Flavor changing neutral current constraints from Kaluza-Klein gluons and quark mass matrices in the Randall-Sundrum I framework

Flavor changing neutral current constraints from Kaluza-Klein gluons and quark mass matricesin the Randall-Sundrum I framework

We-Fu Chang*

Department of Physics, National Tsing Hua University, Hsin Chu 300, Taiwan

John N. Ng† and Jackson M. S. Wu‡

Theory Group, TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, Canada(Received 23 November 2008; published 24 March 2009)

We continue our previous study on what are the allowed forms of quark mass matrices in the Randall-

Sundrum framework that can reproduce the experimentally observed quark mass spectrum and the pattern

of Cabibbo-Kobayashi-Maskawa mixing. We study the constraints the �F ¼ 2 processes in the neutral

meson sector placed on the admissible forms found there, and we found only the asymmetrical type of

quark mass matrices arising from anarchical Yukawa structures remain viable at the few TeV scale

reachable at the LHC. We study also the decay of the first Kaluza-Klein (KK) excitation of the gluon. We

give the decay branching ratios of the first KK gluon into quark pairs, and we point out that measurements

of the decay width and just one of the quark spins in the dominant �tt decays can be used to extract the

effective coupling of the first KK gluon to top quarks for both chiralities. This provides a further probe into

the flavor structure of the Randall-Sundrum framework.

DOI: 10.1103/PhysRevD.79.056007 PACS numbers: 11.30.Hv, 12.15.Ff

I. INTRODUCTION

The use of the warped extra-dimensional model ofRandall and Sundrum (RS) [1] as a framework for flavorphysics has garnered a lot of attention ever since themodel’s introduction. By implementing the split fermionscenario [2], the hierarchy in the standard model (SM)fermion masses can be understood geometrically in termsof the different localization of the SM fermions in the extradimension [3]. In such a setup, the different fermionmasses can be obtained without fine-tuning the Yukawacouplings, in contrast to the usual four-dimensionaltheories.

Having fermions propagating in the extra dimensionrequires that the SM gauge symmetry be promoted to abulk symmetry. Constraints then arise because of the elec-troweak precision tests (EWPTs). In particular, for thesimplest model with just the SM gauge group SUð3Þc �SUð2ÞL �Uð1ÞY , constraints on the S and T parametersand the Zbb couplings are found to be difficult to satisfywithout fine-tuning. Since an SUð2ÞR symmetry is instru-mental in ensuring the very accurate relation � ¼ 1 in theSM, a natural way to satisfy the EWPTs would be topromote the SUð2ÞR to a bulk gauge symmetry, and thiswas done in [4].1

In this work, we continue our study that began in Ref. [7]of the forms of quark mass matrices admissible in a mini-mal RS1 setting with custodial symmetry that can fit all theexperimental data in the quark sector without having hier-archical Yukawa structures. It is well known that tree-levelflavor changing neutral current (FCNC) interactions aregeneric in the RS flavor models. Processes mediated by theKaluza-Klein (KK) excitations of the gauge bosons—inparticular that of the gluons—can give rise to large FCNCeffects, which are tightly constrained by the many low-energy measurements in the neutral meson sector such as�K and B0

q- �B0q transition (q ¼ d, s). We study in this work

the impact these �F ¼ 2 FCNC constraints have on theadmissibility of the forms found in Ref. [7]. In particular,as these FCNC constraints place stringent limits on the

lowest KK gauge boson mass, mð1Þgauge, which sets the scale

of new physics (NP), we investigate which of the forms ofthe quark mass matrices can satisfy all the FCNC con-straints at an NP scale reachable by the LHC. Since thedominant contribution to the FCNCs comes from the KKgluons in the setting we study,2 we concentrate on theireffects below.The paper is organized as follows. In Sec. II, we give a

brief outline of the Minimal Custodial RS (MCRS) modelstudied to set the notation. In Sec. III, we study in theMCRS model the impact FCNCs mediated by the tree-level exchange of KK gluons have on the �F ¼ 2 pro-cesses in the meson sector. These place constraints on thescale of new flavor physics. In Sec. IV, we evaluate thecontribution due to the KK gluon exchanges in the neutralB-meson observables, and we calculate the branching ra-

*[email protected]†[email protected]‡[email protected] having the custodial symmetry is still the surest

way to satisfy the EWPT constraints, Ref. [5] has reportedrecently that they may also be satisfied by having a heavyHiggs boson alone. Interesting scenarios involving extendedbulk gauge symmetries include models with the Higgs as aholographic composite, or with gauge-Higgs unification [6]. 2This is explicitly checked in the calculations below.

PHYSICAL REVIEW D 79, 056007 (2009)

1550-7998=2009=79(5)=056007(11) 056007-1 � 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.79.056007

tios of the first KK gluon decaying into a pair of quarks. Wepoint out that measuring even just one of the quark spins,such as in top decays which are the dominant decay mode,can be very useful in distinguishing the different models inthe RS framework. We conclude in Sec. V. Appendix Acontains asymmetrical quark mass matrices that are typicalrepresentations of the families of the admissible asymmet-rical forms used in this work. In Appendix B, we show thatwith the fermion representation we use in this work, theelectroweak contributions neither displace the dominanceof the KK gluon contributions, nor cause the current FCNCbounds to be violated if they are included as well.

II. THE MCRS MODEL

In this section, we describe briefly the basic setup of theMCRS model to establish notations (see also Ref. [7])relevant for studying the flavor changing processes in themeson sector. A more complete and detailed descriptioncan be found in, e.g., Ref. [4].

The MCRS mode is formulated on a slice of AdS5 spacespecified by the metric

ds2 ¼ GABdxAdxB ¼ e�2�ð�Þ��dx

�dx� � r2cd�2; (1)

where �ð�Þ ¼ krcj�j, �� ¼ diagð1;�1;�1;�1Þ, k is

the AdS5 curvature, and �� . The theory iscompactified on an S1=ðZ2 � Z0

2Þ orbifold, with rc theradius of the compactified fifth dimension, and the orbifoldfixed points at � ¼ 0 and � ¼ � correspond to the UV(Planck) and IR (TeV) branes, respectively. To solve thehierarchy problem, k�rc is set to � 37. The warped down

scale is defined to be ~k ¼ ke�k�rc . Note that ~k sets the scale

of the first KK gauge boson mass, mð1Þgauge � 2:45~k, which

determines the scale of the new KK physics.The MCRS model has a bulk gauge group SUð3Þc �

SUð2ÞL � SUð2ÞR �Uð1ÞX under which the IR brane-localized Higgs field transforms as ð1; 2; 2Þ0. The SMquarks are embedded into SUð2ÞL � SUð2ÞR �Uð1ÞX viathe five-dimensional (5D) bulk Dirac spinors

Qi ¼ uiL½þ;þ�diL½þ;þ�

� �; Ui ¼ uiR½þ;þ�

~diR½�;þ��

;

Di ¼ ~uiR½�;þ�diR½þ;þ�

� �; i ¼ 1; 2; 3;

(2)

where Qi transforms as ð2; 1Þ1=6, and Ui, Di transform as

ð1; 2Þ1=6. The parity assignment � denote the boundary

conditions applied to the spinors on the [UV, IR] brane,with þ (� ) being the Neumann (Dirichlet) boundaryconditions. Only fields with the ½þ;þ� parity containzero-modes that do not vanish on the brane. These survivein the low-energy spectrum of the 4D effective theory, andare identified as the SM fields.

A given 5D bulk fermion field, �, can be KK expandedas

�L;Rðx;�Þ ¼ e3�=2ffiffiffiffiffiffiffiffirc�

pX1n¼0

c ðnÞL;RðxÞfnL;Rð�Þ; (3)

where subscripts L and R label the chirality, and the KKmodes fnL;R are normalized according to

1

�

Z �

0d�fn�L;Rð�ÞfmL;Rð�Þ ¼ mn: (4)

The KK-mode profiles are obtained from solving the equa-tions of motion. For the zero-modes, the RS flavor func-tions are given by

f0L;Rð�; cL;RÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikrc�ð1� 2cL;RÞekrc�ð1�2cL;RÞ � 1

seð1=2�cL;RÞkrc�; (5)

where the c-parameter is determined by the bulk Diracmass parameter, m ¼ ck, and the upper (lower) sign ap-plies to the LH (RH) label. Depending on the orbifoldparity of the fermion, one of the chiralities is projected out.After spontaneous symmetry breaking, the Yukawa in-

teractions localized on the IR brane lead to mass terms forthe fermions on the IR brane

SYuk ¼Z

d4xvW

krc�½ ��uðx; �Þu

5�uðx; �Þ

þ ��dðx; �Þd5�dðx; �Þ� þ H:c:; (6)

where vW ¼ 174 GeV is the vacuum expectation value

acquired by the Higgs field, and u;d5 are the (complex)

dimensionless 5D Yukawa coupling matrices. For zero-modes, this gives the mass matrices for the SM quarks inthe 4D effective theory

ðMRSf Þij ¼ vW

krc�f5;ijf

0Lð�; cLfiÞf0Rð�; cRfjÞ

vW

krc�f5;ijFLðcLfiÞFRðcRfjÞ; f ¼ u; d; (7)

where the label f denotes up-type or down-type quarkspecies. The up and down mass matrices are diagonalizedby a bi-unitary transformation

ðUu;dL ÞyMRS

u;dUu;dR ¼

mu;d1 0 00 mu;d

2 00 0 mu;d

3

0B@

1CA; (8)

where mu;di are the masses of the SM up-type and down-

type quarks. The mass eigenbasis is then defined by c 0 ¼Uyc , and the CKM matrix given by VCKM ¼ ðUu

LÞyUdL.

III. �F ¼ 2 PROCESSES IN THE MESON SECTOR

In extra-dimensional models, tree-level flavor changingneutral currents arising from the KK excitations of gaugebosons are generic. For �F ¼ 2 FCNCs, by virtue of thestrength of the strong coupling constant, the largest andthus the most constrained contribution comes from pro-cesses mediated by the exchange of the KK gluons as

WE-FU CHANG, JOHN N. NG, AND JACKSON M. S. WU PHYSICAL REVIEW D 79, 056007 (2009)

056007-2

depicted in Fig. 1. Effective four-fermion operators aregenerated when the KK gluons are integrated out.

In the gauge (weak) eigenbasis, the coupling of the n-th

level KK gluon, GðnÞ, to zero-mode fermions is given by

GðnÞ�

�Xi

ðgnfÞLii �fiL��fiL þ ðL $ RÞ�; f ¼ u; d; (9)

where ðgnfÞij ¼ diagðgnf1 ; gnf2 ; gnf3Þ is the weak eigenbasis

coupling matrix with

gnfi ¼gs�

Z �

0d�jf0ð�; cfiÞj2�nð�Þ; gs ¼ g5sffiffiffiffiffiffiffiffi

rc�p :

(10)

Here, g5s is the bulk 5D SUð3Þ gauge coupling, gs that inthe SM, and �n the profile of the n-th KK gluon. Note thatthe matching relation between g5s and gs can be changedby the presence of localized brane kinetic terms. As inRef. [8], we have chosen here and for the analysis belowUV boundary terms such that the bare kinetic terms cancelexactly the contribution coming from the one-loop run-ning. The IR brane kinetic terms are small and can beneglected.

Going to the mass eigenbasis f0 ¼ Uyf, the GðnÞf0f0coupling reads

GðnÞ�

�Xa;b

ðgnfÞLab �f0aL��f0bL þ ðL $ RÞ�; f ¼ u; d;

(11)

where

ðgnfÞL;Rab ¼ Xi;j

ðUyL;RÞaiðgnfÞL;Rij ðUL;RÞjb: (12)

The off-diagonal couplings ðgnfÞab appear because the di-

agonal weak eigenbasis couplings, gnfi , are not all equal.

In order to compute the coefficients of the effective four-fermion operators arising from the tree-level KK gluonexchanges, one has to perform (in the mass eigenbasis)sums of the form

S!; ab;cd ¼

X1n¼1

ðgnfÞ!abðgnfÞ cdm2

n

; !; ¼ L; R; (13)

where mn is the mass of the n-th KK gluon, and !, labelthe chirality. The sum over the KK gluon tower can beefficiently calculated with the help of the massive gauge5D mixed position-momentum space propagators[4,9,10].3 It can be computed in terms of the overlapintegral,

Gþþff ðc!i ; c j Þ ¼

1

�

Z �

0d�jf0!ð�; c!i Þj2 ~GðþþÞ

p¼0 ð�;�0Þ

� jf0 ð�0; c j Þj2; !; ¼ L; R; (14)

where ~GðþþÞp¼0 is the zero-mode subtracted gauge propagator

evaluated at zero 4D momentum, and is given by [10]

~GðþþÞp¼0 ð�;�0Þ ¼ 1

4kðkrc�Þ�1� e2krc�

krc�

þ e2krc�<ð1� 2krc�<Þþ e2krc�>½1þ 2krcð��>Þ�

�; (15)

where �< ¼ minð�;�0Þ, �> ¼ maxð�;�0Þ. The sumover the KK tower is then given by

S!; ab;cd ¼ g2s

Xi;j

ðUy!ÞaiðU ÞibGþþ

ff ðc!i ; c j ÞðUy!ÞcjðU Þjd;

!; ¼ L; R: (16)

The most general effective Hamiltonian for the �F ¼ 2processes beyond the SM can be written as

HNPeff ¼

X5i¼1

Cið�ÞQabi þX3

i¼1

~Cið�Þ ~Qabi ; (17)

where � is the scale of new physics, and

Qab1 ¼ �c �

aL��c�bL

�c �aL�

�c �bL;

Qab2 ¼ �c �

aRc�bL

�c �aRc

�bL; Qab

3 ¼ �c �aRc

�bL

�c �aRc

�bL;

Qab4 ¼ �c �

aRc�bL

�c �aLc

�bR; Qab

5 ¼ �c �aRc

�bL

�c �aLc

�bR;

(18)

with �, � the color indices, and a, b the generation

indices.4 The operators ~Qab1;2;3 are obtained from Qab

1;2;3 by

the L $ R exchange. All operators are given in the mass

eigenbasis here. In the MCRS model, only Qab1;4;5 and

~Qab1

arise from the tree-level exchange of KK gluons, and theircoefficients are given by

FIG. 1. Contributions to �F ¼ 2 processes from the tree-levelexchange of KK gluons. The fermions are in the weak eigenba-sis.

3See also Ref. [8] for an equivalent way of summing up thegluon KK tower.

4The so-called supersymmetric (SUSY) basis of operators [11]is used here. Other basis can be obtained via the appropriateFierz identities.

FLAVOR CHANGING NEUTRAL CURRENT . . .. I FRAMEWORK PHYSICAL REVIEW D 79, 056007 (2009)

056007-3

C1ð�Þ ¼ 16S

LLab;ab;

~C1ð�Þ ¼ 16S

RRab;ab;

C4ð�Þ ¼ �SLRab;ab; C5ð�Þ ¼ �1

3C4:(19)

Note that here the NP scale is the scale where the KKexcitations first come in, hence �m1.

Recently, a model-independent global analysis of thephysical observables in the �F ¼ 2 processes have beenperformed by the UTfit Collaboration [12]. Bounds on theNP scale Wilson coefficients Cið�Þ above have been foundwith the Renormalization Group evolution fully taken intoaccount. Given these bounds, an immediate question withregard to the admissible forms of quark mass matricesfound in Ref. [7] is whether they remain viable, as theygovern the form of the rotation matrices, U!, that deter-mine the Wilson coefficients in the MCRS model (seeEqs. (16) and (19)).

Two types of mass matrix structures were found inRef. [7] that reproduce well the observed patterns of quarkmasses and CKM mixings, and are compatible with non-hierarchical and perturbative Yukawa structures (j5j< 4[13]) in the RS framework. In one type, mass matrices havea symmetrical texture that is a slight deformation of theansatz proposed by Koide et. al. [14]. In the other, there areno symmetries a priori. The form of the mass matrices ischaracterized by the localizations of the fermions in the 5Dbulk that are admissible under the electroweak constraints,and each particular realization of the form arises fromYukawa structures that are completely anarchical. Foreach type of the quark mass matrices, we calculate belowthe resulting Wilson coefficients for the �F ¼ 2 processesdue to KK gluon exchanges, and we compare them to theUTfit bounds.

For the symmetrical Koide-type form of quark massmatrix, we begin by focusing on the kaon sector wherethe constraints are most stringent [12]. At � ¼ 4 TeV,while the imaginary part of the resulting kaon sectorWilson coefficients are all very much smaller than thebounds listed, we find the real parts are all larger than therespective bounds by 3 orders of magnitude. As a result,insisting that the symmetrical type pass the UTfit boundswould require one to push the NP scale up toOð100Þ TeV.5

For asymmetrical forms, we demonstrate that each of theasymmetrical configurations discussed in Ref. [7] remainviable at the few TeV scale. In Table I, we list the UTfitbounds on the relevant Wilson coefficients, and we givetheir values for a typical ‘‘solution’’—admissible set of upand down quark mass matrices which give the observedquark masses and mixings, and satisfy all electroweak and

FCNC bounds—at � ¼ 4 TeV (corresponding to ~k ¼1:65 TeV where m1 ’ 4 TeV) in each of the asymmetrical

configurations, and we see that they are all well within thebounds. The details of the specific quark mass matricesused are given in Appendix A. In all calculations, we haveexplicitly checked that the KK gluons do indeed give riseto the dominant contributions in the tree-level �F ¼ 2process under study. We show in Appendix B that thecontributions from the electroweak sector are small asexpected, and would not lead to violations of the UTfitbounds if included with the KK gluon contributions.Note that in Table I, only one of the many admissible

solutions we found are given for each asymmetrical con-figuration. Moreover, the configurations of fermion local-izations themselves are just three of many that we foundwhich lead to admissible solutions. Indeed, we have foundthat parameter space generically exists in the RS1 settingwhere quark mass and mixing data and �F ¼ 2 FCNCbounds can be satisfied at the few TeV scale with asym-metrical quark mass matrices that arise from underlyinganarchical Yukawa structures. This does not, however,contravene the conclusion reached in Ref. [8] that a KKscale of 10 to 20 TeV is necessary to satisfy the �F ¼ 2FCNC constraints in the kaon sector. The higher NP scaleis required if one wants to ensure that the FCNC bounds aregenerically satisfied for any given quark mass matrices thatgive the pattern of the observed quark mass hierarchy andCKM mixings. Our point here is that a subset of theseconsisting of asymmetrical quark mass matrices existssuch that the experimental quark masses and mixings arereproduced to a high accuracy, and at the same time thelower, few TeV scale is still viable.6 We emphasize herethat this subset does not contain isolated singular points inthe parameter space, but generic solutions throughout allthe parameters space.Now one may worry that radiative correction may spoil

our results, as there are loop induced corrections to thebrane-localized Yukawa couplings, and loop induced branekinetic mixing terms that can introduce additional flavorviolations. This is, however, not so. First, we are notcalculating theoretically the Yukawa couplings in the RSframework; to do that requires a UV completion of thetheory. There will be radiative corrections to the Yukawamatrices, but they will not change the form of the 4Deffective mass matrices given in Eq. (7) even if it is derivedat tree-level. Thus if the physical (or renormalized)Yukawa matrices take any of the forms that we found,the FCNC bounds will be satisfied, the form of the massmatrices we give should therefore be viewed as ‘‘physical’’and the corresponding Yukawa matrices renormalized.Next, the brane kinetic mixings, which are loop sup-pressed, lead to a correction to the gauge-fermion cou-plings of order j5Dj2=4�2 as can be estimated from

5As can be seen from Eq. (13), the mass of the lightest modesets the suppression scale for the four-fermion operator. To makeup for a factor of Oð103Þ at m1 ’ � ¼ 4 TeV would require afactor of Oð30Þ increase in m1.

6The few TeV NP scale can also be achieved if one imposesadditional symmetries. For recent works in this direction see,e.g., Refs. [15,16].


056007-4

NDA (naive dimensional analysis) [16]. In the search forsolutions, we have set j5Dj & 2; consequently � 1 andthe flavor violating contribution from the brane kineticmixing terms is small (Oð0:01Þ of the KK gluon contribu-tions), which do not impact the Wilson coefficientscalculated.

IV. EXPERIMENTAL OBSERVABLES

A. B0q- �B

0q mixings

One very sensitive probe to NP in the meson sectorcomes from the B0

q- �B0q mixing (q ¼ d, s), which has re-

ceived much theoretical attention, and has now an exten-sive body of experimental data from the B factories andFermilab. The contribution of NP to �B ¼ 2 transitionscan be parametrized in a model-independent way as theratio of the full (SMþ NP) amplitude to the SM one [12]7:

hB0qjH full

eff j �B0qi

hB0qjH SM

eff j �B0qi

¼ 1þ hB0qjH NP

eff j �B0qi

hB0qjH SM

eff j �B0qi

Cqe2i�q ; q ¼ d; s:

(20)

The SM amplitude arises mainly from the one-loop boxdiagram, which is dominated by the top quark exchanges.It can be written as

hB0qjH SM

eff j �B0qi ¼ G2

Fm2W

6�2�Bm

2Bqf2Bq

BBqðV�

tqVtbÞ2S0ðxtÞ;(21)

where xt m2t =m

2W , �B ¼ 0:552 is a short distance QCD

correction [18], and S0 is an ‘‘Inami-Lim’’ function [19]with mtðmtÞ ¼ 163:6 GeV [20]. We take for the CKMmixings [17]

jV�tdVtbj ¼ 8:6� 10�3; jV�

tsVtbj ¼ 41:3� 10�3;

(22)

for the decay constants [21]

fBd¼ 197 MeV; fBs

¼ 240 MeV; (23)

and for the renormalization invariant bag parameter [22]

fBd

ffiffiffiffiffiffiffiffiBBd

q¼ 244 MeV; fBs

ffiffiffiffiffiffiffiffiBBd

q¼ 295 MeV: (24)

All other input parameters take their values from the PDG[23].In the MCRS model, the NP contribution to the �B ¼ 2

transition amplitude is dominated by the tree-level ex-changes of KK gluons, as the coupling strength for thestrong interactions is much larger than that for the electro-weak interactions. Evolving down from the NP scale � tothe hadronic scale � ¼ mb, the KK gluon contribution isgiven by

hB0qjH NP

eff j �B0qi ¼

�B0qjXr

Crð�ÞQbqr ð�Þ

þXs

~Csð�Þ ~Qbqs ð�Þj �B0

q

; (25)

where

TABLE I. The 95% allowed range of the Wilson coefficients [12] contributing in the �F ¼ 2tree-level gluon exchange processes, and their typical values at � ¼ 4 TeV in each of theasymmetrical configurations given in Ref. [7]. All values are given in units of GeV�2.

Parameter 95% allowed range Config. I Config. II Config. III

ReC1K ½�9:6; 9:6� � 10�13 4:3� 10�17 1:8� 10�15 �4:2� 10�15

ReC4K ½�3:6; 3:6� � 10�15 �1:4� 10�16 �2:8� 10�16 �1:8� 10�15

ReC5K ½�1:0; 1:0� � 10�14 4:6� 10�17 9:4� 10�17 6:0� 10�16

ImC1K ½�4:4; 2:8� � 10�15 2:6� 10�18 1:8� 10�15 �1:0� 10�15

ImC4K ½�1:8; 0:9� � 10�17 1:5� 10�19 8:8� 10�18 �1:8� 10�18

ImC5K ½�5:2; 2:8� � 10�17 �4:9� 10�20 �2:9� 10�18 6:0� 10�19

jC1Dj <7:2� 10�13 1:3� 10�13 3:1� 10�13 1:6� 10�14

jC4Dj <4:8� 10�14 1:7� 10�15 8:8� 10�15 4:0� 10�14

jC5Dj <4:8� 10�13 5:7� 10�16 2:9� 10�15 1:3� 10�14

jC1Bdj <2:3� 10�11 7:5� 10�13 7:7� 10�14 4:8� 10�13

jC4Bdj <2:1� 10�13 1:9� 10�13 4:8� 10�14 1:7� 10�13

jC5Bdj <6:0� 10�13 6:2� 10�14 1:6� 10�14 5:6� 10�14

jC1Bsj <1:1� 10�9 9:0� 10�11 4:1� 10�11 4:0� 10�11

jC4Bsj <1:6� 10�11 9:4� 10�12 7:6� 10�13 5:8� 10�12

jC5Bsj <4:5� 10�11 3:1� 10�12 2:5� 10�13 1:9� 10�12

7See also Ref. [17].


056007-5

Crð�Þ ¼ Xi;j

ðbðr;iÞj þ �cðr;iÞj Þ��jCið�Þ;

� ¼ �sð�Þ�sðmpole

t Þ ;(26)

are the Wilson coefficients at the hadronic scale, with ~Cr

defined similarly with the same coefficients as for Cr, and

mpolet ¼ 171:4 GeV [23]. The magic numbers �j, b

ðr;iÞj ,

cðr;iÞj , and the operator matrix elements can be found in

Ref. [24].8

In Table II, we give the values of the parameters Cq and

�q for each of the three asymmetrical configurations so-

lutions used in Table. I. The values of Cq and�q agree well

with the UTfit values at 95% probability (and mostly at68% as well; see Table III in Ref. [12]) as expected, sincethe physical observables fitted here are the same that gointo the analysis for the meson sector flavor bound on theNPWilson coefficients listed in Table I. As above, we havechecked that the electroweak contributions are small—theyare much less than the standard error given by the UTfitCollaboration at 68% probability—and do not cause largeshifts that would violate the UTfit bounds. We note for theconfigurations of solutions given here, KK gluons are notmanifest in the B0

q- �B0q mixing, and the SM effects are

expected to be dominant.

B. KK gluon top decays

In RS models, a distinguishing property of the KKgluons is that their couplings to the LH and RH fermions(in the mass eigenbasis), gL and gR, are in general not thesame. For all the asymmetrical quark mass matrix solutionsthat we found, this is true. To test this experimentally, oneway is to measure both the decay width and the spin of thetop in the decays of gluons into top pairs as we show below.

We will also be concentrating on the first KK gluon, Gð1Þ,which has the highest potential of being within the reach ofthe LHC.

As Gð1Þ couples strongly to states localized near the IRbrane, and the large top mass requires that either Q3 or tRbe IR localized, top decays are expected to be dominant

modes of decay. The partial width of Gð1Þ decaying intoquarks in the mass eigenbasis, �qaqb, can be written as

�ðGð1Þ ! �qaqbÞ ¼ m1

48�ð1; x2a; x2bÞ1=2

�1

2ðjg1Lj2 þ jg1Rj2Þ½2� x2a � x2b � ðx2a � x2bÞ2� þ 6Re½g1Lðg1RÞ��xaxb

�

! m1

48�ðjg1Lj2 þ jg1Rj2Þ ðxa ¼ xb � 1Þ; xa;b ¼ ma;b

m1

; (27)

where m1 is the mass of Gð1Þ, g1L;R denote the mass eigen-basis couplings, ðg1fÞL;Rab , given in Eq. (12), andðu; v; wÞ ¼ ðu� v� wÞ2 � 4vw. For the three asym-metrical configuration solutions used in Table I, and form1 ¼ 4:0 TeV, the widths into the �tt pairs are

769:3 GeV ðConfig: IÞ;635:4 GeV ðConfig: IIÞ;747:4 GeV ðConfig: IIIÞ:

(28)

In Table III, we give the branching ratios of Gð1Þ into top,bottom, and all other modes involving at least one lightquark (jets) for the same three asymmetrical solutions.

We see from Table III that most decays are into top pairs,with negligible fraction into light quarks. Compared toRef. [27] (see Table I), the branching ratio into top pairsfrom each of our asymmetrical configurations is slightlylower at about 80% instead of around 90%, which is due tothe difference in the quark mass matrices and the localiza-tion parameters used. Note that the branching ratios arestable across the different configurations. This is becausethe couplings of KK gluons to quarks are dominated by that

TABLE II. Parameters determining the NP contributions toB0q- �B

0q mixings in the MCRS model with mass matrices from

the three asymmetrical configurations given in Ref. [7].

Parameter Config. I Config. II Config. III

Cd 1.13 1.02 1.08

�d [�] �2:48 �0:24 �3:02Cs 1.68 1.36 1.29

�s [�] 0.61 0.12 0.04

TABLE III. Branching ratios of Gð1Þ into �qaqb pair in theMCRS model with mass matrices from the three asymmetricalconfigurations given in Ref. [7]. The term ‘‘light quarks’’ heredenotes all modes (flavor changing included) that involve at leastone light quark (jet).

Branching ratios Config. I Config. II Config. III

Top quarks 0.83 0.83 0.84

Bottom quarks 0.16 0.16 0.15

Light quarks 0.01 0.01 0.01

8Note that Ref. [24] works in the Landau RI-MOM scheme[25]; for magic numbers in theMS (NDR) scheme, see Ref. [26].For consistency, all running quark masses used in Eq. (25)should be in the same scheme as the operator matrix elements.The relevant quark masses in the RI-MOM scheme arembðmbÞ ¼ 4:6 GeV, msðmbÞ ¼ 87 MeV, and mdðmbÞ ¼5:4 MeV.


056007-6

to the third-generation quarks, which varied little acrossthe configurations. It is thus fairly robust that in the RSscenario, the KK gluon will decay predominantly into toppairs, and then into b-jets with a much smaller, but non-negligible rate. Other light quark modes are negligible andcertainly no leptons. However, as can be seen fromEq. (28), the top pair width is not small as ��tt=m1 0:2.Thus looking for signals in the resonant productions willrequire good top identification. Detailed discussions of thediscovery potential at the LHC can be found in Ref. [27].We note that the bottom mode should not be overlookedand can be used as a check if not the primary discoverytool.

If a KK gluon is found at the LHC, it will certainly beimportant to measure the spin of the top in its �tt decays.The differential decay rate with only one of the top spinsmeasured but with the other top spin summed over is givenby

d�s

d cos�¼ m1

192�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4x2t

qfðjgLj2 þ jgRj2Þð1� x2t Þ

þ 6ReðgLg�RÞx2t þ 2ðjgRj2 � jgLj2Þ� xt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4x2t

qs pg; (29)

where

s p ¼ cos�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ð1� 4x2t Þcos2�

p ; cos� s p; (30)

with s the measured top spin three-vector, and p the three-

momentum of the same top quark in the rest frame of Gð1Þ.From this we see that a measurement of the angular de-pendence together with the decay width can allow one toextract out gL and gR. The feasibility of doing this at theLHC requires detailed numerical simulations which arebeyond the scope of the present work (see Ref. [28] forwork in this direction).

V. CONCLUSION

Previously in Ref. [7] we have studied the phenomeno-logically allowed form of quark mass matrices in theMCRS model, and we have found admissible both a sym-metrical form, and many distinctive asymmetrical configu-rations with Yukawa structures nonhierarchical andanarchical that satisfy all EWPTs. The benchmark warped

down scale was chosen at ~k ¼ 1:65 TeV implying anequivalently NP scale of � ¼ 4 TeV, since a higher scalewill prevent the KK gauge bosons from being detectable atthe LHC initially at least. A much higher scale will alsocreate its own hierarchy problem which one would like toavoid. We continue the study in this work with the con-straints that �F ¼ 2 processes in the neutral meson sectorimpose. We found from these constraints that for the sym-metrical mass quark matrices, the viable scale is pushed up

toOð100Þ TeV. However, for the asymmetrical quark massmatrices, � ¼ 4 TeV is still viable. This is a consequenceof the fact that in the asymmetrical cases there is freedomin the LH and RH rotations being very different—ratherthan being locked into a specific pattern as in the sym-metrical case—which can supply the suppression requiredto pass the meson sector �F ¼ 2 constraints. This under-scores the importance of the quark mass matrices in the RSframework both phenomenologically and theoretically foridentifying any family symmetries that may be hidden.At the� ¼ 4 TeV scale, discovery of the first KK gluon

state at the LHC is possible. This can be achieved through aresonance search in the �tt channel which we predict to havea branching ratio of � 0:8. Note that the dominance of the�tt decays is a characteristic of the RS1 scenario. The �bbmode has a branching fraction of about 0.15, and shouldnot be overlooked. This mode consists mainly of LH pairsbecause bL is an SUð2Þ partner to tL, which has a largeoverlap with GKK. Thus this channel can be useful as adiagnostic tool if the expected background can be handled.All other decay modes involving light quarks are negli-gible. Finally, if one can also measure in the �tt decays atleast one of the quark spins, it will help to unravel gL andgR, and provide further an invaluable probing into theflavor structure of the RS scenario.

ACKNOWLEDGMENTS

We thank C. Csaki for useful comments. W. F. C. isgrateful to the TRIUMF Theory Group for their hospitalitywhen part of this work was completed. The research ofJ. N. N. and J.M. S.W. is partially supported by the NaturalScience and Engineering Council of Canada. The work ofW. F. C. is supported by the Taiwan NSC under GrantNo. 96-2112-M-007-020-MY3.Note added,—At the time when this work was com-

pleted, Ref. [34] came out which has also consideredsome of the same issues. There the bulk gauge groupcontains an additional discrete left-right parity group, andconsequently the fermion matter contents are embedded ina different representation than the one used in this work,resulting in electroweak contributions to the �F ¼ 2FCNCs that are far larger. We have checked that bothworks agree whenever direct comparisons can be made.

APPENDIX A: TYPICAL SOLUTIONS FOR THEASYMMETRICAL CONFIGURATIONS

In this appendix, we give the details of the quark massmatrices of the typical solution used in Table I in each ofthe three asymmetrical configurations given in Ref. [7].Although the bound on Z �bLbL used there was that given inthe PDG [23], many generic solutions from generic con-figurations exist with localization parameters that caneasily accommodate the more stringent bound found in,e.g., Refs. [29–31].


056007-7

Parametrizing the complex 5D Yukawa couplings as5;ij ¼ �ije

i�ij , admissible mass matrices of the forms

given by Eq. (7) are found with �ij and �ij randomly

and uniformly generated in the intervals ð0; 2Þ and½��;�Þ respectively. In the following, we list the complexmass matrices in the form of Mf ¼ jMfjei�f , the magni-

tude and phase of the 5D Yukawa couplings, and the masseigenvalues for both the up and down sector. All values aregiven to six significant figures. The mass eigenvalues agreewith the quark masses at 1 TeV found in Ref. [20] to within2 standard deviations quoted.

(i) Configuration I:

cQ ¼ f0:633604; 0:556171; 0:256293g;cU ¼ f�0:663816;�0:535621; 0:185413g;cD ¼ f�0:641469;�0:572479;�0:616085g:

(A1)

jMuj ¼0:00136839 0:0770365 1:19782

0:00778813 0:560874 2:93683

0:24404 8:1122 147:741

0BB@

1CCA;

�u ¼1:59621 2:80118 �2:65001

�2:34319 �0:190895 �0:644161

�1:61289 0:584021 0:07447

0BB@

1CCA

(A2)

�u ¼1:52494 1:57620 1:561650:765990 1:01280 0:3379231:46664 0:895098 1:03875

0@

1A;

�u ¼1:39426 1:49660 1:500050:716676 0:984072 0:3321611:39602 0:884794 1:03875

0@

1A

(A3)

mu1 ¼ 0:369308 MeV;

mu2 ¼ 0:409125 GeV;

mu3 ¼ 147:999 GeV:

(A4)

Md ¼0:00205044 0:0096169 0:0025584

0:00702768 0:0985925 0:0173996

0:242765 2:33774 0:76264

0BB@

1CCA;

�d ¼�0:184947 2:04673 1:12293

�1:04910 1:68206 �2:47164

0:00506372 �2:31542 3:06043

0BB@

1CCA

(A5)

�d ¼1:07943 0:555546 0:583531

0:326515 0:502659 0:350252

0:689207 0:728280 0:938063

0BB@

1CCA;

�d ¼0:993588 0:522050 0:541247

0:307557 0:483365 0:332447

0:660453 0:712475 0:905822

0BB@

1CCA

(A6)

md1 ¼ 2:25527 MeV;

md2 ¼ 47:9153 MeV;

md3 ¼ 2:47254 GeV:

(A7)

(ii) Configuration II:

cQ ¼ f0:628524; 0:546221; 0:285007g;cU ¼ f�0:662224;�0:550397; 0:0801805g;cD ¼ f�0:579521;�0:628656;�0:626738g:

(A8)

Mu ¼0:000705160 0:0296351 1:25154

0:00391734 0:303462 4:75543

0:157250 8:57855 148:068

0BB@

1CCA;

�u ¼3:03996 0:107148 2:03582

1:79158 �1:76781 �2:88842

2:07507 �0:648895 3:02998

0BB@

1CCA

(A9)

�u ¼0:576867 0:721635 1:44261

0:259565 0:598525 0:443977

0:908033 1:47451 1:20473

0BB@

1CCA;

�u ¼3:03996 0:107148 2:03582

1:79158 �1:76781 �2:88842

2:07507 �0:648895 3:02998

0BB@

1CCA

(A10)

mu1 ¼ 1:05432 MeV;

mu2 ¼ 0:399582 GeV;

mu3 ¼ 148:398 GeV:

(A11)


056007-8

Md ¼0:00418127 0:000860589 0:00186071

0:0663893 0:0619168 0:0228064

2:43751 0:183510 0:140323

0BB@

1CCA;

�d ¼�2:87101 1:39416 2:70561

�2:90716 1:17362 2:90809

�1:80892 �2:30267 2:33582

0BB@

1CCA (A12)

�d ¼0:237597 0:231857 0:4708860:305562 1:35114 0:4674770:977694 0:348985 0:250662

0@

1A;

�d ¼�2:87101 1:39416 2:70561�2:90716 1:17362 2:90809�1:80892 �2:30267 2:33582

0@

1A

(A13)

md1 ¼ 1:41124 MeV;

md2 ¼ 66:9487 MeV;

md3 ¼ 2:44931 GeV:

(A14)

(iii) Configuration III:

cQ ¼ f0:627322; 0:570679; 0:272429g;cU ¼ f�0:517935;�0:664365; 0:180466g;cD ¼ f�0:576159;�0:610047;�0:638422g;

(A15)

Mu ¼0:147921 0:00223583 0:70694

0:787783 0:00477027 4:06577

8:66604 0:201339 145:112

0BB@

1CCA;

�u ¼�2:80680 2:86302 2:43167

�0:23652 �1:20710 �1:23730

1:00216 0:0966827 0:0

0BB@

1CCA

(A16)

�u ¼1:53467 1:88939 0:7234851:38068 0:680969 0:7028960:641530 1:21401 1:05965

0@

1A;

�u ¼�2:80680 2:86302 2:43167�0:236520 �1:20710 �1:237301:00216 0:0966827 0:0

0@

1A

(A17)

mu1 ¼ 1:49993 MeV;

mu2 ¼ 0:553929 GeV;

mu3 ¼ 145:430 GeV:

(A18)

Md ¼0:0122178 0:00379117 0:00346894

0:0813964 0:0316802 0:0033306

2:33248 0:899976 0:488706

0BB@

1CCA;

�d ¼2:54815 2:37217 �1:79028

0:769324 �0:385483 0:262617

0:348142 2:10335 0:0

0BB@

1CCA

(A19)

�d ¼0:603011 0:537917 1:237890:678640 0:759331 0:2007750:821415 0:911140 1:24436

0@

1A;

�d ¼2:54815 2:37217 �1:790280:769324 �0:385483 0:2626170:348142 2:10335 0:0

0@

1A

(A20)

md1 ¼ 2:38820 MeV;

md2 ¼ 60:8655 MeV;

md3 ¼ 2:54821 GeV:

(A21)

APPENDIX B: ELECTROWEAK CONTRIBUTIONSTO THE TREE-LEVEL �F ¼ 2 FCNCS

The electroweak contributions to �F ¼ 2 FCNCs comefrom the tree-level processes mediated by the KK photons,the Z boson, and the heavy Z0 boson that arise due to theSUð2ÞR in the MCRS model [4].9 As we show below, theelectroweak contributions are small due to the suppressionof the (much) smaller electroweak interaction strengthrelative to that of the strong interaction (at the NP scale�).The electroweak gauge bosons contribute to the�F ¼ 2

processes in two ways. They contribute either directlythrough the four-fermion process analogous to that inFig. 1, or they modify the gauge-fermion vertex throughmixings with gauge and fermion KKmodes as discussed inRef. [7]. In the former case, all electroweak gauge KKmodes can contribute, while the latter only happens via themixing of the Z zero-mode with the Z0 KK modes and theKK fermions.For the direct electroweak contribution, the Wilson co-

efficients have similar forms as those for the KK gluonsgiven in Eq. (19), but with appropriate changes in thenumerical coefficients (no color factor 1=3), the interactionstrengths, and gauge boson wave functions:

9Note unlike the Z field which has ½þ;þ� boundary condi-tions, the Z0 field has ½�;þ� boundary conditions, which giverise to KK excitations only.


056007-9

C1ð�ÞEW ¼ 12S

LLab;abðAKKÞ;

~C1ð�ÞEW ¼ 12S

RRab;abðAKKÞ;

C4ð�ÞEW ¼ 0;

C5ð�ÞEW ¼ �2SLRab;abðAKKÞ;

(B1)

where A ¼ �, Z, Z0. Note that without the color structure,there is no electroweak contribution to the Wilson coeffi-cient C4 at tree-level.

For the photon and the Z (and their respective KKexcitations), their couplings to the fermions are the sameas in the SM. For the Z0, its coupling to the fermionsdepends on gR, the gauge coupling constant of SUð2ÞR.Since it is commonly assumed in the literature that thecoupling constants of SUð2ÞL;R are equal, for the purpose

of comparison we take gR ¼ gL also. The Z0ff coupling isthen given by gZ0QZ0 ðfÞ, where

gZ0 ¼ cgZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� s2=c2

p ; QZ0 ðfÞ ¼ T3RðfÞ �

s2

c2Yf

2; (B2)

with the usual definitions gZ ¼ e=ðscÞ, s ¼ e=gL, and c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� s2

p.

In Table IV, we list the ratio of electroweak contributionto the Wilson coefficients at � ¼ 4 TeV to that due to KKgluons alone for each source of the direct electroweak tree-

level four-fermion processes. Note that because the differ-ence between the overlap integrals for the ½þ;þ� and½�;þ� gauge bosons is very small as the respective bulkprofiles are almost the same in regions where large overlaphappens, the ratios listed in Table IV are essentially justthat of the respective charge factors and electroweak gaugecoupling constants. Note also that the total electroweakcontribution for up-type and down-type quarks is the same,

and that for C1 and ~C1 is the same. This can be seen mosteasily in the gauge interaction basis where the magnitudeof the gauge charges are the same for both up-type anddown-type quarks, and the SUð2ÞL;R quark quantum num-

bers are the same. Since the direct processes depend on thesquare of the gauge charges, the conclusion follows.We remark here that the contributions to the Wilson

coefficients due to KK gluon and KK photon are universalfor all RS models with bulk fermions, but those due to Zand Z0 are not. This is because the coupling of the Z and Z0to fermions depends on the representation in which thefermions are embedded in the gauge group of the model.Throughout this work and in Table IV, the bulk fermionsare embedded such that the SM LH doublets (singlets) areSUð2ÞR singlets (doublets) so that they have SM quantumnumbers (see Eq. (2)). However, ratios different from thoselisted in Table IV would arise if different fermion repre-sentation is used. For example, we list the electroweak toKK gluon ratios in Table V in the case where the SM LH

TABLE IV. Ratio of direct electroweak contributions to KK gluon contributions in the Wilsoncoefficients. The fermion type ‘‘u’’ (‘‘d’’) denotes that up-type (down-type) quarks are involvedin the �F ¼ 2 FCNC process. All quarks have SM quantum numbers.

Wilson coefficient ratio Fermion type KK � KK Z KK Z0 Total

C1ð�ÞEW=C1ð�ÞQCD u 0.13 0.18 0.0054 0.32

d 0.033 0.28 0.0054 0.32~C1ð�ÞEW= ~C1ð�ÞQCD u 0.13 0.044 0.14 0.32

d 0.033 0.011 0.28 0.32

C5ð�ÞEW=C5ð�ÞQCD u �0:27 0.18 0.056 �0:033d �0:067 0.11 �0:077 �0:033

TABLE V. Ratio of direct electroweak contributions to KK gluon contributions in the Wilsoncoefficients. The fermion type ‘‘u’’ (‘‘d’’) denotes that up-type (down-type) quarks are involvedin the �F ¼ 2 FCNC process. Here the SM LH doublets, and u-type and d-type singletstransform as ð2; �2Þ2=3, ð1; 1Þ2=3, and ð1; 3Þ2=3 under SUð2ÞL � SUð2ÞR �Uð1ÞX respectively.

Wilsoncoefficient ratio Fermion type KK � KK Z KK Z0 Total

C1ð�ÞEW=C1ð�ÞQCD u 0.13 0.18 0.36 0.67

d 0.033 0.28 0.56 0.87~C1ð�ÞEW= ~C1ð�ÞQCD u 0.13 0.044 0.087 0.26

d 0.033 0.011 1.43 1.47

C5ð�ÞEW=C5ð�ÞQCD u �0:27 0.18 0.35 0.26

d �0:067 0.11 �1:78 �1:74


056007-10

doublets are embedded as bifundamentals in the SUð2ÞL �SUð2ÞR, and the up-type (down-type) singlets as SUð2ÞRsinglets (triplets) (see, e.g., Ref. [32]) so that there is a left-right parity [33]. Note that only the KK Z0 contributions aredifferent in changing to this representation because onlyQ0

Z is sensitive to the different assignment of the SUð2ÞRquantum numbers.

For the electroweak contributions due to mixings, theeffects are no longer universal—the suppression factors areno longer determined by the electroweak charges andcoupling constants alone—as there is now dependence onthe fermion localization parameters and the quark mixingmatrices. However, they are generically expected to besmall as they are Oðv4=�4Þ compared to the direct con-

tributions,10 although nongeneric enhancement may hap-pen depending on the particular quark mixing matricesinvolved, which typically do not exceed Oð0:01Þ of theKK gluon contributions. We have checked in each case thatthe combined effect of the direct and mixing electroweakcontributions does not appreciably alter the KK gluoncontributions to the Wilson coefficients, and that they arestill well within the UTfit bounds.

[1] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370(1999).

[2] N. Arkani-Hamed and M. Schmaltz, Phys. Rev. D 61,033005 (2000).

[3] Y. Grossman and M. Neubert, Phys. Lett. B 474, 361(2000); T. Gherghetta and A. Pomarol, Nucl. Phys.B586, 141 (2000); S. J. Huber and Q. Shafi, Phys. Lett.B 498, 256 (2001).

[4] K. Agashe, A. Degado, M. J. May, and R. Sundrum, J.High Energy Phys. 08 (2003) 050.

[5] S. Casagrande, F. Goertz, U. Haisch, M. Neubert, and T.Pfoh, J. High Energy Phys. 10 (2008) 094.

[6] E.g., R. Contino, Y. Nomura, and A. Pomarol, Nucl. Phys.B671, 148 (2003); G. Burdman and Y. Nomura, Phys. Rev.D 69, 115013 (2004); Y. Hosotani, S. Noda, Y. Sakamura,and S. Shimasaki, Phys. Rev. D 73, 096006 (2006); A. D.Medina, N. R. Shah, and C. E.M. Wagner, Phys. Rev. D76, 095010 (2007).

[7] W. F. Chang, J. N. Ng, and J.M. S. Wu, Phys. Rev. D 78,096003 (2008).

[8] C. Csaki, A. Falkowski, and A. Weiler, J. High EnergyPhys. 09 (2008) 008.

[9] L. Randall and M.D. Schwarz, J. High Energy Phys. 11(2001) 003.

[10] M. S. Carena, A. Delgado, E. Ponton, T.M. P. Tait, andC. E.M. Wagner, Phys. Rev. D 68, 035010 (2003).

[11] F. Gabbiani, E. Gabrielli, A. Masiero, and L. Silvestrini,Nucl. Phys. B477, 321 (1996).

[12] M. Bona et al. (UTfit Collaboration), J. High Energy Phys.03 (2008) 049.

[13] K. Agashe, G. Perez, and A. Soni, Phys. Rev. D 71,016002 (2005).

[14] Y. Koide, H. Nishiura, K. Matsuda, T. Kikuchi, and T.Fukuyama, Phys. Rev. D 66, 093006 (2002).

[15] J. Santiago, J. High Energy Phys. 12 (2008) 046.

[16] C. Csaki, A. Falkowski, and A. Weiler, arXiv:0806.3757.[17] P. Ball and R. Fleischer, Eur. Phys. J. C 48, 413 (2006).[18] E.g., G. Buchalla, A. J. Buras, and M. E. Lautenbacher,

Rev. Mod. Phys. 68, 1125 (1996).[19] T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981);

65, 1772(E) (1981).[20] Z. Z. Xing, H. Zhang, and S. Zhou, Phys. Rev. D 77,

113016 (2008).[21] C. Bernard et al. (Fermilab Lattice, MILC, and HPQCD

Collaborations), Proc. Sci., LAT2007 (2007) 370.[22] M. Okamoto, Proc. Sci. LAT2005 (2006) 013.[23] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1

(2008).[24] D. Becirevic et al., Nucl. Phys. B634, 105 (2002).[25] G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa, and A.

Vladikas, Nucl. Phys. B445, 81 (1995).[26] A. J. Buras, S. Jager, and J. Urban, Nucl. Phys. B605, 600

(2001).[27] B. Lillie, J. Shu, and T.M. P. Tait, Phys. Rev. D 76, 115016

(2007); B. Lillie, L. Randall, and L. T. Wang, J. HighEnergy Phys. 09 (2007) 074.

[28] F. Hubaut, E. Monnier, P. Pralavorio, K. Smolek, and V.Simak, Eur. Phys. J. C 44S2, 13 (2005).

[29] H. E. Haber and H. E. Logan, Phys. Rev. D 62, 015011(2000).

[30] D. Choudhury, T.M. P. Tait, and C. E.M. Wagner, Phys.Rev. D 65, 053002 (2002).

[31] K. Agashe and R. Contino, Nucl. Phys. B742, 59 (2006).[32] A. Delgado and A. Falkowski, J. High Energy Phys. 05

(2007) 097.[33] K. Agashe, R. Contino, L. Da Rold, and A. Pomarol, Phys.

Lett. B 641, 62 (2006).[34] M. Blanke, A. J. Buras, B. Duling, S. Gori, and A. Weiler,

arXiv:0809.1073.

10The zero mode–KK mode mixing happens through interac-tions with the Higgs, hence the effective coupling of the fermi-ons to the zero mode of Z is Oðv2=�2Þ compared to the directfermion couplings to the KK modes of Z and Z0. More detailscan be found in Ref. [7].


056007-11

Documents

Flavor changing neutral current constraints from Kaluza-Klein gluons and quark mass matrices in the Randall-Sundrum I framework