Flavour Physics beyond the Standard Model ... Flavour Physics beyond the Standard Model:...

Flavour Physics beyond the Standard Model:Phenomenological analyses through rare b-hadron decays

Dottorato di Ricerca in Fisica

XXVII Ciclo – A.A. 2014-2015

Candidate

Pietro Biancofiore

Thesis Advisor

Dr. Pietro Colangelo

A thesis submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics

2015

The image on the cover is taken from:CKMfitter Group (J. Charles et al.), Eur. Phys. J. C41, 1-131 (2005) [hep-ph/0406184],

updated results and plots available at: http://ckmfitter.in2p3.fr

http://ckmfitter.in2p3.fr

UNIVERSITA DEGLI STUDI DI BARI ALDO MORO

DIPARTIMENTO DI FISICA INTERATENEO MICHELANGELO MERLIN

Dottorato di Ricerca in Fisica

CICLO XXVII

Settore Scientifico–Disciplinare:Fis/02 – Fisica teorica, modelli e metodi matematici

Flavour Physics beyond the Standard Model:Phenomenological analyses through rare b-hadron decays

Dottorando:

Dott. Pietro BIANCOFIORE

Supervisore (Tutor):

Dott. Pietro COLANGELO

Coordinatore:

Ch.mo Prof. Gaetano SCAMARCIO

ESAME FINALE 2015

Contents

Preface iii

Abstract v

Sommario vii

Acknowledgements ix

1 Introduction 1

2 Models with TeV scale extra dimensions 7

2.1 Two Universal Extra Dimensions (UED6): compactification on a square 10

2.1.1 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.3 Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.4 Kaluza-Klein parity . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Fields in a slice of AdS5: compactification on an interval . . . . . . . . . 18

2.2.1 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.3 Gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Flavour structure of extra dimensional models 43

3.1 Minimal flavour violation and UED’s . . . . . . . . . . . . . . . . . . . . 44

3.1.1 Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.2 Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Flavour in the RSc model . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 Gauge sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.2 Fermionic sector . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.3 Higgs sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2.4 Yukawa sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2.5 EWSB pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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ii Contents

4 Impact of Universal Extra Dimensions on radiative b→ sγ decays 614.1 Effective theories for rare B decays . . . . . . . . . . . . . . . . . . . . . 62

4.1.1 B → Xsγ decays . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2 Constraints on the UED6 model from exclusive b→ sγ decays . . . . . . 67

4.2.1 Exclusive B(s) → V γ meson decays . . . . . . . . . . . . . . . . . 684.2.2 Exclusive Λb → Λγ decays . . . . . . . . . . . . . . . . . . . . . . 704.2.3 Comments and results . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 The radiative B → Kη(′)γ decays in the SM and in a scenario withuniversal extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3.1 Improved analysis of B → Kη(′)γ in SM . . . . . . . . . . . . . . 744.3.2 B → Kη(′)γ decay rates and photon spectra in SM and UED6 . . 78

5 Manifestation of warped extra dimensions in electroweak B penguins 835.1 The effective Hamiltonian for B → Xs `

+ `− . . . . . . . . . . . . . . . . 835.2 The effective Hamiltonian for B → Xs ν ν . . . . . . . . . . . . . . . . . 855.3 The B → K∗`+`− puzzle in a scenario with one warped extra dimension 85

5.3.1 Angular distributions in B → K∗`+`−: formalism and theory . . 855.3.2 Modification of the Wilson coefficients in RSc model . . . . . . . 895.3.3 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3.4 B0 → K∗0`+`− observables in the RSc model . . . . . . . . . . . 96

5.4 Sensitivity of exclusive b → sνν induced transitions to a warped extradimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.4.1 B → Kνν and B → K∗νν decays . . . . . . . . . . . . . . . . . . 1035.4.2 Bs → (φ, η, η′, f0(980))νν in RSc . . . . . . . . . . . . . . . . . . 111

6 Search of New Physics in semileptonic B → D(∗)τ ντ decays 1156.1 Exclusive b→ c`ν` decays . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.1.1 B → D`ν` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.1.2 B → D∗`ν` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.2 Impact of the tensor operator on R(D(∗)) and other observables . . . . . 1266.3 Tensor operator in B → D∗∗`ν` decays . . . . . . . . . . . . . . . . . . . 131

Conclusions 135

A Flavour in RSc: Feynman rules 139A.1 Quarks mass and flavour mixing matrices . . . . . . . . . . . . . . . . . 139A.2 Feynman rules for neutral current interactions . . . . . . . . . . . . . . . 139

B B → D(∗)(D∗∗)τ ντ : theoretical inputs for the matrix elements 143B.1 Wilson Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143B.2 B → D∗∗ matrix elements and differential semileptonic decay rates . . . 144

References 147

Preface

I declare that the research reported in this thesis is the outcome of an original workdone in collaboration with Pietro Colangelo, Fulvia De Fazio and Egidio Scrimieri, andhas been or will be published as follows:

Papers published in refereed journals

• P. Biancofiore, P. Colangelo, F. De Fazio,“Rare semileptonic B → K∗`+`− decays in the RSc model”.Physical Review D 89, 095018 (2014)http://dx.doi.org/10.1103/PhysRevD.89.095018

ISSN: 1550-7998 (printed version) 1550-2368 (online version)Impact Factor: 4.691 (2012 JCR Data, Thomson Reuters)

• P. Biancofiore“Bounds on the compactification scale of two universal extra dimensions fromexclusive b→ sγ decays”.Journal of Physics G 40, 065006 (2013)http://dx.doi.org/10.1088/0954-3899/40/6/065006


• P. Biancofiore, P. Colangelo, F. De Fazio“On the anomalous enhancement observed in B → D(∗)τ ντ decays”.Physical Review D 87, 074010 (2014)http://dx.doi.org/10.1103/PhysRevD.87.074010


• P. Biancofiore, P. Colangelo, F. De Fazio“B → K(∗)η(′)γ decays in the standard model and in scenarios with universaldimensions”.Physical Review D 85, 094012 (2012)http://dx.doi.org/10.1103/PhysRevD.85.094012

iii

http://dx.doi.org/10.1103/PhysRevD.89.095018

http://dx.doi.org/10.1088/0954-3899/40/6/065006



iv Preface


Preprints

• P. Biancofiore, P. Colangelo, F. De Fazio and E. Scrimieri,“Exclusive b→ sνν induced transitions in RSc model, [arXiv: 1408.5614[hep-ph]].To appear in European Physical Journal C.

Papers published in conference proceedings

• P. Biancofiore,“On the recent anomalies in semileptonic B decays”,To appear in the Conference proceedings in EPJ Web of Conferences

• P. Biancofiore,“On the recently observed tensions in B Decays”,EPJ Web of Conferences 80, 00048 (2014)http://dx.doi.org/10.1051/epjconf/20148000048

ISBN: 978-2-7598-1685-9

• P. Biancofiore,“On the anomalies recently discovered in semileptonic B decays to the third family”.Nuovo Cimento C037, 268-269 (2014)http://dx.doi.org/10.1393/ncc/i2014-11703-9

ISSN: 2037-4909 (printed version) 1826-9885 (online version)

• P. Biancofiore“Rare B(s) decays in the standard model and in a scenario with two universal extradimensions”.AIP Conference Proceedings 1492, 113-116 (2012)http://dx.doi.org/10.1063/1.4763502

ISSN: 0094-243X (printed version)

The copyright of this thesis rests with the author. No quotation from it should bepublished without the prior written consent and information derived from it should beacknowledged.

Pietro Biancofiore

http://arxiv.org/abs/1408.5614

http://dx.doi.org/10.1051/epjconf/20148000048

http://dx.doi.org/10.1393/ncc/i2014-11703-9

http://dx.doi.org/10.1063/1.4763502

Abstract

In this thesis we perform several phenomenological analyses in flavour-changing processesinvolving B mesons, mainly prompted by recent LCHb and BABAR measurements thatshow tensions with respect to the Standard Model (SM), in the flavour changing neutralcurrent (FCNC) B → K∗`+`− decay and the B → D(∗)τ ντ decays, respectively. Weattempt to explain these experimental puzzles in New Physics (NP) scenarios includingextra dimensions for what concerns FCNC processes, or considering an additional tensoroperator in the effective weak Hamiltonian for the weak-charged currents.The thesis is divided in three parts. In the first one we derive a constraint on thecompactification scale in a model with two universal extra dimensions, compactifiedon a square, from exclusive b→ sγ decays. This bound together with similar boundscoming from collider phenomenology and astrophysics, helps to better constrain thismodel, conceived also as a theory of Dark Matter.In the second part of the thesis we carry out an analysis of selected angular observablescharacterizing the B → K∗`+`− decay, in the framework of the custodially protectedRandall-Sundrum model (RSc). In our study we also consider the mode with τ leptonsin the final state. We discuss the deviations of RSc results, found scanning the parameterspace of the model, from SM ones. In the same model, we also study a set of exclusiveB and Bs decay modes induced by the rare b→ sνν transition. We emphasize the roleof correlations among the observables, and their importance for detecting the predictedsmall deviations from the SM expectations.In the last part of the thesis, we consider the possibility of unveil NP in the weak-chargedtransition b→ cτ ντ , motivated by the BABAR measurements of the ratios R(D(∗)) =B(B→D(∗)τ ντ )

B(B→D(∗)µνµ)which deviate from the SM expectation, while new results on the purely

leptonic B → τ ντ mode show a better consistency with the SM, within the uncertainties.In a NP scenario, one possibility to accommodate these two experimental facts consistsin considering an additional tensor operator in the effective weak Hamiltonian. We studythe effects of such an operator in a set of observables, in semileptonicB → D(∗) modesas well as in semileptonic B and Bs decays to excited positive parity charmed mesons.

v

vi Abstract

Sommario

In questa tesi conduciamo diverse analisi fenomenologiche in processi con cambiamentodi sapore riguardanti i mesoni B, principalmente ispirati da recenti misure effettuatedalle collaborazioni LHCb e BABAR, le quelli mostrano delle tensioni rispetto allepredizioni del Modello Standard (MS) nel decadimento mediato da correnti neutrecon cambiamento di sapore (FCNC) B → K∗`+`− e nei decadimenti B → D(∗)τ ντ ,rispettivamente. Tentiamo di spiegare questi rompicapi sperimentali attraverso scenaridi Nuova Fisica (NF) che includono dimensioni extra per quanto riguarda i processiFCNC, o considerando un operatore tensoriale addizionale nell’Hamiltoniana efficace deiprocessi con corrente debole carica.

La tesi e divisa in tre parti. Nella prima deriviamo un vincolo sulla scala di compattifi-cazione di un modello con due dimensioni extra universali, compattificate su una varietaquadrata, tramite decadimenti esclusivi del tipo b → sγ. Tale limite assieme a limitianaloghi provenienti dalla fenomenologia ai collisori adronici e dall’astrofisica, e di aiutonel migliorare i vincoli di esclusione per il modello in questione, proposto anche cometeoria della Materia Oscura.

Nella seconda parte della tesi conduciamo un’analisi su una selezione di osservabiliangolari che caratterizzano il decadimento B → K∗`+`−, nel contesto del modello diRandall e Sundrum con simmetria “custodial” (RSc). Nel nostro studio prendiamo anchein considerazione il modo con leptoni τ nello stato finale. Discutiamo le deviazioni deirisultati nel modello RSc, determinati tramite uno scanning dello spazio dei parametri delmodello, rispetto alle predizioni di MS. All’interno dello stesso scenario, studiamo inoltreun insieme di decadimenti esclusivi dei mesoni B e Bs indotti dal processo elementareb→ sνν, con due neutrini nello stato finale. Sottolineiamo il ruolo delle correlazioni traosservabili, e la loro importanza per la rivelazione delle deviazioni predette rispetto aivalori aspettati in MS.

Nell’ultima parte della tesi, consideriamo la possibilita della rivelazione di NF in processidi corrente debole carica del tipo b→ cτ ντ , motivati da misure condotte all’esperimento

BABAR sul rapporto R(D(∗)) = B(B→D(∗)τ ντ )

B(B→D(∗)µνµ), il quale devia dal valore aspettato in MS,

mentre nuovi risultati sui modi puramente leptoniciB → τ ντ mostrano una sostanzialeaderenza con MS, nei limiti delle incertezze. In uno scenario di NF, e possibile spiegarequeste due dati sperimentali prendendo in considerazione un nuovo operatore tensorialenell’Hamiltoniana efficace. Studiamo gli effetti di questo genere di operatori su un

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viii Sommario

insieme di osservabili di interesse sia teorico che sperimentale, sia nei canali semileptoniciB → D(∗), cosı come nei decadimenti semileptonici del B e del Bs con mesoni eccitati diparita positiva nello stato finale.

Acknowledgements

I would like to thank Pietro Colangelo and Fulvia de Fazio for collaboration in thiswork, and I am grateful to Pietro for having been my Ph.D. supervisor. I thank all mycolleagues for the stimulating and pleasant time that we spent together during these years.I especially thank Leonardo Campanelli for many useful discussions and suggestions,concerning both science and life. Finally, I thank the TH Unit at CERN for hospitality,and in particular Gilad Perez, Leandro da Rold and Sebastian Jager, for the fruitfulcollaboration during my visit at CERN. This thesis is dedicated to my parents, to whomI express my gratitude for their support in the pursue of my scientific achievements.

ix

x Acknowledgements

Chapter 1

Introduction

Now that the scalar sector of the electroweak symmetry breaking (EWSB) mechanismhas been settled down by the discovery of the Higgs boson, the Standard Model (SM)of particle physics seems apparently complete. Although there are good reasons toconsider it as an effective description of Nature at low energy scales, it actually has aprecise structure made by a set of “accidental” (exact and approximate) symmetriesleading to baryon and lepton number conservation, suppression of flavor changing neutraltransitions, and CP violation. Decades of experimental efforts have led to an astonishingconfirmation of the entire flavour paradigm, and did not find flaws in it. However, somelong-standing issues still require a satisfying explanation within this picture. As anexample, one can accommodate the neutrino masses and mixing by a straightforwarddeformation of the effective structure of the SM. At the same time, the well knownsee-saw mechanism (introduced to describe the origin of a mass hierarchy in the leptonicsector) suggests an extremely high mass scale, beyond the reach of the current hadronmachines and too big also to unveil effects in rare processes.

Furthermore, a combined analysis of astrophysical and cosmological data, such asthe galaxy rotation curve, type Ia supenovae and the CMB radiation, firmly pointedthe existence of chargeless, non-baryonic and weakly interacting matter that amount to∼ 25% of the energy budget of the Universe. This (cold) dark matter (DM) necessarilyrequires an explanation that goes beyond the SM, since the most natural SM candidate –massive neutrinos – have been almost excluded 1. Among the best known DM particlecandidates beyond the SM, figure the supersymmetric (SUSY) neutralino and the axion,a pseudoscalar introduced in order to solve the strong CP problem (for an updatedreview on the dark matter see [1]).

Those introduced above are not the only problematic matters concerning the SM;however, by virtue of its unquestionable experimental success one could argue that themicroscopic world resembles a “desert”, which does not contain any kind of new physicsfor decades of energy scales. Despite that, the discovery of a light fundamental scalar letan old question come to light: how to stabilize its mass? It is well known that the masses

1It is also possible a non-particle explanation of the DM problem, basically going through somemodifications of the Genereal Relativity.

1

2 Introduction

of scalars are radiatively unstable, therefore Nature undergoes a highly fine tuning, unlessnew particles coupled both to the SM Higgs and the EW sector live at the TeV scale.This is the so-called “naturalness” (or hierarchy) problem, and a viable solution to it, ina minimal approach, would arise by extending different SM sectors according to the sizeof their individual radiative contributions to the Higgs mass squared. It follows that thelargest contribution comes from the top quark coupling to the Higgs. As a consequence,the top sector should be characterized by the lowest new physics (NP) scale, beside thatof the Higgs itself. Since the NP contributions to rare decays are inversely proportionalto the NP scale, the resulting microscopic theory would roughly possess the same setof accidental symmetries as the SM, if the above inverted hierarchical NP scales areactually realized. Such a scenario could lead to a feasible phenomenological realizationat zeroth order.

It stands to reason that the above pattern is not universal in flavour space, as thesector related to light quark would account a higher NP scale than the correspondingtop-related sector, and this would generally conduct to flavour violation. The size ofNP effects in flavour changing neutral currents (FCNC) should be proportional to theamount of misalignment between the SM Yukawa couplings and the NP ones. A widelyused approach, which goes under the name of Minimal Flavour Violation (MFV) [2,3],dictates that the NP sector should have the same flavour pattern of the SM. The conceptof MFV is extremely useful in the search of NP, since it provides a lower bound onthe flavour effects generated by the new dynamics at a given mass scale. Thus, theMFV represents a sort of “worst-case” scenario, that can be used to exclude NP flavour-blindness. However, other highly not trivial scenarios could actually come true. Amongthe best studied NP scenarios lying within the MFV paradigm figures the so-calledMinimal SuperSymmetric Model (MSSM). It could provide an explanation of not onlythe hierarchy problem introduced above, but also of the DM problem. However, thelong chain of negative experimental results obtained so far led to develop new strategiesbeyond the supersymmetric scenarios.

A viable alternative, explored in the recent years, actually comes from an oldfashioned idea first thought with the aim of unification of the electromagnetism withthe gravitational theory. We are referring to the theory of Kaluza and Klein [4], whoseparately argued that, in an extension of General Relativity (GR) to 4 + 1 dimension,the electromagnetism could naturally emerge as being part of the extra-dimensional GRtheory provided the fourth spatial dimension is compactified on a circle of radius R.However, the unification purpose of the Kaluza-Klein (KK) framework crashed againstthe difficulty to implement chiral fermions. In modern (early 21st century) proposals,the latter trouble is overcome by considering compactifications with some sort of (mild)singularities. Another important question comes with the size of the extra dimensions.In the most common phenomenological realizations, the compactification scale is relatedin some deep way to the weak scale, even playing a role in the breaking of electroweaksymmetry.

As an illustrative case, let us consider the possibility of two universal extra dimensions(UEDs). In a first approximation, UEDs are defined by:

3

(i) the extra dimensions are assumed to be exactly flat,

(ii) all SM fields are promoted to higher-dimensional bulk fields.

The identification of the opposite sides of a rectangular area (a torus compactification)would be the simplest way to compactify the two extra dimensions. This is a straight-forward generalization to 6D of the original Kaluza-Klein theory, however it gives riseto vector-like fermions, and therefore it is not definitively be suitable to embed the SMtheory. Nevertheless, a modification, called the chiral square compactification [5], inwhich the adjacent and not the opposite sides of a square are identified straightforwardlyrealizes chiral fermions propagating in the bulk at low energies. In this case, the compactspace has three conical singularities, and one has to impose boundary conditions whichprovide, up to discrete phases, the fields and their derivatives being continuous acrossthe glued boundaries. As a consequence of the geometry of the compactification, anew discrete Z2 symmetry, named KK-parity, naturally takes place on the square. Thelatter has essential phenomenological implications, since it guarantees the existence ofstable exotic particles as good DM candidates characterized by astrophysical and collidersignatures distinct from the ones coming from SUSY states [6]. We shall discuss a fewof these issues below.

In the rest of the thesis, we consider another possibility, the compactification on ainterval in a 5D warped extra-dimensional space-time [7], whose line element is definedby ds2 = e−2k|y|ηµνdx

µdxν − dy2 , and ηµν = diag(+,−,−,−) being the 4D Minkowskimetric tensor, and y the extra-dimensional coordinate. The warped-down curvaturescale is taken as k ∼ Planck scale. The Higgs mass parameter and hence EWSB, is thennaturally at the TeV scale through a redshift. Such a model has been first proposedby Sundrum and Randall in order to provide a dynamical solution to the Plank/weakhierarchy problem, and as alternative to SUSY. As mentioned, realistic extra-dimensionalscenarios are characterized by certain defects at the boundaries, sometimes called branes(borrowing string terminology). In this case, low energy physics describing our world“lives” on the so-called TeV (or IR) brane, separated along the warped extra dimensionfrom the Planck (or UV) brane, whose reference scale is the grand-unification one.Generally, the branes can support localized fields with different spins, at the same timeother fields may propagate in the bulk of the space. The localization properties of thefields along the extra dimension become part of the definition of the model itself, and inthe case of fermions they have been also used to provide a theory of flavour. In general,one would expect these models to suffer from severe FCNC and CP problems, the reasonwhy several variants have been proposed.

In the following chapters several theoretical aspects concerning the above scenarioswill be touched more deeply. Some preliminary comments are in order:

(i) these extra-dimensional models are conceived to be consistent with all aspects ofthe SM,

(ii) they are genuinely phenomenological, in the sense that these extensions can betested or ruled out at the TeV scale,

4 Introduction

(iii) having extra dimensions at the TeV scale constitutes a good reason for the existenceof a cutoff near that scale. This couldn’t claim by itself a solution of the hierarchyproblem, but it should provide a good starting point for a deeper understandingof the naturalness question. On a speculative side, the discover of an extra-dimensional structure would have an impressive consequence on our understandingof Nature, since a cutoff should exist not far above, and the hierarchy “Planck vsEW scale” should move to the background in force of a new, more urgent, issueconcerning the “little hierarchy” between the EW scale and the cutoff related tothe extra-dimensional reality.

In the sense specified above, TeV scale extra-dimensional models can be considered asmotivated by the hierarchy problem, assuming a compactification scale near the weakscale, in analogy to the SUSY solution to the naturalness issue when the superpartnermasses are around a TeV (although the two approaches are basically different).

In this thesis we consider rare b−hadron decays as viable probes through which to lookfor NP signatures. We perform several phenomenological analyses both model dependentand model independent. In the first case, fall the studies that we conduct in theframework of those TeV scale extra-dimensional models, whose theoretical motivationshave been exposed above. We mainly focus of exclusive FCNC b→ sγ, b→ s`+`− andb→ sνν induced transitions. For some of them new and interesting results came fromthe LHCb experiment in 2013 [8]. The second case concerns an analysis done on theexclusive b→ c`ν induced decays, which would cast new light on the nature of chargedweak currents, and for which updated measurements have been provided in 2012 by theBABAR collaboration [9].

Therefore, motivated by the impressive experimental activities already conducted atthe flavour factories like BABAR at SLAC (Stanford) and Belle at KEK (Tsukuba), andthe wide research program scheduled for the next years by the LHCb collaboration at theCERN LHC and at the Super-flavour factories under construction (e.g. Belle II at theTsukuba Laboratories), we address to those observables (mainly CP conserving) that aretheoretically clean and particularly sensitive to NP. Under the hypotheses (as well thehopes) that the data collected at the upcoming machines will be very precise, we showthe importance of correlations between these observables. As a matter of fact, adoptingthis strategy could help to clearly distinguish models with new left-handed and/or right-handed currents as well as non-MFV interactions. The pattern of deviations from therelations so obtained may address the identification of the correct NP scenario. Of course,the complementary research of direct signals of NP at the LHC would undoubtedlyfacilitate the flavour program, although is well known that the indirect research is morepowerful in terms of the smaller distances that can be probed with respect to hadroncolliders, even shorter than 4× 10−20 m [10].

The thesis is organized as follows:

• In Chapters 2 and 3 we introduce the main theoretical aspects and the flavourstructures of two extra-dimensional models, the UED 6 model with two universal

5

extra dimensions and the custodially protected Randall-Sundrum model (RSc).The custodial protection is realized by enlarging the SM gauge group to the gaugegroup SU(2)c×SU(2)R×SU(2)L×U(1)X×PLR symmetry [11]. Thus, the Z2-typePL,R symmetry implies a mirror action of the two SU(2)L,R groups, preventinglarge Z couplings to down-type left-handed fermions that would be incompatiblewith experiment.

• In Chapter 4 we provide the bound on the compactification scale 1/R for thescenario with two universal extra dimensions from exclusive b → sγ decays,prompted by the example of a single UED in which such processes turned out to beeffective in constraining the compactification scale. Here we focus on the transitionsB → K∗γ, B → K∗2γ, Bs → φγ and Λb → Λγ. Moreover, we discuss the case ofthe B → Kη(′)γ decay, since the three-body modes shake an undeniable theoreticalinterest for a clear understanding of the hadronization in b→ sγ induced channels.For some of the processes listed above, precise experimental data are available,in particular new data from the BABAR collaboration have been provided forB → Kη(′)γ decays.

• Recently, LHCb measurements [8] show small discrepancies with respect to the SMpredictions in selected angular distributions of the mode B0 → K∗0µ+µ−. Thepossibility of explaining such tensions within theories beyond the SM cruciallydepends on the size of the deviations of the Wilson coefficients of the effectiveHamiltonian for this mode, in comparison to their SM values. In Chapter 5 weanalyse this issue in the framework of the RSc model; in our study we also considerthe mode with τ leptons in the final state. We discuss the small deviations of RScresults from SM ones, found scanning the parameter space of the model. In thesame framework, we also study a set of exclusive B and Bs decay modes inducedby the rare b → sνν, which are among the cleanest FCNC processes in the Bsector. We emphasize the role of correlations among the observables, and theirimportance for detecting the predicted small deviations from the standard modelexpectations.

• The decays B → D(∗)τ ντ and B → τ ντ are suitable enough to unveil the effectsof New Physics in charged–current interactions. In fact, the presence of third–generation fermions in both the initial and final states leads to sensitivity to newparticles that couple more strongly to heavy fermions, such as a charged Higgsboson (both SUSY-like and not) [12]. These decays have two or more neutrinosin the final state, and so cannot be fully reconstructed using only the observableparticles. Therefore, their study demands the use of additional constraints relatedto the production of the B meson. Constraints of such a type are accessible at B–factories, where electrons and positrons collide at an average center–of–mass (CM)energy of

√s ≈ 10.58 GeV, corresponding to the mass of the Υ(4S) resonance. As

a result, the B–factories BABAR and Belle have provided the only measurementsof such decays, so far. The B–factory results give evidence for more than 3.4–σdifference between the B → D(∗)τ+ντ decay rates and the expectation of the

6 Introduction

SM [9,13]. Morevover, results on the purely leptonic B → τ ντ mode show a betterconsistency with the SM predictions, within the uncertainties [14, 15]. In a NPscenario, one possibility to accommodate these two experimental facts consistsin considering an additional tensor operator in the effective weak Hamiltonian.In Chapter 6 we study the effects of such an operator in a set of observables,in semileptonic B → D(∗) modes as well as in semileptonic B and Bs decays toexcited positive parity charmed mesons.

• Finally, we shall present our Conclusions. Several technical details are collected inthe Appendices.

Chapter 2

Models with TeV scale extradimensions

Models with extra dimensions (ED), are built by including, beside the usual 3 + 1 di-mensional space-time xµ = (x0, x1, x2, x3), additional spatial dimensions parameterizedby coordinates x4, x5, . . . , x3+N . With N we indicate the number of extra dimensions.Arguments developed in the context of string phenomenology suggest thatN can be aslarge as 6 or 7.

Depending on the type of metric in the bulk, the ED models fall into one of thefollowing categories: flat, also known as “universal” extra dimensions (UED) models, orwarped ED models. In the UED models, all particles of the Standard Model (SM) areallowed to propagate in the bulk, i.e. along any of the x3+i (i = 1, . . . , N) directions [16].The extra dimensions in these cases must be sufficiently small in order to prevent largecorrections to the experimental observables already tested, e.g. at hadron collidersand flavour factories. Therefore, the extra dimensions are conceived to be properlycompactified on some small size manifold (see Figure 2.1). A fascinating possibilityis that extra spatial dimensions would reveal themselves at or near the TeV scale, acase study that has taken place over the past few years, whose origins lie in the workof Arkani-Hamed, Dimopoulos and Dvali (ADD) [17]. Thenceforth, extra dimensionsacquired a more significant role in the model building of phenomenological scenariosbeyond the SM, also addressing some of the most outstanding issues in particle physicsand cosmology. A partial list of some of these hints includes:

• providing a solution to the hierarchy problem within the electroweak symmetrybreaking [7, 17]

• giving hints on the origin of fermion masses, the CKM entries and the CP violation[18–22]

• producing a TeV scale grand unification while suppressing proton decay [23–25]

• determining good dark matter candidates, with proper collider signatures, differentfrom those occurring for analogous SUSY candidates [6, 16,26]

7

8 Models with TeV scale extra dimensions

Figure 2.1: Artistic representation of a flat bidimensional space-time, with two extra dimensionscompactified on a sphere. [Picture from http://www.preposterousuniverse.com/blog/2004/06/30/

extra-dimensions/].

Let us point out a consideration about the sign of the metric tensor for the extradimensions. We mentioned that the latter are always space-like, but we did not motivatethis statement. Let as consider, for the sake of simplicity, a flat space-time (e.g. witha Minkowskian metric) with a single extra dimension, that we parametrize with thecoordinate x4 (occasionally we may use the y coordinate). Therefore, the generic metrictensor is given by:

gαβ = diag(1,−1,−1,−1,±1) . (2.1)

We did not specified the sign for the extra dimension, since it can be space or time-like.Let us consider a massless particle and that Lorentz invariance is still valid in 5D. Wehave

p2 = 0 = gαβpαpβ = p2

0 − p2 ± p24 , (2.2)

where p0 is the energy of the particle, p2 is the particle 3-momentum squared, and p4 isthe component of the momentum along the fifth dimension. We set to zero the squaredtotal momentum as a consequence of the masslessness of the particle. Now, we observethat Eq. (2.2) can be rewritten singling out the squared 4-momentum of the particle:

p20 − p2 = pµp

µ = ∓p24 . (2.3)

On the other hand, assuming 4D Lorentz invariance, pµpµ represents the mass of a

particle propagating in a four dimensional space-time. Hence, a massless particle in 5Dadmits a massive component propagating in 4D. Accepting Lorentz invariance in both5D and 4D space-time, selects a − sign in the metric (2.1), in order to get a positive massfor the particle in the ordinary space-time. The other choice, with a + sign in the metric,would allow the propagation of particle with a negative mass, a.k.a. “tachion” which oneusually avoids in quantum field theory because of dangerous causality problems [27,28].

Kaluza-Klein modes

We may observe that, because of the compactness of the extra dimensions, the extradimensional components of the particle momentum are quantized in 1/R units, where R

http://www.preposterousuniverse.com/blog/2004/06/30/extra-dimensions/

http://www.preposterousuniverse.com/blog/2004/06/30/extra-dimensions/

9

Figure 2.2: (right) Compactification on N = 1 extra dimension on a circle with two opposite endpointsidentified, the orbifold S1/Z2. (left) Compactification on N = 2 extra dimensions, the so-called chiralsquare, with two adjacent sides identified. In both pictures the blue arrows show the correspondingidentification. The marked dots are fixed (boundary) points. Pictures from Ref. [29].

is the parameter which identifies the “compactification modulus” of the extra dimensionalmanifold (in the case of the S1/Z2 manifold, see below, this is exactly the radius of thecircle). From the 4-dimensional point of view, these components must be interpretedas masses. Therefore, in generic extra dimensional models, every predicted particle isusually associated with an infinite tower of fields, a.k.a. Kaluza-Klein modes with a massspectrum proportional to n

R , where n takes into account the quantum units of extradimensional momentum. The lowest mode or the zero mode of the tower is recognizedas one of the SM particles, i.e. quarks, leptons, gauge bosons and the Higgs bosonwhich have been observed at LHC [30, 31]. All particles characterized by a certain nare indicated to belong to the n-th level of the corresponding KK tower, and at leadingorder in perturbative corrections look exactly degenerate. Nevertheless, one has to becareful with radiative corrections that usually break this degeneracy.

Orbifolding

A long-standing issue for extra dimensional models has been the implementation of chiralfermions in the reduced 4-dimensional theory (the chirality is furthermore broken by theEWSB and the Higgs mechanism of the SM). To address this question, one has not only tochoose an appropriate manifold on which ED are compactified, but the selected manifoldmust include endpoints allowing to implement proper boundary conditions (BCs) on thefields; e.g. the orbifold S1/Z2, illustrated in Figure 2.2. In the five-dimensional ACDmodel (by Appelquist, Cheng and Dobrescu) [16], the extra dimension parametrizedby the coordinate x4 is compactified on a manifold with the topology of a circle S1. Inorder to obtain chiral fermions in the reduced 4D Minkowski space-time, two oppositepoints on the circle are identified, leading to a new topology known as the orbifold S1/Z2.In the case of two universal extra dimensions, the compactification is more tricky, andtakes place on a square manifold with two sides identified, the so-called chiral squarerepresented in Figure 2.2.


A further and theoretically challenging aspect of extra dimensional models is relatedto the notion of Kaluza-Klein (KK) parity, whose origin takes place in the geometricalprocedure of compactification and has several phenomenological implications both atcolliders and in astrophysics. For example, in the N = 1 case of Figure 2.2, KK-parityoriginates from the reflection symmetry with respect to the center of the diameterbetween the endpoints. In the case N = 2 of Figure 2.2 the KK-parity corresponds tothe symmetry with respect to the center of the chiral square. Since a SM particle is azero mode and has an even KK-parity, the lightest KK-parity odd particle (LKP) isautomatically stable as it can never decay to zero modes due to KK-parity conservation.The stability of the LKP, makes the UED’s models a theory of dark matter. We discussthis issue in the context of a specific extra dimensional scenario in the next sections.

2.1 Two Universal Extra Dimensions (UED6): compactifi-cation on a square

Extensions of Quantum Field Theory in six dimensions have been proposed in order toprovide explanations for several issues regarding physics beyond the SM, such as a longproton lifetime, the origin of electroweak symmetry breaking, the number of the fermiongenerations and the breaking of grand unified gauge groups (see Ref. [5] and referencestherein for a complete bibliography). A relevant question concerns the manifold on whichthe two extra dimensions are compactified, since if fermions are allowed to propagatein the bulk, only few topologies guarantee the preservation of chirality for the reducedfour-dimensional fermionic fields. Among them, the T 2/Z4 topology, which representsthe compactification on a square, ensures the proton stability through a Z8 symmetryprotection [32].

In this section we examine the BCs and the solutions of the field equations for scalars,fermions and gauge fields on the T 2/Z4 manifold, following the analysis in [5].

2.1.1 Scalars

Let us consider a six dimensional space-time: we denote the four Minkowskian space-time coordinates as xµ, µ = 0, 1, 2, 3, while with x4 and x5 we indicate two flat extradimensions compactified on a square of side L, therefore 0 ≤ x4, x5 ≤ L (see Figure2.2 right). The metric tensor is provided by the extension of the Minkowski metric:diag(1, −1, −1, −1, −1, −1).

The 6D action of a scalar field Φ is given by:

SΦ =

ˆd4x

ˆ L

0dx4

ˆ L

0dx5

(∂αΦ†∂αΦ−M0

2Φ†Φ). (2.4)

With the first greek letters we indicate the hexa-dimensional coordinates α, β =0, 1, . . . , 5, while with the middle letters of the same alphabet we refer to the ordinaryspace-time coordinates µ, ν, · · · = 0, 1, 2, 3. Because of a field variation δΦ(xµ, x4, x5),

2.1 UED6: compactification on a square 11

the action (2.4) undergoes the variation

δSΦ = δSvΦ + δSsΦ (2.5)

where the first term represents a “volume” integral,

δSvΦ = −ˆd4x

ˆ L

0dx4

ˆ L

0dx5

(∂α∂αΦ† +M2

0 Φ†)δΦ , (2.6)

and the second term is a “surface” integral

δSsΦ =

ˆd4x

[ˆ L

0dx4

(∂5Φ†δΦ

∣∣x5=L

− ∂5Φ†δΦ∣∣x5=0

)+

ˆ L

0dx5

(∂4Φ†δΦ

∣∣x4=L

− ∂4Φ†δΦ∣∣x4=0

)]. (2.7)

From the variational principle δSvΦ = 0 it follows that Φ is a solution of the 6D Klein-Gordon equation: (

∂µ∂µ − ∂42 − ∂5

2 +M20

)Φ = 0 . (2.8)

At the same time, the variational principle δSsΦ = 0 for the “surface” piece of the actiongives us a prescription on the BCs for the field Φ. Let us remind that only few of thoseconditions can accomplish for chiral fermions in the effective four dimensional theory. Asimple solution goes through the identification of the adjacent side of the square, Figure2.2 (right):

(y, 0) ≡ (0, y), (y, L) ≡ (L, y), ∀y ∈ [0, L] . (2.9)

We denote this compactification as the “chiral square”. The topology of the extradimensional manifold implies that the Lagrangian density takes the same values onthe identified points. Therefore, since the Lagrangian is symmetric under global U(1)transformations, the values acquired by the scalar fields on the adjacent sides of thesquare may differ for a phase factor:

Φ (xµ, y, 0) = eiθΦ (xµ, 0, y) y ∈ [0, L] . (2.10)

Moreover, the phase chosen for two sides can be different for the opposite couple of sides,hence we also have:

Φ (xµ, y, L) = eiθΦ (xµ, L, y) y ∈ [0, L] (2.11)

with θ 6= θ in general. However the conditions (2.10) do not automatically ensure theLagrangian to be the same at the identified points. By deriving Eq. (2.10) with respectto y one obtains:

L (xµ, 0, y)− L (xµ, y, 0) = ∂5Φ†∂5Φ∣∣(x4,x5)=(y,0)

− ∂4Φ†∂4Φ∣∣(x4,x5)=(0,y)

. (2.12)


Therefore, the identification of the Lagrangian density on the adjacent sides of the squareholds only if, in addition to Eq. (2.10), one imposes

∂5Φ∣∣(x4,x5)=(y,0)

= eiθ′∂4Φ

∣∣(x4,x5)=(0,y)

y ∈ [0, L] , (2.13)

where θ′ is another phase factor in general different from θ. Analogous relations existfor the other couple of sides where the phases θ and θ′ have been chosen. All together,Eqs. (2.10) and (2.13) are the “folding” BCs for the scalar field on the manifold. Byimposing the stationary condition encoded in Eq. (2.7) the following constraints can beeasily obtained:

eiθ′

= −eiθ, or δΦ∂5Φ|(x4,x5)=(y,0) = 0 , (2.14)

and for the opposite sides we similarly get:

eiθ′

= −eiθ, or δΦ∂5Φ|(x4,x5)=(y,L) = 0 . (2.15)

Since the BCs are independent of xµ, we can write the solution of the Klein-Gordonequation (2.8) by a Fourier decomposition:

Φ (xµ, y, L) =1

L

∑j,k

Φ(j,k) (xµ) f (j,k)(x4, x5

). (2.16)

If we insert (2.16) in Eq. (2.8) we get that the modes Φ(j,k) are solutions of(∂µ∂µ +M2

0 +M2j,k

)Φ(j,k) (xµ) = 0 , (2.17)

where (∂4

2 + ∂52 +M2

j,k

)f (j,k)

(x4, x5

)= 0 , (2.18)

whose generic solution is a combination of trigonometric functions provided by

f (j,k) = C+1 e

i(jx4+kx5)

R + C−1 e−i (jx4+kx5)

R + C+2 e

i(jx4−kx5)

R + C−2 e−i (jx4−kx5)

R

+ C+3 e

i(kx4+jx5)

R + C−3 e−i (kx4+jx5)

R + C+4 e

i(kx4−jx5)

R + C−4 e−i (kx4−jx5)

R , (2.19)

where j and k are real numbers such as

M (j,k) =

√M2

0 +j2 + k2

R2. (2.20)

This is the mass spectrum of the KK tower, and we have defined the “compactificationradius”

R ≡ L

π. (2.21)


The coefficients C±i with i = 1, 2, 3, 4 are derived imposing the “folding” conditions(2.10)–(2.13), from which we also obtain the most general BCs for the scalar field thatguarantee a non-trivial solution for the 6D Klein-Gordon equation:

Φ (xµ, y, 0) = einπ2 Φ (xµ, 0, y) ,

Φ (xµ, y, L) = (−1)leinπ2 Φ (xµ, L, y) ,

∂5Φ|(x4,x5)=(y,0) = −einπ2 ∂4Φ|(x4,x5)=(0,y),

∂5Φ|(x4,x5)=(y,L) = −(−1)leinπ2 ∂4Φ|(x4,x5)=(L,y) , (2.22)

where n = 0, 1, 2, 3 and l = 0, 1, since the phases for the two couple of opposite sides ofthe square may differ by lπ. The normalization condition for the f j,k profiles,

1

L2

ˆ L

0dx4

ˆ L

0dx5

[f (j,k)

(x4, x5

)] ∗f (j′,k′)(x4, x5

)= δj,j′δk,k′ , (2.23)

allows us to rewrite these functions as

f (j,k)n =

1

1 + δj,0δk,0

[e−in

π2 cos

[π

(jx4 + kx5

L+n

2

)]+ cos

[π

(kx4 − jx5

L+n

2

)]],

(2.24)with j + l/2 and k + l/2 integer numbers. For n = 0, the solution with j = k = 0 isallowed, this is the so-called “zero mode” of the spectrum, which has a null component ofmomentum along the extra dimensions. Furthermore, substituting the Eq. (2.16) in (2.4)and integrating along the extra dimensions x4, x5 we get the effective 4D Lagrangian forthe scalar theory:

L (xµ) =∑j,k

{∂µΦ†(j,k)∂µΦ(j,k) −

(M2

0 +j2 + k2

R2

)Φ†(j,k)Φ(j,k)

}. (2.25)

2.1.2 Fermions

The Clifford algebra in six dimensions is generated by six 8 × 8 anticommuting Γα

(α = 0, 1, 2, 3, 4, 5) matrices. The generators of the spinorial representation of theLorentz group SO(1, 5) are given by:∑αβ

2=i

4

[Γα,Γβ

]. (2.26)

This reducible representation encodes two irreducible Weyl representations with twodifferent six-dimensional chiralities labeled by the symbol + and −. These chiralityeigenstates are projected by the operators:

P± =1

2

(1± Γ

), (2.27)

where

Γ =1

6!εα0α1...α5Γα0Γα1 ...Γα5 = Γ0Γ1Γ2Γ3Γ4Γ5 (2.28)


commutes with the other Γα.

We are interested in the description of chiral fermions in the effective 4D theory,therefore we may observe that a six-dimensional chiral fermion can be decomposed intotwo components of definite chirality under the 4D Lorentz group SO(1, 3)

Ψ±(xµ, x4, x5

)= Ψ±L

(xµ, x4, x5

)+ Ψ±R

(xµ, x4, x5

)(2.29)

where

Ψ±L,R = PL,RP±Ψ

and PL,R = 12

(1± iΓ0Γ1Γ2Γ3

)are the left(right) four dimensional chirality projectors.

The action for the six-dimensional chiral field Ψ+ is given by1

SΨ =

ˆd4x

ˆ L

0dx4

ˆ L

0dx5 i

2

[Ψ+Γα∂αΨ+ −

(∂αΨ+

)ΓαΨ+

], (2.30)

By a field variation δΨ+

(xµ, x4, x5

)in the action (2.30) we recover the 6D Weyl equation

for the field Ψ+

Γµ∂µΨ+L = −(Γ4∂4 + Γ5∂5

)Ψ+R ,

Γµ∂µΨ−R = −(Γ4∂4 + Γ5∂5

)Ψ−L . (2.31)

Proceeding similarly as in the case of the scalar field, we can fix the “folding” BCs forthe fermions, that take into account the 4D chiralities with different relatives phases forthe left(L)- and right(R)-handed fields:

Ψ±L,R (xµ, y, 0) = eiθL,RΨ±L,R (xµ, 0, y) ,

Ψ±L,R (xµ, y, L) = eiθL,RΨ±L,R (xµ, L, y) , (2.32)

∀y ∈ [0, L]. It can be shown that the BCs (2.32) assure the equality of the Lagrangianat the adjacent sides of the square. Moreover, the same BCs must satisfy the stationaryprinciple for the “surface” piece of the fermionic action:

δSsΨ =i

2

ˆd4x

[ˆ L

0dx4

(Ψ+Γ5δΨ+

∣∣∣x5=L

−Ψ+Γ5δΨ+

∣∣∣x5=0

)+

ˆ L

0dx5

(Ψ+Γ4δΨ+

∣∣∣x4=L

−Ψ+Γ4δΨ+

∣∣∣x4=0

)]= 0 , (2.33)

from which the following prescriptions derive:[1− iei(θR−θL)

]Ψ+L (xµ, 0, y) Γ4Ψ+R (xµ, 0, y) = 0,[

1− iei(θR−θL)]

Ψ+L (xµ, 0, y) Γ4Ψ+R (xµ, 0, y) = 0 . (2.34)

1A similar equation is still valid for the chiral “−” field with trivial substitutions in the notation.


In addition, the Weyl equations (2.31) can be manipulated to get

∂4Ψ+L,R|(x4,x5)=(0,y) = −e−iθL,R∂5Ψ+L,R|(x4,x5)=(y,0) (2.35)

similar to the conditions on the derivatives of the scalar fields(2.13). All together, Eqs.(2.32–2.34–2.35) show that both Ψ+, and Ψ+R satisfy the same BCs of the scalars inEqs. (2.22).

We further observe that both Ψ+L and Ψ+R, are solutions of the 6D Klein-Gordonequation for massless fields, and therefore we can perform a Fourier decomposition ofthese fields as:

Ψ+L

(xµ, x4, x5

)=

1

L

∑j,k

f(j,k)+L

(x4, x5

)Ψ

(j,k)+L (xµ)⊗

(10

),

Ψ+R

(xµ, x4, x5

)=

1

L

∑j,k

f(j,k)+R

(x4, x5

)Ψ

(j,k)+R (xµ)⊗

(01

), (2.36)

where Ψ(j,k)+L,R are four-dimensional Weyl fermions that are solutions of

(iγµ∂µ −Mj,k)(

Ψ(j,k)+L + Ψ

(j,k)+R

)= 0 , (2.37)

and Mj,k is the same of Eq. (2.20). Instead, the extra dimensional profiles f(j,k)+(L,R) are

solutions of the coupled Weyl equations

(∂4 − i∂5) f(j,k)+R = Mj,kf

(j,k)+L ,

(∂4 + i∂5) f(j,k)+L = −Mj,kf

(j,k)+R . (2.38)

With simple algebraic manipulations these equations decouple and reduce to two Klein-Gordon equations of the type:(

∂42 + ∂5

2)

f(j,k)+R = −M2

j,k f(j,k)+R , (2.39)

which admits the same solution seen for the scalar profiles:

f j,kn+R

=1

1 + δj,0δk,0

[e−in

+Rπ2 cos

[π

(jx4 + kx5

L+n+R

2

)]+cos

[π

(kx4 − jx5

L+n+R

2

)]]≡ f

(j,k)+R (2.40)

with n+R = 0, 1, 2, 3 and j, k integer numbers if l+ = 0 or half-integers if l+ = 1. By

inserting the previous solution in the first equation of the system (2.38) one obtains:

f(j,k)+L =

k + ij√j2 + k2

f(j,k)

1+n+R

. (2.41)

We should remind that the BCs for the left-handed field Ψ+L are exactly the same validfor the right-handed ones, except for n+

R being replaced by

n+L = n+R+ 1 mod 4.

The solutions for the“−” chirality fields trivially descend from an analogous treatment.


2.1.3 Gauge Fields

Let us consider a 6D Abelian gauge field, Aα(xβ), with α, β = 0, 1, . . . , 5. The actionin the gauge field strength is

S =

ˆd4x

ˆ L

0dx4

ˆ L

0dx5

(−1

4FαβF

αβ + LGF

), (2.42)

where LGF is a gauge-fixing term chosen such as

LGF = − 1

2ξ[∂µA

µ − ξ (∂4A4 + ∂5A5)] 2 , (2.43)

with ξ a gauge-fixing parameter2.The stationarity principle applied to the action with respect to theAα, in the bulk,

leads to

∂µFµν +1

ξ∂µ∂νA

µ =(∂2

4 + ∂25

)Aν ,(

∂µ∂µ − ξ∂2

4 − ∂25

)A4 = (ξ − 1)∂4∂5A5 ,(

∂µ∂µ − ∂2

4 − ξ∂25

)A5 = (ξ − 1)∂4∂5A4 . (2.44)

The BCs derive directly from the requirements of identicalness of the Lagrangian at theidentified points of the square, compatibly with the gauge symmetry principle. Moreover,we are interested in those BCs that preserve a zero mode, i.e. we avoid the breakingof gauge symmetry by boundary conditions. Therefore, for a generic matter field Φ wehave:

DµΦ|(x4,x5)=(y,0) = eiθDµΦ|(x4,x5)=(0,y) ,

D4Φ|(x4,x5)=(y,0) = eiθD5Φ|(x4,x5)=(0,y) ,

D5Φ|(x4,x5)=(y,0) = −eiθD4Φ|(x4,x5)=(0,y) , (2.45)

where the first two conditions directly follow from a derivation of that condition whichprescribes the identification of matter fields on adjacent sides (modulo a phase factor).The last condition requires a “smoothness” of the derivatives normal to the “edges” ofthe square, and we already encountered it in the case of the scalar field. The covariantderivatives are usually defined as

Dα = ∂α − ig6Aα with α = 0, 1, . . . , 5 (2.46)

and g6 is the 6D gauge coupling, with mass dimension −1.The conditions (2.45), together with the stationarity principle for the action on the

boundaries, allow us to derive a complete set of gauge invariant BCs for the field Aµ

2We neglect in the following kinetic terms that may appear at the fixed points of the manifold, byassuming that they are small enough to be treated perturbatively.


and the linear combinations A± = A4 ± iA5:

A±(y, 0) = ∓iA±(0, y),

∂4A±|(x4,x5)=(y,0) = ∓i∂5A±|(x4,x5)=(0,y) ,

∂5A±|(x4,x5)=(y,0) = ±i∂4A±|(x4,x5)=(0,y) ; (2.47)

similar relations are also valid for the the sides (y, L) and (L, y).By performing the Fourier expansion of the 6D gauge field Aα, α = 0, 1, . . . , 5 (with

j ≥ 1 and k ≥ 0 integers):

Aµ(xν , x4, x5

)=

1

L

A(0,0)µ (xν) +

∑j≥1

∑k≥0

f(j,k)0

(x4, x5

)A(j,k)µ (xν)

,A+

(xν , x4, x5

)= − 1

L

∑j≥1

∑k≥0

f(j,k)3

(x4, x5

)A

(j,k)+ (xν) ,

A−(xν , x4, x5

)=

1

L

∑j≥1

∑k≥0

f(j,k)1

(x4, x5

)A

(j,k)− (xν) , (2.48)

one gets a tower of spin-1 modes, A(j,k)µ , which have a zero mode (j = k = 0), and

two towers of spin-0 modes without a zero mode, A(j,k)+ and A

(j,k)− . While the spin-1

zero mode is associated with the unbroken 4D gauge invariance, the other spin-1 modes

acquire a mass term. A linear combination of A(j,k)± , labeled as A

(j,k)G and given by

A(j,k)± = rj,±k

(A

(j,k)H ∓ iA(j,k)

G

), (2.49)

with rj,±k = j+ik√j2+k2

, plays the role of the Nambu-Goldstone boson eaten by the spin-1

modes A(j,k)µ , providing their longitudinal polarization. The orthogonal combination,

A(j,k)H , is gauge invariant and remains as an additional physical degree of freedom. This

gives rise to a tower of heavy particles in the adjoint representation of the gauge group,called “spinless adjoints”, whose interactions depend on their KK number.

Moreover, the extra dimensional profiles fn with n = 0, 1, 2, 3 are algebraic combi-

nations of trigonometric functions having as arguments jx4+kx5

R or kx4−jx5

R . We do notshow them here, but the complete list is available in Ref. [5]. It is worth to mentionthat the function fn are typically solutions of the Klein-Gordon equation(

∂42 + ∂5

2 +M2j,k

)f (j,k)n

(x4, x5

)= 0 , (2.50)

with M2j,k ≡ j2+k2

R2 , and the normalization condition

1

L2

ˆ L

0dx4

ˆ L

0dx5

[f (j,k)n

(x4, x5

)]∗f (j′,k′)n

(x4, x5

)= δj,j′δk,k′ , (2.51)

still holds.


By integrating on the extra dimensions x4 and x5, the effective 4D Lagrangianbecomes

L (xµ) =∑j,k

{− 1

4F (j,k)µν F (j,k)µν +

1

2M2j,k

(A(j,k)µ

)2 − 1

2ξ

(∂µA(j,k)

µ

)2

+1

2

[(∂µA

(j,k)H

)2 −M2

j,k

(A

(j,k)H

)2 +

(∂µA

(j,k)G

)2 − ξM2

j,k

(A

(j,k)G

)2]}

. (2.52)

One may observe that in the limit ξ →∞, only the fields A(j,k)H propagate with masses

Mj,k.

2.1.4 Kaluza-Klein parity

A KK-parity may be imposed on this field theory and it guarantees that the lightest KKparticle is a possible DM candidate. This parity acts as a Z2 symmetry and distinguishesamong the KK modes as

Φ(j,k)(xµ) 7→ (−1)j+kΦ(j,k)(xµ) , (2.53)

where Φ is a field of generic spin, and j, k are integers labeling the KK level. ThisKK-parity can be geometrically explained as a rotation by π about the center of thechiral square. Moreover, KK-parity requires that localized operators at (0, 0) and (L,L)be the same. This concludes the description of the 6D theory with fields propagating inall dimensions, which represents the simplest extension of the SM. The phenomenologyof this theory will be discussed in Chapter 4.

2.2 Fields in a slice of AdS5: compactification on an inter-val

We now redirect our attention to the case where the space-time curvature is important.Much of the literature has exploited the scenario with a Anti-de Sitter (AdS)5 background,thanks to the Randall-Sundrum (RS) model which has been originally proposed by R.Sundrum and L. Randall in 1999 [7] as an attempt to solve the hierarchy problem3.

The RS framework belongs to the so-called “brane-world” cosmological scenarios,which conceive the universe (characterized by the electroweak symmetry breaking phe-nomenon) as living on a 4D slice, regarded as the TeV (or IR) brane separated along acompactified extra dimension from the Planck (or UV) brane, on which the fundamentalforces are unified. The geometry of the fifth dimension descends from a peculiar Anti-deSitter metric, with a tuned scale factor:

ds2 = e−2k|y|ηµνdxµdxν − dy2 , (2.54)

3More recently, it has been suggested that this kind of models may address the comprehension ofstrongly coupled systems with the help of the AdS/CFT correspondence.

2.2 Fields in a slice of AdS5: compactification on an interval 19

ηµν = diag(+,−,−,−) being the 4D Minkowski metric tensor4. k is of the order of thePlanck scale, moreover the extra dimensional space-like coordinate y, is compactified onan interval [0, L]5, and its values stand for the proper distance along the extra dimension.Therefore, the only physical scale is the “grand unification” or Planck one, while theelectroweak scale should emerge as a pure geometrical effect of the extra dimension.

In recent years, a lot of work has been done in order to extend the SM phenomenologyby including signatures of the extra dimension predicted by RS. The scenarios withthe richest pattern of viable phenomenological effects already measurable at hadronmachines, are those in which all the fundamental fermions and gauge fields are allowedto propagate in the bulk. The spontaneous symmetry breaking is accomplished by aHiggs field, both localized on the TeV brane or allowed to propagate along the fifthdimension, with different phenomenological implications.

In the developing of the free field theory, in the next sections, we follow Ref. [33].

2.2.1 Scalars

Let us consider the scalar field action in 5D:

SΦ =

ˆd4x

ˆ L

0dy√G[(∂MΦ

)†(∂MΦ

)−M2

ΦΦ†Φ], (2.55)

where we set M2Φ ≡ ak2 and

√G = e−4ky is the root of the metric determinant. The

equations of motion (EOMs) directly follow from the stationarity of the action under afield variation in the bulk:

∂M

( ∂L∂(∂MΦ†)

)− ∂L∂Φ†

= ∂M (√G∂MΦ) +

√GM2

ΦΦ = 0 , (2.56)

(with L the integrand of Eq. (2.55)) together with analogous conditions on the boundaryterms of the action, which comes out by setting to zero the surface variation6

δSSΦ =

ˆd5x ∂M

( ∂L∂(∂MΦ)†

δΦ†)

=

ˆd4x

ˆdy ∂µ

( ∂L∂(∂µΦ)†

δΦ†)

+

ˆd4x

ˆdy ∂5

( ∂L∂(∂5Φ)†

δΦ†)

=

ˆd4x

( ∂L∂(∂5Φ)†

δΦ†)∣∣∣∣y=L

y=0

. (2.57)

Therefore, we get the following BCs:(δΦ†∂5Φ

)∣∣∣y=L

y=0= 0 . (2.58)

4In the following, we use the capital latin letters M,N,P, . . . as indices in the 5D curved space, whilewe denote the indices of the flat 4D manifold with the greek letters µ, ν, ρ, . . . .

5A valid alternative would be the compactification on the orbifold S1/Z2. The two strategies areequivalent providing suitable BCs on the fields. As we have already discussed, this is especially crucialin the case of fermions, in order to get chiral zero modes.

6We exploit the hypotheses that the fields vanish at spatial infinity “sufficiently fast”.


By working out the partial derivatives we can reconstruct the EOMs{e2kyηµν∂µ∂ν − e4ky∂5e

−4ky +M2Φ

}Φ = 0 . (2.59)

Let us solve (2.59) by assuming the separation of variables

Φ(xµ, y) =1√L

∞∑n=0

Φ(n)(x)f(n)Φ (y) , (2.60)

where the fields Φ(n)(x) are solutions of the 4D Klein-Gordon equation, ηµν∂µνΦ(n) =

−m2nΦ(n). Therefore, the profiles f

(n)Φ (y) are solutions of the Sturm-Liouville equation7:

− d

dy

(p(y)

d

dyf

(n)Φ (y)

)+ q(y)f

(n)Φ (y) = λn ω(y)f

(n)Φ (y) , (2.61)

where p(y) = e−4ky, q(y) = M2Φe−4ky, ω(y) = e−2ky and eigenvalues, λn = m2

n. From thetheory of Sturm-Liouville differential equations we know that, if p(y) is a differentiablefunction, q(y) and ω(y) are continuous, with p(y) > 0 and ω(y) > 0 in the interval [0, L],then the eigenvalues λn are real and ordered, namely λ0 < λ1 < · · · < λn →∞. Moreover,

f(n)Φ (y) constitute a complete set of eigenfunctions, satisfying an orthonormalization

condition [34]:1

L

ˆ L

0dy ω(y)f

(n)Φ f

(m)Φ = δnm . (2.62)

Let us completely solve the Eq. (2.59), by taking into account the boundary conditions

that can be of two types, f(n)Φ

∣∣ = 0 (Dirichlet), which we denote with the symbol (−), or

∂5f(n)Φ

∣∣ = 0 (Neumann) denoted by (+). Of course, there exist four possible combinationsof them if one considers the two branes. It is convenient to rewrite Eq. (2.59), by

introducing the Bogoliubov transform of the function f(n)Φ (y):

f(n)Φ = e2kyf

(n)Φ .

We get, in this way, the second order equation

− d2

dy2f

(n)Φ + (4 + a)k2f

(n)Φ = m2

ne2kyf

(n)Φ , (2.63)

that we solve in both the massless (mn = 0) and massive (mn 6= 0) case.

Massless solutions: mn = 0

The solution is provided by

f(0)Φ = c

(0)1 e(2−

√4+a)ky + c

(0)2 e(2+

√4+a)ky , (2.64)

7We substituted in Eq. (2.60) the right side of the Klein-Gordon equation for Φ(n)(x), and leftmultiplied for e−4ky.


where c(0)1 and c

(0)2 are arbitrary constants that one determines starting from the BCs

and from the normalization conditions. It is worth observing that, for a = 0, there areno massless solutions for every combinations of BCs.

A nontrivial solution can be obtained by providing a brane localized term in the freeLagrangian, e.g. [34]:

S∂Φ =

ˆd5x√Gbk

[δ(y)− δ(y − L)

]∣∣Φ∣∣2 , (2.65)

where b is a dimensionless parameter that stands for the mass at the boundary, in kunits. In this way, Neumann BCs are modified to8

(∂5 − b k

)∣∣∣y=L

y=0= 0 , (2.66)

and allow us to recover the massless profile:

f(0)Φ = c

(0)1 ebky, (2.67)

if applied on both the branes. In the latter it should be b = 2±√

4 + a if (2.64) is still

valid. The normalization condition let us determine the c(0)1 coefficient:

1 =1

L

ˆ L

0dy e−2ky|f (0)

Φ |2 =1

L|c(0)

1 |2e2(b−1)kL − 1

2(b− 1)kL,

and for b > 1 one has

|c(0)1 | ≈

√2(b− 1)kL

e2(b−1)kL,

and thereforef

(0)Φ ≈

√2(b− 1)kLekLebk(y−L). (2.68)

It is worth focusing on the localization properties of the scalar field, looking at thekinetic term:ˆ

d5x√GGµν∂µΦ†∂νΦ + · · · =

ˆd5xe2(b−1)kyηµν∂µΦ(0)†(x)∂νΦ(0)(x) + . . . .

Then, with respect to flat 5D metric the zero mode profile of the scalar is given by

f(0)Φ (y) ∝ e(b−1)ky = e(1±

√4+a)ky .

Therefore, for b > 1 (b < 1) the zero mode will be localized near to the IR (UV) brane,while for b = 1 the profile is flat. In models with custodial symmetry, one generallychooses an IR localized profile for the Higgs boson. More practically, one assumes thatthe Higgs field does not propagate in the bulk, and its profile is set to a Dirac deltafunction, zero everywhere except at y = L, and Eq. (2.68) is a good approximationprovided b� 1.

8Let us remind that ∂L∂(∂5Φ)†

= −√G∂5Φ.


æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

ò

ò

ò

ò

ò

ò

ò

ò

ò

ò

0 2 4 6 8 100

5

10

15

20

25

30

n

JΑH xn L

Figure 2.3: Validity of the approximation xn ≈ π(n+ α/2− 3/4) for the zeros of J−1+α(xn). The redmarkers are the exact solutions, while the free triangles are the approximated ones changing n.

0.0 0.2 0.4 0.6 0.8 1.00

5000

10 000

15 000

t

f F0HtL

b=10

b=100

b=1000

Figure 2.4: Zero mode profiles for the scalar in the bulk, changing the parameter b. The extradimensional coordinate is t = εky ( see the section 2.2.2 for more details).


Massive solutions: mn 6= 0

The solutions for the massive case are the following:

f(n)Φ (y) =

e2ky

N(n)Φ

[Jα

(eky

mn

k

)+ b

(n)Φ Yα

(eky

mn

k

)], (2.69)

where α =√

4 + a and J, Y are Bessel functions. The coefficients b(n)Φ and N

(n)Φ follow

after imposing the BCs and the normalization condition, respectively. For example, inthe case of the modified Neumann BCs (the only ones that assure the existence of a zeromode), one has:

b(n)Φ =

−mnk J−1+α(mnk ) + (−2 + b+ α)Jα(mnk )

mnk Y−1+α(mnk )− (−2 + b+ α)Yα(mnk )

=−mn

k J−1+α(mnk )mnk Y−1+α(mnk )

.

Let us observe that, in the limits mnk � 1 and kL� 1 we can extract the masses of KK

modes:9

mn ≈ kπe−kL(n+

1

2α− 3

4

). (2.70)

2.2.2 Fermions

Let us consider the most general case of a fermion embedded in a Riemannian manifold10,whose action can be written as

SΨ =

ˆd4x

ˆdy√G{EMA ΨiγA

←→∂MΨ +

1

2EMA ωBCM iΨγ

AσBCΨ}, (2.71)

where EMA are the inverse vielbein and ωBCM is the spin connection. Let us rewrite Eq.(2.71) as

SΨ =

ˆd4x

ˆdy√G{EMA ΨiγA

←→∂MΨ +

1

2EMA ωBCM iΨ

1

2[γA, σBC ]Ψ

+1

2EMA ωBCM iΨ

1

2{γA, σBC}Ψ

}, (2.72)

where←→∂M =

1

2(∂M −

←−∂M ) .

9Indeed, the Neumann BC on the IR brane:

ekLmn/k(J−1+α(xn) + b

(n)Φ (mn/k)Y−1+α(ekLmn/k)

)≈ ekLmn/kJ−1+α(ekLmn/k) = 0

is satisfied if ekLmn/k ≡ xn are zeros of J1−α(xn), for which holds the approximated formula: xn ≈(n+ 1/2(−1 + α)− 1/4

)as shown in Figure 2.3.

10The manifold has a generic number of dimensions, for which the index notation of the previoussection is still valid. The metric tensor G is not yet specified.


Therefore, the Lagrangian density for the fermion is:

L =√GEMA iΨγ

A 1

2(∂M −

←−∂M )Ψ− 1

2

√GEMA ωBCM iΨ

1

2[σBC , γA]Ψ

+1

2

√GEMA ωBCM iΨ

1

2{γA, σBC}Ψ . (2.73)

We use the relation11

∂M (√GEMA ) =

√GEMB ω

BAM , (2.74)

to rewrite the second derivative of the first term of Eq. (2.73)

−1

2(∂M√GEMA iΨ)γAΨ = −1

2(∂M√GEMA )iΨγAΨ− 1

2

√GEMA i(∂MΨ)γAΨ

= −1

2

√GEMB ω

BAM iΨγ

AΨ− 1

2

√GEMA i(∂MΨ)γAΨ . (2.75)

The second term of Eq. (2.73) can be rewritten as12:

−1

2

√G EMA ωBCM iΨ

1

2[σBC , γA]Ψ

= −1

2

√GEMA ωBCM iΨ

1

2(γBηCA − γCηBA)Ψ

= −1

4

√GEMA ωBCM iΨγ

BηCAΨ +1

4

√GEMA ωBCM iΨγ

CηBA

= −1

4

√GEMB ωA

BM iΨγ

AΨ +1

4

√GEMB ω

BAM iΨγ

AΨ

=1

2

√GEMB ω

BAM iΨγ

AΨ .

11Eq. (2.74) directly derives from the metricity condition

ωM = eBN (∂MENA + ΓNMPE

PA )σAB/2

where eBN are the vielbein and the trace relation for the Christoffel symbol, indeed

∂MEMA = EMB ω

BAM − ΓNNKE

KA ,

from which, by multiplying both the sides of the latter equation for√G and by

√G(∂ME

MA ) = ∂M (

√GEMA )− (∂M

√G)EMA ,

one has

∂M (√GEMA ) = (∂M

√G)EMA +

√GEMB ω

BAM −

√GΓP PME

MA =

√GEMB ω

BAM ,

where the relation ΓP PM = 1√G∂M√G has been used.

12In this case we make use of the following identity:

[σBC , γA] = γBηCA − γCηBA ,

that derives from the Dirac algebra {γA, γB} = 2ηAB , and from the antisymmetric property of the spinconnection ωABM = −ωBAM .


The latter term cancels the one obtained from the left derivative of the first term in theLagrangian (2.73). Therefore, the Lagrangian density reduces to

L =1

2

√GEMA ΨiγA

←→∂MΨ +

1

2EMA ωBCM iΨ

1

2{γA, σBC}Ψ . (2.76)

Since

ωM = ωBCMσBC

2,

the last term in Eq. (2.76) can be rewritten as

1

2

√GEMA iΨγ

AωMΨ +1

2

√GEMA iΨωMγ

AΨ . (2.77)

If one specializes to the AdS5 background with a metric given by (2.54)13, one has:

ωM =

(i

2ke−k|y|γµγ

5, 0

),

providing the definition

−iJAB = σAB =1

4[γA, γB]

and γA = (γµ, iγ5) the Dirac matrices in 5D14. Therefore, due to the particular expression

of ωM in the case of an AdS5 metric, the two terms in (2.77) cancel out, and by includinga bulk mass term for the fermion embedded in AdS5, the action turns to

SΨ =

ˆd4x

ˆ L

0dy√G{EMA ΨiγA

←→∂MΨ−mΨΨΨ

}, (2.78)

where we assume from now on, that the operator←→∂M acts only on the spinor.

13This result has been achieved by noticing that the vielbein are given by eAM = (e−kyδAM , 1), whereδAM is the Kronecker delta. The Christoffel connection is

ΓRMN =1

2GRP (∂MGNP + ∂NGMP − ∂PGMN ) ,

and for the metric (2.54) the only non-vanishing terms are

−Γ0(05) = Γ1

(15) = Γ2(25) = Γ3

(35) = k

−Γ500 = Γ5

11 = Γ522 = Γ5

33 = ke−2k|y| .

14The γ5 matrix is defined as iγ0γ1γ2γ3. Matrices γA satisfy the usual Dirac algebra in 5D Minkowskispace

{γA, γB} = 2ηAB .


Equations of motion

Let us expand Eq. (2.78) taking into account the vielbein definition accounted in (2.74):

SΨ =

ˆd4x

ˆ L

0dy e−3ky

{iΨγµ∂µΨ +

1

2e−ky[Ψγ5∂5Ψ− (∂5Ψ)γ5Ψ]− e−kymΨΨΨ

},

(2.79)where we integrated by parts on xµ and eliminated the surface terms assuming the fieldis vanishing at infinity. By varying the action with respect to the field Ψ, one obtains avolume term15

δSVΨ =

ˆd5x e−3kyδΨ

{iγµ∂µΨ + e−kyγ5(∂5 − 2k)Ψ− ekymΨΨ

}, (2.80)

while the integration on the extra coordinate gives the surface term

δSSΨ = −ˆd4x e−4kyδΨγ5Ψ

∣∣∣y=L

y=0. (2.81)

The application of the stationarity principle give us the EOMs{iekyγµ∂µ + (∂5 − 2k)γ5 −mΨ

}Ψ = 0 , (2.82)

together with the BCs for the field Ψ:

− δΨγ5Ψ∣∣∣y=L

+ δΨγ5Ψ∣∣∣y=0

= 0 . (2.83)

Let us rewrite the Eqs. (2.82–2.83) in terms of the chiral components Ψ ≡ PL,RΨ, where

ΨL =

(ψL0

), ΨR =

(0ψR

)(2.84)

and

ΨL =(1− γ5)

2Ψ , ΨR =

(1 + γ5)

2Ψ , ΨL,R = ∓γ5ΨL,R . (2.85)

A suitable choice for the gamma matrices is represented by

γ0 =

(0 11 0

), γi =

(0 σi

−σi 0

), γ5 =

(−1 00 1

).

and σi are the 2× 2 di Pauli matrices. The BCs then become:

δΨLΨR − δΨRΨL

∣∣∣y=L

y=0= 0 . (2.86)

15Let us rewrite the term

e−4ky(∂5Ψ)γ5Ψ = ∂5(e−4kyΨγ5Ψ)−Ψ(∂5e−4kyγ5Ψ) ,

the first member on the right-hand side of the equivalence is a surface term.


Let us be more precise. One may observe that periodic BCs (compactification on acircle) of the type

ΨL

∣∣∣y=L

= ΨL

∣∣∣y=0

& ΨR

∣∣∣y=L

= ΨR

∣∣∣y=0

are unfit to distinguish among the left- and right-handed modes, even if they satisfy thecondition (2.86). On the other hand, one could impose exclusive BCs like

ΨL

∣∣∣ = 0 or ΨR

∣∣∣ = 0 .

Both the above conditions (with four possible combinations taking into account the twobranes) respect the variational principle and are able to distinguish between the L andR modes. One basically cannot exploit Dirichlet BCs for both the chiralities.

Therefore, the EOMs for the chiral fields become:

iekyγµ∂µΨR + [−(∂5 − 2k)−mΨ]ΨL = 0 , (2.87)

iekyγµ∂µΨL + [(∂5 − 2k)−mΨ]ΨR = 0 . (2.88)

It is worth having a look to the implications of the latter BCs. Let us suppose to choose

ΨL

∣∣∣ = 0 (Dirichlet), from (2.88) we get for the right-handed mode ΨR a Neumann BC:

∂5ΨR

∣∣∣ = (2k +mΨ)ΨR

∣∣∣ ,We denote this BC with the symbol (−). Conversely, with the condition ΨR

∣∣∣ = 0,

through Eq. (2.87) one should recover

∂5ΨL

∣∣∣ = (2k −mΨ)ΨL

∣∣∣ .The latter condition will be denoted by (+). Finally, we have four different BCs (and asmany solutions of the EOMs for the L and R modes) that we label by the notation (++),(−−), (+−), (−+). Among them, only the first two combinations allow the existence ofchiral zero modes for the L and R modes, respectively.

Let us observe that the solutions of Eqs. (2.87–2.88) can be obtained by a separationof variables16

ΨL,R(xµ, y) =1√L

∞∑n=0

ψ(n)L,R(x)f

(n)L,R(y) . (2.89)

The KK modes ΨL,R(xµ, y) are solutions of the Dirac equation

iγµ∂µψ(n)L,R = mnψ

(n)L,R .

The EOMs for the profiles are therefore:

∂5f(n)L (y) + (c− 2)kf

(n)L (y) = mne

kyf(n)R (y) , (2.90)

−∂5f(n)R (y) + (c+ 2)kf

(n)R (y) = mne

kyf(n)L (y) , (2.91)

where we set mΨ = ck for the mass of the fermion in the bulk.

16In Eq. (2.89), the extra dimensional profiles f(n)L,R(y) are dimensionless, while the KK modes

ΨL,R(xµ, y) are correctly of dimensions 2.


Massless solution: mn = 0

In this case Eqs. (2.90–2.91) decouple, and the zero mode solution is given by

f(0)L,R = d

(0)L,Re

(2∓c)ky, (2.92)

where d(0)L,R are constant fixed by the normalization condition. It is worth noticing that,

if one chooses (++) BCs, namely fR

∣∣∣ = 0, the profile f(0)R should identically be zero, by

looking at the explicit solution in (2.92). In the last case, we would say that only theleft-hande mode admits a zero mode, whose profile is given by the (2.92). Vice-versa,in the (−−) BCs case, is the left-handed zero mode that vanishes. For the remainingBCs, neither left-handed nor right-handed zero modes exist. Therefore, with a single 5Dfermion in the bulk, one cannot have two opposite chirality zero modes, since the BCswould select only one definite chirality (L or R) mode a time.

Let us turn to (++) BCs. We already know that the zero mode is the left-handed

one, with a profile f(0)L = d

(0)L e(2−c)ky. We can study the localization property of the

fermion ψ(0)L through the kinetic term

ˆd5x√GΨΓµ∂µΨ + · · · =

ˆd5x e2( 1

2−c)kyψ

(0)L γµ∂µψ

(0)L + . . .

The zero mode profile is

f(0)L (y) ∝ e2( 1

2−c)ky (2.93)

with respect to the flat 5D metric. Therefore, if c > 1/2(c < 1/2) the zero mode islocalized near the UV(IR) brane, while it is flat for c = 1/2. Similar results hold in thecase of right-handed fermions, providing the substitution c↔ −c.

Massive case: mn 6= 0

In this case, we can derive a second order equation for each profile f(n)L and f

(n)R . Each

one of the latter is equivalent to a Sturm-Liouville equation, providing f(n)L,R ≡ e−2kyf

(n)L,R:

− d

dy

(e−ky

d

dyf

(n)L,R

)+ c(c± 1)k2e−kyf

(n)L,R = m2

nekyf

(n)L,R . (2.94)

The general solution is given by [34]:

f(n)L,R(y) =

e5/2ky

N(n)Ψ

[Jc±1/2

(mn

keky)

+ b(n)Ψ Yc±1/2

(mn

keky)]. (2.95)

b(n)Ψ and N

(n)Ψ are the same for both fL and fR, and are determined by the orthonormality

condition1

L

ˆ L

0dy e−3kyf

(n)L,Rf

(m)L,R = δnm . (2.96)


In order to extract the exact solutions for every BC, it is useful to perform a changeof variable [35]:

y −→ t = εeky ,

where17 ε = e−kL. The extra dimensional profiles’ EOMs can be rewritten as

{t∂t + (c− 2)}f (n)L (t) = xntf

(n)R (t) , (2.97)

{−t∂t + (c+ 2)}f (n)R (t) = xntf

(n)L (t) , (2.98)

where we have introduced the quantity xn ≡ mnkε . The solutions for the f

(n)L,R profiles are:

√εf

(n)L (t) =

√t{a

(n)L Jc+1/2(xnt) + b

(n)L J−c−1/2(xnt)

}, (2.99)

√εf

(n)R (t) =

√t{a

(n)R J−c+1/2(xnt) + b

(n)R Jc−1/2(xnt)

}, (2.100)

and from Eqs. (2.97–2.98) the coefficients a(n)L,R and b

(n)L,R are tied together by the relations

a(n)L = b

(n)R ,

b(n)L = −a(n)

R . (2.101)

Let us rewrite

√εf

(n)L (t) =

√t{a

(n)L Jc+1/2(xnt)− a(n)

R J−c−1/2(xnt)}, (2.102)

√εf

(n)R (t) =

√t{a

(n)R J−c+1/2(xnt) + a

(n)L Jc−1/2(xnt)

}. (2.103)

The solutions (2.102–2.103) are equivalent to (2.95), once we have performed the changeof variable and multiplied by e2ky, thanks to the following relations between the Besselfunctions Jν and Yν :

Yν(z) =Jν(z) cos(νπ)− J−ν(z)

sin(νπ).

The orthonormality condition among f(n)L,R turns to:

1

L

1

kε

ˆ 1

εdt f

(n)L,Rf

(m)L,R = δnm ,

and by further redefining the functions f(n)L,R(t) −→ 1√

kLεf

(n)L,R(t) ≡ f (n)

L,R(t) one has:

ˆ 1

εdt f

(n)L,Rf

(m)L,R = δnm , (2.104)

17For phenomenological reasons linked to the hierarchy problem between the electroweak and thePlanck scale, one generally sets ε ≈ 10−16.


æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

ò

ò

ò

ò

ò

ò

ò

ò

ò

ò

0 2 4 6 8 100

5

10

15

20

25

30

n

J0HxnL

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

ò

ò

ò

ò

ò

ò

ò

ò

ò

ò

0 2 4 6 8 100

5

10

15

20

25

30

n

J1HxnL

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

ò

ò

ò

ò

ò

ò

ò

ò

ò

ò

0 2 4 6 8 100

5

10

15

20

25

30

n

Jc-1�2HxnL

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

ò

ò

ò

ò

ò

ò

ò

ò

ò

ò

0 2 4 6 8 100

5

10

15

20

25

30

n

Jc-1�2HxnL

Figure 2.5: Plots showing the validity of the approximation xn ≈ π(n+ ν/2− 1/4) as zeros of Jν(xν).The red markers are the exact solutions, the green triangles represent the approximated solutions. ForJc−1/2, the cases in which c is set to 0.45 (at the bottom, left) and to 0.55 (at the bottom, right) havebeen exploited.

and

f(n)L (t) =

√t{a

(n)L Jc+1/2(xnt)− a(n)

R J−c−1/2(xnt)}, (2.105)

f(n)R (t) =

√t{a

(n)R J−c+1/2(xnt) + a

(n)L Jc−1/2(xnt)

}, (2.106)

where we have redefined a(n)L,R −→

a(n)L,R√k

. The profiles f(n)L,R are connected to f

(n)L,R by the

relation

f(n)L,R(t) =

√kL

ε3t2f

(n)L,R(t) . (2.107)

Let us evaluate the exact shape of the profile, that is the coefficients a(n)L,R depending

of the specific BCs.

• BC (++). In this case we have

f(n)R

∣∣∣t=ε

= f(n)R

∣∣∣t=1

= 0 ,

therefore f(0)R (t) ≡ 0. For what concerns the left-handed zero mode, by Eq. (2.107)

one hasf

(0)L (t) ≡ d(0)

L t−c .


The d(0)L coefficient derives from the orthonormality condition

1 =

ˆ 1

εdt∣∣d(0)L

∣∣2t−2c =∣∣d(0)L

∣∣2 t−2c+1

−2c+ 1

∣∣∣∣∣1

ε

.

Hence, one has

d(0)L =

√1− 2c

1− ε1−2c

and

f(0)L =

√1− 2c

1− ε1−2ct−c , (2.108)

while the f(0)L (y) function is given by

f(0)L (y) = e2ky

√kLεf

(0)L (y) =

√(1− 2c)kL

e(1−2c)kL − 1e(2−c)ky . (2.109)

Let us now determine the chiral profiles of the KK modes. We use the limit ε→ 0,to do suitable approximation of the Bessel functions; in particular, we exploit thelimit Jν(x) ≈ xν , for x→ 0. The boundary condition fL,R(ε) = 0,

f(n)R ≈ √ε{a(n)

R (xnε)−c+1/2 + a

(n)L (xnε)

c−1/2} ≈ 0 ,

can be achieved by two possible choices of the parameter c. For c > 1/2 it will be

a(n)R ≈ 0, therefore we rewrite the solutions as

f(n)L ≈

√t a

(n)L Jc+1/2(xnt) , (2.110)

f(n)R ≈

√t a

(n)L Jc−1/2(xnt) . (2.111)

The boundary t = 1 condition for f(n)R (t):

f(n)R (1) = 0 −→ xn zeros os Jc−1/2(xn)

allows us to recover the mass spectrum18:

mn ≈ kπe−kL(n+

1

2

(c− 1

2

)− 1

4

). (2.112)

18In order to identify the zeros of Jν(x), we adopt the approximation

x ≈ π(n− 1

4+ν

2

),

valid for ν � 1, Figure 2.5.


The orthonormality condition determine the coefficient a(n)L in the limit ε→ 0:

1 =

ˆ 1

εdt∣∣f (n)R (t)

∣∣2 =∣∣a(n)L

∣∣2{− ε2

2J2c−1/2(xnε)︸︷︷︸≈0

+ε2

2Jc−3/2(xnε)Jc+1/2(xnε)︸︷︷︸

≈0

− 1

2Jc−3/2(xn)Jc+1/2(xn)︸︷︷︸

≈−J2c+1/2

(xn)

+1

2J2c−1/2(xn)︸︷︷︸

=0

},

hence ∣∣a(n)L

∣∣2 ≈ 2∣∣Jc+1/2(xn)∣∣2 ≡ ∣∣Nc+1/2(xn)

∣∣2 . (2.113)

The choice c < 1/2 involves a(n)L ≈ 0, then the solution is given by

f(n)L ≈ −

√t a

(n)R J−c−1/2(xnt) , (2.114)

f(n)R ≈

√t a

(n)R J−c+1/2(xnt) . (2.115)

This time, the boundary t = 1 condition for f(n)R (t) is correctly addressed if xn are

zeros of J−c+1/2(xn). Therefore:

mn ≈ πe−kL(n+

1

2

(− c+

1

2

)− 1

4

). (2.116)

The normalization of f(n)R (t) in the limit ε→ 0:

1 =

ˆ 1

εdt∣∣f (n)R (t)

∣∣2 =∣∣a(n)R

∣∣2{− ε2

2J2−c+1/2(xnε)︸︷︷︸≈0

+ε2

2J−c+3/2(xnε)J−c−1/2(xnε)︸︷︷︸

≈0

− 1

2J−c+3/2(xn)J−c−1/2(xn)︸︷︷︸

≈−J2−c+3/2

(xn)

+1

2J2−c+1/2(xn)︸︷︷︸

=0

}

gives the coefficient a(n)R :∣∣a(n)R

∣∣2 ≈ 2∣∣J−c+3/2(xn)∣∣2 ≡ ∣∣N−c+3/2(xn)

∣∣2 . (2.117)

• BC (– –). The BCs fL

∣∣∣t=ε

= fL

∣∣∣t=1

= 0 select only the right-handed tower having

a zero mode, with profile

f(0)R = d

(0)R tc ,

and, because of the normalization, one has∣∣d(0)R

∣∣2 =1 + 2c

1− ε1+2c. (2.118)


Hence

f(0)R (t) =

√1 + 2c

1− ε1+2ctc .

The complete profile is given by

f(0)R (y) = e2ky

√kLεf

(0)R (y) =

√(1 + 2c)kL

e(1+2c)kL − 1e(2+c)ky . (2.119)

Following the convention adopted for the “t = ε” BCs, f(n)L (ε) = 0 and from Eq.

(2.105) evaluated in ε, we infer that a(n)R = 0, independently of c. Therefore, the

profiles in this case are

f(n)L ≈ −

√t a

(n)L Jc+1/2(xnt) , (2.120)

f(n)R ≈

√t a

(n)L Jc−1/2(xnt) . (2.121)

The KK mass spectrum follows from the condition f(n)L (1) = 0, with xn the zeros

of Jc+1/2(xn):

mn ≈ kπe−kL(n+

1

2

(c+

1

2

)− 1

4

). (2.122)

From the normalization condition

1 =

ˆ 1

εdt∣∣f (n)L (t)

∣∣2 =∣∣a(n)L

∣∣2{− ε2

2J2c+1/2(xnε)︸︷︷︸≈0

+ε2

2Jc+3/2(xnε)Jc−1/2(xnε)︸︷︷︸

≈0

− 1

2Jc+3/2(xn)Jc−1/2(xn)︸︷︷︸

≈−J2c+3/2

(xn)

+1

2J2c+1/2(xn)︸︷︷︸

=0

}

we have ∣∣a(n)L

∣∣2 ≈ 2∣∣Jc+3/2(xn)∣∣ ≡ ∣∣N3/2+c(xn)

∣∣2. (2.123)

• BC (– +). The BCs f(n)L (ε) = f

(n)R (1) = 0 forbid the existence of any chiral zero

mode. The determination of the KK profiles proceeds similarly to the previouscases. In particular, one has:

f(n)L ≈ −

√t a

(n)L Jc+1/2(xnt) , (2.124)

f(n)R ≈

√t a

(n)L Jc−1/2(xnt) . (2.125)

The mass spectrum directly descends from the boundary t = 1 constraint, and isgiven by

mn ≈ kπe−kL(n+

1

2

(c− 1

2

)− 1

4

), (2.126)


and the a(n)L coefficient is19:

∣∣a(n)L

∣∣2 ≈ 2∣∣Jc+1/2

∣∣2 ≡ ∣∣Nc+1/2

(xn)|2 .

• BC (+ –). No chiral zero modes still exist. We have to distinguish betweenc < 1/2 and c > 1/2, similarly to the (++) case. For c < 1/2, the profiles can bewritten

f(n)L ≈ −

√t a

(n)R J−c−1/2(xnt) , (2.127)

f(n)R ≈

√t a

(n)R J−c+1/2(xnt) . (2.128)

The boundary t = 1 condition selects the xn, which are the zeros of J−c−1/2, hence

mn ≈ kπe−kL(n+

1

2

(− c− 1

2

)− 1

4

). (2.129)

The a(n)R coefficient is given by

∣∣a(n)R

∣∣2 ≈ 2∣∣J−c+1/2

∣∣2 ≡ ∣∣N−c+1/2

(xn)|2 .

In the case of c > 1/2, the profiles get the following expression

f(n)L ≈

√t a

(n)L Jc+1/2(xnt) , (2.130)

f(n)R ≈

√t a

(n)L Jc−1/2(xnt) . (2.131)

This time, xn are zeros of Jc+1/2, then

mn ≈ kπe−kL(n+

1

2

(c+

1

2

)− 1

4

)(2.132)

and ∣∣a(n)L

∣∣2 ≈ 2∣∣Jc−1/2

∣∣2 ≡ ∣∣Nc−1/2

(xn)|2 .

The results of the solutions of the EOMs for the embedded fermion are collected inTable 2.1. In Figure 2.6 the f

(0)L,R profiles are shown for a particular choice of c.

19This coefficient is different from the one derived within (−−) BCs, since this time xn are solutionsof J(c−1/2).


Table

2.1

:K

alu

za-K

lein

pro

file

sfo

ra

ferm

ion

inth

ebulk

,as

emer

ge

by

solv

ing

the

corr

esp

onden

tE

OM

sfo

rdiff

eren

tB

Cs.

BC

sc

valu

em

ass

spectr

um

KK

pro

file

szero

mode

(++

)c>

1/2

mn≈kπe−

kL( n

+1 2

( c−

1 2

) −1 4

)f

(n)

L,R

(y)≈√

2kLe−

2kL

|Jc+

1/2(m

nkekL

)|e5/2kyJc±

1/2(m

n keky)

f(0

)L

(y)

=√ (1

−2c)kL

e(1−

2c)kL−

1e(

2−c)ky

c<

1/2mn≈kπe−

kL( n

+1 2

( −c

+1 2

) −1 4

) f(n

)L,R

(y)≈∓√

2kLe−

2kL

|J−

c+

3/2(m

nkekL

)|e5/2kyJ−c∓

1/2(m

n keky)f

(0)

L(y

)=√ (1

−2c)kL

e(1−

2c)kL−

1e(

2−c)ky

(−−

)any

mn≈kπe−

kL( n

+1 2

( c+

1 2

) −1 4

)f

(n)

L,R

(y)≈∓√

2kLe−

2kL

|Jc+

3/2(m

nkekL

)|e5/2kyJc±

1/2(m

n keky)

f(0

)R

(y)

=√ (1

+2c)kL

e(1+

2c)kL−

1e(

2+c)ky

(−+

)any

mn≈kπe−

kL( n

+1 2

( c−

1 2

) −1 4

)f

(n)

L,R

(y)≈∓√

2kLe−

2kL

|Jc+

1/2(m

nkekL

)|e5/2kyJc±

1/2(m

n keky)

–

(+−

)c>

1/2

mn≈kπe−

kL( n

+1 2

( c+

1 2

) −1 4

)f

(n)

L,R

(y)≈√

2kLe−

2kL

|Jc−

1/2(m

nkekL

)|e5/2kyJc±

1/2(m

n keky)

–

c<

1/2mn≈kπe−

kL( n

+1 2

( −c−

1 2

) −1 4

) f(n

)L,R

(y)≈∓√

2kLe−

2kL

|J−

c+

1/2(m

nkekL

)|e5/2kyJ−c∓

1/2(m

n keky)

–


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

t

f L0HtL�

kLΕ

c =0.45

c =0.55

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

t

f R0HtL�

kLΕ

c =0.45

c =0.55

Figure 2.6: f(0)L,R(t) profiles obtained for c = 0.55 and c = 0.45 respectively. We may notice that for the

left–handed massless mode, in case of c > 1/2 the profile vanishes at the IR boundary (left-panel). Thesame does not happen for the right–handed one, this is direct consequence of the model (right-panel).

2.2.3 Gauge fields

Let us consider an abelian gauge field AM propagating in the bulk of a AdS5 space-time.The action is given by

SA =

ˆd4x

ˆdy√G

1

4g25

FMNFMN , (2.133)

where g5 is 5D coupling constant and FMN = ∂MAN − ∂NAM . We adopt the gaugeA5 = 0 with the constraint ∂µA

µ = 0. The EOMs are

∂M (√GGMOGNPFOP ) = 0 , (2.134)

that is

∂µ[√Ge4kyηµoηνρFoρ −

√Ge2kyηµoFo5

]+ ∂5

[−√Ge2kyηνρF5ρ +

√GF55

]= 0 ,

from which one derives {− ηµν∂µ∂ν + ∂5e

−2ky∂5

}Aρ = 0 . (2.135)

The validity of the Eulero-Lagrange Eq. (2.134) is guaranteed by the vanishing of thesurface terms in the action after a variation with respect to field AM . These terms areof the type:

δSSA =

ˆd4x

ˆdy ∂M

( ∂L∂(∂MAN )

δAN

)=

ˆd4x

ˆdy ∂µ

( ∂L∂(∂µAν)

δAν

)+

ˆd4x

ˆdy ∂5

( ∂L∂(∂5Aν)

δAν

). (2.136)

The first contribution at the right-hand side vanishes when we evaluate the fields atx→∞, the second one becomes:

δSSA =

ˆd4x( ∂L∂(∂5Aν)

δAν

)∣∣∣∣∣y=L

y=0

= −ˆd4x(−√Ge2ky∂5A

ν)δAν

∣∣∣∣∣y=L

y=0

= 0 .


Therefore, the right BCs become:(δAν∂5Aν

)∣∣∣y=L

y=0= 0 . (2.137)

We solve the (2.135) by the separation of variables, as usual20:

Aν(xµ, y) =1√L

∞∑n=0

A(n)ν (xµ)f

(n)A (y) , (2.138)

where A(n)ν (x) are the KK modes that solve ∂µ∂

µA(n)ν (x) = −m2

nA(n)ν (x). The EOMs for

the profiles are

− ∂5(e−2ky∂5f(n)A ) = m2

nf(n)A , (2.139)

Sturm-Liouville type equations with p(y) = e−2ky, q(y) = 0 and ω(y) = 1. Hence, f(n)A (y)

constitute a complete and orthonormalized set of eigenfunctions:

1

L

ˆ L

0dy f

(n)A f

(m)A = δnm . (2.140)

The BCs (2.137) are satisfied if one imposes ∂5Aµ∣∣brane

= 0 (Neumann), denoted as (+),

or Aµ∣∣brane

= 0 (Dirichlet), labeled with (−).

Substituting f(n)A with its Bogoliubov transform f

(n)A = e2kyf

(n)A in Eq. (2.139), we

obtainf

(n)′′

A + 2kf(n)′

A = −m2ne

2kyf(n)A (2.141)

that we solve in both the massless (mn = 0) and the massive (mn 6= 0) case.

Massless case: mn = 0

Solving (2.141) in f(0)A and performing the inverse Bogoliubov transform, one obtains

f(0)A = c

(0)1 + c

(0)2 e2ky , (2.142)

where c(0)1 and c

(0)2 are constants determined imposing suitable BCs the normalization

constraint (2.140). It is worth observing that, with (−−) BCs, both the coefficientsvanish, implying the non existence of the corresponding zero mode. The same happensfor the (−+) and (+−) conditions.

From the BCs (++), it follows that c(0)2 = 0 when one considers the UV brane. From

the orthonormalization condition c(0)1 = 1 ensues, therefore

f(0)A = 1 .

It is not possible to localize the field in the bulk, by observing thatˆd4x

ˆdy√GGµρGνσFµνFρσ + · · · =

ˆd4x

ˆdy

1

LηµρηνσF (0)

µν F(0)ρσ + . . . .

20We observe that the mass dimension of the field Aν(xµ, y) is 1, therefore the profiles f(n)A (y) are

dimensionless.


0 2 4 6 8 10

0.0

0.5

1.0

1.5

x

J 0,1HxL

J1J0

2

Πx-1�2

Figure 2.7: Approximation of J0(xn) via the function√

2/πx−1/2n . The values J0(xn) correspond to

the points of maximum of J0(x), since xn are zeros of J1(x), since the relation J ′0(z) = J1(z) holds.

Massive solution: mn 6= 0

The general solution for the KK modes with mn 6= 0 is

f(n)A (y) =

eky

N(n)A

[J1

(mn

keky)

+ b(n)A Y1

(mn

keky)]

, (2.143)

where N(n)A and b

(n)A arbitrary constants that, as always, can be deduced by the BCs

(2.137) and from the normalization condition. In order to perform a systematic studyfor the distinct BCs, it is still useful to change the variable y −→ t = εeky, as in the caseof fermions. Let us rewrite the solution (2.143):

f(n)A (t) =

t

εN(n)A

[J1(xnt) + b

(n)A Y1(xnt)

], (2.144)

where xn = mnεk and t ∈ [ε, 1], and use the limit ε→ 0. The BCs become

f(n)A

∣∣t=ε, 1

= 0 (Dirichlet) ,

andt ∂tf

(n)A

∣∣t=ε, 1

= 0 (Neumann) .

• BC (– –). From f(n)A (ε) = 0 we get the first constraint on the parameter b

(n)A :

b(n)A = −J1(mnk )

Y1(mnk ). (2.145)

From the IR (t = 1) boundary condition, f(n)A (1) = 0, one obtains

J1(xn)− J1(xnε)

Y1(xnε)Y1(txn) = 0 ,


0 2 4 6 8 10

-1

0

1

2

x

Y1HxL J0HxL

F1HxL

Figure 2.8: Approximation of the zeros of F1(z) = ∂z(z J1(zxn)

)by the zeros of Y1(z) and J0(z).

which, in the limit ε→ 0 is fulfilled by J1(xn) = 0, that allows us to recover themass spectrum of KK modes21

mn ≈ kπe−kL(n+

1

4

). (2.146)

From the orthonormality one gets

1 =1

L

ˆ L

0dy

1

|N (n)A |2

e2ky[J1

(mn

keky)

+ b(n)A Y1

(mn

keky)]2

=1

L

ˆ 1

εdt

1

kt

1

|N (n)A |2

t2

ε2

[J1(xnt)−

J1(xnε)

Y1(xnε)Y1(xnt)

]2,

that, for ε→ 0, gets

|N (n)A |2 ≈

1

ε2

1

kL

ˆ 1

εt∣∣J1(xnt)

∣∣2dt .By making use of the mass spectrum condition J1(xn) = 0, the right integral canbe written asˆ 1

εt∣∣J1(txn)

∣∣2dt =1

2xn

{xnJ

20 (xn)− xnε2J2

0 (xnε)− 2J0(xn)J1(xn)

+ xnJ21 (xn) + 2εJ0(xnε)J1(xnε)− xnε2J2

1 (xnε)}.

Let us exploit the asymptotic behavior Jν(x) ≈ xν , valid for small x, to rewrite

the N(n)A constant in power of ε:

|N (n)A |2 ≈

1

kL

J20 (xn)

2ε2+

1

2− xn

ε2

2−−−→ε → 0

1

kL

J20 (xn)

2 ε2,

21In recovering Eq. (2.146) we make use of the asymptotic property of the zeros of Bessel functionJν(xn): xn ≈ π(n− 1/4 + ν/2).


hence

N(n)A ≈ ekL√

2kL

∣∣J0(xn)∣∣ .

We may observe that it is possible to approximate the values taken by J0(z) inxn, with the values of the function

√2/πz−1/2 in the same points, Figure 2.7,

obtaining [34]

N(n)A ≈ ekL/2√

π Lmn. (2.147)

• BC (++). From the Neumann BC at the UV(t = ε) brane, ∂tf(n)A (ε) = 0, we get

b(n)A = −J1(mnk ) + mn

k J′1(mnk )

Y1(mnk ) + mnk Y

′1(mnk )

. (2.148)

If one imposes a Neumann condition also at the IR(t = 1) brane, in the limit ε→ 0

(so that b(n)A (xnε)→ 0) he obtains

∂t(t J1(txn)

)= 0 ,

further approximated by the request22

J0(xn) = 0 .

Using the approximation formula for the zeros of Jν(xn), xn ≈ π(n− 1/4 + ν/2),we recover the mass spectrum

xn ≈ π(n− 1

4

)=⇒ mn ≈ kπe−kL

(n− 1

4

). (2.149)

The normalization condition is

1 =1

kL

ˆ 1

εdt

1

|N (n)A |2

t

ε2

[J1(txn)− b(n)

A (xnε)Y1(txn)]2

≈ε → 0

1

|N (n)A |2

1

kL

1

ε2

ˆ 1

εt∣∣J1(txn)

∣∣2dt .The right integral turns to23

ˆ 1

εt∣∣J1(txn)

∣∣2dt =1

2xn

{xnJ

20 (xn)− ε2xnJ

20 (xnε)− 2J0(xn)J1(xn)

+ xnJ21 (xn) + 2εJ0(xnε)J1(xnε)− ε2xnJ

21 (xnε)

}≈

ε → 0

J21 (xn)

2. (2.150)

22See Figure 2.8.23We are still using the approximations for the zeros of both Y1(z) and J0(z), see Figure 2.8.


Table 2.2: Solutions of EOMs for a gauge field in the bulk, varying the BCs in the warped space-time.

BCs mass spectrum b(n)A N

(n)A zero mode

(−−) mn ≈ kπe−kL(n+ 1

4

)−J1(mn

k)

Y1(mnk

)≈ ekL/2√

π Lmn–

(++) mn ≈ kπe−kL(n− 1

4

)− J1(mn

k)+mn

kJ ′1(mn

k)

Y1(mnk

)+mnkY ′1(mn

k)≈ ekL/2√

π Lmnf

(0)A = 1

(+−) mn ≈ kπe−kL(n+ 1

4

)− J1(mn

k)+mn

kJ ′1(mn

k)

Y1(mnk

)+mnkY ′1(mn

k)≈ ekL/2√

π Lmn–

(−+) mn ≈ kπe−kL(n− 1

4

)−J1(mn

k)

Y1(mnk

)≈ ekL/2√

π Lmn–

Therefore, the normalization constant, for the present BCs, can be expressed by

ˆ 1

εt∣∣J1(txn)

∣∣2dt =1

2xn

{xnJ

20 (xn)− ε2xnJ

20 (xnε)− 2J0(xn)J1(xn)

+ xnJ21 (xn) + 2εJ0(xnε)J1(xnε)− ε2xnJ

21 (xnε)

}≈

ε → 0

J21 (xn)

2. (2.151)

• BC (+ –). Similarly to previous case, the Neumann BC at the UV brane

constrains the coefficient b(n)A to be

b(n)A = −J1(mnk ) + mn

k J′1(mnk )

Y1(mnk ) + mnk Y

′1(mnk )

.

With a Dirichlet BC at the IR brane, f(n)A (1) = 0, in the limit ε → 0, we get a

constraint on the mass spectrum of the KK modes that resembles the one obtainedin the (−−) case

J1(xn) = 0 =⇒ mn ≈ kπe−kL(n+

1

4

).

The normalization constant, in the limit ε→ 0, descends from the integral:

ˆ 1

εt∣∣J1(txn)

∣∣2dt =1

2xn

{xnJ

20 (xn)− xnε2J2

0 (xnε)− 2J0(xn)J1(xn)

+ xnJ21 (xn) + 2εJ0(xnε)J1(xnε)− xnε2J2

1 (xnε)},

obtaining

N(n)A ≈ ekL/2√

π Lmn.


• BC (– +).We have seen that, in the case of mixed BCs, the constant b(n)A is

determined by the constraint on the UV brane, while the mass spectrum and thenormalization constant descend from the constraint on the IR brane for ε → 0.Therefore we have:

b(n)A = −J1(mnk )

Y1(mnk ),

and

mn ≈ kπe−kL(n− 1

4

),

N(n)A ≈ ekL/2√

π Lmn.

In summary, the normalization constant is always the same regardless of the BCs,

while the constant b(n)A and the mass spectrum of KK modes depend on the influenced

by the BCs. These results are collected in Table 2.2.

Chapter 3

Flavour structure of extradimensional models

One of the most important achievement still needed in particle physics is the under-standing of the flavour problem. This knowledge will reasonably come from very shortdistance scales. The loop-induced or rare processes, like the ones driven by flavourchanging neutral currents, will play a crucial role. They can be mainly studied in B andK decays, however also D and hyperons can provide useful hints. In the SM, the FCNCprocesses are controlled by

• the unitary Cabibbo-Kobayashi-Maskawa (CKM) matrix [36, 37] that governs theweak charged current interactions of quarks;

• the Glashow-Iliopoulos-Maiani (GIM) mechanism [38], forbidding tree-level FCNC,and allowing them at the loop level, with a role for the CKM parameters;

• the asymptotic freedom of QCD [39], that allows to take into account the correctionson the weak decays due to strong interactions;

• the operator product expansion (OPE) [40], which separates in the effective theory,the long-distance contributions contained in the operator matrix elements fromthe short-distance physics encoded in the Wilson coefficients.

The present data on rare SM CP violating K and B decays basically show an astonishingagreement with the above picture. Nevertheless, only several FCNC processes have beenmeasured, therefore more efforts have to be done at the new facilities, both hadroncolliders and B factories with high luminosity, to improve our knowledge. At present,there are no strong indications that could avoid the discovery of exotic tree-level FCNCdecays or new sources of complex phases in CKM matrix; indeed, new physics could stillbe there, and we need more sensitivity in the measurements, or we should give a look tothose processes that did not draw enough attention in the past.

In this chapter, we explore the flavour structure of two different extra dimensionalscenarios. The first one, with universal extra dimensions, belongs to the class of minimal

43

44 Flavour structure of extra dimensional models

flavour violation models; the second one is the RSc model, and it generally predicts newFCNC transitions at tree-level and new sources of CP violation. We do not deal with thelatter issue: in our phenomenological analyses we take into account only CP conservingobservables in rare processes.

3.1 Minimal flavour violation and UED’s

In models with minimal flavour violation (MFV), the CKM matrix rules all flavourchanging transitions, with the CKM phase being the only source of CP violation, andthere are no FCNC processes at tree-level. Furthermore, the only operators in theeffective Hamiltonian below the weak scale are those appearing in the SM [3].

Models with universal extra dimensions, like the UED6, generally fit with the previousscenarios as they simply reproduce the flavour structure of the SM. In the following, themain Feynman rules for the SM fields and the KK modes, are described.

3.1.1 Leptons

Gauge sector

Let us consider in a 6D space-time the standard electroweak gauge structure described bythe group SU(2)L×U(1)Y ; the SU(2)L gauge fields are denoted by W a

α , α = 0, 1, . . . , 5,with the index a running on the adjoint representation of the group, while with Bα wedenote the U(1)Y gauge field. As a consequence of the compactification on the chiralsquare, these fields undergo a Kaluza-Klein decomposition from which a tower of vector

fields Wa(j,k)µ (B

(j,k)µ ), with µ = 0, 1, . . . , 3 and two scalar components W

(j,k)± (B

(j,k)± ),

emerge. To avoid the self-mixing among the vector gauge fields and the scalar ones, asuitable gauge-fixing term in the Lagrangian must be included:

LGF = − 1

2ξ[∂µW a

µ − ξ(∂4Wa4 + ∂5W

a5 − g6v6η

a)]2

− 1

2ξ′[∂µBµ − ξ′(∂4B4 + ∂5B5 + g′6v6η

3)]2 . (3.1)

g6 and g′6 are the 6D coupling constants, while ξ and ξ′ are the gauge-fixing parameters.The fields ηa are components of the 6D Higgs field

H =1√2

(η2 + iη1

v6 + h+ iη3

). (3.2)

The 6D coupling constants and vacuum expectation value (VEV) of the Higgs field arerelated to the effective 4D ones by

g(′)6 = g(′)πR,

v6 = πv

R, (3.3)

3.1 Minimal flavour violation and UED’s 45

with R the compactification radius of the extra dimensional manifold.The chiral fundamental leptons are embedded in opposite 6D chirality multiplets, in

order to provide a complete cancellations of anomalies [5]; a suitable choice is the “+”chirality for the left-handed doublet

Ψ+,L ≡(ψν+ψe+

)L ,

and the “−” one for the right-handed singlet

Ψ−,R ≡ ψe−,R.

We consider only left-handed neutrinos. The zero modes of the corresponding KK tower,for each field, are identified with the SM leptons, hence(

ψν(0,0)+L

ψe(0,0)+L

)≡(νeLeL

),

ψe(0,0)−R ≡ eR . (3.4)

These statements rule also the remaining generations.The gauge Lagrangian for the first generation of leptons is given by

Lgauge(x) = −1

2Tr(Wαβ(x)Wαβ(x)

)− 1

4Bαβ(x)Bαβ(x)+

+(ψν+,L(x) ψe+,L(x) ψe−,R(x)

)iΓαDα

ψν+,L(x)

ψe+,L(x)

ψe−,R(x)

, (3.5)

where the covariant derivative is

Dα =(∂α + ig6W

aα(x)T a + ig′6Bα(x)Y

), (3.6)

with Ta defined as

Ta =

(12τa 00 0

)and τa being the 2× 2 Pauli matrices. The gauge field strength Wαβ is defined as

Wαβ(x) = W aαβ(x)

τa2,

with W aαβ(x) = ∂αW

aβ (x) − ∂βW a

α(x) − g6εabcWbα(x)W c

β(x). Similarly, the U(1) gaugefield strength is given by Bαβ(x) = ∂αBβ(x) − ∂βBα(x) . Moreover, the spinless KK

components W(j,k)± undergo the redefinition already seen in Eq. (2.49):

W(j,k)± = rj,±k

(W

(j,k)H ∓ iW(j,k)

G

), (3.7)


with rj,±k = j+ik√j2+k2

. Analogously, one has:

B(j,k)± = rj,±k

(B

(j,k)H ∓ iB(j,k)

G

). (3.8)

From the Lagrangian (3.5), one can extracts the effective 4D kinetic term, substitutingthe KK expansion of each field and integrating over the extra dimension; as a result onehas:

Lkin (xν) = i(νeL(x) eL(x) eR(x)

)γµ(igW a(0,0)

µ T a + ig′B(0,0)µ Y

) νeL(x)eL(x)eR(x)

, (3.9)

where g and g′ are the 4D standard gauge coupling. Furthermore the hyperchargeoperator Y respects the Gell-Mann–Nishijima relation: Y = −T3 +Q, with Q being theelectric charge of the leptons.

The Lagrangian (3.5) allows us to infer the effective couplings between the left-handedlepton doublets and the KK spin-1 tower:∑

j1,k1

∑j2,k2

∑j3,k3

δ(j1,k1)(j2,k2)(j3,k3)0,0,0

(ψν+,L(x) ψe+,L(x)

)(j1,k1)γµ

×(−gW(j2,k2)

µ − g′yLB(j2,k2)µ

)(ψν+,L(x)

ψe+,L(x)

)(j3,k3)

, (3.10)

where we have used

δ(j1,k1)...(jr,kr)n1,...,nr ≡ 1

L2

ˆ L

0dx4

ˆ L

0dx5f (j1,k1)

n1. . . f (jr,kr)

nr . (3.11)

Similarly, one recovers the couplings of the right-handed singlet:∑j1,k1

∑j2,k2

∑j3,k3

δ(j1,k1)(j2,k2)(j3,k3)0,0,0 δ

(j1,k1)(j2,k2)(j3,k3)0,0,0 ψe−,R

(j1,k1)γµ(−g′yRB(j2,k2)

µ

)×(ψe−,R

)(j3,k3). (3.12)

The couplings with the spinless WH,G and BH,G read∑j1,k1

∑j2,k2

∑j3,k3

δ(j1,k1)(j2,k2)(j3,k3)0,0,0 δ

(j1,k1)(j2,k2)(j3,k3)0,0,0 r∗j2,k2

(ψν+,L(x) ψe+,L(x)

)(j1,k1)

×[+g(W

(j2,k2)H + iW

(j2,k2)G

)+ g′yL

(B

(j2,k2)H + iB

(j2,k2)G

)](ψν+,L(x)

ψe+,L(x)

)(j3,k3)

, (3.13)

and∑j1,k1

∑j2,k2

∑j3,k3

δ(j1,k1)(j2,k2)(j3,k3)0,0,0 δ

(j1,k1)(j2,k2)(j3,k3)0,0,0 ψe−,R

(j1,k1)

×[−g′yR

(B

(j2,k2)H + iB

(j2,k2)G

)] (ψe−,R

)(j3,k3), (3.14)


for the left-handed doublet and the right-handed singlet, respectively. Moreover, followingthe Gell-Mann–Nishijima relation, one deduces that yL = 1/2 and yR = −1.

It is worth observing that, analogously to the SM, the neutral gauge fields W 3µ and

Bµ undergo a linear mixing, from which one gets the physical Z and photon eigenstates.This mixing proceeds through the relations

Zµ =1√

g62 + g

′26

(g6W

3µ − g′6Bµ

),

Aµ =1√

g62 + g

′26

(g′6W

3µ + g6Bµ

), (3.15)

from which one recovers

Zµ = cos θWW3µ − sin θWBµ,

Aµ = sin θWW3µ + cos θWBµ , (3.16)

with

sin θW =g′6√

g26 + g

′26

=g′√

g2 + g′2,

cos θW =g6√

g26 + g

′26

=g√

g2 + g′2. (3.17)

Therefore, the effective Lagrangian describing the standard neutral and charged currents,between the zero modes, naturally emerges:

Lint = −e∑j1,k1

∑j2,k2

∑j3,k3

δ(j1,k1)(j2,k2)(j3,k3)0,0,0

{A(j2,k2)µ J (j1,k1)(j3,k3)µ

em

+1

sin θW cos θWZ(j2,k2)µ J (j1,k1)(j3,k3)µ

NC

+1√

2 sin θW

(W+(j2,k2)µ J (j1,k1)(j3,k3)µ

CC +W−(j2,k2)µ J †(j1,k1)(j3,k3)µ

CC

)}, (3.18)

where

J (j1,k1)(j3,k3)µem = Ψ(j1,k1)γµ

(T 3 + Y

)Ψ(j3,k3) ,

J (j1,k1)(j3,k3)µNC = Ψ(j1,k1)γµT 3Ψ(j3,k3) − sin2 θWJ (j1,k1)(j3,k3)µ

em ,

J (j1,k1)(j3,k3)µCC = Ψ(j1,k1)γµ

(T 1 + iT 2

)Ψ(j3,k3) . (3.19)

Higgs sector

The 6D action for the scalar Higgs field is

SΦ =

ˆd4x

ˆ L

0dx4

ˆ L

0dx5

[(DαΦ) † (DαΦ)−M2

ΦΦ†Φ− λ6

2

(Φ†Φ

)2], (3.20)


where λ6 is a parameter with mass dimension −2. With M2Φ < 0, a spontaneous

symmetry braking occurs, and the above Lagrangian can be written as

LΦ = |DαΦ|2 − λ6

2(Φ†Φ− 1

2v2

6)2 , (3.21)

providing the following prescription on the VEV quantity v6 > 0:

−M2Φ =

1

2λ6v

26 . (3.22)

The Higgs doublet introduced in (3.2) becomes(0

1√2(h(x) + v6)

), (3.23)

once the unitary gauge (ξ → ∞) is chosen. If one substitutes the VEV 〈0|Φ|0〉 =(0

1√2v6

)into the Lagrangian (3.21), the mass terms for the gauge fields

〈0|Φ† |0〉(−ig6W

aµ (x)

τa

2− ig′6Bµ(x)yH

)(ig6W

aµ (x)

τa

2+ ig′6Bµ(x)yH

)〈0|Φ |0〉

(3.24)

come out, and by the definitions Wµ± =W 1µ∓iW 2

µ√2

and the analogous mixing among the

W 3µ and Bµ in (3.16) one obtains:

(0 1√

2v6

)2g6g′6Aµ+

(g26−g

′26

)Zµ

2√g26+g

′26

g6√2W+µ

g6√2W−µ −

√g26+g

′26

2 Zµ

2g6g′6Aµ+(g26−g

′26

)Zµ

2√g26+g

′26

g6√2W+µ

g6√2W−µ −

√g26+g

′26

2 Zµ

(

01√2v6

). (3.25)

This reduces to

g26v

26

4W+µ W

−µ +

(g2

6 + g′26

)8

v26ZµZ

µ . (3.26)

By substituting the KK expansions for each field, and after the integration over thex4, x5 coordinates, we get the mass terms for the KK charged gauge bosons:

M(j,k)W± =

√m2

(j,k) +g2v2

4, (3.27)


where m(j,k) ≡√

j2+k2

R2 and the 4D VEV v and the effective couplings g(′) have been

defined in Eq. (3.3). The neutral bosons acquire masses

M(j,k)Z =

√m2

(j,k) +(g2 + g′2) v2

4. (3.28)

The KK excitations of the photon get their masses only from the compactification, andread:

M(j,k)A = m(j,k) =

√j2 + k2

R2. (3.29)

The KK spinless adjoints acquire mass terms in the same manner:

M(j,k)WH

=

√m2

(j,k) +g2v2

4, (3.30)

and

M(j,k)BH

=

√m2

(j,k) +g′2v2y2

H

4, (3.31)

with yH = 1/2 the hypercharge of the Higgs doublet. Naturally, the KK modes of theHiggs field are also massive, with

M(j,k)h =

√m2

(j,k) + (λv)2 , (3.32)

where λ = λ6/L2 is the effective 4D quartic coupling.

Yukawa couplings

The Yukawa terms for the leptons arise from the Higgs couplings with the matter fields,through the Lagrangian

LYuk = −ce6ψe−,R Φ†(ψν+ψe+

)L

+ h.c. (3.33)

ce6 is a 6D coupling constant (for the first generation), with mass dimension −1, whoseeffective 4D realization is ce = ce6/L. Inserting the KK Fourier decomposition into (3.33)and integrating over the extra coordinates, one gets

LYuk =∑j1,k1

∑j2,k2

∑j3,k3

δ(j1,k1)(j2,k2)(j3,k3)0,0,0

{ce√2vψe−,R

(j1,k1)h†(j2,k2)

vψe+,L +H.c.

}, (3.34)

and the masses of the KK leptons read

M(j,k)è

=

√m2

(j,k) + c2e

v2

2. (3.35)


3.1.2 Quarks

The quark fields are embedded in the fundamental SU(2)L × U(1)Y representations as

Q+,L ≡(Qu+Qd′+

)L

,

(Qc+Qs′+

)L

,

(Qt+Qb′+

)L

, (3.36)

U−,R ≡ Qu−,R,Qc−,R,Qt−,R ,D−,R ≡ Qd

′−,R,Qs

′−,R,Qb

′−,R . (3.37)

Since the flavour structure of the model strictly follows the SM one, we can anticipatethat the down quarks undergo the rotation induced by the standard CKM matrix. Manyparameterization of the CKM matrix have been proposed in the literature. The standardparameterization is given by [41]:

VCKM =

c12c13 s12c13 s13e−iδ

−s12c23−c12s23eiδ c12c23 − s12s23s13e

iδ s23c13

s12s23 − c12c23s13eiδ −s23c12 − s12c23s13e

iδ c23c13

, (3.38)

where ci,j = cos θi,j and si,j = sin θi,j (i, j = 1, 2, 3) and the complex phase δ is necessaryfor CP violation. We denote the rotated down quark fields, in the so-called “mass basis”,with a primed index: Qd′ , Qs′ , Qb′ .

Therefore, the charged current terms in the effective Lagrangian are:

Lcharge = −∑j1,k1

∑j2,k2

∑j3,k3

δ(j1,k1)(j2,k2)(j3,k3)0,0,0

g√2

(J (j1,k1)(j3,k3)hµ W (j2,k2)+

µ

), (3.39)

with

J (j1,k1)(j3,k3)hµ =

(Qu+,L Qc+,L Qt+,L

)(j1,k1)γµ

Qd′

+,L

Qs′+,LQb′+,L

(j3,k3)

(3.40)

the hadronic weak current.Moreover, the effective Yukawa Lagrangian is given by

LYuk = −δ(j1,k1)(j2,k2)(j3,k3)0,0,0

v + h(j2,k2)(x)√2

×

( Qd′−,R Qs′−,R Qb′−,R ) (j1,k1)VCKM

cd 0 00 cs 00 0 cb

V †CKM

Qd′

+,L

Qs′+,LQb′+,L

(j3,k3)

+(Qu−,R Qc−,R Qt−,R

)(j1,k1)

cu 0 00 cc 00 0 ct

Qu+,LQc+,LQt+,L

(j3,k3) + h.c.

, (3.41)

where the Higgs VEV has already been substituted.

3.2 Flavour in the RSc model 51

The terms proportional to the VEV v, in the above Lagrangian, give us the quarkmass terms,

Lmass = −δ(j1,k1)(j2,k2)(j3,k3)0,0,0

×

( Qd′−,R Qs′−,R Qb′−,R ) (j1,k1)VCKM

md 0 00 ms 00 0 mb

V †CKM

Qd′

+,L

Qs′+,LQb′+,L

(j3,k3)

+(Qu−,R Qc−,R Qt−,R

)(j1,k1)

mu 0 00 mc 00 0 mt

Qu+,LQc+,LQt+,L

(j3,k3) +H.c.

, (3.42)

where we setmi =

civ√2. (3.43)

Besides the effective Yukawa couplings and the couplings with the spin-1 gauge fields,there exist also effective couplings among the quarks and the spinless adjoint fields,which can be directly read from the analogous ones already explored in the leptonicsector.

3.2 Flavour in the RSc model

Several variants of the RS model have been proposed, each one adding new featuresto those of the original model. Here, we consider the scenario in which the SM gaugesymmetry group is enlarged to the gauge group

SU(3)c × SU(2)L × SU(2)R × U(1)X × PL,R (3.44)

which, together with the warped metric, defines the Randall-Sundrum model withcustodial protection RSc [11,42,43]. The custodial protection is realized imposing thePL,R symmetry, which implies a mirror action of the two SU(2)L,R groups, preventinglarge Z couplings to left-handed fermions that would be incompatible with experiment.This variant has been proven to be consistent with electroweak precision observables formasses of the lightest KK excitations of the order of a few TeV [44,45], in the reach ofthe LHC.

Two symmetry breakings occur (see Figure 3.1): first, the gauge group (3.44) isbroken to the SM gauge group imposing suitable boundary conditions (BC) on the UVbrane. Afterwards, the spontaneous symmetry breaking occurs, which is Higgs-driven asin SM. All the SM fields are allowed to propagate in the bulk, except for the Higgs fieldwhich is localized close to the IR brane. Here, we consider the case of a Higgs bosoncompletely localized at y = L. The extension of the SM leads to the presence of newparticles, as a consequence of the requirements/assumptions listed below.

• The two SU(2) groups require a larger number of gauge bosons. Those corre-sponding to SU(2)L are W a,µ

L (a = 1, 2, 3), while W a,µR correspond to SU(2)R.


Figure 3.1: Gauge structure of the RSc model.

The gauge conditions W a,5L,R = 0 and ∂µW

a,µL,R = 0 are chosen, as well as for all the

other gauge bosons. The PL,R symmetry imposes the equality gL = gR = g for theSU(2)L,R gauge couplings.

The number of remaining gauge bosons is the same as in SM. In particular, theeight gauge fields corresponding to SU(3)c are still identified with the gluons, whilethe U(1)X gauge field is denoted as Xµ, with coupling gX . The 5D couplings aredimensionful: the relations to their 4D counterparts are g4D = g5D/

√L. We shall

describe below the mixing pattern among the various gauge fields.

• Fitting matter fields in suitable representations of the group (3.44) leads to newfermions, as discussed in the following.

• The presence of a compact fifth dimension implies the existence of a tower ofKaluza-Klein (KK) excitations for all particles. As generically done in extra-dimensional models, the boundary conditions help to distinguish particles havinga SM correspondent from those without SM partners, by requiring the existenceor not of a zero mode in the KK mode expansion of a given field. Two choices forBC are considered: Neumann BC on both branes (++), or Dirichlet BC on theUV brane and Neumann BC on the IR one (−+). Only fields with (++) BC havea zero mode which can be identified with a SM particle.

For each one of the fields listed above we perform a KK decomposition of the genericform:

F (x, y) =1√L

∑k

F (k)(x)f (k)(y) , (3.45)

referring to the functions f (k)(y) (specific for each field F ) as the 5D field profiles, whileF (k)(x) are the corresponding effective 4D fields. Then, we consider the 5D Lagrangiandensities for the various fields, and solve the resulting 5D EOMs to obtain the variousprofiles. Following the strategy outlined in [46], this can be done before the EWSB.After such a symmetry breaking takes place, one can treat the ratio v/MKK of the HiggsVEV v and the mass of the lowest KK modeMKK as a perturbation. The effective 4DLagrangian is obtained after integration over y, and the Feynman rules of the model


are worked out neglecting terms of O(v2/M2KK) or higher. On the same footing, the

mixing occurring between SM fermions and higher KK fermion modes is neglected, sinceit leads to O(v2/M2

KK) modifications of the relevant couplings.

As for gauge bosons, we consider KK modes up to the first excitation (1-mode).Indeed, as observed in [47], the model becomes non perturbative for scales correspondingto the first few KK modes, so that including the whole tower of excitations would leadto unreliable results.

Let us now examine the various sectors of the model, stressing the most relevantfeatures for our analysis.

3.2.1 Gauge sector

The 5D free Lagrangian for each gauge boson reads

Lgauge = −1

4FMNF

MN , (3.46)

where FMN is the 5D field strength. From (3.46) the equation of motion for Vµ can be

obtained. The solution provides us with the bulk profiles of each KK mode, f(n)V (y),

which are different if the (++) or (−+) BCs are imposed, but do not depend on thespecific boson. Here we remind that

• profiles of zero-modes are flat, f(0)V (y) = 1 ,

• the profile of the first KK excitation of a gauge boson has been derived in Eq.(2.143). The solution with a zero mode, denoted by g(y), is determined by theconstants

b1(m1) = −J1

(m1k

)+ m1

k J′1

(m1k

)Y1

(m1k

)+ m1

k Y′

1

(m1k

) , (3.47)

N1 =ekL/2√πLm1

, (3.48)

• similarly, the solution without a zero mode, denoted by g(y), has

b1(m1) = −J1

(m1k

)Y1

(m1k

) . (3.49)

Furthermore, the solution of the equation of motion shows that such masses are

m1 ≡M++ ' 2.45f

and

m1 ≡M−+ ' 2.40f ,


where the dimensionful parameter f is defined as f = k e−kL. We set the scale of newphysics f to 1 TeV in our numerical analyses, consistent with similar analyses in theflavour sector [47,48]. Charged gauge bosons are defined in analogy to SM,

W±L(R)µ =W 1L(R)µ ∓ iW 2

L(R)µ√2

. (3.50)

Mixing occurs between the bosons W 3R and X with a mixing angle φ. The resulting

fields are denoted as ZX and B:

ZXµ = cφW3Rµ − sφXµ ,

Bµ = sφW3Rµ + cφXµ , (3.51)

where

cφ = cosφ =g√

g2 + g2X

,

sφ = sinφ =gX√g2 + g2

X

. (3.52)

In a second step, W 3L mixes with B with an angle ψ, in complete analogy to SM,

providing the Z and A fields:

Zµ = cψW3Lµ − sψ Bµ ,

Aµ = sψW3Lµ + cψ Bµ , (3.53)

with

cψ = cosψ =1√

1 + s2φ

,

sψ = sinψ =sφ√

1 + s2φ

. (3.54)

At the end of the mixing pattern (leaving aside the eight gluons Gµ with BC (++)), weare left with

• four charged bosons: W±L (++) and W±R (−+);

• three neutral bosons: A(++), Z(++) and ZX(−+).

We have specified the BC for these fields. For each vector boson field Vµ(x, y) the KKexpansion is

Vµ(x, y) =1√L

∞∑n=0

V (n)µ (x)f

(n)V (y) . (3.55)


3.2.2 Fermionic sector

Quarks

We still refer to Ref. [46] (and references therein) for the realization of the fermionic sector.It has been observed that, in order to preserve the O(4) ∼ SU(2)L × SU(2)R × PLRsymmetry necessary to avoid large corrections to EW precision observables, a particularset of representations of O(4) must be chosen. Hence, the fundamental quarks areembedded in three O(4) multiplets per each generation i = 1, 2, 3:

ξi1L =

(χuiL,5/3 quiL,2/3χdiL,2/3 q

diL,−1/3

)2/3

, (3.56)

ξi2R = uiR,2/3 , (3.57)

ξi3R = T i3R ⊕ T i4R =

ψ′iR,5/3U ′iR,2/3D′iR,−1/3

2/3

⊕

ψ′′iR,5/3U ′′iR,2/3DiR,−1/3

2/3

. (3.58)

Moreover, the existence of analogous three states per generations with opposite chirality,is easily deducible. The following comments can be made:

• Left-handed doublets are in a bidoublet of SU(2)L × SU(2)R, together with twonew fermions;

• Right-handed up-type quarks are singlets;

• Right-handed down-type quarks are in multiplets that transform as (3, 1)⊕ (1, 3)under SU(2)L × SU(2)R; the multiplets contain additional new fermions;

• The electric charge reads, in terms of the third component of the SU(2)L andSU(2)R isospins and of the charge QX : Q = T 3

L + T 3R +QX .

The 5D Lagrangian in the hadron sector is given by:

Lfermion =1√2

√G

3∑i=1

[(ξi1)aαiΓ

M (D1M )ab,αβ(ξi1)bβ + (ξi1)aα(iΓMωM − ciQk)(ξi1)aα

+ ξi2(iΓMD2M + iΓMωM − ciuk)ξi2

+ (T i3)aiΓM (D3

M )ab(Ti3)b + (T i3)a(iΓ

MωM − cidk)(T i3)a

+ (T i4)αiΓM (D4

M )αβ(T i4)b + (T i4)α(iΓMωM − cidk)(T i4)α]

+ h.c.

(3.59)

More about the Dirac gamma matrices ΓM and the spin connection ωM in curvedspace-time has been derived in Chapter 2. It is worth observing that the terms including


the spin connection ωM cancel out by making the “+h.c.” terms explicit. The covariantderivatives Di

M in (3.59) are

(D1M )ab,αβ = (∂M + igst

AGAM + igXQXXM )δabδαβ

+ig(τ c)abWcL,Mδαβ + ig(τγ)αβW

γR,Mδab , (3.60)

D2M = ∂M + igst

AGAM + igXQXXM , (3.61)

(D3M )ab = (∂M + igst

AGAM + igXQXXM )δab + gεabcW cL,M , (3.62)

(D4M )αβ = (∂M + igst

AGAM + igXQXXM )δαβ + gεαβγW γR,M . (3.63)

tA = λA/2 (A = 1, . . . , 8) are the SU(3)c generators, τa = σa/2 (τα = σα/2) are thegenerators of the fundamental SU(2)L (SU(2)R), respectively, with −iεabc (−iεαβγ)being the generators of the adjoint representations of SU(2)L (SU(2)R), respectively.The SU(2)L and SU(2)R generators act on different internal spaces, in spite of the samematrix structure.

Leptons

For the sake of simplicity the leptons can be embedded in the same representationsof the quarks, with a slightly different notation to discern the fields, even if otherimplementations are allowed and in agreement with the data. Few comments are inorder:

• leptons transform as singlets under SU(3)c, therefore the coupling to gluons in(3.60–3.63) has to be removed,

• QX = 0 in order to get the correct electric charge, hence also the coupling to theX boson is denied in (3.60–3.63),

• The right-handed neutrinos νiR are introduced as complete gauge singlets, hencetheir couplings to neutral gauge bosons are forbidden.

The implementation of a PL,R symmetry in the leptonic sector prevents the couplingsZìL

¯iL and ZνiR

νiR from large corrections, and this has non trivial phenomenologicalconsequences.

Since we consider only the zero-modes of the SM quarks and leptons in our phenomeno-logical analyses, we do not elaborate on the exotic fermions. Solving the EOMs for

ordinary fermions leads to their zero-mode profiles, denoted as f(0)L,R(y, c) and given by

f (0)(y, c) =

√(1− 2c)kL

e(1−2c)kL − 1e−cky . (3.64)

In principle, right and left-handed fermions are treated as distinct fields. The onlydifference among the fermions resides in the parameter c, identified with the fermionmass in the bulk. c is the same for fields belonging to the same SU(2)L × SU(2)Rmultiplet: This is the case of uL and dL, cL and sL, tL and bL, as well as for ν` and `−L(` = e, µ, τ). All the c parameters are chosen real.


3.2.3 Higgs sector

After the gauge group (3.44) has undergone the breaking to the SM SU(3)c × SU(2)L ×U(1)Y , the electroweak symmetry breaking takes place. A Higgs field H(x, y) is intro-duced, which transforms as a bidoublet under SU(2)L × SU(2)R and as a singlet underU(1)X . This field contains two charged and two neutral components:

H(x, y) =

(π+√

2−h0−iπ0

2h0+iπ0

2π−√

2

). (3.65)

Performing the KK decomposition, one writes

H(x, y) =1√L

∑k

H(k)(x)h(k)(y) . (3.66)

For the Higgs localized on the IR brane the choice

h(y) ≡ h(0)(y) ' ekLδ(y − L) (3.67)

is done. As for the components depending on the 4D coordinates, one chooses that onlythe neutral field h0 has a non vanishing vacuum expectation value coinciding with theSM Higgs VEV v = 246.22 GeV. The 5D Lagrangian involving the Higgs field reads:

LHiggs =√GTr

[[DMH(x, y)]†[DMH(x, y)]− V (H)

], (3.68)

with√G = det[gMN ] = e−4ky, gMN being the 5D metric tensor and M,N = 0, 1, 2, 3, 5.

We do not specify the potential V (H) since it is irrelevant for our present purposes. Thecovariant derivative

(DµH)aα = ∂µHaα + ig(τ c)abWcL,µHbα + ig(τγ)αβW

γRµHαβ , (3.69)

with a and α running on the SU(2)L and SU(2)R generators respectively, involves thegauge bosons of the group (3.44) and is the starting point to give mass to a number ofthem.

3.2.4 Yukawa sector

The most general Yukawa Lagrangian which includes the Higgs bidoublet H and thequark fields ξi1,2,3 reads [46]:

LYuk =√G

3∑i,j=1

[√2λuij(ξ

i1)aαHaαξ

j2

− 2λdij [(ξi1)aα(τ c)ab(T

j3 )cHbα + (ξi1)aα(τγ)αβ(T j4 )γHaβ] + h.c.

]. (3.70)

The 5D couplings λu,dij are dimensional, with mass dimension −1.


3.2.5 EWSB pattern

Before the EWSB the zero modes of the gauge fields (when present) are massless, whilehigher KK modes are massive. The Higgs mechanism occurs to partially break thesymmetry. Since the QCD group SU(3) remains unbroken, as well as U(1)em, as inthe SM gluons and photon do not get mass. This means that their zero modes remainmassless, while higher KK modes are massive but they do not get a mass enhancementfrom the Higgs mechanism. For the remaining fields, mass is acquired and depends onthe Higgs VEV. Furthermore, mixing among zero modes and higher KK modes occurs.Neglecting modes with KK number larger than 1, the mixing involves

• the charged bosons W±(0)L , W

±(1)L and W

±(1)R , with the result W±

W±HW ′±

= GW

W±(0)L

W±(1)L

W±(1)R

; (3.71)

• the neutral bosons Z(0), Z(1) and Z(1)X , giving the mass eigenstates as follows: Z

ZHZ ′

= GZ

Z(0)

Z(1)

Z(1)X

. (3.72)

The expression of the mixing matrices GW and GZ , as well as of the masses of the masseigenstates, can be found in Ref. [46].

An important issue concerns the quark mass eigenstates. As in SM, they are obtainedupon rotation of the flavour eigenstates. We adopt the notation UL(R), DL(R) for therotation matrices of the up-type left (right) and down-type left (right) quarks, respectively.The relation

VCKM = U†LDLholds. However, while in the SM the CKM matrix only enters in charged currentinteractions, here the rotation matrices also modify the neutral currents. This happensbecause the integration over the fifth coordinate in the action leads to factors representingoverlap integrals of the profiles of two fermions fi and fj and a gauge boson profile.These integrals are of two kinds:

Rfifj =1

L

ˆ L

0dy eky f

(0)fi

(y, ci) f(0)fj

(y, cj) g(y)

Rfifj =1

L

ˆ L

0dy eky f

(0)fi

(y, ci) f(0)fj

(y, cj) g(y) . (3.73)

Before EWSB the interaction is flavour diagonal, so that the overlap integrals can be col-

lected in two matricesRf = diag (Rf1f1 ,Rf2f2 ,Rf3f3) and Rf = diag(Rf1f1 , Rf2f2 , Rf3f3

).


After the rotation to mass eigenstates, one is left with a typical productM†RfM, whereM = UL,R,DL,R, so that one can no more exploit the relation: M†M = 1 and FCNC areinduced already at tree level. They are mediated by the threee neutral EW gauge bosonsZ, Z ′, ZH as well as by the first KK mode of the photon and of the gluon, although thelatter does not contribute to processes with leptons in the final state. For this reason,the quark rotation matrices appear in the Feynman rules of this theory: we collect inthe Appendix A.2 a few rules needed in our analysis.

The expression of the elements of the rotation matrices are required. We refer to [47]for the list of the various entries. Here we only mention that they all are written interms of the quark profiles and of the 5D Yukawa couplings λu,dij for up-type (down-type)quarks, respectively. On the other hand, the effective 4D Yukawa couplings can bedefined as:

Yu(d)ij =

1√2

1

L3/2

ˆ L

0dy λ

u(d)ij f

(0)

qiL(y)f

(0)

ujR(djR)(y)h(y) . (3.74)

Since the fermion profiles depend exponentially on the bulk mass parameters (seeAppendix A.1), one identifies in the above relation the origin of the hierarchy of fermionmasses and mixing [19, 20]. The assumption of “flavour anarchy” means that the 5DYukawa matrices are generic matrices with all entries of the same order, i.e. with nospecial flavour structure. The hierarchal structure of the observed fermion masses isgeometrically generated. Thus, the first two generation quarks, plus the RH bottom,are localized close to the UV brane, where the overlap with the Higgs field is small.Instead, the top quark is localized close to IR brane, with a larger overlap with the Higgsprofile, so that the top quark mass is naturally of the order of the Higgs VEV. One mayargue that tree-level FCNC effects could easily emerge within this picture, since thelight quarks are not localized in exactly the same way. Therefore, a quite dangerousflavour-dependence in the couplings of fermions to KK gluons could naturally arise, andbreak the current bounds on the K − K mixing observables. However due to the factthat the KK gluon, along with the other gauge bosons, has an almost flat profile, theeffective coupling with fermions has a mild c−dependence. This leads to suppressedflavour-changing KK-gluon vertices, an effect also known as RS-GIM mechanism. Whatoutlined above suggests that RS has the potentiality to be an actual “theory of flavour”,and yet the scenario can roughly fit the actual FCNC bounds. Strictly speaking, RSccould not considered a Minimal Flavour Violation scenario since new FCNC effectsalready at the tree-level as like as new CP violating phases have to be suppressed insome way [49].

As for the matrices UL(R) and DL(R), not all their elements are independent, since

the Yukawa couplings determine the quark masses and since the product VCKM = U†LDL


should be satisfied, as already mentioned. In particular, the following relations hold [46]:

mu =v√2

det(λu)

λu33λu22 − λu23λ

u32

ekL

LfuLfuR ,

mc =v√2

λu33λu22 − λu23λ

u32

λu33

ekL

LfcLfcR , (3.75)

mt =v√2λu33

ekL

LftLftR ,

as well as the analogous relations for down-type quarks with the substitution λu → λd.

We have adopted the short notation: fqL,R = f(0)qL,R(y = L, cqL,R).

Parameter counting

To understand how many among the remaining entries should be considered as indepen-dent ones, we adopt further simplifications. In particular, the entries of the matrices λu,d

are treated as real numbers, since the effects that we are interested to investigate involveCP conserving observables, hence they do not require the introduction of new phasesbesides those present in SM. Therefore, after imposing the quark mass constraints, weare left with six independent entries among the elements of the Yukawa matrices, thatwe choose to be1

λu12 , λu13 , λu23 ,

λd12 , λd13 , λd23 . (3.76)

Together with the bulk mass parameters, these constitute the set of numerical input inour study. The way we treat them is described in Section 5.3.3 devoted to the numericalanalysis.

1A parametrization of the matrices λu,d that considers complex entries can be found in [47].

Chapter 4

Impact of Universal ExtraDimensions on radiative b→ sγdecays

In this and in the next chapters we shall discuss radiative penguin B meson decaysinduced by b → sγ, and electroweak penguin B meson decays induced by b → s`+`−

and b→ sνν transitions. These FCNC B decay modes are among the most interestingchannels for the searches of physics beyond the SM, because they occur at loop levelin the SM and their rates can be accurately predicted. In the SM the decays proceedat lowest order through penguin loop and box diagrams involving heavy virtual topquarks and weak W or Z bosons. Beyond SM, these could also contain additionalheavy particles, e.g. the Kaluza-Klein modes usually predicted in extra-dimensionalscenarios. At the B Factories many of these decays have been investigated. Inclusive andexclusive branching fractions have been accurately determined for b→ sγ transitions,and measured for the first time for b→ s`+`−, so that a phenomenological analysis ofdeviations from the SM can be already carried out.

An inclusive decay is denoted, for example, as B → Xsγ, where Xs is the sum of thehadronic final states formed by the recoiling s quark from b→ sγ and the spectator uor d quark, whereas an exclusive decay specifies the final state hadron(s), for example,B → K∗(892)γ. For b→ s`+`−, the decay amplitude depends on q2, the invariant masssquared of the leptonic pair. Angular analyses, which are sensitive to the interferencebetween different terms in the decay amplitudes, can be carried out as function of q2.

On the theoretical side, the SM predictions are at a similar level of accuracy to theexperimental precision for the inclusive branching fractions; this is due, in addition tothe presence of leptons and photons in the final state, to the possibility of exploitingthe infinite heavy quark limit, and to perform a QCD calculation in terms of quarksand gluons. The SM theoretical prediction for the exclusive branching fractions suffer oflarger uncertainties due to hadronic form factors, which take into account the fact thatquarks and gluons are bound in a hadronic state.

61

62 Impact of Universal Extra Dimensions on radiative b→ sγ decays

4.1 Effective theories for rare B decays

In rare B decays an interplay takes place between weak and strong interactions, andQCD effects both at short- and long-distances modify the amplitudes of quark flavourprocesses. These interactions need to be accurately computed in order to pin down theparameters and mechanisms of quark flavour physics from the weak decays of hadronsobserved in experiments.

The standard framework is provided by the effective weak Hamiltonian

Heff ∼∑i

Ci(µ)Qi(µ) , (4.1)

in which, following the method of the operator product expansion (OPE) [40] we factorizetwo distinct parts: the long-distance contributions are contained in the operator matrixelements (Qi) and the short-distance physics is encoded in the Wilson coefficients (Ci).The operators Qi contain the relevant degrees of freedom below the scale µ. The effectivecouplings for these operators are given by the Wilson coefficients, which are obtained byintegrating out all the fields above the scale µ, considering them as dynamical degrees offreedom [50].

4.1.1 B → Xsγ decays

The flavour structure of the vertex b→ sγ in the SM forbids the process at tree-level.However, with the help of the flavour changing W± vertex one can build 1-loop andhigher order amplitudes for this process. Moreover, the fact that it occurs at loop levelis the main reason for its crucial role in the search of particles beyond the SM.

In the case of b→ sγ the 1-loop process can be described by effective triple verticesknown as “penguin” diagrams, as shown in Figure 4.1. It is worth noticing that diagramswith fictitious Higgs exchanges in place of W± must also to be taken into account. Alsoself-energy corrections on external lines have to be included to make the effective verticesfinite [50]. With the help of the elementary vertices and propagator, one can derive theFeynman rules for the effective vertices shown in Figure 4.2. In the case of B → Xsγ(g)we need magnetic-vertices with on-shell photons or gluons. In this case, it is essential tokeep the mass of the external b-quark, as otherwise the corresponding vertices wouldvanish. The rules for the effective bγs and bGas vertices in the ’t Hooft-Feynman gaugefor the W± propagator are the following:

sγb∣∣∣effective vertex

≡ i V ∗isVibGF√

2

e

8π2D′0(xi)s[iσµνq

νmb(1 + γ5)b]b (4.2)

sGab∣∣∣effective vertex

≡ i V ∗isVibGF√

2

gs8π2

E′0(xi)sα[iσµνqνmb(1 + γ5)b]T aαβbβ , (4.3)

with xi = m2i /M

2W (i = u, c, t) and the sum over the internal quarks is implied; qµ is

the outgoing gluon or photon momentum and ms has been set to zero. The function D′0and E′0 have been calculated by several authors, in particular by Inami and Lim [51].

4.1 Effective theories for rare B decays 63

b s

W

t t

γ

b s

t

W W

γ

Figure 4.1: 1-loop FCNC diagrams giving rise to the “magnetic-penguin” operators.

The subscript “0” indicates that these functions do not include QCD corrections to therelevant penguin diagrams. They are given explicitly as follows:

E′0 =3x3

i − 2x2i

2(xi − 1)4lnxi +

−8x3i − 5x2

i + 7xi12(xi − 1)3

, (4.4)

D′0 =−3x2

i

2(xi − 1)4lnxi +

−x3i + 5x2

i + 2xi4(xi − 1)3

. (4.5)

Taking into account the unitary of the CKM matrix and setting mu = 0, a sum onlyover c and t quarks must be understood in the previous relations.

The effective Hamiltonian for b→ sγ

The effective Hamiltonian for the b → sγ transitions at the scale µ = O(mb) can bewritten as:

Heff(b→ sγ) = −GF√2V ∗tsVtb

[6∑i=1

Ci(µ)Qi(µ) +C7γ(µ)Q7γ(µ) +C8G(µ)Q8G(µ)

], (4.6)

where we have neglected terms proportional to V ∗usVub, as |V ∗usVub/V ∗tsVtb| < 0.02. GF isthe Fermi constant. The list of operators is the following:

Q1 = (sicj)V−A(cjbi)V−A , (4.7a)

Q2 = (sc)V−A(cb)V−A , (4.7b)

Q3 = (sb)V−A∑q

(qq)V−A, (4.7c)

Q4 = (sibj)V−A∑q

(qjqi)V−A , (4.7d)

Q5 = (sb)V−A∑q

(qq)V+A , (4.7e)

Q6 = (sibj)V−A∑q

(qjqi)V+A , (4.7f)


Q7γ =e

8π2mbsiσ

µν(1 + γ5)biFµν , (4.7g)

Q8G =g

8π2mbsiσ

µν(1 + γ5)T aijbjGaµν , (4.7h)

with (V ∓A) we refer to the Lorentz structure γµ(1∓ γ5), with i and j we denote thecolor indices, T ai,j are SU(3) generators and Fµν (Gaµν) is electromagnetic (gluonic) fieldstrength. With mb we indicate the mass of quark b. The operators refer to diagramsin Figure 4.2; they are classified according with the following definitions: Q1 and Q2

are “current-current” operators, Q3...6 are called “QCD penguin”, while Q7γ and Q8G

are the known “magnetic-penguin” operators. To derive the contribution of Q7γ to theHamiltonian (4.6), in absence of the QCD corrections, one multiplies the vertex in Eq.(4.2) by “i” and makes the replacement

2iσµν → −σµνFµν .

Analogous procedure gives the contribution of Q8G .

The magnetic γ-penguin plays the most important role. Nevertheless, contributionsfrom “current-current” operator Q2 could also be significant. In fact, the short-distanceQCD effects, involving in particular the mixing between Q2 and Q7γ , are crucial in thisdecay. In the decay B → Xsγ these effects, induced by hard-gluon exchange betweenthe quark lines of the 1-loop electroweak diagrams, lead to a rate enhancement by afactor larger than two [50].

The Renormalization Group Analysis (RGA) involves also the operators Q1, . . . ,Q6

beside the magnetic-penguins Q7γ and Q8G. A non-trivial achievement of the RGA isthat the mixing under renormalization between the set (Q1, . . . ,Q6) and the operators(Q7γ , Q8G) vanishes at the 1-loop level. Therefore, in order to calculate the coefficientsQ7γ(µb) and Q8G(µb) in the leading logarithms approximation, two loop calculation ofO(eg2

s) and O(g3s) are necessary.

The effective field theory paradigm means that the NP effects beyond SM correspondto specific modifications of the Wilson coefficients, or to the appearance of new operatorsin the Hamiltonian. The constraints on the Wilson coefficients reflect to constraintson the NP models. For instance, the measurements of the FCNC b → sγ branchingfraction [41] severely constrain the flavour structure of many NP scenarios. Moreover, inthe SM, the FCNC decays are suppressed by the elements of the CKM matrix and bythe electroweak scale, e.g by a factor GFV

∗tsVtb/M

2W . There are no particular reasons

to impose an analogous Cabibbo suppression in NP models, nevertheless experimentalevidences [52] seem to privilege models with MFV. As we have already seen, in thesemodels the flavour structure resembles the SM one, where the FCNC suppression comesout from the CKM pattern.

Wilson coefficients beyond leading logarithms

It is convenient at the leading order (LO), to introduce the so-called “effective coefficients”,developed by Buras, Misiak, Munz and Pokorski (1994) [53], for the operatorsQ7γ and

4.1 Effective theories for rare B decays 65

W

u s

d u

W

u s

d u

g

(a)

d s

W

u, c, t u, c, t

g

q q

(b)

b s

W

t t

g, γ

(c)

Figure 4.2: Typical diagrams that give rise to operators in Eqs. (4.7a)–(4.7h). The diagrams onthe top-left give rise to the so-called “current-current” operators, the “QCD penguins” originate fromdiagrams like the one on the top-right. The diagrams on the bottom generate the “magnetic-penguin”operators.

Q8G, which are independent of the regularization scheme:

C(0)eff7γ (µ) = C

(0)7γ (µ) +

6∑i=1

yiC(0)i (µ) , (4.8)

C(0)eff8G (µ) = C

(0)8G(µ) +

6∑i=1

ziC(0)i (µ) . (4.9)

Let us introduce the scheme-independent vector

~C(0)eff(µ) = (C(0)1 , . . . , C

(0)6 (µ), C

(0)eff7γ (µ), C

(0)eff8G (µ)) . (4.10)

From the renormalization-group equations (RGE) for ~C(0)(µ) one can derive the RGEfor ~C(0)(µ), which are given by

µd

dµC

(0)effi (µ) =

αs4πγ

(0)effji C

(0)effj (µ) , (4.11)

where

γ(0)effji =

γ

(0)j7 +

∑6k=1 ykγ

(0)jk − yjγ

(0)77 − zjγ

(0)87 , i = 7, j = 1, . . . , 6

γ(0)j8 +

∑6k=1 zkγ

(0)jk − zjγ

(0)88 , i = 8, j = 1, . . . , 6

γ(0)ji , othewise.

(4.12)


The matrix γ(0)eff is a scheme-independent quantity.The coefficients Ci(µ) in Eq. (4.6) are evaluated using

~C(µ) = U5(µ,MW )~C(MW ) . (4.13)

U5(µ,MW ) is a 8× 8 evolution matrix, with γ the 8× 8 anomalous-dimension matrix 1.In the LO, the initial conditions ~C(0)(MW ) are [54]:

C(0)2 (MW ) = 1 , (4.14)

C(0)7γ (MW ) =

3x3t − 2x2

t

4(xt − 1)4lnxt +

−8x3t − 5x2

t + 7xt24(xt − 1)3

≡ −1

2D′0(xt) , (4.15)

C(0)8G(MW ) =

−3x2t

4(xt − 1)4lnxt +

−x3t + 5x2

t + 2xt8(xt − 1)3

≡ −1

2E′0(xt) , (4.16)

with all remaining coefficients being zero at µ = MW and with xt =m2t

M2W

.

The leading order results for the Wilson coefficients of all operators entering theeffective Hamiltonian (4.6) can be written in analytic form as follows [54]:

C(0)j (µ) =

8∑i=1

kjiηai (j = 1, . . . , 6) , (4.17)

C(0)eff7γ (µ) = η16/23C0

7γ(MW ) +8

3(η14/23 − η16/23)C

(0)8G(MW ) + C

(0)2 (MW )

8∑i=1

hiηai ,

(4.18)

C(0)eff8G (µ) = η14/23C

(0)8G(MW ) + C

(0)2 (MW )

8∑i=1

hiηai , (4.19)

with η = αs(MW )αs(µ) , and C

(0)7γ (MW ) and C

(0)8G(MW ) given respectively by Eq. (4.15) and

Eq. (4.16). The numerical inputs for ai, kji, hi, and hi can be found in Table 4.1.Let us conclude observing that, to evaluate the Wilson coefficients for the decay

B → Xsγ in the LO, the LO approximation for αs should be used. Following Ref. [53],we set αs(MZ) = 0.1184, but we let αµ evolve to µ ≈ O(mb) using the LO evolution

αs(µ) =αs(MZ)

1− β0αs(MZ)/2π ln(MZ/µ).

For instance, for mt = 172.9 GeV, µb = 2.5 GeV and αs(MZ) = 0.1185 one obtains

C(0)eff7γ (µb) ' −0.301 . At present, due to the efforts in the calculations beyond the LO,

the central value of CSM7γ (µb) is known at the NNLO and reads [55]:

CSM7γ (µb) = −0.3523 . (4.20)

1For the expressions of U and the γ we refer to Ref. [50], in particular to Eq. (3.93) and the Eqs.(9.14 – 9.21) of the same reference.

4.2 Constraints on the UED6 model from exclusive b→ sγ decays 67

Table 4.1: Magic numbers that appear in the expressions of C(0)j (µ), C

(0)eff7γ (µ), and C

(0)eff8G (µ) [54].

i 1 2 3 4 5 6 7 8

ai1423

1623

623 −12

23 0.4086 −0.4230 −0.8994 0.1456k1i 0 0 1

2 −12 0 0 0 0

k2i 0 0 12

12 0 0 0 0

k3i 0 0 − 114

16 0.0510 −0.1403 −0.0113 0.0054

k4i 0 0 − 114 −1

6 0.0984 0.1214 0.0156 0.0026k5i 0 0 0 0 −0.0397 0.0117 −0.0025 0.0304k6i 0 0 0 0 0.0335 0.0239 −0.0462 −0.0112hi 2.2996 −1.0880 −3

7 − 114 −0.6494 −0.0380 −0.0185 −0.0057

hi 0.8623 0 0 0 −0.9135 0.0873 −0.0571 0.0209

4.2 Constraints on the UED6 model from exclusive b→ sγ

decays

We are considering a model with two universal extra dimensions compactified on thechiral square [5]. The effects on the radiative b→ sγ decay of such a NP scenario arisefrom the Kaluza-Klein towers of both gauge and fermion fields, propagating in the loopof the corresponding penguin diagrams. The new contributions to the electromagneticpenguin operators are known at the LO approximation [56]. They involve several KKexcitations that comprise the vectorial gauge bosons W a

µ , µ = 0, . . . , 3 (with a runningon the adjoint representation of SU(3)), the would-be Goldstone bosons G±, and thespinless scalars W±H and a± that come out from the mixing of the Higgs-doublet scalarsand the fourth and fifth components of the gauge bosons. Moreover, a contribution fromthe KK towers of SM fermions is also taken into account.

Following Ref. [56] we can decompose the effective Wilson coefficients involved inthe Hamiltonian (4.6) as:

Ceffi (µ) = Ceff

i,SM (µ) + ∆Ceffi (µ), i = 1, . . . , 8 , (4.21)

where

∆Ceffi (µ) =

∞∑n=0

(αs(µ)

4π

)n∆C

eff(n)i (µ) . (4.22)

Therefore, in a scenario with two universal extra dimensions, the NP contributionsdisplay their effects at 1-loop via a correction to the initial conditions of the Wilsoncoefficients. Moreover, it is worth observing that neither new operators beyond the SMones nor additional tree-level FCNC contributions appear in the effective Hamiltonian.This reflects the fundamental assumption of the MFV paradigm, implemented in thisclass of models.

The main diagrams that affect the correction at 1-loop, arising from the 6D Feynmanrules, are shown in Figure 4.3. We stress that the tree-level couplings between SM and


KK field preserve both KK numbers2. Therefore, only diagrams where all particles inthe loop have the same KK index (j, k) have to be taken into account.

At the scale µ0 = O(mt) the results in the LO approximation are the following:

∆Ceff(0)i (µ0) =

0 for i = 1, . . . , 6 ,

−12

∑′j,k A

(0)(xjk) for i = 7 ,

−12

∑′j,k F

(0)(xjk) for i = 8 ,

(4.23)

where the primed index in the sum indicates a restriction to KK indices j ≥ 1 andk ≥ 0. The analytical expressions for the Inami-Lim functions A(0)(F (0))(xj, k), wherexj, k = (j2 + k2)/(R2M2

W ), can be found in the Appendix of Ref. [56], and we alwaysrefer to those formulas in our discussion. It should only be mentioned that the sums overthe KK modes entering in these functions diverge logarithmically in the cut-off scaleΛ, and should be cut in correspondence of some values of NKK = l + k, viewing thistheory as an effective one valid up to a some higher scale. The condition NKK ' 10 hasbeen chosen in Ref. [56] on the basis of NDA arguments. This problem affects all flavourchanging neutral current transitions already at leading order, and it is a peculiar aspectof this model, absent for instance in a scenario with a single universal dimension [57].

4.2.1 Exclusive B(s) → V γ meson decays

To consider the contribution of the effective weak vertex O7γ to the transitions of ourinterest we need the hadronic matrix elements⟨

V (p′, η)∣∣ sσµνqνb ∣∣B(s)(p)

⟩= iεµναβ η

∗νpα(p′)β 2 TB(s)→V1 (q2) (4.24)

⟨V (p′, η)

∣∣ sσµνqνγ5b∣∣B(s)(p)

⟩=[η∗µ(M2

B(s)−M2

V )− (η∗ · q)(p+ p′)µ

]TB(s)→V2 (q2)

+ (η∗ · q)[qµ −

q2

M2B(s)−M2

V

(p+ p′)µ

]TB(s)→V3 (q2) . (4.25)

V stays for K∗ and K∗2(1430) in the case of B decays, and for φ(1020) in Bs decays;q = p − p′ is the photon momentum and η the V meson polarization vector. In thecase of K∗2(1430), which is a spin 2 particle, the polarization vector is described by

a two indices symmetric and traceless tensor, therefore in (4.24)-(4.25) ηα = ηαβpβMB

.

Relations exist among the three form factors TB→Vi , i = 1, 2, 3; in particular, the identity

σµνγ5 = − i2εµναβσ

αβ (with ε0123 = +1) implies TB(s)→V1 (0) = T

B(s)→V2 (0). Due to this

relation, the rate of the processes B(s)(p)→ V (p′, η) γ(q, ε) (ε the photon polarization

vector) can be expressed in terms of a single hadronic parameter TB(s)→V1 (0) for each

2This follows from a Z2-type parity preserved by the tree-level interactions.


b

s

γ

G−(jk)

Qi(jk)

Qi(jk)

b

s

γ

Qi(jk)

G−(jk)

G−(jk)

b

s

γ

W−µ(jk)

Qi(jk)

Qi(jk)

b

s

γ

Qi(jk)

W−µ(jk)

G−(jk)

b

s

γ

Qi(jk)

G−(jk)

W−µ(jk)

b

s

γ

Qi(jk)

W−µ(jk)

W−µ(jk)

b

s

γ

a−(jk)

Qi(jk)

Qi(jk)

b

s

γ

Qi(jk)

a−(jk)

a−(jk)

b

s

γ

W−H(jk)

Qi(jk)

Qi(jk)

b

s

γ

Qi(jk)

W−H(jk)

W−H(jk)

Figure 4.3: 1-loop contributions to the electromagnetic penguins of b → sγ in the model with twouniversal extra dimensions. The particles propagating in the loop are in order: KK modes of thewould-be Goldstone bosons G±(j,k), the vectorial gauge bosons W , W±µ(j,k), and the spinless scalars a±(j,k)

and W±H(j,k). The diagrams where the SU(2) quark doublets Qi(j,k), are replaced by SU(2) singlets U i(j,k)

are not shown. Here i = u, c, t.

200 400 600 800 1000-0.4

-0.3

-0.2

-0.1

0.0

0.1

1

RHGeVL

C7ef

fH0L HΜ

0L

Figure 4.4: Ceff(0)7 (µ0) in UED6 as a function of the compactification scale. The curve is evaluated in

correspondence of NKK ' 10, as explained in the text.


channel:

Γ(B(s) → V γ) =C2

4π

[TB(s)→V1 (0)

]2(m2

b +m2s)M

3B(s)

(1− M2

V

M2B(s)

)3

, (4.26)

Γ(B → K∗2 γ) =C2

32π

[TB→K∗21 (0)

]2(m2

b +m2s)

M5B

M2K∗2

(1−

M2K∗2

M2B(s)

)5

, (4.27)

where C = 4GF√

2VtbV

∗tsC

eff7

e

16π2and the first equation applies both to B → K∗γ and

Bs → φγ. In the numerical analysis we set the particle masses and lifetimes, as well asthe CKM matrix elements to the PDG values; for the quark masses we use mb ' 4.8 GeVand ms ' 0.130 GeV [58]. For the form factors we use results obtained by QCD sumrules [58]: in particular, for B → K∗ we use the results obtained by three-point QCD sumrules [59], based on the short-distance expansion, which provide TB→K

∗1 (0) = 0.38±0.06,

and light-cone QCD sum rules (LCSR) [60], based on the light-cone expansion, which giveTB→K

∗1 (0) = 0.333± 0.028. These values are larger than the lattice QCD result obtained

in quenched approximation [61]. LCSR calculations of B → K∗2 (1430) and Bs → φ form

factors give: TB→K∗21 (0) = 0.17±0.03±0.04 [62] and TBs→φ1 (0) = 0.349±0.033±0.04 [60].

4.2.2 Exclusive Λb → Λγ decays

In the case of Λb → Λγ we define the matrix elements

⟨Λ(p′, s′)

∣∣ siσµνqνb ∣∣Λb(p, s)⟩ = uΛ

[fT1 (q2)γµ + ifT2 (q2)σµνq

ν + fT3 (q2)qµ]uΛb (4.28)

⟨Λ(p′, s′)

∣∣ siσµνqνγ5b∣∣Λb(p, s)⟩ = uΛ

[gT1 (q2)γµγ5 + igT2 (q2)σµνq

νγ5 + gT3 (q2)qµγ5

]uΛb

(4.29)with uΛ and uΛb the Λ and Λb spinors; s denotes the baryon spin. The determinationsof the form factors in Eqs. (4.28)–(4.29) are quite uncertain. However, it is possible toinvoke heavy quark symmetries for the hadronic matrix elements between an initial spin12 heavy baryon comprising a single heavy quark Q and a final spin 1

2 light baryon; dueto the heavy quark symmetries the number of independent form factors is two, since formQ →∞ and a generic Dirac matrix Γ one can write [63]

⟨Λ(p′, s′)

∣∣ sΓb ∣∣Λb(p, s)⟩ = uΛ(p′, s′){F1(p′ · v)+ 6v F2(p′ · v)

}ΓuΛb(v, s) , (4.30)

where v = pMΛb

is the Λb four-velocity. The form factors F1,2 depend on the invariant

p′ · v =M2

Λb+M2

Λ − q2

2MΛb

, but we refer to their q2 dependence for convenience. Using Eq.

(4.30) we can relate the form factors in Eqs. (4.28)–(4.29) to the functions F1,2 in Eq.


Table 4.2: PDG averages for the branching fractions of radiative B, Bs and Λb decay modes [41].

mode BB+ → K∗+ γ (42.1± 1.8)× 10−6

B0 → K∗0 γ (43.3± 1.5)× 10−6

B+ → K∗2 (1430)+ γ (14± 4)× 10−6

B0 → K∗2 (1430)0 γ (12.4± 2.4)× 10−6

Bs → φγ (57±2219)× 10−6

Λb → Λ γ < 1.3× 10−3 (90% C.L.)

(4.30),

fT2 = gT2 = F1 +MΛ

MΛb

F2 ,

fT1 = gT1 = q2 F2

MΛb

,

fT3 = −(

1− MΛ

MΛb

)F2 , (4.31)

gT3 =

(1 +

MΛ

MΛb

)F2 ,

at momentum transfer close to the maximum value q2 ' q2max = (MΛb −MΛ)2. We

assume their validity in the whole phase space, introducing a model dependence in thepredictions.

The Λb → Λγ decay width reads in terms of the form factors F1 and F2:

Γ(Λb → Λ γ) =C2

4π

(F1(0) + F2(0)

MΛ

MΛb

)(m2

b +m2s)M

3Λb

(1− M2

Λ

M2Λb

)3

. (4.32)

A determination of F1 and F2 has been obtained by three-point QCD sum rules inthe mQ → ∞ limit [64]. In the following, we use F1, F2 worked out in [65] updatingsome of the parameters used in [64]: F1(0) = 0.322± 0.015, and F2(0) = −0.054± 0.020.Other determinations of the form factors in the transition Λb → Λγ have been performedin [66] at large recoil and in [67] in the covariant constituent quark model. We set theΛb and Λ masses and the Λb lifetime to their PDG values [41].

4.2.3 Comments and results

The computed branching fractions are functions of 1/R, through the dependence ofthe coefficient C7 in the UED6 model. The results are depicted in Figures 4.5, 4.6,where we have included the uncertainty on the form factor value at q2 = 0, and asecond uncertainty, intrinsic of the model, due to the choice of the matching scale in the


200 400 600 800 10000

1

2

3

4

5

6

7

1�R HGeVL

BRHB+®

K*+ΓL@1

0-

5D

200 400 600 800 10000

1

2

3

4

5

6

7

1�R HGeVL

BRHB+®

K*+ΓL@1

0-

5D

200 400 600 800 10000

1

2

3

4

5

6

7

1�R HGeVL

BRHB

0®

K0*ΓL@1

0-

5D

200 400 600 800 10000

1

2

3

4

5

6

7

1�R HGeVLB

RHB

0®

K0*ΓL@1

0-

5D

Figure 4.5: Predicted branching fractions of B+ → K∗+γ (upper panels) and B0 → K∗0γ (lowerpanels) as a function of the compactification parameter 1/R (in units of GeV), using the form factorin [59] (left) and in [60] (right). The horizontal bands correspond to SM theoretical expectations with1 σ uncertainties (yellow [light]) and experimental measurements with 2 σ uncertainties (blue [dark])respectively.

calculation of C7 and of the value of NKK . These last uncertainty is discussed in [56],together with another one which comes from fixing two boundary couplings h1,2 whichare O(1) and enter in the expression of the masses of the Higgs fields in this model.Altogether these uncertainties do not exceed +17%

−8% [56], and we include this range in ourerror on the branching fractions. The experimental data for the various branching ratiosare collected in Table 4.3 and represent PDG averages [41]. For B → K∗γ modes, theresults represent averages of BABAR [68], Belle [69] and CLEO [70] measurements. ForB → K∗2 , the result is determined on the basis of the analysis in [71], while forBs → φit stems from ref. [72]. The upper bound on the B(Λb → Λγ) has been obtained in [73].Experimental data are represented as horizontal blue [dark] bands (at 95% c.l.) in Figure4.5, and in Figure 4.6 for B0 → K∗02 γ (due the large experimental uncertainty, we onlyshow the neutral mode where the error is smaller) and Bs → φγ.

In the two upper plots in Figure 4.5 B(B+ → K∗+γ) is computed using either theform factor T1 in [59] (left panel) or that derived in [60] (right panel). The same appliesto the two lower plots, where the neutral channel B0 → K∗0γ is considered. There is amodel dependence, with the resulting bounds : 1/R ≥ 397 GeV (charged channel, formfactors in [59]), 1/R ≥ 564 GeV (charged channel, form factors in [60]), 1/R ≥ 433 GeV(neutral channel, form factors in [59]), 1/R ≥ 710 GeV (neutral channel, form factorsin [60]). As for B0 → K∗02 γ, we obtain 1/R ≥ 324 GeV. The other two plots in Figure4.6 refer to Bs and Λb decays.

4.3 The B → Kη(′)γ decays in the SM and in a scenario with UEDs 73

200 400 600 800 10000

1

2

3

4

1�R HGeVLB

RHB

0®

K2*

0ΓL@1

0-

5D

200 400 600 800 10000

1

2

3

4

5

6

7

1�R HGeVL

BRHB

s®ΦΓL@1

0-

5D

200 400 600 800 10000

1

2

3

4

5

6

1�R HGeVL

BRHL

b®LΓL@1

0-

5D

Figure 4.6: Predicted branching fractions of B → K∗2γ, Bs → φγ and Λb → Λγ as a function ofthe compactification parameter 1/R (in units of GeV). The SM results (with 1 σ uncertainties) andexperimental measurements (with 2σ uncertainties) are represented as horizontal yellow [light] and blue[dark] bands, respectively.

Bounds from direct searches of KK modes have been discussed in [74], where thehadron collider phenomenology of (1,0) KK modes in the UED6 model was studied. Thelimit 1/R ≥ 270 GeV was found, set from direct searches at the Tevatron. Our bound ismore restrictive than the one obtained from the inclusive radiative B transition [56], asin the case of a single UED.

4.3 The radiative B → Kη(′)γ decays in the SM and in ascenario with universal extra dimensions

The b→ sγ transitions are particularly relevant as a probe for NP searches. As we knowfrom the previous sections, both the inclusive B → Xsγ mode and several exclusivechannels, namely B → K∗(892)γ, B → K1(1270)γ, B → K∗2(1430)γ, B → Kηγ,B → Kη′γ, B → Kφγ, B → K∗(892)πγ and B → Kππγ, have been measured. Itis worth stressing that observed exclusive modes do not saturate the inclusive rate,therefore the scrutiny of the exclusive transitions is mandatory in view of understandingthe hadronization process for this class of channels. This is one of the motivations of ananalysis of the three-body B → Kη(′)γ modes.

Moreover, there are other features making the multibody decays induced byb→ sγinteresting to be studied. First, the time-dependent CP asymmetry in the neutral modesB0 → K0

S,Lη(′)γ is sensitive to NP, which may also manifest itself in producing right-

handed photons; indeed, in the SM the photons produced in the b→ sγ transition are


Table 4.3: Experimental results for the B → Kη(′)γ branching fractions (×106) from Belle and BABAR.The upper limits are at 90% CL.

mode Belle Collab. BABAR Collab.

B+ → K+ η γ 8.4± 1.5±1.20.9 [79] 7.7± 1.0± 0.4 [82]

B0 → K0 η γ 8.7±3.12.7 ±1.9

2.6 [79] 7.1±2.12.0 ±0.4 [82]

B+ → K+ η′ γ 3.6± 1.2± 0.4 [80] 1.9±1.51.2 ±0.1 (< 4.2) [81]

B0 → K0 η′ γ < 6.4 [80] 1.1±2.82.0 ±0.1 (< 6.6) [81]

mainly left-handed, the amplitude for emitting right-handed photons being suppressedby the quark mass ratio ms/mb [75]. Furthermore, the branching fractions of B → Kηγand B → Kη′γ do not obey the same hierarchy as in the two-body decays B → Kηand B → Kη′, the last process being enhanced with respect to the former one. Theenhancement of two body hadronic transitions with η′ in the final state is common toseveral B and D decays, and is not yet fully understood. In the case of Ds → η(′)π , η(′)ρ,the gluon content of the η′ has been indicated as playing an important role [76]. ForB → Kη and B → Kη′, a possible explanation of the hierarchy between the two decayrates has been found in the destructive interference among the penguins contributions [77],and, modulo large uncertainties, this has been numerically reproduced in the framework ofQCD factorization [78]. On the contrary, the radiative modes B → Kηγ and B → Kη′γshow the opposite trend, as one can infer from the results provided by Belle [79, 80] andBABAR Collaborations [81,82], and collected in Table 4.3: such an outcome deservesinvestigations.

4.3.1 Improved analysis of B → Kη(′)γ in SM

We perform the analysis of the decay B → Kη(′)γ taking into account several possibleunderlying transitions, depicted in Figure 4.7, and observing that, in addition to b→ sγ,the transition b → sqq can contribute to the processes3. In particular, the explicitcalculation shows that the diagram (1) with intermediate virtual K∗ is important,together with diagrams (3) and (4), while the one with intermediate K∗2 (1430) and theother diagrams in Figure 4.7 are smaller.

Moreover, in order to evaluate exactly the vertices with the η(′), a suitable mixingscheme4 must be chosen. To be more precise, the η − η′ mixing is usually described in

3These processes have been studied in ref. [83] considering exclusively the regions of the phase spacewhere one of the two pseudoscalar mesons in the final state is soft, while the photon is hard. Describingthe amplitudes as taking contributions only from virtual intermediate B∗ and B∗s , the Heavy QuarkEffective Theory together with the light meson chiral perturbation theory (χHQET) has been employedto describe the decays in corners of the Dalitz plot; moreover, the η − η′ mixing has been describedin the octet-singlet mixing scheme. As a result, a fraction of about 10% of the measured B → Kηγbranching ratio has been obtained.

4The mixing of the light pseudoscalar (η − η′) or vector (ω − φ) mesons, due to their equal quantumnumbers, is a long standing topic of the quark model and one the first experimental evidences of theSU(3) legacy.


Figure 4.7: Some diagrams contributing to the decays B0 → K0η(′)γ. The dots indicate em and strongcouplings, the squares weak vertices.

two different schemes, adopting either the singlet-octet (SO) or the quark flavour (QF)basis, and in each scheme two mixing angles are involved [84]. Here, we adopt the quarkflavour basis, defining

|ηq〉 =1√2

(|uu〉+

∣∣dd⟩)|ηs〉 = |ss〉 , (4.33)

so that the η-η′ system can be described in terms of the mixing angles ϕq and ϕs:

|η〉 = cos ϕq |ηq〉 − sinϕs |ηs〉∣∣η′⟩ = sinϕq |ηq〉+ cosϕs |ηs〉 . (4.34)

The difference between ϕq and ϕs is due to OZI-violating effects and is experimentallyfound to be small (ϕq − ϕs < 5◦), so that it has been proposed that the approximationof describing the η − η′ mixing in the QF basis and a single mixing angle is suitable [84].The simplification ϕq ' ϕs ' ϕ is supported by a QCD sum rule analysis of the φ→ ηγand φ → η′γ decays [85]. The KLOE Collaboration provided the measurement of

the ratioΓ(φ→ η′γ)

Γ(φ→ ηγ)in the flavour basis with a single mixing angle, with the result:

ϕ =(41.5± 0.3stat ± 0.7syst ± 0.6th

)◦[86]. We set ϕ to this value.

Let us now consider in turn the various diagrams contributing to B → Kη(′)γ andshown in Figure 4.7.


Diagrams 1 and 2

The corresponding amplitudes read:

A1 = A(B → K∗γ)i

s−M2K∗ + iMK∗ ΓK∗

A(K∗ → Kη(′)) , (4.35)

A2 = A(B → K∗2γ)i

s−M2K∗2

+ iMK∗2ΓK∗2A(K∗2 → Kη(′)) , (4.36)

with:

A(B → K∗(2)γ) = Cε∗µ[(mb +ms)

⟨K∗(2)(pK , ε)

∣∣∣ sσµνqνb ∣∣∣B(p)⟩

+(mb −ms)⟨K∗(2)(pK , ε)

∣∣∣ sσµνqνγ5b∣∣∣B(p)

⟩](4.37)

to be computed for an on-shell (q2 = 0) photon, and s = (p− q)2 = M2Kη(′) . The factor

C is defined as C = 4GF√

2VtbV

∗tsC

(eff)7

e

16π2. The variable s introduced in the definition

of the hadronic matrix elements takes into account that the K∗(2) mesons are off-shell,and is needed to ensure gauge invariant amplitudes.

In the same diagrams strong vertices also appear, which are defined as follows:

A(K∗ → Kη(′)) = gK∗Kη(′) ε · pη(′) , (4.38)

A(K∗2 → Kη(′)) = gK∗2Kη(′) εαβpη(′)α pη(′)β . (4.39)

Within the flavour scheme for the η − η′ mixing, the relations gK∗Kη = (cosϕ +√2 sinϕ)gK∗+K+π0 and gK∗Kη′ = (sinϕ −

√2 cosϕ)gK∗+K+π0 can be worked out. As-

suming the width of K∗+ saturated by the two modes K∗+ → K+π0, K0π+, and usingthe relation gK∗+K0π+ =

√2 gK∗+K+π0 , from the measurement Γ(K∗+) = 50.8 ± 0.9

MeV, we obtain gK∗+K+π0 = 6.5± 0.06.

As for the coupling gK∗2Kη, it can be estimated, although with a large uncertainty,using the measurements B(K∗2 → Kη) = (1.5±3.4

1.0) × 10−3 and Γ(K∗2) = 98.5 ± 2.9MeV [41], obtaining: gK∗2Kη = 1.43± 1.60 GeV−1. On the other hand, no information isavailable for gK∗2Kη′ . However, since, as we shall see, the contribution of this diagram issmall in the case of η, we neglect it in the case of the η′ in the final state.

Diagrams 3 and 4

In this case the two amplitudes read:

A3 = A(B → B∗η(′))i

t−m2B∗A(B∗ → Kγ) , (4.40)

A4 = A(B → B∗sK)i

u−m2B∗s

A(B∗s → η(′)γ) , (4.41)


with:

A(B∗ → Kγ) = Cε∗ν[(mb +ms)

⟨K(pK)

∣∣ sσµνqµb ∣∣B∗(p′, ε)⟩ ,+ (mb −ms)

⟨K(pK)

∣∣ sσµνqµγ5b∣∣B∗(p′, ε)⟩] , (4.42)

A(B∗s → η(′)γ) = Cε∗ν[(mb +ms)

⟨η(′)(pη(′))

∣∣∣ sσµνqµb ∣∣∣B∗s (p′, ε)⟩

+(mb −ms)⟨η(′)(pη(′))

∣∣∣ sσµνqµγ5b∣∣∣B∗s (p′, ε)

⟩](4.43)

where (p′, ε) denote the four momentum and the polarization vector of the B∗(s). The

variable t = (q + pK)2 takes into account the off-shellness of the B∗ in diagram (3),while the variable u = (pη(′) + q)2 accounts for the off-shellness of the B∗s in diagram (4).

Obviously, s+ t+ u = M2B +M2

K +M2η(′) .

As for the strong vertices appearing in the two amplitudes, we define

A(B → B∗η(′)) = gB∗Bη(′) ε∗ · pη(′) , (4.44)

A(B → B∗sK) = gB∗sBK ε∗ · pK . (4.45)

The two couplings gB∗Bη(′) and gB∗sBK can be obtained, invoking SU(3)F symmetry,from the analogous quantity gB∗Bπ: gB∗Bη = cosϕgB∗Bηq = cosϕ gB∗Bπ, gB∗Bη′ =sinϕgB∗Bηq = sinϕ gB∗Bπ and gB∗sBK = gB∗Bπ. Regarding gB∗Bπ, it can be related toa low-energy parameter g that describes the coupling of heavy mesons belonging to thedoublet of heavy-light quark states with spin-parity JP = (0−, 1−) to light pseudoscalarstates in the framework of the Heavy Quark Effective Theory with chiral perturbationtheory [87]: gB∗Bπ = 2MB

fπg. There are several theoretical determinations of g spanning

the range [0.2, 0.5] [88]. However, g can be extracted from the measured decay width ofD∗+ → D0π+ [41] which provides us with the value g = 0.59± 0.01± 0.07 [89] used inour analysis.

Diagram 5

The contribution of the intermediate φ(1020) is described by the amplitude:

A5 = A(B → Kφ)i

u−M2φ + iMφ Γφ

A(φ→ η(′)γ) . (4.46)

Adopting factorization, the first amplitude in (4.46) can be written as

A(B → Kφ) =GF√

2VtbV

∗tsaw〈K(pK)|sγµ(1−γ5)b|B(p)〉〈φ(pφ, ε)|sγµ(1−γ5)s|0〉 , (4.47)

where aw is an effective Wilson coefficient that we fix to the value aw = 0.064± 0.009 inorder to reproduce the experimental branching fraction B(B

0 → K0φ) = (8.6±1.3

1.1)×10−6

[41]. Furthermore, we use the parameterizations:

〈K(pK)|sγµ(1− γ5)b|B(p)〉 = fB→K+ (q2)(p+ pK)µ + fB→K− (q2)(p− pK)µ

〈φ(pφε)|sγµ(1− γ5)s|0〉 = fφMφε∗µ . (4.48)


When the two previous definitions are inserted in Eq. (4.47), it turns out that only theform factor fB→K+ contributes. We adopt for it the parameterization in [90]. As for fφ,we fix it from the experimental datum B(φ→ e+e−) = (2.954± 0.030)× 10−4, obtainingfφ = (0.232± 0.002) GeV2.

As for the amplitudes A(φ→ η(′)γ), following [85] we write:

A(φ→ η(′)γ) = −e3F φ→η

(′)γ(q2)εναβδε∗ν(pφ)α(pη)

β εδ . (4.49)

The form factors F φ→η(′)γ(q2) were determined in [85] using QCD sum rules, providing

their values at q2 = 0 (multiplied by the strange quark charge in units of e): |gφηγ | =F φ→ηγ(0)

3= (0.66 ± 0.06) GeV−1 and |gφη′γ | =

F φ→η′γ(0)

3= (1.0 ± 0.2) GeV−1. We

use these results in our analysis.

Diagram 6

It is possible to show that the diagram (6) provides a negligible contribution. Theamplitude can be written in terms of A(B∗ → Bγ) and A(B∗ → Kη). In order tounderstand how large this contribution is, we can invoke naive factorization writingA(B∗ → Kη) = GF√

2V ∗ubVusa

eff2 〈K|sγµ(1 − γ5)|B∗〉〈η|uγµ(1 − γ5)u|0〉, where aeff2 '

−0.286 is an effective Wilson coefficient for colour suppressed decays. The second matrixelements involves (in the flavour basis for the η−η′ mixing) the constant f qη = fq cosφ withfq ' fπ: 〈η|uγµ(1− γ5)u|0〉 = 1√

2i f qη (pη)µ. The matrix element 〈K|sγµ(1− γ5)|B∗〉 can

be decomposed in terms of several form factors, however, when contracted with (pη)µ onlyone of such form factors survives, usually denoted as A0(M2

η ), which in the large energy

limit coincides with TB∗→K

1 . The other ingredient is the radiative amplitudeA(B∗ → Bγ)

that can be written as A(B∗(p′, ε) → B(p)γ(q, ε)) = e

(ebΛb

+eqΛq

)εαβτσ ε

∗α εβ pτ p′σ,

with eb (eq) the b (q = d) quark charge in units of e. The mass parameters Λb andΛq have been estimated in [91]: Λb = 4.93 GeV (very close to the b quark mass) andΛq = 0.59 GeV. As a result, the contribution of the diagram (6) to the branching fractionis O(10−13), hence it can be safely neglected.

As it emerges from the above discussion, the important quantities are the formfactors appearing in the diagrams (1)–(4). In [92] we have computed TB

∗→K1 (q2) by

light-cone QCD sum rules [58], finding TB∗→K

1 (0) = 0.30 ± 0.066. SU(3)F symmetry

and the QF η − η′ mixing scheme allow also to fix: TB∗s→η1 (0) = − sinϕTB

∗→K1 (0) and

TB∗s→η′1 (0) = cosϕTB

∗→K1 (0). Finally, for the relevant hadronic form factors T

B→K∗(2)

1

we set the values of the previous sections.

4.3.2 B → Kη(′)γ decay rates and photon spectra in SM and UED6

We can now discuss the branching fraction and the photon spectrum of the two decaymodes B → Kηγ and B → Kη′γ in the SM and in the scenario with two universal extra


Figure 4.8: Branching ratio B(B → Kηγ) as a function of the strong phase phase between the sum ofthe amplitudes corresponding to the first two diagrams and the sum of the amplitudes of the last twodiagrams. The horizontal band corresponds to the experimental branching ratio.

dimensions. We may observe that both the SM and the NP observables share commonfeatures, namely the hierarchy among the various decay amplitudes and the shape of thephoton spectrum, as UED6 belongs to a MFV-type scenario.

For what concerns the intermediate states appearing in the B → Kη(′)γ decayamplitudes, the most important contributions are from the diagrams (1), (3) and (4) inthe case of the η, while for η′ the first diagram contributes much less that diagrams (3) and(4). This is due to the coupling gK∗Kη which is much larger than gK∗Kη′ . Indeed, fromthe relations in the previous section we get: gK∗Kη = 11±0.1 and gK∗Kη′ = −2.57±0.19.The various amplitudes, in particular those given by Eqs.(5.13), (4.36), (4.40), (4.41)may have a relative strong phase among each other. Since the underlying transition iscommon to the first two diagrams among themselves, and the same for the the thirdand the fourth diagram, we only consider a phase θ between the sum of the first twoamplitudes and the sum A3 + A4. As for the fifth diagram, it turns out to be muchsmaller than the others, so that we assign to it the same phase as to diagrams A3 andA4, having checked that a random variation of its phase does not modify the result. Inthe case of η′, we neglect the contribution of K∗2 , and θ is the phase between A1 andA3 +A4 +A5. In the following we see, from the calculation of B(B → Kηγ), that thereis a range of values of the strong phase θ that allows to reproduce the experimental datain Table 4.3. Let us start from the Standard Model.

From the plot of the computed B(B → Kηγ) as a function of the strong phase θ,depicted in Figure 4.8, we see that the experimental results in Table 4.3 can be obtained,in particular in correspondence to the value θ = 1.29± 0.15 rad (we have considered therange 0 ≤ θ ≤ π since the plot is symmetric with respect to θ = 0). For the central valueof θ, the obtained photon spectrum is depicted in Figure 4.9. The main feature is a quitenarrow peak at large photon energies, and the presence of a structure in B → Kηγ asthe effect of the virtual K∗. The Dalitz plot in the plane (MηK , Eη), displayed in Figure4.10, also shows that the effect of the K∗ in B → Kηγ, at the limit of the kinematicallyaccessible phase space, and it should be observed in the data.


Figure 4.9: Photon spectrum in B0 → K0ηγ (left) and B0 → K0η′γ (right). The relative phasebetween the amplitudes is fixed in order to reproduce the central value of the experimental branchingratio in Table 4.3 .

Figure 4.10: Dalitz plot of B0 → K0ηγ (left) and B0 → K0η′γ (right) in the plane (Mη(′)K , Eη(′)).

The photon and η(′) energy in the B rest frame read as Eγ =M2B −M2

η(′)K

2MBand Eη(′) =

M2B +M2

η(′) − t2MB

.

Lighter colors correspond to higher values of the distributions.

In the case of the η′, since the diagram (1) gives a small contribution with respect to(3) and (4), there is no effect of the strong phase. The prediction for the branching ratiois B(B → Kη′γ) = (0.6± 0.2)× 10−6, with the photon spectrum depicted in Figure 4.9and the Dalitz plot shown in Figure 4.10. The theoretical calculation of the branchingfraction for the neutral mode is compatible with the experimental datum reported inTable 4.3, which is affected by a large uncertainty. The experimental uncertainty issmaller in the charged mode: in this case, while the BABAR result is compatible withthe calculation, the Belle measurement is larger. Before commenting on the chargedcase, it is worth observing that, for these three-body decays, the hierarchy between themode with the η and the η′ in the final state turns out to be reversed with respect tothe two-body η and η′ transitions not only in the experimental result, but also in thetheoretical calculation.

Turning to the differences between the neutral B0 and the charged B± radiative


Figure 4.11: Ratio of the experimental branching fractions [B(B0 → K0ηγ)]/[B(B0 → Xsγ)] as afunction of the phase θ (left panel), and branching fraction B(B0 → K0ηγ) computed in the model withtwo universal extra dimensions as a function of the inverse of the compactification radius (in GeV) andfor the phase in the range fixed in the left picture (right panel). The horizontal bands corresponds tothe experimental data.

decays, we already stated that our analysis refers to the neutral case, and that theanalysis of the charged modes would be the same, except for the contribution of theinner bremsstrahlung (IB) diagrams with the photon coupled to the charged initial B+

and final K+ mesons. The kinematical region in which the bremsstrahlung contributioncould be competitive with the other decay mechanisms is for soft photons, due to thepresence of a pole for the photon energy going to zero. To understand this contributionto the B+ → K+η(′)γ decay, we invoke the Low theorem [93] which, for scalar particles,allows to relate the amplitude of the radiative mode with a soft photon to the amplitudeA(B+ → K+η(′)):

AIB(B+(p)→ K+(PK)η(′)(pη(′))γ(q, ε)) = e

(ε∗ · pKq · pK

−ε∗ · pη(′)

q · pη(′)

)A(B+ → K+η(′)) .

(4.50)The two-body amplitudes above can be experimentally determined from B(B+ →K+η) = (2.33±0.33

0.29)× 10−6 and B(B+ → K+η′) = (7.06± 0.25)× 10−5 [41], leading tothe estimate the contribution of the IB diagram to the decay rate of order O(10−8) inthe case of the mode with the η, and of O(10−7) for the η′. Therefore, we argue that therates of the charged mode would not be significantly affected by the the bremsstrahlungterm, and are analogous to the neutral B0 decays.

Now we can comment on the main modifications occurring in the UED6 scenario,that we are exploring in the present section of the thesis. As we know from the radiativetwo-body B → V decays, previously investigated, the only NP contribution bringingby the Kaluza-Klein particles exchanged in the FCNC loop current characterizing theb→ sγ transition, arises in the coefficient Ceff

7γ .In order to disentangle the dependence of the rate B → Kηγ on the phase θ

and on the Wilson coefficient C7 which encodes the new physics, we consider the ratio[B(B0 → K0ηγ)]/[B(B0 → Xsγ)] versus θ, with the experimental datum of B(B0 → Xsγ)


reported in [94] and the theoretical expression that can be found, e.g., in [95]. In thisratio, the dependence on C7 cancels out, so that we can fix the range of allowed valueson the phase depending on the experimental measurements with their own accuracy.As depicted in Figure 4.11 (left panel), the data allow to determine a range for θ (astrong interacting quantity): θ = 2.19± 0.75 rad, which is compatible with the rangedetermined previously and can be reduced by improved measurements of the decay rates.With θ in this range and the expression of Ceff

7γ dependent on the compactification radius,

we can compute B(B0 → K0ηγ) versus 1/R, which gives the lower bound 1/R > 400GeV, as plotted in Figure 4.11 (right panel). Although such a constraint is weaker thanthe bound established from the inclusive radiative B decay rate, 1/R > 650 GeV [56], itrepresents an additional information that can be made more precise, e.g., improving theexperimental data and the knowledge of the various parameters.

Chapter 5

Manifestation of warped extradimensions in electroweak Bpenguins

5.1 The effective Hamiltonian for B → Xs `+ `−

The effective Hamiltonian for B → Xs `+ `− at scales µ = O(mb) is given by

Heff(b→ s`+`−) = − 4GF√

2VtbV

∗ts

{C1Q1 + C2Q2

+∑

i=3,..,6

CiQi +∑

i=7,..,10,P,S

[CiQi + C ′iQ′i

] }. (5.1)

We neglect doubly Cabibbo suppressed terms proportional to VubV∗us. The current-current

operators Q1 and Q2, and the QCD penguins Qi (i = 3, . . . , 6) have a minor impacton the modes we are considering, therefore we neglect them. Among the remainingoperators, the primed ones have opposite chirality with respect to the unprimed. Onlythe unprimed ones, for i = 7, . . . 10, are present in the SM; the scalar QS and pseudoscalarQP operators, although present in SM, are highly suppressed. Therefore, in addition tothe operators relevant for B → Xsγ, there are four new operators to be considered,

Q9 =e2

16π2(sibj)V−A (¯ )V

Q′9 =e2

16π2(sibj)V+A (¯ )V

Q10 =e2

16π2(sibj)V−A (¯ )A (5.2)

Q′10 =e2

16π2(sibj)V+A (¯ )A .

i and j denote color indices, and V and A refer to γµ and γµγ5 structures, respectively.These operators are the so-called semileptonic electroweak penguins, and can be generated

83

84 Manifestation of warped extra dimensions in electroweak B penguins

b s

W

u, c, t u, c, t

Z

ℓ+ ℓ−

b s

u, c, t

W W

ν

ℓ+ ℓ−

Figure 5.1: EW and Box penguin diagrams for the b→ s`+`− precesses.

from a virtual photon with the same penguin diagram as b→ sγ, or the photon can bereplaced by a virtual Z boson. There is also a contribution from a box diagram involvingvirtual W and a neutrino (Figure 5.1).

The coefficient C10(µ) is given by [54]:

C10(MW ) = − α

2π

Y0(xt)

sin2(θW ), (5.3)

with Y0(x) given by

Y0(x) =x

8

[x− 6

x− 1+

3x+ 2

(x− 1)2 lnx

]− 1

4

[ x

1− x +x lnx

(x− 1)2

],

where xt = m2t /M

2W . Moreover, since Q10 does not renormalize under QCD, its coefficient

does not depend on µ ∼ O(mb).The coefficient C9(µ) can be evaluated in the NDR scheme, obtaining the relation [54]:

C9(µ) =α

2πCNDR

9 (µ) , (5.4)

CNDR9 = PNDR0 +

Y0(xt)

sin2 θW− 4Z0(xt) . (5.5)

We use PNDR0 = 2.6 in our numerical analysis. The Z0(x) function is defined by:

Z0(x) =x

8

[x− 6

x− 1+

3x+ 2

(x− 1)2 lnx

]+

1

4

[− 4

9lnx+

−19x3 + 25x2

36(x− 1)3+x2(5x2 − 2x− 6)

18(x− 1)4lnx].

The decays b→ s`+`− are suppressed relative to b→ sγ by an additional factor of α,which results in branching fractions ofO(10−6). The exclusive channel B → K∗`+`− isof particular interest experimentally, but is theoretically more difficult than the inclusivechannel, since it depends on a set of form factors. A description of the necessarytheoretical tools can be found in the following sections. Moreover, the b→ s`+`− decayshave additional degrees of freedom as compared to b→ sγ decays. First, the amplitudesof the different contributions depend on the invariant mass squared q2 of the dileptonsystem. At small q2, a large contribution from the virtual photon affects the differentialdistribution, whereas for large q2 the weak boson transition dominates. Second, the twoleptons in the final-state provide several additional angular variables, as we show in thenext sections. Particularly relevant is the forward-backward symmetry in the dileptondecay angle, since the shape of this asymmetry as a function of q2 is sensitive to NP.

5.2 The effective Hamiltonian for B → Xs ν ν 85

5.2 The effective Hamiltonian for B → Xs ν ν

The decay B → Xsνν is the theoretically cleanest decay among rare B decays. It isgoverned by an electroweak penguin diagram including a Z boson, or a W+W− boxdiagram with the same topology of those describing the b→ s`+`− transition. Unlikeb → s`+`−, there is no contribution from a virtual photon penguin diagram. Thebranching fraction of this decay provides a probe of NP searches complementary to otherrare B decays.

Within the SM, the process is governed by the effective Hamiltonian [50]:

Heff(b→ sνν) =GF√

2

α

2π sin2(θW )VtbV

∗tsX(xt)(bs)V−A(νν)V−A ≡ cSML QL . (5.6)

QL represents the left-left four fermion operator Q ≡ (bs)V−A(νν)V−A. The masterfunction X depends on the top quark mass mt and on the W mass through the ratioxt = m2

t /M2W :

X(xt) = ηX X0(xt) . (5.7)

The function X0,

X0(xt) =xt8

[xt + 2

xt − 1+

3xt − 6

(xt − 1)2log xt

], (5.8)

results from the calculation of the loop (penguin and box) diagrams at leading order(LO) in αs [51], while the factor ηX = 0.994 accounts for NLO αs corrections [96].X is flavour-universal and real, implying that, in SM, it is possible to relate differentmodes with a neutrino pair in the final state, namely Bd → Xs,dνν and K+ → π+ννor K0 → π0νν. Such relations continue to hold in NP models with minimal flavourviolation.

We only observe that, within SM, the theoretical uncertainty is related to the valueof one Wilson coefficient cSML . Moreover, possible NP effects contributing to (5.6) canmodify the SM value of the coefficient cL, or introduce the new right-right operator,giving

Heff(b→ sνν) ≡ cLQL + cRQR (5.9)

(QR ≡ bγµ(1 + γ5)sνγµ(1 + γ5)ν), with cR receiving contribution only from phenomenabeyond SM. Notice that we only consider massless left-handed neutrinos.

5.3 The B → K∗`+`− puzzle in a scenario with one warpedextra dimension

5.3.1 Angular distributions in B → K∗`+`−: formalism and theory

In the B → K∗`+`− decay mode the angular distribution contains useful informationabout the different amplitudes. This is in contrast to B → K`+`− or B → K∗γ, wherethe angular distributions are fully constrained by angular momentum conservation. TheB → K∗(→ Kπ)`+`− decay is completely described by four independent kinematical


Figure 5.2: Kinematic variables of the B → K∗[→ Kπ]`+`− decay.

variables (see Figure 5.2): the lepton-pair invariant mass squared q2, the angle θK of theK+ relative to the B in the K∗ rest frame, the angle θ` of the `+ relative to the B inthe dilepton rest frame, and the angle φ between the K∗ decay plane and the dileptonplane1. The differential decay distribution can be written as2 [97, 98]:

d4Γ(B → K∗[→ Kπ]`+`−)

dq2 d cos θ` d cos θK dφ=

9

32πI(q2, θ`, θK , φ) , (5.10)

where

I(q2, θ`, θK , φ) = Is1 sin2 θK + Ic1 cos2 θK

+ (Is2 sin2 θK + Ic2 cos2 θK) cos 2θ`

+ I3 sin2 θK sin2 θ` cos 2φ

+ I4 sin 2θK sin 2θ` cosφ

+ I5 sin 2θK sin θ` cosφ

+ (Is6 sin2 θK + Ic6 cos2 θK) cos θ`

+ I7 sin 2θK sin θ` sinφ

+ I8 sin 2θK sin 2θ` sinφ

+ I9 sin2 θK sin2 θ` sin 2φ . (5.11)

In the case of the CP conjugated mode, the B meson decay, one defines analogousfunctions I in which all the weak phases are conjugated. The fully differential decaywidth dΓ can be written in analogy to (5.10), with the function I replaced by I which isobtained from (5.11) by the rule [98]:

I1,2,3,4,7 → I1,2,3,4,7 ,

I5,6,8,9 → −I5,6,8,9 . (5.12)

1LHCb Collaboration uses the `+ direction for B0 decays, `− for B0.2Summing over the spins of the final particles.

5.3 The B → K∗`+`− puzzle in a scenario with one warped ED 87

All the functions Ii(Ii) can be written in terms of eight transversity amplitudes, which inturn are functions of the B → K∗ form factors. We adopt the following parametrizationfor the B → K∗ matrix elements:⟨

K∗(p′, ε)∣∣ sγµ(1− γ5)b

∣∣B(p)⟩

= εµναβε∗νpαp′β

2V (q2)

MB +MK∗

−i[ε∗µ(MB +MK∗)A1(q2)− (ε∗ · q)(p+ p′)µ

A2(q2)

MB +MK∗

− (ε∗ · q)2MK∗

q2

(A3(q2)−A0(q2)

)qµ

], (5.13)

whereas for 〈K∗(p′, ε)| sσµνqν (1+γ5)2 b |B(p)〉 we use the (4.24)–(4.25) previously encoun-

tered in Section 4.2.

We remind that the various form factors are not all independent; A3 can be writtenas

A3(q2) =MB +MK∗

2MK∗A1(q2)− MB −MK∗

2MK∗A2(q2) (5.14)

with A3(0) = A0(0), as like as T1(0) = T2(0). Using these definitions, the transversityamplitudes entering in the Ii structures, and the Ii functions themselves, can be foundin Ref. [98] (their Eqs. (3.28)–(3.45)), with the only change that the three Ti formfactors in that paper have to be divided by a factor of 2 to match our definition in Eqs.(4.24)–(4.25).

From the functions Ii(q2) and Ii(q

2), CP conserving quantities (Si) and CP asymme-tries (Ai) can be built:

Si =Ii + IidΓdq2 + dΓ

dq2

, (5.15)

Ai =Ii − IidΓdq2 + dΓ

dq2

. (5.16)

Starting from these quantities, several observables can be introduced. In particular, wemay consider

• the lepton forward-backward (FB) asymmetry: AFB = −38(2Ss6 + Sc6);

• the longitudinal K∗ polarization fraction: FL = −Sc2;

• binned observables Si, with their numerators and denominators separately inte-grated over q2 bins of the kind [q2

1, q22]: < Si >[q2

1 ,q22 ]

3.

The lepton FB asymmetry is of great interest, since the first analyses by BABAR,Belle and CDF Collaborations seemed to contradict the SM expectation for the sign

3It has been usual to divide the branching fraction results into a set of six bins in q2, three below theJ/ψ mass, two above the ψ′ mass, and one in the gap between the two charmonium jet veto regions.


of AFB(q2) in the low q2 region: q2 ∈ [1 − 6] GeV2 [99–101]. The presently availableimproved analysis by LHCb shows better agreement with SM [102], as we discuss in moredetails below. On general grounds, AFB might or not have a zero in the kinematicallyaccessible q2 range. The position of such a zero, in terms of the B → K∗ form factorsand of the Wilson coefficients in the effective Hamiltonian (5.1), is given by the value ofq2 for which the following equation holds:

Re

{2mb

q2

[(MB +MK∗)

T1(q2)

V (q2)(C7 + C ′7)(C10 − C ′10)∗ + (MB −MK∗)

× T2(q2)

A1(q2)(C7 − C ′7)(C10 + C ′10)∗

]+ C9C

∗10 − C ′9C ′∗10

}= 0 .

(5.17)

In the SM, this equation becomes:

Re

{2mb

q2

[((MB +MK∗)

T1(q2)

V (q2)+ (MB −MK∗)

T2(q2)

A1(q2)

)C7C

∗10

]+ C9C

∗10

}= 0 .

(5.18)In the large energy limit of K∗ and in the heavy quark limit the q2 dependence ofthe form factor ratios in (5.17), (5.18) cancels out, and T1/V = MB/(MB + MK∗),T2/A1 = (MB + MK∗)/MB (modulo radiative corrections); hence, the position of thezero of AFB is, to a large extent, a quantity which only depends on the structure ofthe interactions, i.e. it is independent of the form factors, and at NLO for the Wilsoncoefficients it is given by q2

0|SM = 4.39±0.380.35 GeV2 [103]. This value must be compared

with the LHCb determination: q20|LHCb = 4.9± 0.9 GeV2 [8].

Results that have raised interest are those reported by the LHCb Collaboration [8],with the measurement of the observables [104]

P ′i=4,5,6,8 =Si=4,5,7,8√FL(1− FL)

(5.19)

related to FL and Si defined above. The measurement is carried out in 6 bins of q2 foreach one of the four observables in (5.19): a discrepancy is found in the case of P ′5 inthe third q2 bin, where the datum is sensibly lower than the SM prediction, as we showbelow. A small deviation is also found in P ′4 for another value of q2. We shall displaythe expected distributions during our phenomenological analysis below. Efforts havebeen devoted to identify the kind of new physics effects which may explain the full setof data without altering the observables in agreement with SM predictions. The generalidea is to try to understand which one of the Wilson coefficients (and how many ofthem) should be modified (increased/suppressed), including those not present in SM, toreproduce the data [105–107].

In the following sections we do not adopt the phenomenological approach of lookingseparately at the various Wilson coefficients, but rather we make use of a specific newphysics scenario, the custodially protected Randall-Sundrum model, in which the weak


effective Hamiltonian emerges from a well defined theory of elementary interactions. Theresulting Wilson coefficients are therefore correlated, and such a correlation has precisephenomenological consequences to be considered in the various observables, namelythose in (5.19).

Before performing the phenomenological analysis, we need to evaluate a correctionto the Wilson coefficients in RSc.

5.3.2 Modification of the Wilson coefficients in RSc model

In the RS model the Wilson coefficients in the effective Hamiltonian (5.1) are modifiedwith respect to SM:

C(′)i = C

(′)SMi + ∆C

(′)i , i = 7, 9, 10 . (5.20)

We neglect the tiny SM contribution to the primed coefficients, when present, while forthe unprimed coefficients CSM

9 and CSM10 we use:

CSM9 (µb) = 4.07

CSM10 (µb) = −4.31 . (5.21)

For CSM7 (µb) we refer to the value quoted in (4.20).

In the RSc model, the results for ∆C(′)9,10, derived in [108] at the high scale µ = MKK ,

read:

∆C9 =

[∆Ys

sin2(θW )− 4∆Zs

],

∆C ′9 =

[∆Y ′s

sin2(θW )− 4∆Z ′s

],

∆C10 = − ∆Ys

sin2(θW ), (5.22)

∆C ′10 = − ∆Y ′ssin2(θW )

,

where

∆Ys = − 1

VtbV∗ts

∑X

∆``L (X)−∆``

R(X)

4M2Xg

2SM

∆bsL (X) ,

∆Y ′s = − 1

VtbV∗ts

∑X

∆``L (X)−∆``

R(X)

4M2Xg

2SM

∆bsR (X) ,

∆Zs =1

VtbV∗ts

∑X

∆``R(X)

8M2Xg

2SM sin2(θW )

∆bsL (X) , (5.23)

∆Z ′s =1

VtbV∗ts

∑X

∆``R(X)

8M2Xg

2SM sin2(θW )

∆bsR (X) .


The sums run over the neutral bosonsX = Z, ZH , Z′ andA(1), with g2

SM =GF√

2

α

2π sin2(θW ),

and θW the Weinberg angle. The functions ∆fifjL,R(X), encoding the couplings of the X

bosons to the fermions fi, fj , are collected in Appendix A.1. ∆C(′)9,10 do not need to be

evolved to µb.

Calculation of ∆C(′)7,8 at the high scale MKK

The case of ∆C(′)7 is different. In Ref. [109] a determination of this coefficient in the RS

model with and without custodial protection has been carried out directly in 5D, usingthe mixed position/momentum formalism. This approach includes the contribution ofthe whole tower of KK excitations. However, since the other coefficients used in thisthesis have been computed in the effective 4D model, we work out the calculation forC7(MKK) in 4D, with a set of assumptions concerning the contributions of the KK

excitations consistent with the calculation of ∆C(′)9,10.

To compute ∆C(′)7,8 at the scale MKK one has to consider penguin diagrams with new

particles in addition to the SM ones. For the intermediate fermions, consistently with

the procedure adopted in the evaluation of ∆C(′)9,10 we include only the zero modes. The

dominant diagrams (shown in Figure 5.3) are of the following type [109,110]:

• penguin diagrams mediated by a charged Higgs in which a photon (for C(′)7 ) or a

gluon (for C(′)8 ) is emitted from an internal up-type quark, with a mass insertion

on the internal fermion line. Diagrams contributing to the primed coefficients differfrom those for the unprimed for the chirality of the external quarks;

• penguin diagrams mediated by a gluon with an internal down-type quark. For

C(′)7 we have the contributions (∆C

(′)7 )2 and (∆C

(′)7 )3 as in Figure 5.3, for C

(′)8 the

contributions (∆C(′)8 )2 and (∆C

(′)8 )3 involve the three-gluon vertex.

We list below our results:

(∆C7)1 = iQu r∑

F=u,c,t

[A+ 2m2

F (A′ +B′)] [D†LY u(Y u)†Y dDR

]23,

(∆C7)2 = −iQd r8

3(g4Ds )2

∑F=d,s,b

[I0 +A+B + 4m2

F (I ′0 +A′ +B′)]

(5.24)

×[D†LRLY dRRDR

]23,

(∆C7)3 = iQd r8

3(g4Ds )2

∑F=d,s,b

mF

mb[I0 +A+B]

×{[D†LRLRLY dDR

]23

+mb

ms

[D†LY dRRRRDR

]23

},


URUR

QuL

bR sL

H+

Y †u

YuYd

QdLQd

L

DRbR sL

G

Yd

QdLQd

L

bR sL

GYd

(∆C(′)7 )1 (∆C

(′)7 )2 (∆C

(′)7 )3

URUR

QuL

bR sL

H+

Y †u

YuYd

bR sL

Y †dQd

L DRYd Yd

bR sL

YdDR QdL

(∆C(′)8 )1 (∆C

(′)8 )2 (∆C

(′)8 )3

Figure 5.3: New penguin diagrams contributing to the Wilson coefficients C(′)7,8 in the RSc model.

Diagrams with specular mass insertions must also be considered.

(∆C ′7)1 = iQu r∑

F=u,c,t

[A+ 2m2

F (A′ +B′)] [D†R(Y d)†Y u(Y u)†DL

]23,

(∆C ′7)2 = −iQd r8

3(g4Ds )2

∑F=d,s,b

[I0 +A+B + 4m2

F (I ′0 +A′ +B′)]

(5.25)

×[D†RRR(Y d)†RLDL

]23,

(∆C ′7)3 = iQd r8

3(g4Ds )2

∑F=d,s,b

mF

mb[I0 +A+B]

×{[D†RRRRR(Y d)†DL

]23

+mb

ms

[D†R(Y d)†RLRLDL

]23

},

(∆C8)1 = i r∑

F=u,c,t

[A+ 2m2

F (A′ +B′)] [D†LY u(Y u)†Y dDR

]23,

(∆C8)2 = −i r9

8(g4Ds )2 v2

mbmsT3

∑F=d,s,b

[A+ B + 2m2

F (A′ + B′)]

(5.26)

×[D†LY dRR(Y d)†RLY dDR

]23,

(∆C8)3 = −i r 9

4(g4Ds )2 T3

∑F=d,s,b

[A+ B + 2m2

F (A′ + B′)] [D†LRLY dRRDR

]23,


(∆C ′8)1 = i r∑

F=u,c,t

[A+ 2m2

F (A′ +B′)] [D†R(Y d)†Y u(Y u)†DL

]23,

(∆C ′8)2 = −i r9

8(g4Ds )2 v2

mbmsT3

∑F=d,s,b

[A+ B + 2m2

F (A′ + B′)]

(5.27)

×[D†R(Y d)†RLY dRR(Y d)†DL

]23,

(∆C ′8)3 = −i r 9

4(g4Ds )2 T3

∑F=d,s,b

[A+ B + 2m2

F (A′ + B′)] [D†RRR(Y d)†RLDL

]23.

We have defined r =v

GF4π2 Vtb V

∗tsmb

and T3 is the overlap of the profiles of two KK

1-mode and one KK 0-mode gluons: T3 =1

L

ˆ L

0dy[g(y)]2. Qu = 2

3 and Qd = −13 are

the up- and down-type quark electric charges in units of the positron charge e. Thematrices DL,R and the functions RL,R are collected in Appendix A.1. The quantities

I(′)0 , A(′) and B(′) correspond to the loop integrals, and they are listed below:

I0(t) =i

(4π)2

1

M2KK

(− 1

t− 1+

ln(t)

(t− 1)2

),

I ′0(t) =i

(4π)2

1

M4KK

(1 + t

2t(t− 1)2− ln(t)

(t− 1)3

),

A(t) = B(t) =i

(4π)2

1

4M2KK

(t− 3

(t− 1)2+

2 ln(t)

(t− 1)3

),

A′(t) = 2B′(t) =i

(4π)2

1

M4KK

(− t

2 − 5t− 2

6t(t− 1)3− ln(t)

(t− 1)4

), (5.28)

A(t) = B(t) =i

(4π)2

1

4M2KK

(− 3t− 1

(t− 1)2+

2t2 ln(t)

(t− 1)3

),

A′(t) = B′(t) =i

(4π)2

1

4M4KK

(5t+ 1

(t− 1)3− 2t(2 + t) ln(t)

(t− 1)4

),

with t = m2F /M

2KK .

For the evolution at the scale µb, we use the master formula [109]:

∆C(′)7 (µb) = 0.429 ∆C

(′)7 (MKK) + 0.128 ∆C

(′)8 (MKK) . (5.29)

5.3.3 Numerical analysis

The results for the observables considered within the RSc model are obtained adding thenew contributions to the Wilson coefficients, computed scanning the parameter space ofthe model. In particular, we focus on the elements of the two Yukawa matrices λd,u, andon the bulk mass parameters for quarks and leptons.


The diagonal elements of λd,u are fixed from the relations (3.75) (and the analogousones for down-type quarks), so that, under the assumptions described in Section 3.2.5,we scan over the two sets in Eq.(3.76). As customary for these scenarios, the requirementof perturbativity of the model up to the scale of the first three KK modes sets therange: |λd,uij | ≤ 3/k. However, not all the values in this range are acceptable, since also

the CKM matrix elements should be reproduced after the constraint VCKM = U†LDL isimposed. The first step in the parameter selection consists in fixing the bulk mass termsci for down- and up-type quarks. Several analyses have been devoted to this purposein the literature. We adopt the strategy outlined in [47] and the consequent choice ofparameters [111]. It consists in imposing that quark mass parameters and CKM elementsare reproduced within 2σ. An exception is represented by the bulk mass parameterof the left-handed doublet of the third generation of quarks. We slightly vary it in arange which also satisfies the constraints derived in [112] exploiting the experimentalmeasurements of several quantities related to Z decays to b quarks, i.e. the couplingZbb, the b-quark left-right asymmetry parameter and the forward-backward asymmetryfor b quarks [113]. The set used in our analysis is 4:

cu,dL = 0.63 , cc,sL = 0.57 , cb,tL ∈ [0.40, 0.45] ,

cuR = 0.67 , ccR = 0.53 , ctR = −0.35 , (5.30)

cdR = 0.66 , csR = 0.60 , cbR = 0.57 .

For leptons, c` are set to c` = 0.7 in all cases, motivated by the observation thatlepton flavour-conserving couplings are almost independent of the choice of their bulkmass parameter provided that c` > 0.5 [108]. Other determinations can be foundin [112,114–119].

Fixed such values, we generate the six λ parameters in (3.76) which also satisfy theCKM constraints. In particular, we impose |Vcb| and |Vub| in the largest range foundfrom their experimental determinations from inclusive and exclusiveB decays [94], andimpose that |Vus| lies within 2% of the central value quoted by PDG [41]:

|Vcb| ∈ [0.038, 0.043] ,

|Vub| ∈ [0.00294, 0.00434] , (5.31)

|Vus| ∈ [0.22, 0.23] .

For the quark masses we use

md = 4.9 MeV, ms = 90 MeV, mb = 4.8 GeV . (5.32)

These constraints are the starting point of our analysis. The generated values of theparameters fulfilling all the constraints are used to compute the RS contributions to

4The fermion profile given in Eq. (3.64) corresponds to the case of a left-handed fermion with bulkmass parameter cL. For right-handed fermions one should use the same function reversing the sign of thebulk mass parameter cR. Since cL and cR are independent of each other and vary in the range [−1, 1],we can choose to adopt the same profile for both fermions. However, we reverse the sign of the numericalsolution found in [111] for the parameters cR.


!0.06 !0.04 !0.02 0.00 0.02 0.04 0.06!0.4

!0.3

!0.2

!0.1

0.0

0.1

!C7!

C9"

102

Figure 5.4: ∆C7(mb) vs ∆C9 obtained implementing sequentially the constraints described in thetext. The light green points correspond to the constraints from |Vus| and |Vub|, the red points to theconstraint from |Vcb| and |Vub|, the blue points to the further constraints from B(B → K∗µ+µ−)exp andB(B → Xsγ)exp.

the Wilson coefficients. Two further conditions on the computed B → K∗µ+µ− andB → Xsγ branching fractions are imposed, requiring that they are less than 2σ fromthe experimental measurements

B(B → K∗µ+µ−)exp = (1.02±0.140.13 ±0.05)× 10−6 , (5.33)

B(B → Xsγ)exp = (3.55± 0.24± 0.09)× 10−4 . (5.34)

The result in (5.33) is the average performed by BABAR Collaboration of the branchingfractions of the four modes B+,0 → K∗+,0µ+µ−(e+e−) [120], while the result in (5.34) isthe HFAG Collaboration average over the charged and neutral mode [94]. An exampleof the sequence of the effects of the constraints is shown in Figure 5.4. After imposingthe constraints on Vub and Vus, a set of values of ∆C7(mb) and ∆C9 is computed, and∆C7(mb) spans a quite broad range of positive and negative values, the green region inthe figure. Implementing the constraint on Vcb reduces the possibilities to two isolatedregions, the red spots in the figure, and the region of negative values, the blue one,survives after the constraints (5.33) and (5.34) are imposed. We have checked that theselected points reproduce also the other CKM elements within their uncertainty, exceptfor |Vtd| which lies in the 3σ range around its central value [41]. With the selected set ofpoints in the parameter space it is also possible to reproduce in the RSc model, usingthe expressions in [47], the mass difference of the neutral Bs mesons ∆Ms within 20% ofthe central value of the experimental measurement ∆Ms = 17.69 ps−1 [94]. We depict in

Figure 5.5 the obtained values and correlations for ∆C(′)i . The largest deviations from the

SM are |∆C7|max ' 0.046 , |∆C ′7|max ' 0.05 , |∆C9|max ' 0.0023 , |∆C ′9|max ' 0.038 ,|∆C10|max ' 0.030 , |∆C ′10|max ' 0.50 . As shown in the panel (f) of the figure, ∆C9

and ∆C10 are linearly correlated, and the same happens for each pair ∆C(′)i , i = 9, 10.

Indeed, in the large set of parameters the most relevant input for these coefficients is


-0.05 -0.04 -0.03 -0.02 -0.01 0.00-0.04

-0.02

0.00

0.02

0.04

DC7

C7'

-0.05 -0.04 -0.03 -0.02 -0.01 0.00-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

DC7

DC

9´10

2

(a) (b)

-0.05 -0.04 -0.03 -0.02 -0.01 0.00-0.01

0.00

0.01

0.02

0.03

0.04

DC7

DC

' 9

-0.05 -0.04 -0.03 -0.02 -0.01 0.00

0.00

0.01

0.02

0.03

DC7

DC

10

(c) (d)

-0.05 -0.04 -0.03 -0.02 -0.01 0.00

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

DC7

DC

' 10

-0.0020 -0.0015 -0.0010 -0.0005 0.00000.000

0.005

0.010

0.015

0.020

0.025

0.030

DC9

DC

10

(e) (f)

Figure 5.5: Correlations between the RSc contribution to the Wilson coefficients C(′)7,9,10. The coefficients

∆C(′)7 are evaluated at the scale µb = mb. No correction corresponds to the red dot.

λd23, and the relations approximately hold (for cb,tL fixed to the central value):

∆C9 ' −7.18 10−4 λd23 k ,

∆C ′9 ' 1.22 10−2 λd23 k ,

∆C10 ' 9.55 10−3 λd23 k , (5.35)

∆C ′10 ' −1.62 10−1 λd23 k ,

There have been attempts to understand what would be the required size of deviationsfrom SM values for the Wilson coefficients that could explain the anomalies observedin B → K∗µ+µ− distributions (on which we shall elaborate below) and, consequently,


0 5 10 15 200

2.´ 10-8

4.´ 10-8

6.´ 10-8

8.´ 10-8

1.´ 10-7

1.2´ 10-7

1.4´ 10-7

q2 @GeV2D

dB

r�dq

2@G

eV-

2 DB®K* {

+{-

0 5 10 15 20-0.4

-0.2

0.0

0.2

0.4

q2 @GeV2D

AF

B@G

eV-

2 D

B®K* {+

{-

(a) (b)

Figure 5.6: (left panel) Differential B0 → K∗0µ+µ− decay rate. The light (green) band corresponds tothe SM result, including the uncertainty of the form factors. The red and blue vertical bars correspondto the RSc result, without or with the uncertainty in form factors. The black dots, with their error bars,are the LHCb measurements in [8]. (right panel) Lepton FB asymmetry in B0 → K∗0µ+µ−.

which NP scenario might provide such deviations [105, 107]. Although there is not aunique answer to this question, a possible conclusion is that NP should should provide alarge negative value of ∆C9. For example, in [106] various possibilities are considered inwhich NP affects just one Wilson coefficient, a pair of them, or all of them simultaneously.In the last case, to which the RS scenario belongs, it is found that the required deviationof C9 from its SM value should be ∆C9 ' −0.9 for a real coefficient, or even |∆C9| ' 2.25for a complex one. We anticipate that the result of the analysis performed in this thesisshows that such a huge deviation is not reached in RSc, as in the NP models consideredso far for this purpose5.

5.3.4 B0 → K∗0`+`− observables in the RSc model

It is now possible to compute the set of B → K∗µ+µ− observables measured by LHCb,and compare the outcome with data. The results are collected in Figures 5.6–5.7. Theresults obtained in SM include the uncertainty in the hadronic form factors; for suchnon-perturbative quantities we use the light-cone QCD sum rule determination in [60].The hadronic errors have an impact mainly on the B0 → K∗0µ+µ− differential decayrate and on the K∗ longitudinal polarization distribution, while to position of the zeroin AFB(q2) and of the maximum of FL are less affected, as expected.

In the analysis of the modifications in RSc, for the various observables, we separatelyconsider the changes due to the new Wilson coefficients, and the changes which alsoinclude the hadronic form factor uncertainties. The results show that the deviationsinduced in RSc are small, since the corrections ∆C9,10 are tiny fractions of CSM9,10 andthat also the coefficients of operators absent in SM, ∆C ′9,10 are small. A little effect is

found at small q2, where the changes due to ∆C(′)7 are slightly larger. The comparison

of data with predictions confirms the agreement, excluding the measurement of AFB(q2)

5The largest deviations (still not sizable enough) are found in models introducing a new neutral gaugeboson Z′ with suitable FCNC couplings to quarks [121,122].


0 5 10 15 200.0

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0.8

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2 D

B®K* {+

{-

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-0.8

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B®K* {+

{-

(a) (b)

0 5 10 15 20-1.0

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0.5

1.0

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{-

0 5 10 15 20-0.4

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-0.1

0.0

0.1

0.2

q2 @GeV2D

S 3@G

eV-

2 D

B®K* {+

{-

(c) (d)

Figure 5.7: (a) K∗ longitudinal polarization fraction in B0 → K∗0µ+µ−. (b) Observable P ′4 inB0 → K∗0µ+µ−. The sign is fixed to make the definition (5.19) and the one in Refs. [8] compatible. (c)Observable P ′5 in B0 → K∗0µ+µ−. (d) Observable S3 in B0 → K∗0µ+µ−. The symbols and the colorshave the same meaning as in Figure 5.6.

in the first bin of q2 where the predictions are larger than the experimental result. Inthe high q2 range the hadronic uncertainty for the lepton FB asymmetry is about 20%.

The results for the observables P ′4, P ′5 and S3 are shown in Figure 5.7. In P ′5 thehadronic uncertainty is at the level of 10% in all the q2 range. The modification of theprediction obtained in RSc is similar or larger at low q2, up to q2 ' 7 GeV2, therefore thisis a favorable kinematical range where to investigate this observable. The discrepancywith the measurement in the third q2 bin still persists, while there is agreement in theother bins.

The hadronic uncertainties turn out to be smaller in P ′4, and the changes in thepredictions in RSc seem promising to be observed at low q2. A deviation observed in thefifth q2 bin of the series of measurements is at the level of less than 2σ.

At odds with P ′5, in the observable S3 the RSc result is systematically above theSM for the largest part of the parameter space, in particular in the large q2 range,as shown in Figure 5.7(d). The size of such effect is comparable with the hadronicuncertainty; therefore, one can envisage the possibility of using this observable for a better


Table 5.1: χ2 values for the various observables, defined as χ2 =1

N

N∑i

(Oexpi −Oth

i

)2δ2i + σ2

i

. i runs over

the N experimental bins, Oexpi and Oth

i are the LHCb measurements and the theoretical results in theRSc model, with errors δi and σi, respectively.

dB/dq2 FL AFB P ′4 P ′5 S3

χ2 0.49 0.19 0.79 0.91 2.09 0.53

characterization of the deviations obtained in this new physics scenario. The experimentalresults follow the predictions, but the errors are too large to draw conclusions. All theobservations can be quantified as done in Table 5.1, which confirms that the largestdeviation in the measured observables occurs in P ′5.

The case of B0 → K∗0τ+τ− decay

Motivated by the experimental results of semileptonic and leptonic B decays to τ leptons,on which we shall focus in the next chapter, we also study observables for the case ofmassive final leptons, namely the τ polarization asymmetries. To define polarizationasymmetries, let us consider the spin vector s of the τ− lepton having momentum k1,with s2 = −1 and k1 · s = 0. In the τ− rest frame three orthogonal unit vectors can bedefined, ~eL, ~eN and ~eT , corresponding to the longitudinal sL, normal sN and transversesT polarization vectors:

sL = (0, ~eL) =

(0,~k1

|~k1|

),

sN = (0, ~eN ) =

(0,

~p′ × ~k1

|~p′ × ~k1|

), (5.36)

sT = (0, ~eT ) = (0, ~eN × ~eL) .

In Eq.(5.36) ~p′ and ~k1 are the K∗ and τ− three-momenta in the rest frame of thelepton pair. Choosing the z-axis directed as the τ− momentum in the lepton pair restframe we have k1 = (E1, 0, 0, |~k1|), and boosting the spin vectors s in (5.36) in the restframe of the lepton pair, the normal and transverse polarization vectors sN , sT remainunchanged, sN = (0, 1, 0, 0) and sT = (0, 0,−1, 0), while the longitudinal polarizationvector becomes:

sL =1

mτ(|~k1|, 0, 0, E1) . (5.37)

For each value of the squared momentum transfered to the lepton pair, q2, the polarizationasymmetry for the negatively charged τ− lepton can be defined as:

AA(q2) =

dΓ

dq2(sA)− dΓ

dq2(−sA)

dΓ

dq2(sA) +

dΓ

dq2(−sA)

, (5.38)


13 14 15 16 17 18 19 200

1.´ 10-8

2.´ 10-8

3.´ 10-8

4.´ 10-8

q2 @GeV2D

dB

r�dq

2@G

eV-

2 D

B®K* Τ+ Τ-

13 14 15 16 17 18 19 20-0.1

0.0

0.1

0.2

0.3

q2 @GeV2D

AF

B@G

eV-

2 D

B®K* Τ+ Τ-

(a) (b)

Figure 5.8: (a) Differential B → K∗τ+τ− decay rate. The light (green) band corresponds to the SMresult, including the uncertainty in the form factors. The red and blue vertical bars correspond to theRSc result, without or with the uncertainty in form factors. (b) Lepton FB asymmetry in B → K∗τ+τ−.

with A = L, T and N . These quantities have been analyzed in the SM (e.g. in [123,124]and in the references therein). In particular, in Ref. [124] it has been pointed out thatthe τ polarization asymmetries are also form factor independent quantities in the q2

region where the large energy limit can be applied. These results can be generalizedincluding the new primed operators considered here. In particular, one can exploit theexpressions in [124] for the observables in B → K∗τ+τ− with the substitutions

C7 T1 → (C7 + C ′7)T1 ,

C7 T2,3 → (C7 − C ′7)T2,3 ,

C9 V → (C9 + C ′9)V ,

C10 V → (C10 + C ′10)V , (5.39)

C9A1,2 → (C9 − C ′9)A1,2 ,

C10A1,2,0 → (C10 − C ′10)A1,2,0 .

The results for the case of τ+τ− final state are shown in Figures 5.8–5.9. Thekinematically accessible q2 range starts at q2 ' 12.628 GeV2, so that the small effectsin the muon mode at low q2 do not appear in this case. In the decay rate distributionand in the lepton FB asymmetry the results in RSc systematically deviate from SM inthe full parameter space, but the effect is smaller than the hadronic uncertainty. Sucha systematic deviation also appears in P ′5 and S3, Figures 5.9(c) and 5.9(d), while thelongitudinal and transverse τ polarization asymmetries essentially coincide with the SMones, Figure 5.10, and seem not suitable for characterizing the considered new physicsmodel.

Remarks on Bs,d → µ+µ− decays

A final remark concerns the rare Bs,d → µ+µ− modes. Since the theoretical predictionsfor the branching fractions of these modes depend on a subset of the Wilson coefficients


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2 DB®K* Τ+ Τ-

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2 D

B®K* Τ+ Τ-

(a) (b)

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-0.5

-0.4

-0.3

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0.0

0.1

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eV-

2 D

B®K* Τ+ Τ-

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-0.15

-0.10

-0.05

0.00

0.05

q2 @GeV2D

S 3@G

eV-

2 D

B®K* Τ+ Τ-

(c) (d)

Figure 5.9: (a) K∗ longitudinal polarization fraction in B → K∗τ+τ−. (b) Observable P ′4 in B →K∗τ+τ−. (c) Observable P ′5 in B → K∗τ+τ−. (d) Observable S3 in B → K∗τ+τ−.The symbols havethe same meaning as in Figure 5.8.

that have been considered, we can derive predictions using the same set of parameters,see Figure 5.11. The SM result depends only on the coefficient C10, and in RS one has toreplace C10 → C10−C ′10. In Figure 5.11 we show the correlation between the two modes,comparing the RS prediction to SM and to the experimental data. There is a region of theparameter space in which the SM result for both branching ratios is reproduced. However,the allowed range in RSc is larger than in SM: B(Bs → µ+µ−)|RS ∈ [2.64, 3.83] × 10−9

and B(Bd → µ+µ−)|RS ∈ [0.70, 1.16] × 10−10, in the right direction in the case of Bswhen comparing with data, but still lower than the datum for Bd. A similar resultwas already found in [108], although with a smaller deviation with respect to SM, inparticular in the Bs case.

5.4 Sensitivity of exclusive b→ sνν induced transitions to a warped ED 101

13 14 15 16 17 18 19 20-0.8

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AL@G

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2 D

B®K* Τ+ Τ-

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-0.6

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0.0

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AT@G

eV-

2 D

B®K* Τ+ Τ-

Figure 5.10: Longitudinal (left) and transverse (right) τ polarization asymmetries. The symbols havethe same meaning as in Figure 5.8.

2.0 2.5 3.0 3.5 4.0

1

2

3

4

5

BHBs�Μ+ Μ-L ´ 109

BHB

d�Μ+Μ-L´

1010

Figure 5.11: Correlation between the branching ratios of Bs → µ+µ− and Bd → µ+µ−. The green barsrepresent the experimental data [125,126]: B(Bs → µ+µ−) = (2.9± 0.7)× 10−9 and B(Bd → µ+µ−) =(3.6±1.6

1.4) × 10−10. The red ones the SM predictions [127]: B(Bs → µ+µ−)SM = (3.65 ± 0.23) × 10−9

and B(Bd → µ+µ−)SM = (1.06± 0.09)× 10−10. The blue region is the prediction in RSc derived in thispaper.

5.4 Sensitivity of exclusive b→ sνν induced transitions toa warped extra dimension

Rare b decays with neutrino pairs in the final states are experimentally challenging.Nevertheless, the advent of new high-luminosity B factories opens the possibility toaccess these modes which, on the other hand, present remarkable features of theoreticallyclearness, as we discuss below. We are mainly interested in the exclusive B → Kννand B → K∗νν decays, the branching fractions of which were predicted in SM ofO(10−6) [128,129]. Since the results are affected by the uncertainty of the form factorsparametrizing the hadronic matrix elements, particular attention has to be paid tosuch an issue. Using form factors from light-cone sum QCD sum rules together withexperimental informations on the B → K∗γ decay rate [98], new predictions were


obtained in SM [130],

B(B+ → K+νν) = (4.5± 0.7)× 10−6 ,

B(B → K∗νν) = (6.8±1.01.1)× 10−6 , (5.40)

(considering in the final state the sum over the three neutrino species), that must becompared to the present experimental upper bounds. The Belle Collaboration hasestablished the limits, at 90% C.L. [131],

(B+ → K+νν) < 5.5× 10−5 , (5.41)

(B0 → K0Sνν) < 9.7× 10−5 , (5.42)

(B+ → K∗+νν) < 4× 10−5 , (5.43)

(B0 → K∗0νν) < 5.5× 10−5 . (5.44)

The bounds (at 90% C.L.) obtained by the BABAR Collaboration [132],

(B+ → K+νν) < 1.6× 10−5 , (5.45)

(B0 → K0νν) < 4.9× 10−5 , (5.46)

(B+ → K∗+νν) < 6.4× 10−5 , (5.47)

(B0 → K∗0νν) < 12× 10−5 , (5.48)

are derived combining the results of the semileptonic tag reconstruction method andof the hadronic tag reconstruction method. In addition to B → K(∗)νν, other modesare induced by the b→ sνν transition, namely Bs → (φ, η, η′, f0(980))νν that we alsodiscuss in the following. At present, the experimental upper bounds for their rates arestill quite high [41,94], however they are also expected to be sizeably reduced at the newhigh-luminosity B facilities.

The importance of the rare b → sνν process relies on its particular sensitivityto new interactions. In [133] the effects of scalar and tensor interactions have beendiscussed with particular attention to the distortion of the q2 spectra (with q2 thedilepton squared four momentum) with respect to SM. In other interesting papers, therole of new right-handed operators [129] and the effects of non standard Z coupling to band s quarks [134] have also been investigated. An overview of the effects predicted inseveral NP scenarios can be found in Refs. [130,135]. In [136] has been performed ananalysis of correlations between the branching ratios, as well as between these modesand the decay Bs → µ+µ−, under the hypotheses of new couplings with a neutral gaugeboson Z′. In extensions of SM based on extra dimensions, predictions have been givenfor the decay rates and distributions in minimal models with a single universal extradimension [137]. In the following, we consider the case of a single warped extra dimension,in that phenomenological realization exposed in Section 3.2, and already managed forthe analysis of the B → K∗µ+µ− decay. We remind that already in [108], a range for theB → K(∗)νν branching fractions has been predicted within this framework. We extendthe analysis focusing on the other observables, such as several differential distributions,and on various correlations.


With this purpose in mind, it is useful to introduce two parameters [129],

ε2 =|CL|2 + |CR|2|CSML |2 , η = − Re (CLC

∗R)

|CL|2 + |CR|2, (5.49)

which probe deviations from SM where (ε, η)SM = (1, 0). In particular, η is sensitiveto the right-handed operator in the effective Hamiltonian, while ε mainly measures thedeviation from SM in the coefficient CL.

5.4.1 B → Kνν and B → K∗νν decays

In order to discuss the differential branching fractions and the other observables for theprocesses B → K∗νν, let us introduce the dimensionless neutrino pair invariant masssB = q2/M2

B and the ratio MK = MK/MB. In SM the decay distribution for B → Kννin sB reads:

dΓSM

dsB= 3|CSML |2

96π3M5Bλ

3/2(1, sB, M2K)|F1(sB)|2 , (5.50)

with CSML in (5.6) and λ(x, y, z) the triangular function. In the NP case this expressionis generalized to

dΓ

dsB= 3|CL + CR|2

96π3M5Bλ

3/2(1, sB, M2K)|F1(sB)|2 . (5.51)

As for the hadronic form factor F1, we use the one worked out in [65]. In both Eqs. (5.50)and (5.51) the factor 3 accounts for the sum over the three final neutrino flavours.Modulo a factor of two, the distributions coincide with the distributions in Emiss, the(missing) energy of the neutrino pair, since sB = 2x− 1 + M2

K , with x = Emiss/MB, and

dΓ

dsB=

1

2

dΓ

dx. (5.52)

In the case of B → K∗ transition, three transversity amplitudes can be defined,which depend either on CL − CR or on CL + CR:

A0(sB) = −N(sB)(CL − CR)

MK∗√sB

[(1− M2

K∗ − sB)(1 + MK∗)A1(sB)

− λ(1, M2K∗ , sB)

A2(sB)

(1 + MK∗)

],

A⊥(sB) = 2√

2N(sB)λ1/2(1, M2K∗ , sB)(CL + CR)

V (sB)

(1 + MK∗), (5.53)

A‖(sB) = −2√

2N(sB)(CL − CR)(1 + MK∗)A1(sB) ,

with MK∗ = MK∗/MB and the function N(sB) defined as

N(sB) =

[M3BsBλ

1/2(1, M2K∗ , sB)

3 · 27 π3

]1/2

.


The differential distributions in sB for a longitudinally or transversely polarized K∗

(with helicity h = +1 or h = −1) can be written in terms of these amplitudes. Exploitingthe definitions (5.49), one finds for the sum over the three neutrino flavours:

dΓLdsB

= 3M2BA2

0 =

(dΓLdsB

)SM

ε2 (1 + 2η) ,

dΓ±dsB

=3

2M2B|A⊥ ∓A‖|2 , (5.54)

dΓTdsB

=dΓ+

dsB+dΓ−dsB

= 3M2B

(A2⊥ +A2

‖

)=

(dΓTdsB

)SM

ε2 (1 + 2η fT (sB)) ,

dΓ

dsB= 3M2

B

(A2

0 +A2⊥ +A2

‖

)=

(dΓ

dsB

)SM

ε2 (1 + 2η f(sB)) ,

with

fT (sB) =(1 + MK∗)

4[A1(sB)]2 − λ[V (sB)]2

(1 + MK∗)4[A1(sB)]2 + λ[V (sB)]2,

f(sB) =Num[f(sB)]

Den[f(sB)], (5.55)

and

Num[f(sB)] =[(1 + MK∗)

2(1− sB − M2K∗)A1(sB)− λA2(sB)

]2

+ 8M2K∗sB

[(1 + MK∗)

4[A1(sB)]2 − λ[V (sB)]2], (5.56)

Den[f(sB)] =[(1 + MK∗)

2(1− sB − M2K∗)A1(sB)− λA2(sB)

]2

+ 8M2K∗sB

[(1 + MK∗)

4[A1(sB)]2 + λ[V (sB)]2]. (5.57)

In Eq. (5.55) we use the notation λ = λ(1, M2K∗ , sB). The factor 3 in Eqs. (5.55) accounts

for the sum over the neutrino species. Also in this case, the distributions in sB can beconverted in neutrino missing energy distributions using Eq. (5.52).

Starting from the above defined quantities, several observables can be constructed.The polarization fractions FL,T can be considered [130],

dFL,TdsB

=dΓL,T /dsBdΓ/dsB

, (5.58)

in which several hadronic and parametric uncertainties are reduced or even canceled(namely the overall quantities, like the CKM elements in SM). The integrated polarizationfractions can be obtained, integrating separately the numerator and the denominator inEq. (5.58):

FL,T =1

Γ

ˆ 1−M2K∗

0dsB

dFL,TdsB

. (5.59)


Another observable is the ratio of branching fractions involvingK and the transverselypolarized K∗ [129],

RK/K∗ =B(B → K νν)

B(B → K∗h=−1 νν) + B(B → K∗h=+1 νν),

(5.60)

which is sensitive to η.In [129] the transverse asymmetry has been proposed

AT =B(B → K∗h=−1 νν)− B(B → K∗h=+1 νν)

B(B → K∗h=−1 νν) + B(B → K∗h=+1 νν), (5.61)

for which a reduced hadronic uncertainty is expected. However, its measurement wouldrequire the determination of the lepton pair polarization, therefore we consider it onlyfor a theoretical analysis.

The observables can probe NP effects, as the ones envisaged in warped five-dimensionalextensions of the standard model.

Effects of a warped extra dimension on CL and CR Wilson coefficients

In SM the Wilson coefficients of the left- and right-handed operators OL and OR in theeffective Hamiltonian (5.6), (5.9) are given by

CSML =GF√

2

α

2π sin2 θWV ∗tbVtsX(xt) , (5.62)

CSMR = 0 .

These coefficients are modified in the RSc, in which a right-handed operator OR ispresent:

CRSL =GF√

2

α

2π sin2 θWV ∗tbVtsX

RSL , (5.63)

CRSR =GF√

2

α

2π sin2 θWV ∗tbVtsX

RSR , (5.64)

with XRSL = X(xt) + ∆XL and

∆XL =1

VtbV∗ts

∑X=Z,Z′, ZH

∆bsL (X)∆νν(X)

4M2Xg

2SM

, (5.65)

XRSR =

1

VtbV∗ts

∑X=Z,Z′, ZH

∆bsR (X)∆νν(X)

4M2Xg

2SM

. (5.66)

The constant g2SM is defined as g2

SM =GF√

2

α

2π sin2(θW ). We remind that the functions

∆fifjL,R(X), representing the couplings of the X bosons to the fermions fi, fj , can be read

in Appendix A.1.


2.58 2.59 2.60 2.61 2.62 2.63 2.64-0.1

0.0

0.1

0.2

0.3

��⨯ �� [��-�]

��⨯��[��

-�]

0.990 0.995 1.000 1.005 1.010-0.15

-0.10

-0.05

0.00

ϵ

η

Figure 5.12: (left panel) Correlation between CRSR and CRSL in the RSc model (blue curve). (rightpanel) Correlation between the parameters η and ε, defined in (5.49), in the RSc model. The red dotcorresponds to the central SM values.

Scanning the parameter space of this model resulting from all the constraint, as alreadydone in Section 5.3.3, we obtain the coefficients CRSL and CRSR and their correlation, asshown in Figure 5.12 (left panel). The resulting parameters η and ε, defined in (5.49),are depicted in Figure 5.12 (right panel). The first observation concerns the right-handedcoupling: we find that a deviation from SM is predicted, with the maximum valueCRSR = 0.186 × 10−9 GeV−2. For the left-handed coupling we obtain the maximum∆CL = CRSL − CSML = −0.011× 10−9 GeV−2. The largest deviation of η from its SMvalue η = 0 is η = −0.075. We find that CL and CR are anticorrelated, as shown Figure5.12, and this has a definite impact on the various observables that we are going todiscuss in details. A reason for such a behaviour can be traced to the quite large valuereachable for CR, a point that we are going to discuss in details.

B → Kνν and B → K∗νν observables in RSc model

To compare the RSc predictions to the SM results for the exclusive B → Kνν andB → K∗νν decay observables defined in Section 5.4.1 we need the B → K(∗) formfactors. In according with the other phenomenological analysis performed in this thesis,here we use the light-cone QCD sum rule determination [60]. Moreover, lattice QCDresults are now available [138,139], and we comment below on the differences.

In Figure 5.13 we depict the differential distribution dBdsB

(B0 → K0νν) in the whole

kinematical range 0 ≤ sB ≤(

1− mKmB

)2in SM, including the uncertainty on the form

factor F1(0) quoted in [60] and using the measured lifetime τ(B0) = 1.519±0.005 ps [94].

The predicted branching fraction

B(B0 → K0νν)SM = (4.6± 1.1) × 10−6 (5.67)

has a larger uncertainty than the one in (5.40), due to our more conservative errors onthe form factors. The modifications in RSc, obtained for the central value of F1(0) andaccounting for the uncertainty on F1(0), are also shown in Figure 5.13 and produce a


Figure 5.13:dBdsB

(B0 → K0νν) distribution in SM, including the uncertainty on the form factor F1(0)

(green region), and in RSc for the central value of F1(0) (red points) and including the uncertainty ofthe form factor at sB = 0 (blue bars).

Figure 5.14: DistributionsdBLdsB

(B0 → K∗0νν) (top) anddBTdsB

(B0 → K∗0νν) (bottom). The green

region corresponds to SM including the uncertainties on the form factorsA1(0), A2(0) (top) and A1(0),V (0) (bottom). The red dots and the blue bars correspond to RSc, for the central value of the formfactors and including their uncertainty at sB = 0, respectively.

prediction for the branching fraction spanning a somewhat wider range,

B(B0 → K0νν)RS ∈ [3.45− 6.65] × 10−6 . (5.68)

A similar result is obtained for the charged mode. Hence, the present experimental upperbounds require an improvement by a factor of 3-4 in the case of BABAR, Eq. (5.45),and of about one order of magnitude in the case of Belle, Eq. (5.41), to become sensitiveto these processes, a task within the possibilities of high-luminosity facilities such asBelle II.

For B → K∗νν we separately consider the longitudinally and transversely polarizedK∗, with distributions in Figure 5.14. In RSc a small deviation from SM is found in thelongitudinal distribution. The SM prediction, obtained including the errors on the formfactors in quadrature,

B(B0 → K∗0νν)SM = (10.0± 2.7)× 10−6 (5.69)

becomes in RSc the range

B(B0 → K∗0νν)RS ∈ [6.1− 14.3]× 10−6 . (5.70)


0 2 4 6 8 100

2

4

6

8

10

12

14

BHB0�K0 Υ ΥL� 106

BHB

0�

K*0ΥΥL�

106

Figure 5.15: (left panel) Correlation between B(B0 → K0νν) and B(B0 → K∗0νν) obtained varyingthe RSc parameters and including the uncertainty on the form factors at sB = 0 (lighter blue region).The SM prediction corresponds to the lighter red region. The darker blue curve and the darker reddot correspond to the RSc and SM prediction, respectively, obtained for the central value of the formfactors at sB = 0. (right panel) Correlation between B(B0 → K0νν) and B(B0 → K∗0νν) (blue curve)normalized to the corresponding SM values (red dot) obtained for the central value of the form factors.

For the charged mode the predictions are similar. Hence, the required improvement ofthe current upper bound to reach the expected signal is about a factor of 4 in the caseof the Belle upper bounds (5.41), and about one order of magnitude in the case of theBaBar bounds (5.45), within the reach of new facilities. Also in the case of K∗ our resulthas a more conservative error than the one quoted in (5.40). The difference is due to thechoice in [98] of exploiting additional information on the measured radiative B → K∗γdecay rate, which results in a reduction of the central value and of the error of the formfactors.

Differently from the mode into the pseudoscalar K, the K∗ channel allows toaccess other observables as the polarization fractions FL,T in (5.58). Moreover, themeasurements of both the K and K∗ modes permit the construction of the fractionRK/K∗ in (5.60), and to study the correlations among the various observables predictedin SM and in RSc. Such correlations are important to disentangle different NP scenariosfrom the one we are investigating. In Figure 5.15 (left panel) we show the correlationbetween the rates of B → Kνν and B → K∗νν, with the inclusion of the hadronicuncertainty. Although the effects of the form factor errors are at present noticeable, theSM and the RSc predictions already have a non-overlapping region, which is interestingin view of the envisaged possibility of reducing the hadronic uncertainty. In particular,the K and K∗ modes are anticorrelated, hence a reduction of the B → K∗νν decay rategoes in RSc with an increase of the rate of B → Kνν with respect to SM, as it is visiblein Figure 5.15 (right panel) in the ideal case of an exact knowledge of the hadronicmatrix elements.

As for the longitudinal K∗ polarization fraction, the differential distribution inFigure 5.16 has a small deviation and can be below the SM; the correlation of theintegrated fraction with the branching rate is depicted in Figure 5.17. A precisecorrelation pattern hence exists in RSc among the three observables B(B → Kνν),B(B → K∗νν) and FL: the first one can be above, the other one below its SM values.


Figure 5.16: Differential longitudinal K∗ polarization fraction dFLdsB

(B0 → K∗0νν) in SM including the

uncertainties on A1(0), A2(0) and V (0) (green region), and in RSc for the central values of A1(0), A2(0)and V (0) (red points) and with the error on the form factors (blue bars).

We also show illustratively the correlation between the transverse asymmetry AT in(5.61) in B → K∗νν and the branching fraction in SM and RSc, Figure 5.17, for whichthe two models, with the present hadronic uncertainty, have a big overlap.

The observable RK/K∗ defined in Eq. (5.60) and obtained from the K and K∗

measurements is depicted in Figure 5.18 versus FL. A sizable form factor uncertaintyis still present, at odds with the expectation that such a variable should be quite safe;nevertheless, a region where SM and RSc results do not overlap can be observed, togetherwith the anticorrelation with FL.

An important issue concerns the hadronic error, the reliability of which cannot beasserted without the comparison among form factors obtained by independent non-perturbative methods. Recent lattice QCD determinations of the B → K(∗) formfactors [138, 139] can be used to estimate the size of the hadronic uncertainties affectingthe various observables we have considered. Using the set in [139] we have analyzed, e.g.,the correlation among the B → K∗νν decay rate and FL and AT . The results reportedin Figure 5.17 show that the predictions already obtained are robust within the quotederrors.

Role of the right-handed operators in RSc

The correlation between ε and η is of particular interest, in light of general analyses wherethe effects of Z ′ neutral gauge bosons are considered with no reference to the underlyingNP theory [136]. In such analyses, several possible non-diagonal couplings to left- andright-handed quarks lead to models that can be distinguished by the relative weight ofthe couplings. As an example, a left-right symmetric scenario (LRS) corresponds to Z ′

left- and right-handed couplings equal in size and sign; the ε− η correlation is differentin the various cases.

Comparing our result in Figure 5.12 with the various possibilities considered in thegeneral analysis, Figure 20 of [136], we infer that the RSc model looks similar to theRHS scenario, with a Z ′ mainly coupled to right-handed quarks. Indeed, the differenceCRSL − CSML and the coefficient CRSR , playing the role of the left- and right-handed


0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

FL

BHB

0�

K*0ΥΥL�

106

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

FL

BR@B

0 ®K*

0ΝΝD´

106

0.5 0.6 0.7 0.8 0.9 1.00

2

4

6

8

10

12

14

AT

BHB

0�

K*0ΥΥL�

106

0.5 0.6 0.7 0.8 0.9 1.00

2

4

6

8

10

12

14

AT

BR@B

0 ®K*

0ΝΝD´

106

Figure 5.17: B(B0 → K∗0νν) versus FL(B0 → K∗0νν) (top) and AT (B0 → K∗0νν) (bottom), varyingthe RSc parameters in the allowed ranges and including the uncertainty on the form factors at sB = 0(lighter blue regions) from LCSR (left) and from lattice QCD [139] (right). The SM predictions correspondto the lighter red regions. The darker blue curves and the darker red dots correspond to the RSc andSM predictions, respectively, for the central value of the form factors.

Figure 5.18: RK/K∗ , defined in Eq. (5.60), versus FL(B0 → K∗0νν). The color code is the same as inFigure 5.17.

quark couplings to a new gauge boson, have opposite sign, and CRSR � CRSL − CSML ,Figure 5.12. Although in RSc there are several additional gauge boson, the effect issimilar to the case of one new boson.

The correlation of B(B → Kνν) and B(B → K∗νν) with B(Bs → µ+µ−) provides a


Figure 5.19: Correlation between B(B0 → K0νν) and B(Bs → µµ) (top), and between B(B0 → K∗0νν)and B(Bs → µµ) (bottom) normalized to their central SM values. The hadronic uncertainty is notincluded. The blue lines correspond to RSc, the red dots to SM.

deeper insight. In NP models one has

B(Bs → µ+µ−)

B(Bs → µ+µ−)|SM=C10 − C ′10

CSM10

, (5.71)

where C10 and C ′10 are the Wilson coefficients of the semileptonic electroweak penguinoperators with axial vector leptonic current and V − A and V + A structure of thequark current in the effective b → s`+`− Hamiltonian. In SM only CSM10 is relevant(the contribution of O′10 is negligible). Evaluating CL,R, C10 and C ′10 in the RScparameter space, the correlations in Figure 5.19 are found. The rates of B → Kννand Bs → µ+µ− are anticorrelated: in RSc a lower B(Bs → µ+µ−) than in SM impliesa larger B(B → Kνν). The opposite happens for B → K∗νν: B(B → K∗νν) andB(Bs → µ+µ−) are correlated, therefore finding one of them below its SM value wouldrequire a diminution also of the other one. This again characterizes RSc as an RHSscenario, as one can infer from a comparison with the general result of Figure 21 in [136].

5.4.2 Bs → (φ, η, η′, f0(980))νν in RSc

Several Bs decay modes of great phenomenological interest are driven by the transitionb → sνν. Here we focus on Bs → (η, η′)νν, on the decay Bs → φνν and on Bs →f0(980)νν with the scalar f0(980) meson in the final state, all of them accessible at thenew facilities. The modes into η and η′ must be considered altogether, due to the η − η′mixing. We adopt the mixing scheme already shown in Section 4.3.1 for the analysisof the B → Kη(′)γ decay. The flavour symmetry permits to relate the Bs → η, η′ formfactors to the B → K ones. For a form factor F one has FBs→η = − sinϕFB→K andFBs→η

′= cosϕFB→K . On the other hand, for the Bs → φνν mode we use the LCSR

Bs → φ form factors in Ref. [60].


0 1 2 3 4 5 60

5

10

15

20

BHBs�Η Υ ΥL� 106

BHB

s�ΦΥΥL�

106

0 1 2 3 4 5 60

5

10

15

20

BHBs�Η' Υ ΥL� 106

BHB

s�ΦΥΥL�

106

0 1 2 3 4 5 60

5

10

15

20

BHBs� f0H980L Υ ΥL� 106

BHB

s�ΦΥΥL�

106

Figure 5.20: Correlation of B(Bs → φνν) with B(Bs → ηνν) (top), B(Bs → η′νν) (center) andB(Bs → f0(980)νν) (bottom). The color code is the same as in Figure 5.17.

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

FL

BHB

s�ΦΥΥL�

106

Figure 5.21: B(Bs → φνν) versus FL(Bs → φνν). The color code is the same as in Figure 5.17.

The SM predictions, obtained for τ(Bs) = 1.512± 0.007 ps [94],

B(Bs → ηνν)SM = (2.3± 0.5)× 10−6 , (5.72)

B(Bs → η′νν)SM = (1.9± 0.5)× 10−6 , (5.73)

B(Bs → φνν)SM = (13.2± 3.3)× 10−6 , (5.74)

are modified in RSc:

B(Bs → ηνν)RS ∈ [1.7− 3.3]× 10−6 , (5.75)

B(Bs → η′νν)RS ∈ [1.5− 2.8]× 10−6 , (5.76)

B(Bs → φνν)RS ∈ [8.4− 18.0]× 10−6 . (5.77)


This result is particularly relevant in the case of the Bs → φ mode, which should be thefirst one accessible for the Bs meson: the rate is within the reach of the new facilities,the φ can be easily identified and its decay modes allow to construct, e.g., the FLobservable. The various correlation patterns are shown in Figure 5.20: anticorrelation isfound between the rates of Bs → η(′)νν and Bs → φνν. For FL the results are depictedin Figure 5.21. The last mode in our analysis involves the scalar f0(980) meson. TheBs → f0(980) form factors have been determined assuming for f0(980) a dominantquark-antiquark ss structure [140]. With the updated value of τ(Bs) we find that theSM prediction is modified in RSc:

B(Bs → f0(980)νν)SM = (8.95±2.92.5)× 10−7 , (5.78)

B(Bs → f0(980)νν)RS ∈ [5− 17]× 10−7 . (5.79)

This channel should be accessible through the f0(980)→ ππ transition, providing anothertest of the RSc scenario.


Chapter 6

Search of New Physics insemileptonic B → D(∗)τ ντ decays

The decays B → D(∗)τ ντ and B → τ ντ are suitable to unveil the effects of New Physicsin charged-current interactions. Recent experimental measurements, performed by theBABAR Collaboration, about the rates of B− and B0 semileptonic decays into D(∗) anda τ lepton seem to indicate a deviation from the SM expectation. The measurements,collected in Table 6.1 (together with the previous Belle results), have been estimated todeviate at the global level of 3.4 σ with respect to SM predictions [9, 13]. These resultslead us to consider the possibility that semileptonic processes involving heavy quarksand the τ lepton are unveiling the effects of particles with large couplings to the heavierfermions, as it is natural for charged scalars which could contribute to the tree-levelb→ c`ν transitions [13,142–148].

Before the observation of these possible hints of new physics in semileptonic b→ cdecays, the first experimental analyses of the purely leptonic B− → τ−ντ mode alsoreported an excess of events. In SM the branching fraction B(B− → τ−ντ ) is given by

B(B− → τ−ντ ) =G2FMBm

2τ

8π

(1− m2

τ

M2B

)2

f2B |Vub|2 τB− , (6.1)

neglecting a tiny electromagnetic radiative correction. Using the lattice QCD averagefor the B decay constant fB = (190.6 ± 4.7) MeV quoted in [149], and varying theCabibbo-Kobayashi-Maskawa (CKM) matrix element |Vub| in the range determinedfrom inclusive and exclusive B decays: |Vub| = 0.0035± 0.0005, the prediction follows:B(B− → τ−ντ ) = (0.79± 0.23)× 10−4, in agreement with the outcome of CKM matrixfits [150,151]. This value is smaller by about a factor of 2 than the experimental resultsreported in [152–155] and compiled in [156]: B(B− → τ−ντ ) = (1.68 ± 0.31) × 10−4.However, the latest Belle [14] and BABAR [15] measurements, obtained using thehadronic tagging method,

B(B− → τ−ντ ) =(0.72+0.27

−0.25 ± 0.11)× 10−4 (Belle)

B(B− → τ−ντ ) =(1.83+0.53

−0.49 ± 0.24)× 10−4 (BABAR) , (6.2)

115

116 Search of New Physics in semileptonic B → D(∗)τ ντ decays

Table

6.1

:Sum

mary

of

mea

surem

ents

ofB→D

(∗)τν.NBB

:num

ber

ofBB

pairs

inth

edata

sam

ple

used

for

the

analy

sis,B:

bra

nch

ing

fractio

n(th

efi

rsterro

ris

statistica

l,th

eseco

nd

system

atic,

an

dth

eth

irdd

ue

toth

eb

ran

chin

gfra

ction

un

certain

tyin

the

norm

aliza

tion

mod

e);Σ

:sig

nifi

can

ceof

the

signal

inclu

din

gsy

stematic

errors,

withR

(D(∗

))th

era

tioB

(B→D

(∗)τν)/B

(B→D

(∗)`ν

).

Exp

erimen

tT

ag

NBB

(106)

B(10−

4)Σ

R(D

(∗))

Referen

ces

B0→

D∗−τ

+ντ

Belle

inclu

sive535

2.02+

0.4

0−

0.3

7 ±0.37

5.2[141]

BA

BA

Rh

ad

ronic

4711.74±

0.19±

0.1210.4

0.355±0.039±

0.021[9]

B+→D∗0τ

+ντ

Belle

inclu

sive657

2.12+

0.2

8−

0.2

7 ±0.29

8.1[141]

BA

BA

Rh

adro

nic

4711.71±

0.17±

0.139.4

0.322±0.032±

0.022[9]

B0→

D−τ

+ντ

BA

BA

Rh

ad

ronic

4711.01±

0.18±

0.125.2

0.469±0.084±

0.053[9]

B+→D

0τ+ντ

Belle

inclu

sive657

0.77±

+0.22±

0.12

3.5[141]

BA

BA

Rh

adro

nic

4710.99±

0.19±

0.134.7

0.429±0.082±

0.052[9]

B→Dτ

+ντ

(isosp

incon

strained

)

BA

BA

Rh

ad

ronic

4711.02±

0.13±

0.116.8

0.440±0.058±

0.042[9]

B→D∗τ

+ντ

(isosp

incon

strained

)

BA

BA

Rh

ad

ronic

4711.76±

0.13±

0.1213.2

0.332±0.024±

0.018[9]

117

BaBar semileptonic

BaBar hadronic

Belle semileptonic

Belle hadronic

0.0 0.5 1.0 1.5 2.0 2.5 3.0

BHB- ® τ-

ντL ´ 104

Figure 6.1: Experimental results for B(B− → τ−ντ ) [14,15,153,155] together with the SM expectationcorresponding to |Vub| = 0.0035± 0.0005 (vertical band).

are more consistent with SM, and draw the average B(B− → τ−ντ ) to a smaller value:B(B− → τ−ντ ) = (1.12 ± 0.22) × 10−4, after the combination with the semileptonictagging method results, see Figure 6.1. The different trend of the measurementsinvolving τ in B leptonic and semileptonic decay modes poses two questions. Thefirst one concerns the level of accuracy of the SM predictions for the ratios R(D(∗)) =B(B → D(∗)τν)/B(B → D(∗)`ν). The second question is which kind of new physicseffects, if any, could modify those ratios without affecting the purely leptonic mode.Indeed, several analyses devoted to try to explain the anomalies in B → D(∗)τ ντ withinNP scenarios, have considered as possible candidates models with new scalars whosecouplings to leptons are proportional to the lepton mass, to guarantee the enhancementof the τ modes. This is the case of models with two Higgs doublets (2HDM), the bestknown example being the MSSM in which two Higgs doublets are required to give massto down-type quarks and charged leptons in one case, and up-type quarks in the other.In this framework, the ratios R(D(∗)) depend on the mass of the charged Higgs H± andthe ratio β of the two Higgs doublet VEVs, and no choice of such parameters allowsto simultaneously reproduce the experimental data on R(D(∗)) [9]. Variants of the2HDM [144,145], together with other models providing explicit flavour violation [142],might explain the measurements in Table 6.1. However, an enhancement of the purelyleptonic B decay rate is generally implied in such models.

In this chapter we consider both the above mentioned issues. We analyze the SMprediction for B → D(∗)`ν`, specifying the main sources of uncertainties and possiblerefinements. Then, we scrutinize the effects of possible NP contributions in the effectiveweak Hamiltonian having a structure able to affect the ratios R(D(∗)) but leaving thepure B leptonic modes unchanged. In particular, we focus on a NP operator constructedfrom tensor quark and lepton currents. In our analysis of semileptonic B decays, we firstconsider D and D∗ mesons in the final state, and then we turn to the interesting case offinal states with excited charmed mesons.


6.1 Exclusive b→ c`ν` decays

We consider the b→ c`ν` effective Hamiltonian comprising the SM term and an additionaltensor operator:

Heff = HSMeff +HNP

eff =GF√

2Vcb[cγµ(1− γ5)b ¯γµ(1− γ5)ν`

+ε`T cσµν(1− γ5)b ¯σµν(1− γ5)ν`

]. (6.3)

GF is the Fermi constant and Vcb the CKM matrix element. ε`T is the relative complexcoupling of the new tensor term with respect to the SM one. It is assumed that themain coupling is to the heaviest lepton, hence we set ε`T = 0 for ` = e, µ and εT ≡ ετT .This coupling can be bound experimentally, so that the effects of the new operator canbe scrutinized in physical observables which, in general, are expressed as a SM, a newphysics and an interference contribution.

Such a kind of operators have been also considered in [143] and [148]; we devotethe main attention to differential distributions, namely the lepton forward-backwarddifferential asymmetries, in which the sensitivity to the new Dirac structure is maximal, asemphasized in [146] for different operators. Although there are scenarios in which tensoroperators are generated, in our analysis we do not rely on explicit models: our purpose isto identify physical observables having a mild sensitivity to hadronic uncertainties, whichtherefore can be used to unveil effects easier to interpret. It is only worth mentioningthat these operators emerge (with the help of the Fierz identities [157]), for example, inmodels with new coloured bosons carrying both lepton and baryon quantum number(referred to as leptoquarks, LQ): SU(5)GUT [158], Pati-Salam SU(4) [159], composite [160],superstrings [161] and technicolor models [162]. Leptoquarks couple to quarks and leptonsand, from limits on flavour changing neutral currents, preferably to those within thesame SM generation. Searches for leptoquarks decaying to 2τ and 2b jets, performedby the CMS Collaboration at the CERN LHC, bound (preliminarly) the mass of apossible scalar leptoquark to M(LQ) > 525 GeV, and to M(LQ) > 760 GeV for a vectorleptoquark [163]; other bounds can be found in [164].

The differential B(p)→Mc(p′)`(p1)ν`(p2) decay rate, with Mc a charmed meson,

reads:

dΓ

dq2(B →Mc`ν`) = C(q2)

[dΓ

dq2(B →Mc`ν`)

∣∣∣SM

+dΓ

dq2(B →Mc`ν`)

∣∣∣NP

+dΓ

dq2(B →Mc`ν`)

∣∣∣INT

], (6.4)

with q = p− p′ and C(q2) defined as

C(q2) =G2F |Vcb|2λ1/2(M2

B,M2Mc

, q2)

192π3M3B

(1− m2

`

q2

)2

, (6.5)

6.1 Exclusive b→ c`ν` decays 119

and where λ(x, y, z) is the triangular function. To compute the three terms in (6.4) weneed the relevant hadronic matrix elements.

6.1.1 B → D`ν`

The hadronic matrix elements in B → D`ν` can be parametrized in a standard way,⟨D(p′)

∣∣ cγµb |B(p)〉 = F1(q2)(p+ p′)µ +M2B −M2

D

q2

[F0(q2)− F1(q2)

]qµ , (6.6)

⟨D(p′)

∣∣ cσµν(1− γ5)b |B(p)〉 =FT (q2)

MB +MDεµναβp

′αpβ

+ iGT (q2)

MB +MD(pµp

′ν − pνp′µ) , (6.7)

(with FT = GT from the relation σµνγ5 = i2εµναβ σ

αβ), so that the three terms in (6.4)read:

dΓ

dq2(B → D`ν`)

∣∣∣SM

= λ(M2B,M

2D, q

2)

(1 +

m2`

2q2

)[F1(q2)

]2+M4

B

(1− M2

D

M2B

)23m2

`

2q2

[F0(q2)

]2, (6.8)

dΓ

dq2(B → D`ν`)

∣∣∣NP

=|εT |2

2

q2

(MB +MD)2λ(M2

B,M2D, q

2)

(1 + 2

m2`

q2

)×[FT (q2) +GT (q2)

]2, (6.9)

dΓ

dq2(B → D`ν`)

∣∣∣INT

= −3Re[εT ]m`

MB +MDλ(M2

B,M2D, q

2)F1(q2)

×[FT (q2) +GT (q2)

]. (6.10)

In the infinite heavy quark mass limit, formalized by the heavy quark effective theory(HQET), the form factors in (6.6-6.7) can all be related to the Isgur-Wise function ξ [165].The result is known [166,167]: expressing F1(q2) and F0(q2) in terms of two other formfactors h+(w) and h−(w):

F1(q2) =1

2√MBMD

[(MB +MD)h+(w)− (MB −MD)h−(w)] , (6.11)

M2B −M2

D

q2

[F0(q2)− F1(q2)

]=

1

2√MBMD

[(MB +MD)h−(w)

− (MB −MD)h+(w)] , (6.12)


and defining the meson momenta in terms of four-velocities, p = MBv and p′ = MDv′,

with w = v · v′ and q2 = M2B + M2

D − 2MBMDw. At the leading order in the heavyquark and αs expansion one has

h+(w) = ξ(w) , h−(w) = 0 , (6.13)

with ξ(w) the Isgur-Wise function. Also the form factors in (6.7) are related toξ(w) atthe same order expansion:

FT (q2) = GT (q2) =MB +MD√MBMD

ξ(w) . (6.14)

At the next-to-leading order, corrections must be taken into account, which at first areneeded for the study of the decay in SM. We elaborate a determination of the functionsh+, h− and ξ based on a combination of experimental and theoretical information. Theexperimental input comes from the BABAR analysis of B → Dµνµ [168], the differentialrate of which, neglecting the lepton mass, reads:

dΓ

dw(B → D`ν`) =

G2F |Vcb|248π3

M5Br

3(1 + r)2(w2 − 1)3/2[FD(w)]2 , (6.15)

with

FD(w) =

[h+(w)− 1− r

1 + rh−(w)

], (6.16)

and r =MD

MB. Using the parametrization [169]

FD(w) = FD(1){

1− 8ρ21z + (51ρ2

1 − 10)z2 − (252ρ21 − 84)z3

}(6.17)

in terms of the variable

z =

√w + 1−

√2√

w + 1 +√

2, (6.18)

from the fit of the product GBABAR(w) = FD(w)|Vcb| the BABAR Collaboration providesthe parameters GBABAR(1) = FD(1)|Vcb| and ρ2

1. The outcome of the fit is slightlydifferent for B− or B0 modes; we consider for definiteness the B0 case [168]: 1

GBABAR(1) = 44.9± 3.2± 1.6 , ρ21 = 1.29± 0.14± 0.05 . (6.19)

This result can be translated into a determination of ξ(w), expressing the form factorsh±(w) in terms of the Isgur-Wise function and including the αs and 1/mb,c correctionsworked out by M. Neubert in [166] and by I. Caprini et al., in [169]:

h+(w) =

[C1 +

w + 1

2(C2 + C3) + (εb + εc)L1

]ξ(w) = h+(w) ξ(w) , (6.20)

h−(w) =

[w + 1

2(C2 − C3) + (εc − εb)L4

]ξ(w) = h−(w) ξ(w) , (6.21)

1The average between the charged and neutral B decay modes is quoted asGBABAR(1) = 42.3±1.9±1.4,ρ2

1 = 1.20± 0.09± 0.04.


1.0 1.1 1.2 1.3 1.4 1.5 1.61

2

3

4

5

w

ΞHwLB

aBarÈV

cbÈ´

102

1.0 1.1 1.2 1.3 1.4 1.51

2

3

4

5

w

ΞHwLB

elleÈV

cbÈ´

102

Figure 6.2: Isgur-Wise function ξ(w) (times |Vcb| × 102) obtained using the BABAR data on B0 →D+`−ν` (left) and the Belle data on B0 → D∗+`−ν` (right) . The width of the curves is due to theerrors in the parameters fitted in the two cases and to the uncertainty on Λ and αs in the determinationof the form factor.

with εb =1

2mb, εc =

1

2mc. The coefficients C1,2,3 and Li are collected in Appendix B.1.

Ci account for the perturbative corrections, Li for the heavy quark mass corrections anddepend on the hadronic parameter Λ, the difference between the heavy meson (B, D)and the heavy quark (b, c) mass in the heavy quark limit. We use mb = 4.8 GeV andmc = 1.4 GeV and a conservative value Λ = 0.5± 0.2 GeV [166], so that the uncertaintyin Λ encompasses the error on Λ/mb and Λ/mc. The Isgur-Wise function ξ(w) resultingfrom the expression

|Vcb| ξ(w) =GBABAR(w)[

h+(w)− 1−r1+r h−(w)

] (6.22)

is depicted in Figure 6.2 (left panel).The form factors needed for analysis of the mode with τ can be separately derived

using again Eqs. (6.20, 6.21):

|Vcb|h+(w) =1

1− 1−r1+r A(w)

GBABAR(w) , (6.23)

|Vcb|h−(w) =A(w)

1− 1−r1+r A(w)

GBABAR(w) , (6.24)

with A = h−/h+. For the matrix elements of the tensor operator, we use ξ(w) also in(6.14). In the standard model, the results for the semileptonic B0 → D+ branchingfractions can be quoted as

B(B0 → D+`−ν`)∣∣∣SM

= (2.15± 0.45)× 10−2 , (6.25)

B(B0 → D+τ−ντ )∣∣∣SM

= (0.70± 0.12)× 10−2 , (6.26)

and, taking the correlation between the predictions for ` and τ into account,

R0(D)∣∣∣SM

=B(B0 → D+τ−ντ )

B(B0 → D+`−ν`)

∣∣∣SM

= 0.324± 0.022 . (6.27)


The SM prediction for R0(D) deviates from the measurement collected in Table 6.1(with statistic and systematic uncertainties combined in quadrature) by about 1.5 σ .The deviation is smaller in the charged R−(D) case.

The stability of (6.27) against changes of the input information on form factors isnoticeable: sensitivity to 1/mQ corrections can be estimated varying Λ, and this modifiesthe central value at a few per mille level. Sensitivity to the radiative corrections can beassessed changing the scale in αs as indicated in Appendix B.1, and also these correctionsare not effective. Since the value at zero recoil GBABAR(1) cancels out in the ratio, themain uncertainty in (6.27) comes from the error on the parameter ρ2

1 experimentallydetermined. The value of R0(D) coincides with the one obtained using the form factorsF1 and F0 from lattice QCD with finite quark masses [143].

6.1.2 B → D∗`ν`

While the results for R0(D) and R−(D) do not display a statistically significant deviationfrom the SM expectation, the case ofR0(D∗), R−(D∗) is quite different. The standardparameterization of the B → D∗ matrix element in terms of form factors is

⟨D∗(p′, ε)

∣∣ cγµ(1− γ5)b∣∣B(p)

⟩= − 2V (q2)

MB +MD∗iεµναβε

∗νpαp′β

−{

(MB +MD∗)

[ε∗µ −

(ε∗ · q)q2

qµ

]A1(q2)− (ε∗ · q)

MB +MD∗

×[(p+ p′)µ −

M2B −M2

D∗

q2qµ

]A2(q2) + (ε∗ · q)2MD∗

q2qµA0(q2)

}, (6.28)

(with the condition A0(0) =MB +MD∗

2MD∗A1(0)− MB −MD∗

2MD∗A2(0)) and

⟨D∗(p′, ε)

∣∣ cσµν(1− γ5)b∣∣B(p)

⟩= T0(q2)

ε∗ · q(MB +MD∗)2

εµναβpαp′β

+ T1(q2)εµναβpαε∗β + T2(q2)εµναβp

′αε∗β + i[T3(q2)(ε∗µpν − ε∗νpµ)

+ T4(q2)(ε∗µp′ν − ε∗νp′µ) + T5(q2)

ε∗ · q(MB +MD∗)2

(pµp′ν − pνp′µ)

], (6.29)

with ε, the D∗ polarization vector. We choose the helicity basis for D∗

εµL =1

MD∗

(|~p′|, 0, 0, E′

), εµ± =

1√2

(0, 1,∓i, 0) , (6.30)

with E′ and ~p′ the D∗ energy and three-momentum in the B rest frame (E′ =√M2D∗ + |~p′|2 and |~p′| = λ(M2

B,M2D∗ , q

2)/2MB). The conditions εµa · p′ = 0 and

εµa · εµ,b = −δab, with a, b = L,±, are fulfilled. The differential decay rates for the


longitudinal and the transverse D∗ polarization in terms of form factors are obtainedfrom

dΓLdq2

(B → D∗`ν`)∣∣∣SM

=1

4M2D∗

{6λ(M2

B,M2D∗ , q

2)M2D∗m2`

q2[A0(q2)]2

+

(1 +

m2`

2q2

)[(MB +MD∗)(M

2B −M2

D∗ − q2)A1(q2)− λ(M2B,M

2D∗ , q

2)

MB +MD∗A2(q2)

]2}

,

(6.31)

dΓLdq2

(B → D∗`ν`)∣∣∣NP

= |εT |2q2

8

(1 +

2m2`

q2

)[λ(M2

B,M2D∗ , q

2)

MD∗(MB +MD∗)2T0(q2)

+ 2M2B +M2

D∗ − q2

MD∗T1(q2) + 4MD∗ T2(q2)

]2

, (6.32)

dΓLdq2

(B → D∗`ν`)∣∣∣INT

= −Re(εT )3m`

4(MB +MD∗)

[(MB+MD∗)

2(M2B−M2

D∗−q2)A1(q2)

−λ(M2B,M

2D∗ , q

2)A2(q2)]×[

λ(M2B,M

2D∗ , q

2)

M2D∗(MB +MD∗)2

T0(q2)+2(M2

B +M2D∗ − q2)

M2D∗

T1(q2)+4T2(q2)

],

(6.33)

dΓ±dq2

(B → D∗`ν`)∣∣∣SM

= q2

(1 +

m2`

2q2

){(MB+MD∗)

2[A1(q2)]2+λ(M2

B,M2D∗ , q

2)

(MB +MD∗)2[V (q2)]2

},

(6.34)

dΓ±dq2

(B → D∗`ν`)∣∣∣NP

= |εT |2(

1 +2m2

`

q2

){λ(M2

B,M2D∗ , q

2)[T1(q2) + T2(q2)]2

+ 2q2[M2B[T1(q2)]2 +M2

D∗ [T2(q2)]2 + (M2B +M2

D∗ − q2)T1(q2)T2(q2)]

}, (6.35)

dΓ±dq2

(B → D∗`ν`)∣∣∣INT

= −Re(εT )3m`

{2q2(MB +MD∗)A1(q2)T1(q2)

+

[(MB +MD∗)(M

2B −M2

D∗ − q2)A1(q2)− λ(M2B,M

2D∗ , q

2)

(MB +MD∗)V (q2)

][T1(q2)+T2(q2)]

},

(6.36)


to be multiplied by the factor C(q2) in (6.5). We have used the combinations

T0(q2) = T0(q2)− T5(q2) ,

T1(q2) = T1(q2) + T3(q2) , (6.37)

T2(q2) = T2(q2) + T4(q2) .

At the leading order in the heavy quark expansion, the form factors in (6.28) and(6.29) are related to the Isgur-Wise function, while other contributions appear at thenext-to-leading order. Analogously to the decay to D, one expresses V and Ai in termsof form factors hV and hAi ,

V (q2) =MB +MD∗

2√MBMD∗

hV (w) ,

A1(q2) =√MBMD∗

w + 1

MB +MD∗hA1(w) ,

A2(q2) =MB +MD∗

2√MBMD∗

[hA3(w) +

MD∗

MBhA2(w)

],

A0(q2) =1

2√MBMD∗

[MB(w + 1)hA1(w)− (MB −MD∗w)hA2(w)

− (MBw −MD∗)hA3(w)], (6.38)

with q2 = M2B + M2

D∗ − 2MBMD∗w. Including αs and1

mband

1

mccorrections, the

relations have been worked out [166,169]:

hV (w) = [C1 + εc(L2 − L5) + εb(L1 − L4)] ξ(w) , (6.39)

hA1(w) =

[C5

1 + εc

(L2 −

w − 1

w + 1L5

)+ εb

(L1 −

w − 1

w + 1L4

)]ξ(w) , (6.40)

hA2(w) =[C5

2 + εc(L3 + L6)]ξ(w) , (6.41)

hA3(w) =[C5

1 + C53 + εc(L2 − L3 − L5 + L6) + εb(L1 − L4)

]ξ(w) . (6.42)

The expressions of Ci, which incorporate the radiative corrections, andLi are collectedin Appendix B.1: the Li terms account for the O(1/mQ) corrections in the heavy quarkexpansion, and are determined from QCD sum rule analyses of the subleading formfactors [166]. On the other hand, the relations of the form factors Ti in (6.29) to ξ(w) inthe heavy quark limit are:

T0(q2) = T5(q2) = 0 ,

T1(q2) = T3(q2) =

√MD∗

MBξ(w) , (6.43)

T2(q2) = T4(q2) =

√MB

MD∗ξ(w) .


We use these expressions in the analysis of the tensor operator.Let us focus on the SM. Due to the heavy quark spin symmetry a unique form factor

describes both B → D and B → D∗ transitions, so that we could use the Isgur-Wisefunction found in the previous section. To partially take into account the differentexperimental systematics, we choose to use the determination of ξ obtained by BelleCollaboration from the analysis of B0 → D∗+µνµ [170], for which the differential decayrate, neglecting the lepton mass, is

dΓ

dw(B → D∗`ν`) =

G2F |Vcb|248π3

(MB −MD∗)2M3

D∗G(w)F2(w) , (6.44)

with

G(w)F2(w) = h2A1

(w)√w2 − 1 (w + 1)2

{2[

1− 2wr∗ + r∗2

(1− r∗)2

] [1 +R1(w)2w − 1

w + 1

]+

[1 + (1−R2(w))

w − 1

1− r∗]2

} . (6.45)

In (6.45) r∗ =MD∗

MB, and G, R1 and R2 are given by

G(w) =√w2 − 1(w + 1)2

[1 + 4

w

w + 1

1− 2wr∗ + r∗2

(1− r∗)2

],

R1(w) = (R∗)2w + 1

2

V (w)

A1(w), (6.46)

R2(w) = (R∗)2w + 1

2

A2(w)

A1(w),

with R∗ = 2

√MBMD∗

MB +MD∗. The three unwnown functions in (6.45, 6.46) have been

determined by Belle adopting the parametrization [169]

hA1(w) = hA1(1)[ 1− 8ρ2z + (53ρ2 − 15)z2 − (231ρ2 − 91)z3] , (6.47)

R1(w) = R1(1)− 0.12 (w − 1) + 0.05 (w − 1)2 , (6.48)

R2(w) = R1(1) + 0.11 (w − 1)− 0.06 (w − 1)2 , (6.49)

(with z defined in (6.18)). The fit of the parameters in (6.47–6.49) is quoted as [170]

F(1)|Vcb| = (34.6± 0.2± 1.0)× 10−3 ,

ρ2 = 1.214± 0.034± 0.009 ,

R1(1) = 1.401± 0.034± 0.018 , (6.50)

R2(1) = 0.864± 0.024± 0.008 .

From these expressions one can reconstruct ξ(w):

hA1(w) = hA1(w) , ξ(w) (6.51)


with hA1 defined through Eq. (6.40). The fit provides us with the determination depictedin Figure 6.2 (right panel). Through Eqs. (6.39, 6.41, 6.42) the form factors hV , hA2

and hA3 can be reconstructed including the NLO 1/mQ and αs corrections, and alsoB(B0 → D∗+τ−ν`) can be computed. The results are:

B(B0 → D∗+`−ν`)∣∣∣SM

= (4.62± 0.33)× 10−2 ,

B(B0 → D∗+τ−ντ )∣∣∣SM

= (1.16± 0.08)× 10−2 , (6.52)

and, taking the correlation between the predictions for the ` and τ mode into account,

R0(D∗)∣∣∣SM

=B(B0 → D∗+τ−ντ )

B(B0 → D∗+`−ν`)

∣∣∣SM

= 0.250± 0.003 . (6.53)

The result (6.53) deviates from the measurement in Table 6.1 (with statistic andsystematic errors combined in quadrature) by 2.3 σ. It coincides with the one in[13, 144, 148], due to the stability of the ratio R0(D∗) against changes of the inputparameters: varying the central value of Λ and of the quark masses by 30% producesless than 1% variation in the result. The radiative corrections, changing the scale inαsas mentioned in Appendix B.1, do not produce an appreciable variation of the result.On the other hand, in the individual branching fractions there is a mild sensitivity to Λ:setting this parameter to zero (i.e. ignoring 1/mQ corrections) the branching fractionsin (6.52) are reduced by about 5%. In the charged case, there is a deviation of 1.8 σbetween the SM prediction for R(D∗) and the measurement in Table 6.1.

6.2 Impact of the tensor operator on R(D(∗)) and otherobservables

If the tensions in R(D) and R(D∗) are due to NP effects, it is interesting to investigatethe new operator in the effective Hamiltonian (6.3) which affects the observables inB → D(∗)τντ transitions, focusing on the signatures with minimal dependence onhadronic quantities. As done in [13, 142–146, 148], R(D) and R(D∗) data allow toconstrain the values of the new effective dimensionless coupling. In our case εT isbounded as shown in Figure 6.3. Using the parameterization

εT = |aT |eiθ + εT0 , (6.54)

the tightest bound to εT0 and |aT | is obtained from the measurement of R(D∗), whilethe combination of R(D) and R(D∗) data fixes the range of the phase θ; in the

overlap region, the function χ2(εT ) =(R(D,ε)−R(D)exp

∆R(D)exp

)2+(R(D∗,ε)−R(D∗)exp

∆R(D∗)exp

)2has

values running between 1.51 and 1.75. The permitted range of εT is represented as

Re[εT0] = 0.17 , Im[εT0] = 0 ,

|aT | ∈ [0.24, 0.27] , (6.55)

θ ∈ [2.6, 3.7] rad ,

6.2 Impact of the tensor operator on R(D(∗)) and other observables 127

-1.0 -0.5 0.0 0.5 1.0-1.0

-0.5

0.0

0.5

1.0

Re@ΕTD

Im@Ε TD

-0.2 -0.1 0. 0.1-0.2

-0.1

0.0

0.1

0.2

Re@ΕTDIm@Ε TD

ø

Figure 6.3: (left) Regions in the (Re(εT ), Im(εT )) plane determined from the experimental data (to1 and 2σ) on R(D) (large rings) and R(D∗) (small rings). (right) Region corresponding to values ofχ2 between the minimum (indicated by the star), 1 .55 (yellow, light) and 1.65 (orange, gray) and 1.75(brown, dark).

and is also depicted in Figure 6.3. Varying the effective coupling in this region we cananalyze the impact of the new operator on various differential distributions.

We start with the longitudinal and transverse D∗ polarization distributions inB → D∗τ ντ . We consider the decay to a D∗ with definite helicity, with differential

decay widthdΓL,±dq2

for the three cases L,±. We definedΓTdq2

=dΓ+

dq2+dΓ−dq2

, and show

in Figure 6.4 the differential branching fractions. The uncertainty in the distributionsreflects the uncertainty on the parameters of the Belle Isgur-Wise function, on Λ and, inthe case of NP, on εT . While the shape of the distributions are slightly modified fromSM to NP, the maxima increase, a consequence of the increase of the branching fractions.

Other observables are the longitudinal and transverse D∗ polarization distributionsin B → D∗τ ντ normalized to B → D∗`ν`. They are defined as

RD∗

L,T (q2) =dΓL,T (B → D∗τ ντ )/dq2

dΓL,T (B → D∗`ν`)/dq2. (6.56)

The SM predictions are shown in Figure 6.5 together with the modifications inducedby the tensor operator. At large q2 the observables are enhanced by 30 − 50 %, anoticeable effect. Furthermore, at odds with scenarios in which only RL is affectedby new physics [144], in the case of the tensor operator both the longitudinal and thetransverse RL and RT distributions are modified.

The longitudinal and transverse polarization fractions of the D∗ meson

FL,T (q2) =dΓL,T (B → D∗τ ντ )

dq2×(dΓ(B → D∗τ ντ )

dq2

)−1

, (6.57)


4 6 8 100.

0.04

0.08

0.12

0.16

q2 @GeV2D

dBL

dq2´

102@G

eV-

2 DB

0®D*+ Τ ΝΤ

SM

4 6 8 100.

0.05

0.1

0.15

0.2

0.25

q2 @GeV2D

dBT

dq2´

102@G

eV-

2 D

B0®D*+ Τ ΝΤ

SM

Figure 6.4: Differential branching ratios with polarized D∗:dB(B → D∗τ ντ )L

dq2(left) and

dB(B → D∗τ ντ )Tdq2

(right). The lower (blue) bands are the SM prediction, the upper (orange) bands

include NP effects. In SM, the uncertainties on the parameters of the Isgur-Wise function in Eq.(6.50),together with the errors on Λ and αs are included. In the case of the NP curves, the uncertainty on εTis also considered.

3 4 5 6 7 8 9 100.0

0.2

0.4

0.6

0.8

q2 @GeV2D

RLHq

2 L

B0®D*+ Τ ΝΤ

SM

3 4 5 6 7 8 9 100.0

0.2

0.4

0.6

0.8

q2 @GeV2D

RTHq

2 L

B0®D*+ Τ ΝΤ

SM

Figure 6.5: D∗ polarization ratios RD∗

L (q2) (left) and RD∗

T (q2) (right) defined in (6.56). Notations arethe same as in Figure 6.4.

are shown in Figure 6.6. Both the SM and NP predictions are affected by a small error,since in the heavy quark limit the observables in (6.57) are free of hadronic uncertainties,due to the cancellation of the form factor ξ(w) in the ratio. The residual uncertaintyreflects that on Λ which controls the 1/mQ corrections. The uncertainty on Λ alsoenters in the curves obtained in the NP scenario in combination with εT . In SM, FL(q2)ranges between 0.75 at low q2 and about 0.35 at high squared momentum transfer; inNP in the allowed region of εT , FL(q2) is between 0.35 and about 0.65 at low q2, whilethis observable converges to the SM value at high q2. The SM predicts a dominantlongitudinal polarization at small q2, in NP the longitudinal and transverse polarizationshave similar fractions up to q2 = 6 GeV2.

An important observable is the forward-backward AFB(q2) asymmetry in B → Dτντ

6.2 Impact of the tensor operator on R(D(∗)) and other observables 129

4 6 8 10

0.4

0.5

0.6

0.7

q2 @GeV2D

FL

B0®D*+ Τ ΝΤ

SM

4 6 8 10

0.3

0.4

0.5

0.6

q2 @GeV2D

FT

B0®D*+ Τ ΝΤ

SM

Figure 6.6: Polarization fractions FL(q2) (left) and FT (q2) (right) for B → D∗τ ντ defined in (6.57).Notations are the same as in Figure 6.4.

and B → D∗τ ντ , defined as

AFB(q2) =

´ 10 d cos θ`

dΓdq2d cos θ`

−´ 0−1 d cos θ`

dΓdq2d cos θ`

dΓdq2

, (6.58)

where θ` is the angle between the direction of the charged lepton and the D(∗) meson inthe lepton pair rest frame. We use the notation

AFB(q2) =1dΓdq2

3C(q2)

16

{ASMFB (q2) + ANPFB (q2) + AINTFB (q2)

}, (6.59)

with C(q2) defined in (6.5) and the three terms in the parentheses given for D and D∗:

• D

ASMFB (q2) = 8F0(q2)F1(q2)(M2B −M2

D)m2`

q2

(1− m2

`

q2

)λ1/2(M2

B,m2, q2) , (6.60)

ANPFB (q2) = 0 , (6.61)

AINTFB (q2) = −8Re(εT )F0(q2)[FT (q2) +GT (q2)](MB −MD)m`

(1− m2

`

q2

)× λ1/2(M2

B,m2, q2) . (6.62)

• D∗

ASMFB (q2) =4

MD∗(MB +MD∗)q2

(1− m2

`

q2

)λ1/2(M2

B,M2D∗ , q

2){m2À0(q2)[A1(q2)(MB +MD∗)

2(M2B −M2

D∗ − q2)

− λ(M2B,M

2D∗ , q

2)A2(q2)]− 4MD∗(MB +MD∗)q4A1(q2)V (q2)} , (6.63)


4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

q2 @GeV2D

AF

BHq

2 LB

0®D+ Τ ΝΤ

SM

4 6 8 10-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

q2 @GeV2D

AF

BHq

2 L

B0®D*+ Τ ΝΤ

SM

Figure 6.7: Forward-backward asymmetry AFB(q2) for B → Dτντ (left) and B → D∗τ ντ (right). Thelower (blue) curves are the SM predictions, the upper (orange) bands the NP expectations. Uncertaintyon Λ has been included and, in the case of NP, also on the parameters |aT | and θ.

ANPFB (q2) = 16|εT |2m2`

q2

(1− m2

`

q2

)λ1/2(M2

B,M2D∗ , q

2)(T1(q2) + T2(q2))

× [(M2B −M2

D∗)(T1(q2) + T2(q2)) + q2(T1(q2)− T2(q2))] , (6.64)

AINTFB (q2) = −4Re(εT )m`

(1− m2

`

q2

)λ1/2(M2

B,M2D∗ , q

2)

{4(MB +MD∗)A1(q2)

× (T1(q2) + T2(q2))

+A0(q2)

[λ(M2

B,M2D∗ , q

2)

MD∗(MB +MD∗)2T0(q2) + 2

M2B +M2

D∗ − q2

MD∗T1(q2) + 4MD∗ T2(q2)

]− V (q2)

MB +MD∗

[q2(T1(q2)− T2(q2)) + (M2

B −M2D∗)(T1(q2) + T2(q2))

]}. (6.65)

In Figure 6.7 we plot AFB(q2) for B → Dτντ and B → D∗τ ντ . The SM prediction isaffected by almost no theoretical uncertainty, because of a nearly complete cancellationof the hadronic parameters in the ratio. In the case of NP, we have taken into accountalso the uncertainty on θ and |aT |. The SM curve lies in both cases below the NPdistribution for all values of q2. The most interesting deviation concerns the D∗ mode:the SM predicts a zero for AFB at q2 ' 6.15 GeV2, in the NP case the zero is shiftedtowards larger values q2 ∈ [8.1, 9.3] GeV2. Even though the experimental determinationof the zero of the forward-backward asymmetry is challenging, this observable effectivelydiscriminates SM from the NP model. The integrated asymmetries, obtained integratingseparately the numerator and the denominator in (6.58), are collected in Table 6.2: forD∗, in the NP scenario the integrated asymmetry has opposite sign with respect to SM.

6.3 Tensor operator in B → D∗∗`ν` decays 131

Table 6.2: Integrated forward-backward asymmetry for the considered decay modes. The first linereports the SM results, in the second line the effect of the tensor operator is included.

B0 → D+τ ντ B0 → D∗+τ ντ B0 → D∗+0 τ ντ B0 → D′+1 τ ντ B0 → D+1 τ ντ B0 → D∗+2 τ ντ

ASMFB 0.357± 0.002 −0.040± 0.003 0.315 0.026 0.24 0.07

AFB 0.40± 0.005 0.048± 0.013 0.30± 0.005 0.08± 0.01 0.21± 0.003 0.14± 0.01

6.3 Tensor operator in B → D∗∗`ν` decays

The new operator in the effective Hamiltonian (6.3) affects other exclusive decay modesthat are worth investigating. Of peculiar interest are the semileptonicB and Bs transi-tions into excited charmed mesons. The lightest multiplet of such hadrons, correspondingto the quark model p-wave (` = 1) mesons and generically denoted D∗∗(s), comprises fourpositive parity states which, in the heavy quark limit, fill two doublets labeled by the(conserved) angular momentum ~s` = ~sq + ~ (~sq is spin of the light antiquark), hences` = 1/2 or s` = 3/2. The two mesons belonging to the first doublet, (D∗(s)0, D

′(s)1), have

spin-parity JP = (0+, 1+); the mesons in the second doublet have JP = (1+, 2+) and arenamed (D(s)1, D

∗(s)2). All the members of the doublets, with and without strangeness,

have been observed, and the two sP` = 1/2+ states without strangeness are found to bebroad, as expected [171].

In the heavy quark limit also the semileptonic B transitions to mesons belonging tothe same charmed doublet can be described in terms of a single form factor. B decays to(D∗0, D

′1) are governed by a universal function denoted as τ1/2(w), B decays to (D1, D

∗2)

by the τ3/2(w) form factor (the matrix elements are collected in Appendix B.2). Thereare several determinations of the τi(w) parametrized in terms of the zero-recoil value τi(1)(contrary to the Isgur-Wise function, τi(w) are not normalized to unity at w = 1), of theslope ρ2

i and of the curvature ci. In the ratios of branching fractions and asymmetriesthe zero-recoil value does not play any role, and this reduces the main dependence ofthe observables on the hadronic parameters. The present experimental situation needsto be settled, since the semileptonic B decay rates into (D∗0, D

′1) exceed the predictions

obtained using computed τi(1); the origin of the discrepancy is still unknown, and couldbe related to the broad widths of the final charmed mesons, which determine a difficultyin the exclusive reconstruction, and to a possible pollution from other (e.g. radial)excited states. Semileptonic Bs decays to sP` = 1/2+ cs mesons could clarify the issue,due to the narrow width of the strange charmed resonances [172]. On the other hand,the tensor operator produces precise correlations among various observables, thereforeits effects could be distinguished from others.

For definiteness, we use a QCD sum rule determination of τ3/2(w) at leading orderin αs [173,174], and of τ1/2(w) at O(αs) [175]:

τ3/2(w) = τ3/2(1)[1− ρ2

3/2(w − 1)], (6.66)

τ1/2(w) = τ1/2(1)[1− ρ2

1/2(w − 1) + c1/2(w − 1)2], (6.67)


ææ

òò

0.08 0.1 0.12 0.14 0.160.08

0.1

0.12

0.14

0.16

0.18

RH DHsL 0* L

RHDHsL

1'L

ææ

òò

0.06 0.07 0.08 0.09

0.05

0.07

0.09

0.11

RHDHsL 1L

RHDHsL

2*

L

Figure 6.8: (left) Correlations between the ratiosR(D∗(s)0) and R(D′(s)1) for mesons belonging to the(D∗(s)0, D

′(s)1) doublet without (orange, dark) and with strangeness (green, light). (right) Correlation

between R(D(s)1) and R(D∗(s)2) for mesons in the (D(s)1, D∗(s)2) doublet. The dots (triangles) correspond

to the SM results for mesons without (with) strangeness.

with

τ3/2(1) = 0.28 , ρ23/2 = 0.9 , (6.68)

τ1/2(1) = 0.35 ,±0.08 ρ21/2 = 2.5± 1.0 , c1/2 = 3± 3 . (6.69)

The differential decay rates for B → D∗∗`ν` can be written as in (6.4), see AppendixB.2. The ratios

R(D∗0) =B(B → D∗0τ ντ )

B(B → D∗0` ν`)(6.70)

and the analogous R(D′1), R(D1) and R(D∗2) depend on the effective coupling εT . Thisalso happens in Bs → D∗∗s `ν` transitions, in the SU(3)F symmetry limit for the formfactors.

In Figure 6.8 for each meson doublet we show the correlation between the ratios(6.70) for B and Bs, together with the SM predictions (R(D∗0),R(D′1)) = (0.077, 0.100),(R(D∗s0),R(D′s1)) = (0.107, 0.112), (R(D1),R(D∗2) = (0.065, 0.059) and (R(Ds1),R(D∗s2)= (0.060, 0.055). The tensor operator produces a sizeable increase in the ratios R, whichis correlated for the two members in each doublet. The hadronic uncertainty is mild:using the τi functions in [176], the results remain almost unchanged in the case of thes` = 3/2 doublet, while for s` = 1/2 they are smaller by about 25% in SM and in theNP case. The same effect is found using the form factors obtained by lattice QCD [177].The differential forward-backward asymmetries in the case of the four positive paritycharmed mesons are collected in Figure 6.9, and the integrated ones in Table 6.2. Whilein B → (D∗0, D1)τ ντ the forward-backward asymmetry does not discriminate betweenSN and NP, in the modes with D′1 and D∗2 it is a sensitive observable: The inclusion ofthe tensor operator produces an enhancement ofAFB with respect to SM for all valuesof q2. Moreover, in SM there is a zero which, in the case ofB → D′1τ ντ moves towardslarger values of q2, and disappears in B → D∗2τ ντ once NP is included.

6.3 Tensor operator in B → D∗∗`ν` decays 133

3 4 5 6 7 8-0.1

0.0

0.1

0.2

0.3

0.4

0.5

q2 @GeV2D

AF

BHq

2 L

B0®D0

*+ Τ ΝΤ

SM

3 4 5 6 7 8-0.1

0.0

0.1

0.2

0.3

0.4

0.5

q2 @GeV2D

AF

BHq

2 L

B0®D'1

+ Τ ΝΤ

SM

3 4 5 6 7 8-0.1

0.0

0.1

0.2

0.3

0.4

q2 @GeV2D

AF

BHq

2 L

B0®D1

+ Τ ΝΤ

SM

3 4 5 6 7 8-0.1

0.0

0.1

0.2

0.3

0.4

q2 @GeV2D

AF

BHq

2 L

B0®D2

*+ Τ ΝΤ

SM

Figure 6.9: Forward-backward asymmetry AFB for the decays B → D∗0τ ντ (top, left), B → D′1τ ντ(top, right), B → D1τ ντ (bottom, left) and B → D∗2τ ντ bottom, (right) as function of q2. The solid(blue) curves are the SM predictions, the dotted (orange) bands the NP expectations.

Finally, we remark that, while the tensor operator in (6.3) does not affect the purelyleptonic Bc → τ−ντ mode, it can have an impact on the transitions Bc → (ηc, J/ψ)τ−ντand Λb → Λcτ

−ντ ; therefore, sets of other observables can be identified and investigated,with precise correlated deviations from the SM predictions.


Conclusions

The search of physics beyond the Standard Model (SM) is the main goal of elementaryparticle physics in next years, assuming the effectiveness of the theory up to the TeVscale in according to the naturalness principle. A comprehensive research program hasbeen scheduled at the CERN LHC, motivated by the expectation of a new scale emergingat energies around 1 TeV. However, the direct detection of new particles is not theonly way to look for New Physics (NP). New states may virtually affect B, D and kaondecays, and since quantum effects typically become smaller with the increasing of themass of the virtual particles, high-precision measurements are required for an extendingmass reach.

Flavour physics is the best place for indirect NP search, since Flavour ChangingNeutral Currents (FCNC), neutral meson-antimeson mixing and CP violation, arepotentially sensitive to O(1) NP virtual corrections. The latter depend on a parameterspace that includes the new dynamical scale and NP flavour and CP-violating couplingsrelated, on a general ground, to the masses and the Yukawa couplings of the newparticles. For instance, small flavour effects could originate either/both from large NPscale or/and small couplings. Therefore, a possible flavour-blindness of the NP structurecould in principle undermine the entire search program. Minimal Flavour Violation(MFV) analysis on a wide sample of flavour-violating processes reassures us that even inabsence of new sources of flavour and CP violation in the NP framework, the existing SMflavour-violating processes (although rare) are enough to produce a new phenomenologywhich turns out to be sensitive to exotic particles. It is also worth reminding that,beyond the MFV scenario, a highly non trivial flavour pattern could actually come true.

The astonishing achievements coming from the B Factories together with the Tevatronresults on Bs physics, have already disclosed their effectiveness in constraining NPscenarios. A few discrepancies exist in the current data, although several measurementsalone do not approach 10% accuracy [178]. To this end, an important message has tobe learned: precision is crucial in this kind of studies. With the precision reachable atnew B factories (e.g. Belle II at KEK, Tsukuba) and the LHCb experiment, the currentdiscrepancies would indicate the presence of NP in the flavour sector. It is worth notingthat LHCb and the flavour factory program, with their aim to measure different decayprocesses much better than before, are complementary in the effort to observe NP effectsfrom large scales.

135

136 Conclusions

The present thesis should be thought as being part of the research program, promptedby new and stimulating anomalies measured at the LHCb and flavour factories1. We havefocused our efforts in the investigation of several NP scenarios on a set of CP conservingobservables (mainly branching ratios and angular asymmetries) in different FCNC andrare charged-weak currents processes. To this end, we have shown the impact of the NPeffective scale and the new couplings, both in specific extra-dimensional models and ingeneric model-independent scenarios, on several correlation plots of such observables. Insome cases, the resulting pattern will help to discern among different scenarios the rightone, when improved data will be available.

Here is the summary of the main results obtained by our analyses:

(i) In Chapter 4 we have studied several exclusive b → sγ induced transition, inorder to constrain the effective scale of an extra-dimensional model with twoUniversal Extra Dimensions (UEDs) compactified on a square. Such a model hasbeen developed as an alternative to SUSY for the solution to the Dark Matter(DM) problem. In fact, it predicts natural DM candidates among the lighteststable Kaluza-Klein modes. To this end, it is worth constraining the NP scalecharacterizing such a scenario, which has a relation to the compactification radius(R) of the extra-dimensional manifold, and to the masses of the exotic particles.Flavour physics help to put solid constraints, especially through rare B decaysmediated by FCNC transitions. We found the strongest bound coming fromB0,+ → K∗0,+γ, though a certain dependence from hadronic uncertainties is alsopresent. Here are the results [179]: 1/R ≥ 397 GeV (charged channel, form factorsin [59]), 1/R ≥ 564 GeV (charged channel, form factors in [60]), 1 /R ≥ 433 GeV(neutral channel, form factors in [59]), 1/R ≥ 710 GeV (neutral channel, formfactors in [60]). As for B0 → K∗02 γ, we obtain 1/R ≥ 324 GeV. It is also worthnoting that, in the simplest case of a single UED, a combination of analogousflavour bounds together with some astrophysical measurements, severely constrainthe model, for a Higgs mass close to 125 GeV. Stronger constraints can be alsoprovided for the scenario with two UEDs.

(ii) In the same framework, we have also considered the three-body B → Kη(′)γdecays [92]. B decays to two light pseudoscalar mesons and a photon are interesting,as witnessed by the experimental efforts to determine the branching fractions aswell as the mixing-induced and direct CP asymmetry parameters. Furthermore,these three-body channels have also been proved to be suitable enough to unveil NPeffects in the polarization of the emitted photon. We have studied such channelsconsidering the contribution of several intermediate states, K∗(892) and K∗2 (1430),as well as B∗, B∗s and φ(1020). To carry out this calculation, an explicit light-conesum rule determination of the form factor TB

∗→K1 (q2) has been performed: this

form factor is of interest, since it enters in other amplitudes involving virtual B∗

mesons. Introducing a strong phase θ between the sum of the first two contributionsand the sum of the other three, we found that the experimental branching fraction

1Even if the BABAR experiment ceased operation on 7 April 2008, data analysis is still ongoing.

137

of the B → Kηγ mode can be reproduced. On the other hand, the experimentaluncertainties in the datum for B(B → Kη′γ) are large, so that comparison of thetheoretical result with the experimental outcome does not provide constraints, atpresent. The modes with η′ in the final state are not enhanced with respect to thosewith the η, as experimentally observed, at odds with the two-body nonleptonicB → Kη(′) decays. The photon spectrum, as well as the Dalitz plots, are useful todetermine the structure of the intermediate contributions.

The same transitions have been studied in NP scenarios with one and two universalextra dimensions. In particular, we have found that, in the case of two UEDscompactified on the chiral square, the bound 1/R > 450 GeV.

(iii) In Chapter 5 we carried out two analyses onB0 → K∗0`+`− and several b→ sννdecay modes in the framework of the custodially protected Randall-Sundrum model(RSc). Such a scenario emerged as a viable alternative to SUSY as a solution tothe hierarchy problem, become even more challenging with the discover of a lightHiggs boson. Recently, it has been also argued that RS models can be thought asdual descriptions of strongly coupled dynamical systems (in some modern versionsof compositeness models) via the well-known AdS/CFT conjecture. We have beenalso prompted by new puzzling LHCb measurements which show discrepancies withrespect to the Standard Model (SM) predictions in selected angular distributionsof the mode B0 → K∗0µ+µ−. We have studied several observables of the raredecays B0 → K∗0`+`− within the RSc model, in the case where measurementsare available and in the case of massive leptons [180]. We have carried out a

new calculation of the contributions to the Wilson coefficients C(′)7,8 and a new

scan of the model parameters, imposing the experimental constraints of CKMand quark mass values. The obtained deviations with respect SM are small (anon trivial prediction of the model, due to the small correction beyond the SM

of the C(′)9,10 Wilson coefficients), and at present they are generally hidden by the

uncertainties on the hadronic form factors for several observables. However, infew cases the deviations from SM are systematic in the full q2 range, and aresimilar to the presently accepted hadronic uncertainties. This fact renders suchobservables of great interest in view of searching signals of this possible extension ofthe SM. We have also evaluated the reduction of hadronic uncertainties necessaryfor even more precise theoretical predictions. To this end, the efforts of latticeQCD simulations indicate that an accuracy of O(1%) is achievable on some ofthe hadronic parameters of interest for the Super Flavour factory program, evenwithout considering progress in theory and in algorithms, difficult to anticipate.

(iv) In the second part of Chapter 5 we have analyzed several FCNC exclusive b-hadrontransitions into νν pairs which are of great interest (although experimentallychallenging) as they can provide the evidence of possible deviations from SMthrough signals of remarkable theoretical significance. We have examined a setof B and Bs decay modes in the RSc model, with particular emphasis on thecorrelations among the observables that are features of the model. In the planned

138 Conclusions

experimental programs these modes can be accessible, and the predictions willbecome testable [181].

(v) In Chapter 6 we have considered charged weak precess driven by the b →cτ ντ current, for which an anomalous enhancement of the ratios R(D(∗)) =

B(B → D(∗)τ ντ )

B(B → D(∗)µνµ)with respect to SM has been measured by the BABAR Col-

laboration. It is worth mentioning that the LHCb Collaboration is planning to dosimilar measurements in the next years, although challenging for the intimidatinghadronic background in the reconstruction of the τ [182]. The analyses of R(D(∗))in specific models evidentiate the enhancement the purely leptonic B → τ ντrate, for which data are better compatible with SM. A mechanism enhancing thesemileptonic modes B → D(∗)τ ντ with respect to B → D(∗)µνµ, leaving B → τ ντunaffected, can be based on a tensor operator in the effective Hamiltonian. Wehave bound the relative weight of this operator, and studied the impact on severalobservables, the most sensitive one being the forward-backward asymmetry inB → D∗τ ντ with a shift in the position of its zero. If the anomaly in B → D(∗)τ ντis due to this NP effect, analogous deviations should be found in B to excited Dtransitions. The ratios R for these mesons are enhanced with respect to SM, andthe forward-backward asymmetry is a sensitive observable in channels involvingD′1 and D∗2. These signatures in exclusive semileptonic b→ c τ ντ modes make theunderstanding of the role of the new contribution to the effective weak Hamiltonianfeasible, a step towards a possible disclosure of new interactions through flavourphysics measurements.

Appendix A

Flavour in RSc: Feynman rules

A.1 Quarks mass and flavour mixing matrices

The flavour mixing matrices DL,R are defined as [108]:

DL =

ωdij

fQifQj

(i < j)

1 (i = j)

ωdijfQj

fQi(i > j)

, DR =

ρdij

fdifdj

(i < j)

1 (i = j)

ρdijfdjfdi

(i > j)

(1)

The following notation are introduced:

ωdii = 1 , ωd12 =λd33λ

d12 − λd13λ

d32

λd22λd33 − λd23λ

d32

, ωd13 =λd13

λd33

, ωd13 =λd23

λd33

, (2)

ωd21 = −(ωd12)∗ , ωd21 = −(ωd13)∗ − (ωd23)∗ωd21 , ωd32 = −(ωd23)∗ . (3)

Similarly one has:

ρdii = 1 , ρd12 =

(λd33λ

d21 − λd31λ

d23

λd22λd33 − λd23λ

d32

)∗, ρd13 =

(λd31

λd33

)∗, ρd13 =

(λd32

λd33

)∗, (4)

ρd21 = −(ρd12)∗ , ρd21 = −(ρd13)∗ − (ρd23)∗ρd21 , ρd32 = −(ρd23)∗ . (5)

A.2 Feynman rules for neutral current interactions

In this section we consider the neutral current interactions mediated by the gauge bosonsZ, ZH , Z

′ and A(1). As mentioned, such interactions can be either flavour conservingor flavour violating. We collect in four triplets the up-type quarks, the down-type

quarks, the neutrinos and the charged leptons: f =

f1

f2

f3

, with f = ui, di, νì , `−i , and

139

140 Flavour in RSc: Feynman rules

fi,L(R)

fk,L(R)

−ifk∆ikL(R)γµPL(R)fiX

µX

Figure A.1: Couplings of neutral gauge bosons X = Z, ZH , Z′, A(1) to fermions f with flavour indices

i, k. In addition to flavour conserving couplings, i = k, flavour violating couplings i 6= k are possible.

i = 1, 2, 3 a generation index. We define the effective coupling of a generic gauge bosonX to a pair of fermions fi fk:

LXint = −iXµ(

∆fifkL fk,LγµPLfi,L + ∆fifk

R fk,LγµPRfi,L

), (6)

where PL,R =1∓ γ5

2. Figure A.1 shows this vertex. In the expression of the couplings

∆fifkL,R , two more overlap integrals are needed:

I+1 =

1

L

ˆ L

0dy e−2ky[h(y)]2 g(y) ,

I−1 =1

L

ˆ L

0dy e−2ky[h(y)]2 g(y) , (7)

together with the quantity

Dfifk = RfifkI−1 −RfifkI+1 . (8)

Df is the diagonal matrix with elements Dfifi . We consider separately the case of thefour neutral gauge bosons listed above. In the following,M indicates one of the matricesUL, UR, DL, DR, for up or down-type left- and right-handed quarks. The obtainedrules also hold in the case of leptons with M being the unit matrix. In the case ofthe couplings of Z, Z ′ and ZH the Feynman rules are obtained expanding in the small

parameter ε =g2v2

4LM2, with g the 5D SU(2)L gauge constant and M2 = (m2

1 + m21)/2.

Since, neglecting corrections of O(v2/M2) the mixing angle ψ in (3.53-3.54) coincideswith the Weinberg angle θW , in the following Feynman rules we put sψ = sW = sin θWand cψ = cW = cos θW .

• Couplings of the photon 1-mode A(1)

The couplings are given by

∆fifkL (A(1)) = ∆fifk

R (A(1)) = Qf e(M†RfM

)ki

(9)

where Qf is the electric charge of the fermion f (in units of the positron charge e).

A.2 Feynman rules for neutral current interactions 141

• Z couplings

In terms of the SM couplings of the Z boson:

[gSMZ (f)] = − e

sW cW

(T3 − s2

WQf), (10)

with T3 the third component of the weak SU(2)L isospin, we find:

∆fifk(Z) = [gSMZ (f)]

{δki +

ε

c2W

(M†DfM

)ki

}(11)

for f = uR, cR, tR, for f = dL, sL, bL, and for f = `−L , and

∆fifk(Z) = [gSMZ (f)]

{δki +

ε

c2W

(M†DfM

)ki

}+

εe

sW cWI−1

(M†RfM

)ki

(12)

for f = uL, cL, tL, for f = dR, sR, bR, and for f = `−R, νe, µ, τ .

• ZH couplings

At the order O(ε0) for the heavier bosons ZH and Z ′ we find:

∆fifk(ZH) = cξ(∆cf )ki(ZH) + sξ(∆

sf )ki(ZH) , (13)

where cξ = cos(ξ) and sξ = sin(ξ), and

(∆cf )ij(ZH) = gSMZ (f)

(M†RfM

)ij,

(∆sf )ij(ZH) =

(M†RfM

)ij√

1− 2s2W

[gSMZ (f) + gcW

], (14)

for f = uL, cL, tL, and for f = dR, sR, bR, and

(∆cf )ij(ZH) = gSMZ (f)

(M†RfM

)ij

,

(∆sf )ij(ZH) =

(M†RfM

)ij√

1− 2s2W

gSMZ (f) , (15)

for f = uR, cR, tR, and for f = dL, sL, bL. For the leptons we have:

∆``(ZH) = gSMZ (`)[cξ∆

c,``(ZH) + sξ∆s,``(ZH)

], (16)

142 Flavour in RSc: Feynman rules

where

∆c,``(ZH) = R`` (` = ν, `−L , `−R)

∆s,νν(ZH) = −Rνν√

1− 2s2W

∆s,`−L `+L (ZH) =

R`−L `+L√1− 2s2

W

(17)

∆s,`−R`+R(ZH) = −R`−R`+R

√1− 2s2

W

s2W

.

• Z ′ couplings

Defining the couplings with the same structure as in Eq.(13), we find:

(∆cf )ij(Z

′) = (∆sf )ij(ZH)

(∆sf )ij(Z

′) = −(∆cf )ij(ZH) . (18)

For the leptons we have

∆c,``(Z ′) = ∆s,``(ZH)

∆s,`` = −∆c,``(ZH) . (19)

Appendix B

B → D(∗)(D∗∗)τ ντ : theoreticalinputs for the matrix elements

B.1 Wilson Coefficients

With the aim of providing the information useful to reconstruct the various B → D(∗)

matrix elements, we collect here the expressions of the αs and 1/mQ corrections inEqs.(6.20,6.21) and (6.39-6.42) worked out by M. Neubert and by I. Caprini et al.in [166,169]. The functions Li(w) read as

L1 ' 0.72 (w − 1) Λ

L2 ' −0.16 (w − 1) Λ

L3 ' −0.24 Λ

L4 ' 0.24 Λ (1)

L5 ' −Λ

L6 ' −3.24

w + 1Λ .

The coefficients Ci are expressed in terms of C1,

C51

C1= 1− 4αs

3πrf (w)

C(5)2

C1= −2αs

3πH(5)

(w,

1

zm

)(2)

C(5)3

C1= ∓2αs

3πH(5)(w, zm) ,

with zm = mcmb

and

rf (w) =1√

w2 − 1log[w +

√w2 − 1

], (3)

143

144 B → D(∗)(D∗∗)τ ντ : theoretical inputs for the matrix elements

H(5)(w, zm) =zm(1− log zm ∓ zm)

1− 2wzm + z2m

+zm

(1− 2wzm + z2m)2

[2(w ∓ 1)zm(1± zm) log zm

− [(w ± 1)− 2w(2w ± 1)zm + (5w ± 2w2 ∓ 1)z2m − 2z3

m]rf (w)

]. (4)

In Eqs. (2)–(4) the lower signs refer to the index 5 (corresponding to the axial current).C1 reads:

C1 =

(αs(mc)

αs(µ)

)ahh(w)(1− αs(µ)

πZhh(w)

)

×(

1 +αs(mc)

π

[log(mb

mc

)+ Zhh(w) +

2

3

[f(w) + rf (w) + g(w)

])], (5)

with

ahh(w) =8

27

[w rf (w)− 1

], (6)

Zhh(w) =8

81

(94

9− π2

)(w − 1)− 4

135

(92

9− π2

)(w − 1)2 +O((w − 1)3) , (7)

f(w) = wrf (w)− 2− w√w2 − 1

[L2(1− w2

−) + (w2 − 1)r2f (w)

], (8)

g(w) =w√

w2 − 1

[L2(1− zmw−)− L2(1− zmw+)

]− zm

(1− 2wzm + z2m)

×[(w2 − 1)rf (w) + (w − zm) log(zm)

], (9)

and w± = w ±√w2 − 1. In the numerical analysis we set the scale µ =

√mcmb, and

investigate the sensitivity to higher order corrections varying this scale between µ/2 and2µ.

B.2 B → D∗∗ matrix elements and differential semileptonicdecay rates

In the infinite heavy quark mass limit the B → D∗∗ matrix elements can be defined interms of two universal τ1/2(w) and τ3/2(w) form factors:⟨

D∗0(p′)∣∣ cγµ(1− γ5)b

∣∣B(p)⟩

= −2√MBMD∗0

τ1/2(w)(v − v′

)µ

; (10)

⟨D∗0(p′)

∣∣ cσµν(1− γ5)b∣∣B(p)

⟩= 2√MBMD∗0

τ1/2(w)

×[− εµναβvαv′β + i(vµv

′ν − vνv′µ)

]; (11)

B.2 B → D∗∗ matrix elements and differential semileptonic decay rates 145

⟨D′1(p′, ε)

∣∣ cγµ(1− γ5)b∣∣B(p)

⟩= −2

√MBMD′1

τ1/2(w)

×[− iεµαβσε∗αvβv′σ − (w − 1)ε∗µ + (ε∗ · v)v′µ

]; (12)

⟨D′1(p′, ε)

∣∣ cσµν(1− γ5)b∣∣B(p)

⟩= −2

√MBMD′1

τ1/2(w)

×{− εµναβε∗α(v − v′)β + i[ε∗µ(v − v′)ν − ε∗ν(v − v′)µ]

}; (13)

⟨D1(p′, ε)

∣∣ cγµ(1− γ5)b∣∣B(p)

⟩=

√MBMD1√

2τ3/2(w)

×{i(1 + w)εµαβσε

∗αvβv′σ + (w2 − 1)ε∗µ + (ε∗ · v)[3vµ − (w − 2)v′µ

] }; (14)

⟨D1(p′, ε)

∣∣ cσµν(1− γ5)b∣∣B(p)

⟩=

√MBMD1√

2τ3/2(w)

×{− (w − 1)εµναβε

∗α(v + v′)β + (ε∗ · v)εµναβvαv′β + 2ετσαβε

∗σvαv′β

×[gτµvν − gτνvµ

]+[(1 + w)

(ε∗ν(v − v′)µ − ε∗µ(v − v′)ν

)− 3(ε∗ · v)(vµv

′ν − vνv′µ)

] };

(15)

⟨D∗2(p′, ε)

∣∣ cγµ(1− γ5)b∣∣B(p)

⟩=√MBMD∗2

√3 τ3/2(w)

×{− iεµβτσ

(ε∗αβvα

)vτv′σ +

(ε∗αβvα

)vβv′µ − (1 + w)

(ε∗αµ vα

)}; (16)

⟨D∗2(p′, ε)

∣∣ cσµν(1− γ5)b∣∣B(p)

⟩=√MBMD∗2

√3 τ3/2(w)

×{− εµνβτ

(ε∗αβvα

)(v + v′)τ + i (ε∗ατvα)

[gτµ(v + v′)ν − gτν (v + v′)µ

] }. (17)

In the previous formulae we have set p = MB v, p′ = MD∗∗ v′ and w = v · v′; ε is the

polarization vector (tensor) of the spin 1 (spin 2) D∗∗ meson.The results for the SM, NP and interference contribution to the differential distribu-

tions in (6.4) are given below for each of the four excited mesons. The relation betweenthe squared momentum transfer q2 and w is q2 = M2

B +M2D∗∗ −2MBMD∗∗w, with MD∗∗

the mass of the charmed meson produced in the decay. The lepton mass has been takeninto account, hence the formulae also hold for τ .

• B → D∗0`ν`:

dΓ

dq2(B → D∗0`ν`)

∣∣∣SM

= 4MBMD∗0[τ1/2(w)]2(w − 1)

×{q2

(1− M2

`

q2

)+

(1 +

2m2`

q2

)[(M2

B +M2D∗0

)w − 2MBMD∗0

]}(18)


dΓ

dq2(B → D∗0`ν`)

∣∣∣NP

= 32|εT |2MBMD∗0

× [τ1/2(w)]2(w2 − 1)

(1 +

2m2`

q2

)(M2

B +M2D∗0− 2MBMD∗0

w) (19)

dΓ

dq2(B → D∗0`ν`)

∣∣∣INT

= −48Re(εT )MBMD∗0

× [τ1/2(w)]2(w2 − 1)m`(MB −MD∗0) (20)

• B → D′1`ν`:

dΓ

dq2(B → D′1`ν`)

∣∣∣SM

= 4MBMD′1[τ1/2(w)]2(w − 1)

{q2

(1− m2

`

q2

)

× (2w − 1) +

(1 +

2m2`

q2

)[(M2

B +M2D′1

)3w − 2MBMD′1(2w2 + 1)

]}(21)

dΓ

dq2(B → D′1`ν`)

∣∣∣NP

= 32|εT |2MBMD′1[τ1/2(w)]2(w − 1)

(1 +

2m2`

q2

){

(M2B +M2

D′1)(5w − 1)− 2MBMD′1

[4 + w(w − 1)]

}(22)

dΓ

dq2(B → D′1`ν`)

∣∣∣INT

= 48Re(εT )MBMD′1

×[τ1/2(w)]2(w − 1)m`[MB(w − 5) +MD′1

(5w − 1)]

(23)

• B → D1`ν`:

dΓ

dq2(B → D1`ν`)

∣∣∣SM

= MBMD1 [τ3/2(w)]2(w − 1)(1 + w)2

{q2

(1− m2

`

q2

)

× (w − 2) +

(1 +

2m2`

q2

)[(M2

B +M2D1

)3w − 2MBMD1(w2 + 2)]}

(24)

dΓ

dq2(B → D1`ν`)

∣∣∣NP

= 16|εT |2MBMD1 [τ3/2(w)]2(w − 1)(1 + w)2

×(

1 +2m2

`

q2

){[(M2

B +M2D1

)(2w − 1)− 2MBMD1(w2 − w + 1)]}

(25)

B.2 B → D∗∗ matrix elements and differential semileptonic decay rates 147

dΓ

dq2(B → D1`ν`)

∣∣∣INT

= 24Re(εT )MBMD1

× [τ3/2(w)]2(w − 1)(1 + w)2m`[MB(w − 2) +MD1(2w − 1)] (26)

• B → D∗2`ν`:

dΓ

dq2(B → D∗2`ν`)

∣∣∣SM

= MBMD∗2[τ3/2(w)]2(w − 1)(1 + w)2

{q2

(1− m2

`

q2

)

× (3w + 2) +

(1 +

2m2`

q2

)[(M2

B +M2D∗2

)5w − 2MBMD∗2(3w2 + 2)

]}(27)

dΓ

dq2(B → D∗2`ν`)

∣∣∣NP

= 16|εT |2MBMD∗2[τ3/2(w)]2(w − 1)(1 + w)2

×(

1 +2m2

`

q2

){[(M2

B +M2D∗2

)(1 + 4w)− 2MBMD∗2(3 + w + w2)

]}(28)

dΓ

dq2(B → D∗2`ν`)

∣∣∣INT

= −24Re(εT )MBMD∗2

× [τ3/2(w)]2(w − 1)(1 + w)2m`[MB(4 + w)−MD∗2(1 + 4w)] (29)

The differential decay rates are obtained multiplying the above functions by the coefficientC(q2) in (6.5).


Bibliography

[1] Annika H. G. Peter [astro-ph/1201.3942].

[2] G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B 645, 155(2002) [hep-ph/0207036].

[3] A. J. Buras, Acta Phys. Polon. B 34, 5615 (2003) [hep-ph/0310208].

[4] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) (1921).O. Klein, Z. Phys. 37 (1926).

[5] B. A. Dobrescu and E. Ponton, JHEP 0403 (2004) 071 [hep-th/0401032]. G. Burd-man, B. A. Dobrescu and E. Ponton, JHEP 0602 (2006) 033 [hep-ph/0506334].

[6] G. Servant and T. M. P. Tait, Nucl. Phys. B 650, 391 (2003) [hep-ph/0206071].

[7] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999) [hep-ph/9905221].Phys. Rev. Lett. 83, 4690 (1999) [hep-th/9906064].

[8] R. Aaij et al. [LHCb Collaboration], JHEP 1308, 131 (2013) [arXiv:1304.6325[hep-ex]]. R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 111, no. 19, 191801(2013) [arXiv:1308.1707 [hep-ex]].

[9] J. P. Lees et al. [BABAR Collaboration], Phys. Rev. Lett. 109, 101802 (2012)[arXiv:1205.5442 [hep-ex]]. J. P. Lees et al. [The BABAR Collaboration], Phys. Rev.D 88, no. 7, 072012 (2013) [arXiv:1303.0571 [hep-ex]].

[10] A. J. Buras and J. Girrbach, Rept. Prog. Phys. 77, 086201 (2014) [arXiv:1306.3775[hep-ex]].

[11] K. Agashe, R. Contino, L. Da Rold and A. Pomarol, Phys. Lett. B 641, 62 (2006)[hep-ph/0605341].

[12] A. Soffer, Mod. Phys. Lett. A 29, no. 7, 1430007 (2014) [arXiv:1401.7947 [hep-ex]].

[13] S. Fajfer, J. F. Kamenik and I. Nisandzic, Phys. Rev. D 85, 094025 (2012)[arXiv:1203.2654 [hep-ex]].

[14] I. Adachi et al. [Belle Collaboration], Phys. Rev. Lett. 110, no. 13, 131801 (2013)[arXiv:1208.4678 [hep-ex]].

149

http://arxiv.org/abs/1201.3942?context=astro-ph

http://arxiv.org/abs/hep-ph/0207036


http://arxiv.org/abs/hep-th/0401032





http://arxiv.org/abs/arXiv:1304.6325











150 Bibliography

[15] J. P. Lees et al. [BABAR Collaboration], Phys. Rev. D 88, no. 3, 031102 (2013)[arXiv:1207.0698 [hep-ex]].

[16] T. Appelquist, H. C. Cheng and B. A. Dobrescu, Phys. Rev. D 64, 035002 (2001)[hep-ph/0012100].

[17] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 436,257 (1998) [hep-ph/9804398]. N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali,Phys. Rev. D 59, 086004 (1999) [hep-ph/9807344]. N. Arkani-Hamed, S. Dimopoulosand G. R. Dvali, Phys. Lett. B 429, 263 (1998) [hep-ph/9803315].

[18] N. Arkani-Hamed, S. Dimopoulos, G. R. Dvali and J. March-Russell, Phys. Rev. D65, 024032 (2002) [hep-ph/9811448].

[19] T. Gherghetta and A. Pomarol, Nucl. Phys. B 586, 141 (2000) [hep-ph/0003129].

[20] Y. Grossman and M. Neubert, Phys. Lett. B 474, 361 (2000) [hep-ph/9912408].

[21] B. Lillie, JHEP 0312, 030 (2003) [hep-ph/0308091].

[22] K. Agashe, G. Perez and A. Soni, Phys. Rev. D 71, 016002 (2005) [hep-ph/0408134].

[23] K. R. Dienes, E. Dudas and T. Gherghetta, Nucl. Phys. B 537, 47 (1999) [hep-ph/9806292].

[24] L. Randall and M. D. Schwartz, Phys. Rev. Lett. 88, 081801 (2002) [hep-th/0108115].

[25] M. S. Carena, A. Delgado, E. Ponton, T. M. P. Tait and C. E. M. Wagner, Phys.Rev. D 68, 035010 (2003) [hep-ph/0305188].

[26] H. C. Cheng, K. T. Matchev and M. Schmaltz, Phys. Rev. D 66, 056006 (2002)[hep-ph/0205314].

[27] T. G. Rizzo, AIP Conf. Proc. 1256, 27 (2010) [arXiv:1003.1698 [hep-ex]].

[28] E. Recami, Riv. Nuovo Cim. 9N6, 1 (1986).

[29] K. Kong, K. Matchev and G. Servant, In *Bertone, G. (ed.): Particle dark matter*306-324 [arXiv:1001.4801 [hep-ex]].

[30] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716, 30 (2012)[arXiv:1207.7235 [hep-ex]].

[31] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716, 1 (2012) [arXiv:1207.7214[hep-ex]].

[32] T. Appelquist, B. A. Dobrescu, E. Ponton and H. U. Yee, Phys. Rev. Lett. 87,181802 (2001) [hep-ph/0107056].

[33] T. Gherghetta, [arXiv:1008.2570 [hep-ex]].























Bibliography 151

[34] T. Gherghetta and A. Pomarol, Nucl. Phys. B 586, 141 (2000) [hep-ph/0003129].

[35] Y. Grossman and M. Neubert, Phys. Lett. B 474, 361 (2000) [hep-ph/9912408].

[36] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).

[37] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).

[38] S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D 2, 1285 (1970).

[39] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973). H. D. Politzer, Phys.Rev. Lett. 30, 1346 (1973).

[40] K. G. Wilson, Phys. Rev. 179, 1499 (1969).

[41] J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D 86, 010001(2012).

[42] M. S. Carena, E. Ponton, J. Santiago and C. E. M. Wagner, Nucl. Phys. B 759,202 (2006) [hep-ph/0607106].

[43] G. Cacciapaglia, C. Csaki, G. Marandella and J. Terning, Phys. Rev. D 75, 015003(2007) [hep-ph/0607146].

[44] R. Contino, L. Da Rold and A. Pomarol, Phys. Rev. D 75, 055014 (2007) [hep-ph/0612048].

[45] M. S. Carena, E. Ponton, J. Santiago and C. E. M. Wagner, Phys. Rev. D 76,035006 (2007) [hep-ph/0701055].

[46] M. E. Albrecht, M. Blanke, A. J. Buras, B. Duling and K. Gemmler, JHEP 0909,064 (2009) [arXiv:0903.2415 [hep-ph]].

[47] M. Blanke, A. J. Buras, B. Duling, S. Gori and A. Weiler, JHEP0903, 001 (2009)[arXiv:0809.1073 [hep-ph]].

[48] A. J. Buras, B. Duling and S. Gori, JHEP 0909, 076 (2009) [arXiv:0905.2318[hep-ph]].

[49] E. Ponton, [arXiv:1207.3827 [hep-ph]].

[50] A. J. Buras, [arXiv:9806471 [hep-ph]].

[51] T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981) [Erratum-ibid. 65, 1772(1981)].

[52] G. Isidori, Y. Nir and G. Perez, Ann. Rev. Nucl. Part. Sci. 60 (2010) 355[arXiv:1002.0900 [hep-ph]].

[53] A. J. Buras, M. Misiak, M. Munz and S. Pokorski, Nucl. Phys. B 424 (1994) 374[arXiv:9311345 [hep-ph]].
















152 Bibliography

[54] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996)[hep-ph/9512380].

[55] M. Misiak, H. M. Asatrian, K. Bieri, M. Czakon, A. Czarnecki, T. Ewerth, A. Fer-roglia and P. Gambino et al., Phys. Rev. Lett. 98, 022002 (2007) [hep-ph/0609232].P. Gambino and M. Misiak, Nucl. Phys. B 611, 338 (2001) [hep-ph/0104034].M. Misiak and M. Steinhauser, Nucl. Phys. B 764, 62 (2007) [hep-ph/0609241].

[56] A. Freitas and U. Haisch, Phys. Rev. D 77 (2008) 093008 [arXiv:0801.4346 [hep-ph]].

[57] U. Haisch and A. Weiler, Phys. Rev. D 76, 034014 (2007) [hep-ph/0703064].

[58] For a review see: P. Colangelo and A. Khodjamirian, [hep-ph/0010175].

[59] P. Colangelo, F. De Fazio, P. Santorelli and E. Scrimieri, Phys. Rev. D 53, 3672(1996) [Erratum-ibid. D 57, 3186 (1998)]. [hep-ph/9510403].

[60] P. Ball and R. Zwicky, Phys. Rev. D 71, 014029 (2005). [hep-ph/0412079].

[61] D. Becirevic, V. Lubicz and F. Mescia, Nucl. Phys. B 769, 31 (2007). [hep-ph/0611295].

[62] W. Wang, Phys. Rev. D 83, 014008 (2011). [arXiv:1008.5326 [hep-ph]].

[63] T. Mannel, W. Roberts and Z. Ryzak, Nucl. Phys. B 355, 38 (1991).

[64] C. S. Huang and H. G. Yan, Phys. Rev. D 59, 114022 (1999) [Erratum-ibid. D 61,039901 (2000)]. [hep-ph/9811303].

[65] P. Colangelo, F. De Fazio, R. Ferrandes and T. N. Pham, Phys. Rev. D 77, 055019(2008). [arXiv:0709.2817 [hep-ph]].

[66] T. Mannel and Y. -M. Wang, JHEP 1112, 067 (2011). [arXiv:1111.1849 [hep-ph]].

[67] T. Gutsche, M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij and P. Santorelli, Phys.Rev. D 87, 074031 (2013) [arXiv:1301.3737 [hep-ph]].

[68] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 103, 211802 (2009).[arXiv:0906.2177 [hep-ex]].

[69] M. Nakao et al. [BELLE Collaboration], Phys. Rev. D 69, 112001 (2004). [hep-ex/0402042].

[70] T. E. Coan et al. [CLEO Collaboration], Phys. Rev. Lett. 84, 5283 (2000). [hep-ex/9912057].

[71] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 70, 091105 (2004). [hep-ex/0409035].


















http://arxiv.org/abs/hep-ex/0402042






Bibliography 153

[72] J. Wicht et al. [Belle Collaboration], Phys. Rev. Lett. 100, 121801 (2008).[arXiv:0712.2659 [hep-ex]].

[73] D. Acosta et al. [CDF Collaboration], Phys. Rev. D 66, 112002 (2002). [hep-ex/0208035].

[74] B. A. Dobrescu, K. Kong and R. Mahbubani, JHEP 0707, 006 (2007); [hep-ph/0703231]. A. Freitas and K. Kong, JHEP 0802, 068 (2008); [arXiv:0711.4124[hep-ph]]. K. Ghosh and A. Datta, Nucl. Phys. B 800, 109 (2008); [arXiv:0801.0943[hep-ph]]. Phys. Lett. B 665, 369 (2008). [arXiv:0802.2162 [hep-ph]].

[75] D. Atwood, T. Gershon, M. Hazumi and A. Soni, Phys. Rev. D 71 (2005) 076003[hep-ph/0410036].

[76] P. Colangelo and F. De Fazio, Phys. Lett. B 520, 78 (2001). [hep-ph/0107137].

[77] H. J. Lipkin, Phys. Lett. B 254, 247 (1991).

[78] M. Beneke and M. Neubert, Nucl. Phys. B 651, 225 (2003). [hep-ph/0210085].

[79] S. Nishida et al. [Belle Collaboration], Phys. Lett. B 610, 23 (2005). [hep-ex/0411065].

[80] R. Wedd et al. [Belle Collaboration], Phys. Rev. D 81, 111104 (2010).[arXiv:0810.0804 [hep-ex]].

[81] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 74, 031102 (2006). [hep-ex/0603054].

[82] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 79, 011102 (2009).[arXiv:0805.1317 [hep-ex]].

[83] S. Fajfer, T. N. Pham and N. Kosnik, Phys. Rev. D 78, 074013 (2008).[arXiv:0806.2247 [hep-ph]].

[84] T. Feldmann, P. Kroll and B. Stech, Phys. Rev. D 58, 114006 (1998); [hep-ph/9802409]. Phys. Lett. B 449, 339 (1999); [hep-ph/9812269]. T. Feldmann,Int. J. Mod. Phys. A 15, 159 (2000). [hep-ph/9907491].

[85] F. De Fazio and M. R. Pennington, JHEP 0007, 051 (2000). [hep-ph/0006007].

[86] F. Ambrosino et al. [KLOE Collaboration], Phys. Lett. B 648, 267 (2007). [hep-ex/0612029].

[87] M.B. Wise, Phys. Rev. D 45, R2188 (1992); G. Burdman and J.F. Donoghue,Phys. Lett. B 280, 287 (1992); P. Cho, Phys. Lett. B 285, 145 (1992); H.-Y.Cheng et al., Phys. Rev. D 46, 1148 (1992); R. Casalbuoni, A. Deandrea, N. DiBartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Rept. 281, 145 (1997).[hep-ph/9605342].





























154 Bibliography

[88] P. Colangelo, G. Nardulli, A. Deandrea, N. Di Bartolomeo, R. Gatto and F. Feruglio,Phys. Lett. B 339, 151 (1994); P. Colangelo, F. De Fazio and G. Nardulli, Phys. Lett.B334, 175 (1994); V. M. Belyaev, V. M. Braun, A. Khodjamirian and R. Ruckl,Phys. Rev. D 51, 6177 (1995); P. Colangelo, F. De Fazio, G. Nardulli, N. DiBartolomeo and R. Gatto, Phys. Rev. D 52, 6422 (1995); P. Colangelo and F. DeFazio, Eur. Phys. J. C 4, 503 (1998); D. Becirevic, B. Blossier, E. Chang andB. Haas, Phys. Lett. B 679, 231 (2009). [arXiv:0905.3355 [hep-ph]].

[89] P. Colangelo and F. De Fazio, Phys. Lett. B 532, 193 (2002). [hep-ph/0201305].

[90] P. Ball and R. Zwicky, Phys. Rev. D 71, 014015 (2005). [hep-ph/0406232].

[91] P. Colangelo, F. De Fazio and G. Nardulli, Phys. Lett. B 334, 175 (1994). [hep-ph/9406320].

[92] P. Biancofiore, P. Colangelo and F. De Fazio, Phys. Rev. D 85, 094012 (2012)[arXiv:1202.2289 [hep-ph]].

[93] F. E. Low, Phys. Rev. 110, 974 (1958).

[94] Y. Amhis et al. [Heavy Flavor Averaging Group Collaboration], [arXiv:1207.1158[hep-ex]].

[95] M. Misiak, Acta Phys. Polon. B 40, 2987 (2009) [arXiv:0911.1651 [hep-ph]].

[96] G. Buchalla and A. J. Buras, Nucl. Phys. B 548, 309 (1999) [hep-ph/9901288].

[97] A. Faessler, T. Gutsche, M. A. Ivanov, J. G. Korner and V. E. Lyubovitskij, Eur.Phys. J. C 4, 18 (2002) [hep-ph/0205287].

[98] W. Altmannshofer, P. Ball, A. Bharucha, A. J. Buras, D. M. Straub and M. Wick,JHEP 0901, 019 (2009) [arXiv:0811.1214 [hep-ph]].

[99] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 79, 031102 (2009)[arXiv:0804.4412 [hep-ex]].

[100] J. -T. Wei et al. [BELLE Collaboration], Phys. Rev. Lett. 103, 171801 (2009)[arXiv:0904.0770 [hep-ex]].

[101] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 108, 081807 (2012)[arXiv:1108.0695 [hep-ex]].

[102] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 108, 181806 (2012)[arXiv:1112.3515 [hep-ex]].

[103] M. Beneke, T. Feldmann and D. Seidel, Nucl. Phys. B 612, 25 (2001) [hep-ph/0106067].

[104] S. Descotes-Genon, J. Matias, M. Ramon and J. Virto, JHEP 1301, 048 (2013)[arXiv:1207.2753 [hep-ph]].




















Bibliography 155

[105] S. Descotes-Genon, J. Matias and J. Virto, Phys. Rev. D 88, 074002 (2013)[arXiv:1307.5683 [hep-ph]]; [arXiv:1311.3876 [hep-ph]].

[106] W. Altmannshofer and D. M. Straub, Eur. Phys. J. C 73, 2646 (2013)[arXiv:1308.1501 [hep-ph]].

[107] F. Beaujean, C. Bobeth and D. van Dyk, Eur. Phys. J. C 74, 2897 (2014)[arXiv:1310.2478 [hep-ph]].

[108] M. Blanke, A. J. Buras, B. Duling, K. Gemmler and S. Gori, JHEP 0903, 108(2009) [arXiv:0812.3803 [hep-ph]].

[109] M. Blanke, B. Shakya, P. Tanedo and Y. Tsai, JHEP 1208, 038 (2012)[arXiv:1203.6650 [hep-ph]].

[110] C. Csaki, Y. Grossman, P. Tanedo and Y. Tsai, Phys. Rev. D 83, 073002 (2011)[arXiv:1004.2037 [hep-ph]].

[111] B. Duling, PhD thesis: “The custodially protected Randall-Sundrum model: Globalfeatures and distinct flavor signatures.”, Munich, Tech. U., 2010

[112] S. Casagrande, F. Goertz, U. Haisch, M. Neubert and T. Pfoh, JHEP 0810,094 (2008) [arXiv:0807.4937 [hep-ph]]; M. Bauer, S. Casagrande, U. Haisch andM. Neubert, JHEP 1009, 017 (2010) [arXiv:0912.1625 [hep-ph]].

[113] S. Schael et al. [ALEPH and DELPHI and L3 and OPAL and SLD and LEPElectroweak Working Group and SLD Electroweak Group and SLD Heavy FlavourGroup Collaborations], Phys. Rept. 427, 257 (2006) [hep-ex/0509008].

[114] S. J. Huber and Q. Shafi, Phys. Lett. B 498, 256 (2001) [hep-ph/0010195]; S. J. Hu-ber, Nucl. Phys. B 666, 269 (2003) [hep-ph/0303183].

[115] G. Burdman, Phys. Rev. D 66, 076003 (2002) [hep-ph/0205329]; Phys. Lett. B590, 86 (2004) [hep-ph/0310144].

[116] K. Agashe, G. Perez and A. Soni, Phys. Rev. D 71, 016002 (2005) [hep-ph/0408134].

[117] K. Agashe, G. Perez and A. Soni, Phys. Rev. Lett. 93, 201804 (2004) [hep-ph/0406101].

[118] A. L. Fitzpatrick, G. Perez and L. Randall, Phys. Rev. Lett. 100, 171604 (2008)[arXiv:0710.1869 [hep-ph]].

[119] P. R. Archer, S. J. Huber and S. Jager, JHEP 1112, 101 (2011) [arXiv:1108.1433[hep-ph]].

[120] J. P. Lees et al. [BABAR Collaboration], Phys. Rev. D 86, 032012 (2012)[arXiv:1204.3933 [hep-ph]].






















156 Bibliography

[121] A. J. Buras and J. Girrbach, JHEP 1312, 009 (2013) [arXiv:1309.2466 [hep-ph]].

[122] A. J. Buras, F. De Fazio and J. Girrbach, JHEP 1402, 112 (2014). [arXiv:1311.6729[hep-ph]].

[123] F. Kruger and L. M. Sehgal, Phys. Lett. B 380, 199 (1996) [hep-ph/9603237].

[124] P. Colangelo, F. De Fazio, R. Ferrandes and T. N. Pham, Phys. Rev. D 74, 115006(2006) [hep-ph/0610044].

[125] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 111, 101805 (2013)[arXiv:1307.5024 [hep-ex]].

[126] S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. Lett. 111, 101804 (2013)[arXiv:1307.5025 [hep-ex]].

[127] C. Bobeth, M. Gorbahn, T. Hermann, M. Misiak, E. Stamou and M. Steinhauser,Phys. Rev. Lett. 112, 101801 (2014) [arXiv:1311.0903 [hep-ph]].

[128] P. Colangelo, F. De Fazio, P. Santorelli and E. Scrimieri, Phys. Lett. B 395, 339(1997) [hep-ph/9610297].

[129] D. Melikhov, N. Nikitin and S. Simula, Phys. Lett. B 428 (1998) 171 [hep-ph/9803269].

[130] W. Altmannshofer, A. J. Buras, D. M. Straub and M. Wick, JHEP 0904 (2009)022 [arXiv:0902.0160 [hep-ph]].

[131] O. Lutz et al. [Belle Collaboration], Phys. Rev. D 87 (2013) 11, 111103[arXiv:1303.3719 [hep-ex]].

[132] J. P. Lees et al. [BABAR Collaboration], Phys. Rev. D 87 (2013) 11, 112005[arXiv:1303.7465 [hep-ex]].

[133] C. S. Kim, Y. G. Kim and T. Morozumi, Phys. Rev. D 60 (1999) 094007 [hep-ph/9905528].

[134] G. Buchalla, G. Hiller and G. Isidori, Phys. Rev. D 63 (2000) 014015 [hep-ph/0006136].

[135] A. J. Buras, J. Girrbach-Noe, C. Niehoff and D. M. Straub, [arXiv:1409.4557[hep-ph]].

[136] A. J. Buras, F. De Fazio and J. Girrbach, JHEP 1302 (2013) 116 [arXiv:1211.1896[hep-ph]].

[137] P. Colangelo, F. De Fazio, R. Ferrandes and T. N. Pham, Phys. Rev. D 73 (2006)115006 [hep-ph/0604029].
























Bibliography 157

[138] C. Bouchard et al. [HPQCD Collaboration], Phys. Rev. D 88, no. 5, 054509 (2013)[Erratum-ibid. D 88, no. 7, 079901 (2013)] [arXiv:1306.2384 [hep-lat].

[139] R. R. Horgan, Z. Liu, S. Meinel and M. Wingate, Phys. Rev. D 89, 094501 (2014)[arXiv:1310.3722 [hep-lat].

[140] P. Colangelo, F. De Fazio and W. Wang, Phys. Rev. D 81, 074001 (2010)[arXiv:1002.2880 [hep-ph]].

[141] A. Bozek et al. [Belle Collaboration], Phys. Rev. D 82, 072005 (2010)[arXiv:1005.2302 [hep-ex]].

[142] S. Fajfer, J. F. Kamenik, I. Nisandzic and J. Zupan, Phys. Rev. Lett. 109, 161801(2012) [arXiv:1206.1872 [hep-ph]].

[143] D. Becirevic, N. Kosnik and A. Tayduganov, Phys. Lett. B 716, 208 (2012)[arXiv:1206.4977 [hep-ph]].

[144] A. Celis, M. Jung, X. -Q. Li and A. Pich, JHEP 1301, 054 (2013) [arXiv:1210.8443[hep-ph]].

[145] A. Crivellin, C. Greub and A. Kokulu, Phys. Rev. D 86, 054014 (2012)[arXiv:1206.2634 [hep-ph]].

[146] A. Datta, M. Duraisamy and D. Ghosh, Phys. Rev. D 86, 034027 (2012)[arXiv:1206.3760 [hep-ph]].

[147] D. Choudhury, D. K. Ghosh and A. Kundu, Phys. Rev. D 86, 114037 (2012)[arXiv:1210.5076 [hep-ph]].

[148] M. Tanaka and R. Watanabe, [arXiv:1212.1878 [hep-ph]].

[149] J. Laiho, E. Lunghi and R. S. Van de Water, Phys. Rev. D 81, 034503 (2010)[arXiv:0910.2928 [hep-ph]]. and the average quoted in http://latticeaverages.org/ .

[150] M. Bona et al. [UTfit Collaboration], Phys. Lett. B 687, 61 (2010) [arXiv:0908.3470[hep-ph]].

[151] A. Lenz, U. Nierste, J. Charles, S. Descotes-Genon, A. Jantsch, C. Kaufhold,H. Lacker and S. Monteil et al., Phys. Rev. D 83, 036004 (2011) [arXiv:1008.1593[hep-ph]].

[152] K. Ikado et al. [Belle Collaboration], Phys. Rev. Lett. 97, 251802 (2006) [hep-ex/0604018].

[153] K. Hara et al. [Belle Collaboration], Phys. Rev. D 82, 071101 (2010)[arXiv:1006.4201 [hep-ex]].























158 Bibliography


[156] J. L. Rosner and S. Stone, [arXiv:1201.2401 [hep-ex]].

[157] A derivation of identities for rewriting any scalar contraction of Dirac bilinearscan be found in 29.3.4 of L. B. Okun (1980). Leptons and Quarks North-Holland,ISBN 978-0-444-86924-1.

[158] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32, 438 (1974); S. Chakdar, T. Li,S. Nandi and S. K. Rai, Phys. Lett. B 718, 121 (2012) [arXiv:1206.0409 [hep-ph]].

[159] J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974) [Erratum-ibid. D 11, 703(1975)].

[160] B. Schrempp and F. Schrempp, Phys. Lett. B 153, 101 (1985); W. Buchmuller,R. Ruckl and D. Wyler, Phys. Lett. B 191, 442 (1987) [Erratum-ibid. B 448, 320(1999)].

[161] J. L. Hewett and T. G. Rizzo, Phys. Rept. 183, 193 (1989).

[162] S. Dimopoulos and L. Susskind, Nucl. Phys. B 155, 237 (1979); S. Dimopoulos,Nucl. Phys. B 168, 69 (1980); E. Eichten and K. D. Lane, Phys. Lett. B 90, 125(1980).

[163] [CMS Collaboration], report CMS-PAS-EXO-12-002.

[164] S. Rolli and M. Tanabashi, review Leptoquarks in [41].

[165] N. Isgur and M. B. Wise, Phys. Lett. B 237, 527 (1990).

[166] M. Neubert, Phys. Rept. 245, 259 (1994).

[167] F. De Fazio, in At the Frontier of Particle Physics/Handbook of QCD, ed. byM. Shifman (World Scientific, Singapore, 2001), page 1671, [hep-ph/0010007];A. V. Manohar and M. B. Wise, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol.10, 1 (2000).

[168] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 104, 011802 (2010)[arXiv:0904.4063 [hep-ex]].

[169] I. Caprini, L. Lellouch and M. Neubert, Nucl. Phys. B 530, 153 (1998) [hep-ph/9712417].

[170] W. Dungel et al. [Belle Collaboration], Phys. Rev. D 82, 112007 (2010)[arXiv:1010.5620 [hep-ex]].

[171] A comprehensive analysis of the open charm meson spectrum can be found inP. Colangelo, F. De Fazio, F. Giannuzzi and S. Nicotri, Phys. Rev. D 86, 054024(2012) [arXiv:1207.6940 [hep-ph]].










Bibliography 159

[172] D. Becirevic, A. Le Yaouanc, L. Oliver, J. -C. Raynal, P. Roudeau and J. Serrano,[arXiv:1206.5869 [hep-ph]].

[173] P. Colangelo, G. Nardulli and N. Paver, Phys. Lett. B 293, 207 (1992).

[174] P. Colangelo, F. De Fazio and N. Paver, Nucl. Phys. Proc. Suppl.75B, 83 (1999)[hep-ph/9809586].

[175] P. Colangelo, F. De Fazio and N. Paver, Phys. Rev. D 58, 116005 (1998) [hep-ph/9804377].

[176] V. Morenas, A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Phys. Rev. D56, 5668 (1997) [hep-ph/9706265].

[177] B. Blossier et al. [European Twisted Mass Collaboration], JHEP 0906 (2009) 022[arXiv:0903.2298 [hep-lat]].

[178] M. Bona et al. [SuperB Collaboration], Pisa, Italy: INFN (2007) 453 p.www.pi.infn.it/SuperB/?q=CDR [arXiv:0709.0451 [hep-ex]].

[179] P. Biancofiore, J. Phys. G 40, 065006 (2013) [arXiv:1302.7240 [hep-ph]].

[180] P. Biancofiore, P. Colangelo and F. De Fazio, Phys. Rev. D 89, 095018 (2014)[arXiv:1403.2944 [hep-ph]].

[181] P. Biancofiore, P. Colangelo, F. De Fazio and E. Scrimieri, [arXiv:1408.5614[hep-ph]].

[182] P. Biancofiore, P. Colangelo and F. De Fazio, Phys. Rev. D 87, no. 7, 074010(2013) [arXiv:1302.1042 [hep-ph]].













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