoid
magnet is installed outside of the hadron calorimeter.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS S. ASAI
111013-2
3.3 H0SM ð! ZZ ! ‘þ‘�‘þ‘�Þ in GF
Dominant decay modes of the heavier H0SM are ZZ
and WþW�. The four-lepton channel (H0SM ! ZZ!
‘þ‘�‘þ‘�) is very clean and called ‘‘the gold-plated’’.Although the branching fraction of ZZ! ‘þ‘�‘þ‘� issmall, a sharp mass peak is expected as shown in Fig. 6, ifthe Higgs boson is heavier than 140 GeV. Mass resolutionof the four lepton system is typically 1%.11,12) As shown inFig. 6, the main background process is ZZ production,which gives continuous distribution above 200 GeV. Smallcontaminations come from t�tt and Z0b�bb, in which semi-
leptonic decays of bottom quark are identified as isolatedleptons. In order to reduce these contaminations, leptonsare required to be well isolated from hadron activitiesand their track impact parameters should be consistentwith zero. The ‘þ‘�‘þ‘� channel has a good performancein the wide mass range from 130 to 800 GeV, exceptaround 170 GeV, where the WþW� is the dominate decaymode.
3.4 H0SM ð! WþW� ! ‘��‘��Þ in VBF
When the Higgs mass is around 170 GeV, the branchingfraction of H0
SM!WþW� is almost 100%. The whole massrange of 130 – 200 GeV is well covered by the analysis ofVBF H0
SM !WþW� ! ‘þ�‘��. The transverse mass, MT,is defined as
, in which � is theazimuthal angle between the 6ET and PTð‘þ‘�Þ. Figure 7shows the MT distribution and a clear Jacobian peak isobserved above smooth background distributions. The mainbackground process is t�tt, and this can be suppressed by usingthe azimuthal angle correlation between the dileptons. Sincethe Higgs boson is a spin zero particle, the helicities of theemitted W bosons are opposite. The leptons are then emittedpreferably in the same direction due to the 100% Parityviolation in W decays.
3.5 Overall discovery potential of H0SM
Discovery potential of H0SM are summarized in Fig. 8
as a function of the Higgs mass with an integratedluminosity of 10 fb�1. H0
SM! �� in GF and VBF and�þ�� channels have good potential in the mass regionlighter than 130 GeV. For the heavy mass case (� 130 GeV),decay to ZZ (! ‘þ‘�‘þ‘�) and WþW� have an excellentperformance even above 10�. When both ATLAS andCMS significances are combined, we can perform todiscover (5� C.L.) and exclude (98% C.L.) the Higgs bosonup to 1 TeV mass with the integrated luminosities of 5and 1 fb�1, respectively. Therefore the crucial test of theHiggs mechanism of the symmetry braking can be per-formed within 2009.
Fig. 6. The invariant mass distributions of ‘þ‘�‘þ‘� for MðH0SMÞ ¼ 140
(upper) and 200 (lower) GeV with luminosities of 9.2 and 5.8 fb�1,
respectively (CMS). With these luminosities, a significance of 5� is
obtained.
0
0.25
0.5
0.75
1
0 50 100 150 200 250
MT (GeV/c2)
dσ/d
MT (
fb/1
0 G
eV/c
2 )
Fig. 7. The transverse mass distribution of 6ET and ‘þ‘�, in which
MðH0SMÞ ¼ 160 GeV is assumed (ATLAS). Open histogram shows the
Higgs signal and the hatched histogram show the background processes
(t�tt and WþW�).
0
5
10
15
20
25
40 60 80 100 120 140 160 180 200 220
Mττ(GeV)
Num
ber
of e
vent
s
L=30fb-1
Fig. 5. The invariant mass distribution of �þ�� with high pT forward jets
(L ¼ 30 fb�1 at ATLAS). One tau decays leptonically and another decays
hadronically in this analysis. Open histogram shows the signal with
MðH0SMÞ ¼ 120 GeV. The similar significance is obtained when both taus
decay into leptons.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS S. ASAI
111013-3
3.6 Measurement of mass and couplingsof the Higgs boson
Measurements on the properties of the discovered Higgsboson give further insights to the origin of masses. Higgsmass can be measured precisely in H0
SM ! �� and H0SM !
ZZ (! ‘þ‘�‘þ‘�). Accuracy of less than 0.2% error can beachieved with L ¼ 300 fb�1, if the mass is smaller than500 GeV. When the Higgs boson is heavier than 500 GeV,the resonance becomes too broad, and the precision becomesworse.
Measurements of the couplings between the Higgs bosonand fermions/Gauge bosons will give the direct informa-tions of the origin of ‘‘Mass’’, and fermion’s couplings willgive the first evidence of Yukawa couplings. Figure 9 showsthe accuracy of the coupling measurements between theHiggs boson and fermion/Gauge bosons. We can measurewithout an assumption the relative magnitudes of thecouplings normalized to the coupling between the W andthe Higgs boson. We can determine the coupling betweenthe Z and the Higgs boson precisely, where the accuracy of5 –10% can be achieved in all the mass region. Couplingsbetween the Higgs and the 3rd generation fermions (top andtau) are determined with accuracies of 10 –15%, but can notdetermine well for the bottom quark.
4. Supersymmetry
Supersymmetric (SUSY) standard models16) are mostpromising extensions of the SM, because the SUSY cannaturally explain the weak boson mass scale. Furthermore,the SUSY models provide a natural candidate of the colddark matter,17) and they have given a hint of the GrandUnification in which three gauge couplings of the SM areunified at around 2� 1016 GeV. In these theories, eachelementary particle has a superpartner whose spin differsby 1/2 from that of the particle. Discovery of the SUSYparticles should open a new epoch of the fundamentalphysics, which is another important purpose of the LHCproject.
4.1 Introduction of Super-Gravity modelThere are, in general, more than 100 free parameters to
describe soft SUSY breaking,16) but there are strongconstraints among them from the smallness of the flavor-changing neutral current. Among various models of theSUSY breaking that can naturally satisfy these constraints,Super-Gravity model,18) Gauge-mediated model,19) andAnomaly-mediated model20) are predictable and promising.Performance of the ATLAS experiments based on the Super-Gravity model is summarized in this section.
Minimal Super-Gravity Model (mSUGRA)18) is a specialcase of the Minimal Supersymmetric Model (MSSM), inwhich the soft SUSY breaking terms are assumed to becommunicated from the SUSY breaking sector by gravityonly and that these terms are universal at the GUT scale.There are only five free parameters in this model; m0
(universal mass of all scalar particles at the GUT scale), m1=2
(universal mass of all gauginos at the GUT scale), A0
(common scale of trilinear couplings at the GUT scale),tan � (� v2=v1) (ratio of VEV of two Higgs fields at theElectroweak scale) and the sign of � (Higgsino mass term).
In this model, masses of supersymmetric particles aremainly determined by m1=2 and m0; ~gg becomes heavy due tolarge radiative corrections, and its mass is approximately 2.5m1=2 at the energy scale of the LHC. Higgsino mass (j�j)becomes larger than the weak Gaugino masses at the EWscale, except for the case of m0 � m1=2. Then the lighterstates of the neutralinos, ~0
1 and ~02, become almost pure
Bino- and Wino-states ( ~01 � ~BB0, ~0
2 � ~WW0), and the lighterstate of the charginos, ~�1 , is also Wino-like ( ~�1 � ~WW�).Scalar lepton masses are determined mainly by m0 and subdominantly by m1=2. On the other hand, scalar quark massesdepend on both m0 and m1=2. Here is a typical spectrum ofthe SUSY particles;
pMasses of the lighter state of third generation scalar
fermions (~tt1, ~bb1, and ~��1) depend also on A and tan �,21) andthey are generally lighter than the first and second generationscalar fermions because of the following two reasons.Firstly, one loop radiative corrections due to their Yukawa
[GeV]Hm110 120 130 140 150 160 170 180 190
(H,W
)2
(H,X
) / g
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)2
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) / g
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Δ
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(H,Z) / g2g
(H,W)2
) / gτ(H,2g
(H,W)2
(H,b) / g2g
(H,W)2
(H,t) / g2gwithout Syst. uncertainty
-1 L dt=300 fb∫
ATLAS
Fig. 9. Relative precision of the H0SM coupling squared as a function of the
Higgs mass. Luminosity of 300 fb�1 is assumed. Solid lines shown the
results including the systematic errors.
Sig
nal
Sig
nif
ican
ce
1
10
10-1 L =10 fb
σ5
Combined
(GeV)HM100 200 300 400 500
VBH H->WW->lnulnuVBH H->WW->lnulnu
GF+VBF H->ZZ->4lGF+VBF H->ZZ->4l
GF+VBF H->ZZ->4lGF+VBF H->ZZ->4l
H-> > (inclusive: no K factor)
VBF H-> VBF H-> tautautautau
Fig. 8. The H0SM discovery potential with L ¼ 10 fb�1 in the various
modes (ATLAS). The horizontal axis shows the mass of Higgs boson, and
the vertical gives the significance of the Higgs signal. Open black circles
show the combined performance of all the modes. The horizontal dotted
line shows the 5� discovery level. The exclusive analysis of Fig. 4 is not
yet combined.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS S. ASAI
111013-4
couplings are always negative. Secondly, the supersymmet-ric partners of the right- and left-handed states mix, and theresultant lower mass eigenstate becomes lighter. This mixingcontribution depends on both A0 and tan �.
Dominant SUSY production processes at the LHC are ~gg~gg,~gg~qq, and ~qq~qq through the strong interaction. These productioncross-sections, �, do not strongly depend on the SUSYparameters except for the masses of ~gg and ~qq.22) When theirmasses are lighter than 1 TeV, ~gg~gg is the main productionprocess, and the total SUSY production cross-section, �(~gg~gg,~gg~qq, and ~qq~qq), is about 100 pb for m~qq ¼ m~gg ¼ 500 GeV, and3 pb for 1 TeV. Even when their masses are 2 TeV, sizableproduction cross-section of about 20 fb is expected. ~uu~uu and~uu~dd are the main production processes for such a heavy case,since u and d quarks are the valence quarks of the incomingprotons.
Decay modes of ~gg and ~qq are controlled by the mass-relation between each other, and are summarized in Fig. 10.If kinematically possible, they decay into 2-body through thestrong interactions. Otherwise, they decay into an electro-weak gaugino plus quark(s).
4.2 Event topologies of SUSY events and discoverypotential
~gg and/or ~qq are copiously produced at the LHC. High pT
jets are emitted from the decays of ~gg and ~qq as presented inFig. 10. If the R-parity23) is conserved, each event containstwo ~0
1’s in the final state, which is stable, neutral andweakly interacting and escapes detection. This ~0
1 is anexcellent candidate of the cold dark matter. The missingtransverse energy ( 6ET), which is carried away by the two~01’s, plus multiple high pT jets is the leading experimental
signature of the SUSY at the LHC. Additional leptons, ��
(decaying hadronically) and h (! b�bb), coming from thedecays of ~0
2 and ~�1 , can also be detected in the some part ofevent. Typical simulated SUSY event is shown in Fig. 11 forthe ATLAS detector.
Inclusive searches will be performed with the large 6ET andhigh pT multi-jet topology (no-lepton mode). One additionalhard lepton (one-lepton mode) or two same-sign (SS-dilepton) and opposite-sign leptons (OS-dilepton) can be
added to the selections. These four are the promising modesof the SUSY searches.
The following four SM processes can potentially have 6ET
event topology with jets:. W�ð! ‘�Þ þ jets,. Z0ð! � ���; �þ��Þ þ jets,. t�ttþ jets,. QCD multi-jets with mis-measurement and semi-
leptonic decays of b�bb and c�cc with jetsWe require at least four jets with pT � 50 GeV, and pT ofthe leading jets and 6ET are required to be larger than100 GeV. The azimuthal angles between 6ET and theleading three jets are required to be larger than 0.2 for theno-lepton mode to reduce the QCD background. Effectivemass, Meff ¼6ET þ
Pjets pT, is a good variable to distinguish
the signal from the background listed above. Excess comingfrom the SUSY signals can be clearly seen as shown inFig. 12 for both no-lepton and one-lepton modes. t�tt is thedominant background process for the one-lepton mode, andall of the four processes listed above contribute to no-leptonmode. The QCD processes contribute only to the small 6ET
and Meff . The Meff distribution has steeper slope for the SMbackground processes as shown in these figures. On the otherhand, the distribution of the SUSY signals has a broad peakat large value (around 1.5 TeV in these figures), which isproportional to minðmð~ggÞ;mð~qqÞÞ.
The SM background contributions can be suppresseddramatically by requiring di-leptons as shown in Fig. 13.They show the 6ET distributions for the OS- and SS-dileptonmodes, where t�tt is the dominant background process for bothmodes. Although the background contributions is seriouslysuppressed (the SS dilepton mode is the almost backgroundfree), the statistics of the SUSY signal is also typically a few% of no-lepton mode. Discovery potentials of dileptonmodes are limited in the early stage of the collision, butthese modes are important to reconstruct the decay chainmentioned later.
Figure 14 show the 5�-discovery potential in the m0–m1=2
and the m~qq–m~gg plane for tan � ¼ 10 with an integratedluminosity of 1 fb�1. And the lowest shows the same figure
( 2/3)
( 2/7)
( 1/7)
( 4/7)
( 1/3)
Fig. 10. Summary of decays of the colored Sparticles: Bino/Wino-
eigenstates presented in the table can be regarded as mass-eigenstates,
( ~BB0 � ~01, ~WW0 � ~0
2, and ~WW� � ~�1 ), when m0 is not too larger than m1=2.
In this case, Higgsino mass (j�j) becomes much larger than gaugino
masses at the EW scale, and the Higgsino component decouples from the
lighter mass-eigenstates as explained in the text.
Fig. 11. Expected SUSY signal (simulated with the ATLAS detector); ~gg
and ~qq are associate-produced and the ~gg decays into a quark and a ~qq. The
central colored tracks show the observed charged particles, and the
yellows towers in the gray and orange parts show the energies deposited
in the EM and hadron calorimeters, respectively.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS S. ASAI
111013-5
with an integrated luminosity of 30 fb�1. The results withthree modes of no-lepton, one- and OS-dilepton are shown,and the hatched regions below each line show the systematicerrors coming from the background estimations. The similardiscovery potentials are obtained for no-lepton and one-lepton modes. As shown in these figures, ~gg and ~qq can bediscovered up to the mass of �1:5 TeV with a luminosityof 1 fb�1, which corresponds to just one month run with1033 cm�2 s�1. The discovery reach does not depend stronglyon tan �. The interesting region for the relic density of thedark matter is almost covered with just 1 fb�1. ~gg and ~qq canbe discovered up to �2:5 TeV with a luminosity of 30 fb�1.~gg and ~qq, whose masses are about 2.7/3.0 TeV, can bediscovered/excluded finally with a luminosity of 100 fb�1.This luminosity is corresponding to one year run with designluminosity (1034 cm�2 s�1), and will be achieved in about2011.
4.3 Measurements of masses of SUSY particlesSince two undetected LSP’s exist in each event, there are
six unknown momentum components in addition to the ~01
mass. No mass peak is expected in general. However it ispossible to use kinematic end points of various distributionsas follows.11,24,25)
. Select specific decay chain exclusively. For example,
~qqL! ~02q! ð ~‘‘
�R ‘Þq! ðð ~0
1‘�Þ‘Þq
. Make various distributions of invariant masses and pT.
. Kinematic constraints are obtained from the end pointsof these distributions for the selected chain. These endpoints can be determined by the masses of the SUSYparticles. SUSY events become background itself fordetailed study, since there are many cascade decaypatterns in ~qq and ~gg. It is critical point that we can findout or not a useful decay-pattern in the SUSY events.
If there are three 2-body decay chains like the aboveexample, which is a dominant mode in the parameter spaceof m0 0:8m1=2, full reconstruction of masses is possiblemodel-independently. The invariant mass distributions of‘‘, ‘‘þ jet, and ‘þ jet can be calculated, and the threekinematic end points and one production threshold of the4-body system (‘þ‘�q ~0
1) are obtained. On the other hand,there are four unknown masses (~qqL, ~0
2, ~‘‘�R , and ~01). Then all
the four unknown masses can be determined model-independently. Although the errors of the determined massesare strongly correlated, accuracies of these masses are about3 –10% for mð~qqLÞ ¼ 800 GeV. Mass of the missing ~0
1
can be determined with an accuracy of about 10%, whichis an extremely important result, since it can be the darkmatter.
Reconstructions of the SUSY particle masses for the otherdecay-patterns are also possible11,12,25) by using the similartechniques. When we cannot identify a successive decay
Fig. 13. Missing ET distributions of the SUSY signal and background
processes with a luminosity of 10 fb�1. Upper and lower figures show the
Opposite–Sign-dilepton (OS) and the Same–Sign-dilepton (SS) cases,
respectively. In both figures, the open histogram shows the SUSY signal
for m0 ¼ 70 GeV, m1=2 ¼ 350 GeV and tan� ¼ 10. The hatched histo-
gram shows the sum of all the SM backgrounds, while blue circles (t�tt) and
green-triangles (Z0 þ jets) show each component.
Effective Mass [GeV]0 500 1000 1500 2000 2500
-1ev
ents
/ 1f
b
1
10
210
310
410SUSY(1TeV)all BGttWZQCD
ATLAS Preliminary
Effective Mass [GeV]0 500 1000 1500 2000 2500
-1ev
ents
/ 1f
b
-110
1
10
210
SUSY(1TeV)all BGttWZ
ATLAS Preliminary
Fig. 12. Effective mass distributions of the SUSY signal and background
processes with an integrated luminosity of 1 fb�1. Upper and lower
figures show the no-lepton and one-lepton modes, respectively. Magenta
histogram shows the SUSY signal, in which mass of squark and gluino
are 1 TeV (tan� ¼ 10). The black histogram show the sum of all SM
backgrounds. It includes the followings: t�tt (blue), W� þ jets (red),
Z0 þ jets (green), and QCD jets (light blue). These results are obtained
with the full simulation using GEANT4, in which all the materials in the
detector are taken into account.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS S. ASAI
111013-6
chain listed above, the number of the observed constraints isless than that of the unknown masses. Therefore, someassumptions on SUSY breaking pattern are necessary todetermine the mass spectrum. All the massed of ~gg–~qqL=R, ~‘‘�–~02, and ~0
1 can determined within the assumed model, andwe will be able to test various SUSY models by using thesereconstructed mass spectra.26)
5. New Gauge Bosons
Many theories beyond the SM predict the existence ofadditional U(1) or SU(2) gauge groups and a discovery ofnew gauge bosons, Z0 and W0, should be its clear evidence ofextension of the SM. Current mass limits on Z0 and W0 areabout 800 GeV obtained at the Tevatron.
Z0 and W0 are produced with the Drell–Yan (q�qq
annihilation) process at the LHC, and they decay intofermion pairs. Figure 15 shows the transverse massdistribution between the �� and 6ET momenta, MT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2PTð��Þ 6ETð1� cos ���missingÞp
. Clear Jacobian peak will
be observed at the mass of W0, and the backgroundcontribution is small in such a high MT region. The resultsare obtained by assuming that the W0 coupling are the sameas the W� coupling and no decay mode into new particles.The W0 can be discovered or excluded up to mass of 4.5 and5 TeV, respectively12) with an integrated luminosity of10 fb�1, by combining the both �� and e� channels. Z0 !‘þ‘� makes a resonance peak in the high mass region, andZ0 can be discovered up to mass of 3.8 TeV with anintegrated luminosity of 10 fb�1.
6. Large Extra Space-Dimension and Black Holes
There is much recent theoretical interests that an extra-dimension exists in addition to the 3þ 1 normal space–timedimensions, and it is compactified to the size of a few TeV,which is related to the true ‘‘Planck scale’’ of the completetheory. This new fundamental scale is denoted as ‘‘MP’’. Theidea is introduced as a possible solution of the hierarchyproblem of the SM.27) The scatter processes at a few TeVscale should be treated within multi-dimensions, and thegravity interaction becomes as strong as the other inter-actions. For example, gravitons (G) are emitted in thehard scatter of gg! gG and escapes the detection. Thismakes an event with a single jet and large 6ET (monojet).We have the discovery-potential up to MP ¼ 7 TeV (addi-tional dimension 3) with an integrated luminosity of100 fb�1.12)
If the true Plank scale is in the order of TeV, theShwarzschild radius, Rs, is also sizable,
Rs ¼1ffiffiffip
MP
MBH
MP
8�
�nþ 3
2
�
nþ 2
0B@
1CA
264
375
1=ð1þnÞ
for n extra space-dimensions. Mini black holes of a few TeVmass can be produced with parton–parton collisions withinRs at the LHC. Large production cross-section of orderRs
2 � 100 pb is expected by classical arguments,28) theproduced black hole decays through Hawking evaporationwith the temperature:
Fig. 15. Transverse mass distribution of �� and � with L ¼ 10 fb�1
(CMS): The steep green histogram shows the distributions of the W�
background process. Red and black open histograms show the signal of
W0 with mass of 1 and 5 TeV, respectively.
Fig. 14. 5�-discovery potential (ATLAS preliminary) in the m0–m1=2
and m~qq–m~gg planes: The upper two figures show the performance with
L ¼ 1 fb�1, and the lowest figure shows the results with L ¼ 30 fb�1. In
these figures, black, red and blue lines show the 5�-discovery regions
with no-lepton, one-lepton and OS-dilepton analyzes, respectively. The
hatched region below each line shows the systematic uncertainty of the
background estimations. Gray hatched regions are theoretically forbidden
(no EWSB or slepton LSP).
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS S. ASAI
111013-7
TH ¼ MP
MP
MBH
nþ 2
8�
�nþ 3
2
�264
375
2=ðnþ1Þ
As the produced black hole is heavier, the TH becomessmaller (but still higher than 100 GeV), and many particlesare emitted as shown in Fig. 16. These particles areenergetic as the same orders of TH . High pT multi-particlesare emitted spherically, and this event topology is quitedifferent from the SM background processes. We candiscover mini black holes up to mass of several TeV,29)
but we need more theoretical studies on the production anddecay processes of the mini black holes at the LHC.
7. Summary
The LHC is the first experiment that probes directlyphysics at the TeV scale. The SM Higgs boson (H0
SM) shouldbe discovered with an integrated luminosity of 10 fb�1, if itexists. H0
SM ! �� in GF (gluon fusion) and VBF (vector-boson fusion) and H0
SM ! �þ�� in VBF are importantprocesses for the Higgs lighter than 140 GeV. The decaysinto ZZ (! ‘þ‘�‘þ‘�) and WþW� (! ‘�‘�) play impor-tant role if it is heavier. The Higgs boson mass can bedetermined precisely, and the couplings between Higgs andfermions/Gauge bosons can be measured with accuracies ofabout 5 – 20%. We can perform a critical test on the Higgsmechanism and will understand the origin of the elementaryparticle masses.
Supersymmetry should be discovered at the LHC, ifgluino (~gg) and squarks (~qq) are lighter than about 2.7 TeV.Signals will be found not only in the ( 6ET þ jets) channel butalso in [ 6ET + jets + lepton(s)] channels, where 6ET standsfor the missing transverse momentum carried away by thelightest supersymmetric particle (LSP). The SUSY particlesmasses, including that of the LSP, can be determined by
exclusive studies model-independently, when a three suc-cessive two-body decay chain is identified. In more generalcases, they can still be determined within various super-symmetry models.
The LHC can also discover new physics other thansupersymmetry, such as new gauge boson symmetry andextra space dimensions.
Graduate School of Science, Tohoku University, Sendai 980-8578
(Received June 20, 2007; accepted August 27, 2007; published November 12, 2007)
International Linear Collider (ILC) is an electron–positron collider with the initial center-of-massenergy of 500 GeV which is upgradable to about 1 TeV later on. Its goal is to study the physics at TeVscale with unprecedented high sensitivities. The main topics include precision measurements of theHiggs particle properties, studies of supersymmtric particles and the underlying theoretical structure ifsupersymmetry turns out to be realized in nature, probing alternative possibilities for the origin of mass,and the cosmological connections thereof. In many channels, Higgs and leptonic sector in particular, ILCis substantially more sensitive than LHC, and is complementary to LHC overall. In this short article, wewill have a quick look at the capabilities of ILC.
KEYWORDS: ILC, Higgs boson, New Physics, supersymmetry, extra dimensions, cosmology, LHCDOI: 10.1143/JPSJ.76.111014
1. Introduction
The standard model is an astonishingly successful theoryin describing what have been observed in the field ofelementary particles. The Higgs particle, which gives massto all massive particles, is at the core of the standard model,but so far has not been found. Furthermore, if one tries tocalculate the radiative correction to the mass squared ofHiggs, it diverges quadratically with the cut off energy,and if one assumes that the standard model is correct up tothe energy scale of the grand unification (� 1016 GeV), thecorrection to the Higgs mass becomes the order of the grandunification scale itself. Since precision measurements so farshows that the standard model Higgs should lie below�200 GeV, this is only possible if the original mass and thecorrection are canceling out to an astonishing precision. Thisunpleasant situation is referred to as the fine-tuning problem,or the naturalness problem. The problem is in part caused bythe large difference in energy scale from the Higgs mass tothe grand unification scale, and in this context, it is refered toas the hierarchy problem.1) Also, the standard model doesnot include the gravitational force.
A theoretical solution to the fine-tuning problem isprovided by supersymmetry (SUSY)2) which postulates thatevery particle in the standard model has its so-called super-partner (called a super particle or a s-particle) whose spindiffers by one half from that of the original particle. Not onlythe inclusion of the super particles naturally cancels out thequadratic divergence of the Higgs mass correction, in SUSYthe gauge coupling constants converges to a single value atthe grand unification scale. Furthermore, the gravitationalforce can also be naturally incorporated in SUSY. As abonus, SUSY has candidates for the dark matter which isthought to consist of unknown stable massive particles andaccounts for one quarter of the energy of the universe.
Even though SUSY is an attractive theory with manymerits for us, nature of course would not care about ourconveniences. There are several alternative models thataddress the fine-tuning problem, and some of them may have
connection to the reality of nature. Examples are the modelswith extra dimensions which postulate the existence of spacedimensions more than our 3 (space) + 1 (time) dimen-sions,3) and the little Higgs model4) where the Higgs particleis considered to be composite.
The physics potential of ILC has been extensively studiedand documented.5–9) As we will see below, the standardmodel Higgs particle will have distinctive signals at ILC,and SUSY and other alternative models also have manypossibilities of being found and studied at ILC. The advan-tage of ILC with respect to LHC is in the general cleanlinessof the events where two elementary particles (an electronand a positron) with known kinematics and spin define theinitial state, and the high resolutions of the detector that aremade possible by the relatively low absolute rate of back-ground events. The capability of ILC is further enhancedby the options such as the �� collision, e�e� collision, andZ-pole running (‘‘Giga-Z’’).
2. ILC Machine Parameters and Detectors
The basic parameters, such as energy and luminosity, ofILC are described in the parameter report.10) The baselinemachine allows for a center-of-mass energy range between200 and 500 GeV and luminosity of 500 fb�1 in the first fouryears of running not counting the year zero. The energy scanis possible at any energy within the range, and the electronpolarization is at least 80%. Two detectors are expectedwhich may be in a push–pull configuration.
For each of the two beams, a bunch is �y ¼ 5:7 nm high,�x ¼ 655 nm wide, and �z ¼ 300 mm long, and contains2� 1010 particles. About 3000 bunches with 308 ns bunchsepartion form a train of about 1 ms which comes with 5 Hzrepetition rate. The collision occurs with crossing angle of14 mrad.
The highest priority beyond the baseline is the energyupgrade to approximately 1 TeV, and the upgraded machineshould be able to collect 1 ab�1 in 3 to 4 years after thebaseline running. The options include: running at 500 GeVto double the luminosity to 1 ab�1, e�e� collision, positronpolarization of 50% or more, Z-pole running, WW thresholdrunning, and �� and e�� collisions using backscattered laser
beams. The priorities of these options will depend on theresults of LHC and the baseline ILC.11) In the following,the baseline machine with 200 to 500 GeV center-of-massenergy is assumed unless stated otherwise.
The physics of ILC is realized through synthesis ofunprecedented performances of both machine and detectors.ILC detectors can take advantage of the relatively low ratesand low radiation doses to achieve momentum resolutionthat is order of magnitude better, jet energy resolution factorof two better, and the vertex resolution several times betterthan those at the previous electron–positron colliders. As wewill see below, these performances are not overkill; rather,they are needed to realize the physics potential of ILC.
3. Standard Model Particles
We start from the particles that are ingredients of thestandard model. Their properties and interactions with otherparticles, however, may reveal physics beyond the standardmodel. The goal is to look at the behavior of the members ofthe standard model to see if there is any hint of new physics.Production cross sections for some standard model particlesas well as those for particles beyond the standard model areshown in Fig. 1.
3.1 Gauge bosonsNon-Abelian nature of gauge group leads to couplings
among gauge bosons, and their pattern reflects the structureof the underlying gauge group. W-pair creation eþe� !WþW� is highly sensitive to the triple gauge couplikngsWW� and WWZ which can be separated by beam polar-izations. With 90 and 60% for electron and positron
polarizations, respectively, and 500 fb�1 atffiffisp¼ 500 GeV
and 1 ab�1 atffiffisp¼ 800 GeV, anomalous couplings can be
measured with typical errors of 10�3 relative.12) The WW�magnetic dipole coupling �� , in particular, can be measuredto 10�4, which is more than order of magnitude better thanLHC with the same years of running. The triple gaugecoupling WW� can also be studied by the single gauge bosonproductions eþe� ! e��Wþ, ��Z, and also by the e� and�� options; namely, e�� ! W�� and �� ! WþW� wherethe WWZ coupling does not contribute.
If Higgs is not found at LHC or ILC, it may indicate thatW-pair can form a bound state which could be found in theWW scattering process e�eþ ! � ���WþW� as a resonance oranomalous quartic gauge couplings. Quartic gauge couplingscan generally be probed by gauge boson scattering processesof the type e�eþ ! VV f �ff where V is W or Z and f is e or �,or by triple gauge boson productions e�eþ ! VVV . At ILC,one can tell the initial and final states of the gauge bosonscatterings, which is often difficult at LHC.
If no Higgs or no new particles are found, precisionmeasurements on Z become important. The Giga-Z optioncan collect 1 billion Z’s in a few months, and can improveby more than one order of magnitude those measurementsthat use b-tagging and/or beam polarizations.13) The im-proved b-tagging is realized by the excellent vertexingcapability of ILC detectors.
Couplings of fermions and gauge boson can also be studiedby eþe� ! f �ff ( f stands for a fermion), where anomalouscouplings may be parametrized by ð1=�2
ijÞð �eei��eiÞð �ff j�� fjÞ(ij ¼ L;R). ILC is sensitive to �ij of typically 20 to100 TeV.14)
The eþe� ! f �ff modes are also sensitive to existenceof an extra Z boson (Z 0) even when the mass of Z 0 is abovethe CM energy. Such extra gauge bosons appear in manyextensions of the standard model. Some examples are the E6
� model (�), left–right symmetric model (LR), Littlest Higgsmodel (LH), Simplest Little Higgs model (SLH), and modelwith extra dimensions where Z 0 particles are actually spin-2Kaluza–Klein excitations of gravitons (KK). The signaturesappear in the forward-backward asymmetry of the f �ff
production and in the dependence of the cross section onthe beam polarization. The resolving power of ILC in thetwo-dimensional space of C‘L and C‘R is shown in Fig. 2 foreþe� ! �þ��, where C‘L;R are the left- and right-handedZ 0‘ coupling coefficients where the lepton universality isassumed. Electron and positron polarizations of 80 and 60%respectively, are assumed. There are quadratic ambiguitiesdue to the sign-independence of coupling coefficients. LHCmay find a Z 0 resonance, but it would take ILC to identify theunderlying theory.
3.2 Top quarkThe top quark is the heaviest elementary particle observed
so far, and its mass �174 GeV is in the range of theelectroweak symmetry breaking. Its large mass indicates thatit couples to Higgs strongly and thus should be sensitive tothe structures in the Higgs sector, or whatever is responsiblefor creation of masses. In many models beyond the standardmodel, the Higgs mass strongly depends on the top mass. InMSSM (the minimal supersymmetric standard model), forexample, an error in the top mass corresponds to a similar
0 200 400 600 800 1000
σ (fb
)
√s (GeV)
ZZ|cosθ|<0.8
|cosθ|<0.8W+W–
tt 175GeV
Zh120GeV
μ+μ–
μ+Rμ–
R
140GeV
μ+Lμ–
L
230GeV
~ ~
χ+χ–
220GeV
~~ HA400GeV
H+H–
410GeV
~ ~
H+H–
190GeV
1
106
10–2
103
Σ qqq=t
JLC
Fig. 1. (Color online) Production cross sections for some standard model
particles as well as for new physics particles at eþe� collider as functions
of CM energy.6)
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS H. YAMAMOTO
111014-2
error in the Higgs mass, which means that precision meas-urements of the top and Higgs masses serve as a stringenttest of theoretical models. In some cases, non-standard topcouplings may be the only area new physics can be found.
The top mass mt is best measured by the eþe� ! t�tt
threshold scan, taking about 5 fb�1 each at several points ofCM energy. Since the top quark decays before it hadronizes,the excitation curve, i.e., the cross section as a function ofCM energy, around the threshold can reliably be calculated.It is affected by the beam energy spread, initial-stateradiation, beamstrahlung (radiation from a beam particleunder the coherent electromagnetic field of the incomingbunch), as well as the higher order corrections which hasbeen performed up to including some of the next-to-next-to-leading logarithms (NNLL).15) The experimental and theo-retical uncertainties are of the same order, and the resultingoverall error on mt is expected to be 100 to 200 MeV whichcan be compared to 1 to 2 GeV at LHC. The threshold scanalso yield the top width to a few percent of its value which isaround 1.5 GeV.
The production and decay of top quark in eþe� ! t�tt,t! bW can be studied near the threshold, well above thethreshold, or below the threshold (where one of the top quarkis off-shell). The production is sensitive to ttZ and tt�couplings and the decay is sensitive to tbW coupling. Manybeyond-the-standard models predict deviations in thesecouplings from the standard-model values. The models withfourth generation with large mixing between fourth and thirdgenerations of quarks would have the tbW coupling smallerthan that of the standard model while the ttZ coupling wouldbe the same. The little Higgs models with T-parity and thetop flavor models would have both tbW and ttZ couplingssmaller than those of the standard model. Figure 3 showsthe sensitivities of ILC and LHC on the axial ttZ couplingand the left-handed tbW coupling as well as the expecteddeviations for the top-flavor model and the little-Higgsmodel with T-parity, and the model with fourth generation.The numbers shown on the line for T-parity are the strength
of the Higgs-top- (top partner) coupling and those on the linefor the top flavor model are the mass of the extra Z boson.Furthermore, the KK mode of graviton with mass 10 to100 TeV in Randall–Sundrum models18) may be indirectlydetected as anomalous t�tt production.
3.3 Higgs particleOur current knowledge on the mass of the Higgs particle
mainly comes from the LEP experiments.19) Within theframework of the standard model, Higgs mass mH isbounded as 114:4 < mH < 166 GeV at 95% confidencelevel, where the lower limit is from direct searches andthe upper limit is by an overall fit of the standard modelparameters to the data. On the other hand, Higgs in theMSSM is constrained to be less than 135 GeV, which islower than the upper limit in the standard model. TheseHiggs particle, if they exist, will be found at LHC withinthe first few years of running. At ILC, even though the startwould be many years later than LHC, the same level ofdiscovery sensitivity can be obtained by one day of runningat the design luminosity. With its clean initial and finalstates, and high resolutions of the ILC detectors, ILC will beable to perform measurements on spin and parity of theHiggs particle, and determine coupling strengths to variousparticles model-independent ways.
The primary production channels of the standard modelHiggs are eþe� ! Z� ! ZH (Higgs-strahlung) andeþe� ! � ���H (WW fusion) as shown in Fig. 4. The Higgs-strahlung dominates at low CM energies (<500 GeV) andthe WW fusion dominates at high CM energies (� 1 TeV).For mH of 120 GeV, an integrated luminosity of 500 fb�1 atCM energy of 500 GeV will generate ð3{4Þ � 104 Higgsparticles in each of the two production channels. The decaybranching fractions of Higgs are shown in Fig. 5. If the
Fig. 2. (Color online) The 95% confidence level regions for various
models using eþe� ! lepton pair.16) The regions correspond to mZ 0 ¼ 1,
2, 3, 4 TeV with the smallest region is for 1 TeV.
Fig. 3. (Color online) Sensitivities of ILC and LHC on the axial ttZ
coupling and the left-handed tbW coupling.17) The expected deviations
for the top-flavor model and the little-Higgs model with T-parity, and the
model with fourth generation are also shown.
Fig. 4. The main Higgs production mechanisms at ILC: the Higgs-
strahlung (left) and the WW fusion (right).
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Higggs mass is below around 140 GeV, it decays primarilyto b �bb with a few % each for c �cc, � ���, and gg branchingfranctions. The width of Higgs in this mass range is less than10 MeV. For mH larger than around 150 GeV, it decaysprimarily to WW with the ZZ channel following at 20%level. The t�tt final state opens for mH larger than around350 GeV and peaks for mH � 500 GeV at 20% branchingfraction. At mH of around 500 GeV, the Higgs is quite broadwith �H � 100 GeV.
Figure 6 shows the recoil mass distribution for eþe� !ZH, Z ! �� with 500 fb�1 at CM energy of 300 GeV.Peaks corresponding to different values of mH are showntogether with the background from eþe� ! ZZ followed byone or both of the Z’s decaying to ��. Since the Higgsparticle is not reconstructed, the method is independent ofthe Higgs decay modes including the case where the decay isinvisible. The range of detectable Higgs mass reaches closeto the CM energy itself; more precisely, up to CM energyminus mZ .
The Higgs mass is obtained from the recoil masssdistribution itself. Under the same conditions used forFig. 6, the error in mH is �70 MeV which improves to�40 MeV if hadronic decays of Z are included. The spinand parity of the Higgs particle can be determined by thethreshold excitation curve and the angular distribution of theHiggs production in the Higgs-strahlung process. If the riseof the cross section just above the threshold is � / H , theZH pair is in a S-wave. Then the parity conservation inZ� ! ZH indicates that the parity of Higgs is plus. At wellabove threshold, Z in the final state is mostly helicity 0.Since the intermediate Z� is polarized along the beamdirection, the angular distribution of spin-0 Higgs is givenby jd1
1;0ðÞj2 / sin2 . The spin parity of Higgs can also be
checked in eþe� ! ZH! f �ff f �ff or in H! WW�, ZZ� !f �ff f �ff where f stands for a fermion.22) One can also study thespin correlation of the final state �’s in H! �þ�� to extractthe CP of Higgs.23)
The Higgs-strahlung process allows one to measure theZZH coupling independently of the Higgs decay modes.On the other hand, the WW fusion process gives the WWH
coupling. At low CM energy, the WW fusion processeþe� ! � ���H has a substantial background coming fromeþe� ! ZH, Z ! � ��� which can be removed by lookingat the recoil mass of Higgs. Also, the WW fusion processcan be turned off and on by switching the beam polar-izations to identify the contribution from the WW fusionprocess. The WWH coupling can also be extracted from theH! WW� branching fraction. The statistical errors onWWH and ZZH couplings for mH of 120 GeV are 1– 2%.For the Higgs mass below 150 GeV, the couplings of Higgsto b, c, and � are measured by reconstructing the Higgsdecays to b �bb, c �cc, and �þ�� in the Higgs-strahlung process.Here, the branching ratios are proportional to the squareof the fermion mass, and the excellent vertexing capabilityof ILC detectors is essential in separating c �cc from b �bb.The t�ttH Yukawa coupling is measured by eþe� ! t�tt� !t�ttH at 1 TeV. The process eþe� ! t�tt is itself sensitive tothe t�ttH coupling through the H-loop vertex correction.The gluonic decay H! gg as well as the decays H! ��,�Z are sensitive to the t�ttH coupling though top loop, andalso sensitive to new heavy particles that may contributein the loop. For high Higgs masses, the gauge boson pairfinal states dominate. Still, with 1 ab�1 at 1 TeV, the b �bb
branching fraction can be measured to 12 and 28% formH ¼ 180 and 220 GeV, respectively. Invisible final statecan also be found by the recoil mass technique, with 5�confidence down to 2% branching fraction for 120 < mH <160 GeV.
The total Higgs width for mH less than �200 GeV istoo narrow to be measured directly, but can be indirectlymeasured by �H ¼ �ðH ! WW�Þ=BrðH! WW�Þ whereBrðH! WW�Þ is directly measured and �ðH! WW�Þ isestimated from the measurement of the WWH coupling by,say, the WW fusion process. For 120 < mH < 160 GeV, thetotal Higgs width can be measured with an error of 4 to 13%.
The trilinear Higgs coupling, or the Higgs self coupling,can be measured by eþe� ! ZH� ! ZHH or by eþe� !� ���H� ! � ���HH. The cross section is quite small and the finalstate ðb �bbÞðb �bbÞð‘þ‘�Þ challenges the capability of detector.Here, a superb vertexing resolution is critical for the b
Fig. 5. (Color online) Higgs decay branching fractions as functions of
Higgs mass.21)
Fig. 6. (Color online) The �� recoil mass distribution in eþe� ! ��Xwith 500 fb�1 at CM energy of 300 GeV.6) The peaks correspond to
eþe� ! ZH followed by Z ! �� with different values of mH , together
with the background from eþe� ! ZZ, Z ! ��.7)
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tagging, and an excellent jet energy reconstruction is neededfor calculating the invariant masses of b jet pairs. With1 ab�1 at 500 GeV CM energy and for mH ¼ 200 GeV, theerror on the Higgs self coupling constant �HHH is estimatedto be about 20% using the eþe� ! ZH� ! ZHH modeonly.24) If one combines eþe� ! ZHH and eþe� ! � ���HHat 1 TeV CM energy, the error in �HHH becomes 12% for thesame Higgs mass with 1 ab�1 and 80% electron polar-ization.25)
Expected results for Higgs coupling measurements areplotted in Fig. 7 as functions of mass of the particle thatHiggs couples to. Coupling constants of Higgs to fermions,weak bosons W and Z, and Higgs itself are given by mf =a,gmW , gmZ=2 cos W , and m2
H=2a, respectively, where g �0:65 is the SUð2Þ coupling constant, and a � 246 GeV is thevacuum expectation value of Higgs. Thus, when properlynormalized, the Higgs couplings of the standard modelshould be proportional to the mass of the particle it couplesto. The pattern of deviation from the standard model servesas a powerful probe of the mechanism of mass generation.For example, for a two-Higgs-doublet model where up-typefermion masses are generated by one doublet and down-typefermion masses by another (so-called Type-II two-Higgs-doublet models), the Higgs couplings to all the up-typefermiosns are shifted by a factor, and those to all the down-type fermions are shifted by another factor. And in modelswith Radion-Higgs mixing, the Higgs couplings may bereduced uniformly with respect to the standard modelvalues.
4. New Physics Particles
Among the extensions of the standard model, the SUSYmodels occupy a special place due to their theoreticalvirtues, the primary one of which is to make the Higgs massstable in the weak scale. There are also other models thataddress the same problem, and these models usually containparticles that do not appear in the standard model. Oneshould keep in mind, however, that Nature may have in storefor us something that have nothing to do with any of these,and we may be lucky enough to encounter them at LHC/ILC.
4.1 SUSY particlesThe MSSM is the most economical model with R-parity
conservation which makes the lightest superparticle (LSP)stable. The LSP thus becomes a candidate for the darkmatter. The two complex Higgs doublets and the fourmassless gauge bosons have 8 charged degrees of freedomand 8 neutral degrees of freedom. After breaking of SUSYand gauge symmetries, their super partners mix to form twocharginos ��1;2 (8 degrees of freedom) and four neutralinos�0
1;2;3;4 (8 degrees of freedom) all with spin 1/2. Theneutralinos are self-conjugate; namely, they are Majoranaparticles. For each fermion f , there are two spin-0 super-partners corresponding to two helicities of the fermion: ~ff Rand ~ff L which could in general mix, particularly for the thirdgeneration fermions. Since the actual masses of each particleand its super-partner are clearly different, the supersymme-try is broken by some mechanism. One popular model isa minimal model with gravity-mediated SUSY breaking(mSUGRA) in which there are only four free parametersand a sign, which may be taken as the mass parameters ofscalers and winos: m0 and M2, the trilinear Higgs couplingA0, the ratio of vacuum expectation values of the two Higgsdoublets, tan, and signð�Þ where � is a Higgs massparameter. For concreteness, we look for SUSY particles inthis section with mSUGRA as a guide.
In many scenarios of SUSY, the super-partners of leptons(sleptons) are light enough to be produced at ILC. Inaddition, they tend to decay to the corresponding leptonplus the LSP neutralino. In the scenario called SPS1a ofmSUGRA, all sleptons decay dominantly as ~‘‘! ‘�0
1,where ‘ is a lepton and ~‘‘ is its super-partner. Decays andinteractions of right-handed sleptons are particularly simplesince they are SUð2ÞL singlets and thus do not interact withSUð2ÞL gauge particles. Figure 8 demonstrates simultaneousmass determination of the right-handed smuon ~��R and theLSP neutralino �0
1 in eþe� ! ~��þR ~���R followed by ~��þR !�þ�0
1 and its charge conjugate mode. The data is taken wellabove the threshold with 100 fb�1 at 350 GeV CM energy.The smuon and the LSP masses are assumed to be 142 and118 GeV, respectively. The high and low end points of the
Fig. 7. (Color online) The Higgs coupling constants as functions of mass
of the particle that Higgs couples to.7) The couplings with gauge bosons
and the self coupling are normalized differently from those with fermions.
Fig. 8. (Color online) The muon energy distribution in smuon pair
production at well above threshold,7) eþe� ! ~��þR ~���R , ~��þR ! �þ�01.
The high and low end points gives both m� and m ~��.
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muon energy distribution gives both masses to a few �10�3
of themselves. This is in contract to the LHC case wherethe mass of LSP is difficult to measure directly. This modealso illustrates the effectiveness of beam polarization inbackground reduction. The muon acoplanarity distributionin eþe� ! ~��þR ~���R , ~��þR ! �þ�0
1 is shown in Fig. 9 for noelectron polarization and with 90% electron polarization.Here, the acoplanarity angle is the angle between the muonpair projected to a plane perpendicular to the beam line. Bypolarizing the electron right-handedly, one can eliminate thebackground caused by eþe� ! WþW�, Wþ ! �þ� and itscharge conjugate. This is because for the s-channel the initialstate e�R limits the intermediate state to B (the gauge boson ofhypercharge Y) which does not couple to W in the final state,and the t-channel neutrino exchange is a V–A interactionwhich does not couple to e�R .
The angular distribution of the smuon production shouldbe sin2 since smuon is spin 0 and the intermediate Z=�state is polarized as j1;�1i along the beam line since theelectron coupling to the intermediate state is a linearcombination of vector and axial vector. The productionangle can be reconstructed with a quadratic ambiguity wherethe wrong solution has a flat distribution that can besubtracted. The resulting angular distribution can be checkedto be consistent with the expected shape.
The smuon mass can also be determined at the threshold,where an energy scan gives the threshold excitation curvewhich should rise slowly as 3
~�� due to the P wave nature ofthe smuon pair.
A large mixing effect is expected for the stau sector and~��R and ~��L would mix to form mass eigenstates ~��1 and ~��2where ~��1 is defined to be the lighter of the two. The mixingangle can be determined by two or more measurementsof eþe� ! ~��þ1 ~���1 with different beam polarizations. In theSPS1a scenario mentioned earlier, ~��1 is the lightest of thesleptons with its mass around 100 GeV, and the dominantdecay is ~��1! ��0
1. In this case, the mixing angle (cos 2)can be determined at the percent level.
The situation for the chargino pair production is similar tothat of smuon pair production: eþe� ! �þ1 �
�1 followed by
��1 ! �01W�, where the energy distribution of W� simulta-
neously determines the masses of the chargino ��1 and theLSP neutralino. With the mass of the LSP obtained in the
smuon study, the mass of the lightest chargino ��1 can bedetermined to 1% level.
For neutralinos, the invariant mass distribution of thelepton pair in eþe� ! �0
2�01 followed by �0
2! �01‘þ‘� can
determine the mass difference between �02 and �0
1 to betterthan 1%. This mode may also demonstrate a sizable CPviolation effects for some parameter space of MSSM. Forexample, the sign asymmetry of the T-odd triple productpe� � ðp‘þ � p‘�Þ can be as large as 20%.27) Similar T-oddtriple products can be formed for other modes such aseþe� ! �þ1 �
�1 .
The ability to select the beam polarization allows usto probe into the structures of the SUSY models. Forexample, the charginos ��1;2 are the mass eigen states of thesystem composed of the charged Higgsinos ð ~HHþu ; ~HH�d Þ andcharged gauginos ( ~WW�) where the mass matrix term can bewritten as
~WWþ ~HHþu� � M2
ffiffiffi2p
mW cos ffiffiffi2p
mW sin �
!~WW�
~HH�d
!:
By using a right-handed electron beam for eþe�R ! �þ1 ��1 ,
the intermediate s-channel state is purely B which is thegauge boson for hypercharge Y . On the other hand, B
couples only to the Higgsino component of chargino; thus,one can obtain information on the mixing parameters ofthe charginos. Together with cross section measurementsof eþe�R ! ~eeþR ~ee�R which is sensitive to the mass parameterM1 which is the mass parameter for Bino (superpartnerof B), one can perform a global fit to the parametersðM1;M2; �; tan Þ. Figure 10 shows the result of the globalfit. If the masses M1 and M2 are to converge to a single valueat the GUT scale, they would satisfy the GUT relation
M1 ¼5
3tan2 WM2;
which is tested in a highly model independent way.In most SUSY scenarios, squarks are in general heavier
than sleptons and many of them are beyond the reach ofILC even with the energy upgrade to 1 TeV. Still, due to thelarge mixing effects expected for the third generationsquarks ~tt and ~bb, the lighter ones, ~tt1 and ~bb1, can be withinreach of ILC. When they can be pair produced as in
Fig. 9. (Color online) The muon acoplanarity distri-
butions for the smuon production7) eþe� ! ~��þR ~���R ,
~���R ! ���01. With no electron polarization (left)
and with 90% electron right-handed polarization.
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eþe� ! ~tt1 ~tt1, then multiple measurements of cross sectionswith different beam polarizations can determine the mixingangle just as in the case of the stau pair production with asimilar precision.
4.2 KK mode gravitonsIn the models with large extra dimensions where only
gravitons can propagate in the extra dimensions, thefundamental gravity mass scale MD can be as small as theTeV scale.3) When the wave function of the graviton has acertain number of nodes in the direction of the extradimensions (KK modes), it can have mass as a function ofthe number of nodes. When the number of the extradimension � is 2 to 6, the size of extra dimension can be verylarge and is around 0.1 mm to 1 fm, for which the KK modegraviton GKK has effectively a continuous mass spectrum. AtILC, one may search for emission of KK mode gravitonin eþe� ! �GKK where GKK escapes the detector andappear as missing energy. Here again, the beam polarizationis a powerful handle to suppress the main backgroundeþe� ! � ����. With 1 ab�1 at 800 GeV and with the electronand positron beam polarizations of 80 and 60% respectively,the 95% confidence level lower limit of MD is 10 (3) TeV for� of 2 (6). This is similar to the sensitivities at LHC. At ILC,however, one can utilize the angular distribution of � toverify the spin of GKK which should be two. In addition, thenumber of extra dimension � can be measured at ILC by theenergy dependence of the cross section, say at 500 GeV vs at800 GeV, and the missing mass distribution.
4.3 Little Higgs modelsIn the Little Higgs models, the Higgs particle is
composite, and there exist extra gauge bosons and toppartners.4) Most new particles are too heavy to be directlydetected at ILC, but indirect search for extra gauge bosonsis possible with eþe� ! f �ff as described earlier (Fig. 2).Furthermore, in the model with T-parity, there could be apseudo-axion below 1 TeV. In such cases, eþe� ! ZHH
can be substantially enhanced by ZH coupling: eþe� !Z� ! �H, � ! ZH which should be easily detectable withthe TeV upgrade of the machine.
4.4 Cosmological connectionsThe WMAP satellite data indicates that the cold dark
matter density of the universe is given by �DMh2 ¼ 0:113�
0:009 and makes up about 1/4 of the energy of theuniverse.26) The error on �DM will be reduced significantlyby the Planck measurements expected around 2010. Inthe MSSM, the lightest neutralino �0
1 serves as a candidatefor the cold dark matter. In order to predict the relic densityof the cold dark matter, however, all interactions contribu-ting to �0
1 annihilation should be known. Figure 11 showsthe result of a study within the mSGURA SPS1a scenario.The sensitivities of LHC and ILC in the two-dimensionalspace of m�0
1and the estimated error on �DM are shown
together with the uncertainties on �DM by WMAP andPlanck. ILC can determine the mass of �0
1 much moreaccurately than LHC, and the error on the estimate of �DM iscomparable to the error expected for the future measurementby Planck.
5. Options: �� and e�e� Colliders
The eþe� mode of ILC can accomodate e�e� and ��colliders with relatively minor modifications. The ��collider requires a pair of powerful lasers that are aimed atthe interaction point from both sides along the beam line.The photons that are Compton back-scattered by incomingbeams collide at the interaction point. The maximum CMenergy of the �� collision is only slightly lower than that ofthe eþe� collision, and the luminosity is also comparable.The disrupted beams, however, need to be extracted withouthitting the sensitive detector parts, and this necessitatesa crossing angle greater than 25 mrad (compared to thenominal 14 mrad). Also, the original beams also collide ontop of the �� collisions, and this favors the e�e� mode overthe eþe� mode which has larger total cross section. Thee�e� is suited for the �� collision also because it is easier toproduce polarized electrons than polarized positrons.
The Higgs particle can be produced by the s-channel�� ! H process which involves loop diagrams of chargedparticles. It allows a precision measurement of the Higgscoupling to photon and is sensitive to new particles that cancontribute in the loop. The Higgs mass reach is close to the
Fig. 11. (Color online) The error in �DM vs the lightest neutralino mass
as constrained by LHC and ILC data within the framework of mSUGRA
with parameter set SPS1a.28)
Fig. 10. (Color online) The result of a global fit7) to eþe�R ! ~eeþR ~ee�R and
eþe�R ! �þ1 ��1 . The line shows the GUT relation between M1 and M2.
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CM energy of the e�e� beams itself. Higgs below 140 GeVwould be detected in the b �bb final state. With 410 fb of ��luminosity at the beam CM energy of 210 GeV, and formH ¼ 120 GeV, �ðH! ��Þ � BrðH! b �bbÞ can be deter-mined to a statistical error of 2%.29) Even for heavy Higgs of200 to 350 GeV, the two photon width can be determinedwith errors of 3 to 10%. The total Higgs decay width can beobtained by combining the �ðH! ��Þ measurement withBrðH! ��Þ measured at the eþe� collider at high CMenergy. The expected error on the Higgs total width is about5% for mH ¼ 120 to 140 GeV, which is competitive with themethod using �H ¼ �ðH ! WW�Þ=BrðH ! WW�Þ men-tioned earlier.
The e�e� collider can generate exotic charge twoparticles in s-channel. It is also sensitive to Majorananeutrino exchange in e�e� ! W�W�. The neutralinoexchange interactions e�e� ! e�L;Re
�L;R allows one to study
the quantum numbers of selectrons through beam polar-izations.
6. Summary
The clean environment and the well-defined initial stateof eþe� collision, including the spin states, as well as thesuperb resolutions of the ILC detectors make the ILCphysics program very attractive. ILC can study the particlesfound at LHC in detail to uncover the underlying theoreticalstructures, and in some cases discover new particles andreactions that are buried in backgrounds at LHC.
Acknowledgment
The author would like to thank the editorial panel of thedetector concept report (DCR) who has put together anexcellent summary of the ILC physics studies performed sofar, and all those who have contributed to these studies. Hewould like to thank, in particular, Professor Komamiya whohas entrusted him to write this article. This work is supportedin part by a Grant-in-Aid for Scientific Research 18GS0202from the Japan Society for the Promotion of Science.
1) E. Gildener: Phys. Rev. D 14 (1976) 1667, and references therein.
2) See, for example, S. Martin: hep-ph/9709356.
3) N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali: Phys. Rev. D 59
(1999) 086004, and references therein.
4) H.-C. Cheng and I. Low: J. High Energy Phys. JHEP08 (2004) 061,
and references therein.
5) WWS DCR panel, the detector concept report (DCR) (http://
www.linearcollider.org/wiki/doku.php).
6) Particle Physics at Experiments at JLC, KEK-Report 2001-11.
7) ‘‘GLC project’’, KEK Report 2003-7, September 2003.
Cosmic Acceleration, Dark Energy, and Fundamental Physics
Michael S. TURNER1;2;3� and Dragan HUTERER1;2y
1Kavli Institute for Cosmological Physics, The University of Chicago, Chicago, IL 60637, U.S.A.2Department of Astronomy and Astrophysics, The University of Chicago, Chicago, IL 60637, U.S.A.
3Department of Physics, The University of Chicago, Chicago, IL 60637, U.S.A.
(Received April 17, 2007; accepted May 14, 2007; published November 12, 2007)
A web of interlocking observations has established that the expansion of the Universe is speeding upand not slowing, revealing the presence of some form of repulsive gravity. Within the context of generalrelativity the cause of cosmic acceleration is a highly elastic (p � ��), very smooth form of energycalled ‘‘dark energy’’ accounting for about 75% of the Universe. The ‘‘simplest’’ explanation for darkenergy is the zero-point energy density associated with the quantum vacuum; however, all estimates forits value are many orders-of-magnitude too large. Other ideas for dark energy include a very light scalarfield or a tangled network of topological defects. An alternate explanation invokes gravitational physicsbeyond general relativity. Observations and experiments underway and more precise cosmologicalmeasurements and laboratory experiments planned for the next decade will test whether or not darkenergy is the quantum energy of the vacuum or something more exotic, and whether or not generalrelativity can self consistently explain cosmic acceleration. Dark energy is the most conspicuous exampleof physics beyond the standard model and perhaps the most profound mystery in all of science.
KEYWORDS: cosmology, gravitation, early universe, cosmological parametersDOI: 10.1143/JPSJ.76.111015
1. Quarks and the Cosmos
The final 25 years of the 20th century saw the rise of twohighly successful mathematical models that describe theUniverse at its two extremes, the very big and the very small.The standard model of particle physics (detailed in thisvolume) provides a fundamental description of almost allphenomena in the microscopic world. The standard hot bigbang model describes in detail the evolution of the Universefrom a fraction of a second after the beginning, when it wasjust a hot soup of elementary particles, to the present some13.7 billion years later when it is filled with stars, planets,galaxies, clusters of galaxies and us.1) Both standard modelsare consistent with an enormous body of precision data,gathered from high-energy particle accelerators, telescopesand laboratory experiments. The standard model of particlephysics and the hot big bang cosmology surely rank amongthe most important achievements of 20th century science(see Fig. 1).
Both models raise profound questions. Moreover, the ‘‘bigquestions’’ about the very small and the very large areconnected, both in their asking and ultimately in theiranswering. This suggests that the deeper understanding thatlies ahead will reveal even more profound connectionsbetween the quarks and the cosmos. The big questionsinclude
. How are the forces and particles of nature unified?
. What is the origin of space, time and the Universe?
. How are quantum mechanics and general relativityreconciled?
. How did the baryonic matter arise in the Universe?
. What is the destiny of the Universe?
. What is the nature of the dark matter that holds the
Universe together and of the dark energy that is causingthe expansion of the Universe to speed up?
The last question illustrates the richness of the connec-tions between quarks and the cosmos: 96% of the matter andenergy that comprises the Universe is still of unknown form,is crucial to its existence, and determines its destiny. Darkmatter and dark energy are also the most concrete andpossibly most important evidence for new physics beyondthe standard model of particle physics.
The solution to the dark matter problem seems withinreach: we have a compelling hypothesis, namely that itexists in the form of stable elementary particles left overfrom the big bang; we know that a small amount of darkmatter exists in the form of massive neutrinos; we have twogood candidates for the rest of it (the axion and neutralino)and an experimental program to test the particle dark matterhypothesis.2)
The situation with cosmic acceleration and dark energy isvery different. While we have compelling evidence that theexpansion of the Universe is speeding up, we are far from aworking hypothesis or any significant understanding ofcosmic acceleration. The solution to this profound mysterycould be around the corner or very far away.
2. Evidence for Cosmic Acceleration
2.1 Cosmology basicsFor mathematical simplicity Einstein assumed that the
Universe is isotropic and homogeneous; today, we have goodevidence that this is the case on scales greater than 100 Mpc(from the distribution of galaxies in the Universe) and that itwas at early times on all scales (from the uniformity of thecosmic microwave background). Under this assumption, theexpansion is merely a rescaling and is described by a singlefunction, the cosmic scale factor, RðtÞ. (By convention, thevalue of the scale factor today is set equal to 1.) Thewavelengths of photons moving through the Universe scale
with RðtÞ, and the redshift that light from a distant objectsuffers, 1þ z ¼ �rcvd=�emit, directly reveals the size of theUniverse when that light was emitted: 1þ z ¼ 1=RðtemitÞ.
The key equations of cosmology are
H2 � ð _RR=RÞ2 ¼8�G�
3�
k
R2þ
�
3ð1Þ
€RR=R ¼ �4�G
3ð�þ 3pÞ þ
�
3ð2Þ
wi �pi
�i�i / ð1þ zÞ3ð1þwiÞ ð3Þ
qðzÞ � �€RR
RH2¼
1
21þ 3wð Þ; ð4Þ
where � is the total energy density of the Universe (sum ofmatter, radiation, dark energy) and p is the total pressure.For each component the ratio of pressure to energy densityis the equation-of-state wi which, through the conservationof energy, dðR3�Þ ¼ �pdR3, determines how the energydensity evolves. For constant w, � / ð1þ zÞ3ð1þwÞ: Formatter (w ¼ 0) �M / ð1þ zÞ3 and for radiation (w ¼ 1=3)�R / ð1þ zÞ4. The first of these equations, known as theFriedmann equation, is the master equation of cosmology.
The quantity k is the 3-curvature of the Universe andRcurv � R=
ffiffiffiffiffijkjp
is the curvature radius; k ¼ 0 corresponds toa spatially flat Universe, k > 0 a positively curved Universeand k < 0 a negatively curved Universe. Because of theevidence from the cosmic microwave background that theUniverse is spatially flat (see Fig. 1), unless otherwise notedwe shall assume k ¼ 0.
� is Einstein’s infamous cosmological constant; it isequivalent to a constant energy density, �� ¼ �=8�G, withpressure p� ¼ ��� (w ¼ �1). The quantity qðzÞ is thedeceleration parameter, defined with a minus sign so thatq > 0 corresponds to decelerating expansion.
The energy density of a flat Universe (k ¼ 0), �C �3H2=8�G, is known as the critical density. For a positivelycurved Universe, �TOT � �=�C > 1 and for a negativelycurved Universe �TOT < 1. Provided the total pressure isgreater than �1=3 times the total energy density, gravityslows the expansion rate, i.e., €RR < 0 and q > 0. Because ofthe (�þ 3p) term in the €RR equation (Newtonian gravitywould only have �), the gravity of a sufficiently elastic formof energy (p < ��=3) is repulsive and causes the expansionof the Universe to accelerate. In Einstein’s static solution(H ¼ 0, q ¼ 0) the repulsive gravity of � is balanced againstthe attractive gravity of matter, with �� ¼ �M=2 and Rcurv ¼1=
Astronomers determine the luminosity distance from thedifference between the apparent magnitude m of the object(proportional to the log of the flux) and the absolutemagnitude M (proportional to the log of the intrinsicluminosity), m�M ¼ 5 log10ðdL=10 pcÞ (where 5 astronom-ical magnitudes correspond to a factor of 100 in flux or afactor of 10 in luminosity distance).
The use of ‘‘standard candles’’ (objects of known intrinsicluminosity L) and measurements of the energy flux F
constrain the cosmological model through this equation.In particular, the Hubble diagram (or magnitude-redshiftdiagram) is the simplest route to probing the expansionhistory. In terms of the deceleration parameter the equationis deceptively simple:
H0dL ¼ zþ1
2ð1� q0Þz2 þ � � � ð6Þ
where the subscript ‘‘0’’ denotes the value today. While thisTaylor expansion of eq. (5), valid for z� 1, is of historicalsignificance and utility, it is not useful today since objectsas distant as redshift z � 2 have been used to probe theexpansion history. However, it does illustrate the generalprinciple: the first term on the r.h.s. represents the linearHubble expansion, and the deviation from a linear relationreveals the deceleration (or acceleration).
2.2 �’s checkered historyBefore discussing the evidence for cosmic acceleration,
we will recount some of the history of the cosmologicalconstant. Realizing that there was nothing to forbid such aterm and that it could be used to obtain an interestingsolution (a static and finite Universe), Einstein introducedthe cosmological constant in 1917. While his static solutionwas consistent with astronomical observations at that time,Hubble’s discovery of the expansion of the Universe in 1929led Einstein to discard the cosmological constant in favor ofexpanding models without one, calling the cosmologicalconstant ‘‘my greatest blunder’’.
Fig. 1. (Color online) Multipole power spectrum of the CMB temperature
fluctuations from WMAP and other CMB anisotropy experiments.
Position of the first peak at l ’ 200 indicates the flatness of the Universe;
height of the first peak determines the matter density, and the ratio of the
first to second peaks determines the baryon density. Together with SDSS
large-scale structure data, the CMB measurements have determined the
shape and composition of the Universe: �TOT ¼ 1:003� 0:010, �M ¼0:24� 0:02, �B ¼ 0:042� 0:002, and �� ¼ 0:76� 0:02.18) The curve
is the theoretical prediction of the ‘‘concordance cosmology’’, with a band
that indicates cosmic variance. Figure adopted from ref. 19.
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In 1948, Bondi, Gold, and Hoyle put forth the ‘‘steadystate cosmology’’, with �� > 0 and �M ’ 0. The model wasmotivated by the aesthetics of an unchanging universe and aserious age problem (the measured value of the Hubbleconstant at the time, around 500 km s�1 Mpc�1 implied anexpansion age of only 2 Gyr, less than the age of Earth). Theredshift distribution of radio galaxies, the absence of quasarsnearby and the discovery of the cosmic microwave back-ground radiation in 1960s all indicated that we do not livein an unchanging Universe and ended this revival of acosmological constant.
The cosmological constant was briefly resurrected in thelate 1960s by Petrosian et al.3) to explain the preponderanceof quasars at redshifts around z � 2 (as it turns out, this is areal effect: quasar activity peaks around z � 2). In 1975weak evidence for a cosmological constant from a Hubblediagram of elliptical galaxies extending to redshifts of z �0:5 was presented.4) Significant concerns about whetheror not elliptical galaxies were good standard candles led tothe demise of � once again. Shortly thereafter came the riseof the standard cosmology with � ¼ 0.
The current attempt at introducing a cosmologicalconstant (or something similar), which is backed up bymultiple lines of independent evidence, traces its roots to theinflationary universe scenario and its prediction of a spatiallyflat Universe. In the early 1980s when inflation wasintroduced, the best estimate of the average mass densityfell short of the critical density by almost a factor of 10(�M � 0:1); the saving grace for inflation was the largeuncertainty associated with measuring the mean matterdensity. From 1980 to the mid 1990s, as measurementtechniques took better account of dark matter, �M rose to oforder 0.5 or so. However, as the uncertainties got smaller,�M began converging on a value of around 1/3, not 1.Moreover, the predictions of the cold dark matter scenarioof structure formation matched observations if �M wasaround 1/3, not 1.
Starting in 1984 and continuing to just before thediscovery of cosmic acceleration, a number of paperssuggested the solution to inflation’s ‘‘� problem’’1) was acosmological constant.5) Owing to its checkered history,there was not much enthusiasm for this suggestion at first.However, with time the indirect evidence for � grew,6–8)
and in 1998 when the supernova evidence for acceleratedexpansion was presented the cosmological constant wasquickly embraced — this time, it was the missing piece ofthe puzzle that made everything work.
2.3 Discovery and confirmationTwo breakthroughs enabled the discovery that the
Universe is speeding up and not slowing down. The firstwas the demonstration that type Ia supernovae (SNe Ia), thebrightest of the supernovae and the ones believed to beassociated with the thermonuclear explosions of 1.4 Mwhite-dwarf stars pushed over the Chandrasekhar mass limitby accretion, are (nearly) standard candles.10) The secondbreakthrough involved the use of large (of order 100megapixel) CCD cameras to search big regions of the skycontaining thousands of galaxies for these rare events (theSN Ia rate in a typical galaxy is of the order of one per 100 to200 years). By comparing images of thousands of galaxies
taken weeks apart the discovery of SNe could be reliably‘‘scheduled’’ on a statistical basis.
Two teams working independently in the mid- to late-1990s took advantage of these breakthroughs to determinethe expansion history of the Universe. They both found thatdistant SNe are dimmer than they would be in a deceleratingUniverse, indicating that the expansion has actually beenspeeding up for the past 5 Gyr;11,12) see Fig. 2. Analyzed fora Universe with matter and cosmological constant, theirresults provide evidence for �� > 0 at greater than 99%confidence; see Fig. 3.
Since this work, the two teams have discovered andstudied more SNe, as have other groups.13–16) Not onlyhas the new data confirmed the discovery, but it has alsoallowed measurements of the equation-of-state of darkenergy w ¼ p=� (assuming constant w), and even constrainsthe time variation of w, with the parametrization w ¼ w0 þwað1� RÞ.
Especially important in this regard are SNe with redshiftsz > 1 which indicate that the universe was decelerating atearlier times (see Fig. 4), and hence that dark energy startedits domination over the dark matter only recently, at redshiftz ¼ ð�M=�DEÞ1=3w � 1 0:5. This finding is an importantreality check: without a long, matter-dominated, slowingphase, the Universe could not have formed the structure wesee today.
Evidence for dark energy comes from several otherindependent probes. Measurements of the fraction of X-rayemitting gas to total mass in galaxy clusters, fgas, also
Supernova Cosmology Project
=0.3, =0.7
=0.3, =0.0
=1.0, =0.0
m-M
(m
ag)
High-Z SN Search Team
0.01 0.10 1.00z
(m-M
) (m
ag)
Fig. 2. (Color online) Discovery data: Hubble diagram of SNe Ia meas-
ured by the Supernova Cosmology Project and High-z Supernova Team.
Bottom panel shows magnitudes relative to a universe with �TOT ¼�M ¼ 0:3. Figure adopted from ref. 9.
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indicates the presence of dark energy. Because galaxyclusters are the largest collapsed objects in the universe,the gas fraction in them is presumed to be constant and equalto the overall baryon fraction in the universe, �B=�M (mostof the baryons in clusters reside in the gas). Measurementsof the gas fraction fgas depend not only on the observedX-ray flux, but also on the distance to the cluster; therefore,only the correct cosmology will produce distances whichmake the apparent fgas constant in redshift. Using data fromthe Chandra X-ray Observatory, Allen et al.17) have deter-mined �� to an accuracy of about �0:2; see Fig. 3.
Cosmic microwave background (CMB) anisotropies pro-vide a record of the Universe at simpler time, beforestructure had developed and when photons were decouplingfrom baryons, z 1100.20) The multipole power spectrum isdominated by the acoustic peaks that arise from gravita-tionally driven photon–baryon oscillations (see Fig. 1). Thepositions and amplitudes of the acoustic peaks encode muchinformation about the Universe, today and at earlier times. In
particular, they indicate that the Universe is spatially flat,with a matter density that accounts for only about a quarterof the critical density. However, the presence of a uniformlydistributed energy density with large negative pressurewhich accounts for three-quarters of the critical densitybrings everything into good agreement, both with CMB dataand the large-scale distribution of galaxies in the Universe.The CMB data of WMAP together with large-scale structuredata of the Sloan Digital Sky Survey (SDSS) provides thefollowing cosmic census:18) �TOT ¼ 1:003� 0:010, �M ¼0:24� 0:02, �B ¼ 0:042� 0:002, and �� ¼ 0:76� 0:02.
The presence of dark energy also affects the large-angleanisotropy of the CMB (the low multipoles) and leads to theprediction of a small correlation between the galaxydistribution and the CMB anisotropy. This subtle effecthas been observed;21) while not detected at a level ofsignificance that could be called independent confirmation,its presence is a reassuring cross check.
Baryon acoustic oscillations (BAO), so prominent in theCMB anisotropy (see Fig. 1), leave a smaller characteristicsignature in the clustering of galaxies that can be measuredtoday and provide an independent geometric probe of darkenergy. Measurements of the BAO signature in the corre-lation function of SDSS galaxies constrains the distance toredshift z ¼ 0:35 to a precision of 5%.22) While this alonedoes not establish the existence of dark energy, it serves as asignificant complement to other probes, cf. Fig. 5.
Weak gravitational lensing23) — slight distortions of gal-axy shapes due to gravitational lensing by intervening large-scale structure — is a powerful technique for mapping darkmatter and its clustering. Currently, weak lensing shedslight on dark energy by pinning down the combination�8ð�M=0:25Þ0:6 0:85� 0:07, where �8 is the amplitudeof mass fluctuations on the 8 Mpc scale.25) Since othermeasurements put �8 at � 0:9, this implies that �M ’ 0:25,consistent with a flat Universe whose mass/energy densityis dominated by dark energy. In the future, weak lensingwill also be very useful in probing the equation-of-state ofdark energy;24) see §4.
0 0.5 1 1.5 2Redshift z
-0.5
0
0.5
1
Δ (m
-M)
always accelerates
accelerates nowdecelerates in the past
always decelerates
Fig. 4. (Color online) Evidence for transition from recent acceleration to
deceleration in the past. The Hubble diagram with measured distances to
SNe Ia is not fitted well by either a purely accelerating or purely
decelerating cosmological model, but rather one with early deceleration
and recent acceleration. SN data are binned in redshift and come from
ref. 14.
Fig. 5. (Color online) 68 and 95% C.L. constraints on the matter density
�M and dark energy equation-of-state w, assuming a flat Universe.
Constraints come from the Supernova Legacy Survey,15) WMAP,27) and
SDSS detection of BAO.22) The combined constraint is shown by the
central dark contours.
Fig. 3. (Color online) Three independent lines of evidence for cosmic
acceleration: galaxy clusters, CMB anisotropy, and SNe. Flatness is not
assumed, but w ¼ �1. Note the concordance of the independent methods;
evidence for �� > 0 is greater 99.9% C.L. Figure adopted from ref. 17.
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Finally, because the time back to the big bang, t0 ¼Rdz=ð1þ zÞHðzÞ, depends upon the expansion history,
the comparison of this age with other independent ageestimates can be used to probe dark energy. The ages ofthe oldest stars in globular clusters constrain the age ofthe Universe: 11 . t0 . 15 Gyr.26) CMB anisotropy is verysensitive to the expansion age, and WMAP data determineit accurately: t0 ¼ 13:84þ0:39
�0:36 Gyr.27) Figure 6 shows that aconsistent age is possible if �2 . w . �0:75. Agreementon the age of the Universe provides an important consistencycheck as well as confirmation of a key feature of darkenergy, its large negative pressure.
3. Understanding Cosmic Acceleration
Sir Arthur Eddington is quoted as saying, ‘‘It is (also) agood rule not to put too much confidence in observationalresults until they are confirmed by theory’’. While this mayseem a bit paradoxical (or worse yet, an example of blatanttheoretical arrogance), the point is well taken: science isnot just a collection of facts, it is also understanding; if theunderstanding does not eventually follow new facts, perhapsthere is something wrong with the facts.
Cosmic acceleration meets the Eddington criterion and atthe same time presents a stunning opportunity for theorists:General relativity (GR) can accommodate acceleratedexpansion, but GR has yet to provide a deeper understandingof the phenomenon.
Within GR, a very elastic fluid has repulsive gravity, and,if present in sufficient quantity, can lead to the observedaccelerated expansion. This then is the definition of darkenergy: the mysterious, elastic and very smooth form ofenergy which is responsible for cosmic acceleration and ischaracterized by an equation-of-state w ¼ p=� � �1.8)
Vacuum energy is a concrete example of dark energy.General covariance requires that the stress energy associatedwith the vacuum take the form of a constant times the metrictensor. This implies that it has a pressure equal to minus itsenergy density, is constant both in space and time, and ismathematically equivalent to a cosmological constant.
The stress energy associated with a homogeneous scalarfield � can also behave like dark energy. It takes the form ofa perfect fluid with
� ¼ _��2=2þ Vð�Þp ¼ _��2=2� Vð�Þ;
ð7Þ
where Vð�Þ is the potential energy of the scalar field, dotdenotes time derivative, and the evolution of the field � isgoverned by
€��þ 3H�þ V 0ð�Þ ¼ 0: ð8Þ
If the scalar field evolves slowly, that is _��2 � V , then p ’�� and the scalar field behaves like a slowly varyingvacuum energy.
While cosmic acceleration can be accommodated withinthe GR framework, the fundamental explanation could benew gravitational physics. With this as a prelude, we nowbriefly review the present theoretical situation.
(a) Vacuum energy. Vacuum energy is both the mostplausible explanation and the most puzzling possibility.For almost 80 years we have known that there shouldin principle be an energy associated with the zero-pointfluctuations of all quantum fields. Moreover, pVAC ¼��VAC. However, all attempts to compute the value of thevacuum energy lead to divergent results. The so-calledcosmological constant problem was finally articulated aboutthirty years ago.28) However, because of the success of thestandard hot big bang model (where � ¼ 0) and the absenceof good (or any) ideas, the problem was largely ignored.With the discovery of cosmic acceleration, the cosmologicalconstant problem is now front and center and can no longerbe ignored.
To be more quantitative, the energy density requiredto explain the accelerated expansion is about three quar-ters of the critical density or about 4� 10�47 GeV4 (3� 10�3 eV)4. This is tiny compared to energy scales inparticle physics (with the exception of neutrino mass dif-ferences). Such a small energy precludes solving the problemby simply cutting off the divergent zero-point energy integralat some energy beyond which physics is not yet known. Forexample, a cutoff of 100 GeV would leave a 54 orders-of-magnitude discrepancy. If supersymmetry were an unbrokensymmetry, fermionic and bosonic zero-point contributionswould cancel. However, if supersymmetry is broken at ascale of order M, one would expect that imperfect cancella-tions leave a finite vacuum energy of the order M4, which forthe favored value of M � 100 GeV to 1 TeV, would leave adiscrepancy of 50 or 60 orders-of-magnitude.
One approach to the cosmological constant probleminvolves the idea that the value of the vacuum energy is arandom variable which can take on different values indifferent disconnected pieces of the Universe. Because avalue much larger than needed to explain the observedcosmic acceleration would preclude the formation ofgalaxies, we could not find ourselves in such a region.29)
This very anthropic approach finds a home in the landscapeversion of string theory.30)
(b) Scalar fields, etc. While introducing a new dynam-ical degree of freedom can also provide a very elastic formof energy density, it does not solve the cosmologicalconstant problem. In order to roll slowly enough the mass
0 0.1 0.2 0.3 0.4 0.5 0.6Ω
M = 1 - Ω
DE
8
10
12
14
16
18
Age
of
the
univ
erse
(G
yr)
-2.0-0.75-1.0
w=-0.5
Globular clusters
WMAP
Fig. 6. (Color online) Age of the universe as a function of the matter
energy density, assuming a flat universe and four different values of
the dark energy equation-of-state. Also shown are constraints from
globular clusters26) and from WMAP27) and the range of �M favored by
measurements of the matter density. Age consistency holds for �2 .
w . �0:75.
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of the scalar field must be very light, m . H0 � 10�42 GeV,and its coupling to matter must be very weak to be consistentwith searches for new long-range forces.31) Unlike vacuumenergy, scalar-field energy clusters gravitationally, but onlyon the largest scales and with a very small amplitude.32)
The equation-of-state w for a scalar field can take onany value between �1 and 1 and in general varies withtime. (It is also possible to have w < �1, though at theexpense of ghosts, by changing the sign of the kinetic energyterm in the Lagrangian.) Scalar field models also raise newquestions and possibilities: Is cosmic acceleration relatedto inflation? Is dark energy related to dark matter orneutrino mass? No firm or compelling connections havebeen made to either, although interesting possibilities havebeen suggested.
Scalar fields in a very different form can also explaincosmic acceleration. The topological solitons that arise inbroken gauge theories, e.g., strings, walls, and textures, arevery elastic, and tangled networks of such defects can onlarge scales behave like an elastic medium with w ¼ �N=3,where N is the dimensionality of the network (N ¼ 1 forstrings, 2 for walls, and 3 for textures). In this case w is afixed, rational number.
(c) Modified gravity. A very different approach holdsthat cosmic acceleration is a manifestation of new gravita-tional physics and not of dark energy. Assuming that our 4-dspacetime can still be described by a metric, the operationalchanges are twofold: (1) a new version of the Friedmannequation governing the evolution of the background space-time; (2) modifications to the equations that govern thegrowth of the small matter perturbations that evolve into thestructure seen in the Universe today. A number of ideas havebeen explored, from models motivated by higher-dimen-sional theories and string theory33,34) to generic modifica-tions of the usual gravitational action.35)
An aside: One might be concerned that when theassumption of general relativity is dropped the evidencefor accelerated expansion might disappear. This is not thecase; using the deceleration qðzÞ as a kinematic descriptionof the expansion, the SNe data still provide strong evidencefor a period of accelerated expansion.36)
Changes to the Friedmann equation are easier to derive,discuss, and analyze. In order not to spoil the success of thestandard cosmology at early times (from big-bang nucleo-synthesis to the CMB anisotropy to the formation ofstructure), the Friedmann equation must reduce to the GRform for z� 1. Because the matter term scales as ð1þ zÞ3and the radiation term as ð1þ zÞ4, to be safe any modifica-tions should decrease with redshift more slowly than this.As a specific example, consider the DGP model, whicharises from a five-dimensional gravity theory,33) and has a4-d Friedmann equation,
H2 ¼8�G�M
3þ
H
rc; ð9Þ
where rc is an undetermined scale and �M is the matter enegydensity. As �M! 0, there is a (self) accelerating solution,with H ¼ 1=rc. The additional term in the Friedmannequation, H=rc, behaves just like dark energy with anequation-of-state that evolves from w ¼ �1=2 (for z� 1) tow ¼ �1 in the distant future.
4. Prospects for Revealing the Nature of Dark Energy
We divide the probes of dark energy into three broadcategories: kinematical and dynamical cosmological probes,and laboratory/astrophysical probes. Kinematical tests relyon the measurement of cosmological distances and volumesto constrain the evolution of the scale factor and thus thebackground cosmological model. Specific techniques in-clude SNe Ia, CMB, and baryon acoustic oscillations.
The dynamical tests probe the effect of dark energy onperturbations of the cosmological model, including theevolution of the small inhomogeneities in the matter densitythat give rise to structure in the Universe. Specifictechniques include the use of gravitational lensing to directlydetermine the evolution of structure in the dark matter andthe study of the growth of the abundance of galaxy clustersto indirectly probe the growth of structure. A potential probeof dark energy, which at the present seems beyond reach, isto study the clustering of dark energy itself. Since vacuumenergy does not cluster, detection of such would rule outvacuum energy as the explanation for cosmic acceleration.
In general relativity, for both kinematical and dynamicalcosmological probes, the primary effect of dark energyenters through the Friedmann equation, cf. eq. (1),
HðzÞ2 ¼8�G
3�M þ �DE
� �
¼ H20 �Mð1þ zÞ3 þ ð1��MÞð1þ zÞ3ð1þwÞ� �
; ð10Þ
where a flat Universe and constant w have been assumed. Inturn, the expansion rate affects the luminosity distance, dL ¼ð1þ zÞ
Rdz=HðzÞ, the number of objects seen on the sky,
d2N=ðd� dzÞ ¼ nðzÞd2L=½ð1þ zÞ2HðzÞ (n is the comoving
density of objects), and the evolution of cosmic structurevia the growth of small density perturbations. In GR thegrowth of small density perturbations in the matter, and onsubhorizon scales, is governed by
€��k þ 2H _��k � 4�G�M�k ¼ 0; ð11Þ
where density perturbations in the cold dark matter havebeen decomposed into their Fourier modes of wavenumberk. Dark energy affects the growth through ‘‘the drag term’’,2H _��k. The equations governing dark energy perturbationsdepend upon the specific dark energy model.
The kinematical and dynamical tests probe complemen-tary aspects of the effect of dark energy on the Universe: theoverall expansion of the Universe (kinematical) and theevolution of perturbations (dynamical). Together, they cantest the consistency of the underlying gravity theory. Inparticular, different values of the dark energy equation-of-state obtained by the two methods would indicate aninconsistency of the underlying gravity theory.
The kinematical tests are easier to frame because theyonly depend upon knowing the effect of dark energy on thebackground cosmological model. Further, cosmologicalvariables (such as q) can even be formulated withoutreference to a particular theory of gravity. The dynamicaltests are both harder to frame — they require detailedknowledge of how dark energy clusters and affects thegrowth of density perturbations — and also harder to imple-ment — they rely upon the details of describing and measur-ing the distribution of matter in the Universe. Both the
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kinematical and dynamical tests have their greatest probativepower at redshifts between about z ¼ 0:2 and z ¼ 2, for thesimple reason that at higher redshifts dark energy becomesincreasingly less important, �DE=�M / ð1þ zÞ3w.37)
While the primary probes of dark energy are cosmolog-ical, laboratory experiments may be able to get at theunderlying physics. If dark energy couples to matter therewill be long-range forces that are in principle detectable; if itcouples to electromagnetism, polarized light from distantastrophysical sources should suffer rotation.31) It is alsopossible that accelerator-based experiments will have some-thing to say about dark energy. For example, if evidence forsupersymmetry is found at the Large Hadron Collider,understanding how supersymmetry is broken could shedlight on the vacuum energy puzzle.
Observations to date have established the existenceof dark energy and have begun to probe its nature; e.g.,by constraining w �1� 0:1. Future experiments willfocus on testing whether or not it is vacuum energy and theconsistency of GR to accommodate dark energy. TheSupernova/Acceleration Probe (SNAP),38) a proposedspace-based telescope to collect several thousand SNeout to z 2, would significantly reduce uncertainties (bothstatistical and systematic) on dark-energy parameters.SNAP, together with the planned wide-field surveys fromthe ground, the Dark Energy Survey (DES)39) and the LargeSynoptic Survey Telescope (LSST),40) would map the weaklensing signal from one arcminute out to the largestobservable scales on the sky and accurately determine theeffect of dark energy on the growth of structure. Large BAOsurveys are also planned, both from the ground and space.The just-completed South Pole Telescope (SPT)41) and theAtacama Cosmology Telescope (ACT)42) will soon beginstudying dark energy by determining the evolution of theabundance of galaxy clusters. In 2008, ESA’s PlanckSurveyor CMB satellite43) will be launched and will extendprecision measurements of CMB anisotropy to l � 3000
(i.e., down to angular scales of about 5 arcmin), moreaccurately pinning down the matter density and providing animportant prior constraint for other dark energy measure-ments.44) Theoretical forecasts for future constraints on theparameters (w0;wa) are shown in Fig. 7.
5. Dark Energy and Destiny
One of the first things one learns in cosmology is thatgeometry is destiny: a closed (positively curved) Universeeventually recollapses and an open (flat or negatively curved)Universe expands forever. Provided that the Universe onlycontains matter and � ¼ 0, this follows directly from eq. (1).If k > 0, the Universe achieves a maximum size when H2 isdriven to zero by the inevitable cancellation of �M and k=R2.If k ¼ 0, R always grows as t2=3 and q ¼ 1=2. For k < 0, theUniverse ultimately reaches a coasting phase where R growsas t and q ¼ 0. Adding radiation only changes the storyat early times (see Fig. 8): because the radiation densityincreases as ð1þ zÞ4, for z� 3� 103 the Universe isradiation dominated and during this epoch, R / t1=2 andq ¼ 1 [this is a manifestation of the fact that gravity issourced by �þ 3p ¼ 2� for radiation, cf. eqs. (2) and (4)]. Itis only during the matter-dominated phase that small densityinhomogeneities are able to grow and form bound structures.
Dark energy provides a new twist: because the darkenergy density varies slowly if at all, it eventually becomesthe dominant form of matter/energy (around z � 0:5); seeFig. 8. After that, the expansion accelerates and structureformation ceases, leaving in place all the structure that hasformed. The future beyond the present epoch of acceleratedexpansion is uncertain and depends upon understanding darkenergy.
In particular, if dark energy is vacuum energy, acceler-ation will continue and the expansion will become expo-nential, leading inevitably to a dark Universe. (In a hundredbillion years, the light from all but a few hundred nearbygalaxies will be too redshifted to detect.) On the other hand,if dark energy is explained by a scalar field, then eventuallythe field relaxes to the minimum of its potential. If theminimum of the potential energy is zero, the Universe again
-1.5 -1 -0.5w
0
-2
-1
0
1
2
wa
Weak lensing
CMB (Planck)
SNe
All
Fig. 7. (Color online) Forecasts of future constraints to the dark energy
equation-of-state and its time evolution using SNe Ia, weak gravitational
lensing, and CMB anisotropy (measured by Planck). Future SNe Ia and
weak lensing estimates are both modeled on the SNAP experiment.
The large improvement in combining the various data sets is due to
breaking of the parameter degeneracies in the full (eight-dimensional)
parameter space.
-101234Log [1+z]
-48
-44
-40
-36
Log
[en
ergy
den
sity
(G
eV4 )]
radiation matter
vacuum energy
Fig. 8. (Color online) Evolution of dark matter, dark energy, and radia-
tion. Earlier than z � 3000 radiation dominates the mass/energy density
of the Universe; between z � 3000 and z � 0:5 dark matter dominates,
and thereafter dark energy dominates and the expansion accelerates.
Structure only grows during the matter dominated epoch.
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becomes matter dominated and returns to deceleratedexpansion. If the minimum of the scalar potential hasnegative energy density, the energy of dark matter and ofscalar field energy will eventually cancel, leading to arecollapse. Finally, if the potential energy at the minimumis positive, no matter how small, accelerated expansioneventually ensues again.
Absent dark energy geometry and destiny are linked. Thepresence of dark energy severs this relation7) and linksinstead destiny to an understanding of dark energy.
6. Summary
We end our brief review with our list of the ten mostimportant facts about cosmic acceleration
1. Independent of general relativity and based solely uponthe SN Hubble diagram, there is very strong evidence thatthe expansion of the Universe has accelerated recently.36)
2. Within the context of general relativity, cosmic accel-eration cannot be explained by any known form of matter orenergy, but can be accommodated by a nearly smooth andvery elastic (p � ��) form of energy (‘‘dark energy’’) thataccounts for about 75% of the mass/energy content of theUniverse.
3. Taken together, current data (SNe, galaxy clustering,CMB and galaxy clusters) provide strong evidence for theexistence of dark energy and constrain the fraction of criticaldensity contributed by dark energy to be 71� 5% and theequation-of-state to be w �1� 0:1 (stat) �0:1 (sys), withno evidence for variation in w. This implies that theUniverse decelerated until z � 0:5 and age � 10 Gyr, whenit began accelerating.
4. The simplest explanation for dark energy is the zero-point energy of the quantum vacuum, mathematicallyequivalent to a cosmological constant. In this case, w isprecisely �1, exactly uniformly distributed and constant intime. All extant data are consistent with a cosmologicalconstant; however, all attempts to compute the energy of thequantum vacuum yield a result that is many orders-of-magnitude too large (or is infinite).
5. There is no compelling model for dark energy.However there are many intriguing ideas including a newlight scalar field, a tangled network of topological defects, orthe influence of additional spatial dimensions. It has alsobeen suggested that dark energy is related to cosmicinflation, dark matter and neutrino mass.
6. Cosmic acceleration could be a manifestation ofgravitational physics beyond general relativity rather thandark energy. While there are intriguing ideas about correc-tions to the usual gravitational action or modifications to theFriedmann equation that can give rise to the observedaccelerated expansion, there is no compelling, self-consis-tent model for the new gravitational physics that explainscosmic acceleration.
7. Even assuming the Universe has precisely the criticaldensity and is spatially flat, the destiny of the Universedepends crucially upon the nature of the dark energy. Allthree fates — recollapse or continued expansion with andwithout slowing — are possible.
8. Cosmic acceleration is arguably the most profoundpuzzle in physics. Its solution could shed light on or becentral to unraveling other important puzzles, including the
cause of cosmic inflation, the vacuum-energy problem,supersymmetry and superstrings, neutrino mass, new grav-itational physics, and dark matter.
9. Today, the two most pressing questions about darkenergy and cosmic acceleration are: Is dark energy some-thing other than vacuum energy? Does general relativity selfconsistently describe cosmic acceleration? Establishing thatw 6¼ �1 or that it varies with time would rule out vacuumenergy; establishing that the values of w determined by thekinematical and dynamical methods are not equal wouldindicate that GR cannot self consistently accommodateaccelerated expansion.
10. Dark energy affects the expansion rate of theUniverse, which in turn affects the growth of structure andthe distances to objects. (In gravity theories other than GR,dark energy may have more direct effects on the growth ofstructure.) Upcoming ground- and space-based experimentsshould probe w at the percent level and its variation at theten percent level. These measurements should dramaticallyimprove our ability to discriminate between vacuum energyand something more exotic as well as testing the selfconsistency of general relativity. Laboratory- and acceler-ator-based experiments could also shed light on dark energy.
Because of its brevity, this review could not do justice tothe extensive literature that now exists; for readers interestedin a more thorough treatment of dark energy and/or a moreextensive review, we refer them to ref. 45.
1) E. W. Kolb and M. S. Turner: The Early Universe (Addison-Wesley,
Reading, MA, 1990).
2) N. J. C. Spooner: J. Phys. Soc. Jpn. 76 (2007) 111016.
3) V. Petrosian, E. E. Salpeter, and P. Szekeres: Astrophys. J. 147 (1967)
1222.
4) J. E. Gunn and B. M. Tinsley: Nature 257 (1975) 454.
5) M. S. Turner, G. Steigman, and L. M. Krauss: Phys. Rev. Lett. 52
(1984) 2090; P. J. E. Peebles: Astrophys. J. 284 (1984) 439; G.
Efstathiou, W. J. Sutherland, and S. J. Maddox: Nature 348 (1990)
705; M. S. Turner: Proc. IUPAP Conf. Primordial Nucleosynthesis and
the Early Evolution of the Universe, Tokyo, Japan, Sept. 4 – 8, 1990
(A93-17626 05-90), pp. 337–350.
6) J. P. Ostriker and P. J. Steinhardt: Nature 377 (1995) 600.
7) L. Krauss and M. S. Turner: Gen. Relativ. Gravitation 27 (1995)
1137.
8) M. S. Turner and M. White: Phys. Rev. D 56 (1997) 4439.
9) S. Perlmutter and B. P. Schmidt: in Supernovae and Gamma-Ray
Bursters, ed. K. Weiler (Springer, Heidelberg) Lecture Notes in
Physics, Vol. 598, pp. 195– 217.
10) M. M. Phillips: Astrophys. J. 413 (1993) L105.
11) A. Riess et al.: Astron. J. 116 (1998) 1009.
12) S. Perlmutter et al.: Astrophys. J. 517 (1999) 565.
13) R. A. Knop et al.: Astrophys. J. 598 (2003) 102.
14) A. Riess et al.: Astrophys. J. 607 (2004) 665; A. Riess et al.:
Astrophys. J. 659 (2007) 98.
15) P. Astier et al.: Astron. Astrophys. 447 (2006) 31.
16) W. M. Wood-Vasey et al.: astro-ph/0701041.
17) S. W. Allen, R. W. Schmidt, H. Ebeling, A. C. Fabian, and L. van
Speybroeck: Mon. Not. R. Astron. Soc. 353 (2004) 457.
18) M. Tegmark et al.: Phys. Rev. D 74 (2006) 123507.
19) WMAP web site: http://lambda.gsfc.nasa.gov/product/map/
20) W. Hu and S. Dodelson: Annu. Rev. Astron. Astrophys. 40 (2002)
171.
21) S. Boughn and R. Crittenden: Nature 427 (2004) 45; P. Fosalba and
E. Gaztanaga: Mon. Not. R. Astron. Soc. 350 (2004) 37; R. Scranton
et al.: astro-ph/0307335; N. Afshordi, Y.-S. Loh, and M. Strauss:
Phys. Rev. D 69 (2004) 083524.
22) D. J. Eisenstein et al.: Astrophys. J. 633 (2005) 560.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS M. S. TURNER and D. HUTERER
Institute of Underground Science, Department of Physics and Astronomy, University of Sheffield,
Hicks Building, Sheffield S3 7RH, U.K.
(Received May 23, 2007; accepted July 23, 2007; published November 12, 2007)
For many working in particle physics and cosmology successful discovery and characterisation of thenew particles that most likely explain the non-baryonic cold dark matter, known to comprise the majorityof matter in the Universe, would be the most significant advance in physics for a century. Reviewed hereis the current status of direct searches for such particles, in particular the so-called weakly interactingmassive particles (WIMPs), together with a brief overview of the possible future direction of the fieldextrapolated from recent advances. Current best limits are at or below 10�7 pb for spin-independentneutralino coupling, sufficient that experiments are already probing SUSY models. However, newdetectors with tonne-scale mass and/or capability to correlate signal events to our motion through theGalaxy will likely be needed to determine finally whether WIMPs exist.
KEYWORDS: dark matter, WIMPs, radiation detectors, bolometers, scintillators, TPCDOI: 10.1143/JPSJ.76.111016
1. Introduction
Building from the earliest measurements of galaxyclusters, through decades of observational cosmology,ground-based observation of the cosmic microwave back-ground (CMBR) and on in recent times to precisioncosmology measurements with the CMBR satellite WMAPand observations of distant supernovae,1,2) the remarkableconclusion we have is that the Universe is geometrically flat(� � 1� 0:04) but contains only �4% ordinary baryonicmatter. Even of this 4% only about 1/10th is actually visibleto us, as stars mainly, the rest is likely composed of cold gas,sub-solar mass ‘‘dead’’ stars and other forms of non-luminous baryonic matter. The great majority of theUniverse then is found to be a mix of mysterious darkenergy (�� � 73%), the nature of which is unknown, andnon-luminous, non-baryonic dark matter (� � 23%).3,4)
What form this dark matter takes is also so far unknown.However, a generic class of relic particles producedthermally in the early Universe and termed weakly interact-ing massive particles (WIMPs), has emerged as a leadingpossibility.5) Such, non-relativistic particles would constitutea cold dark matter (CDM) population that appears requiredto explain galaxy formation. The observed density requiredof them in the galaxy (� � 0:1), is consistent with thefreeze-out relic density appropriate if the mass and cross-section of the particles is determined by the weak scale.6,7)
The CDM model itself provides significant motivation tosearch for such weakly interacting neutral particles, theappropriate mass range being several GeV to �TeV.However, two leading theories in particle physics phenom-enology greatly enhance this motivation by also, independ-ently, predicting new particles with these features. Firstly, insupersymmetric (SUSY) extensions of the standard model,7)
the lightest SUSY particle (LSP), stable in models where R-parity is conserved, provides the required population, knownas neutralinos. Theories of universal extra dimensions in
which Kaluza–Klein parity is conserved provides a secondpossible class known as the lightest Kaluza–Klein particles(LKP).8–10)
As an alternative to WIMPs, axions form a furtherpotential candidate, motivated here by extensions to theStandard Model through Peccei–Quinn symmetry as asolution to the strong CP problem.11) The PQ symmetry isspontaneously broken at a scale fa, with the axion as theassociated pseudo-Goldstone boson produced in the earlyUniverse.12–15) Though outside the scope of this review wenote the rapid progress being made by the ADMX axionsearch. This experiment is currently setting stringent limits,excluding at >90% c.l. the KSVZ halo axion mass of 1.9 –3.3 eV and indicating that the local axion dark matter halomass density is greater than 0.45 GeV cm�3 for KSVZ DFSZaxions.16,17) For a full review on axions see for instance.18)
In this rich context, research aimed at an explanation ofnon-baryonic dark matter encompasses a huge worldwideeffort. This includes: searches for SUSY at acceleratorexperiments, such as the upcoming ATLAS/CMS at theLarge Hadron Collider and elsewhere; indirect searchesthrough attempts to detect the products of neutralino self-annihilation in astrophysical objects, such as neutrinos fromthe Galactic centre, the Sun and Earth; and direct particlesearches for both axions and WIMPs using experiments inthe laboratory. Focus will be placed here on the latter case—arguably the best motivated candidate studied by the bestgeneric technique. That is, efforts toward detection of relicWIMPS in the galaxy via their direct interactions in detectortarget materials on Earth.19,20) For a recent review of indirectsearches for neutralinos see for instance21) and for accel-erators see for example.22,23) The firmest indirect neutralinolimits from high energy neutrinos coming from the Sun arecurrently set by Super-Kamiokande.24)
2. Requirements for Direct Detection
For direct detection our starting point is that theGalaxy contains a halo of WIMPs normally assumed to beof spherical isothermal form with local density 0.3
GeV c�2 cm�3, an escape velocity of 650 km s�1, with rmsvelocity 279 km s�1 and relative Halo-Earth velocity of235 km s�1.20) The basis for detection is then elasticscattering of these neutral, non-relativistic particles, offtarget nuclei in a suitable detection medium, such that theenergy transferred as the resulting nuclear recoil passesthrough the material can be observed, usually as eitherionisation, scintillation or heat (phonons). Kinematics andthe likely mass range and velocity of the particles implies anuclear recoil spectrum with energy below �100 keV, withexponential form rising to low energies and with no spectralfeatures. This characteristic, together with the expected lowinteraction rate of likely 1–10�6 event kg�1 d�1, dictatesthree core requirements of WIMP detector technology: lowenergy threshold (<10 keVrecoil); potential for target massesof >10 kg; and low particle background of all types. Thelatter implies the need for a deep underground site to reducecosmic ray muon-induced neutrons, that could otherwiseproduce nuclear recoils indistinguishable from WIMPs;use of additional passive and active gamma and neutronshielding; and detector construction using materials withgreatly reduced radioactive U, Th, and K content.
The coupling of these non-relativistic WIMPs has twoterms, a scalar, spin-independent (SI) part and an axial spin-dependent (SD) part.25) For most SUSY models SI providesthe dominant coupling and hence highest rate. This isbecause although neutralino-nucleon cross-sections aremainly much smaller for the SI case,26) coherence acrossthe nucleus results in constructive interference which greatlyenhances the WIMP-nucleus elastic cross-section for high Atargets. The opposite is true for SD where the axial couplingto nucleons with opposite spins interferes destructively.Effectively, sensitivity to SD interactions thus requires atarget isotope with an unpaired nucleon, either proton orneutron. Although generally lower sensitivity is implied forthe SD case this is not true for all neutralino models. SDtargets are certainly required if the full WIMP parameterspace is to be studied and the widest investigation of anysignals undertaken.27–29)
The stringent requirement for low background, bearing inmind, for instance, that typical ambient environmentalgamma fluxes can produce event rates >105 times higherthan the expected WIMP signal rate in an unprotecteddetector, has focussed world attention on technologies thatcan actively reject electron recoil events, whilst maintaininghigh sensitivity to nuclear recoils. This is possible inprinciple because the latter have typically �10 higherdE=dx values.20) In practice, few technologies can make useof this physics, the prime ones being: (1) low temperatureionisation/phonon or scintillation/phonon detectors inwhich the ratio of event-produced ionisation or scintillationto phonons is measured in suitable cryogenic materials suchas Ge or Si (ionisation) and CaWO4 (scintillation); and (2)noble liquid gases, notably xenon and argon, in whichscintillation and ionisation is measured simultaneously. Amoderate level of discrimination can also be achieved inspecific scintillators such as NaI(Tl), CsI(Tl), liquid Ar, andliquid Xe by pulse shape analysis (see §4).
Although recoil discrimination, and background reduc-tion, appears feasible in such technologies there remains theissue, given the lack of spectral features in the recoil
spectrum, of how to determine in a clear way whether anyremaining counts are due to WIMPs from the galaxy and noteither nuclear recoils from an unaccounted for background(such as neutrons or surface interactions), or a detectorartefact. There are two prime possibilities for addressing thisusing galactic dynamics. Firstly, at least for the standardhalo model, the Earth’s motion through the Galaxy impliesan expected seasonal modulation in the recoil spectrum(flux and shape).30,31) This is because the component of theEarth’s solar orbital velocity in the direction of our galacticmotion (orbital plane inclined at 60�) either adds to orsubtracts from the galactic orbital velocity depending on theseason. Secondly, thanks also to our galactic orbital motion(�235 km s�1), we would expect the direction of the WIMP-induced nuclear recoil tracks themselves within a target to bedominantly opposite to our direction of motion (in galacticcoordinates)32,33) (see §8). Information may also be gleanedby comparing different targets, [since WIMPs interactdifferently with target nuclei of different A (see §8)]different technologies and different sites.20)
Unfortunately, the annual modulation effect is very small,typically a few %, requiring already at least tonne-scaledetectors to obtain sufficient event statistics for a viablesearch.20) Such small effects are also at risk of being maskedby detector characteristics always vulnerable to naturalseasonal changes in the environment. The recoil directioneffect is far more powerful, in principle. Only of order afew 10 s of WIMP events may be required in such a recoildirection sensitive detector to identify them as of galacticorigin (see §8). Furthermore, the angular distribution ofWIMP-induced recoil tracks can not be mimicked by anyterrestrial backgrounds since we would expect a sidereal(not daily) modulation of the WIMP-induced track directionsand an average ‘‘washed-out’’ isotropic distribution of anybackground in the galactic frame. The challenge here,however, is the likely need to use low pressure gasdetectors that would then need to be very large in volume(100 s m3).
Built on these basics a wide variety of experiments haveand are being run worldwide. Figure 1 provides a summaryof results from recent key examples, given here as anexclusion plot of WIMP-nucleon cross-section vs WIMPmass for the SI case, assuming the standard halo model asabove. Referring to this the following sections outline thecurrent status and possible future scenarios. Note this isnecessarily selective and likely tinged by personal bias—fora wider view we refer to recent workshop proceedings suchas.34) Note also that some of these results are yet to bepublished. They are shown here for the sake of completenessas reported in workshops and preprints, leaving discussion oftheir validity out of the scope of this review.
3. Semiconductors
Ionization detectors, in the form of low backgroundgermanium (HPGe) and silicon diodes used for double betadecay searches, provided the first limits on WIMP inter-actions.35,36) Such experiments were vital to ruling out earlycandidates for WIMPs including, Cosmions and heavy Diracneutrinos.37) However, as a technology they suffer from aninability to distinguish between gamma background eventsand the nuclear recoil events of interest. This is partly
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compensated for by the possibility of high radio-purity in Gewhich has allowed more recent experiments such as HDMSand IGEX to set interesting limits (see Fig. 1).38,39) TheIGEX experiment at Canfranc used 2.1 kg of purifiedGe with a 20 cm thick Pb gamma shield inside a muonveto. The detector achieved an eventual background of 0.21keV�1 kg�1 d�1 at 4 –10 keV.39)
Next generation HPGe detectors aim at further reductionin activity, for instance by possibly �1000 using noveltechniques such as crystal growth underground to reducecosmic-ray spallation activity. There is also prospect foractive rejection of Compton scatter events using segmenta-tion to provide position sensitivity and use of activecoincidence Compton vetos. Key ideas have been proposedby GEDEON,40) following from IGEX; the GERDA experi-ment at Gran Sasso;41) and MAJORANA.42) All these areprimarily aimed at neutrinoless double beta decay detection.The GERDA detector incorporates the novel prospect ofusing direct submersion in liquid argon or nitrogen (an ideatested by GENIUS-TF43)) with, for argon, use of the liquidas a possible active veto.
4. Scintillators
Pulse shape analysis (PSA) in certain organic andinorganic scintillators has been known to allow discrim-ination against low dE=dx events (electron recoils) for manydecades.44) NaI in particular was turned to advantage forWIMP searches by the UKDM collaboration, using cooledundoped NaI and later NaI(Tl), and by BPRS/DAMA.45–47)
Unfortunately, the light output [�40 photons per keVelectron
in NaI(Tl)] is too low for event by event discrimination atlow energy even though the quench factors (�9% for I and�25% for Na) are relatively high.48) Statistical methods,combined with material purification to reduce intrinsicactivity,49) can be used and were successfully implementedin, for instance, NAIAD to produce significant new limits.50)
Nevertheless, the discrimination power with statisticalanalysis is limited. The DAMA experiment thus later turnedto using NaI(Tl) in simple counting mode as a means of
searching instead for an annual modulation signal, withno nuclear recoil identification applied. They have claimedevidence for a modulation, reporting the discovery ofWIMPs in 1997.51)
The final DAMA result from a total of 107,731 kg dayaccumulated with 9 low background 9.7 kg NaI(Tl) crys-tals52) remains the only claimed direct observation ofWIMPs, corresponding to a mass of �52 GeV and cross-section �7:2� 10�6 pb (for standard halo model assump-tions). However, the result appears in contradiction withseveral other experiments including the bolometric Geexperiments of EDELWEISS and CDMS, and the liquidxenon experiment ZEPLIN53–55) (see §5 and §6). Thiscontradiction appears to hold regardless of the halo model orif SD interactions dominate,56–58) though there remainsdebate as to whether fine tuning of models can allowcompatibility, particularly for the SD case. The DAMAgroup is now running an expanded array, the 250 kg LIBRAexperiment.59,60) Following closure of NAIAD no direct testis being made of the result with an independent NaI-baseddetector, although the Zaragosa/Canfranc group is buildinga 107 kg NaI experiment, ANAIS, to address this gap61,62)
and the KIMS experiment is producing competitive limitswith CsI.63)
More recently there has been interest in other inorganicscintillators, notably CsI(Tl), CaF2(Eu), and, for instance,CaWO4. The former, now developed for the KIMS experi-ment in South Korea,63) has intrinsically better pulse shapediscrimination than NaI(Tl)64) but potentially higher intrinsicbackground, in particular due to 137Cs from nuclear fallout.Nevertheless, encouraging results have been obtained bytaking care in material selection and purification. CaF2(Eu)has relatively poor discrimination.65) CaWO4 and similarcompound inorganics are not efficient scintillators at roomtemperature but operate well at mK temperature. CaWO4
has become an integral part of the CRESST bolometricexperiments in which scintillation light is measured simul-taneously with heat66) (see §5). For recent measurementsof the quench factors here see.67)
Wimp mass [GeV]10 210 310 410
WIM
P-n
ucl
eon
cro
ss-s
ecti
on
[p
b]
-810
-710
-610
-510
-410
-310
IGEX - 2002
EDELWEISS - 2002
CRESST - W - 2004
DAMA - 2002
NAIAD - 2004
KIMS 3409 kg days
WARP
ZEPLIN I - 2004
ZEPLIN II - 2007
CDMS - 2006
XENON10 - 2006
XENON10 - 2006 (bg subtract)
Fig. 1. (Color online) Summary of current spin-
independent WIMP-nucleon limits (for references
and details see text).
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Certain organic crystal scintillators such as stilbene,plastics and liquids also demonstrate pulse shape discrim-ination.44) They have the advantage of potentially relativelylow cost per kg and high radio-purity. However, theircomposition is dominated by H, C, and possibly F or otherlight elements. This results in poor quench factors, typically2% or less,68,69) and poor kinematic coupling to WIMPsleading to very poor sensitivity relative to the inorganics likeNaI(Tl). Nevertheless, there has been significant interest inthe organic crystals, in particular because some of these, forinstance stilbene and anthracene, yield a response that isdependent on the direction of the contained recoiling nucleus,at least as measured using alpha particles. This yields a rareexample of a technology relevant to the possibility of adirection sensitive WIMP detector70,71) (see §8).
Whilst the lack of powerful recoil discrimination is adisadvantage for experiments based purely on scintillationdetection there is a potential advantage for SD sensitivitydue to the greater possibility of using spin nuclei, partic-ularly iodine (e.g., in NaI, CsI) and fluorine (e.g., in CaF2).This has allowed NAIAD to maintain competitive limitsfor WIMP-proton coupling.72) Finally, a remaining class ofscintillators of interest is the liquid noble gases, notablyliquid xenon and argon. These are covered in §6.
5. Bolometers
At low temperature the heat capacity of a dielectric crystalgoes as T3. Thus at mK temperatures the small energydeposition from a nuclear recoil can yield a measurableproportional increase in crystal temperature.20) Some ofthe earliest techniques investigated for WIMP dark matterdetection were based on this, where energy released byparticle interactions can be observed as phonons or quanta oflattice vibrations. Work started on this idea in the 1980’s(see for instance73)), the original motivation being in part theprospect of obtaining very low recoil energy thresholds andhigh energy resolution, due to the meV level of quantisationinvolved.74) However, it was soon demonstrated, first in Si75)
and then in Ge,76) that phonon detection could be combinedwith simultaneous detection of ionisation to provide also apowerful means of discrimination against electron recoils,on an event by event basis. This arises because theproportion of energy observed in the two channels isdependent on the event dE=dx—a high dE=dx event, suchas a recoiling nucleus, produces proportionally more heatthan ionisation (the ionisation is quenched). For instance, theratio of ionization to recoil energy (the ionisation yield) forGe recoils in Ge is �0:3 of the value for electron recoilsabove 20 keV.53)
Whilst bolometers without collection of ionisation haveproven quite useful for dark matter searches, the hybridtechnique of simultaneous ionisation and phonon collectionwith its capability for background rejection has been pushedharder. Most notable is the CDMS collaboration (at Soudanmine) and EDELWEISS-I (at Frejus)53,77–79) (see Fig. 1).The latter used 320 g Ge crystals operated at 17 mK withNTD-Ge thermometric sensors attached for the heat signaland Al electrodes used to collect the charge. 10 cm of Cu and15 cm of Pb where used to shield the cryostat from rockgamma-ray background with an additional 7 cm Pb insideand a total of 30 cm paraffin outside the entire setup to
reduce rock neutrons. A variety of detectors were tried inEDELWEISS-I with several runs completed from 2000 untilMarch 2004. These yielded a total exposure of 62 kg days,the main results coming from three crystals with recoilenergy threshold of 13 keV or better over 4 months of stableoperation. Figure 1 shows the limits produced. After cuts atotal of 40 nuclear recoil candidates were recorded in therange 15 – 200 keV with 3 events between 30 and 100 keV,most likely due to remaining background neutrons or surfaceelectrons.
The CDMS experiment operates towers of Ge and Sicrystals each 1 cm thick and respectively of mass 250 and100 g. These are mounted in a dilution fridge and shieldedmainly by 22.5 cm of external Pb and 50 cm of polyethylene.A 5 cm layer of plastic scintillator is used to veto any eventscoincident with cosmic muons (necessary here due to therelative shallowness of the Soudan site at 2080 m.w.e.).Charge electrodes are used for ionisation collection as inEDELWEISS but here athermal phonons are detected usingsuperconducting transition edge sensors, applied by photo-lithography to the crystal surfaces. This design has thepotential advantage of providing depth position sensitivity,via measurement of the phonon pulse risetime, and hence thepossibility of rejecting surface electron events that couldotherwise contaminate the signal region, as suspected inEDELWEISS-I. Two towers were operated in 2004 yieldingan exposure for 10 –100 keVrecoil of 34 kg days Ge and 12kg days Si. No events were observed in the Si and only oneevent, consistent with the expected surface event back-ground, in the Ge, yielding a 90% c.l. SI upper limit in Geof 1:6� 10�7 pb at 60 GeV c�2 WIMP mass (see Fig. 1).
As an alternative ROSEBUD80) and CRESST66,81) havedeveloped detectors in which scintillation light is measuredin coincidence with heat, in particular using CaWO4.81) Herea silicon wafer of 30� 30� 0:4 mm3 with tungsten ther-mometer is used to detect the photons and a 8� 8 mm2, 200-nm-thick superconducting evaporated film used as the heatsensor. Although only 1% or less of the energy deposited isdetected as photons this is much higher than feasible at roomtemperature and is sufficient to produce an energy resolutioncomparable to NaI(Tl). Results so far have been obtainedwith two 300 g crystals at the Gran Sasso undergroundlaboratory with a total exposure of 20.5 kg days. Thisrevealed 16 events in the range 12 – 40 keV consistent withthe expected background from neutrons given that theexperiment did not have a neutron shield. The resultinglimit, due to W recoils (see Fig. 1), is comparable to othersin the field, including EDELWEISS.
Notable in the pure cryogenic detector field is the work ofthe Milan group through the CUORE/COURICINO experi-ment at Gran Sasso.82,83) CUORE is designed primarily forneutrinoless double beta decay searches. CUORE demon-strates one particular advantage of pure cryogenic detectorsover the hybrid types. The latter is essentially restricted toGe and Si because only these are found to have sufficientlyhigh electron–hole transport at mK temperatures. The non-hybrid technique, at the expense of throwing away recoildiscrimination, is open to a much greater variety of targetcrystals, for instance TeO2 for CUORE, LiF, Sapphire, andothers have been demonstrated. CUORE aims to build anarray of 988 TeO2 cryogenic crystals with total mass
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�750 kg, building on the first stage CUORICINO experi-ment already operated with 62 (�40:7 kg) crystals. AlthoughCUORE will have exceptionally high target mass, compet-itive WIMP limits will only come through more work tosuppress intrinsic crystal backgrounds.84)
All these cryogenic experiments are now progressingtowards significant upgrades. CDMS is proposing 25 kg anda possible move to the deeper SNOLAB site. EDELWEISSis progressing towards a more ambitious phase II with up to120 detectors and CRESST is upgrading to allow 33 CaWO4
detectors, totalling 10 kg. However, as outlined in §7, it islikely that even greater target mass will be needed, possiblyat the tonne-scale or larger.
6. Liquid Noble Gases
Whilst a large world effort has been devoted to cryogenicbolometers over many years, linked now to quite an industryin alternative applications, there has been recent rapidgrowth in liquid noble gas technology for WIMP searches.Most notable has been liquid xenon (LXe), started byDAMA/Xe,85,86) but also recently liquid neon87) and, inparticular, liquid argon. A prime motivation here has beenimproved low background discrimination combined withprospects for tonne-scale target mass at reasonable cost (see§7). LXe has particularly good intrinsic properties for WIMPdetection including: high mass (Z ¼ 54, A ¼ 131:3) yield-ing a good kinematic match to likely WIMP candidates; highscintillation and ionisation efficiency (�46 photons/keV at178 nm); and high radiopurity, enhanced further by theavailability of liquid gas purification techniques. However,of greater importance is the recoil discrimination achievable.This is possible firstly, as in NaI(Tl), by simple PSA ofthe scintillation light. This is the basis for the single phaseLXe XMASS experiments in Japan88) and of ZEPLIN I.55)
The latter detector, comprising 3.2 kg of active LXe viewedby 3 PMTs, accumulated 293 kg days during operation atBoulby mine until 2002, producing significant limits withthis technology (see Fig. 1).
The XMASS group have also run a 100 kg prototype PSAdetector and are aiming to achieve higher sensitivity byconstructing an 800 kg experiment for operation in Kamiokamine.88) However, more powerful discrimination is feasiblein LXe by recording also the ionisation produced and hencethe ionisation/scintillation ratio. This arises because fornuclear recoils the ionisation signal (termed S2) is quenchedsignificantly more than the primary scintillation (S1) relativeto electron recoils of the same energy. This is beingimplemented by the ZEPLIN II/III, XENON 10/100 andXMASS II (two-phase) experiments, all aiming to achievehigher sensitivity with lower fiducial mass than likelyrequired with PSA alone.89–93) Collection of event ionisationcan in principle be achieved in the liquid phase of xenon (see§7), but obtaining stable operation with the high avalanchefields required (�1 MV cm�1) is a challenge. The currentgeneration therefore uses two-phase operation, the chargefirst drifted out of the liquid into a gas phase amplificationregion, used to produce an electroluminescence observableby PMTs as the second (S2) light pulse.
Exciting progress has been made recently with this two-phase LXe technology with both ZEPLIN II (Boulby) andXENON 10 (LNGS) announcing new leading limits (see
Fig. 1).94) As the largest two-phase detector so far the 30 kgZEPLIN II successfully accumulated over 1 tonne.day ofdata during 2006 (before cuts). This detector, using arelatively simple design comprising a PTFE lined LXechamber viewed by 7 PMTs, has pioneered operation ofbulk two-phase xenon. XENON10, with 15 kg activevolume, has a more complex design involving PMTsviewing both top and bottom (48 and 41 HamamatsuR8520s respectively). This detector achieved light collection>2 p.e./keV and excellent stability over 9 months in 2006/7, accumulating 136 kg.d of data after cuts. This recentlyallowed new world best limits to be derived, set provision-ally at 5:5� 10�8 pb at 100 GeV c�2 assuming backgroundsubtraction.92) Figure 2 gives a plot of ionisation yield vsenergy from XENON10 showing the gamma backgroundregion around log S2/S1 ¼ 2:5 and the signal region belowused to set the current limits.
Despite the current interest in LXe the intrinsic discrim-ination may not match that of bolometers. In this respectliquid argon (LAr) may provide better prospects. Someproperties of LAr are inferior to LXe for WIMP searches,notably the lower Z, A (18, 40) and the need to usewavelength shifter for the VUV scintillation light(�40 photons/keV at 135 nm). However, both pulse shapediscrimination and two phase primary/secondary discrim-ination are now known to be more powerful,95,96) capable inprinciple of combined discrimination factors up to �108.LAr is also a factor ��400 lower in cost. Based on this theWARP collaboration has built and deployed a 3.2 kg LArexperiment at LNGS and recently reported a sensitivity near10�6 pb at 100 GeV c�2 for an exposure of 96.5 kg days.97)
WARP is currently constructing a larger 140 kg detectorwith a full active Compton veto. Other LAr experiments arealso under construction including ArDM, CLEAN andDEAP.98–101) ArDM involves two-phase operation but withionisation readout using direct collection by large electronmultipliers (LEMs) in the gas phase. The other designsare based on single phase PSA plus self-shielding, akin tothe XMASS concept with liquid xenon.
Although ZEPLIN I, II, XENON 10, and WARP haveshown excellent progress, significant issues remain with
Fig. 2. (Color online) Plot of ionisation yield from recent XENON10
results showing the gamma background region (black dots) and the
nuclear recoil selection region below (for details see ref. 92).
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liquid noble gases. Firstly, the quench factor for nuclearrecoils remains poorly determined. Measurements for LXein zero field have indicated 0.13 – 0.23 (10 – 56 keV)102) buthigher values have been claimed.103) For argon there areseveral conflicting results, for instance.97,100) Secondly, LXehas not yet demonstrated recoil discrimination competitivewith cryogenic technology. Populations of events areobserved to spread from the gamma region into the signalregion. This may reflect the youthfulness of current detectordesigns but may be intrinsic. It is possible that spontaneoussingle electron emission occurs in the liquid, producingsecondary electroluminescence with minimal ionisationsignal and yielding events indistinguishable from nuclearrecoils. The ZEPLIN III detector, currently being installed atBoulby, has improved light collection and higher drift fieldsthan ZEPLIN II and will be used in part to investigate thisprospect.91)
The background issue above may also be present in LAr.However, for argon104) there is a more important issue toresolve, the presence of radioactive 39Ar. Produced bycosmic ray spallation in the atmosphere, this yields inindustrial argon a beta background of �1 bq/lt. Discrim-ination of 1010 would be sufficient in principle to cope butthen data acquisition deadtimes in a tonne-scale detectorwould likely be unmanageable. Calculations show that argonfrom deep gas wells, shielded from cosmic rays, couldprovide an economic source of, so called, dead-argon.Activation times on the surface are long enough that oncebrought to the surface there is sufficient time to constructand deploy an experiment.105)
7. Tonne-scale Concepts and Alternative Techniques
After over two decades of development, WIMP experi-ments with target masses of kg-scale are reaching sensitiv-ities improved by about 4 orders of magnitude, probing wellinto SUSY favoured parameter space. This achievement hasbeen accompanied by a continuing rise in the number ofexperimental scientists involved, now >300. There has beenan expansion of interest in new and emerging technologies,not just liquid noble gases but others not detailed here,including superheated droplet detectors (SSDs), specificallySIMPLE and PICASSO, and the MACHe3 detector that usessuperheated He.106–110) The SSD experiments, through use ofFluorine-loaded targets, show particular promise for SDsensitivity and are producing interesting limits. However,whilst all this activity together reflects substantial maturity, acrossroads has likely been reached in the field.
Firstly, it is pretty certain, setting aside claims by DAMA,that favoured SI coupled dark matter does not exist withcross-sections >�2� 10�7 pb (see Fig. 1). Meanwhile,theoretical predictions for both neutralinos and LKPs reach<10�11 pb.8,111–113) Thus next generation experiments mustnot only achieve further background suppression but also becapable of tonne/multi-tonne masses, simply to ensure astatistically observable signal rate. This represents a majorleap, implying significantly higher costs and likely largercollaborations. Secondly, for such large detectors it can beargued that though active gamma discrimination remainsimportant, greater emphasis is needed on material purifica-tion, passive shielding of external backgrounds and onsearches for additional features in the data to show that
remaining events are non-terrestrial signals and not, inparticular, neutrons.
The latter argument arises as follows: assuming nextexperiments are at sufficient depth to avoid muon-inducedneutrons, then gammas and neutrons from U/Th chains inthe environment and detector will dominate background. Forthe relevant energy range, <200 keV, such contaminationproduces typically 105–106 more gammas than neutroninduced nuclear recoils.20) The levels of detector sensitivityrequired for tonne-scale experiments now imply that gammabackgrounds must be comparable with or lower than theneutron rates, such as could be achieved by neutron/gammadiscrimination of 105–106 [the rate for fast neutrons from therock at Boulby, for instance, has recently been measured tobe 1:72� 0:61ðstat.Þ � 0:38ðsyst.Þ] � 10�6 cm�2 s�1 above0.5 MeV.114) Thus neutron induced recoils, which can not bedistinguished from WIMP interactions, naturally will be thedominant particle background. Detector position sensitivitymay help, by allowing rejection of multi-scatter events.115)
However, ultimately reliance will be needed on passiveneutron shielding and material purification plus WIMPsignal identification via: (1) use of at least two targets/technologies with different A and different systematics; and/or (2) correlation of events with Galactic motion byobservation of annual modulation or a directional signal.The former relies on the different behaviour of WIMP andneutron scattering cross-section vs A to deduce that a signalis not neutrons. The latter allows direct identification ofevents as of extra-terrestrial origin.
Following these notions, similar to arguments adopted inneutrino physics by, for instance, Borexino and SNO,116,117)
it is natural to consider larger WIMP detectors with (near)spherical design, a central fiducial zone containing minimaldetector components other than the target material, and anintegral passive outer shield. This is the basis of the XMASS(Xe), CLEAN (Ar and Ne), and DEEP (Ar) scale-upprogrammes (see §6). Single phase liquid noble gases areused here with photons recorded by photomultipliers in theouter region, pointing inwards (see for instance Fig. 3). Thisallows both position information (fiducialisation) and somerecoil discrimination via PSA, but the dominant theme isbulk passive shielding for gammas and neutrons.
Fig. 3. (Color online) Schematic design for the proposed miniCLEAN
100 kg detector, a precursor to a potential tonne-scale experiment (for
details see text and ref. 100).
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The reliance on PSA and the presence of PMTs (withrelatively high radioactivity) is a potential limitation forthese experiments. Replacement of PMTs is a possibility byusing an internal photocathode, such as CsI, to convertphotons to electrons for collection by charge readout in thegas phase, for instance using micromegas or gas electronmultipliers (GEMs).118) Current two-phase programmes,with greater discrimination potential, are also being de-veloped for scale-up, for instance WARP,97) LUX,119)
XENON100,120) and ArDM.98) The latter is already at thetonne-scale and uses LEM charge readout. However, use ofgas-phase electroluminescence is not well suited to thebenefits of a pure spherical concept due to the need for a topgas layer. An alternative hybrid design has been suggested,termed CORE, in which the ionisation signal is recordeddirectly in the liquid phase at a point gain region central to asphere.121) Such a spherical TPC concept has in fact alreadybeen realised in the gas phase by NOSTOS.122) Use of highpressure noble gas, as developed by SIGN, may itselfprovide an alternative class of scale-up technique.123)
Cryogenic technology is not so well suited to the massiveself-shielding spherical concepts above. Nevertheless, scale-up to tonne-scale is planned here also, making best use of thehigh discrimination power demonstrated notably by CDMS,EDELWEISS, and CRESST. Two particular efforts areforeseen, SuperCDMS124) and EURECA (European Under-ground Rare Event search with Calorimeter Array).125) Theformer will use Ge and Si ionisation/thermal technologylike CDMS in a staged expansion from 27 to 145 kg andeventually to 1100 kg by 2015, either at the US DUSEL,if built, or SNOLAB in Canada. EURECA represents amerger of EDELWEISS, CRESST with further new groupsto develop a 100 –1000 kg array using various targets,possibly both ionisation/thermal and scintillation/thermalideas. For both experiments a priority will be the need todevelop improved detectors, in particular to allow betterrejection of surface events, for instance through eventposition reconstruction or improved analysis, and to reduceunit costs.
8. Directional Detectors and Proof of a Galactic Signal
The scale-up programmes, assuming more than onebecomes reality, in part address the signal identificationissues noted in §7 by opening the way to examining the A-dependence of a potential WIMP signal. However, definitiveproof that a signal is of galactic, and not terrestrial, origincan only be achieved by correlating in some way the eventswith our motion through the Galactic WIMP halo (see §1).32)
This has been the objective of DAMA/LIBRA by makinguse of the small predicted annual modulation in flux/energy.59) However, a much more powerful, though techno-logically challenging, possibility is to correlate in 3D thephysical direction of nuclear recoils in a target with ourmotion. This is the motivation behind the DRIFT, MIMAC,NEWAGE, and other low pressure gas time project chamber(TPC) r&d programmes.126–130) Calculations show that inprinciple only a few 10 s of WIMP events are needed toprove a Galactic origin. Furthermore, a powerful siderealday modulation of the signal is expected in the laboratoryframe, impossible to be mimicked by any terrestrial back-ground.33,131–133)
Much progress has been made here by the US-UK DRIFTcollaboration using negative ion CS2 TPCs at Boulbymine. The latest version, DRIFT II, comprises 3 units of1 m3 of CS2 at 40 Torr (each 70 g fiducial mass).127) Thereduced pressure is needed so that nuclear recoil tracks areextended to a few mm, sufficient for observation by themulti-wire proportional counters (MWPC) readout used.Negative ion gas is used to minimise track diffusion withoutthe need for expensive magnets, the CS2 (and possibleadditives) providing also a multi-A target. Each detectorcontains a 1 m2 central high voltage cathode plane and twoback to back drift regions of 50 cm depth, each read out by a1 m2 MWPC comprising planes of 20 mm wires at 2 mmpitch.
DRIFT II demonstrated stable, neutron shielded, long-term running during 2005/6. Operation is by remotecontrol at room temperature, with no cryogenics or complexservices. The in-built sensitivity of the TPC technology toparticle dE=dx (ionisation charge density) allows exceptionalelectron track rejection (>105), sufficient that no gammashielding is required for DRIFT II. More importantly analy-sis of event drift time, MWPC anode hits and inducedsignals on the orthogonal grid planes allows in principle fullx; y; z 3D reconstruction of ionisation tracks down to 300NIPs (number of ionising pairs) or <20 keVrecoil. Figure 4shows 3D reconstruction of a typical S-recoil event resultingfrom a neutron elastic scatter.134)
As currently the only known route to significant recoildirection sensitivity, TPC technology holds exceptionalpotential for WIMP physics and is possibly the only routeto a definitive galactic signal. However, there are severalchallenges to address. The requirement for use of lowpressure gas implies large volume detectors will eventuallybe needed, possibly 1000 s m3. For this, new charge readouttechnologies such as bulk Micromegas and GEM planesare under development to reduce spatial resolution andhence allow high pressure, lower volume, operation.135,136)
A further issue is the desirability of achieving track headto tail discrimination. Orientation of a recoil track aloneprovides significant directional information but a factor��10 greater sensitivity can be achieved in principle ifthe head can be distinguished form the tail.33) Whilstmore careful data analysis is required to demonstratefeasibility it is clear from recent detailed simulations thatthis also may be possible.137,138) Shown in Fig. 5 is anexample simulated track for a 100 keV S recoil in 40 TorrCS2 together with dE=dX curves. Firstly, the predictedranges for the simulated S tracks are found to agree to <10%with experimental data. The dE=dx for the observableelectronic channel suggests, for the energy range of interestbelow 100 keV, a rapid decrease in ionisation per unitlength toward the end of the track (we are well below theBragg curve peak at these energies). This would indicate ahead-tail asymmetry favouring more signal at the beginningof the track (head) than the end. However, simulations alsoshow, as indicated in the example track, that straggling (ball-up) increases at the end. Therefore, a charge readoutmechanism that involves projection of the track ionisationonto a single axis, for instance, will likely be sensitive to thistopological issue, increasing the charge measured for thetail. Work is underway to determine whether any asymmetry
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is observable in a practical detector set-up and how thisdepends on dE=dx and the track topology in relation toreadout systems. A new international cooperation, CYG-NUS, has formed to address this and to study the designchallenge of building a very large directional dark matterdetector.139)
9. Conclusion
In summary, great progress has been made towarddetection of particle dark matter in recent years, notablythrough development of cryogenic detectors but alsoliquid noble gas experiments now beginning to set thebest limits. Much effort has been placed on producingtechnology with recoil discrimination against gammas.However, it is not necessarily clear that gamma discrim-ination alone will be sufficient to prove the presence ofWIMPs. Proof that any remaining signal is in fact fromextra-terrestrial dark matter and not neutrons or some otherun-determined terrestrial background or artefact will bevital. Multi-tonne detectors with optimal passive shieldingto achieve sufficient count rate are being developed butthere is also a potential route toward the needed defini-tive galactic signal through new recoil direction sensitivetechnology.
Acknowledgments
The author wishes to thank E. Daw, V. Kudryavtsev, andP. Majewski (University of Sheffield) for many usefuldiscussions. Thanks also to the ILIAS integrated activity(contract R113-CT-2004-506222) workpackage N3-LX, N3-AD, N3-BSNS; the INTAS programme (grant number 04-78-6744); and the UK’s Particle Physics and AstronomyResearch Council (PPARC) for financial support.
1) D. N. Spergel et al.: Astrophys. J. Suppl. 170 (2007) 377.
2) S. Perlmutter et al.: Astrophys. J. 517 (1999) 565.
3) D. N. Spergel et al.: Astrophys. J. Suppl. 148 (2003) 175.
4) M. Tegmark et al.: Phys. Rev. D 69 (2004) 103501.
5) G. Steigman and M. S. Turner: Nucl. Phys. B 253 (1985) 375.
6) J. R. Primack, D. Seckel, and B. Sadoulet: Annu. Rev. Nucl. Part.
Sci. 38 (1988) 751.
7) G. Jungman, M. Kamionkowski, and K. Griest: Phys. Rep. 267
(1996) 195.
8) G. Servant et al.: New J. Phys. 4 (2002) 9.
9) D. Majumdar: Phys. Rev. D 67 (2003) 095010.
10) J. D. Vergados: Proc. DSU 2006, Madrid, 2006, hep-ph/0610017.
11) R. D. Peccei and H. R. Quinn: Phys. Rev. Lett. 38 (1977) 1440.
12) S. Weinberg: Phys. Rev. Lett. 40 (1978) 223.
13) F. Wilczek: Phys. Rev. Lett. 40 (1978) 279.
14) G. G. Raffelt: Phys. Rep. 198 (1990) 1.
Fig. 5. (Color online) Example simulation of a 100
keV S recoil in 40 Torr CS2 together with curves of
dE=dx vs energy for electronic and nuclear channels.
Results indicate head-tail asymmetry may be pos-
sible though experimental proof is needed that this
can be observed in practice.134)
Fig. 4. Example 3D reconstruction of a 100 keV S recoil track obtained with the DRIFT II directional dark matter detector. Circles are indicative
of the energy deposited along the track.
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Stanford Linear Accelerator Center, Stanford University, Menlo Park, CA 94025, U.S.A.
(Received May 1, 2007; accepted June 7, 2007; published November 12, 2007)
Astrophysicists now know that 80% of the matter in the universe is ‘‘dark matter’’, composed of neutraland weakly interacting elementary particles that are not part of the Standard Model of particle physics.I will summarize the evidence for dark matter. I will explain why I expect dark matter particles to beproduced at the CERN LHC. We will then need to characterize the new weakly interacting particles anddemonstrate that they the same particles that are found in the cosmos. I will describe how this might bedone.
KEYWORDS: dark matter, WIMP, LHC, ILCDOI: 10.1143/JPSJ.76.111017
1. Introduction
One of the themes of the history of physics has been thediscovery that the world familiar to us is only a tiny part ofan enormous and multi-faceted universe. From Copernicus,we learned that the earth is not the center of the universe,from Galileo, that there are other worlds. More recently,Hubble’s extragalactic astronomy taught us that our galaxyis a tiny part of an expanding universe, and the observationof the cosmic microwave background by Penzias andWilson revealed an era of cosmology before the formationof structure. Over the past 10 years, astronomers haverecognized another of these shifts of perspective. They haveshown that the stuff that we are made of accounts for only4% of the total content of the universe. As I will describe, wenow know that about 20% of the energy in the universe takesthe form of a new, weakly interacting form of matter, called‘‘dark matter’’. The remaining 75% of the energy of theuniverse is found in the energy content of empty space,‘‘dark energy’’.
Dark energy is the most mysterious of these components.Its story is described by Turner and Huterer.1) Dark matter,though, is the component that most worries the imaginationsof particle physicists. What particle is this dark matter madeof? Why have we not discovered it at our accelerators? Howdoes it fit together with the quarks, leptons, and bosons thatwe have spent our lives studying?
And, conversely, dark matter is the component that mostexcites us by the possibility of its discovery. There are strongarguments that the next generation of particle accelerators,beginning next year with the Large Hadron Collider (LHC)at CERN, will produce the elementary particles of whichdark matter is made. How can we recognize them? How canwe prove that these particles are the ones that are present inthe cosmos? And, finally, how can we use this knowledgeto image the dark matter structure of the universe? I willaddress all of these questions in this article.
2. Evidence for Dark Matter
Although the astronomical picture of dark matter hasbecome much clearer in the last ten years, the evidence for
dark matter goes back to the early days of extragalacticastronomy. The evidence for dark matter is summarized in abeautiful 1988 review article by Virginia Trimble.2) I willdescribe the most telling elements here.
In 1933, Fritz Zwicky measured the mass of the Comacluster of galaxies, one of the nearest clusters of galaxiesoutside of our local group.3) Zwicky’s technique was tomeasure the relative velocities of the galaxies in this clusterfrom their Doppler shift, use the virial theorem to infer thegravitational potential in which these galaxies were moving,and compute the mass that must generate the potential.He found this mass to be 400 times the mass of the visiblestars in galaxies in the cluster. The observation was soonconfirmed by similar measurements of the Virgo cluster bySmith.4)
We now know that most of the atoms in clusters ofgalaxies are not seen in observations with visible light.Because these clusters generate enormously deep gravita-tional potential wells, it is easy for hydrogen gas from thegalaxies to leak out and fill the whole volume of the cluster.These atoms acquire large velocities and emit X-rays whenthey collide. X-ray images show the clusters as glowing ballsof gas. This does not remove the mystery, however, TheX-ray emitting gas accounts for at most 20% of the massof the cluster and cannot explain the origin of the deeppotential well.5) For this, we must postulate that the clustersare also filled with a new, invisible, weakly interacting formof matter.
In the 1970’s, astonomers began to systematically measurethe rotataional velocity profiles or rotation curves, for manygalaxies. One would expect that the mass of a galaxy isconcentrated in the region where the stars are visible.Then, outside this region, Kepler’s law would predict that thevelocities should fall off as 1=
ffiffirp
. In fact, the velocities areseen to be constant or even slightly increasing.6) In the galaxyNGC 3067, using hydrogen gas lit up by a backgroundquasar, Rubin, Thonnard, and Ford showed that the rotationalvelocity profile maintains its large value at a distance of40 kpc (105 light-years) from the center of the galaxy,even though the visible stars become rare outside of 3 kpc.7)
From measurements of the velocities of globular clusters,it was found that the rotation curve of our own galaxy isalso flat out to distances of 100 kpc from the center.2)
Detailed measurements of cosmic microwave back-ground, including not only the averaged intensity of thisbackground radiation but also its fluctation spectrum, giveadditional information on dark matter. The microwavebackground was emitted at the time of recombination, whenthe hydrogen filling the universe, at a temperature of about1 eV, converted from an ionized plasma to a transparentneutral gas. From the Fourier spectrum of fluctuations of thebackground radiation, it is possible to measure the dissipa-tion of this medium. The most recent measurements from theWMAP satellite require a medium in which only 20% of thematter is hydrogen gas and 80% is composed of a veryweakly interacting species in nonrelativistic motion.8) Thesemeasurements can be converted to the current fractions ofatomic and dark matter in the total energy of the universe,�i ¼ �i=�tot
9)
�at ¼ 0:042� 0:003 �DM ¼ 0:20� :02 ð1Þ
In all of these systems, dark matter is observed onlythrough its gravitational influence. One might wonder, then,whether it is possible to explain the observations bymodifying the law of gravity rather than by introducing anew form of matter.10) The interpretation in terms of a newform of matter was recently boosted by the observationsshown in Fig. 1. These picture show three images of thegalaxy cluster 1E0657-558.11) The first is the optical image,showing the galaxies that, as we have discussed, make uponly a few percent of the mass of the cluster. The secondpicture shows the X-ray image from the Chandra telescope.This image shows where the bulk of the gas in the cluster islocated. The superimposed contours show the total densityof the mass in the cluster, as measured by gravitationallensing. It is remarkable that the peaks of the massdistribution occur where there are very few atoms. In thisobject, which probably arose from a collision of two clustersof galaxies, the atomic matter and the dark matter havebecome spatially separated. The observations cannot beexplained by an altered law of gravity centered on the atoms.They require dark matter as a new and distinct component.
3. The WIMP Model of Dark Matter
Thus, dark matter exists. What is it made of? In theStandard Model of particle physics, we know no neutralheavy elementary particles that are stable for the lifetime ofthe universe. Let us postulate a new species of elementaryparticle to fill this role. Bahcall called this a Weakly
Interacting Massive Particle (WIMP). I would like to addone more assumption: Although it is stable, the WIMP canbe produced in pairs (perhaps with its antiparticle), and itwas produced thermally at an early time when the temper-ature of the universe was very high. WIMPs must alsoannihilate in pairs. I will assume that these processesestablished a thermal equilibrium.
These assumptions lead to an attractive theory of darkmatter whose consequences I will explore in the remainderof this article. There are other models of dark matter that donot fit into this paradigm. A comprehensive review of darkmatter models has recently been given by Bertone, Hooper,and Silk.12)
Using the WIMP model, we can build a quantitativetheory of the density of dark matter in the universe. As theuniverse expanded and cooled, the reactions energeticenough to produce WIMPs became more rare. But at thesame time, WIMPs had more difficulty finding partners toannihilate. Thus, at some temperature kBTf , they dropped outof equilibrium. A small density of WIMPs was left over. Atthis era, the energy density of the universe was dominatedby a hot thermal gas of quarks, gluons, leptons, and photons,with a total number of degrees of freedom g� � 80. Usingthis thermal density to fix the expansion rate of the universeas a function of time, we can determine the evolution of theWIMP density by integrating the Boltzmann equation. It isconvenient to normalize the WIMP density to the density ofentropy, since in standard cosmology the universe expandsapproximately adiabatically. Then one finds13)
�DM ¼s0
�tot
�
45g�
� �1=2 kBTf=mc2
mPl=h�2 � h�vi
; ð2Þ
where s0 and �tot are the current densities of entropy andenergy in the universe, mPl is the Planck mass, equal toh�c=
ffiffiffiffiffiffiffiGN
p, and h�vi is the thermally averaged annihilation
cross section of WIMP pairs multiplied by their relativevelocity. In the equation that determines Tf , this temperatureappears in a Boltzmann factor e�mc
2=kBTf , where m is theWIMP mass. Taking the logarithm, one finds kBTf=mc
2 �1=25 for a wide range of values of the annihilation crosssection.
With this parameter determined, we know all of the termsin (2) except for the value of the cross section, and so wecan solve for this factor. The result is
h�vi ¼ 1 pb ð3Þ
Fig. 1. (Color online) Images of the bullet cluster, 1E0657-558: optical image from the Hubble Space Telescope, X-ray image from the Chandra telescope,
and mass density contours from gravitational lensing reconstruction.11)
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A particle physicist would recognize this value as the typicalsize of the production cross sections expected for newparticles at the LHC. More generally, if we assume that thecoupling constant in the WIMP interactions is roughly samesize as the dimensionless coupling � that gives the strengthof weak and electromagnetic interactions, this cross sectionresults from an interaction mediated by a particle whosemass is of the order of 100 GeV.
This result is remarkable for two reasons. First, it allowsus to transform our astrophysical knowledge of the cosmicdensity of dark matter into a prediction of the mass ofthe dark matter particle. Second, that prediction picks outa value of the mass that is very close to the mass scaleassociated with the Higgs boson and the symmetry breakingin the weak interactions. In an earlier article in this volume,Okada has argued that we should expect to find newelementary particles at that mass scale.14) Perhaps these newparicles are in some way responsible for the dark matter.
In fact, explicit models of symmetry breaking in elec-troweak interactions often provide a natural setting fordark matter. Supersymmetry, discussed by Yamaguchi,15)
predicts a new boson for each known fermion in Nature,and vice versa. It is natural that the fermionic partner ofthe photon is its own antiparticle, so that it is stablebut annihilates in pairs. This particle is then a perfectcandidate for the WIMP. Other models of electroweaksymmetry breaking also contain new neutral weakly-inter-acting particles that can be made stable by natural symmetryprinciples.
4. Production and Dectection of WIMPs at the LHC
If the mass of the WIMP should be about 100 GeV, weshould be able to produce WIMPs if we can build anaccelerator that provides elementary particle collisions atenergies higher than 100 GeV. However, it is not sostraightforward. A WIMP, being as weakly interacting asa neutrino, passes through a typical elementary particledetector unseen. It is only from the properties of the otherparticles produced in association with the WIMP that wecan recognize these events and select them for analysis.Particle physicists have analyzed in some detail how to dothis. Most of the specific analysis has been done in modelsof supersymmetry, so, for concreteness, I will use thatpicture here. The general conclusions apply to WIMPs inmany other models of weak interaction symmetry breaking.
Supersymmetry predicts many new elementary particlesin addition to the WIMP. In particular, the gluon of QCD hasa fermionic partner, the gluino, and the quarks have bosonicpartners, called squarks. Gluinos and squarks carry the sameconserved quantum number that keeps the WIMP stable.They are expected to be heavier than the WIMP and to decayto the WIMP by emitting quarks, leptons, and StandardModel bosons. Events with squark or gluino pair production,then, will have a characteristic form. Many energetic quarksand leptons will be emitted, but also each event will end withthe production of two WIMPs that make no signal in thedetector. The observable particles in the event will show animbalance of total momentum. The missing momentum isthat carried off by the WIMPs.
We have not yet seen events of this type at currentlyoperating accelerators. The highest-energy accelerator now
operating is the Tevatron collider at Fermilab, and theexperiments there put lower limits of about 300 GeV on themasses of gluinos and squarks.16) In 2008, however, the LHCwill begin operation with proton-proton collisions at a centerof mass energy of 14000 GeV. Not all of this energy isavailable for production of supersymmetry particles. Theproton, after all, is a bound state of quarks and gluons.Gluinos and squarks are produced in collisions of individualquarks and gluons, which typically carry 10% or less of thetotal energy of the proton. Still, we expect to see collisionswith total energy above 2000 GeV at a significant rate. Thisimplies that squark and gluon pair production, leading toevents with WIMPs, can be seen over almost all of theparameter space of the model. Figure 2 shows a simulationof a characteristic event of this type, as it would be observedby the CMS detector at the LHC.17)
5. Recognizing the Mass of the WIMP
The discovery of events at the LHC with apparentunbalanced momentum will signal that this accelerator isproducing weakly interacting massive particles. However,it would be far from clear that this particle is the same onethat is the dominant form of matter in the universe. Todemonstrate this, we would need to correlate properties ofthe WIMP that we observe at the LHC with astrophysicalobservations. This will probably first be done throughmeasurements of the mass of the dark matter particle. Usingdetailed measurements of the kinematics of quarks andleptons in the LHC events, it is expected that the mass of theWIMP produced there will be measured to 10% accuracy.18)
We then must compare this value with measurements of themass of the cosmic WIMP. To do this, it is necessary todetect the dark matter in the galaxy, not as a distribution ofgravitating mass, but as individual particles.
There are two strategies to make this detection. The first,reviewed by Spooner,19) is to place very sensitive detectorsin ultra-low background environments and look for rareevents in which a WIMP in our cosmic neighborhood fallsto earth and scatters from an atomic nucleus in the detector.
Fig. 2. (Color online) Simulated LHC event, with pair production of
gluinos and the decays of these particles to WIMPs, as would be observed
by the CMS experiment at the LHC.17) The apparent imbalance of
momentum transverse to the beam axis is due to the WIMPs (denoted ~��01
in the figure), which produce no signals in the detector.
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The cross section for this process is expected to have theremarkably small value of 1–10 zeptobarns, but in thenext few years semiconductor and liquid noble gas detec-tors in deep mines are expected to reach this level ofsensitivity. The mean energy deposited in these eventsdepends on the WIMP mass m and the target nucleus massmT roughly as
hEi ¼2v2mT
ð1þ mT=mÞ2: ð4Þ
Then, for a 100 GeV WIMP, detection of 100 scatteringevents would lead to a mass determination at roughly 20%accuracy.20,21)
The second strategy is to look for WIMP annihilations inour galaxy. Although the density of WIMPs is sufficientlysmall that WIMPs cannot annihilate frequently enough toaffect the overall mass density of the universe, WIMPs stillshould annihilate at a low rate, especially in places wheretheir density is especially high. Astrophysicists understandthe formation of galaxies and larger structures in theuniverse as arising from the clumping of dark matter as aresult of its gravitational attraction. So our galaxy, andespecially the center of the galaxy, should be a place with arelatively high density of WIMPs and thus a higher rate ofWIMP annihilations. In an individual WIMP annihilation,the two WIMPs produce two showers of quarks, which areobserved mainly as pions and photons. The pions and othercharged particles are bent by the galactic magnetic field. Butthe photons, energetic gamma rays, fly outward in a straightline from the annihilation point. A gamma ray telescope canobserve these photons and measure their energy spectrum.The spectral shape is characteristic, with a sharp cutoff inenergy at the mass of the WIMP.22) The galaxy is expectedto contain clumps of dark matter that should show up asspots bright in gamma rays with no counterpart in opticalradiation. These spots should be intense enough to be seenwith the gamma ray telescope satellite GLAST, and, if theWIMP mass is greater than several hundred GeV, by newground-based gamma ray telescopes. Measurement of theendpoint of the energy spectrum should give a secondastrophysical determination of the WIMP mass to 20%accuracy.
If the mass of the WIMP seen at the LHC is the same asthe mass from astrophysical detection experiments, this willprovide strong evidence that the LHC is producing the trueparticle of dark matter.
6. Predicting the Properties of the WIMP
To provide additional evidence on the identity of theWIMP observed at the LHC, we would like to assembleenough data about this particle to predict its pair annihilationcross section. From (2), we see that knowledge of this crosssection from particle physics would give a prediction ofthe cosmic density of dark matter. It will be very interestingto compare that prediction to the value of the dark matterdensity obtained from cosmic microwave backgroundmeasurements. Agreement of these values would not onlyconfirm the identity of the WIMP. It would also verify thestandard picture of the early universe up to the temperatureTf , corresponding to a time in the early universe about 10�9 safter the Big Bang.
It is quite a challenge to predict the WIMP pairannihilation cross section. At the minimum, this requiresmeasuring the masses and couplings of the heavier particlesthat are exchanged in the process of WIMP annihilation. Insupersymmetry, WIMP annihilation is often dominated bythe exchange of the bosonic partners of leptons, which mustbe identified through their decays to leptons and missingmomentum. An alternative mechanism for WIMP annihila-tion is the exchange of the fermionic partners of the weakinteraction bosons W and Z. These cross sections dependsensitively on the mixing angles that determine the exactmass eigenstates of these particles. If several differentreactions can contribute, the parameters of each must bemeasured or bounded.
Detailed studies of this program in a variety of super-symmetry models show that it requires more preciseknowledge of the parameters of the model than can beobtained from the LHC. Fortunately, there is anothertechnique for producing and studying new elementaryparticles that is capable of achieving higher precision.Electron-positron annihilation at high energy can createpairs of the new particles in a controlled setting, throughreactions that are much simpler than those that we expect atthe LHC. This process will be studied at the future electron-positron collider ILC discussed in the contribution ofYamamoto.23) A simulated supersymmetry production eventat the ILC is shown in Fig. 3.
Once we have measured the masses of supersymmetricparticles with high precision and also measured the crosssections that determine their couplings and mixing angles,we will be able to put forward a prediction of the cosmicdark matter density from particle physics data that can becompared to astrophysical measurements. Recently, Baltz,Battaglia, Wizansky, and I discussed quantitatively howaccurate such microscopic predictions could be. Startingfrom a set of supersymmetry models with a variety ofdifferent mechanisms for WIMP annihilation, we analyzedthe accuracy of measurements on supersymmetric particlesthat could realistically be expected from the LHC andfrom the ILC and derived from these the accuracy of theprediction to be expected for the dark matter density.25)
Fig. 3. (Color online) Simulated ILC event, with pair production of the
supersymmetric partners of W bosons and subsequent decay to quarks,
leptons, and WIMPs.24) Only the visible products are shown in the figure.
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Figure 4 shows our results for two of these models,expressed as the likelihood distribution for �DM predictedfrom the collider data that would be expected from LHC,from ILC measurements at the design energy of 500 GeV,and from an upgraded ILC running at an energy of 1000GeV. Other groups have found similar results for first ofthese models.26,27) These predictions will be compared tothe cosmic microwave background results from the next-generation experiments, which should determine �DM to thepercent level.28) It will take some time to collect all of thedata required, but eventually we will have this sharp test ofthe WIMP identity of dark matter.
7. The WIMP Profile of the Galaxy
Once we have established the identity and properties ofthe WIMP, these results should feed back into astronomy.I noted in §2 that it is possible to detect dark matter oncluster scales and to map its distribution using gravitationlensing. However, for dark matter in the galaxy, the gravi-tational bending of light is not a large enough effect toprovide structure information. To see where the dark matteris in our galaxy, we need to map dark matter particles.
The distribution of dark matter in the galaxy is stillmysterious, and in fact is one of the most controversialquestions in astrophysics. In the cold dark matter model ofstructure formation, a galaxy as large as ours must be builtfrom the assembly of smaller clusters of dark matter. Thesmaller clusters merge through their gravitational interac-tion, disrupt one another tidally, and eventually smooth outto form the halo of the galaxy. The time required for thisevolution is on the order of the current age of the universe.Thus, most cold dark matter theories predict that the haloof the galaxy is inhomogeneous. A model of the densitydistribution of dark matter in a model galaxy, based on theclustering model of Taylor and Babul29) is shown in Fig. 5.An especially large clustering of dark matter should occur atthe center of the galaxy. Some models predict caustics withlarge, almost singular dark matter densities; other modelspredict smoothing of the dark matter below some scale.Understanding the true situation will bring us closer tounderstanding how our galaxy and the others in the universewere born and evolved.31)
The determination of the properties of the dark matterparticle will give us the information that is needed to predict
Fig. 4. (Color online) Predictions of �DM from collider data.25) Each figure was generated by assuming a specific supersymmetry model of the WIMP,
working out the set of measurements that would be made to determine the spectrum of supersymmetry particles—at the LHC, at the ILC, and at an
upgraded ILC at 1000 GeV in the center of mass—and determining the cosmic density of WIMPs from this data. The figure gives the likeihood
distribution of the prediction in each model; the accuracy of the collider measurements determines the spread in the predictions.
Fig. 5. Dark matter distribution in a model galaxy, according to the simulation of Taylor and Babul.29) This visualization, done by Baltz,30) shows a map of
the column density of dark matter along each line of sight. This quantity gives the brightness with which each cluster of dark matter shines in
annihilation gamma rays.
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the interaction rates of dark matter particles with ordinarymatter and with one another. This in turn will allow us tointerpret detection signals in terms of the absolute density ofdark matter both here and elsewhere in the galaxy. Bydividing the underground detection rate for dark matter bythe interaction cross section determined from collider data,we will be able to measure the absolute flux of dark matter atour position in the galaxy. By measuring the luminosity ofclumps of dark matter in the galaxy and dividing by the darkmatter annihilation cross section determined from colliderdata, we wll be able to map at least the largest clumps ofdark matter in terms of their absolute density.
8. Conclusions
Today, dark matter is one of the great mysteries of physicsand astronomy. But I have argued in this article that the timeis approaching for its solution. I have motivated the idea thatdark matter is composed of a new elementary particle, theWIMP, whose mass is about 100 GeV. If this is true, thenover the next five years we should produce the WIMP at theLHC, and we should see signals of astrophysical WIMPsin several different detection experiments. This will set inmotion a campaign to determine the properties of darkmatter by measurements both in high-energy colliderexperiments and through mapping of astrophysical signals.Over the next fifteen years, we will learn the story of thismajor constituent of the universe, its identity, its properties,and its role in our cosmic origin.
Acknowledgment
I am grateful to Marco Battaglia, Ted Baltz, JonathanFeng, Mark Trodden, Tommer Wizansky, and many otherswhose insights were essential in creating the pictureexplained here. I thank David Griffiths for instructivecomments on the manuscript. This work was supported bythe U.S. Department of Energy under contract DE-AC02-76SF00515.
1) M. S. Turner and D. Huterer: J. Phys. Soc. Jpn. 76 (2007) 111015.
2) V. Trimble: Annu. Rev. Astron. Astrophys. 25 (1987) 425.
3) F. Zwicky: Helv. Phys. Acta 6 (1933) 124.
4) S. Smith: Astrophys. J. 83 (1936) 23.
5) P. Rosati, S. Borgani, and C. Norman: Annu. Rev. Astron. Astrophys.
40 (2002) 539; astro-ph/0209035.
6) Y. Sofue and V. Rubin: Annu. Rev. Astron. Astrophys. 39 (2001) 137;
astro-ph/0010594.
7) V. C. Rubin, N. Thonnard, and W. K. Ford, Jr.: Astrophys. J. 87 (1982)
477.
8) D. N. Spergel et al. (WMAP Collaboration): astro-ph/0603449.
9) O. Lahav and A. R. Liddle, in W. M. Yao et al. (Particle Data Group):
J. Phys. G 33 (2006) 1.
10) R. H. Sanders and S. S. McGaugh: Annu. Rev. Astron. Astrophys. 40
(2002) 263; astro-ph/0204521.
11) D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall,
C. Jones, and D. Zaritsky: astro-ph/0608407; M. Bradac et al.:
Astrophys. J. 652 (2006) 937; astro-ph/0608408.
12) G. Bertone, D. Hooper, and J. Silk: Phys. Rep. 405 (2005) 279;
hep-ph/0404175.
13) R. J. Scherrer and M. S. Turner: Phys. Rev. D 33 (1986) 1585 [Errata;
34 (1986) 3263].
14) Y. Okada: J. Phys. Soc. Jpn. 76 (2007) 111011.
15) M. Yamaguchi: J. Phys. Soc. Jpn. 76 (2007) 111012.
16) D. Toback (CDF Collaboration): AIP Conf. Proc. 753 (2005) 373; hep-
Department of Physics, Okayama University, 3-1-1 Tsushima-naka, Okayama 700-8530
(Received February 2, 2007; accepted February 19, 2007; published November 12, 2007)
The advent of the standard gauge theory of particle physics is a story of remarkable suceess. Yet it isthe view of most particle physicists that it is not the end for the quest of the ultimate microscopic world.Direct experimental hints for physics beyond the standard theory have recently been provided by thediscovery of neutrino oscillation. On the theoretical side, there are many unsolved problems, one ofwhich we will explain in this article.
KEYWORDS: baryogenesis, leptogenesis, grand unified theory, cosmology, Majorana neutrinoDOI: 10.1143/JPSJ.76.111018
Problem The big bang cosmology has undergoneenormous developments via varieties of precision measure-ments. Along with a negative search, in our vicinity of theuniverse, for anti-matter (such as He and annihilation � from�0 decay of �NNN annihilation), the result pertaining to thematter content is summarized by saying that the presentuniverse is baryon dominated and that the degree of theasymmetry is given by the important quantity, the baryon tothe photon ratio, nB=n� � 6� 10�10.1) This value is con-sistent with the old value provided by comparing cosmicabundance observations with the calculation of the nucleo-synthesis theory that predicts the primordial abundance ofthe elements He, D, 7Li.
With the expansion of the cosmic scale factor, this impliesthat the early universe prior to nucleosynthesis must havebeen baryon asymmetric; one excess out of �1010 pairs ofnuleon + anti-nucleon created the present baryon dominateduniverse. As will be described shortly, this number is notsmall from the physics viewpoint, and it is a numberphysicists should explain.
Early history of baryogenesis theory in brief Immedi-ately after the theoretical revolution related to the standardmodel in the early 70’s, there had been intriguing proposalsof grand unified theories (GUT)2) based on the gaugeprinciple. These theories predict baryon number non-conservation, which was believed otherwise until thosedays. Proponents of these theories asserted that the gaugesymmetry and its associated conservation law alone areexactly preserved, and other empirical conservation laws aredoomed to be violated, although they may be suppressed bysome inverse power of a new physics scale. Electric chargeconservation belongs to the sacred class, but both the baryon(B) and the lepton (L) numbers may be violated. To theseproponents it is only the magnitude of violation that isquestioned.
Experimental evidence confirming the detailed structureof the electroweak theory, one of the backbones of thestandard theory, has accumulated towards 1978. It was inthis atmosphere that the first idea3,4) of applying baryon non-conservation to cosmology was put forward to explain the
observed baryon asymmetry of the universe. Immediatelyafter this suggestion, there appeared many alternativescenarios such as evaporating black hole as the seed ofthe asymmetry generation. It is particularly appropriate tomention that the possibility of baryogenesis at high temper-atures in the electroweak theory was proposed5) among thesealternatives.
Basic ingredients Three necessary conditions for bar-yogenesis were written earlier.6) These are (1) baryonnumber non-conservation, (2) C and CP violation, (3)departure from thermal and chemical equilibria. Intricateinterplay of these conditions has been appreciated only muchlater.7) Furthermore, the first condition was replaced by B�L non-conservation, as will be explained later.
In most sensible extended unified theories, it is automaticto expect the first two conditions. Let us take an example ofa heavy X-boson that violates the baryon number by havingdecay modes of different baryon numbers. Typically, acolored X-boson can simultaneously decay into two quarksand an anti-quark + anti-lepton, without violating the SUð3Þcolor gauge symmetry;
X! qþ q; X! �llþ �qq; ð1Þ
with their respective rates �q and �l. The coexistence of thesetwo channels that differ by 1 in the baryon number of finalstates is the origin of baryon non-conservation. Matterinstability occurs due to the exchange of X-bosons, since thetwo quarks uu in proton can annihilate into eþ �dd, making thedecay p! eþ þ �0 possible.
Suppose that equal amounts of the pair, X and its anti-particle �XX, existed in the early universe. When this pairdecays independently, a non-vanishing baryon number �B
may be created according to the rate formula,
d�B
dt¼
2
3�q �
1
3�l �
2
3���q þ
1
3���l: ð2Þ
On the other hand, the CPT theorem, an exact consequenceof the local field theory, tells that the total decay rate �X isequal for particle X and its anti-particle �XX. In our context,this means that �X ¼ �q þ �l ¼ ���q þ ���l. Hence d�B=dt ¼�q � ���q. In other words, a non-vanishing asymmetry isgenerated if �q 6¼ ���q. Despite the equality of the total decay
rate, the partial decay rates of X and �XX, �q and ���q, can differwhen the C and CP symmetry is violated.
The third condition, the need for the arrow of time, alsoseems obvious, but its relation to unified theories is rathernon-trivial. In thermal and chemical equilibria a process thatcan generate baryon asymmetry is precisely balanced by theinverse process that eliminates this asymmetry. The onlypossible way for finite asymmetry generation is to block theinverse process energetically. Suppose that this is realized,because when the decay of heavy X-bosons occurs, theuniverse is sufficiently cool to recreate the massive X-boson.Denoting the temperature of early hot universe by T , thiscondition is expressed as
T < mX ; at HðTÞ ¼ �X ð� �XmXÞ; ð3Þ
where HðTÞ is the Hubble expansion rate �ffiffiffiffiNp
T2=mpl, withmpl ¼ 1=
ffiffiffiffiGp� 1019 GeV the gravitational energy scale and
N the number of particle species contributing to the energydensity of the hot universe of O½100{200�. We used thenatural unit h� ¼ c ¼ 1 along with the unit Boltzmannconstant kB ¼ 1. The decay rate �X has a weak temperaturedependence /
ffiffiffiffiTp
due to the time dilatation effect, whichwas ignored here. The constant �X is the unified couplingconstant at the energy scale of grand unification, and istypically O½1=45�.
This out-of-equilibrium condition leads to a lower boundon the X-boson mass, mX > O½�Xmpl=
ffiffiffiffiNp�. An experimental
indication of coupling unification in supersymmetric theoriesis roughly consistent with this large unification mass scale.8)
An actual computation of baryon to photon ratio nB=n�involves a numeircal integration of a simplified set ofBoltzmann equations for Yi ¼ ni=n� , the respective numberdensitties ni of X, �XX, B, �BB divided by the photon numberdensity.9) These quantities are invariant with cosmic ex-pansion unless physical processes change them, and for thebaryon to the photon ratio YB ¼ nB=n� ,
dYB
dt¼ ��XðYþ � YthÞ þ �X�Y�; ð4Þ
Y� ¼ n�=n� with n� ¼ nX � n �XX obeying similar equations.Here, Yth ¼ ðnX=n�Þth refers to the thermal value. Thefundamental asymmetry � is the baryon number created bythe decay of the pair X and �XX; � ¼ ð�q � ���qÞ=ð�q þ ���qÞ, and �is a known quantity of the order of unity. The outcome ofnumerical integration may be summarized as
nB
s¼ O½10�2�
�
1þ ð16 KÞ1:3; ð5Þ
where K ¼ �X=HðmXÞ is the ratio of the X-decay rate to theHubble rate at the temperature mX , and the entropy densitys � 7n� in the present universe.
The important asymmetry parameter � that appears in thisequation can be calculated using a perturbation theory of thegrand unified theory. It is the sum over the phase space ofdifferences in the rates of the form,
� /Xjg1 f1 þ g2 f2 þ j2 � jg�1 f1 þ g�2 f2 þ j
2
¼ �4=ðg1g�2Þ=ð f1 f
�2 Þ þ ;
ð6Þ
where gi is the product of coupling constants in basicLagrangian, and fi the amplitude determined by dynamics.
The computed asymmetry is thus necessarily small due tohigher order effects of interference terms of differentperturbative orders.
Other important basic ingredients of baryogenesis havebeen summarized in my early review article.7)
Broader perspective Since the original suggestion ofGUT baryogenesis,3) many interesting works that bridgemicrophysics and the universe have appeared. Among them,the inflationary universe scenario10) became a paradigm ofthe modern time. Inflation has also given an important bonusto baryogenesis; the initial condition of the vanishing baryonnumber is automatically satisfied in an inflationary universe.
Inflation essentially depletes all conserved and non-conserved quantum numbers including the baryon number.What it left behind is a cold oscillation of the inflaton field.The hot big bang that has connection with our presentuniverse is created when the inflaton field decays, and decayproducts interact fast enough to thermalize. How hot it mustbe after inflation will be discussed later with regard to theproblem of gravitino overproduction.
The basic picture of inflation has been confirmed byrecent observations including COBE and WMAP. They alsoposed a big conundrum of the dark energy, which faces thefundamental physics in the coming years.
Electroweak baryon non-conservation ’t Hooft in hisinfluential paper11) pointed out among other things that thebaryon number may be violated in the standard electroweaktheory. This is due to tunneling effect, which is suppressedby a factor like e�1=� with � � 1=137, and hence completelynegligible under a normal circumstance.
Already in their 1978 paper5) Dimopoulos and Susskindsuggested that this suppression might be removed at hightemperatures of the early universe by barrier crossing insteadof tunneling. The necessary ingredients for electroweakbaryon non-conservations have steadily accumulated; (1)discovery of sphaleron solution12) which is an unstableobject of half a baryon number made of Higgs and gaugefields, and has a mass of O½1�TeV, (2) analysis of quark andlepton propagation under a non-trivial field configuration ofsphalerons and alike,13) and (3) the mechanism of how thebaryon number is violated at high temperatures.14)
A concrete scenario of electroweak baryogenesis has beenproposed.15) It utilizes a non-equilibrium first-order electro-weak phase transition. This scenario has been criticized.16)
Even disregarding the likely possibility of the second-orderphase transition, a correct treatment gives an estimatednB=n� as a combination of CP violation (CKM) parameters,which is much smaller than the observed nB=n� . It is ourconsensus that the standard theory cannot explain theproblem of baryon asymmetry.
Electroweak processes, both at low and high temperatures,exactly conserve B� L, the baryon minus lepton number.Although unsucessful in generating baryon asymmetry,electroweak processes at high temperatures are very efficientin redistributing already existing B� L asymmetry toindivisual thermal B and L values. Assuming B� L gener-ation at higher temperatures, the electroweak process givesfinite baryon and lepton numbers according to the physics ofthermal and chemical equilibria;
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111018-2
B ¼ a��ðB� LÞ; a ¼8ng þ 4nH
22ng þ 13nH
¼28
79: ð7Þ
Here ng ¼ 3 is the number of generationa or families andnH ¼ 1 the number of Higgs doublets. B� L generation ofamount �ðB� LÞ prior to electroweak baryon non-conserv-ing processes thus becomes the prime objective of themicroscopic theory irrespective of whether the first gener-ation is B or L.
Leptogenesis Baryogenesis according to GUT, howeverattactive it may appear, has an important missing element;observation of proton decay. On the other hand, lepto-genesis17) is attractive since it might be related to theobserved finiteness of neutrino masses confirmed by neu-trino oscillation experiments. According to oscillationparameters, neutrino masses are fit by a formula of theseesaw mechanism,19) m� ¼ m2
t =M, where mt ¼ O½100 GeV�is a typical quark and charged lepton mass, and M is a newphysics mass scale such as GUT.
Neutrinos are unique in the sense that they alone arecharge-neutral fundamental fermions unlike all other quarksand leptons. Moreover, they participate in the weakinteraction with a definite handedness; they have the left-handed chirality. This diparity between the left and the rightis presumably the most important hint towards furtherunification. It is true that without interaction all chargedfermions are described by the 4-component Dirac spinor. It ishowever possible to describe neutral fermions, even massiveones, by the 2-component spinor, and this is exactly whatwas proposed by Majorana, a long time ago in the 1930’s.
The Majorana particle without interaction is described bythe Majorana equation,
ði@t � i� rÞ’ ¼ im�2’�; ð8Þ
where m is the neutrino mass. The most salient feature ofthis Majorana equation is that it contains ’ as well as itsconjugate ’�, thus the lepton number is violated, or moreproperly, one cannot define the lepton number. In theconventional 4-component ( ) description the 2-component’ ¼ ð1� �5Þ =2 is the left-handed chiral projection.
The right-handed partner of the left-handed neutrino, theother chiral projection, ð1þ �5Þ =2 may or may not exist innature; indeed some GUT theory like the one based on theSUð5Þ gauge group lacks this partner. But most othertheories such as those based on SOð10Þ have right-handedpartners. The important point is that they are insensitiveto quantum numbers of the low energy gauge symmetrySUð3Þ � SUð2Þ � Uð1Þ. It is then natural to expect thatright-handed partners have masses, much larger than theelectroweak scale of O½100 GeV� and necessarily of theMajorana type.
It is customary to write ordinary neutrinos by �L and theirchiral partners by NR. In the Lagrangian field theory, theyare described by the diagonal Majoana NR mass M and theoff-diagonal Dirac mixing mass mt as
1
2ð�LNRÞT i�2
0 mt
mt M
� ��L
NR
� �þ ðh:c:Þ: ð9Þ
Diagonalization leads to two Majorana types of masses ofthe order, M and m2
t =M.
The Majorana type neutral lepton NR is a promissingagent for generating cosmological lepton asymmetry. Thisreplaces the role of X-boson in GUT baryogenesis. Theasymmetry computation is quite similar to that in the GUTscenario.18)
Problem to solve Both X-boson and NR lepton are verymassive, much beyond ordinary electroweak energy scale.To create them with a large quantity in the early universemeans that the universe must have been very hot right afterinflation; the temperature after inflation TR M, with M asthe heavy particle mass.
One serious problem against a very hot universe exists;gravitino overproduction. In supersymmetric (SUSY) theo-ries, the super-partner of the graviton has the spin 3=2 and iscalled gravitino denoted by g3=2. In popular SUSY models,the gravitino mass is of the order (0.1–1) TeV, reflectingSUSY breaking at the TeV scale. The gravitino decays at therate Gm3
3=2 ¼ m33=2=m
2pl (G being the gravitational constant).
Numerically, this lifetime is O½105� secð1TeV=m3=2Þ3. It issufficiently long to cause concern over the destruction oflight elements synthesized at �3 min after the big bang.
Gravitinos may be produced in the hot universe rightafter inflation. Its production rate is again dictated by thegravitational strength, and is compared with the Hubblerate
ffiffiffiffiNp
T2=mpl. The produced thermal abundance relative tothe photon density is then �T=ð
ffiffiffiffiNp
mplÞ. This abundanceimmediately tells how much entropy may be created whengravitinos decay.
The gravitino abundance is limited by the alloweddestruction of light elements that proceeds by the quark jetproduced by the gravitino decay.20) This argument gives anallowed maximum temperature after inflation of the order(106–108) GeV. This appears too small compared with theNR Majorana mass required by a detailed analysis ofleptogenesis computation.18)
There are ways to overcome this difficulty; for instance,an entirely different baryogenesis idea exists that does notassume a very high temperature, known as the Affleck–Dinemechanism.21)
Looking ahead for further experimental evidenceWhether it is B-genesis or L-genesis, it is important toverify their fundamental ingredients by experiments. Weshall discuss experimental implications of L-genesis anddescribe new types of experimental approach.22,23)
Leptogenesis requires (1) lepton number non-conservationalong with the Majorana nature, (2) a new source of CPviolation that is not related to a neutrino oscillationexperiment. Examples of lepton number non-conservingprocesses are
AZX!
AZþ2Y
þþ þ e� þ e�; AZX!
AZ�2Y þ eþ þ e�: ð10Þ
The first one is the neutrinoless double beta decay, the primeongoing experimental target, and the second e� ! eþ
conversion via laser irradiation has been discussed recent-ly.22) These processes clearly violate the lepton number by2 units, and it is natural to expect that their transitionamplitude to be proportional to the Majorana neutrino mass,or their weighted average of 3 mass eigenvalues. But theMajorana nature is conceptually independent of lepton
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111018-3
number non-conservation, and indeed it is possible to haveother sources of lepton number non-conservation that giverise to eq. (10), although the main term of violation in manyGUT models tends to be given by the Majorana mass term.
Whether neutral fermions belong to the Majorana classor the Dirac class is one of the most important questionsfacing fundamental science, much more important than anyparticular particle theory. A direct method of distinguishingthe Majorana neutrino from the Dirac neutrino has beenworked out recently.23) The Majorana nature may show uponly via interference terms due to identical fermions. andonly in this way can one distinguish the Majorana particlefrom the Dirac particle. There are only few processes of thisnature; neutrino pair emission from excited atoms.
A variety of energy differences between atomic levels areideal for the determination of small neutrino massesindicated by the neutrino oscillation experiment. If a masshierarchy is assumed, these masses are �O½50�meV. Let usdiscuss how atomic transitions via a neutrino pair �1�2 helpsin the determination of the Majorana nature. The neutrinopair emission current (in the charge retention order after theFierz transformation) is given as
where uð p; hÞ’s are 2-component spinor solutions of theMajorana equation, of momentum p and helicity h. The pairemission rate is proportional to jj�ð p1h1; p2h2Þj2, the paircurrent squared, and this should be summed over the phasespace. The interference of two terms in eq. (11) is due totwo identical Majorana fermions, which is missing in theDirac pair emission. It can be proved that the interferencerate / m1m2, the product of paired neutrino masses.23)
We hope that varieties of atomic transitions, when therate is enhanced by laser or microwave irradiation, willopen a new experimental method of mass spectroscopy. Themethod works both for the Majorana and the Dirac cases.
1) W. M. Yao et al.: J. Phys. G 33 (2006) 1; Reviews of Particle Physics,
Section of Cosmological Parameters.
2) H. Georgi and S. L. Glashow: Phys. Rev. Lett. 32 (1974) 438; J. C.
Pati and A. Salam: Phys. Rev. D 10 (1974) 275.
3) M. Yoshimura: Phys. Rev. Lett. 41 (1978) 281 [Errata; 42 (1979) 746];
M. Yoshimura: Phys. Lett. B 88 (1979) 294.
4) D. Toussaint, S. B. Treiman, F. Wilczek, and A. Zee: Phys. Rev. D 19
(1979) 1036; S. Weinberg: Phys. Rev. Lett. 42 (1979) 850.
5) S. Dimopoulos and L. Susskind: Phys. Rev. D 18 (1978) 4500.
6) A. D. Sakharov: Pis’ma Zh. Eksp. Teor. Fiz. 5 (1967) 32 [JETP Lett. 5
(1967) 24].
7) M. Yoshimura: Cosmological Baryon Production and Related Topics
in Proc. 4th Kyoto Summer Institute on Grand Unified Theories and
Related Topics, ed. M. Konuma and T. Maskawa (World Scientific,
Singapore, 1981).
8) U. Amaldi, W. de Boer, and H. Furstenau: Phys. Lett. B 260 (1991)
447; P. Langacker and M. Luo: Phys. Rev. D 44 (1991) 817.
9) E. W. Kolb and S. Wolfram: Nucl. Phys. B 172 (1980) 224; J. N. Fry,
K. A. Olive, and M. S. Turner: Phys. Rev. D 22 (1980) 2953; J. N. Fry,
K. A. Olive, and M. S. Turner: Phys. Rev. D 22 (1980) 2977.
10) A. Guth: Phys. Rev. D 23 (1981) 347; A. D. Linde: Phys. Lett. B 129
(1983) 177.
11) G. t’ Hooft: Phys. Rev. Lett. 37 (1976) 8.
12) F. R. Klinkhamer and N. S. Manton: Phys. Rev. D 30 (1984) 2212.
13) N. Christ: Phys. Rev. D 21 (1980) 1591; P. Arnold and L. McLerran:
Phys. Rev. D 36 (1987) 581; M. Dine, O. Lechtenfeld, B. Sakita, W.
Fischler, and J. Polchinski: Nucl. Phys. B 342 (1990) 381.
14) V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov: Phys. Lett. B
155 (1985) 36.
15) G. R. Farrar and M. E. Shaposhnikov: Phys. Rev. D 50 (1994) 774.
16) P. Huet and E. Sather: Phys. Rev. D 51 (1995) 379; M. B. Gavela, M.
Lozano, J. Orloff, O. Pene, and C. Quimbay: Nucl. Phys. B 430 (1994)
345; M. B. Gavela, M. Lozano, J. Orloff, O. Pene, and C. Quimbay:
Nucl. Phys. B 430 (1994) 382.
17) M. Fukugita and T. Yanagida: Phys. Lett. B 174 (1986) 45.
18) W. Buchmuller and M. Plumacher: Int. J. Mod. Phys. A 15 (2000)
5047; G. F. Giudice, A. Notari, M. Raidal, A. Riotto, and A. Strumia:
Nucl. Phys. B 685 (2004) 89.
19) T. Yanagida: in Proc. Workshop Unified Theory and the Baryon
Number in the Universe, Tsukuba, Ibaraki, Japan, ed. A. Sawada and
A. Sugamoto (KEK Report, No. KEK-79-18, 1979); M. Gell-Mann,
P. Ramond, and R. Slansky: in Supergravity, ed. D. Z. Freedman and
P. Van Niewenhuizen (North-Holland, Amsterdam, 1979).
20) M. Kawasaki, K. Khori, and T. Moroi: Phys. Rev. D 71 (2005)
083502.
21) I. Affleck and M. Dine: Nucl. Phys. B 249 (1985) 361.
22) M. Ikeda, I. Nakano, M. Sakuda, R. Tanaka, and M. Yoshimura:
hep-ph/0506062.
23) M. Yoshimura: Phys. Rev. D 75 (2007) 113007; hep-ph/0611362.
Motohiko Yoshimura was born in Shanghai,
China in 1942. He obtained his B.S (1965) and
M.S. (1967) degrees from University of Tokyo
(UT), and his Ph. D. (1971) from University of
Chicago. He was a research associate at University
of California, Berkeley (1970–1972), at University
of Pennsylvania (1972–1974), and at Ecole Nor-
male Superieur, Paris (1974–1975). He joined the
faculty member at Tohoku University (TU) from
1975 and moved to National Laboratory for High
Energy Physics as an associate professor in 1979 and was a professor since
1983. After serving as a professor at TU during 1988–2001, he became a
professor and director of Institute for Cosmic Ray Research at TU during
2001–2004. After retirement from TU, he has been a professor at Okayama
University since 2004. His research interest covers particle physics
phenomenology of the standard model and beyond that, and various areas
of cosmology. He has initiated application of grand unified theories to the
early universe and proposed a scenario of baryogenesis in 1978.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS M. YOSHIMURA
Fermi National Accelerator Laboratory, Batavia, IL 60510, U.S.A.
(Received June 20, 2007; accepted July 23, 2007; published November 12, 2007)
Modern lattice gauge theory calculations are making it possible for lattice QCD to play an increasinglyimportant role in the quantitative investigation of the Standard Model. The fact that QCD is stronglycoupled at large distances has required the development of nonperturbative methods and large-scalecomputer simulations to solve the theory. The development of successful numerical methods for QCDcalculations puts us in a good position to be ready for the possible discovery of new strongly coupledforces beyond the Standard Model in the era of the Large Hadron Collider.
The Standard Model is amazingly successful (madden-ingly so), yet its many gaps and puzzles make it clear that itis simply the low energy manifestation of new, higher-energy physics yet to be discovered. Its many parameters aresimply arbitrary, having their observed values as a result ofas yet undiscovered physics at higher energies than obtainedup until now. Its three similar generations of fermions andthree similar forces are undoubtedly related in some way yetto be discovered. To help understand what lies beyond theStandard Model, the fundamental parameters of the StandardModel are being pinned down with greater and greaterprecision in heavy flavor experiments around the world.New particles and forces are being sought in very highenergy experiments at the Tevatron and will be soon at theLarge Hadron Collider (LHC).
Lattice calculations are essential to this program in twoways. First, they are required to extract properties of quarksfrom properties of hadrons (particles that contain quarks).Unlike leptons, such as the electron or neutrino, quarkscannot be observed directly, but are confined permanentlywithin hadrons. Their properties must be inferred usinglattice gauge theory calculations.
Secondly, lattice gauge theory calculations are essential toprepare for possible new nonperturbative phenomena incoming experiments. Lattice gauge theory is the first andonly general tool for solving nonperturbative quantum fieldtheories. Of the four interactions known to particle physics,only one (quantum electrodynamics) is known to bedescribed by a perturbative theory, whose properties canbe expressed as a power series in the electromagneticcoupling constant, �em, over all energy scales. Stronginteractions are known to be described by a nonperturbativetheory, quantum chromodynamics or QCD. String theory,the current best candidate for a theory of gravity, musthave nonperturbative effects in it, or it would produce aspace-time quite different from the four dimensional worldthat we live in. In the theory of the weak interactions,consider the ‘‘Higgs’’, the particle that generates particlemasses. Is it
. an elementary, perturbative Higgs?
. a bound state of a new strong interactions (technicolor,topcolor, . . .)?
. accompanied by very high energy gluino condensates(as in many models of supersymmetry with stronglycoupled sectors)?
It is likely that whatever new physics is discovered by theLHC, it will contain some nonperturbative effects. QCD isproviding an excellent test bed to sharpen our nonperturba-tive tools to prepare for such questions.
2. Quarks, Gluons, and Lattice QCD
Asymptotic freedom and quark confinement. In theearly 60s, the classification properties of the observedhadrons led Gell-Mann and Zweig to note that the hadronswere arranged in multiplets as if they were composed ofsmaller particles, which Gell-Mann called quarks. In thesame decade, deep inelastic scattering experiments atSLAC showed that in high energy electron–proton colli-sions, protons behaved as if they were composed ofweakly interacting, almost-free constituents. Bjorken andFeynman called these entities partons. It was not immedi-ately clear whether to regard quarks as actual particles,or whether they were merely a convenient classificationtool. Furthermore, no one had ever seen a quark, so theyseemed to be strongly confined inside hadrons. This seemedinconsistent with the weakly coupled nature observed inpartons, so the relation between quarks and partons was notclear. Why should such almost-free constituents be perma-nently confined?
This paradox was resolved in 1973 with the discoveryof the ‘‘asymptotic freedom’’ of QCD. The self-coupling ofthe gluons mediating the strong force caused the effectivevalue of the strong coupling ‘‘constant’’ to become larger andlarger at long distances (long compared with the protonradius), contrary to the well-known behavior of the electro-magnetic coupling constant. This meant that even though thequarks were indeed weakly interacting at short distances andhigh energies, the force between them did not die off at longdistances, leading to their permanent confinement. Gross,Politzer, and Wilczek shared the 2004 Nobel Prize for thisdiscovery.
The consequence for particle physics is that, even though
perturbation theory may be used to analyze quark-quarkscattering at high energies, to infer the properties of quarksfrom the relatively low energy dynamics of hadron con-stituents, the nonperturbative methods of lattice QCD arerequired.
Lattice gauge theory calculations. Quantum fieldtheories are defined by their path integrals. For gaugetheory, this may be written schematically as
Z ¼Z
d Ax� x; x
� �exp �SðA; ; Þ
� �; ð1Þ
where A and are the field variables of the gluons andquarks, and S is the classical action of the theory. Thequantum amplitude for a state of quarks and gluons at agiven time to evolve into another state at a later time isobtained by integrating over all possible intervening classi-cal field configurations. In principle, one integrates overindependent fields defined at each space-time point. Aquantum field theory is in principle defined by an infinitedimensional integral (not a very well-defined object).Quantum field theories must therefore be ‘‘regulated’’.
A lattice quantum field theory regulates the continuumtheory by defining the fields on a four-dimensional space-time lattice. Quarks are defined on the sites of the lattice,and gluons on the links. Continuum quantum field theoryis obtained in the zero lattice spacing limit. This limit iscomputationally very expensive, which is why large-scalecomputer simulations are required.
Operationally, lattice QCD calculations consist of severalsteps. First, sets of gauge configurations are computedthat form a representative sample of the infinite set ofpossible configurations. They are constructed in longMarkov chains with Monte Carlo methods, such as thevenerable Metropolis method, or the more modern hybridMonte Carlo algorithm. Configurations are accumulated atseveral lattice spacings, and at several values of the massesof the light quarks in the fermi sea, which are heavierthan the physical light quark masses. Final physics resultsmust ultimately be extrapolated to the continuum and lightquark mass limits.
Second, the propagation of quarks through the gaugeconfigurations is calculated. This means solving the Diracequation on each gauge configuration. On the lattice, this is asparse-matrix problem, solved with relaxation methods, suchas the biconjugate gradient algorithm. This step can consumecompute power that approaches that of the first step if manydifferent physical processes are analyzed.
Third, hadron correlation functions and amplitudes arecomputed from the quark propagators. This is a computa-tionally cheap step, consisting mostly of I/O.
State-of-the-art price/performance for computing hard-ware for this type of calculation is currently under $1/MF.Larger projects are of order a few Teraflop-years. (Thatis, computing power of several delivered Teraflops, dedi-cated for a year.) Many types of large computers are usedin lattice calculations,1) such as the purpose-built QCDOCat Brookhaven National Laboratory (Fig. 1), large clustersof commodity computers such as the ones at Fermilab(Fig. 2) and many other places, and the IBM Blue Gene atKEK (Fig. 3), currently the largest computer in the worldpredominantly dedicated to lattice QCD calculations.
Progress in numerical science comes from both largercomputers and from improvement of methods. A methodo-logical improvement that has been particularly important forthe work I will discuss is improved discretization. Numericalanalysis tells us that if a derivative is approximated by adiscrete difference, the resulting discretization errors vanishas the square of the lattice spacing:
Fig. 1. (Color online) A motherboard of the Columbia/Brookhaven
QCDOC, a purpose built computer for lattice QCD.
Fig. 2. (Color online) Clusters at Fermilab devoted to lattice QCD.
Fig. 3. (Color online) The IBM Blue Gene computer at KEK, currently
the largest computer predominantly dedicated to lattice QCD calcula-
tions.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS P. MACKENZIE
111019-2
@ ðxiÞ@x¼ �x ðxiÞ þOða2Þ; ð2Þ
where �x ðxiÞ � ½ ðxi þ aÞ � ðxi � aÞ�=ð2aÞ. By incorpo-rating next-to-nearest neighbor interactions, we can writedown an approximation to the derivative whose errors vanishas a higher power of the lattice spacing:
@ ðxiÞ@x¼ �x ðxiÞ �
a2
6�3
x þOða4Þ: ð3Þ
This allows control of discretization errors with far lesscomputing power than the simpler derivative.
It is relatively unambiguous how to remove the Oða2Þerrors in the gluon action,2) and the various improvementsin use are closely related to each other. The situation isdramatically different with lattice fermions. There areseveral families of discretization methods, that each havevery different virtues and drawbacks. Staggered fermions3–5)
can be calculated much more rapidly than the other methods.They have therefore been the first to produce reasonablyprecise unquenched results, and many of the results inStandard Model phenomenology in the next section usethem. They have some ugly theoretical properties,6) how-ever, that lead some physicists to look at alternatives. Wilsonfermions were the original fermions used introduced by KenWilson. They break chiral symmetry badly, and for thatreason have had trouble getting to quark masses as light asthe ones in nature. Recent algorithmic advances7–9) havealtered this situation much for the better. Domain wallfermions10) and overlap fermions11) have the nicest theoret-ical properties of all. They do not suffer the complications ofstaggered fermions, and have clean chiral structure unlikeWilson fermions. In the past, they have been by far the mostexpensive with which to calculate of all the methods, sophenomenological calculations are just beginning. Rapidalgorithmic advance in the last few years have greatly spedup all the fermion methods, and it is not known at presentwhich of these methods will ultimately prove superior. Atpresent, lattice theorists around the world are hard at workon all of them, making sure that all methods give the samephysical answers.
3. Lattice QCD Confronts Experiment
Progress in unquenched lattice QCD. In comparingQCD with experiment, we have several different types oftasks. One is comparing the results of QCD calculations withknown experimental results and verifying that we canreproduce experiment. Since QCD is by now a very solidlyestablished theory, this serves more to verify lattice gaugetheory methods, rather than to test QCD. A second taskis, where possible, to make predictions of physical resultsbefore they have been determined by experiment. Sincehadron physics has been going on for decades, it is rare whenopportunities can be found, but a few have been. A third,different type of task is to use verified lattice methods,combined with experiment, to obtain results that cannotbe obtained by experiment alone. Examples of these includethe quark masses and the strength of transformations ofone quark to another under weak interactions (the Cabibbo–Kobayashi–Maskawa, or CKM, matrix elements). Since onlyhadrons are observed in experiment, and never quarks, these
cannot be directly determined from experiment. Instead,quark masses and CKM matrix elements are used asparameters in lattice calculations and chosen so that theresults of the calculations agree with experiment. Thesequark masses and CKM matrix elements are among thefundamental parameters of the Standard Model that mustultimately be postdicted by future Beyond-the Standard-Model theories. Their determination is one of the mostimportant tasks of lattice QCD as far as particle physics as awhole is concerned.
In the last few years, there has been dramatic progressin our ability to perform precise calculations of simplequantities. For twenty years after the first lattice Monte Carlocalculations appeared around 1980, almost all lattice QCDphenomenology was done in the quenched theory, meaningignoring the effects of light quark-antiquark pairs. Althoughcomputationally much cheaper than correct unquenchedcalculations, this method introduces errors of unknown sizeinto the results. The left-hand graph of Fig. 4 shows variouscombinations of particle masses and decay constantscalculated in the quenched theory and shows around 10%discrepancies with experiment. There is no theory, however,that allows us to estimate in advance what size errorsquenching introduces for any given quantity. Great strides,however, have been made in methods, in algorithms, andin computational power. They have now brought us to anera when unquenched calculations are becoming the norm,with all three light flavors of quark, u, d, and s, includeddynamically.
The calculations in the graph of Fig. 4 were performedwith improved staggered fermions (called ‘‘asqtad’’ fermionsin the jargon). The right-hand graph of Fig. 4 shows thesame quantities as on the left, but unquenched and nowshowing good agreement with experiment at the few percentlevel.12) For these calculations, the masses of some quantities
Fig. 4. (Color online) Lattice predictions compared with experiment for
simple quantities in quenched (left) and unquenched (right) lattice
QCD.12)
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like the pion and kaon masses are used as inputs to fix thefundamental parameters of QCD, the quark masses and thestrong coupling constant. Three different groups using thismethod, Fermilab, MILC, and HPQCD, then compared noteson their predictions for the simplest quantities they werecalculating, with the results shown. These results are for thesimplest physical quantities we know how to calculate inlattice QCD. The calculations are now being extended tomore and more complicated quantities. Likewise, the resultsshown are obtained with staggered fermions, the leastcomputationally costly of the fermion methods, and it willbe interesting (and necessary) to verify that one obtains thesame answers with more costly methods.
The prediction of a particle mass: the Bc. Most of theparticle masses and other simple quantities that are to be‘‘predicted’’ by lattice QCD have been well known for fiftyyears, so that only postdiction is possible. An exception hasbeen the mass of the Bc meson, a meson made of a bottomquark and a charm antiquark. Bottom quarks were discov-ered only in the 1970’s, and since they are rarely produced inassociation with charm quarks, Bc mesons had not beenobserved as of a few years ago. Figure 5 shows thepredictions of unquenched lattice calculations, before theobservation of the Bc.
13) In December of 2004, the CDFexperiment at Fermilab announced the discovery of the Bc.Their result for the mass is shown in the gold bar across thegraph, in good agreement with the lattice prediction.
The strong coupling constant. The effective coupling‘‘constant’’, �sðEÞ, governs the strength of the strong interac-tions of QCD. It is one of the best measured parameters ofQCD. Asymptotic freedom means that �sðEÞ is small incollisions at high energy, E. Therefor, perturbation theorycan be used to analyze high energy collisions in terms of apower series in �sðEÞ. The strong coupling constant can bemeasured in a large number of high-energy processes,some of which are shown in the plot in Fig. 6. One can alsoobtain the strong coupling constant with lattice methods.One should obtain the same results if lattice methods arecorrect. One obtains �s on the lattice by using it as aparameter in particle spectroscopy calculations, as in ref. 12.One then uses perturbation theory to convert the latticecoupling constant to the form used in conventional contin-uum perturbation theory analyses. The result is shown in thenext-to-bottom point in Fig. 6. It incorporates three-loop
lattice perturbation theory. It agrees well with the worldaverage of continuum results, as it should, and the latticeresult is now the most precise of the individual determi-nations. The result is15)
�sðMZÞ ¼ 0:1170ð12Þ:
The light quark masses. The value of the strongcoupling constant was well known before lattice calcula-tions. Its confirmation by lattice calculations is a welcomevalidation of lattice methods. By contrast, without thelattice, the values of the light quark masses can only beestimated approximately. The mass of the strange quarkin particular plays an important role in analysis of weakinteraction phenomenology, so a good determination of itsvalue is a pressing concern. Quark models and a variety ofphenomenological methods yielded conventional wisdomestimates of ms � 150 MeV for the strange quark mass, andml � 6 MeV for the average of the up and down quarkmasses. That conventional wisdom, we now know, is far offthe mark. With lattice QCD, we can determine these masseswith first-principles calculations, for example, by tuning thequark masses to obtain the correct masses for pions andkaons. Last year, the MILC collaboration using improvedstaggered fermions reported16,17)
ms ¼ 90ð6ÞMeV; ð4Þml ¼ 3:3ð3ÞMeV: ð5Þ
A recent paper by the CP-PACS and JLQCD collaborationsreported a result using OðaÞ improved Wilson fermions:
ms ¼ 91þ15�6 MeV; ð6Þ
ml ¼ 3:5þ0:6�0:3 MeV: ð7Þ
The two results are very compatible, giving necessaryevidence that the results of lattice calculations are notdependent on the quark method.
Fig. 5. (Color online) The mass of the Bc meson observed by the CDF
collaboration (gold bar across the figure) compared with predictions of
lattice QCD made before the observation (rightmost two data points).
Fig. 6. (Color online) Determinations of the strong coupling constant
from a variety of high energy processes, evaluated by convention at the
mass of the Z boson, MZ . (From the Review of Particle Properties.14)) The
lattice determination (second from bottom) agrees well with the world
average and is the most precise individual determination.
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Golden quantities and the CKM matrix elements.Most of the results discussed so far are for a particularlysimple kind of quantity for lattice QCD: stable mesons (thatis, ones that only decay weakly and not hadronically), inprocesses with a single meson present at a time. These aregolden quantities for lattice QCD, with uncertainties that aresmaller and easier to understand than for most quantities.Although this is a restricted set, many of the most importanttasks of lattice gauge theory can be accomplished withquantities of this type. In particular, almost all of the CKMmatrix elements and quark masses can be determined withlattice calculations in this category.
CKM matrix elements are measured in decay processes inwhich a quark of one flavor turns into a quark of anotherflavor. Fig. 7 illustrates B meson ‘‘semileptonic’’ decay, thatis, decay into two leptons plus one or more hadrons. In theexperimentally observed process, a B meson decays into twoleptons, for example the electron and a neutrino, plushadrons (labeled X), for example a pion. The complicatedstrong interactions of gluons with quark-antiquark pairsresponsible for confining the valence b and u quarks in theB meson is represented schematically in the figure by thecurly red lines (gluons) and green circles (quark = antiquarkpairs). The experimental rate depends on a QCD amplitude,which must be supplied by lattice QCD, and on the CKMmatrix element Vub, which is the coupling between an ‘‘up’’quark (u) and a ‘‘bottom’’ quark (b). Purely leptonic decays,such as a pion decaying into an electron plus a neutrino, areparameterized by decay constants such as f�. Pion leptonicdecay depends on the QCD amplitude f� and on Vud, theCKM matrix element connecting up and down quarks. Theamplitudes for mesons like K, B, and Bs to mix with theirantiparticles, K, B, Bs, are proportional to other combina-tions of CKM matrix elements. In all, eight of the nineCKM matrix elements can be determined from relativelysimple lattice QCD calculations combined with experiment,as shown in Table I.
Semileptonic decays. In semileptonic decay, the shapeof the decay amplitude as a function of the momentum ofthe decay products is predicted by lattice QCD and can bemeasured in experiment. Figure 8 shows the form factor thatparameterizes the decay amplitude for D! Kl� semilep-tonic decay as a function of t, the square of the four-momentum transfered to the leptons, l and �. The circles
(green) are lattice QCD predictions, the diamonds (blue) arefrom the experiment of the Focus collaboration whichappeared after the lattice predictions.20–22) As can be seen,the agreement is excellent. Having tested the calculation byverifying that the predicted shape is correct, by comparingthe normalization between theory and experiment, the CKMmatrix element describing the quark coupling (in this case,Vcs) may be obtained.
BB and BsBs mixing. To illustrate the challenge ahead,consider the �� � plane, shown in Fig. 9. In the StandardModel, the CKM matrix may be parameterized by fourparameters, two of which are called � and �. � and � havethe form �� i� / Vub. By determining these parameters inmany different ways, one can test whether or not consistentresults are obtained. Inconsistent results would be a signalof contributions to quark mixing from Beyond-the-Standard-Model theories, rather from the Standard Model alone.� and � parameterize the CP violation in the Standard
Model. CP is a symmetry relating the properties of particlesto those of their antiparticles. Understanding the source ofCP violation in nature is key to understanding the abundanceof matter over antimatter in the visible universe. The plotis one of the most famous graphs in particle physics at themoment, and reducing its uncertainties is an important goalof particle physics.
The plot shows the bounds on the �–� arising fromvarious physical processes, with the small red circleillustrating the combined bound. Several of the uncertaintiesin the plot arise from estimates of the uncertainties in latticeQCD calculations. For example, the bounds in the darkgreen curves, labeled �K , arise from measuring the mixingbetween K mesons and their antiparticles, analyzed with
Fig. 7. (Color online) The decay of a B meson into leptons plus hadrons
X, where X might be a pion or a D meson.19)
Table I. (Color online) The Cabibbo–Kobayashi–Maskawa matrix ele-
ments, with particle processes by which they can be measured.
Vud Vus Vub
f� fK fBK ! �l� B! �l�
Vcd Vcs Vcb
fD fDs
D! �l� D! Kl� B! Dl�Vtd Vts Vtb
hBdjBdi hBsjBsi |
0BBBBBBBBBB@
1CCCCCCCCCCA
Fig. 8. (Color online) Shape of the form factor describing the semi-
leptonic decay D! Kl�, as a function of the momentum transfer t ¼ q2.
The predicted shape (green circles) agrees well with the observed shape
(blue diamonds).
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lattice QCD. Similarly, the bounds in the yellow circles,labeled md, arise from BB mixing. The orange circles showthe constraint arising from the combination BB and BsBs
mixing from before (Fig. 9) and after (Fig. 10) the discoveryof BsBs mixing at the Tevatron last year. These constraintsare only possible due to the existence of good lattice gaugetheory calculations. The experimental errors on the mixingsthat have been measured are of order 1%. The 10 or 20%uncertainties in the quantities shown in the graph areestimates of the uncertainties of lattice calculations. Thecurrent round of calculations aims at reducing these tosomething of order 5%. Clearly, to profit fully from theexperiments that have been done, one needs to aim at latticeuncertainties of around 1%, So challenging work remainsahead for lattice gauge theorists if the experimental resultsare to be fully exploited.
Decay constants. To show that accuracies of 1– 2% inthis type of calculation are not an unreasonable goal, wecan consider a new paper from HPQCD on decay constants(which parameterize the amplitudes for decay of a mesoninto a pair of leptons). They employ several improvementsover previous calculations. Most importantly, they use astaggered fermion actions for the quarks that is morehighly improved than previously (HISQ, or Highly ImprovedStaggered Quarks25)). They employ several other improve-ments to reduce the size of the uncertainties. Their resultsare:26)
fD ¼ 241ð3ÞMeV; ð8ÞfDs¼ 208ð4ÞMeV; ð9Þ
fDs= fD ¼ 1:162ð9Þ; ð10Þ
fK= f� ¼ 1:189ð7Þ: ð11Þ
The accuracies for the D and Ds decay constants are a factorof 4 – 5 improved over previous results,27) an impressive stepforward.
Figure 11 shows their results as a function of light quarkmass, extrapolated down to the physical light quark limit,and compared there with experiment (black lines). In thecase of the D and Ds decay constants, the theory calculationshave moved quite a bit beyond experiment in accuracy (in
Fig. 9. (Color online) Current bounds on � and �, which parameterize CP
violation in the CKM matrix.24) The plot shows the status before the
discovery of BsBs mixing at the Tevatron.
Fig. 10. (Color online) Current bounds on � and �, which parameterize
CP violation in the CKM matrix.24) The plot shows the status after the
discovery of BsBs mixing at the Tevatron.
Fig. 11. (Color online) The decay constants of the Ds, D, K, and �
mesons, as a function of the light quark mass. When the results are
extrapolated to the physical light quark masses (black dashed lines), the
results agree well with experiment (black circles at the left of the graphs).
For the D and Ds mesons, the theory results are much more accurate than
the experimental results.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS P. MACKENZIE
111019-6
contrast to the reverse situation in BB and BsBs mixing). Thechallenge now for lattice calculations is to extend this levelof accuracy to many quantities.
4. The Future
I have emphasized a small set of well-done quantities thathave a strong connection to particle physics experiment.However, the current reach of lattice QCD is much broaderthan this. It is possible to study processes with multiplehadrons present at the same time, although more difficultthan for single-hadron processes. The case of K ! �� hasbeen worked out very clearly.28) Lattice calculations caninvestigate QCD in the realms of high temperatures andpotentially of high densities that are of interest in neutronstars and the early universe.29) Well-developed investiga-tions of nuclear structure are underway.30)
Calculations continue to become more and more powerfulthrough improved methods, better algorithms, and morepowerful computers. This is allowing us to improve theprecision of existing calculations, to verify that all fermionformulations give the same answers, and to extend our reachto new QCD quantities.
More exciting times could await lattice gauge theory inBeyond-the-Standard-Model physics, depending on what isdiscovered at the LHC. Such new physics could present avariety of challenges. For theories that are like QCD, butwith larger numbers of colors or flavors, the same methodsthat are proving successful for QCD can be used. For simplesupersymmetric theories, promising methods are underdevelopment for their solutions.31) For other conceivablenew theories, major algorithmic advances will be required,for example in the very interesting case of theories in whichright and left handed quarks do not come in pairs with thesame color charge. The era of the LHC and Beyond-the-Standard-Model physics is likely to prove as eventful forlattice gauge theory as the current one has been.
1) T. Wettig: PoS LAT2005 (2006) 019.
2) M. Luscher and P. Weisz: Nucl. Phys. B 266 (1986) 309.
3) S. Naik: Nucl. Phys. B 316 (1989) 238.
4) D. Toussaint and K. Orginos (MILC Collaboration): Nucl. Phys. B:
Proc. Suppl. 73 (1999) 909.
5) G. P. Lepage: Phys. Rev. D 59 (1999) 074502.
6) S. R. Sharpe: PoS LAT2006 (2006) 022.
7) M. Luscher: PoS LAT2005 (2006) 002.
8) M. J. Peardon and J. Sexton (TrinLat Collaboration): Nucl. Phys. B:
Proc. Suppl. 119 (2003) 985.
9) M. Hasenbusch: Phys. Lett. B 519 (2001) 177.
10) D. B. Kaplan: Phys. Lett. B 288 (1992) 342.
11) H. Neuberger: Phys. Lett. B 417 (1998) 141.
12) C. T. H. Davies et al. (Fermilab, HPQCD, MILC, UKQCD Collab-
orations): Phys. Rev. Lett. 92 (2004) 022001.
13) I. F. Allison, C. T. H. Davies, A. Gray, A. S. Kronfeld, P. B.
Mackenzie, and J. N. Simone (HPQCD Collaboration): Phys. Rev.
Lett. 94 (2005) 172001.
14) W. M. Yao et al. (Particle Data Group): J. Phys. G 33 (2006) 1.
15) Q. Mason et al. (HPQCD Collaboration): Phys. Rev. Lett. 95 (2005)
052002.
16) C. Aubin et al. (HPQCD, MILC Collaborations): Phys. Rev. D 70
(2004) 031504.
17) C. Bernard et al. (MILC Collaboration): hep-lat/0609053.
18) T. Ishikawa et al. (JLQCD Collaboration): arXiv:0704.1937.
19) Graphic from talk of M. Luke at CKM 2005.
20) M. Okamoto et al.: Nucl. Phys. B: Proc. Suppl. 140 (2005) 461.
21) C. Aubin et al. (Fermilab Lattice Collaboration): Phys. Rev. Lett. 94
(2005) 011601.
22) For a recent review of heavy quark physics on the lattice, see, T.
Onogi: PoS LAT2006 (2006) 017.
23) M. Ciuchini, E. Franco, F. Parodi, V. Lubicz, L. Silvestrini, and A.
Stocchi: eConf C0304052 (2003) WG306.
24) D. Buchholz: presented at Flavor Physics and CP Violation Confer-
ence (FPCP 2006), Vancouver, 9 –12 April, 2006.
25) E. Follana et al. (HPQCD Collaboration): Phys. Rev. D 75 (2007)
054502.
26) E. Follana, C. T. H. Davies, G. P. Lepage, and J. Shigemitsu:
arXiv:0706.1726.
27) C. Aubin et al.: Phys. Rev. Lett. 95 (2005) 122002.
28) L. Lellouch and M. Luscher: Commun. Math. Phys. 219 (2001) 31.
29) For recent reviews, see, U. M. Heller: PoS LAT2006 (2006) 011;
C. Schmidt: PoS LAT2006 (2006) 021.
30) K. Orginos: PoS LAT2006 (2006) 018.
31) J. Giedt: PoS LAT2006 (2006) 008.
Paul Mackenzie was born in Illinois in 1950. He
received his B.S. in physics and mathematics from
the University of Illinois in 1975, and his Ph. D. in
physics from Cornell University in 1981. He was a
research associate at Fermilab and the Institute for
Advanced Study. He has been a staff member at
Fermilab since 1986. He has worked in various
areas of perturbative and nonperturbative Quantum
Chromodynamics (QCD). His recent research has
focused on using nonperturbative QCD calculations
to extract the fundamental parameters of the Standard Model from particle
physics experiments.
J. Phys. Soc. Jpn., Vol. 76, No. 11 SPECIAL TOPICS P. MACKENZIE
Institute of Physics, University of Tokyo, Komaba, Meguro, Tokyo 153-8902
(Received February 1, 2007; accepted February 15, 2007; published November 12, 2007)
This article is devoted to a nontechnical review on the present status of string theory towards anultimate unification of all fundamental interactions including gravity. In particular, we emphasize theimportance of string theory as a new theoretical framework in which the long-standing conflict betweenquantum theory and general relativity is resolved.
The standard model has provided us a good understandingof the basic properties of present-day elementary particles,quarks, leptons and gauge bosons. Given various numericaldata, we can in principle compute the probability amplitudesof every possible process involving these particles. How-ever, it seems needless to emphasize that the model isstill quite incomplete from a theoretical standpoint. Inaddition to the fact that the standard model has to assume alarge number of input parameters, the model regarded as afundamental theory of matter and its interactions is yet at avery unsatisfactory level, since it says nothing aboutquantum gravitational interaction of elementary particles.
As is well known, the mathematical framework of thestandard model, gauge-field theory, has been developed inour endeavor towards unification of fundamental interac-tions: the structure of the standard model is governed bynon-Abelian gauge symmetries. Even putting aside universalgravitational force, however, the standard model still has notreally achieved desired unification between electroweak andstrong nuclear forces. We often expect that the idea ofunified gauge theory could be extended to a unification,‘‘grand’’ unification, of these two fundamental forces. Inregard to gravity, however, a majority of us now agree,after intensive efforts of many years, on that the ultimateunification of general relativity with quantum gauge-fieldtheory would require a totally new mathematical framework.
The reason for this view is that there remains a deepconflict between quantum theory and general relativity.Quantum mechanics and special theory of relativity togetherare two basic physical laws on which the gauge-field theoryand hence the standard model are based. But once we takeinto account general theory of relativity, we encounterserious troubles, both technically and conceptually. If weapply the standard method of quantization to generalrelativity, the difficulty of non-renormalizable infinitiesimmediately mars our attempts. Even if gravitational fieldis left temporarily as a classical field, the quantum theory ofmatter with black hole backgrounds leads us into aconceptual problem of contradicting the principle of con-servation of probabilities, namely unitarity, one of the basicprinciples of quantum mechanics, caused by the famous
Hawking radiation. Of course, in terms of classical physics,the effect of gravity is negligible at present experimentalscales when it is compared with other forces. However, theexistence of such fundamental difficulties lying beneath theextremely successful framework of modern quantum physicsshould never be discarded. The situation is analogous towhat physicists in the early 20th century were faced with inexploring microscopic laws of physics at atomic scale. Therecent development of string theory1) strengthens our hopethat string theory contains crucial ingredients for achiev-ing a reconciliation between quantum theory and generalrelativity.
String theory has a quite curious history. It started outfrom something which was nothing to do with unifiedtheories of interactions. From the 1950s to the 60s, evenafter a spectacular success of quantum electrodynamics, alarge group of high-energy physicists at that time tendedto believe that quantum field theory might not be theappropriate framework for describing strong nuclear force.Therefore the so-called ‘‘S-matrix approach’’ became amajor stream during this period, and string theory actuallyemerged from this development in the late 60s. However, asour understanding on its nature was becoming deepened,various facets as an ideal theory of all interactions includinggravity have gradually been uncovered. Even after almost 40years since its first discovery, we are still in the midst of thisprocess of exploring true meanings and new outcome ofstring theory. It is very important to recognize such peculiarevolution of our understanding in order to assess the presentstatus of string theory impartially.
As another historical analogy, we may recall a stage in thedevelopment of non-Abelian (Yang–Mills) gauge theoryfrom the mid 1950s to the whole 60s. During this period,precise method for quantizing non-Abelian gauge theory wasnot yet established, nor whether could its structure fit intoobserved properties of weak and strong nuclear forces.However, there have been many attempts proposing Yang–Mills type theories as attractive models for nuclear forces, onthe ground of fundamental symmetry requirement which wasinherited to theoretical physicists from the time of Einsteinand Weyl. This attitude turned out to be right in the 70s,after appropriate understanding of its dynamics had beenachieved, such as spontaneous symmetry breaking, Higgsmechanism, renormalizabilty, quark confinement, and quan-tum anomaly.
In the case of string theory, we have not yet arrived at anysatisfactory non-perturbative definition of string theory, norat primordial principles governing its structure. In spite ofsuch an obscure status with regard to its ultimate fate, itseems fair to say that string theory has already provided usan entirely new perspective on how gravity could be unifiedwith other interactions on the basis of quantum theory ofstrings and associated branes. It also suggested a newviewpoint on the dynamics of gauge-field theories in a waywhich has never been envisaged without unification withgeneral relativity via string theory.
In this article, we try to convey the present situation ofstring theory to physicists who are working in other researchareas than particle physics, explaining several key ingre-dients of string theory and reviewing some of importantdevelopments without technical details. For mathematicalexpressions, we use the natural units in which h� ¼ 1 andc ¼ 1 throughout this article.
2. Perturbative Formulation of String Theory
2.1 Discovery of relativistic stringsString theory evolved from a proposal made in the late
60s for a particular 2$ 2 scattering amplitude, called the‘‘Veneziano formula’’,2) of mesons which satisfies a specialsymmetry requirement called s–t ‘‘channel’’ duality. Thelatter demands that the amplitude is composed of elementssuch as the formula
Vðs; tÞ ¼Z 1
0
dx x��0s��0�1ð1� xÞ��
0t��0�1 ð2:1Þ
which can equally be described by exchanges of particlesbetween two interacting particles
Vðs; tÞ ¼X1n¼0
rnðsÞt � m2
n
¼X1n¼0
rnðtÞs� m2
n
ð2:2Þ
(first equality, ‘‘t-channel’’ description) or through formationof resonance-like states (second equality, ‘‘s-channel’’description). Here, s and t are Lorentz invariant combina-tions of energy-momenta s ¼ �ðp1 þ p2Þ2, t ¼ �ðp2 þ p3Þ2,and �0, �0 are two parameters. It soon turned out3) that thisamplitude and its various generalizations can be interpretedin terms of the dynamics of relativistic open stringspropagating space–time, provided �0 ¼ 1. The analogousamplitudes4) which correspond to closed strings were alsoconstructed.
For example, the pole singularities at s or t ¼ m2n ¼
ðn� 1Þ=�0 are interpreted as representing possible states ofstrings with definite (mass)2. There are an infinite numberof them corresponding to various vibrational and rota-tional modes of strings. Actually, it also turned that forcompletely consistent formulations of quantum stringtheory,5) it is necessary that the space–time dimensions mustbe at some particular value (critical dimensions), 26, or if wewant to include space–time (and world-sheet) fermions6)
consistently, at 10. In the latter case we can eliminate thetachyonic ground state with negative (mass)2 with n ¼ 0, bydemanding space–time supersymmetries.7) This is the originof the naming, superstring theory. It was also understoodthat closed strings can actually be generated by open strings,since one-loop amplitudes of open strings necessarilycontain singularities corresponding to the propagation of
closed strings. In other words, the s–t channel dualityextended to loop amplitudes of strings implies that closedstrings are channel-dual to both open and closed strings.
2.2 World-sheet quantum mechanics of stringsWe can formulate quantum string dynamics using a path-
integral over all possible configurations of world sheetsswept out by strings in space–time. In a symbolic andabbreviated notation, the amplitudes are expressed as
Xf�g
g��ð�Þs
ZM½dX d � exp �
1
4��0S�½X; �
� �ð2:3Þ
where the symbol f�g denotes the set of all in-equivalent(two-dimensional) Riemann surfaces, and M is the set ofconfigurations of world sheets, described by fields X; ; . . .defined on the Riemann surface. The manner of how theconstant �0 appears in this expression shows that 1=�0 isessentially proportional to the energy, or tension, of thestring per unit of length. As in the usual path integrals, wehave to specify some boundary conditions corresponding tothe initial and final states, which are suppressed in thepresent symbolic notation. The action S� is an integralover a given Riemann surface � and takes the form
Z�
d2�LðX; @�X; ; @� ; . . .Þ
with
L ¼ g��ðXÞ@�zzX�@zX
� þ � � � ð2:4Þ
where ð�1; �2Þ with z ¼ �1 þ i�2; �zz ¼ �1 � i�2 are two-dimen-sional coordinates parametrizing the Riemann surface �.The space–time coordinates of strings are represented byfields X�ð�Þ (� ¼ 1; 2; . . . ; d � 1; 0 with last index 0 beingthe time direction) on �, and g��ðXÞ is the metric tensor oftarget space–time. The additional field variable in (2.3)designates all other necessary fields, which are used todescribe non-orbital degrees of freedom, such as spins,associated with strings. The constant gs, called stringcoupling constant, specifies the weight of Riemann surfaceswith various different topologies. It is well known that thetopologies of Riemann surfaces are classified by the numbersof handles and boundaries, (h and b respectively). Thesymbol �ð�Þ � 2� 2h� b� pc � po=2 is the Euler numberof Riemann surface fixed by topology, with additionalinformation about the numbers, pc and/or po, of ‘‘punc-tures’’ inserted in the bulk of � and/or on the boundaries,respectively. The punctures essentially amount to attachinginfinite Riemann surfaces of cylinder topology (pc) or ofstrip topology (po), which correspond to (initial and final)external states of closed or open strings, respectively, ontheir mass shell.
This description would look abstract at first sight, but itis not difficult to capture basic concept if one imagines ananalogy with the notion of a particle propagator in ordinaryquantum mechanics. In the latter case, one considers pathintegrals over all possible configurations of particle trajec-tories, world lines, in space–time. A world line is regarded asa one-dimensional base space, parametrized by a singlecoordinate � for the particle coordinates X�ð�Þ. In stringtheory, the role of particle picture in local field theory isreplaced by strings. The one-dimensional base spaces
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composed of particle trajectories are elevated to special two-dimensional spaces, Riemann surfaces, and the particlecoordinates are to fields on Riemann surfaces with theirtarget space being the space–time. In particle theories whichcan be derived as perturbative rules of calculation inordinary local-field theories, we have to sum up over allpossible Feynman graphs corresponding to different top-ologies of particle trajectories, by combining propagatorsand vertices. Similarly, in string theory, we have to sum overall possible configurations of Riemann surfaces, on whichstring coordinates and their generalizations are defined.Just as the weight of each Feynman graph is essentiallydetermined by the number of vertices and external lines, thestring amplitudes are weighted by a single power of thestring coupling constant. The outcome of particle Feynmanrules is reduced to a series of expressions represented asmultiple integrals with respect to Feynman’s auxiliaryparameters, or equivalently Schwinger’s proper-time varia-bles. The dimensions of such multiple integrals equal to thenumber of propagators involved. Similarly, the stringamplitudes can be reduced to multiple integrals over theparameters called the ‘‘moduli’’ parameters of Riemannsurfaces, which classify in-equivalent Riemann surfaceswith given topology. The summation symbol
Pf�g in (2.3) is
meant to include such integrals too.The Veneziano amplitude (2.1) corresponds to a single
Riemann surface with one boundary (� disk) with fouropen-string punctures (h ¼ 0, b ¼ 1, pc ¼ 0, po ¼ 4), wherethe four external states are the ground state correspondingto n ¼ 0, which is tachyonic with a pure imaginary mass.The integration variable x appearing in (2.1) is the simplestexample of the moduli parameters.
The choice of Riemann surfaces as the base space forworld sheets of strings is a crucial ingredient for gettingconsistent theory, in that it leads to a mechanism by whichthe physical on-shell states of strings are interpreted interms of a positive definite Hilbert space of infinitenumber of particle-like modes of strings, unitarity, andalso to ultraviolet finiteness (also known as modularinvariance8)) at each order with respect to expansion interms of the string coupling gs. Riemann surfaces as thebase space of generic two-dimensional field theories canequivalently be characterized by a symmetry requirementthat we only allow conformally invariant theories. From theviewpoint of renormalization theory, conformally invariantfield theories are very special, corresponding to fixed pointsof renormalization group. In particular, two-dimensionalconformal group is an infinite dimensional group andconsequently puts strong constraints. In string theory, onedoes not allow its violation even by quantum anomaly(conformal anomaly). Recall that in usual two-dimensionalconformal field theories one allows the conformal anomaly.This stringent requirement of the world-sheet conformalinvariance explains why in string theory we have manyunusual constraints, such as critical space–time dimensions,possible gauge groups and so forth. Apart from the basicdimensional constant �0, there is no free continuousparameter.
2.3 Connection with local field theoriesThe similarity of these rules of world-sheet dynamics with
particle Feynman rules is not a mere analogy. If we take thelimit �0 ! 0, all of excitation modes other than the masslessstates corresponding to poles at s or t ¼ 0 [with n ¼ 1 in(2.2)] go infinity, and hence become invisible from thespectrum of strings. As the expression (2.3) clearly shows,the constant �0 controls the extendedness of strings. In thislimit, only world sheets shrinking into infinitely thin tubes orstrips contribute, and the string amplitudes reduce to sum ofFeynman-graph amplitudes for massless particles of ordi-nary local-field theories. In particular, multiple integralsover the moduli parameters of Riemann surfaces reduce tothose of Feynman–Schwinger parameters. The reducedtheories are nothing but gauge theories9) and generalrelativity10) with suitable inclusion of massless fields otherthan ordinary vector gauge fields and tensor graviton. Thisdiscovery started the possibility of string theory as a unifiedtheory as explicitly claimed first in ref. 11.
Basically, open strings can only lead to massless gaugefields and their cousins, while closed strings necessarilycontain graviton as well as gauge fields, together with theirsupersymmetry partners. The mechanism why the stringnecessarily contains graviton and gauge bosons in closed andopen string sector, respectively, owes again to conformalsymmetry of the world-sheet dynamics. The infinite-dimen-sional nature of the conformal group serves the reduction ofparticle degrees of freedom, as required for the presenceof massless spinning states. The graviton can be identifiedwith the one represented by the space–time metric tensorg��ðXÞ appearing in the world-sheet action in (2.3), throughthe standard state-operator correspondence in conformalfield theories.
In the mid 80s, it was established that there are fiveconsistent superstring theories in this sense. These fivetheories are classified depending on their spectra withrespect to gauge groups, space–time supersymmetries, andchiralities of massless fermion states: type I with chiralgauge group SO(32), two ‘‘heterotic’’ string theories con-sisting only of closed strings with chiral gauge group SO(32)or E8 � E8, all with one supersymmetry in 10-dimensionalsense, and type IIA and IIB theories, with two super-symmetries, which again consist of only closed strings withchiral or non-chiral massless fermion states, respectively,but do not have gauge fields in 10 dimensions. The type Itheory is unique in the sense that its world sheet must beregarded as non-orientable. The strange naming ‘‘heterotic’’string is originated from the fact that strings in these casesare constructed by pairing bosonic strings with criticaldimensions 26 with 10-dimensional fermionic strings. Thedifference 26� 10 ¼ 16 is responsible for the emergence ofgauge groups SO(32) or E8 � E8 of rank 16. In the low-energy limit �0 ! 0, these five perturbatively definedtheories reduce to corresponding local field theories, calledsupergravities in 10-dimensional space–time. Constructionof generic supergravity theories were started from the mid70s, and its perturbative quantization has been studiedvigorously in the early 80s. Even aside from the problem ofnon-renormalizability, it turned out that chiral supergravitytheories other than those which were obtained as the low-energy limits of string theory suffered from difficulty ofquantum anomalies with respect to gauge invariance and/orgeneral coordinate invariance. Recognition of such status of
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supergravity theories and its resolution12) in terms of stringsstrongly motivated to reconsider string theory as final seriousmodel for the unification of gravity with gauge theories.
3. Duality Relations among Perturbative StringTheories
3.1 T-dualityAt present, we do not know any definition of string theory
which goes beyond the perturbative rules. However, webelieve that seemingly different perturbative string theoriesactually correspond to a set of particular local stationarypoints in the space of all classical solutions in someunknown nonperturbative formulation of a single deepertheory. Here, ‘‘classical’’ means an approximation whereonly tree-type (h ¼ 0 and/or b ¼ 1) contributions are kept.Each perturbative string theory is seeing only infinitesimallysmall oscillations around one of these classical solutions.There are many good reasons for this belief. Even in a semi-perturbative framework, there are links between differenttheories.
The oldest among such relations is known as T-duality.Owing to one-dimensional spatial extension of strings, thereare excitations of strings which can never exist in particletheory. Imagine that one among 9 (or 25) spatial dimensionsis curved into a circle of radius R (circle compactification).The center of motion of strings along this direction givesstates (momentum modes) with quantized momenta pn ¼n=R (n ¼ 0;1;2; . . .), which contribute to the single-bodyenergy as E ¼ jnj=R. We can also consider strings stretchingalong this direction such that they wind up the circle m
times. This leads to states (winding modes) with single-bodyenergy as E ¼ jmjR=�0. It turned out13) that these twodifferent sets of states play a symmetrical role in genericstring amplitudes, and a string perturbation theory with agiven radius R can be mapped into a string perturbationtheory with the opposite radius RT � �0=R. The manner howthis arises is essentially the same as in the well-knownKramers–Wannier duality of the Ising model in statisticalmechanics. The radius R is the analogue of temperature T inthe latter case.
In the case of bosonic string theory, this interchange ofradius R$ RT is realized within the single theory. Hence, itis actually a symmetry of the theory. In the case of fermionicstring theories, it relates different perturbative string theo-ries. For example, type IIA theory is transformed intotype IIB theory combining with the transformation of stringcoupling as gBs ¼ gAs
ffiffiffiffi�0p
=R. Similarly, under the circlecompactification, it is known that the SO(32) and E8 � E8
heterotic theories give the same physical spectrum in 9(¼ 8þ 1) dimensional sense. This implies that two different10-dimensional heterotic theories should be regarded asdifferent limits with infinitely large circle R!1 of asingle theory.
3.2 S-dualityOne of the surprising developments of string theory
achieved from the mid 90s is the recognition that differentperturbative string theories may be related to each other byinterchanging the string coupling between strong and weakregimes. We have no reasonable definition of string theorybeyond perturbation theory, and hence we cannot directly
study strong coupling regimes in general. However, in thelow-energy limit, it is not unreasonable to assume thatsupergravity theories provide effective descriptions even instrong-coupling regime at least when one restricts attentionto some particular physics, which are not strongly affectedby massive degrees of strings, especially their symmetries.
As a typical example, take the type IIB theory. Theeffective equations of motion for massless fields in thiscase turn out to be invariant under the transformationgs! g0s ¼ 1=gs, called by ‘‘S-duality’’ transformation,14) ifone simultaneously redefines the space–time metric g��ðXÞof the world-sheet action by
where �ðXÞ is a massless scalar field called ‘‘dilaton’’. Thestring coupling is actually given by the vacuum expectationof the dilaton as gs ¼ eh�i. The transformation must beaccompanied by some additional redefinition of other fields,most notably, the interchange of two second-rank anti-symmetric tensor fields as
B0��
C0��
!¼�C��B��
� �: ð3:2Þ
The anti-symmetric tensor B�� ¼ �B�� is a gauge fieldwhose source is the strings themselves through an importantterm in the world-sheet action
Z�
d2� abB��ðXÞ@aX�@bX�; ð3:3Þ
which was suppressed for brevity in (2.4). This field togetherwith the graviton and the dilaton belongs to a sector ofclosed strings called ‘‘N(eveu)S(chwarz)–NS’’ sector. How-ever, the other anti-symmetric tensor field C�� which is in adifferent category called ‘‘R(amond)–R(amond)’’ sector doesnot directly couple to strings. Two different sectors R andNS are originated from the different boundary conditions(periodic or anti-periodic) for the fermionic degrees offreedom living on the world sheet. The repetitions such asR–R or NS–NS correspond to the existence of twoindependent propagating modes (left and right) along thestring. Clearly, in order for the S-duality transformation (3.2)to be a symmetry of the type IIB theory, we needan object which can be the source for fields in the R–Rsector. It turned out that there are such objects whichprecisely play the required role.
3.3 Dirichlet-branesThey are known as Dirichlet (D)-branes.15) To understand
the origin of the D-branes, we have to go back to T-duality.The open strings are defined by the free boundary condition@nX
�ð�Þ ¼ 0 at their end points, where @n is the derivativealong normal directions of world sheets. But T-dualitytransformation maps a normal derivative to a tangentialderivative. This means that the open strings after T-dualityobey essentially the boundary condition of Dirichlet type attheir ends along the direction of T-duality. As we havealready emphasized, the open strings can generate closedstrings. In the case of open strings with Dirichlet boundaryconditions, it turns out that closed strings generated in thisway contain states in the R–R-sector. This implies that the
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new objects at which open strings can end serve as thesource for R–R-gauge fields. They are the D-branes. Thestring coordinates at the ends of open strings with Dirichletboundary conditions can be regarded as a sort of collectivecoordinates of the D-branes, and as such the dynamics of theD-branes can be formulated as collective motions of openstrings, in an analogous way as various solitons are describ-ed in non-linear local field theories.
The D-branes can have various different dimensions withrespect to spatial extension, depending on the number ofremaining extended directions obeying the Neumann con-dition. If the number of such dimensions is p, we call thecorresponding D-branes by ‘‘Dp-brane’’. In type IIA or IIBtheories, stable Dp-branes can exist only for even or odd p,respectively. Each time we make a T-duality transformationin a particular direction, the Neumann and Dirichletboundary conditions are interchanged, and hence D-branesin respective theories are changed by one.
We can now go back to the S-duality in the type IIBtheory. The additional second-rank anti-symmetric gaugefield C�� has D1-branes as its source. In other words, by theS-duality, the original ‘‘F’’(undamental)-string of the typeIIB theory is actually transformed into the D1-brane (oftencalled ‘‘D-string’’). The collective coordinate interpretationof end points of open strings coupled to D1-branes leads tothe fact that the tension of D-string is proportional to 1=gs,because the dependence on the string coupling of a simplestD-string amplitude is determined by � ¼ �1 with h ¼ 0,b ¼ 1, po ¼ 0 ¼ pc. This is consistent with (3.1), since itrequires that the tension of D1-string must be the same 1=�0
as that of the fundamental string in terms of the new unitusing the transformed space–time metric ds0 � ds=gs. Otherelements of the closed-string states in the R–R-sector and D-branes with other dimensions naturally fit into this inter-pretation of S-duality for D-branes. Thus we are convincedthat the type IIB theory is self-dual with respect to S-duality,in a similar way that the bosonic string theory is self-dualwith respect to T-duality. Although we cannot go intodetails, other D-branes also play their respective roles in S-duality. In particular, there exist D3-branes which actuallybe regarded as self-dual objects under S-duality.
In other theories than the type IIB, the S-duality in generalgives various connections among different theories. Surpris-ingly, it turned out that the type IIA theory in the strongcoupling regime actually leads to a new theory which is notlisted among five possible perturbative string theories.
The lowest dimensional D-branes in the type IIA theoryare the D0-branes, or D-particles, since the open stringsassociated to them obey the Dirichlet condition with respectto all spatial directions. They are sources for the 1-formgauge field A� in the RR-sector. From the viewpoint of low-energy effective theory, namely, 10-dimensional type IIAsupergravity, it has long been known that this gauge field cannicely be interpreted geometrically if one starts from 11-dimensional supergravity theory16) and make a dimensionalreduction to 10 dimensions. This is an application of the oldidea of Kaluza–Klein theory. In 11 dimensions, the space–time metric g�� has 11� 12=2 ¼ 66 independent compo-nents. On the other hand, in 10 dimensions one has 10�11=2 ¼ 55 components. Supposing that 10-dimensionalspace–time is a result of compactification of one of the
spatial dimensions in 11-dimensional space–time along acircle of infinitely small radius R11, one can regard theremaining 66� 55 ¼ 11 components as composed of a 10-component vector field g11� and a scalar field g11;11. Each ofthem can be related to the 1-form gauge field A� and thedilaton �, respectively. In this interpretation, the chargeassociated to A� is identified with the momentum alongthe compactified direction. The excitations with the lowestunit of quantized momentum along this direction have adefinite mass 1=R11 when interpreted in 10 dimensions.Under this 11-dimensional interpretation of the type IIAtheory, the D0-branes as the source for A� should beregarded as such excitations along the compactified direc-tion. This demands that
1=R11 ¼ 1=gAsffiffiffiffi�0p
: ð3:4Þ
Thus we expect that the strong coupling regime, R11!1,of the type IIA theory should actually be formulated in 11-dimensional space–time.
It is interesting to note that with the help of T-dualitybetween IIA and IIB, we can express the string couplingof the type IIB theory as gBs ¼ R11=R with R being the radiusof circle compactification along one of spatial directions inthe remaining 10-dimensional space–time. This gives anattractive 11-dimensional interpretation of S-duality as thesymmetry under the interchange, R$ R11, of the twodirections of circle compactifications.
3.4 M-theory and the unity of all string theoriesThe above observation strongly suggests that there exists
some unknown theory as an extension of 11-dimensionalsupergravity such that under compactification to 10 (orlower) dimensions it reduces to type II string theories.17) Theexpected theory is called ‘‘M-theory’’. There are variousindications that the objects which then replace strings are(super)membranes (hence the naming ‘‘M’’) whose spatialextensions are two. In fact, at least classically, it is easy toshow that the action for such supermembrane reduces to thatof the superstring when it is compactified along the circlein 11th dimension. The existence of D2-brane in type IIAtheory is also consistent with this interpretation, since itshould be the supermembrane of the M-theory immersedcompletely within 10 dimensions. Unfortunately, however,the quantum theory of (relativistic) membranes has longbeen an unsolved open problem, and it is difficult to confirmthese ideas at the present stage of development.
Recognition of the existence of D-branes implies that thedistinction between theories with only closed strings andtype I theory is not quite fundamental. D-branes can exit inboth theories, though there is some speciality in the case ofthe type I reflecting a characteristic that the world sheets oftheir open strings are non-orientable. The latter propertyafter T-duality leads to special (hyper)planes known as‘‘orientifolds’’ which play a sort of mirrors reversing bothspatial directions and world-sheet parity simultaneously.Detailed studies of, say, D-strings in T-dualized type Itheory show that the spectrum of oscillations around theground state of a D-string coincides with that of a longheterotic strings with SO(32) gauge group. In the low-energylimit, the supergravity actions of the type I and SO(32)heterotic theories are connected to each other by relating the
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metric tensors and dilatons as gI�� ¼ e��h
gh��;�I ¼ ��h,
together with some relations of gauge fields on both sides.These observations indicate that the strong-coupling regimeof the type I theory is equivalent to the weak couplingregime of the SO(32) heterotic theory. Furthermore, theassumed M-theory can provide a new link including theE8 � E8 heterotic theory. As we have already mentioned,with a circle compatification, two heterotic theories in ninedimensions are regarded as two different limits of a singletheory. This together with the above link between thetype I and SO(32) heterotic theory makes us possible toconnect the theory to the type IIA theory when the latter isconsidered with special compactification on a line S1=Z2
which is a circle divided by Z2 reflection around a point.With the help of the relation between the type IIA and Mtheories, one finally arrives at a set of links by which all fiveperturbative string theories are connected owing to theassumed existence of the M-theory. In this way, we nowbelieve that there must exist a single unified and back-ground-independent framework of string/M theory fromwhich all five string theories and M-theory in 11 dimensionsare formulated as different limiting cases.18)
4. Facets of String Theory towards the UltimateUnification
Thus far, we have explained what the perturbativeformulation of string theory is and how we are trying toachieve the unity of different possibilities of the theories.The attentive readers would ask; ‘‘All these developmentsare indeed impressive as themselves, but why do we have tobelieve that this is the theory which you are seeking forbehind the standard model? After all, the theory cannot yetprovide any decisive experimental predictions by which onewould check evidence for the theory directly . . .’’ Of course,only time will be able to answer this question.
4.1 Merits of string theoryBefore proceeding to issues relevant in making connec-
tions of string theory to the real world, let us summarizehere several characteristic properties of string theory as anew framework toward the ultimate unification, whichdiscriminate the string/M theory from any other attemptswithin the ordinary framework of local field theory.String theory(1) encompasses almost all relevant ideas and/or methods
devised in the past, towards unification of particleinteractions:Such ideas include gauge invariance, Kaluza–Kleinmechanism, induced gravity, composite models, andvarious higher symmetries including supersymmetry.Methods include bootstrap, current algebra, topologicalexcitations, various duality transformations, and soon. In a sense, string theory has already achieved aspectacular unification of ideas;
(2) provides a conceptually satisfying scheme of unifyingall interactions including gravity:One of the merits of the world-sheet formulation ofstrings on the basis of Riemann surfaces is thatinteraction and motion become a completely unifiedconcept. Note that locally a Riemann surface is alwaysa single sheet. The interaction vertices of various
different particle modes of strings automatically re-sulted from the difference of global topologies of thesurfaces as an artifact of taking the low-energy limit. Itautomatically generates gravity and gauge interactionsas inevitable consequences from its mathematicalstructure;
(3) resolves the ultraviolet difficulty which is inherent toall perturbative theories of particle-field theories withlocal interactions:Previous attempts to unify gravity suffered from theultraviolet difficulty. Removal of the ultraviolet diffi-culty within the usual framework of local field theoryor in an extended framework allowing non-localinteractions usually induces violation of unitarity. Instring theory, by contrast, the resolution of ultra-violetinfinities is achieved by conformal invariance ofRiemann surface in complete conformity with unitar-ity. The singular limit of a Riemann surface corre-sponds, in terms of space–time geometry, only tophysical unitarity singularities of Feynman graphs atlong distances, while, in local-field theories, Feynmangraphs contain also unphysical singularities associatedto short-distance limit which lead to the ultra-violetdifficulty;
(4) provides for the first time a microscopic explanation,19)
albeit only for some special cases protected by super-symmetry, of Black hole entropy in terms of quantumstatistical language:This is based upon the interpretation of the extremaland near extremal black holes in terms of D-branes;
(5) provides several new perspectives for understandingthe dynamics of ordinary gauge field theories:The most recent and remarkable example of this is theso-called AdS/CFT correspondence (or more generallygravity-gauge correspondence).
Perhaps we should add to this list that string/M theorywould also provide a possibility of explaining why our worldhas four dimensions, starting from the critical dimensions,provided that one could finally achieve a proper understand-ing on the dynamics of reduction of space–time dimensions.
Among others, the importance of resolving the difficultyof non-renomalizability which has been marring attemptsquantum gravity cannot be overemphasized. For example, ifone tries to compute the black-hole entropy using local fieldtheories, one necessarily encounters ultraviolet infinities.Not only that, the renormalization also forces us to intro-duce infinitely many other dimensionful constants to themicroscopic theory than the Newton constant. One mightthink that different approaches such as, say, loop-quantumgravity20) which uses instead of the metric tensor a set ofvariables defined on extended loop regions of space–time,would also be able to evade the ultraviolet infinities. In suchattempts, however, there is unfortunately no microscopicprinciple in writing down the action. It seems to fair to saythat they are useful only as possible languages for effectivedescription of some particular aspects of quantum gravity.The importance of this aspect of string theory has beenmotivating various proposals on the interpretation of short-distance properties of string theory, such as a generalizeduncertainty principle or a novel uncertainty principle ofspace–time itself.21)
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As regards to the black-hole entropy, it is interesting torecall that in the genesis of quantum theory the statisticalinterpretation of the entropy of black body radiation playedan indispensable role in identifying the correct microscopicdegrees of freedom. The ultraviolet catastrophe whichphysicists had to struggle 100 years ago has never beencompletely resolved when gravity is taken into account.Certainly, string theory provides the first (and only known)promising direction toward its resolution.
4.2 Attempts towards non-perturbative formulationsWe now briefly describe some representative attempts
towards non-perturbative formulation of string/M theory.The leading among them is the string field theory. Non-perturbative effects of strings must involve various non-trivial processes of creation, annihilation and condensationof strings and D-branes. However, the world-sheet formu-lation explained in §2 is simply an extension of perturbativeFeynman rules in ordinary field theory, and as such itsapplicability is in principle restricted to processes involvingonly a small number of strings. In order to treat morecomplicated phenomena without such restriction, it is naturalto reformulate the world-sheet formalism into a field theoryof strings by introducing string fields which create andannihilate strings. The string fields are then functional fields�½XðÞ; . . .� on the base space of all possible stringconfigurations fX�ðÞ; . . .g. We can in principle write downaction principle for the string fields, such that it reproducesthe world-sheet string amplitudes (2.3) at each order withrespect to power-series expansion in the string couplingconstant gs. If we represent the general string configurationin terms of Fourier expansion X�ðÞ ¼ x�0 þ
Px�n expðinÞ
with x�0 being the center-of-mass coordinates of string, sucha string field theory can be regarded as a special version ofinfinite-component field theories.
Construction of string field theory was started alreadyfrom the mid 70s.22) The most successful among suchattempts is the one proposed by Witten23) in the mid 80sfor bosonic open string theory. But its applications to non-perturbative effects have become possible only in recentseveral years, owing mainly to technical obstacles. Forexample, we can discuss the problem of instability ofbosonic string theory. The existence of the tachyonic groundstate indicates that the usual vacuum of bosonic string theoryis unstable against creation of strings in the tachyonicmodes. If one treats the string field in classical approxima-tion, one can discuss the fate of this unstable vacuum bystudying the possibility of stable vacua which shouldmanifest themselves as local minima of effective potentialfor the string field. Such stable vacua must correspond tonon-trivial classical (static) solutions of the string-fieldequation of motion. Quite recently, an example of such exactclassical solutions has been constructed,24) and led to a proofof a conjecture25) that the unstable vacuum can be regardedin the classical approximation as the excitation of anunstable D-brane on the background of a newly foundstable vacuum.
Unfortunately, however, fully satisfactory formulation ofopen-string field theory has never been achieved in the caseof superstrings with fermionic strings. Also, the status of thisapproach for closed strings has been much more subtle.
From the viewpoint of Riemann surface, the string fieldtheory requires a decomposition of its moduli space suchthat each element in the decomposition corresponds to asingle Feynman graph in the ordinary sense of particletheory. Taking the example of the Veneziano amplitude(2.1), the integration region ½0; 1� of the variable x isdecomposed into two segments ½0; 1=2� and ½1=2; 1�, relatedby a mapping x$ 1� x; s$ t, each of which leads tocontribution with poles with respect only to either s or t,corresponding to particle Feynman graphs with s- or t-channel particle exchange, respectively. A possible stringfield theory gives such a particular decomposition of thewhole moduli space of Riemann surfaces. In the case ofclosed strings,26) however, to achieve this in a covariant waybecomes an extremely complicated problem as one goes tohigher orders in gs. It should also be noted that, by so doing,various nice characteristics of string theory, especially theduality between open and closed strings, resulting fromglobal properties of Riemann surfaces become largelyinvisible in terms of string fields.
4.3 Matrix modelsIn the framework of string field theory, dealing with
multi-body processes of D-branes is not easy, since they aresort of collective excitations of strings. Here it is useful torecall that the low-energy approximations of open stringfield theories are (super) Yang–Mills gauge theories definedon (pþ 1)-dimensional base space–time, in which the sizeof gauge group corresponds to the number N of Dp-branes.There are some situations where the Yang–Mills modelscan even be regarded exact in certain sense. For example,in the case of D0-brane (D-particles), large N means largemomentum P � N=R11 along the compactified circle fromthe viewpoint of 11-dimensional M-theory. We can then goto an infinite-momentum frame which is boosted along thisdirection. All motions in this frame become infinitely slowand hence we can rely on a low-energy non-relativisticapproximation. Motivated by this, it has been proposedthat an appropriate large N limit of (0þ 1)-dimensionalsuper Yang–Mills theory (Yang–Mills quantum mechanics)could be a version, known as M(atrix) theory,27) of non-perturbative description of the M-theory. The Yang–Millsquantum mechanics for large N can also be regarded asa particular regularized version of membrane theory.28)
Unfortunately, the quantum dynamics of the theory forlarge N is not accessible to our present technology, sincethe Yang–Mills quantum mechanics is necessarily stronglycoupled in the infra-red regime. It is, however, remark-able that this theory in perturbation theory can reproducenon-linear general-relativistic effects29) of graviton self-interactions.
A similar proposal known as IIB matrix model30) hasbeen given from the viewpoint of the type IIB string theoryand its lowest dimensional D(-1)-branes, which are essen-tially ‘‘instanton’’-like objects whose open strings obeyDirichlet condition with respect to all directions includingtime direction. It is simply a large N limit of 10-dimensionalsuper Yang–Mills theory defined on one point, or dimen-sional reduction from 10 to 0, in close analogy with the so-called Eguchi–Kawai reduction31) of lattice gauge theoriesproposed in the early 80s.
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In any of these and related proposals, we had not madetruely new insight with respect to the most importantcharacteristics of string/M theory such as the S-duality, andthe (channel) duality between open and closed strings,unfortunately. It seems too early to assess whether the stringfield theory or its matrix model versions could be the rightlanguages for non-perturbative formulation of fully quan-tized string/M theory.
4.4 Gravity-gauge correspondencePerhaps, one of the most surprising and potentially useful
outcome of string theory whose impact might go down toother research fields than fundamental physics is theexistence of duality between gauge theory and generalrelativity. Of course, string theory unifies both theories: thisis the main point of our discussions above. However, weusually regard these two theories as two different fieldtheories formulated on the basis of entirely different fields,vector gauge fields and metric tensors, and on differentprinciples, gauge principle and general covariance, respec-tively. In recent 10 years or so, it turned out that there can bespecial circumstances where these two theories embeddedin string theory actually describe the same single physicsusing two different languages, in an analogous way as welearned in the wave-particle duality of quantum mechanics.The key ingredient to understand this remarkable phenom-ena is again the D-brane.
On one hand, D-branes can be formulated as collectivemodes of open strings attached to them. In an appropriatelow-energy limit, they can be described by gauge fieldtheory. On the other hand, they act as sources for closedstrings, and thus their dynamics is reflected in the behaviorof metric tensors and their partners including those in theR–R sector. If one can have some regions of space–timewhere massive string states on both theories are ignored,this implies that these two different field theories candescribe one and the same physics. The most typical case ofthis conjecture is known as AdS/CFT correspondence32)
between the four-dimensional super SU(N) Yang–Millstheory with maximum possible supersymmery, known asN ¼ 4 supersymmetry, and supergravity corresponding tothe type IIB theory around a special background geometrywhich is the product of a five-dimensional anti-deSitterspace–time (AdS5) and a five-dimensional (hyper) sphere(S5). The former is the low-energy effective theory of N
D3-branes, while the latter corresponds to the geometryproduced by them in the near horizon region. The symbolCFT indicates that the gauge-theory side is a conformallyinvariant field theory in four dimensions. D-branes ingeneral generate black-hole type geometries with eventhorizons. The AdS geometry appears when we study thisblack D3-brane geometry by approaching the horizonsufficiently closely, comparing to the string scale
ffiffiffiffi�0p
. Thecurvature radius in the near horizon limit of this geometry isof order L � ðgsNÞ1=4
ffiffiffiffi�0p
for both five-dimensional ele-ments of this product space–time, whose metric is
ds2 ¼ L2 z�2ðdz2 þ dx24Þ þ ds2S5
� �ð4:1Þ
with dx24 and ds2
S5 being the metrics of flat four-dimensionalMinkowski space–time and a unit five-sphere, respectively.The low-energy approximation of supergravity is valid when
gsN 1 and gs � 1, the latter of which is necessary tosuppress the quantum loop effects of closed strings. TheYang–Mills coupling constant gYM is related to the stringcoupling as gs / g2
YM.One might wonder how the correspondence could be
realized between 4-dimensional and 10-dimensional fieldtheories. The mapping between them is as follows. There are10 bosonic fields on the Yang–Mills side. Among them, sixscalar fields describe the collective motions of D3-branesalong the six transverse spatial directions, while theremaining four-component vector gauge field correspondsto the lowest modes of open strings propagating along thebase four-dimensional space–time of D3-branes. The formersix directions are separated into one radial direction and fiveangles, since D3-branes are point-like with respect to thesesix directions. The radial direction and the four-dimensionsof the D3-brane base space together are related to the fivecoordinates ðz; x�4 Þ of AdS5 geometry, while the five anglesare to the S5. The conformal invariance of the gauge theoryis translated to the isometry of the metric (4.1) underSOð4; 2Þ coordinate transformations, the simplest amongwhich is the scale transformation ðz; x�4 Þ ! �ðz; x�4 Þ.
Because of the common conformal symmetry togetherwith supersymmetry, we can classify the spectra of boththeories using conformal dimensions. In particular, there arecertain classes of states, called BPS states, whose spectra donot depend on any continuous parameters. They can bereliably enumerated on both sides using weak-couplinganalysis, and an explicit mapping of states and operatorsbetween them can be established. Non-BPS states are moredifficult to make correspondence, since they necessarilydepend on the strong effective coupling constant g2
YMN
( 1) of the large N gauge theory. In recent years, we aremaking important progress along this direction,33) includingsome explicit relations of correlation functions of the gaugetheory and the corresponding amplitudes on the supergravityside. These new developments suggest how to interpretmassive closed-string states in terms of the gauge-theoryvariables. This leads to a conjecture that the whole contentof the type IIB closed string theory may be encoded in thegauge theory.
In the above correspondence, the four-dimensional gaugetheory can be regarded as living on the four-dimensionalspace–time at the boundary z! 0 of the five-dimensionalbulk space–time, AdS5. For example, correlation functionsof the gauge theory can be interpreted34) as amplitudes ofgravity theory observed at the boundary, examining re-sponses from the bulk against small disturbances added atz � 0. In this sense, the AdS/CFT correspondence providesa concrete example of the idea of ‘‘Holography’’.35)
The latter has originated from an interpretation of theBekenstein–Hawking entropy of black holes, which indi-cates that the information on micro-states of three-dimen-sional black holes is encoded in the degrees of freedomon the two-dimensional black-hole horizons. Some generalconsiderations suggest that this decrease of physicaldimensions is a universal feature of the correct quantumtheory of gravity.
There are many attempts and proposals for possibleextensions of the gravity-gauge correspondence. One caneven hope that the complicated non-perturbative dynamics
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of QCD may be rephrased holographically into someappropriate gravity theory in higher dimensions, generaliz-ing the AdS/CFT correspondence to non-conformal andnon-supersymemtric theories.
5. Roads to the Real World?
As emphasized above, we are still in the midst of theprocess of exploring hidden meanings of string theory,without knowing the real definition of the theory. Thereforewe cannot make any definite statement with respect toprospect on various issues related to scenarios how thetheory gives predictions for the real four-dimensional world.
However, attempts to construct 4-dimensional stringmodels by reducing 10 dimensions into products of 4-dimensional Minkowski space–time M4 and small unob-servable 6-dimensional spaces K have been naturally a bigenterprise of string physicists, ever since the first explosionof string theory in the mid 80’s. Such works were withinthe framework of five possible perturbative formulations.In these approaches, one requires in view of the hierarchyproblem that reduced models should have N ¼ 1 space–timesupersymmetry in four dimensions, which sets some specialgeometrical constraints for the six-dimensional compactifiedmanifolds K. A dominant role for satisfying this require-ments is played by the so-called Calabi–Yau manifolds, andstudies have been performed under fruitful interactions ofstring physicists with mathematicians. In particular, variousduality relations on the string-theory side suggest usefulinsights on some unexpected relations, such as ‘‘mirrorsymmetry’’ which can be regarded as a cousin of T-duality,between mathematically different characterizations of thesemanifolds.
It is impossible here to summarize very extensive studiesover 20 years along this direction. A general consensus isthat although there are a small class of models which mimicthe features of the standard model or grand-unified theories,there are a huge number of different possibilities of suchschemes of compactifications. Even if one starts fromsome possibilities of standard model-like string vacua, theyusually have many unwanted fields, such as dilatons andother perturbatively massless excitations called ‘‘moduli’’fields corresponding to infinitesimal deformations of thegeometry allowed under the constraints for the Calabi–Yaucompactifications.36) These additional degrees of freedommust be lifted to massive fields in order to be consistentwith various experimental and cosmological constraints. Theissues together with the problem of breaking supersymmetrynecessarily become non-perturbative.
Recently, with the advent of D-branes and related objects,there have been proposed various interesting partially non-perturbative mechanisms37) to resolve some of these prob-lems. For example, we can assume presence of D-branes ortheir associated fluxes of R–R-gauge fields extending in thecompactified manifolds. They can be used to generate somedesirable masses for unwanted fields. Myriads (10500 ormore in the case of the type IIB model) of those possibilitiesare expected. Total space consisting of all such possiblevacua is now called ‘‘String Landscape’’. Majority of suchvacua would have finite cosmological constant characterizedby the string scale. There are various conceivable mecha-nisms to obtain vacua with different cosmological constant
by considering excitations of D-branes and anti D-branes.Such arguments are however based on low-energy effectivefield theories. It is not clear whether these discussions ofcompactifications survive when we hopefully arrive at rightstringy languages for non-perturbative string physics.
Finally, D-branes have also suggested a different scenariofor approaching 4 dimensions. As we have explained above,the system of N Dp-branes in type II theories are describedin a low-energy approximation by SU(N) gauge fieldtheories. For p ¼ 3, we have thus four-dimensional gaugetheories. This inspired the idea38) that our universe couldactually be confined on a four-dimensional ‘‘world-brane’’.The five-dimensional AdS part of the metric (4.1) providessimplest such a model. If one assumes that the universe sitsat y ¼ ‘ with z ¼ ey, there exist different length scales,separated exponentially by a ‘‘warp factor’’ e‘ from that ofbulk (gravitational) physics, and one could hope that thismight effectively explain the hierarchy of mass scales inunified models. This and related possibilities have been oneof hottest topics in phenomenological approaches both inparticle physics and cosmology in recent several years.We leave this subject to some of other articles in the presentspecial issue.
Acknowledgments
The present work is supported in part by Grants-in-Aidfor Scientific Research [No. 13135205 (Priority Areas) andNo. 16340067 (B)] from the Ministry of Education, Culture,Sports, Science, and Technology.
1) The following bibliography is very incomplete, since we cannot
mention all relevant original works in view of the nature of this
article. The author apologizes anyone whose important works are not
listed here. For extensive text-book accounts of string theory with
more references, see M. B. Green, J. H. Schwarz, and E. Witten:
Superstring Theory (Cambridge Univ. Press, 1987) Vols. 1 and 2;
J. Polchinski: String Theory (Cambridge Univ. Press, 1998) Vols. 1
and 2; K. Becker, M. Becker, and J. Schwarz: String Theory and
M-Theory: A Modern Introduction (Cambridge Univ. Press, 2007).
2) G. Veneziano: Nuovo Cimento A 57 (1968) 190.
3) L. Susskind: Phys. Rev. Lett. 23 (1969) 545; Y. Nambu: in Proc. Int.
Conf. Symmetries and Quark Models (Gordon and Breach, 1970)
p. 269; H. B. Nielsen: submitted to 15th Int. Conf. High Energy
Physics, Kiev, 1970.
4) M. Virasoro: Phys. Rev. 177 (1969) 2309.
5) P. Goddard, J. Goldstone, J. Rebbi, and C. B. Thorn: Nucl. Phys. B 56
(1973) 109.
6) P. Ramond: Phys. Rev. D 3 (1971) 2415; A. Neveu and J. H. Schwarz:
Nucl. Phys. B 31 (1971) 86.
7) F. Gliozzi, J. Scherk, and D. Olive: Nucl. Phys. B 122 (1977) 253.
8) J. A. Shapiro: Phys. Rev. D 5 (1972) 1945.
9) A. Neveu and J. Scherk: Nucl. Phys. B 36 (1972) 155.
10) T. Yoneya: Prog. Theor. Phys. 51 (1974) 1907; J. Scherk and J. H.
Schwarz: Nucl. Phys. B 81 (1974) 118.
11) J. Scherk and J. H. Schwarz: Phys. Lett. B 57 (1975) 463.
12) M. B. Green and J. H. Schwarz: Phys. Lett. B 149 (1984) 117.
13) K. Kikkawa and M. Yamasaki: Phys. Lett. B 149 (1984) 357.
14) For a review of S-duality, J. Polchinski: Lectures on Superstring
and M Theory Dualities (World Scientific, Singapore, 1996) Fields,
Strings, and Duality, TASI 1996, pp. 359– 418; J. Polchinski:
hep-th/9607201.
15) J. Polchinsiki: Phys. Rev. Lett. 75 (1995) 4724.
16) E. Cremmer, B. Julia, and J. Scherk: Phys. Lett. B 76 (1978) 409.
17) C. M. Hull and P. K. Townsend: Nucl. Phys. B 438 (1995) 109;
E. Witten: Nucl. Phys. B 443 (1995) 85.
18) For more detailed discussions on the unity of perturbative string
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