Flavor Physics in the Randall-Sundrum Model: I. Theoretical Setup and Electroweak Precision Tests
MMZ-TH/08-18October 20, 2008
Flavor Physics in the Randall-Sundrum Model
I. Theoretical Setup and Electroweak Precision Tests
S. Casagrande, F. Goertz, U. Haisch, M. Neubert and T. Pfoh
Institut f¨ur Physik (THEP), Johannes Gutenberg-Universit¨atD-55099 Mainz, Germany
Abstract
A complete discussion of tree-level flavor-changing effects in the Randall-Sundrum (RS)model with brane-localized Higgs sector and bulk gauge and matter fields is presented.The bulk equations of motion for the gauge and fermion fields, supplemented by boundaryconditions taking into account the couplings to the Higgs sector, are solved exactly.For gauge fields the Kaluza-Klein (KK) decomposition is performed in a covariant R ξ gauge. For fermions the mixing between different generations is included in a completelygeneral way. The hierarchies observed in the fermion spectrum and the quark mixingmatrix are explained naturally in terms of anarchic five-dimensional Yukawa matricesand wave-function overlap integrals. Detailed studies of the flavor-changing couplings ofthe Higgs boson and of gauge bosons and their KK excitations are performed, includingin particular the couplings of the standard W ± and Z bosons. A careful analysisof electroweak precision observables including the S and T parameters and the Z b ¯ b couplings shows that the simplest RS model containing only Standard Model particlesand their KK excitations is consistent with all experimental bounds for a KK scaleas low as a few TeV, if one allows for a heavy Higgs boson ( m h (cid:46) W ± bosons, tree-level flavor-changing neutral current couplingsof the Z and Higgs bosons, the rare decays t → c ( u ) Z and t → c ( u ) h , and the flavormixing among KK fermions. The results obtained in this work form the basis for generalcalculations of flavor-changing processes in the RS model and its extensions. a r X i v : . [ h e p - ph ] O c t Introduction
The Standard Model (SM) of particle physics has passed every direct experimental test withflying colors. Yet it is not an entirely satisfactory theory, because it raises but leaves unan-swered many fundamental questions. In particular, there are many hierarchies built into theSM that have a priori no explanation. The most famous of these is the huge separation be-tween the electroweak and the Planck scales. Various solutions have been proposed to explainthis hierarchy. One particularly appealing possibility is the Randall-Sundrum (RS) scenario[1]. In this model, four-dimensional (4D) Minkowskian space-time is embedded into a slice offive-dimensional (5D) anti de-Sitter (AdS ) space with curvature k . The fifth dimension is a S /Z orbifold of size r labeled by a coordinate φ ∈ [ − π, π ], such that the points ( x µ , φ ) and( x µ , − φ ) are identified. In its original form, the metric of the RS geometry is given by ds = e − σ ( φ ) η µν dx µ dx ν − r dφ , σ ( φ ) = kr | φ | , (1)where x µ denote the coordinates on the 4D hyper-surfaces of constant φ with metric η µν =diag(1 , − , − , − e σ is called the warp factor. Three-branes are placed at the orbifoldfixed points φ = 0 as well as φ = π and its reflection at φ = − π . The brane at φ = 0 is calledPlanck or ultra-violet (UV) brane, while the brane at φ = π is called TeV or infra-red (IR)brane. The parameters k and 1 /r are assumed to be of the order of the fundamental Planckscale M Pl , with the product kr ≈
12 chosen such that the inverse warp factor (cid:15) = Λ IR Λ UV ≡ e − krπ ≈ − (2)explains the hierarchy between the electroweak and the Planck scales.In the RS framework the question about the origin of the gauge hierarchy is thus trans-formed into the question why the logarithm of the warp factor, L ≡ − ln (cid:15) = krπ ≈ (cid:29) , (3)is moderately large compared to its natural size of the order of a few. Indeed, this smallhierarchy can be generated dynamically with reasonable tuning using the Goldberger-Wisestabilization mechanism [2], and we will take for granted that such a mechanism is at work.The warp factor also sets the mass scale for the low-lying Kaluza-Klein (KK) excitations ofthe SM fields to be of order of the “KK scale” M KK ≡ k(cid:15) = k e − krπ = O (few TeV) . (4)For instance, the masses of the first KK gluon and photon are approximately 2 . M KK .While the original RS model aimed at solving the hierarchy problem up to the Planck scale,there may be good arguments in favor of lowering the UV scale to a value significantly below M Pl . Higher-dimensional spaces with warp factors arise naturally in flux compactifications ofstring theory [3, 4, 5, 6], and it is thus not unlikely that the RS model will have to be embeddedinto a more fundamental theory (a UV completion) at some scale Λ UV (cid:28) M Pl . From a purelyphenomenological point of view, it would be possible to lower this cutoff to a scale only few1rders of magnitude above the TeV scale, even though in this case the true solution to thehierarchy problem is only postponed to larger energy. Such a scenario has been called the“little RS” model [7], in analogy with “little Higgs” models, which stabilize the Higgs massup to scales of order (10–100) TeV [8, 9]. While less appealing from a conceptual point ofview, the possibility that the parameter L in (3) could be less than about 37 should not bediscarded. As we will see, many amplitudes in the RS model are enhanced by this parameter.By virtue of the AdS/CFT correspondence [10, 11, 12], 5D gravitational theories in anti de-Sitter space have a dual description in terms of strongly coupled 4D conformal field theories.For the case of the RS scenario, in which the conformal symmetry is broken on the IR brane,implications of this correspondence have been explored in [13, 14, 15, 16] (for a review, see[17]). Among other things, the holographic dictionary implies that 5D fields living near theIR brane in the RS setup correspond to composite objects in the conformal field theory, whilefields living near the UV brane correspond to elementary particles. In a setup where the Higgssector is localized on (or near) the IR brane, it is thus natural to think of the Higgs boson asa composite object [18, 19], whose mass is naturally of the order of the KK scale. The factthat a light Higgs boson with m H (cid:28) M KK is (in this sense) unnatural in the RS model willbecome important for parts of the discussion in this work.It will often be convenient to introduce a coordinate t = (cid:15) e σ ( φ ) , which equals (cid:15) on the UVbrane and 1 on the IR brane [21]. Integrals over the orbifold are then obtained using (cid:90) π − π dφ → πL (cid:90) (cid:15) dtt , (cid:90) π − π dφ e σ ( φ ) → πL(cid:15) (cid:90) (cid:15) dt , etc . (5)Another widely used form of the RS background is the conformally flat metric [22] ds = (cid:18) Rz (cid:19) (cid:0) η µν dx µ dx ν − dz (cid:1) , (6)restricted to the interval z ∈ [ R, R (cid:48) ], where R and R (cid:48) denote the positions of the UV and IRbranes, respectively. To facilitate the comparison of our formulae with existing results, werecall that the variables in the two reference frames are related by z = tM KK , R = 1 k , R (cid:48) = 1 M KK , ln R (cid:48) R = L . (7)In the original formulation of the RS model [1], all SM fields were constrained to resideon the IR brane. It was soon realized that, while the Higgs field has to be localized on (ornear) the IR brane in order to solve the hierarchy problem, gauge [23, 24, 25, 26] and matterfields [21, 26] can live in the bulk of AdS . This possibility furnishes both challenges andopportunities for model building. With bulk gauge fields, the compatibility of the model withelectroweak precision measurements at first seemed in jeopardy, since delocalized W ± and Z bosons were found to induce harmful corrections to the Peskin-Takeuchi [27, 28] parameters S and T [22, 23], which however are tightly constrained by experiments [29]. Placing fermions inthe bulk allows to significantly relax the constraint from the S parameter [26, 30, 31, 32]. The This may be different in gauge-Higgs unification models [20], which will not be considered here. T parameter can be cured, e.g. , by extendingthe bulk hypercharge group to SU (2) R × U (1) X and breaking it to U (1) Y on the UV brane[33]. An embedding of the SM fermions into the custodially symmetric SU (2) L × SU (2) R model, under which the left-handed bottom quark is symmetric under the exchange of SU (2) L and SU (2) R , allows one to protect the left-handed Z b ¯ b coupling from vast corrections [34].Delocalized fermions have the further virtue of admitting a natural explanation of the flavorstructure of the SM by harnessing the idea of split fermions [35]. In fact, it is perhaps notan overstatement to say that the RS scenario offers the best theory of flavor we have to date.Starting from anarchic 5D Yukawa couplings, the large mass hierarchies of the SM fermionscan be generated without flavor symmetries by localizing the SM fermions at different points inthe fifth dimension [21, 26, 36, 37]. Given the large hierarchy of quark masses in the SM, smallmixing angles in the Cabibbo-Kobayashi-Maskawa (CKM) matrix are a natural consequenceof this scenario [37]. This way of generating fermion mass hierarchies also implies a certainamount of suppression of dangerous flavor-changing neutral currents (FCNCs) [26], which goesby the name of RS-GIM mechanism [38, 39]. This mechanism successfully suppresses almostall flavor-changing transitions in the quark sector below their experimental limits.During the past years various studies of the flavor structure of the RS model have beenperformed. Properties of the (generalized) CKM matrix, neutral-meson mixing, and CP vio-lation were studied in [37]. Z -mediated FCNCs in the kaon system were considered in [40],and effects of KK gauge bosons on CP asymmetries in rare hadronic B -meson decays inducedby b → s transitions were explored in [41]. A rather detailed survey of ∆ F = 2 and ∆ F = 1processes in the RS framework was presented in [38, 39]. In particular, the second paperexplores a variety of possible effects and analyzes several different rare decay processes. Thebranching ratios for the flavor-changing top-quark decays t → cZ ( γ, g ) were examined in [42].The first complete study of all operators relevant to K – ¯ K mixing was presented in [43]. Somegeneral, model-independent approaches for studying new physics contributions to ∆ F = 2 and∆ F = 1 operators were developed in [44, 45]. It has been recognized that the only observableswhere some fine-tuning of parameters appears to be unavoidable are CP-violating effects inthe neutral kaon system [43, 45, 46] and the neutron electric dipole moment [38, 39], whichfor generic choices of parameters turn out to be too large unless the masses of the lightestKK gauge bosons lie above (10–20) TeV. In view of this problem, several modifications of thequark flavor sector of warped extra-dimension models have been proposed. Most of themtry to implement the notion of minimal flavor violation [47, 48] into the RS framework byusing a bulk flavor symmetry [49, 50, 51, 52]. In another approach the idea of textures ofthe 5D Yukawa matrices is explored [53]. The problem of too large electric dipole momentshas been addressed using the idea of spontaneous CP violation in the context of warped extradimensions [54].This paper is the first in a series of articles devoted to a thorough analysis of flavor physicsin RS models. It collects a large body of known results in the literature and extends them invarious aspects. In a companion paper we apply these results to present a complete discussionof tree-level flavor-violating ∆ F = 2 and ∆ F = 1 effects in the quark sector [55, 56]. As ourbenchmark scenario we consider the simplest implementation of the RS model, in which all SMgauge and matter fields are allowed to propagate in the bulk. We thus assume that the bulktheory is symmetric under the SM gauge group SU (3) C × SU (2) L × U (1) Y , which is broken3o SU (3) C × U (1) EM on the IR brane by coupling the SM fields to a minimal, brane-localizedHiggs sector. It would be rather straightforward to extend the discussion of tree-level effectsto more complicated setups with an enlarged gauge symmetry or the Higgs sector living inthe bulk. At the loop level, the predictions for FCNC processes will become model-dependentand the baroque fermionic content of many extensions of the original RS scenario might leadto more stringent flavor constraints when compared to the minimal model.This article is organized as follows. In Section 2 we discuss the KK decomposition of thebulk gauge fields in the presence of the brane-localized Higgs sector, working in a covariant R ξ gauge. We derive exact results for the masses of the SM gauge bosons and their KKexcitations, as well as for the profiles of these fields. We also discuss how sums over KKtowers of gauge bosons arising in tree-level diagrams at low energy can be evaluated in closedform. The analogous discussion for bulk fermions is presented in Section 3, where we solvethe exact bulk equations of motion in presence of the brane-localized Yukawa sector, whichmixes the N fermion generations. In particular, we point out that the exact equations ofmotion imply that the bulk profiles belonging to different mass eigenstates are not orthogonalon each other. This observation will have important implications for flavor physics. Wealso comment on dimensional-analysis constraints on the scale of the 5D Yukawa matrices.Section 4 is devoted to a review of the implications of the RS setup for the hierarchicalstructures of fermion masses and mixings. We emphasize the invariance of the results forthese masses and mixings under two types of reparametrization transformations of the 5Dbulk mass parameters and Yukawa matrices. In Section 5 we present the main results ofour work by analyzing the structure of gauge boson interactions with SM fermions and theirKK excitations. In particular, we study in detail the flavor-violating couplings of the W ± and Z bosons to SM fermions. Some phenomenological implications of these results arediscussed in Section 6. We begin by studying the constraints on the model imposed by variouselectroweak precision measurements. Even though these constraints have been explored inthe past by many authors, generally concluding that they force an extension of the minimalmodel, we find that even the simplest model provides a consistent framework if one gives upthe requirement of a light Higgs boson. As mentioned earlier, this is indeed natural in modelswith a brane-localized Higgs sector. We also show that this model can be made consistentwith the experimental constraints on the Z b ¯ b couplings. Allowing again for a heavy Higgsboson, we obtain a significantly better description of the data than in the SM. In our workwe concentrate on the leading contributions to the electroweak precision observables, ignoringpossible effects of brane-localized kinetic terms [57, 58, 59]. Although the UV dynamics is notspecified, it is natural to assume that these terms are loop suppressed, so that they can beneglected to first order. We also point out that the KK excitations of the SM fermions havegenerically large mixings between different generations as well as between SU (2) L singlets anddoublets. The reason is that without the Yukawa couplings the spectra of the KK towersof different fermion fields are nearly degenerate because of the closely spaced 5D bulk massparameters, so that even small Yukawa couplings can lead to large mixing effects. Finally, weexplore predictions for the tree-level FCNC decays t → c ( u ) Z and t → c ( u ) h , which might bedetectable at the LHC. A comprehensive study of flavor effects in the B -, D -, and K -mesonsystems will be presented in [56]. Section 7 contains our conclusions and an outlook. In aseries of Appendices we collect details on the textures of the various flavor mixing matrices, our4nput values for SM parameters, and some numerical results for mixing matrices correspondingto a reference RS parameter point. In this and the following sections we derive the KK decompositions of bulk gauge and matterfields in the presence of a brane-localized Higgs sector. While most authors treat the couplingsof bulk fields to the Higgs sector as a perturbation and expand the theory in powers of v /M , we instead construct the exact solutions to the bulk equations of motion subjectto the boundary conditions imposed by the couplings to the Higgs sector. In that way weobtain exact results for the masses and profiles of the various SM particles and their KKexcitations. This approach is more elegant than the perturbative one and avoids the necessityof diagonalizing infinite-dimensional mass matrices.We begin our discussion with the gauge sector, considering the simplest case for which thebulk gauge group is that of the SM. We will discuss the KK decomposition for the electroweaksector only. The extension to the strong interaction is straightforward. We consider bulk gauge fields W aM and B M of SU (2) L × U (1) Y , coupled to a scalar sector onthe IR brane. We choose the vector components W aµ and B µ to be even under the Z orbifoldsymmetry and the scalar components W aφ and B φ to be odd. This ensures that the light masseigenstates (often called “zero modes”) correspond to the SM gauge bosons. The action canbe split up as S gauge = (cid:90) d x r (cid:90) π − π dφ (cid:16) L W , B + L Higgs + L GF + L FP (cid:17) , (8)where L W , B = √ Gr G KM G LN (cid:18) − W aKL W aMN − B KL B MN (cid:19) (9)is the Lagrangian for the 5D gauge theory, while the Higgs-sector Lagrangian L Higgs = δ ( | φ | − π ) r (cid:2) ( D µ Φ) † ( D µ Φ) − V (Φ) (cid:3) , V (Φ) = − µ Φ † Φ + λ (cid:0) Φ † Φ (cid:1) (10)is localized on the IR brane. After electroweak symmetry breaking (EWSB), we decomposethe Higgs doublet in terms of real scalar fields ϕ i asΦ( x ) = 1 √ (cid:32) − i √ ϕ + ( x ) v + h ( x ) + iϕ ( x ) (cid:33) , (11)where v ≈
246 GeV is the Higgs vacuum expectation value, and ϕ ± = ( ϕ ∓ iϕ ) / √
2. We alsoperform the usual field redefinitions of the gauge fields W ± M = 1 √ (cid:0) W M ∓ iW M (cid:1) , M = 1 (cid:112) g + g (cid:48) (cid:0) g W M − g (cid:48) B M (cid:1) , (12) A M = 1 (cid:112) g + g (cid:48) (cid:0) g (cid:48) W M + g B M (cid:1) , where g and g (cid:48) are the 5D gauge couplings of SU (2) L and U (1) Y , respectively. This diago-nalizes the 5D mass terms resulting from EWSB, in such a way that the W ± and Z bosonsget “masses” (with mass dimension 1/2) M W = vg , M Z = v (cid:112) g + g (cid:48) , (13)while the photon remains massless ( M A = 0).The kinetic terms for the Higgs field give rise to mixed terms involving the gauge bosonsand the scalar fields ϕ ± and ϕ , which can be read off from D µ Φ = 1 √ (cid:32) − i √ (cid:0) ∂ µ ϕ + + M W W + µ (cid:1) ∂ µ h + i ( ∂ µ ϕ + M Z Z µ ) (cid:33) + terms bi-linear in fields . (14)In addition, the kinetic terms for the gauge fields in (9) contain mixed terms involving thegauge bosons and the scalar components W ± φ , Z φ , and A φ . All of these mixed terms can beremoved with a suitable choice of the gauge-fixing Lagrangian. We adopt the form L GF = − ξ (cid:18) ∂ µ A µ − ξ (cid:20) ∂ φ e − σ ( φ ) A φ r (cid:21)(cid:19) − ξ (cid:18) ∂ µ Z µ − ξ (cid:20) δ ( | φ | − π ) r M Z ϕ + ∂ φ e − σ ( φ ) Z φ r (cid:21)(cid:19) − ξ (cid:32) ∂ µ W + µ − ξ (cid:34) δ ( | φ | − π ) r M W ϕ + + ∂ φ e − σ ( φ ) W + φ r (cid:35)(cid:33) × (cid:32) ∂ µ W − µ − ξ (cid:34) δ ( | φ | − π ) r M W ϕ − + ∂ φ e − σ ( φ ) W − φ r (cid:35)(cid:33) . (15)More generally, each term could be written with a different gauge-fixing parameter ξ i . Notethat, despite appearance, there is no problem in squaring the δ -functions in this expression.We will see below that the derivatives of the scalar components of the gauge fields W ± φ and Z φ also contain a δ -function contribution, which precisely cancels the δ -functions from the Higgssector. As a result, contrary to the treatment in [60], we do not need to introduce separategauge-fixing Lagrangians in the bulk and on the IR brane.Using integration by parts, we now obtain for the quadratic terms in the action S gauge , = (cid:90) d x r (cid:90) π − π dφ (cid:26) − F µν F µν − ξ ( ∂ µ A µ ) e − σ ( φ ) r [ ∂ µ A φ ∂ µ A φ + ∂ φ A µ ∂ φ A µ ] − ξ (cid:20) ∂ φ e − σ ( φ ) A φ r (cid:21) − Z µν Z µν − ξ ( ∂ µ Z µ ) + e − σ ( φ ) r [ ∂ µ Z φ ∂ µ Z φ + ∂ φ Z µ ∂ φ Z µ ] − W + µν W − µν − ξ ∂ µ W + µ ∂ µ W − µ + e − σ ( φ ) r (cid:2) ∂ µ W + φ ∂ µ W − φ + ∂ φ W + µ ∂ φ W − µ (cid:3) (16)+ δ ( | φ | − π ) r (cid:20) ∂ µ h∂ µ h − λv h + ∂ µ ϕ + ∂ µ ϕ − + 12 ∂ µ ϕ ∂ µ ϕ + M Z Z µ Z µ + M W W + µ W − µ (cid:21) − ξ (cid:20) δ ( | φ | − π ) r M Z ϕ + ∂ φ e − σ ( φ ) Z φ r (cid:21) − ξ (cid:34) δ ( | φ | − π ) r M W ϕ + + ∂ φ e − σ ( φ ) W + φ r (cid:35) (cid:34) δ ( | φ | − π ) r M W ϕ − + ∂ φ e − σ ( φ ) W − φ r (cid:35) + L FP (cid:27) . The form of the Faddeev-Popov ghost Lagrangian L FP will be discussed after the KK decom-position.Before proceeding, a comment is in order concerning the precise meaning of the aboveexpressions. For the consistency of the theory it is important that one can integrate byparts in the action without encountering boundary terms. Otherwise the Lagrangian is nothermitian. Yet, the presence of δ -function terms on the IR brane gives rise to discontinuitiesof some of the fields at | φ | = π , which appears to jeopardize this crucial feature. In order todefine the model properly, we will always understand the δ -functions via the limiting procedure δ ( | φ | − π ) ≡ lim θ → + (cid:104) δ ( φ − π + θ ) + δ ( φ + π − θ ) (cid:105) . (17)In this way the discontinuities are moved into the bulk, and the fields can be assigned properboundary conditions on the branes, which are consistent with integration by parts. We performall calculations at small but finite θ and find that at the end the limit θ → + is smooth, givingrise to well-defined jump conditions for the fields and their derivatives on the IR brane. Whennecessary, we will use the notation f ( π − ) ≡ lim θ → + f ( π − θ ) to indicate the value of a function f that is discontinuous at | φ | = π . We write the KK decompositions of the various 5D fields in the form A µ ( x, φ ) = 1 √ r (cid:88) n A ( n ) µ ( x ) χ An ( φ ) , A φ ( x, φ ) = 1 √ r (cid:88) n a An ϕ ( n ) A ( x ) ∂ φ χ An ( φ ) ,Z µ ( x, φ ) = 1 √ r (cid:88) n Z ( n ) µ ( x ) χ Zn ( φ ) , Z φ ( x, φ ) = 1 √ r (cid:88) n a Zn ϕ ( n ) Z ( x ) ∂ φ χ Zn ( φ ) ,W ± µ ( x, φ ) = 1 √ r (cid:88) n W ± ( n ) µ ( x ) χ Wn ( φ ) , W ± φ ( x, φ ) = 1 √ r (cid:88) n a Wn ϕ ± ( n ) W ( x ) ∂ φ χ Wn ( φ ) , (18)7here A ( n ) µ etc. are the 4D mass eigenstates, and the various χ an profiles form complete sets ofeven functions on the orbifold, which can be taken to obey the orthonormality condition (cid:90) π − π dφ χ am ( φ ) χ an ( φ ) = δ mn . (19)The 4D scalar fields can also be expanded in the basis of mass eigenstates, and we write theseexpansions in the form ϕ ± ( x ) = (cid:88) n b Wn ϕ ± ( n ) W ( x ) , ϕ ( x ) = (cid:88) n b Zn ϕ ( n ) Z ( x ) . (20)We will denote the masses of the 4D vector fields by m an ≥ a = A, Z, W ). The massesof the scalar fields ϕ ( n ) a will be related to these by gauge invariance.Inserting these decompositions into the action, one finds that the profiles χ an obey theequation of motion [23, 24] − r ∂ φ e − σ ( φ ) ∂ φ χ an ( φ ) = ( m an ) χ an ( φ ) − δ ( | φ | − π ) r M a χ an ( φ ) . (21)The boundary conditions are ∂ φ χ an (0) = 0 , (UV brane) ∂ φ χ an ( π − ) = − rM a (cid:15) χ an ( π ) . (IR brane) (22)From these conditions one derives the eigenvalues m an .We find that the action takes the desired form S gauge , = (cid:88) n (cid:90) d x (cid:26) − F ( n ) µν F µν ( n ) − ξ (cid:0) ∂ µ A ( n ) µ (cid:1) + ( m An ) A ( n ) µ A µ ( n ) − Z ( n ) µν Z µν ( n ) − ξ (cid:0) ∂ µ Z ( n ) µ (cid:1) + ( m Zn ) Z ( n ) µ Z µ ( n ) − W +( n ) µν W − µν ( n ) − ξ ∂ µ W +( n ) µ ∂ µ W − ( n ) µ + ( m Wn ) W +( n ) µ W − µ ( n ) + 12 ∂ µ ϕ ( n ) A ∂ µ ϕ ( n ) A − ξ ( m An ) ϕ ( n ) A ϕ ( n ) A + 12 ∂ µ ϕ ( n ) Z ∂ µ ϕ ( n ) Z − ξ ( m Zn ) ϕ ( n ) Z ϕ ( n ) Z + ∂ µ ϕ +( n ) W ∂ µ ϕ − ( n ) W − ξ ( m Wn ) ϕ +( n ) W ϕ − ( n ) W (cid:27) + (cid:90) d x (cid:18) ∂ µ h∂ µ h − λv h (cid:19) + (cid:88) n (cid:90) d x L ( n )FP , (23)if and only if a an = − m an , b an = M a √ r χ an ( π − ) m an . (24)8he resulting theory contains a tower of massive gauge bosons with masses m an , accompaniedby a tower of massive scalars with masses √ ξ m an , as well as the Higgs field h with mass √ λv .Note that with (24) the 4D gauge-fixing Lagrangian derived from (15) takes the simple form r (cid:90) π − π dφ L GF = (cid:88) n L ( n )GF , (25)with L ( n )GF = − ξ (cid:16) ∂ µ A ( n ) µ − ξm An ϕ ( n ) A (cid:17) − ξ (cid:16) ∂ µ Z ( n ) µ − ξm Zn ϕ ( n ) Z (cid:17) − ξ (cid:16) ∂ µ W +( n ) µ − ξm Wn ϕ +( n ) W (cid:17) (cid:16) ∂ µ W − ( n ) µ − ξm Wn ϕ − ( n ) W (cid:17) . (26)For each KK mode these expressions are identical to those of the SM. It follows that the formof the Faddeev-Popov ghost Lagrangians L ( n )FP in (23) is analogous to that of the SM, with theonly generalization that a ghost field is required for every KK mode. The tree-level exchange of a SM gauge boson accompanied by its KK excitations in a genericFeynman diagram leads to a combination of propagator and vertex functions, which in thelow-energy limit, i.e. , for small momentum transfer q , can be expanded as (cid:88) n χ n ( φ ) χ n ( φ (cid:48) ) m n − q = ∞ (cid:88) N =1 (cid:0) q (cid:1) N − (cid:88) n χ n ( φ ) χ n ( φ (cid:48) )( m n ) N . (27)The gauge-boson profiles χ n are integrated with other profiles at each vertex. The sums overprofiles weighted by inverse powers of m n can be evaluated in closed form by generalizing amethod developed in [61].The simplest sum is obtained using that the bulk profiles χ n form a complete set of or-thonormal, even functions on the orbifold, subject to the boundary conditions (22). Relation(19) then implies that (cid:88) n χ n ( φ ) χ n ( φ (cid:48) ) = 12 [ δ ( φ − φ (cid:48) ) + δ ( φ + φ (cid:48) )] , (28)where n runs from 0 to ∞ . Here and below, the index n = 0 refers to the light SM particle,which may or may not be massless.To proceed, we integrate the equation of motion (21) twice, accounting for the boundarycondition ∂ φ χ n (0) = 0 on the UV brane. This yields χ n ( φ ) − χ n (0) = − r m n (cid:90) φ dφ (cid:48) e σ ( φ (cid:48) ) (cid:90) φ (cid:48) dφ (cid:48)(cid:48) χ n ( φ (cid:48)(cid:48) ) . (29) From now on we omit the superscript a on the gauge-boson profiles unless it is required for clarity. x n determining the gauge-boson masses, but it does not affect the functional form of the profilesbeyond that change. The appearance of the profile χ n (0) on the UV brane in (29) is a newelement, which has not been considered in [61].Using the result (29) along with the completeness relation (28), the sums over profilesin (27) can be evaluated using an iterative procedure, which can be generalized to arbitraryorder. For the first and most important sum we obtain (cid:88) n χ n ( φ ) χ n ( φ (cid:48) ) m n = (cid:88) n χ n (0) m n − L(cid:15) πM (cid:2) e σ ( φ > ) − (cid:3) , (30)where φ > ≡ max( | φ | , | φ (cid:48) | ). As it stands, this relation holds only for the case where the gaugesymmetry is spontaneously broken, so that the lowest mass eigenvalue satisfies m > χ n (0) on the UV brane can be performed by integrating χ ( φ ) times the expression on the left-hand side of (30) over the entire orbifold and using theorthonormality condition (19). We find (cid:88) n χ n (0) m n = 12 πm + 14 πM (cid:20)(cid:18) − L (cid:19) − (cid:15) (cid:18) L − L (cid:19)(cid:21) − m πM (cid:20)(cid:18) L −
52 + 218 L − L (cid:19) − (cid:15) (cid:18) L − L (cid:19) − (cid:15) (cid:18) L + 1 L (cid:19)(cid:21) + O (cid:18) m M (cid:19) . (31)For the case where a massless zero mode exists ( m = 0), one must subtract the contri-bution of the ground state from the sum over states. This can either be done by subtractingthis contribution from both sides of the completeness relation (28), or by inserting the explicitform of the ground-state profile (see Section 2.4 below) and then taking the limit m → (cid:88) n (cid:48) χ n ( φ ) χ n ( φ (cid:48) ) m n = 14 πM (cid:20) L − (cid:15) (cid:18) L + 1 + 12 L (cid:19)(cid:21) − L(cid:15) πM (cid:2) e σ ( φ > ) − (cid:3) + (cid:15) πM (cid:20) e σ ( φ ) (cid:18) L | φ | π − (cid:19) + e σ ( φ (cid:48) ) (cid:18) L | φ (cid:48) | π − (cid:19) + 1 (cid:21) , (32)where the prime on the sum indicates that n runs from 1 to ∞ .Note that O ( (cid:15) n ) terms on the right-hand sides of (30), (31), and (32) not accompanied by n powers of a warp factor are Planck-scale suppressed and can be dropped for all practicalpurposes, in which case the relations simplify considerably. Introducing the variables t = (cid:15) e σ ( φ ) and t (cid:48) = (cid:15) e σ ( φ (cid:48) ) , we obtain for m > m = 0 the important results (cid:88) n χ n ( φ ) χ n ( φ (cid:48) ) m n = 12 πm + 14 πM (cid:20) L t < − L (cid:0) t + t (cid:48) (cid:1) + 1 − L + O (cid:18) m M (cid:19)(cid:21) , (33)and (cid:88) n (cid:48) χ n ( φ ) χ n ( φ (cid:48) ) m n = 14 πM (cid:20) L t < − t (cid:18) − ln t (cid:19) − t (cid:48) (cid:18) − ln t (cid:48) (cid:19) + 12 L (cid:21) , (34)10espectively, where t < ≡ min( t, t (cid:48) ). The neglected terms in the first relation only affect theconstant, i.e. , t -independent contribution. Notice that the terms proportional to t and t (cid:48) inthe first sum are enhanced by a factor L , whereas this is not the case for the second sum. Thisfact has important consequences for the phenomenology of flavor-violating processes [55, 56].It implies that in the RS model new physics contributions to ∆ F = 2 processes such as B – ¯ B or K – ¯ K mixing are dominated by tree-level KK gluon exchange, whereas those to ∆ F = 1processes such as rare B -meson decays arise predominantly from the FCNC couplings of the Z boson to fermions.It is possible to extend the above procedure in an iterative way to sums with N > N = 2, neglecting O ( (cid:15) ) terms. They read (cid:88) n χ n ( φ ) χ n ( φ (cid:48) ) m n = 12 πm + 14 πm M (cid:18) − L (cid:19) − πM (cid:18) L − L − L (cid:19) − (cid:20) πm M + 18 πM (cid:18) − L (cid:19)(cid:21) (cid:20) t (cid:18) L −
12 + ln t (cid:19) + t (cid:48) (cid:18) L −
12 + ln t (cid:48) (cid:19)(cid:21) + L πM (cid:20) t > + 4 t t (cid:48) (cid:18) L −
12 + ln t < (cid:19)(cid:21) + O (cid:18) m M (cid:19) , (35)and (cid:88) n (cid:48) χ n ( φ ) χ n ( φ (cid:48) ) m n = 132 πM (cid:18) L − L (cid:19) − πM (cid:20) t (cid:18) L −
54 + ln t (cid:19) + 2 t L (cid:18) L −
12 + ln t (cid:19)(cid:21) − πM (cid:20) t (cid:48) (cid:18) L −
54 + ln t (cid:48) (cid:19) + 2 t (cid:48) L (cid:18) L −
12 + ln t (cid:48) (cid:19)(cid:21) + 132 πM (cid:20) L t > + 4 t t (cid:48) (cid:20) L ln t < − (cid:18) L − (cid:19) (cid:18) ln tt (cid:48) − (cid:19) − ln t ln t (cid:48) (cid:21)(cid:21) . (36)For completeness, we note that for the case without EWSB the gauge-boson propagator in(27) coincides with the 5D mixed position/momentum-space propagator derived in [62, 63].Our relations (34) and (36) can also be derived by expanding this quantity about q = 0. The explicit form of the profiles χ n was first obtained in [23, 24] for the case of an unbrokengauge symmetry. The same solution remains valid in the case of a spontaneously brokensymmetry. However, the boundary conditions on the IR brane must be modified in this caseso as to obey (22). In terms of the variable t , which takes values between t = (cid:15) (UV brane)and t = 1 (IR brane), we write the solution in the form χ n ( φ ) = N n (cid:114) Lπ t c + n ( t ) , (37)11here c + n ( t ) = Y ( x n (cid:15) ) J ( x n t ) − J ( x n (cid:15) ) Y ( x n t ) ,c − n ( t ) = 1 x n t ddt (cid:2) t c + n ( t ) (cid:3) = Y ( x n (cid:15) ) J ( x n t ) − J ( x n (cid:15) ) Y ( x n t ) . (38)Here x n ≡ m n M KK (39)are dimensionless parameters related to the masses of the gauge bosons and their KK excita-tions in the 4D theory. The normalization condition (19) fixes the constant N n to obey N − n = (cid:2) c + n (1) (cid:3) + (cid:2) c − n (1) (cid:3) − x n c + n (1) c − n (1) − (cid:15) (cid:2) c + n ( (cid:15) ) (cid:3) . (40)Obviously, equation (37) satisfies the boundary condition ∂ φ χ n (0) = 0 on the UV brane,since c − n ( (cid:15) ) = 0. The boundary condition (22) on the IR brane imposes the relation x n c − n (1) = − g v M L c + n (1) , (41)from which the eigenvalues x n can be derived. Here we have introduced the 4D gauge coupling g , which is related to the 5D coupling g via g = g √ πr . (42)Analogous relations hold for all gauge couplings [23].Without EWSB, i.e. , for v = 0, there is a zero mode ( m = 0) with flat profile χ γ,g ( φ ) = 1 √ π . (43)In the case where m (cid:54) = 0, the results for the SM gauge bosons can be simplified, since x (cid:28) x leads to m W = g v (cid:20) − g v M (cid:18) L − − (cid:15) L (cid:19) + O (cid:18) v M (cid:19)(cid:21) , (44)and an analogous relation with g replaced by ( g + g (cid:48) ) holds for the Z -boson mass. Thisresult may be compared with the SM relations m W = g v / m Z = ( g + g (cid:48) ) v /
4. Itwill also be useful to have an approximate expression for the ground-state profile χ valid for x (cid:28)
1. We find χ W,Z ( φ ) = 1 √ π (cid:20) m W,Z M (cid:18) − − (cid:15) L + t (1 − L − t ) (cid:19) + O (cid:18) m W,Z M (cid:19)(cid:21) . (45) This relation holds for the profiles of W ± bosons and their KK excitations. For the case of the Z boson,one must replace g by ( g + g (cid:48) ). χ W,Z ( π ) = 1.We finish this section by collecting useful expressions for the bulk profiles of KK gaugebosons. The fact that (cid:15) ≈ − is extremely small allows us to replace J ( x n (cid:15) ) ≈ , Y ( x n (cid:15) ) ≈ − π (cid:16) L − ln x n − γ E (cid:17) (46)in (38) and (40), where γ E ≈ . χ n ( φ ) = − ¯ N n (cid:114) Lπ t (cid:104)(cid:16) L − ln x n − γ E (cid:17) J ( x n t ) + π Y ( x n t ) (cid:105) , (47)where¯ N − n = (cid:104)(cid:16) L − ln x n − γ E (cid:17) J ( x n ) + π Y ( x n ) (cid:105) (cid:34) − g v M Lx n + (cid:18) g v M Lx n (cid:19) (cid:35) − x n . (48)As before this relation holds for the KK excitations of the W ± bosons. For the Z -boson caseone replaces g by ( g + g (cid:48) ). The KK masses follow from the solutions to (41). To goodapproximation they are given by the zeros of the Bessel function J ( x n ). We consider N generations of 5D fermions in the bulk. They are grouped into SU (2) L doublets Q and singlets u c and d c , each of which is an N -component vector in flavor space. After EWSBon the IR brane, these fields are coupled by Yukawa matrices Y u and Y d , like in the SM. The quadratic terms in the 5D action can be written in the form [21, 26] S ferm , = (cid:90) d x r (cid:90) π − π dφ (cid:26) e − σ ( φ ) (cid:18) ¯ Q i / ∂ Q + (cid:88) q = u,d ¯ q c i / ∂ q c (cid:19) − e − σ ( φ ) sgn( φ ) (cid:18) ¯ Q M Q Q + (cid:88) q = u,d ¯ q c M q q c (cid:19) − e − σ ( φ ) r (cid:34) ¯ Q L ∂ φ e − σ ( φ ) Q R − ¯ Q R ∂ φ e − σ ( φ ) Q L + (cid:88) q = u,d (cid:18) ¯ q cL ∂ φ e − σ ( φ ) q cR − ¯ q cR ∂ φ e − σ ( φ ) q cL (cid:19)(cid:35) − δ ( | φ | − π ) e − σ ( φ ) v √ r (cid:104) ¯ u L Y (5D) u u cR + ¯ d L Y (5D) d d cR + h.c. (cid:105) (cid:27) , (49)13here M Q,q are diagonal matrices containing the (real) bulk masses, and Y (5D) q are the 5DYukawa matrices. We define the dimensionless 4D Yukawa matrices via Y (5D) q ≡ Y q k , q = u, d . (50)Note that the bulk masses can be positive or negative. Unlike in 4D, the sign of the Diracmass term cannot be reversed by a field redefinition in the 5D theory. In fact, phenomenologyrequires that the bulk masses are clustered around the values M Q i ≈ − k/ M q i ≈ + k/ Q = Q L + Q R etc. Theleft-handed (right-handed) components of the SU (2) L doublet Q are even (odd) under the Z orbifold symmetry. Likewise, the right-handed (left-handed) components of the singlets u c and d c are even (odd). This assignment of Z parities is such that the zero modes of the even fieldscorrespond to the SM particles. Without the Yukawa interactions, each 5D fermion wouldgive rise to a massless Weyl fermion in the 4D effective theory, accompanied by a tower ofmassive KK excitations [21]. After EWSB, the Yukawa couplings remove the massless modesand replace them by light (compared with the KK scale) SM fermions, each accompanied bytwo towers of heavy KK states.The choice of diagonal bulk masses in (49) can be justified as follows. In a more generalsetup, the action contains positive, hermitian kinetic matrices Z A and hermitian bulk massmatrices M A , with A = Q, u, d . Unitary transformations can be used to bring the kinetic termsinto diagonal form, i.e. , U † A Z A U A = diag( Z A , . . . , Z A N ) ≡ D A with Z A i >
0. The fields arethen rescaled to obtain the canonical normalization. In the process, the bulk mass matricesget transformed into M (cid:48) A = D − / A U − A M A U A D − / A . These new mass matrices can be diag-onalized by another set of unitary transformations, i.e. , U (cid:48)† A M (cid:48) A U (cid:48) A = diag( M A , . . . , M A N ).It follows that, without loss of generality, it is always possible to switch to a basis in whichthe bulk mass terms are diagonal in flavor space. We will refer to this as the bulk mass basis.If not stated otherwise, from now on Y u and Y d will always denote the Yukawa matrices inthis specific basis. We will discuss the solution of the eigenvalue problem to find the KK decomposition for thecase of the up-type quark sector. An analogous discussion holds for the down-type quarksector. Let m n > u ( n ) = u ( n ) L + u ( n ) R the corresponding spinor fields. We write the KK decomposition of the 5Dfields in the form u L ( x, φ ) = e σ ( φ ) √ r (cid:88) n C ( Q ) n ( φ ) a ( U ) n u ( n ) L ( x ) , u R ( x, φ ) = e σ ( φ ) √ r (cid:88) n S ( Q ) n ( φ ) b ( U ) n u ( n ) R ( x ) ,u cL ( x, φ ) = e σ ( φ ) √ r (cid:88) n S ( u ) n ( φ ) b ( u ) n u ( n ) L ( x ) , u cR ( x, φ ) = e σ ( φ ) √ r (cid:88) n C ( u ) n ( φ ) a ( u ) n u ( n ) R ( x ) , (51)where C ( Q,u ) n are even profiles under Z , while S ( Q,u ) n are odd. The index n labels the masseigenstates with fermion masses m n (in this case m u , m c , m t as well as KK excitations in the14p-type quark sector) and spinor fields u ( n ) . The spinor fields on the left-hand side of theequations as well as the objects a ( U,u ) n and b ( U,u ) n are N -component vectors in flavor space, andthe profiles C ( Q,u ) n , S ( Q,u ) n are diagonal N × N matrices, where each entry refers to a differentbulk mass parameter (in the bulk mass basis). Note that while the bulk profiles C ( Q ) n and S ( Q ) n are the same for the up- and down-type quarks belonging to the same SU (2) L doublet,the vectors a ( U ) n and a ( D ) n associated with these profiles in the KK decomposition of the up-and down-type 5D quark fields will be different objects, as indicated by our notation.Inserting these relations into the action, one derives the equations of motion (cid:18) r ∂ φ − M Q sgn( φ ) (cid:19) C ( Q ) n ( φ ) a ( U ) n = − m n e σ ( φ ) S ( Q ) n ( φ ) a ( U ) n , (cid:18) − r ∂ φ − M Q sgn( φ ) (cid:19) S ( Q ) n ( φ ) b ( U ) n = − m n e σ ( φ ) C ( Q ) n ( φ ) b ( U ) n + δ ( | φ | − π ) e σ ( φ ) √ vkr Y u C ( u ) n ( φ ) a ( u ) n , (52)and similarly (cid:18) − r ∂ φ − M u sgn( φ ) (cid:19) C ( u ) n ( φ ) a ( u ) n = − m n e σ ( φ ) S ( u ) n ( φ ) a ( u ) n , (cid:18) r ∂ φ − M u sgn( φ ) (cid:19) S ( u ) n ( φ ) b ( u ) n = − m n e σ ( φ ) C ( u ) n ( φ ) b ( u ) n + δ ( | φ | − π ) e σ ( φ ) √ vkr Y † u C ( Q ) n ( φ ) a ( U ) n . (53)In the bulk, i.e. , for | φ | (cid:54) = π , and for the special case of a single generation, these relationsreduce to the equations first obtained in [21], where it has been shown that the general solutionscan be written as linear combinations of Bessel functions (see Section 3.3 below). The presenceof the brane-localized terms only affects the boundary conditions for the solutions. At φ = 0one has S ( Q,u ) n (0) = 0 , (UV brane) (54)and integrating the equations of motion over an infinitesimal interval around | φ | = π we find S ( Q ) n ( π − ) b ( U ) n = v √ M KK Y u C ( u ) n ( π ) a ( u ) n , − S ( u ) n ( π − ) b ( u ) n = v √ M KK Y † u C ( Q ) n ( π ) a ( U ) n . (IR brane) (55)Without the brane-localized Yukawa terms, the profiles C ( Q,u ) n and S ( Q,u ) n form completesets of even and odd functions on the orbifold, which can be chosen to obey orthonormalityconditions with respect to the measure dφ e σ [21]. However, it is not difficult to show that15he δ -function terms in the equations of motion are inconsistent with these orthonormalityrelations. We thus impose the generalized orthonormality conditions (cid:90) π − π dφ e σ ( φ ) C ( Q,u ) m ( φ ) C ( Q,u ) n ( φ ) = δ mn + ∆ C ( Q,u ) mn , (cid:90) π − π dφ e σ ( φ ) S ( Q,u ) m ( φ ) S ( Q,u ) n ( φ ) = δ mn + ∆ S ( Q,u ) mn . (56)We then find that the 4D action reduces to the desired form S ferm , = (cid:88) n (cid:90) d x (cid:2) ¯ u ( n ) ( x ) i / ∂ u ( n ) ( x ) − m n ¯ u ( n ) ( x ) u ( n ) ( x ) (cid:3) , (57)if and only if, in addition to the boundary conditions (54) and (55), the relations a ( U,u ) n = b ( U,u ) n , a ( U ) † n a ( U ) n + a ( u ) † n a ( u ) n = 1 , (58)and a ( U,u ) † m ∆ C ( Q,u ) mn a ( U,u ) n + a ( u,U ) † m ∆ S ( u,Q ) mn a ( u,U ) n = 0 (59)hold. With the help of the relations (58) it is straightforward to show that the equations ofmotion imply m m ∆ C ( Q,u ) mn − m n ∆ S ( Q,u ) mn = ± r C ( Q,u ) n ( π ) S ( Q,u ) m ( π − ) . (60)Using the symmetry of the relations (56) in m and n , we obtain for m (cid:54) = n ∆ C ( Q,u ) mn = ± r m m C ( Q,u ) n ( π ) S ( Q,u ) m ( π − ) − m n C ( Q,u ) m ( π ) S ( Q,u ) n ( π − ) m m − m n , ∆ S ( Q,u ) mn = ∓ r m m C ( Q,u ) m ( π ) S ( Q,u ) n ( π − ) − m n C ( Q,u ) n ( π ) S ( Q,u ) m ( π − ) m m − m n . (61)Finally, using the explicit results for the bulk profiles derived in Section 3.3, one finds thatthe correct expressions for m = n are ∆ C ( Q,u ) nn = − ∆ S ( Q,u ) nn = ± rm n C ( Q,u ) n ( π ) S ( Q,u ) n ( π − ) . (62)One would naively expect that the extra terms in the generalized orthonormality conditions(56) are small corrections of order v/M KK . However, as we will see below, these terms are infact O (1) for the profiles of the light SM fields.The boundary conditions (55) on the IR brane can now be simplified as S ( Q ) n ( π − ) a ( U ) n = v √ M KK Y u C ( u ) n ( π ) a ( u ) n , − S ( u ) n ( π − ) a ( u ) n = v √ M KK Y † u C ( Q ) n ( π ) a ( U ) n . (63)16hese relations can be written in the form of system of 2 N linear equations for the componentsof the vectors a ( U,u ) n , and the eigenvalues are thus determined by the zeros of the determinantof a (2 N ) × (2 N ) matrix. The problem can be simplified by noting that the matrices C ( Q,u ) n , S ( Q,u ) n are non-singular, so that the inverse matrices exist. We can thus replace (63) by thedecoupled equations S ( Q ) n ( π − ) a ( U ) n = − v M Y u C ( u ) n ( π ) (cid:104) S ( u ) n ( π − ) (cid:105) − Y † u C ( Q ) n ( π ) a ( U ) n , S ( u ) n ( π − ) a ( u ) n = − v M Y † u C ( Q ) n ( π ) (cid:104) S ( Q ) n ( π − ) (cid:105) − Y u C ( u ) n ( π ) a ( u ) n . (64)The mass eigenvalues follow from the solutions to the equationdet (cid:18) − v M (cid:104) S ( Q ) n ( π − ) (cid:105) − Y u C ( u ) n ( π ) (cid:104) − S ( u ) n ( π − ) (cid:105) − Y † u C ( Q ) n ( π ) (cid:19) = 0 . (65)Once they are known, the eigenvectors a ( U,u ) n can be determined from (64). Note that, whileit is always possible to work with real profiles C ( Q,u ) n and S ( Q,u ) n , these eigenvectors are, ingeneral, complex-valued objects. The explicit form of the profiles ( C ( Q,q ) n ) i and ( S ( Q,q ) n ) i associated with bulk mass parameters M Q i ,q i (with q = u, d ) was obtained in [21, 26]. We will drop the index i for the purposes ofthis discussion. In terms of the variable t = (cid:15) e σ , one finds C ( Q,q ) n ( φ ) = N n ( c Q,q ) (cid:114) L(cid:15)tπ f + n ( t, c Q,q ) ,S ( Q,q ) n ( φ ) = ±N n ( c Q,q ) sgn( φ ) (cid:114) L(cid:15)tπ f − n ( t, c Q,q ) , (66)where c Q,q ≡ ± M Q,q /k are dimensionless parameters derived from the bulk mass terms, and f ± n ( t, c ) = J − − c ( x n (cid:15) ) J ∓ + c ( x n t ) ± J + c ( x n (cid:15) ) J ± − c ( x n t ) , (67)where as before x n = m n /M KK . The orthonormality relations (56) imply the normalizationconditions 2 (cid:90) (cid:15) dt t (cid:2) f ± n ( t, c ) (cid:3) = 1 N n ( c ) ± f + n (1 , c ) f − n (1 , c ) x n , (68)from which we derive N − n ( c ) = (cid:2) f + n (1 , c ) (cid:3) + (cid:2) f − n (1 , c ) (cid:3) − cx n f + n (1 , c ) f − n (1 , c ) − (cid:15) (cid:2) f + n ( (cid:15), c ) (cid:3) . (69)For the special cases where c + 1 / x n (cid:28)
1, since even the top-quark mass is much lighter than the KK scale. We find C ( Q,q ) n ( φ ) ≈ (cid:114) L(cid:15)π F ( c Q,q ) (cid:112) δ n ( c Q,q ) (cid:2) t c Q,q − δ n ( c Q,q ) t − c Q,q (cid:3) ,S ( Q,q ) n ( φ ) ≈ ± sgn( φ ) (cid:114) L(cid:15)π x n F ( c Q,q ) (cid:112) δ n ( c Q,q ) t c Q,q − (cid:15) c Q,q t − c Q,q c Q,q , (70)where we have introduced the “zero-mode profile” [21, 26] F ( c ) ≡ sgn[cos( πc )] (cid:114) c − (cid:15) c , (71)and the parameters (valid for c (cid:54) = 1 / δ n ( c ) ≡ x n c − (cid:15) c . (72)The sign factor in (71) is chosen such that the signs in (70) agree with those derived from theexact profiles (66).The quantities F ( c ) and δ n ( c ) strongly depend on the values of c and x n . For − / < c < /
2, one has to an excellent approximation F ( c ) ≈ √ c , δ n ( c ) ≈ . (73)For − / < c < − /
2, on the other hand, F ( c ) ≈ −√− − c (cid:15) − c − (74)is exponentially small, while δ n is strictly positive and can be large in magnitude providedthat x n (cid:29) O ( (cid:15) − − c ). For the case of one fermion generation this can never happen, since theeigenvalue equation (65) ensures that x n (cid:15) c can be at most of O ( v /M ) for the lightestmass eigenstate, which we would identify with the SM fermion. For more than one generation,however, some of the δ n parameters become large, of order (cid:15) a v /M with a negative coefficient a . It is therefore not justified to drop the corresponding terms in (70). However, we find thatafter the fermion profiles are combined with the mixing parameters a ( U,u ) n in (51), the δ n termsalways have a very small effect on the fermion masses and, in particular, on the flavor-changingcouplings of the RS model.For the KK excitations of the SM fermions the general relations for the bulk profiles givenabove can be simplified using that (cid:15) ≈ − is extremely small, so that we can take the limit (cid:15) → c > − / N n ( c ) f ± n ( t, c ) = ¯ N n ( c ) J ∓ + c ( x n t ) , (cid:2) ¯ N n ( c ) (cid:3) − = J + c ( x n ) + J − + c ( x n ) − cx n J + c ( x n ) J − + c ( x n ) , (75)18hereas for c < − / N n ( c ) f ± n ( t, c ) = ± ¯ N n ( c ) J ± − c ( x n t ) , (cid:2) ¯ N n ( c ) (cid:3) − = J − c ( x n ) + J − − c ( x n ) + 2 cx n J − c ( x n ) J − − c ( x n ) . (76) Our definition of the 4D Yukawa couplings in (50) was based on making the 5D Yukawacouplings dimensionless with the help of the AdS curvature k . In that regard we followed theconvention most frequently adopted in the literature and used in the original papers [21, 26].It is important to emphasize that this convention is neither unique nor particularly natural.While the relation between the 5D and 4D gauge couplings in (42) is dictated by the fact thatthe couplings of the zero-mode gauge bosons to fermions take the familiar form (see relation(118) in Section 5.1 below), no such argument can be given for the Yukawa couplings. Inthe SM the couplings of the Higgs boson to fermions are proportional to the masses of thefermions. However, in the RS framework the fermion masses arise from products of Yukawacouplings with strongly hierarchical fermion profiles.Given the form of the Yukawa interactions in (49) and the fact that the KK decomposition(51) of the 5D fermion fields introduces an additional factor 1 /r , it would appear logical todefine dimensionless 4D Yukawa couplings by using the scale r instead of k in (50), e.g. ,¯ Y q ≡ Y (5D) q πr = 1 L Y q , q = u, d , (77)which resembles (42). The so-defined 4D Yukawa couplings are more than an order of mag-nitude smaller than the ones defined in (50) and used in most of the literature. The questionarises: should we expect that Y q = O (1), or ¯ Y q = O (1), or something else?Notice that with the conventional definition (50) the 4D Yukawa matrices absorb a factor of L arising from the fermion profiles evaluated on the IR brane, which appear in the Higgs-bosoncouplings to fermions. However, the same factor of L appears in the Higgs-boson couplingsto KK gauge bosons, after the 5D gauge coupling is rescaled as in (42). More generally, allinteraction terms in the effective 4D Lagrangian that result from interactions on (or close) tothe IR brane are enhanced by powers of √ L . For instance, it is well known that the couplingsof KK gauge bosons to heavy fermions contain such a factor [23, 24]. It seems ad hoc toremove the factor L in the case of the Yukawa couplings, when similar factors are present andcannot be removed for all other interactions near the IR brane.Consider, for example, the Feynman rule for the Higgs coupling to a pair of neutral weakgauge bosons of KK levels n and n , which can be written as h Z ( n ) Z ( n ) = 2 im Z v z n z n g µν , (78)19 hq ( n ) q ( n ) h q ( n ) q ( n ) q ( n ) q ( n ) h h q ( n ) q ( n ) q ( n ) q ( n ) g ( n ) Figure 1: Examples of one-loop contributions to the renormalization of the Higgs-boson mass (left) and of the Yukawa couplings, involving a Higgs-boson (middle) anda gluon as well as its KK excitations (right).where to very good approximation z = 1 , z n ≥ = ( − n √ L . (79)For comparison, the Feynman rule for the Higgs coupling to a pair of quarks of KK levels n and n reads h q ( n ) q ( n ) = − i √ (cid:20)(cid:0) f Qn (cid:1) ∗ Y q f qn γ (cid:0) f qn (cid:1) ∗ Y ∗ q f Qn − γ (cid:21) , (80)where to good approximation (for A = Q, q ) f A = F ( c A ) √ a ( A )0 , f An ≥ = ( − n sgn[cos( πc A )] √ a ( A ) n . (81)The complex coefficients a ( A ) n are O (1) parameters determined by the conditions (58) and (63),with √ | a ( A )0 | ≈ f Q ) ∗ Y q f q = √ m q /v with the effective Yukawa coupling of the SM fermion, which is closeto 1 for the top quark. For the Higgs-boson couplings to two (or one) KK particles, on theother hand, the vertex (78) is enhanced by a factor of L (or √ L ), while no such factor appearsin (80). However, if we were to replace the Yukawa couplings Y q by L ¯ Y q according to (77),then this would make (80) look more similar to (78).One may try to derive an upper bound on the scale of the Yukawa couplings by invokingperturbativity of the effective 4D theory up to a cutoff scale of a few TeV. For instance,naive dimensional analysis shows that at one-loop order the Yukawa interactions receive amultiplicative correction of order [43] | Y q | π Λ M , (82)where Λ UV is the cutoff scale of the theory on the IR brane. The graph in the middle inFigure 1 shows an example of a diagram giving rise to such a correction. Requiring that this We neglect flavor mixing for simplicity here, i.e. , we assume a single fermion generation. O (1) for a cutoff a factor of 4 above the KK scale gives the upperbound | Y q | < π , which has been used in [43, 51].A potential weakness of this argument is that it is not clear a priori if the theory shouldbe weakly coupled in the Yukawa sector (or in any other sector), and up to what cutoff scaleweak coupling should hold. We note in this context that an explicit calculation of the graphon the right in Figure 1 suggests that there is a QCD correction to the Yukawa interaction oforder Lα s π Λ M , (83)where the factor L reflects the enhanced strength of the coupling of KK gluons to KK fermions.There is no point in requiring that the correction (82) be smaller than (83), and therefore | Y q | < √ πLα s ≈ √ L appears to be the strongest reasonable bound one should impose.Even this bound may be too restrictive, however. Using once more naive dimensionalanalysis, we expect a correction to the Higgs mass from Yukawa interactions that scales like δm h ∼ n f Λ | Y q | π Λ M , (84)which stems from the graph on the left in Figure 1. Here n f = 6 is the number of quark flavors.Additional corrections will arise from the lepton sector. We emphasize that this correctiongrows like the fourth power of the cutoff scale, not like the second power as in 4D. The “littlehierarchy problem” is thus more severe than in 4D extensions of the SM. Choosing the cutofffar above the KK scale would inevitably induce a very large correction to the Higgs-bosonmass, which would reintroduce a fine-tuning problem to the RS model. In order to avoid thisproblem, we must require that the cutoff be of order the KK scale, as is the natural expectationfor the Higgs-boson mass. For Λ UV ∼ M KK , however, relation (82) allows Yukawa couplingsas large as | Y q | < π . In order to analyze the implications of the RS setup for the masses and mixings of the SMfermions, it is instructive to use approximate formulae for the bulk profiles. The approachmost commonly adopted in the literature is to employ a “zero-mode approximation” (ZMA),in which one first solves for the fermion bulk profiles without the Yukawa couplings and thentreats these couplings as a perturbation [21, 26, 36]. At the technical level, this corresponds toan expansion of the exact profiles in powers of v /M , in which also the quantities δ n in (72)are treated as O ( v /M ) parameters. In this approximation the values of the SM fermionprofiles on the IR brane simplify considerably. From (70) we obtain C ( Q,q ) n ( π ) → (cid:114) L(cid:15)π F ( c Q,q ) , S ( Q,q ) n ( π − ) → ± (cid:114) L(cid:15)π x n F ( c Q,q ) . (85)Using these expressions, the system of equations (63) can be recast into the form √ m n v ˆ a ( U ) n = Y eff u ˆ a ( u ) n , √ m n v ˆ a ( u ) n = ( Y eff u ) † ˆ a ( U ) n , (86)21here (cid:0) Y eff u (cid:1) ij ≡ F ( c Q i ) ( Y u ) ij F ( c u j ) (87)are effective Yukawa matrices, and the rescaled vectors ˆ a ( A ) n ≡ √ a ( A ) n obey the normalizationconditions ˆ a ( U ) † n ˆ a ( U ) n = ˆ a ( u ) † n ˆ a ( u ) n = 1 . (88)Furthermore, we obtain from (86) the equalities (cid:18) m n − v Y eff u ( Y eff u ) † (cid:19) ˆ a ( U ) n = 0 , (cid:18) m n − v Y eff u ) † Y eff u (cid:19) ˆ a ( u ) n = 0 , (89)and the mass eigenvalues are the solutions to the equationdet (cid:18) m n − v Y eff u ( Y eff u ) † (cid:19) = 0 . (90)Notice that in the ZMA, but not in general, the vectors a ( A ) n and ˆ a ( A ) n belonging to different n are orthogonal on each other.The eigenvectors ˆ a ( Q ) n and ˆ a ( q ) n of the matrices Y eff q (cid:0) Y eff q (cid:1) † and (cid:0) Y eff q (cid:1) † Y eff q (with n = 1 , , Q = U, D , q = u, d ) form the columns of the unitary matrices U q and W q appearing inthe singular-value decomposition Y eff q = U q λ q W † q , (91)where λ u = √ v diag( m u , m c , m t ) , λ d = √ v diag( m d , m s , m b ) . (92)It follows that in this approximation the relations between the original 5D fields and the SMmass eigenstates involve the matrices U q and W q . In particular, the CKM mixing matrix isgiven by V CKM = U † u U d . (93) Concentrating on the well-motivated case of anarchic 5D Yukawa couplings, i.e. , complex-valued matrices Y q with random elements, it turns out that the up- and down-type quarkmass hierarchies can be reproduced by assuming a hierarchical structure of the elements ofthe zero-mode profiles of the form | F ( c A ) | < | F ( c A ) | < | F ( c A ) | . (94)In the RS framework such a hierarchy is natural, since it results from small differences in thebulk mass parameters c A i . The quarks are assumed to be ordered in such a way that these relations hold. In the case of degeneratevalues | F ( c A i ) | = | F ( c A i +1 ) | , the following discussion is only order-of-magnitude wise correct. m u m c m t = v √ | det ( Y u ) | (cid:89) i =1 , , | F ( c Q i ) F ( c u i ) | ,m d m s m b = v √ | det ( Y d ) | (cid:89) i =1 , , | F ( c Q i ) F ( c d i ) | . (95)Since | F ( c A i ) | < | F ( c A i +1 ) | , one can consistently evaluate all the eigenvalues to leading orderin hierarchies. We obtain m u = v √ | det( Y u ) || ( M u ) | | F ( c Q ) F ( c u ) | , m d = v √ | det( Y d ) || ( M d ) | | F ( c Q ) F ( c d ) | ,m c = v √ | ( M u ) || ( Y u ) | | F ( c Q ) F ( c u ) | , m s = v √ | ( M d ) || ( Y d ) | | F ( c Q ) F ( c d ) | ,m t = v √ | ( Y u ) | | F ( c Q ) F ( c u ) | , m b = v √ | ( Y d ) | | F ( c Q ) F ( c d ) | . (96)Here ( M q ) ij denotes the minor of Y q , i.e. , the determinant of the square matrix formed byremoving the i th row and the j th column from Y q .The elements of the matrices U q and W q are given to leading order in hierarchies by( U q ) ij = ( u q ) ij F ( c Q i ) F ( c Q j ) , i ≤ j ,F ( c Q j ) F ( c Q i ) , i > j , ( W q ) ij = ( w q ) ij e iφ j F ( c q i ) F ( c q j ) , i ≤ j ,F ( c q j ) F ( c q i ) , i > j . (97)Expressed through the elements ( Y q ) ij of the original Yukawa matrices and their minors ( M q ) ij ,the coefficient matrices u q and w q read u q = M q ) ( M q ) ( Y q ) ( Y q ) − ( M q ) ∗ ( M q ) ∗ Y q ) ( Y q ) ( M q ) ∗ ( M q ) ∗ − ( Y q ) ∗ ( Y q ) ∗ , w q = M q ) ∗ ( M q ) ∗ ( Y q ) ∗ ( Y q ) ∗ − ( M q ) ( M q ) Y q ) ∗ ( Y q ) ∗ ( M q ) ( M q ) − ( Y q ) ( Y q ) . (98)The phase factors e iφ j entering W q are given by e iφ j = sgn (cid:2) F ( c Q j ) F ( c q j ) (cid:3) e − i ( ρ j − ρ j +1 ) , (99)with ρ = arg (det( Y q )) , ρ = arg (( M q ) ) , ρ = arg (( Y q ) ) , (100)23nd ρ = 0. Note that to leading order the matrices U q and therefore also the CKM mixingmatrix do not depend on the right-handed profiles F ( c q i ). This feature has first been pointedout in [37]. Exploiting the invariance of the singular-value decomposition (91) under fieldredefinitions allows to make either the diagonal elements ( U q ) ii or ( W q ) ii real. In (97) we havechosen ( U q ) ii to be real, so that all phase factors e iφ j appear in the elements ( W q ) ij .Recalling the definitions of the Wolfenstein parameters of the CKM matrix, λ = | V us | (cid:113) | V ud | + | V us | , A = 1 λ (cid:12)(cid:12)(cid:12)(cid:12) V cb V us (cid:12)(cid:12)(cid:12)(cid:12) , ¯ ρ − i ¯ η = − V ∗ ud V ub V ∗ cd V cb , (101)it is straightforward to derive from (93), (97), and (98) the leading-order expressions for λ , A ,¯ ρ , and ¯ η . We find λ = | F ( c Q ) || F ( c Q ) | (cid:12)(cid:12)(cid:12)(cid:12) ( M d ) ( M d ) − ( M u ) ( M u ) (cid:12)(cid:12)(cid:12)(cid:12) , A = | F ( c Q ) | | F ( c Q ) | | F ( c Q ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( Y d ) ( Y d ) − ( Y u ) ( Y u ) (cid:20) ( M d ) ( M d ) − ( M u ) ( M u ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ¯ ρ − i ¯ η = ( Y d ) ( M u ) − ( Y d ) ( M u ) + ( Y d ) ( M u ) ( Y d ) ( M u ) (cid:20) ( Y d ) ( Y d ) − ( Y u ) ( Y u ) (cid:21) (cid:20) ( M d ) ( M d ) − ( M u ) ( M u ) (cid:21) . (102)Notice that ¯ ρ and ¯ η are to first order independent of the zero-mode profiles F ( c A i ). Like inthe case of the Froggatt-Nielsen mechanism [64], the RS setup predicts that these parametersare thus not suppressed by any small parameters, while their precise O (1) values remainunexplained.The relations given in (96) and (101) do not allow one to determine the zero-mode profilessolely in terms of the quark masses and Wolfenstein parameters. Expressing them through F ( c Q ), one finds for the left-handed quark profiles | F ( c Q ) | = λ (cid:12)(cid:12)(cid:12)(cid:12) ( M d ) ( M d ) − ( M u ) ( M u ) (cid:12)(cid:12)(cid:12)(cid:12) | F ( c Q ) | , | F ( c Q ) | = (cid:12)(cid:12)(cid:12)(cid:12) ( Y d ) ( Y d ) − ( Y u ) ( Y u ) (cid:12)(cid:12)(cid:12)(cid:12) Aλ | F ( c Q ) | . (103)In the case of the right-handed up- and down-type quark profiles, one obtains | F ( c u ) | = √ m u v | ( M u ) | (cid:12)(cid:12)(cid:12)(cid:12) ( M d ) ( M d ) − ( M u ) ( M u ) (cid:12)(cid:12)(cid:12)(cid:12) λ | det( Y u ) | | F ( c Q ) | , | F ( c u ) | = √ m c v | ( Y u ) || ( M u ) | | F ( c Q ) | , | F ( c u ) | = √ m t v Aλ | ( Y u ) | (cid:12)(cid:12)(cid:12)(cid:12) ( Y d ) ( Y d ) − ( Y u ) ( Y u ) (cid:12)(cid:12)(cid:12)(cid:12) | F ( c Q ) | , (104)24nd | F ( c d ) | = √ m d v | ( M d ) | (cid:12)(cid:12)(cid:12)(cid:12) ( M u ) ( M u ) − ( M d ) ( M d ) (cid:12)(cid:12)(cid:12)(cid:12) λ | det( Y d ) | | F ( c Q ) | , | F ( c d ) | = √ m s v | ( Y d ) || ( M d ) | | F ( c Q ) | , | F ( c d ) | = √ m b v Aλ | ( Y d ) | (cid:12)(cid:12)(cid:12)(cid:12) ( Y u ) ( Y u ) − ( Y d ) ( Y d ) (cid:12)(cid:12)(cid:12)(cid:12) | F ( c Q ) | . (105)These relations imply a hierarchical structure among the various quark profiles [37]. For theleft-handed profiles one finds | F ( c Q ) || F ( c Q ) | ∼ λ , | F ( c Q ) || F ( c Q ) | ∼ λ , | F ( c Q ) || F ( c Q ) | ∼ λ . (106)The values of the right-handed up- and down-type quark profiles are then fixed by the observedquark-mass hierarchies. One obtains | F ( c u ) || F ( c u ) | ∼ m u m t λ , | F ( c u ) || F ( c u ) | ∼ m c m t λ , | F ( c d ) || F ( c u ) | ∼ m d m t λ , | F ( c d ) || F ( c u ) | ∼ m s m t λ , | F ( c d ) || F ( c u ) | ∼ m b m t . (107)It is now straightforward to deduce the hierarchical structures of the flavor mixing matrices U q and W q in (97). The matrices in the left-handed quark sector have the same structure asthe CKM matrix, i.e. , U u,d ∼ V CKM ∼ λ λ λ λ λ λ ∼ .
23 0 . .
23 1 0 . .
01 0 .
05 1 , (108)while in the right-handed quark sector one has W u ∼ m u m c λ m u m t λ m u m c λ m c m t λ m u m t λ m c m t λ ∼ .
012 0 . .
012 1 0 . .
001 0 .
077 1 , W d ∼ m d m s λ m d m b λ m d m s λ m s m b λ m d m b λ m s m b λ ∼ .
26 0 . .
26 1 0 . .
12 0 .
44 1 . (109)25he default values for the quark masses and CKM parameters used to obtain these estimatesare collected in Appendix B.It is worth emphasizing at this point that while the fermion profiles are strongly hierarchicalin the left-handed quark sector as well as in the right-handed up-quark sector, the hierarchyin the right-handed down-quark sector is much weaker, namely | F ( c d ) || F ( c d ) | ∼ m d m s λ ∼ . , | F ( c d ) || F ( c d ) | ∼ m s m b λ ∼ . . (110)In this particular sector (and only there), it is thus a viable possibility to assume equal profiles F ( c d i ) and explain the required modest splittings in terms of O (1) variations of the Yukawacouplings. While such a choice might seem ad hoc , it has the effect of strongly suppressingsome particularly dangerous FCNC couplings in K – ¯ K mixing [51]. The results for quark masses and mixing matrices obtained in the ZMA and discussed inthe previous section are invariant under a set of reparametrization transformations, whichchange the values of the 5D bulk mass parameters and Yukawa couplings. The first type ofreparametrization invariance (RPI-1) refers to a simultaneous rescaling of the fermion pro-files for SU (2) L doublet and singlet fields by opposite factors, while leaving the 5D Yukawacouplings invariant. Specifically, F ( c Q i ) → e − ξ F ( c Q i ) , F ( c q i ) → e + ξ F ( c q i ) , Y q → Y q . (RPI-1) (111)Provided the c i are all below − /
2, these transformations approximately correspond to thefollowing shifts of the bulk mass parameters: c Q i → c Q i − ξ/L and c q i → c q i + ξ/L . The secondtype of reparametrization invariance (RPI-2) refers to a simultaneous rescaling of all fermionprofiles by a common factor, while the 5D Yukawa couplings are rescaled with an oppositefactor. Specifically, F ( c Q i ) → η F ( c Q i ) , F ( c q i ) → η F ( c q i ) , Y q → η Y q . (RPI-2) (112)The corresponding shifts of the bulk mass parameters are approximately universal and givenby c Q i → c Q i + L − ln η and c q i → c q i + L − ln η , provided again the c i are all below − /
2. Astronger form of this relation is F ( c Q i ) → η F ( c Q i ) , F ( c q i ) → η q F ( c q i ) , Y q → ηη q Y q (RPI-2 (cid:48) ) (113)with different parameters η u (cid:54) = η d . Of course, it is possible to combine (111) and (113) inarbitrary ways.While the masses and mixing angles are not affected by these transformations (in theZMA), the shapes of the fermion bulk profiles change, and so do the results obtained forflavor-changing interactions (apart from the CKM matrix) derived in later sections.26 .3 Basis Transformations In a recent paper [50], the authors expand the 5D bulk mass matrices under the assumptionof minimal flavor violation. Their ansatz is M Q = m Q + m (cid:48) Q (cid:16) r Y u Y † u + Y d Y † d (cid:17) + . . . , M q = m q + m (cid:48) q Y † q Y q + . . . , q = u, d , (114)where the ellipses stand for subdominant higher-order terms. The quantity r is a small pa-rameter, which is adjusted to reproduce the observed mixings in the quark sector.It is instructive to work out to what ansatz for the Yukawa matrices the model (114)corresponds after we transform it to the bulk mass basis, i.e. , the basis in which the bulkmass terms are diagonal in flavor space. Using unitary transformations on the 5D fields, it iseasy to show that the 5D Yukawa matrix Y u remains anarchic in this case, while Y d becomesdiagonal (with hierarchical diagonal elements) up to terms of O ( r ). In the limit r →
0, flavorviolation in the down-quark sector is thus eliminated by hand, and flavor-changing effectsincluding those described by the CKM matrix only arise from the up-quark sector. Obviously,to explain the hierarchical structure of Y d in the bulk basis either requires an additional flavorsymmetry or mere fine tuning. Unless a dynamical mechanism giving rise to relations of theform (114) can be found, one thus gives up the natural explanation of the quark mass andmixing hierarchies offered by the RS framework. The results of the previous sections allow us to derive exact expressions for all Feynman rulesin the RS model. Of particular importance are, of course, the couplings of gauge bosons tofermions. They will be discussed in Sections 5.1 to 5.3. The couplings of the Higgs boson tofermions are the subject of Section 5.4.
Consider, for example, the coupling of gluons to up-type quarks. Analogous relations hold fordown-type quarks. The 4D Lagrangian contains the terms L (cid:51) (cid:88) n ,n ,n (cid:26) (cid:104) a ( U ) † n I C ( Q ) n n n a ( U ) n + a ( u ) † n I S ( u ) n n n a ( u ) n (cid:105) ¯ u ( n ) L g s / A ( n ) a t a u ( n ) L + (cid:104) a ( u ) † n I C ( u ) n n n a ( u ) n + a ( U ) † n I S ( Q ) n n n a ( U ) n (cid:105) ¯ u ( n ) R g s / A ( n ) a t a u ( n ) R (cid:27) , (115)where g s = g s, / √ πr is the 4D gauge coupling of QCD, and we have defined the overlapintegrals I C ( A ) n n n = (cid:90) π − π dφ √ π χ n ( φ ) e σ ( φ ) C ( A ) n ( φ ) C ( A ) n ( φ ) , A = Q, u, d , (116) The universal term and O (1) coefficients were omitted in [50] for simplicity. I S ( A ) n n n in terms of integrals over S ( A ) n profiles. These objects are N × N diagonal matrices in generation space. For the gluon zero mode, we obtain using (43) and (56) I C ( A )0 n n = δ n n + ∆ C ( A ) n n , I S ( A )0 n n = δ n n + ∆ S ( A ) n n . (117)The relations (58) and (59) then imply that these couplings are flavor diagonal and take thesame form as in the SM, i.e. , L (cid:51) (cid:88) n ¯ u ( n ) g s / A (0) a t a u ( n ) , (118)with u ( n ) = u ( n ) L + u ( n ) R . Note that the couplings of KK gluons are not flavor diagonal and mustbe worked out from the general relation (115). It follows from the structure of the overlapintegrals (116) that the effective coupling strength of KK gluons to heavy fermions, which livenear the IR brane, is not g s but √ L g s [23, 24].The couplings of photons and their KK excitations to fermions are obtained from the aboverelations by replacing the strong coupling constant and color matrices by the appropriateelectromagnetic couplings, i.e. , g s t a → eQ f , where f can be any charged fermion species. Z Boson
For the weak interactions, overlap integrals similar to those in (115) arise, but in this case thegauge bosons couple differently to the SU (2) L doublet and singlet fermions, and so differentcombinations of overlap integrals contribute. Interestingly, in this case even the couplings ofthe light W ± and Z bosons to fermions get corrected compared with their standard form.These non-universal corrections are not diagonal in flavor space, giving rise to tree-level FCNCcouplings of the Z boson. Indeed, these couplings are parametrically enhanced by a factor L compared to the FCNC couplings of KK gauge bosons [55, 56].FCNC couplings of the Z boson arise from two effects: first, the bulk profiles (45) ofthe lowest-lying massive gauge bosons are not flat, giving rise to non-trivial overlap integralswith the fermion profiles; secondly, as seen from (56), the overlap integrals of profiles of onlythe SU (2) L doublet fermions do not obey exact orthonormality conditions. The latter effecthas so far not been studied in great detail in the literature. In the perturbative approach, inwhich the Yukawa couplings are treated as a small correction, it would be interpreted as an SU (2) L singlet admixture in the wave functions of the SU (2) L doublet SM fermions due tomixing with their KK excitations [37, 65, 66] (see also [67]). This effect is often neglected,since it is proportional to the masses of the light SM fermions. However, we will see that itis parametrically as well as numerically as important as the first one. For the first time wewill present exact expressions for the corresponding corrections and compact analytical resultsvalid at first non-trivial order in the ZMA.Including corrections up to O ( m Z /M ), the Z -boson couplings to fermions and theirKK excitations can be written in the form L (cid:51) g cos θ W (cid:20) m Z M (cid:18) − L (cid:19)(cid:21) Z µ × (cid:88) f (cid:88) n ,n (cid:104)(cid:0) g fL (cid:1) n n ¯ f L,n γ µ f L,n + (cid:0) g fR (cid:1) n n ¯ f R,n γ µ f R,n (cid:105) , (119)28here g is the 4D weak SU (2) L gauge coupling defined in (42). As in the SM, the weak mixingangle is given by cos θ W = g (cid:112) g + g (cid:48) , sin θ W = g (cid:48) (cid:112) g + g (cid:48) . (120)The factor in brackets in the first line of (119) accounts for a universal correction due tothe constant terms in the bulk profile (45). The left- and right-handed couplings g fL,R areinfinite-dimensional matrices in the space of fermion modes, which can be parametrized as g fL = (cid:16) T f − sin θ W Q f (cid:17) (cid:20) − m Z M ( L ∆ F − ∆ (cid:48) F ) (cid:21) − T f (cid:20) δ F − m Z M ( L ε F − ε (cid:48) F ) (cid:21) , g fR = − sin θ W Q f (cid:20) − m Z M (cid:0) L ∆ f − ∆ (cid:48) f (cid:1)(cid:21) + T f (cid:20) δ f − m Z M (cid:0) L ε f − ε (cid:48) f (cid:1)(cid:21) . (121)Here T f and Q f denote the weak isospin and electric charge (in units of e ) of the fermion f . On the matrices ∆ , ∆ (cid:48) , δ , ε , and ε (cid:48) , a subscript F refers to a fermion from an SU (2) L doublet( F = U, D in the quark sector, and F = ν, E in the lepton sector), while f refers to a singlet( f = u, d or f = ν R , e , respectively). The matrices ∆ ( (cid:48) ) and ε ( (cid:48) ) arise due to the effects of the t -dependent terms in the gauge-boson profile (45). The elements of the former are given by(∆ F ) mn = 2 πL(cid:15) (cid:90) (cid:15) dt t (cid:104) a ( F ) † m C ( F ) m ( φ ) C ( F ) n ( φ ) a ( F ) n + a ( f ) † m S ( f ) m ( φ ) S ( f ) n ( φ ) a ( f ) n (cid:105) , (∆ f ) mn = 2 πL(cid:15) (cid:90) (cid:15) dt t (cid:104) a ( f ) † m C ( f ) m ( φ ) C ( f ) n ( φ ) a ( f ) n + a ( F ) † m S ( F ) m ( φ ) S ( F ) n ( φ ) a ( F ) n (cid:105) , (∆ (cid:48) F ) mn = 2 πL(cid:15) (cid:90) (cid:15) dt t (cid:18) − ln t (cid:19) (cid:104) a ( F ) † m C ( F ) m ( φ ) C ( F ) n ( φ ) a ( F ) n + a ( f ) † m S ( f ) m ( φ ) S ( f ) n ( φ ) a ( f ) n (cid:105) , (cid:0) ∆ (cid:48) f (cid:1) mn = 2 πL(cid:15) (cid:90) (cid:15) dt t (cid:18) − ln t (cid:19) (cid:104) a ( f ) † m C ( f ) m ( φ ) C ( f ) n ( φ ) a ( f ) n + a ( F ) † m S ( F ) m ( φ ) S ( F ) n ( φ ) a ( F ) n (cid:105) . (122)Recall that on the profiles of SU (2) L doublet fermions no distinction is made between Q = U, D and L = ν, E . The matrices ε ( (cid:48) ) are given in terms of analogous expressions, in which thecontributions from the even profiles C ( A ) n are omitted. Finally, the matrices δ arise from thefact that the fermion profiles are not orthonormal on each other. They are defined as( δ F ) mn = a ( f ) † m (cid:16) δ mn + ∆ S ( f ) mn (cid:17) a ( f ) n , ( δ f ) mn = a ( F ) † m (cid:16) δ mn + ∆ S ( F ) mn (cid:17) a ( F ) n . (123)Since our main focus in this work is on tree-level processes, we are particularly interested inthe couplings of the gauge bosons to the SM fermions. These are described by the upper-left3 × g fL,R for f = u (up-type quarks), f = d (down-type quarks), f = ν (neutrinos), and f = e (charged leptons). With the exception of the heavy top quark The discussion of this section refers to quarks as well as leptons. In the remainder of the paper we will,however, focus on the quark sector of the theory. t = O ( (cid:15) ). On the other hand, thepresence of the weight factor t in the overlap integrals in (122) emphasizes the region nearthe IR brane, where t = O (1). As a result, these overlap integrals are strongly suppressed forthe light SM fermions, giving rise to a corresponding suppression of FCNC processes in theRS model [26]. This mechanism is referred to as the RS-GIM suppression [38, 39, 44], eventhough its dynamical origin is rather different from the GIM mechanism of the SM.It is very instructive to have approximate formulae for the overlap integrals at hand, fromwhich interesting information about the structure and magnitude of flavor-changing effectscan be read off. To this end, we employ the ZMA for the fermion profiles given in (124), whichcorresponds to working to leading order in v /M . The even fermion profiles in (70) thenreduce to the familiar zero-mode profiles obtained in a theory without Yukawa couplings [21],while the odd profiles are suppressed by an extra factor of x n = m n /M KK and thus can beneglected to first approximation. This yields C ( Q,u ) n ( φ ) a ( U,u ) n → (cid:114) L(cid:15) π diag( F ( c Q i ,u i ) t c Qi,ui ) ˆ a ( U,u ) n , S ( Q,u ) n ( φ ) a ( U,u ) n → , (124)and similarly for the down-type quarks. It is then straightforward to find the expressions ∆ F → U † f diag (cid:20) F ( c F i )3 + 2 c F i (cid:21) U f , ∆ f → W † f diag (cid:20) F ( c f i )3 + 2 c f i (cid:21) W f , ∆ (cid:48) F → U † f diag (cid:20) c F i c F i ) F ( c F i ) (cid:21) U f , ∆ (cid:48) f → W † f diag (cid:20) c f i c f i ) F ( c f i ) (cid:21) W f , (125)where all quantities are 3 × ∆ A ≈ ∆ (cid:48) A for A = F, f , since all c i parameters are near − /
2. The approximate relations (125) arenot new. They have been presented first in [33, 40]. Corresponding results valid for a generalwarped metric can be found in [68].The matrices ε ( (cid:48) ) A vanish at leading order in the ZMA, meaning that they are suppressedby an extra factor of v /M . The same is true for the matrices δ A , which are also given interms of overlap integrals containing S ( A ) n profiles. However, since the contributions of thesematrices in (121) are not suppressed by v /M , it is necessary to go beyond the leading orderin the ZMA. From (70) we then obtain S ( Q,u ) n ( φ ) a ( U,u ) n → ± sgn( φ ) (cid:114) L(cid:15) π x n diag (cid:18) F ( c Q i ,u i ) t c Qi,qi − (cid:15) c Qi,qi t − c Qi,qi c Q i ,q i (cid:19) ˆ a ( U,u ) n , (126)30nd using these results gives δ F → x f W † f diag (cid:20) − c f i (cid:18) F ( c f i ) − F ( c f i )3 + 2 c f i (cid:19)(cid:21) W f x f , δ f → x f U † f diag (cid:20) − c F i (cid:18) F ( c F i ) − F ( c F i )3 + 2 c F i (cid:19)(cid:21) U f x f , (127)where x f = diag( m f , m f , m f ) /M KK is a diagonal matrix containing the masses of the SMfermions. To the best of our knowledge these important expressions have not been presentedbefore. Approximate results for the singlet admixture in the wave functions of the SU (2) L doublet SM fermions due to mixing with their KK excitations can be found in [43, 53, 65, 66],but this effect has never been discussed systematically in connection with flavor-changingeffects. Using the scaling relations derived in Section 4.1, it is straightforward to find that toleading power in hierarchies(∆ F ) ij ∼ (∆ (cid:48) F ) ij ∼ F ( c F i ) F ( c F j ) , (∆ f ) ij ∼ (cid:0) ∆ (cid:48) f (cid:1) ij ∼ F ( c f i ) F ( c f j ) , ( δ F ) ij ∼ m f i m f j M F ( c f i ) F ( c f j ) ∼ v Y f M F ( c F i ) F ( c F j ) , ( δ f ) ij ∼ m f i m f j M F ( c F i ) F ( c F j ) ∼ v Y f M F ( c f i ) F ( c f j ) , (128)where Y f represents an element (or a combination of elements) of the Yukawa matrix Y f .These relations make the RS-GIM suppression factors explicit. Note that the contributions ofthe δ A matrices in (121), which have often been neglected in the literature, are of the sameorder as the effects proportional to the ∆ ( (cid:48) ) A matrices. More accurate expressions for the mixingmatrices, in which the relevant combinations of Yukawa matrices are included, can be foundin Appendix A.It is interesting to study how the various flavor-changing couplings transform under thetwo types of reparametrizations, which leave the fermion masses and CKM mixing anglesunchanged in the ZMA. From the relations (111), we obtain for a RPI-1 transformation ∆ F → e − ξ ∆ F , ∆ f → e +2 ξ ∆ f , δ F → e − ξ δ F , δ f → e +2 ξ δ f . (129)This redistributes effects between the left- and right-handed sectors. For a RPI-2 (cid:48) transfor-mation, on the other hand, we find from (113) ∆ F → η ∆ F , ∆ f → η q ∆ f , δ F → η q δ F , δ f → η δ f . (130)31his type of reparametrization acts in a similar way on the left- and right-handed couplings,while reshuffling effects between the two sources of flavor violations: those arising from thegauge-boson profiles ( ∆ A ) and those arising from the fermion profiles ( δ A ). When one typeof effect is enhanced, the other is reduced. W ± Bosons
The couplings of the charged weak bosons W ± to SM fermions and their KK excitations canbe derived in an analogous way. In this case, of course, flavor-changing effects are unsup-pressed already in the SM. Focusing only on the quark sector, and including corrections upto O ( m W /M ), we find L (cid:51) g √ W + µ (cid:88) n ,n (cid:104)(cid:0) V L (cid:1) n n ¯ u L,n γ µ d L,n + (cid:0) V R (cid:1) n n ¯ u R,n γ µ d R,n (cid:105) + h.c., (131)where in terms of the overlap integrals defined in (116) one has( V L ) n n = a ( U ) † n I C ( Q )0 n n a ( D ) n , ( V R ) n n = a ( U ) † n I S ( Q )0 n n a ( D ) n . (132)The couplings to the SM fermions are encoded in the upper-left 3 × V L → U † u U d ≡ V CKM , V R → . (133)At O ( v /M ) corrections to the two matrices arise, which give rise to a non-unitarity ofthe CKM matrix and to right-handed charged currents. The matrix V R can be estimatedusing the ZMA as described in the previous section, i.e. , by including the leading contributionto the S ( A ) n profiles shown in (70) but ignoring the δ n terms. In this way we obtain V R → x u U † u diag (cid:20) − c Q i (cid:18) F ( c Q i ) − F ( c Q i )3 + 2 c Q i (cid:19)(cid:21) U d x d . (134)The scaling relations from Section 4.1 imply that( V R ) ij ∼ v M F ( c u i ) F ( c d j ) ∼ m u i m d j M F ( c Q i ) F ( c Q j ) . (135)The deviations of the CKM matrix from a unitary matrix are sensitive to the t -dependentterms in the gauge-boson profile (45), and to the deviations of the even fermion profiles andthe eigenvectors a ( A ) n from the expressions valid in the ZMA. These effects are best studiedusing the exact results for the fermion profiles and eigenvectors derived in Section 3.3. Willwe present numerical estimates in Section 6.3. The non-unitarity of the CKM matrix hasbeen analyzed previously in [37, 54]. However, a thorough discussion of all relevant effects ismissing in these articles. An analogous discussion holds for leptons. .4 Fermion Couplings to the Higgs Boson Within the SM, the tree-level interactions of fermions with the Higgs-boson are flavor diagonalin the mass eigenstate basis of the fermion fields. Due to the mixing of fermion zero-modesand their KK excitations, this is not the case in the RS scenario [42]. The flavor-changing hf ¯ f couplings can be expressed in terms of the Yukawa matrices Y q , the even profiles C ( A ) n evaluated at the IR brane, and certain combinations of the eigenvectors a ( A ) n . Working inunitary gauge, the relevant terms in the 4D Lagrangian describing the coupling of the Higgsboson to up-type quarks read L (cid:51) − (cid:88) n ,n ( g uh ) n n h ¯ u ( n ) L u ( n ) R + h . c . , (136)where we have defined the couplings( g uh ) mn = √ π(cid:15)L a ( U ) † m C ( Q ) m ( π ) Y u C ( u ) n ( π ) a ( u ) n ≡ δ mn m u m v − (∆ g uh ) mn . (137)The first term in the last expression gives the SM couplings, while (∆ g uh ) mn contains theRS contributions. Analogous expressions describe the interactions of the Higgs boson anddown-type quarks.The couplings (∆ g qh ) mn are not independent from the flavor matrices derived earlier in thiswork. Using relations (58), (59), (60), (63), and (123), it is not difficult to show that (for q = u, d and Q = U, D ) (∆ g qh ) mn = m q m v ( δ q ) mn + ( δ Q ) mn m q n v . (138)The flavor-changing Higgs-boson couplings are thus parametrized in terms of the matrices δ A entering also the Z -boson couplings to SM fermions. The expressions for these matrices validin the ZMA have been given in (127). Note that the extra suppression by factors of m q /v implies that FCNC processes mediated by Higgs-boson exchange are suppressed comparedwith those mediated by the exchange of a Z boson. Assuming N quark generations, one starts out with N Y = 2 ( N , N ) real moduli and CP-oddphases for the 5D Yukawa matrices Y (5D) u,d , plus N c = 3 ( N ( N + 1) / , N ( N − /
2) parametersfor the hermitian bulk mass matrices c Q,u,d . The Yukawa matrices break the global bulkflavor symmetry G = U ( N ) Q × U ( N ) u × U ( N ) d with N G = 3 ( N ( N − / , N ( N + 1) / H = U (1) B with N H = (0 ,
1) parameters. One thus ends up with N phys = N Y + N c − N G + N H = ( N (2 N + 3) , ( N − N − N = 3 quark generations this leads to 27 moduli and ten phases [39]. In the ZMAthe real parameters consist of six quark masses, twelve mixing angles appearing in the ma-trices U u,d and W u,d , and the nine eigenvalues of the zero-mode profiles F Q,u,d . Out of theten phases, one is the phase of the CKM matrix, and nine new phases enter (in different33ombinations) the various matrices in (122) and (123), which parametrize the flavor-changinginteractions in the RS model.For N = 2 quark generations one has 14 moduli and three phases, all of which can bechosen to reside in the new mixing matrices, since the two-generation CKM matrix can bemade real by phase redefinitions. It is instructive to illustrate this point using the explicit(approximate) forms for the mixing matrices collected in Appendix A. For N = 2 thesematrices depend on the four complex quantities ( Y q ) / ( Y q ) and ( Y q ) / ( Y q ) for q = u, d .Adopting a phase convention in which the two-generation Cabibbo matrix is real imposes theconstraint ( Y u ) / ( Y u ) = ( Y d ) / ( Y d ) .We emphasize that the RS model allows for CP-violating effects which do not involve allthree fermion generations, and which can be restricted to either the up- or down-quark sectors.In that sense CP violation is much less suppressed that in the SM. It would be interesting towork out the implications of this observation for baryogenesis.As a final comment, let us mention that the new CP-odd phases encountered in the minimalRS model induce electric dipole moments of the electron and neutron at the one-loop level. Toavoid the stringent experimental limits on the neutron electric dipole moment, the KK gauge-boson masses typically have to be larger than 10 TeV [39]. Several solutions [50, 51, 54] to this“CP problem” have been proposed, all of which postpone the appearance of a non-vanishingneutron electric dipole moment to the two-loop level by reducing the number of CP-violatingphases in the two-generation case to one. We now discuss some simple applications of the results derived so far, beginning with thedefinition and determination of important SM parameters. We then study the constraintsimposed by the electroweak precision measurements encoded by the S , T , and U parametersand the Z b ¯ b couplings. In this context, we point out that the minimal RS scenario providesa consistent framework in the case of a heavy Higgs boson for fairly low KK gauge-bosonmasses. Next we deal with the mixing matrices in the charged- and neutral-current sectors,paying special attention to the non-unitarity of the CKM matrix and the potential size of theanomalous right-handed W tb coupling. Then follows a discussion of the mass spectrum of KKfermions. We find that due to the near degeneracy of 5D fermionic bulk mass parameters, evensmall Yukawa couplings generically lead to large mixing effects among the fermion excitationsof the same KK level. We close this section by analyzing FCNC top-quark decays. Afterdetermining the preferred chirality of the Z tc ( u ) interactions we perform a detailed studyof the t → c ( u ) Z and t → c ( u ) h branching ratios. Comments on correlations betweenflavor-diagonal and non-diagonal Z vertices as well as anomalous ht ¯ t couplings round off ourphenomenological survey. Since the couplings of the photon and gluon zero modes to fermions are universal and have thesame form as in the SM, the low-energy extractions of the fine-structure constant α (defined in34 − e − ν e ν µ W − ( n ) Figure 2: Tree-level contributions to µ − → e − ν µ ¯ ν e arising from the exchange of a W − boson and its KK excitations.the Thomson limit) and of the strong coupling α s are, to very good approximation, unaffectedfrom higher-dimensional effects in the RS model. The weak mixing angle θ W is related to the4D gauge couplings as usual by (120). It follows that g = 4 πα sin θ W . (139)The mixing angle defined in this way can be extracted from measurements of the left-rightpolarization asymmetries of light SM fermions on the Z pole. In this case the RS-GIMmechanism ensures that the modifications of the Z f ¯ f couplings are given to excellent ap-proximation by the universal prefactor in (119), which cancels in the standard formula for theasymmetries, A f = Γ( Z → f L ¯ f R ) − Γ( Z → f R ¯ f L )Γ( Z → f L ¯ f R ) + Γ( Z → f R ¯ f L ) = ( g fL ) − ( g fR ) ( g fL ) + ( g fR ) ≈ (1 / − | Q f | sin θ W ) − ( Q f sin θ W ) (1 / − | Q f | sin θ W ) + ( Q f sin θ W ) . (140)We next turn to the determination of the Fermi constant G F from muon decay. As shownin Figure 2, at tree level in the RS model this process is mediated by the exchange of theentire tower of the W − boson and its KK excitations. We have calculated the relevant sumover these intermediate states in (33). The terms proportional to t or t (cid:48) in this relation giverise to non-universal effects suppressed by the fermion profiles near the IR brane, which toexcellent approximation can be neglected for the light leptons involved in muon decay. Thisleaves a universal correction given by the constant terms in (33). We obtain G F √ g m W (cid:20) m W M (cid:18) − L (cid:19) + O (cid:18) m W M (cid:19)(cid:21) . (141)The correction term receives a contribution (1 − /L ) from the ground-state W − boson and1 / (2 L ) from the tower of KK excitations.The SM relation between the weak mixing angle and the masses of the weak gauge bosonsreceives important corrections in the RS model. From (44) it follows that m W m Z = cos θ W (cid:20) θ W m Z M (cid:18) L − L (cid:19) + O (cid:18) m Z M (cid:19)(cid:21) . (142)35 (cid:3) CL95 (cid:3)
CL99 (cid:3)
CL60 GeV 300 GeV m h (cid:4) M KK (cid:4)
150 175 20080.380.480.5 m t (cid:2) GeV (cid:3) m W (cid:2) G e V (cid:3) Figure 3: Regions of 68%, 95%, and 99% probability in the m t – m W plane followingfrom the direct measurements of m W and m t at LEP2 and the Tevatron. The blackdot corresponds to the SM prediction based on the value of G F for our reference inputvalues, while the green (medium gray) shaded band shows the SM expectation forvalues of the Higgs-boson mass m h ∈ [60 , M KK ∈ [1 ,
10] TeV. See text for details.Notice that in this case the correction term is enhanced by a factor of L . It is customaryto encode the corrections to the SM (tree-level) relation between m W /m Z and cos θ W in aparameter (cid:37) ≡ m W / ( m Z cos θ W ). The prediction for this parameter obtained in the RSmodel depends on how the W ± -boson mass is determined. If m W is obtained from a directmeasurement, then the (cid:37) parameter is given by the expression in brackets on the right-handside of the relation (142). Alternatively, m W can be derived from precise measurements of α , G F , and sin θ W using the SM relation( m W ) indirect ≡ πα √ G F sin θ W . (143)In the RS model we have( m W ) indirect = m W (cid:20) − m W M (cid:18) − L (cid:19) + O (cid:18) m W M (cid:19)(cid:21) . (144)When this is done, one obtains instead (cid:37) = 1 + m Z M (cid:18) L sin θ W − L (cid:19) + O (cid:18) m Z M (cid:19) . (145)The comparison between the indirect constraint and the direct measurements of m W and m t is shown in Figure 3. The regions of 68%, 95%, and 99% probability following from thedirect measurements performed at LEP2 [29] and the Tevatron [69] are shown by the ellipses.36he shaded band shows the SM prediction based on (143) for Higgs-boson masses in the range m h ∈ [60 , W ± -boson mass determinations are given by m W = (80 . ± . , ( m W ) indirect = (80 . ± . . (146)The value of ( m W ) indirect has been derived using ZFITTER [70, 71]. The quoted central valueand error correspond to the SM reference values for ∆ α (5)had ( m Z ), α s ( m Z ), m Z , and m t collectedin Appendix B, as well as the Higgs-boson mass m h = 150 GeV. The vertical line and pointscorrespond to the RS prediction of m W following from (144) for L = ln(10 ) and M KK ∈ [1 ,
10] TeV. It is evident from the plot that the RS corrections allow to explain the shift ofaround 50 MeV between the direct measurement and the indirect determination for KK scalesslightly above M KK = 1 . M KK = 1 . M KK = 3 TeV.Notice also that even for a heavy Higgs boson, agreement between m W and ( m W ) indirect at the99% CL can always be reached for KK scales above 1 TeV. Taking for example m h = 400 GeV( m h = 1000 GeV) would allow for KK scales as low as 1.5 TeV (1.0 TeV). S , T , and U Parameters
The S , T , and U parameters measure deviations from the electroweak radiative correctionsexpected in the SM from new physics effects in universal electroweak corrections, i.e. , thoseentering through vacuum polarization diagrams [27, 28, 72, 73, 74, 75, 76]. These parametersare defined as shifts relative to a fixed set of SM values, so that S , T , and U are identicalto zero at that point. In the entire discussion of the electroweak precision measurements wefocus on the leading contributions to the S , T , and U parameters and thus restrict ourselvesto tree-level effects in the framework of the minimal RS model. We further assume that SMloop effects are not significantly modified by the presence of KK excitations. A calculation ofthe complete one-loop contributions to S , T , and U , extending the work of [77, 78], would beworthwhile, but is beyond the scope of this paper.Using the relations (44) and (45), it is a simple exercise to derive the S , T , and U parametersin the minimal RS scenario at tree-level. In agreement with [59, 68], we find the positivecorrections S = 2 πv M (cid:18) − L (cid:19) , T = πv θ W M (cid:18) L − L (cid:19) , (147)while U vanishes. Placing the fermion fields in the bulk greatly softens the strong constraintfrom S present in the RS models with bulk gauge fields and brane-localized fermions, for which S, T ∼ −
Lπv /M are both large and negative [22].The experimental 68% CL bounds on the S and T parameters, corrected to the presentworld average of the top-quark mass [79], and their correlation matrix are given by [29] S = 0 . ± . ,T = 0 . ± . , ρ = (cid:32) .
00 0 . .
85 1 . (cid:33) . (148)37 m t m h m t (cid:3) (cid:2) (cid:4) (cid:3) GeV M KK LM KK (cid:5) (cid:4)
1, 10 (cid:5)
TeV L (cid:5) (cid:4)
5, 37 (cid:5) m h (cid:5) (cid:4)
60, 1000 (cid:5)
GeV68 (cid:6)
CL95 (cid:6)
CL99 (cid:6) CL U (cid:3) (cid:7) (cid:7) (cid:7) (cid:7) S T (cid:2) m t m h m t (cid:3) (cid:2) (cid:4) (cid:3) GeV M KK LM KK (cid:5) (cid:4)
1, 10 (cid:5)
TeV L (cid:5) (cid:4)
5, 37 (cid:5) m h (cid:5) (cid:4)
60, 1000 (cid:5)
GeV68 (cid:6)
CL95 (cid:6)
CL99 (cid:6) CL U (cid:3) (cid:7) (cid:7) (cid:7) (cid:7) S T Figure 4: Regions of 68%, 95%, and 99% probability in the S – T plane. The green(medium gray) shaded stripes in both panels indicate the SM predictions for m t =(172 . ± .
4) GeV and m h ∈ [60 , M KK ∈ [1 ,
10] TeV and L ∈ [5 , S – T plane are shown in the left panel ofFigure 4. The light-shaded stripe indicates the SM predictions for different values of m h and m t , while the dark-shaded area represents the RS corrections in (147) for different values of M KK and L . In the global fit to the LEP and SLC measurements the parameter U is set to zero,and the subtraction point corresponds to the SM reference values compiled in Appendix B and m h = 150 GeV. The S – T error ellipses show that there are no large unexpected electroweakradiative corrections from physics beyond the SM, as the values of the S and T parametersare in agreement with zero.Requiring that the RS contributions (147) satisfy the experimental constraint from S and T yields for the reference point ( i.e. , for the SM parameters given above) M KK > . . (149)Since the lightest KK modes have masses of around 2 . M KK in the RS framework, theconstraint from T forces the first KK gauge-boson excitations to be heavier than about 10 TeV.Interestingly, one can show that the problem with T persists in any 5D warped model withSM gauge symmetry in the bulk, in which the solution to the gauge hierarchy problem isassociated with a moderately large volume factor [68]. This strong constraint introduces a“little hierarchy problem” and calls for a cure.There are at least four possibilities that can mitigate the strong constraint from T inwarped 5D scenarios. The first one consists in canceling the large positive corrections (147)to the T parameter by a large negative correction associated with a massive Higgs boson[80, 81, 82, 83, 84]. Keeping only the leading logarithmic loop effects in the SM, the shifts in38 (cid:2) CL95 (cid:2)
CL99 (cid:2) CL L (cid:3) ln (cid:2) (cid:3) (cid:2) CL95 (cid:2)
CL99 (cid:2) CL L (cid:3) ln (cid:2) (cid:3) m h (cid:4) GeV (cid:5) M KK (cid:4) T e V (cid:5) (cid:2) CL95 (cid:2)
CL99 (cid:2) CL M KK (cid:3) (cid:2) CL95 (cid:2)
CL99 (cid:2) CL M KK (cid:3) m h (cid:2) GeV (cid:3) L Figure 5: Regions of 68%, 95%, and 99% probability in the m h – M KK (left panel) and m h – L (right panel) plane in the RS scenario without custodial protection. The upper(lower) area in the left and right panel corresponds to L = ln(10 ) ( L = ln(10 )) and M KK = 3 TeV ( M KK = 1 . S and T due to a Higgs-boson mass different from the reference value m ref h = 150 GeV read[28] ∆ S = 16 π ln m h m ref h , ∆ T = − π cos θ W ln m h m ref h , (150)while U remains unchanged. These relations imply that the bound on the KK gauge-bosonmasses following from T may be relaxed by taking large values of m h . A large Higgs-bosonmass also induces a positive correction to the parameter S . However, since ∆ T / ∆ S ≈ − M KK is alleviated. Forexample, taking m h = 1 TeV, the shifts due to the Higgs boson amount to ∆ S ≈ .
10 and∆ T ≈ − .
30, and the lower bound (149) is relaxed to M KK > . . (151)This feature is illustrated by the upper sets of bands in the panels of Figure 5, which show theregions of 68%, 95%, and 99% probability in the m h – M KK and m h – L planes for L = ln(10 )(left plot) and M KK = 3 TeV (right plot). Since, as explained in Section 3.4, the Higgs-bosonmass in warped models with the Higgs field residing on the IR-brane is naturally of order theKK rather than the electroweak scale, the possibility that the constraint from T is temperedby the presence of a heavy Higgs boson should be considered seriously. In addition, a smallervalue of the top-quark mass induces a small negative shift in T without affecting S and wouldthus help in relaxing the constraint from T further. For example, the present total error on the Unitarity of longitudinal W ± -boson scattering requires the SM Higgs-boson mass to satisfy the bound m h (cid:46)
870 GeV. In the RS model, unitarity can be satisfied for substantially higher values of m h [85]. ± . T ≈ ± .
02. A heavy Higgs boson incombination with a somewhat lighter top quark thus might allow for KK gauge-boson massesas low as 6 TeV without invoking a symmetry that protects T from unacceptably large positivecorrections.The second way to protect T from vast corrections gives up on the solution to the fullgauge hierarchy problem by working in a volume-truncated RS background [7]. Taking forexample L = ln(10 ) to address the hierarchy between the electroweak scale and 10 TeV, theRS bound (149) changes into the “little RS” limit M KK > . . (152)In the “little RS” scenario with L = ln(10 ), the lightest KK modes have masses of approx-imately 2 . M KK , so that T pushes the mass of the lowest-lying KK gauge-boson excitationto around 4 TeV. This lower limit is weaker by about a factor 2.5 than the one obtained from(149). The bound (152) relaxes further for larger Higgs-boson mass. This feature is illus-trated by the lower sets of bands in the panels of Figure 5, which show the regions of 68%,95%, and 99% probability in the m h – M KK and m h – L planes for L = ln(10 ) (left plot) and M KK = 1 . m h = 500 GeV instead of the reference valueof 150 GeV turns the limit (152) into 1.1 TeV.A third possibility to obtain a consistent description of the experimental data, while al-lowing for masses of the first KK gauge-boson modes of the order of 5 TeV, utilizes largebrane-localized kinetic terms for the gauge fields [57, 58, 59]. Since such terms are needed ascounterterms to cancel divergences appearing at the loop level [86, 87], they are expected ongeneral grounds to be present in any realistic orbifold theory. The bare contributions to thebrane-localized kinetic terms encode the unknown UV physics at or above the cutoff scale. Toretain the predictivity of the model, we simply assume that these bare contributions are small.As we concentrate on the leading contributions to the electroweak precision observables, wefurthermore ignore possible effects of brane-localized kinetic terms appearing at the loop level.Even if these assumptions would be relaxed, the bound on the KK gauge-boson masses thatderives from the constraints on the S and T parameters would remain anti-correlated withthe mass of the Higgs boson. For example, Ref. [59] finds that having a light KK spectrumwould require a value of the Higgs-boson mass in the range of several hundred GeV.The fourth cure for an excessive T parameter is the custodial SU (2) R symmetry [33]. Inthe context of 5D theories with warped background the custodial symmetry is promoted to agauge symmetry. The hypercharge group is extended to SU (2) R × U (1) X , such that the bulkgauge symmetry is SU (3) c × SU (2) L × SU (2) R × U (1) X . The gauge group is then broken to SU (3) c × SU (2) L × U (1) Y on the UV brane. The tree-level S and T parameters in the RSscenario with this extended electroweak sector are given by [33] S = 2 πv M (cid:18) − L (cid:19) , T = − πv θ W M L , (153)while U remains unaffected. Notice that the S parameter is given by the same expression asin the case without the custodial symmetry. For the T parameter the custodial symmetry isat work, since T turns out to be suppressed, rather than enhanced, by the logarithm L of the40arp factor. Thus the tree-level T parameter ends up being tiny for the RS background witha large warp factor. The lower bound on the KK scale then follows from the experimentalconstraint on S . Numerically, one finds for the reference point M KK > . , (154)which translates into a lower bound on the first KK gauge-boson mass of about 6 TeV. Thislimit is only marginally better that the bound (151) obtained in the original RS model withoutcustodial symmetry, but with a heavy Higgs boson. Notice also that in the case of the RSscenario with extended electroweak sector, the existence of a heavy Higgs boson would spoil theglobal electroweak fit, since the corrections (153) are generically too small to compensate forthe negative shift ∆ T due to a large value of m h . This feature is illustrated in the right panelof Figure 4. On the other hand, it has been shown in [33] that in this model fermionic loopcorrections to T are UV-finite and positive and allow to lower the KK scale for a sufficientlylight Higgs boson. As a word of caution we should mention that, without a custodial bulksymmetry, the T parameter could also be affected by unknown UV dynamics. It is thuspossible that loop and cutoff effects might raise the bounds (151) and (152) to significantlyhigher values of the KK scale.In [77, 78] one-loop corrections to the T parameter arising from KK excitations that couplevia the top Yukawa coupling have been studied in the context of the custodially symmetric SU (2) L × SU (2) R model. The interesting observation made by these authors is that thepresence of quark bi-fundamentals under SU (2) L × SU (2) R , introduced to protect the left-handed Z b ¯ b coupling [34], typically induces a negative shift in the T parameter at the one-loop level, although a positive correction is possible in some regions of parameter space. It isapparent from the right plot in Figure 4 that obtaining a negative contribution to T , togetherwith a positive value of the S parameter, would be rather problematic. In the RS scenariowithout SU (2) R one-loop corrections to T have not been calculated. The precise impact ofhigher-order corrections on (149), (151), and (152) therefore remains unknown. We now turn our attention to the sources of flavor violation in the couplings of the weakgauge bosons to fermions. To this end we need to specify a large number of model parameters,namely the bulk mass parameters of the 5D fermion fields and the 5D Yukawa matrices. Theirchoice is restricted by the fact that one should reproduce the known values of the quark massesand CKM matrix elements within errors, but this information still leaves some freedom.It will be useful for many considerations to have a default set of input parameters, whichis consistent with all experimental constraints concerning the quark masses and CKM param-eters. We use M KK = 1 . M KK = 3 TeV, as a second reference point. Unless otherwise noted, we set L = ln(10 ) for41he logarithm of the warp factor. Our default set of bulk mass parameters is c Q = − . , c Q = − . , c Q = − . ,c u = − . , c u = − . , c u = +0 . ,c d = − . , c d = − . , c d = − . . (155)For the 4D Yukawa matrices in the normalization (50) we take Y u = . − . i .
102 + 1 . i . − . i . − . i . − . i − . − . i − .
729 + 0 . i .
682 + 0 . i . − . i , Y d = . − . i .
358 + 0 . i − . − . i . − . i − . − . i − .
043 + 0 . i .
968 + 1 . i .
697 + 0 . i − .
853 + 0 . i . (156)These matrices have been obtained by random choice, subject to the constraints that theabsolute value of each entry is between 1 / ρ and¯ η in (102) agree with experiment within errors. The bulk mass parameters c A i have then bedetermined using the scaling relations described in Section 4.1.With these parameters, the exact values for the quark masses obtained from the eigenvalueequation (65) are m u = 1 .
45 MeV , m c = 563 MeV , m t = 136 GeV ,m d = 2 .
98 MeV , m s = 49 . , m b = 2 .
23 GeV . (157)These values should be interpreted as the running quark masses in the MS scheme evaluatedat the scale µ = M KK . After renormalization-group evolution, they agree with the massvalues derived from low-energy measurements (see Appendix B). Note that essentially thesame values are obtained using relations (91) and (92) valid in the ZMA. The only exceptionis the top-quark mass, which comes out 5.5 GeV too large in this approximation.In parts of our analysis below, specifically in Sections 6.3.1, 6.4, 6.5, and 6.6, we willperform a scan over the parameter space of the RS model. Whenever this is done, we generate3000 randomly chosen points using uniform initial distributions for the input parameters.The parameter ranges are M KK ∈ [1 ,
10] TeV for the KK scale and | ( Y u,d ) ij | ∈ [1 / ,
3] forthe Yukawa couplings. We further require in a somewhat ad hoc way that | F ( c A i ) | ≤ √ c A i < /
2. The bulk mass parameters are then chosen such that all pointsreproduce the quark masses and CKM parameters with a global χ / dof of better than 11.5/10(corresponding to 68% CL). This large set of points provides a reasonable range of predictionsthat can be obtained for a given observable. Only for a subset of the 3000 scatter pointsthe RS predictions for the Z b ¯ b couplings are consistent at the 99% CL with experimental Here and below, results are given to at least three significant digits. Z → b ¯ b “pseudo observables” is deferred toSection 6.4.) We will generally show two plots for a given observable, one made with all 3000parameter points, and one restricted to those points that are compatible with the measured Z b ¯ b couplings. In the latter case, we add further scatter points that pass all constraints, sothat the number of shown parameter points also amounts to 3000 in total. We next consider the results for the flavor-mixing matrices, starting with the CKM matrix.In all cases we adjust the phases of the SM quark fields according to the standard CKM phaseconvention [69], which is defined by the requirements that the matrix elements V ud , V us , V cb , V tb are real, and that Im V cs = V us V cb V ud + V us Im V ub . (158)These five conditions fix the phase differences between the six quark fields uniquely.With our default parameters, the exact expression for the left-handed charged-currentmixing matrix V L defined in (132) is V L = .
975 0 .
225 0 . e − i . ◦ − . e i . ◦ . e − i . ◦ . . e − i . ◦ − . e i . ◦ . . (159)For the purposes of this work, we define the CKM matrix as V CKM ≡ V L , i.e. , as the matrixgoverning the charged-current couplings of the lightest W ± bosons in units of g/ √
2, see(131). From this matrix we extract for the Wolfenstein parameters the values λ = 0 . , A = 0 . , ¯ ρ = 0 . , ¯ η = 0 . . (160)They are all in good agreement with experiment (see Appendix B). Unlike the CKM matrixin the SM, the left-handed mixing matrix in the RS model is not a unitary matrix. As twomeasures of this effect we consider the deviation of the sum of the squares of the matrixelements in the first row from 1, and the lack of closure of the unitarity triangle [88]. A muchmore detailed analysis of new physics effects on determinations of CKM and unitarity-triangleparameters will be presented in [56]. With our default parameters, we obtain1 − (cid:0) | V ud | + | V us | + | V ub | (cid:1) = − . , V ud V ∗ ub V cd V ∗ cb + V td V ∗ tb V cd V ∗ cb = − . . i . (161)The first number should be compared with the one following from a global fit performed in[89], which yields1 − (cid:0) | V ud | + | V us | + | V ub | (cid:1) = 0 . ± . V ud ± . V us . (162) An alternative, more physical definition is based on the effective four-fermion interactions induced by theexchange of the entire tower of W ± bosons and their KK excitations [56]. V ud and V us might thus allow to detect the non-unitarity of the CKM matrix induced in the RS framework.The second prediction in (161) can be compared with the current precision on the values ofthe Wolfenstein parameters, which are ¯ ρ = 0 . +0 . − . and ¯ η = 0 . ± .
016 according to theCKMfitter Group [90], and ¯ ρ = 0 . ± .
029 and ¯ η = 0 . ± .
016 according to the UT fit
Collaboration [91]. Again, the predicted unitarity violation is of the same order as the presenterrors.Our default result for the right-handed mixing matrix V R in (132) is given in Appendix C.Its entries are extremely small compared to the corresponding entries of the CKM matrixdue to the suppression proportional to two powers of light quark masses, which is evidentfrom (134). The only two elements exceeding the level of 10 − in magnitude are ( V R ) = − . · − e − i . ◦ and ( V R ) = 1 . · − . However, these couplings are still too small togive rise to any observable effect. Consider the right-handed W tb coupling as an example. Toleading power in hierarchies only the profile F ( c Q ) and certain combinations of c Q , c Q , andthe elements of the up- and down-type Yukawa matrices enter in the formula for this quantity.Explicitly, we find in the ZMA v R ≡ ( V R ) → m b m t M (cid:34) − c Q (cid:18) F ( c Q ) − F ( c Q )3 + 2 c Q (cid:19) + (cid:88) i =1 , ( Y u ) ∗ i ( Y d ) i ( Y u ) ∗ ( Y d ) − c Q i F ( c Q ) (cid:35) . (163)Interestingly, the inclusive B → X s γ decay provides a stringent bound on the potential sizeof the anomalous W tb coupling proportional to v R . Due to the chiral m t /m b enhancementpresent in B → X s γ [92, 93], the bound on this specific effective coupling turns out to bemore than an order of magnitude stronger than the limit expected from future measurementsof top-quark production and decay at the LHC [94]. A recent careful analysis arrives at [95] v R ∈ [ − . , . . (164)In Figure 6 we show the RS predictions for v R as a function of M KK for 3000 randomlygenerated parameter points, see Section 6.3. The range (164) disfavored by B → X s γ isindicated by the horizontal band. In the right plot we have excluded scatter points thatdo not lead to an agreement with the measured Z b ¯ b couplings at the 99% CL. The plotsillustrate that although values of v R at the level of 10 − are possible for light KK masses, theRS corrections to v R are generically too small to lead to an observable effect in B → X s γ . Thepossibility that an anomalous right-handed W tb coupling changes the top-quark productionand decay in a way that would be detectable at the LHC thus seems highly unlikely in theminimal RS framework.Note that the correlation between the
W tb and Z b ¯ b couplings is in general different inwarped models with extended electroweak sector, because the custodial symmetry cannotsimultaneously protect the W tb and Z b ¯ b couplings [34]. We therefore expect that, depending44 (cid:3) (cid:2) CL B (cid:3) X s Γ without Z (cid:3) bb (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) M KK (cid:2) TeV (cid:3) V R (cid:2) (cid:2) CL B (cid:3) X s Γ with Z (cid:3) bb (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) M KK (cid:2) TeV (cid:3) V R Figure 6: Coefficient v R as a function of M KK . The red (dark gray) band represents theparameter region disfavored by B → X s γ at the 95% CL. All scatter points reproducethe correct quark masses and CKM parameters within errors. In the left (right) plot,points that violate the constraints from the Z → b ¯ b “pseudo observables” are included(excluded). The values of v R for our RS reference points with M KK = 1 . M KK = 3 TeV are indicated by the yellow (light gray) triangle and the orange (mediumgray) stars. See text for details.on the exact realization of the fermionic sector, the corrections to the W tb couplings could bemore pronounced, which might allow for an indirect detection of the anomalous right-handed
W tb coupling through a deviation in the B → X s γ branching ratio. The largest flavor-changing effects in the neutral-current sector arise from the Z -boson cou-plings to SM fermions. They are encoded in the left- and right-handed couplings g fL,R given in(121). The dominant effects arise from the matrices ∆ Q,q (whose contributions are enhancedby a factor L ) and δ Q,q . We now list the results for these matrices obtained with our defaultparameters. Note that the ∆ A terms are multiplied by a factor m Z L/ (2 M ) ≈ . ≈ / δ A terms. This needs to be taken into account when comparing the numericalvalues of the various matrices. 45eft-handed down-quark sector: ∆ D = 10 − .
290 0 . e − i . ◦ − . e i . ◦ . e i . ◦ . − . e − i . ◦ − . e − i . ◦ − . e i . ◦ . , δ D = 10 − .
214 0 . e i . ◦ − . e i . ◦ . e − i . ◦ . − . e i . ◦ − . e − i . ◦ − . e − i . ◦ . . (165)Right-handed down-quark sector: ∆ d = 10 − . . e − i . ◦ . e i . ◦ . e i . ◦ . . e i . ◦ . e − i . ◦ . e − i . ◦ . , δ d = 10 − . − . e − i . ◦ . e i . ◦ − . e i . ◦ . . e − i . ◦ . e − i . ◦ . e i . ◦ . . (166)Left-handed up-quark sector: ∆ U = 10 − .
775 1 . e − i . ◦ − . e i . ◦ . e i . ◦ . − . e − i . ◦ − . e − i . ◦ − . e i . ◦ . , δ U = 10 − .
419 1 . e − i . ◦ − . e i . ◦ . e i . ◦ . − . e i . ◦ − . e − i . ◦ − . e − i . ◦ . . (167)Right-handed up-quark sector: ∆ u = 10 − . − . e − i . ◦ . e − i . ◦ − . e i . ◦ . − . e − i . ◦ . e i . ◦ − . e i . ◦ . , δ u = 10 − . − . e − i . ◦ . e i . ◦ − . e i . ◦ .
76 276 . e − i . ◦ . e − i . ◦ . e i . ◦ . (168)The mixing matrices ∆ (cid:48) A take similar values (both in magnitudes and phases) as the corre-sponding matrices ∆ A . They are collected in Appendix C. Also given there are our resultsfor the matrices ε ( (cid:48) ) A , which are more than an order of magnitude smaller in magnitude than46 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) Y (cid:2) Y (cid:3) t c u Q Q Q Q Q c Q u t (cid:4) (cid:4) (cid:4) n f l a vo r c on t e n t m n (cid:2) T e V (cid:3) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) Y (cid:2) Y (cid:3) b s d Q Q Q Q Q b Q s d (cid:4) (cid:4) (cid:4) n f l a vo r c on t e n t m n (cid:2) T e V (cid:3) Figure 7: Mass spectrum of the first KK excitations of the up- (left) and down-type(right) quarks. Black lines show the exact masses, while gray lines show the massesobtained by switching off the Yukawa couplings. The mixings of the mass eigenstatesare visualized by the bar charts at the bottom of each panel. See text for details.the corresponding matrices ∆ ( (cid:48) ) A . Their contributions to the Z -boson couplings in (121) arenegligible. In fact, these contributions are formally of O ( v /M ).It is instructive to compare the above exact results for the various mixing matrices withthe approximations (125) and (127) obtained using the ZMA. We find that in all cases theZMA reproduces the magnitudes and phases of the matrix elements with very good accuracy.Deviations typically arise at the percent level.While the various mixing matrices presented above refer to a particular point in parameterspace, they are nevertheless useful in order to obtain a picture of “typical” flavor-changingeffects that can be expected in the RS model. The input parameters have been chosen such thatthe resulting values of the fermion masses and CKM parameters are in good agreement withthe experimental values of these quantities. Apart from O (1) effects on the various entries,the freedom one has in varying these matrices is thus restricted to the reparametrizationtransformations (129) and (130). Finally, the overall scale of all flavor-changing effects inthe neutral-current sector can be tuned by changing the KK scale, since these effects areproportional to v /M . The equations derived in Section 3 allow us to study also the mass spectrum and mixingsof the KK excitations of the SM fermions. These mixings can give rise to flavor-violatingtransitions among the light SM quarks when inserted into loop diagrams. As an example,we study the first level of KK quarks numerically using the default parameters given earlierin this section and assuming M KK = 1 . Y u,d = 0 correspond to pure SU (2) L doublets and singlets, labeled by Q , Q , Q and u , c , t or d , s , b , respectively. Introducing the Yukawa couplings leads to mixings between fieldswith different flavor and SU (2) L quantum numbers, which are visualized by the bar charts atthe bottom of each panel. The area of each colored region is proportional to the square of theabsolute value of the corresponding entry in the mixing vectors a ( U,D )4 − and a ( u,d )4 − (altogether36 numbers), which appear in the KK decomposition (51) of the fermion fields.As the KK scale is much larger than the Higgs vacuum expectation value, one might expectthe flavor mixings between KK fermions to be small. To the contrary, however, one finds verylarge mixing effects especially in the down-type quark sector due to the near degeneracy of the5D bulk masses of the corresponding fermion fields. The mass splittings of the undisturbedKK states are typical of order 100 GeV, which is not large compared to v . Thus the Yukawacouplings generically induce O (1) mixings among the KK excitations of the same KK level.The generation mixing of KK modes provides an important example where the approachof treating the Yukawa couplings as a small perturbation is inadequate. On the other hand,we have checked that good approximations to the exact results for the masses and mixings ofKK fermions can be obtained by working with the undisturbed states, treating the Yukawacouplings as interactions, and diagonalizing the resulting mass matrices on a truncated basisof KK states [96]. Z b ¯ b Couplings
From (119) and (121), it follows that the flavor-diagonal Z -boson couplings to down-typequarks can be written in the form L (cid:51) (cid:16) √ G F m Z (cid:17) / (cid:20) m Z M (cid:18) L sin θ W − L (cid:19) + O (cid:18) m Z M (cid:19)(cid:21) Z µ × (cid:2)(cid:0) g dL (cid:1) ii ¯ d L,i γ µ d L,i + (cid:0) g dR (cid:1) ii ¯ d R,i γ µ d R,i (cid:3) , (169)where i = 1 , ,
3. For the light down and strange quarks the left- and right-handed couplingsare, to excellent approximation, given by the SM expressions, ( g dL ) ≈ − / θ W / g dR ) ≈ sin θ W /
3. For the bottom quark, the non-universal corrections in (121) canbe significant. Using the approximate expressions (125) and (127) valid in the ZMA, we canderive compact analytical expressions for these corrections. From the explicit results for themixing matrices collected in Appendix A, we see that to leading power in hierarchies only thetwo profiles F ( c Q ) and F ( c d ) and simple combinations of c Q , , c d , , and the elements of thedown-type Yukawa matrix appear. Denoting c b L ≡ c Q and c b R ≡ c d , we obtain g bL ≡ (cid:0) g dL (cid:1) → (cid:18) −
12 + sin θ W (cid:19) (cid:20) − m Z M F ( c b L )3 + 2 c b L (cid:18) L − c b L c b L ) (cid:19)(cid:21) + m b M (cid:34) − c b R (cid:18) F ( c b R ) − F ( c b R )3 + 2 c b R (cid:19) + (cid:88) i =1 , | ( Y d ) i | | ( Y d ) | − c d i F ( c b R ) (cid:35) , (170)48 bR ≡ (cid:0) g dR (cid:1) → sin θ W (cid:20) − m Z M F ( c b R )3 + 2 c b R (cid:18) L − c b R c b R ) (cid:19)(cid:21) − m b M (cid:34) − c b L (cid:18) F ( c b L ) − F ( c b L )3 + 2 c b L (cid:19) + (cid:88) i =1 , | ( Y d ) i | | ( Y d ) | − c Q i F ( c b L ) (cid:35) . Note that the non-universal corrections always reduce the couplings with respect to theirSM values in magnitude. Using the freedom of reparametrization invariance, one can rescale F ( c b L ), F ( c b R ), and the Yukawa couplings in such a way that the values of the quark massesand the CKM mixing angles remain unaffected, see (111) and (113). This has the effect ofredistributing contributions between the left-handed and right-handed couplings. In practice,however, the value of F ( c b L ) cannot be made too small if one wants to reproduce the largetop-quark mass with reasonable Yukawa couplings.The ratio of the width of the Z -boson decay into bottom quarks and the total hadronicwidth, R b , the bottom quark left-right asymmetry parameter, A b , and the forward-backwardasymmetry for bottom quarks, A ,b FB , are given in terms of the left- and right-handed bottomquark couplings as [97] R b = (cid:34) (cid:80) q = u,d [( g qL ) + ( g qR ) ] η QCD η QED (cid:2) (1 − z b )( g bL − g bR ) + ( g bL + g bR ) (cid:3) (cid:35) − ,A b = 2 √ − z b g bL + g bR g bL − g bR − z b + (1 + 2 z b ) (cid:18) g bL + g bR g bL − g bR (cid:19) , A ,b FB = 34 A e A b , (171)where η QCD = 0 . η QED = 0 . z b ≡ m b ( m Z ) /m Z = 0 . · − describes the effects of the non-zero bottomquark mass. To an excellent approximation we can neglect the RS contributions to the left-and right-handed couplings of the light quarks, g qL,R , and to the asymmetry parameter of theelectron, A e , and fix these quantities to their SM values. In what follows we will employ g uL = 0 . g uR = − . g dL = − . g dR = 0 . A e = 0 . g bL = − . g bR = 0 . R b = 0 . , A b = 0 . , A ,b FB = 0 . . (172)These SM expectations should be compared to the experimentally extracted values for thethree “pseudo observables”. They are [29] R b = 0 . ± . ,A b = 0 . ± . ,A ,b FB = 0 . ± . , ρ = . − . − . − .
08 1 .
00 0 . − .
10 0 .
06 1 . , (173)49 (cid:3) CL95 (cid:3)
CL99 (cid:3) CL (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) g Lb g R b (cid:2) (cid:2)(cid:3) (cid:3) CL95 (cid:3)
CL99 (cid:3) CL m h (cid:4) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) g Lb g R b Figure 8: Regions of 68% , 95% , and 99% probability in the g bL – g bR plane. Thehorizontal stripe in the left plot consists of a large number of points in parameterspace. The triangle and star in the right panel correspond to the results for ourreference RS point with M KK = 1 . M KK = 3 TeV, assuming a Higgs-bosonmass of m h = 400 GeV. The black dot is the SM expectation for the reference point,and the green (medium gray) line in the right panel indicates the SM predictions for m h ∈ [60 , ρ is the correlation matrix. We see that while the R b and A b measurements agreewithin +0 . σ and − . σ with their SM predictions, the A ,b FB measurement is almost − . σ away from its SM value. Whether this is an experimental problem, a statistical fluctuationor an effect of new physics in the bottom-quark couplings is up to date unresolved. In fact,the relative experimental error is much larger in A ,b FB than in R b and A b , where no anomaliesare observed. Furthermore, the value of the left-handed bottom-quark coupling, which isessentially determined by the measurement of R b ∝ ( g bL ) + ( g bR ) due to the smallness of g bR , shows no discrepancy. The data therefore invite an explanation in terms of a possibledeviation of the right-handed bottom-quark coupling from its SM value. This would affect A b and A ,b FB , which both depend linearly on the ratio g bR /g bL , more strongly than R b .The results of our fit to the Z → b ¯ b “pseudo observables” are shown in Figure 8. In theleft panel we have superimposed the RS predictions obtained for 3000 randomly generatedpoints in parameter space. It is evident that the RS prediction for g bL is always larger thanthe SM reference value indicated by the black dot, while g bR is essentially unaffected. Thisimplies that the RS corrections to the Z b ¯ b couplings necessarily shift the values g bL,R furtheraway from the best fit g bL = − . g bR = 0 . g bR needed to explain the anomaly in A ,b FB .The apparent large positive corrections to g bL imply that the R b , A b , and A ,b FB measurements For m h = 115 GeV the discrepancy in A ,b FB would amount to around − . σ . The tiny RS shifts in g bR are always negative. (cid:3) Y d (cid:4) (cid:2) (cid:2) M KK (cid:2) N Y (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) c b L c b R (cid:2)(cid:3) Y d (cid:4) (cid:2) (cid:2) M KK (cid:2) N Y (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) c b L c b R Figure 9: Regions of 99% probability in the c b L – c b R plane for M KK = 1 . M KK = 3 TeV (right). We set c Q , = c d , = − / N Y ≡ | ( Y d ) ij | / | ( Y d ) | = 1. Thecolored contours indicate the value of | ( Y d ) | necessary to reproduce the value of thebottom-quark mass. The RS results with (without) the m b /M terms are indicatedby bright (faint) colors. Only the parameter region inside the dashed rectangles leadsto consistent values for the quark masses and mixings. See text for details.impose stringent constraints on the parameter space of the RS scenario. The distribution ofpoints depends strongly on the bulk mass parameters, while the exact values of the KK scaleand the elements of the down-type Yukawa matrices have only a minor impact on the overallpicture. Notice that the dependence of g bL,R on c b L,R is much more pronounced than the oneon c Q , and c d , , because the parameters c b L,R enter (170) through their corresponding zero-mode profiles F ( c b L,R ). As a result, the allowed values of c b L,R are strongly constrained bythe Z → b ¯ b “pseudo observables”, while the bulk mass parameters c Q , and c d , are onlyweakly bounded. The former feature is illustrated in Figure 9, which shows the regions of99% probability in the c b L – c b R plane for M KK = 1 . M KK = 3 TeV (right).The colored contours in both plots indicate the magnitude | ( Y d ) | necessary to achieve thecorrect value of the bottom-quark mass. We find that under the restriction | ( Y d ) | <
12 theallowed c b L,R parameters all lie in the intervals c b L ∈ [ − . , − .
45] ( c b L ∈ [ − . , − . c b R ∈ [ − . , .
30] ( c b R > − .
60) for M KK = 1 . M KK = 3 TeV). Notice that increasingthe magnitude | ( Y d ) | decreases the available parameter space. Requiring a consistent fit ofall quark masses and CKM parameters restricts the allowed region in the c b L – c b R plane evenfurther. This feature is illustrated by the dashed rectangles in Figure 9, which represent theallowed c b L,R ranges that lead to a global χ / dof of better than 11.5/10 for | ( Y u,d ) ij | restrictedto the range [1 / , c u < / m b /M terms are indicated by bright(faint) colors. Comparing the allowed regions makes clear that the m b /M terms in (170),51hich result from the matrix elements ( δ D ) and ( δ d ) in (121), are numerically important,especially if | ( Y d ) | is allowed to take values larger than a few. Neglecting these contributionsis in general not justified. Notice also that the terms in (170) proportional to m Z /M dependlinearly on L , whereas the terms proportional to m b /M are independent of the logarithmof the warp factor. These latter terms thus cannot be removed by truncating the volume ofthe RS background. In turn, they are typically dominant for moderate and small values of L .After a proper adjustment of bulk mass parameters, the constraint on M KK coming fromthe combined R b , A b , and A ,b FB measurements is in general weaker than the one stemmingfrom the electroweak precision measurements, encoded in S and T . For example, for the RSreference input values spelled out in Section 6.3, we find M KK > . . (174)Like in the case of the S and T parameters, it turns out to be instructive to study theHiggs mass dependence of the observables in question. The leading logarithmic Higgs-masscorrections to the Z → b ¯ b “pseudo observables” are well approximated by∆ R b = 3 . · − ln m h m ref h , ∆ A b = − . · − ln m h m ref h , (175)∆ A ,b FB = − . · − ln m h m ref h . These relations have been derived with the help of
ZFITTER [70, 71]. Interestingly, theshifts in R b and A b are only moderate compared to the experimental errors, while the shiftin A ,b FB is rather pronounced, due to the strong Higgs-mass dependence of A e , and negative.These features allow to significantly improve the quality of the fit to the Z → b ¯ b “pseudoobservables” by taking large values of m h . For example, a Higgs-boson mass of m h = 400 GeVwould bring the predictions of g bL,R very close to the best fit value. This feature is illustrated bythe triangle and star in the right panel of Figure 8, which correspond to the predictions (170)for our two RS reference points. Numerically, we obtain g bL = − . g bR = 0 . g bL = − . g bR = 0 . M KK = 1 . M KK = 3 TeV). The line in the sameplot indicates the SM predictions for m h ∈ [60 , A ,b FB anomaly, sincein these setups the Higgs boson is expected to be heavy, which leads to a good agreementbetween Z → b ¯ b data and theory. Needless to say, even for a heavy Higgs boson the tightconstraints on c b L,R arising from the measurements of R b , A b , and A ,b FB persist.The strong bulk-mass parameter dependence of the RS corrections to the Z b ¯ b vertex can begreatly reduced [77] by using an embedding of the SM fermions into the custodially symmetric SU (2) L × SU (2) R model, under which the left-handed bottom quark is symmetric under theexchange of SU (2) L and SU (2) R [34]. The simplest implementation of this Z symmetry is The default flags of
ZFITTER version 6.42 are used, except for setting
ALEM=2 to take into account theexternally supplied value of ∆ α (5)had ( m Z ). SU (2) L × SU (2) R ,while the right-handed top quark is chosen to be a singlet. In these variations of the originalRS scenario, mixing between the Z boson and the KK excitations allows to explain the A ,b FB anomaly without affecting the agreement of the other precision electroweak observables withexperimental data for moderate KK masses of the order of 3 TeV [98]. t → cZ A particularly interesting class of FCNC processes in the RS model involves the flavor-violatingcouplings of the top quark. As the heaviest fermion in the SM, the top quark is located closestto the IR brane and hence couples most strongly to the KK excitations of the gauge bosons.Sizable flavor-changing effects involving the top quark are hence expected in the RS scenario.Because flavor transitions in the up-type quark sector are far less constrained by kaon and B -meson physics than those in the down-type quark sector, the presence of non-negligibleanomalous flavor-changing top-quark couplings is not ruled out experimentally. This makessearches for radiative and rare ∆ F = 1 processes involving the top quark unique probes of theRS framework.The flavor-changing couplings of quarks to the Z boson in (119) allow the top quark todecay via the process t → cZ . The branching ratio is given to excellent approximation by B ( t → cZ ) = 2 (1 − r Z ) (1 + 2 r Z )(1 − r W ) (1 + 2 r W ) × (cid:26) | ( g uL ) | + | ( g uR ) | − r c r Z (1 − r Z ) (1 + 2 r Z ) Re (cid:2) ( g uL ) ∗ ( g uR ) (cid:3)(cid:27) ≈ . (cid:104) | ( g uL ) | + | ( g uR ) | (cid:105) − .
048 Re (cid:2) ( g uL ) ∗ ( g uR ) (cid:3) , (176)where r i ≡ m pole i /m pole t , and for simplicity we have only kept terms up to first order in v /M and the charm-quark mass ratio r c ≈ . · − . The flavor-changing couplings are given by( g uL ) = − m Z M (cid:18) −
23 sin θ W (cid:19) (cid:104) L (∆ U ) − (∆ (cid:48) U ) (cid:105) −
12 ( δ U ) , ( g uR ) = m Z M
23 sin θ W (cid:104) L (∆ u ) − (∆ (cid:48) u ) (cid:105) + 12 ( δ u ) . (177)The branching ratio and the flavor-changing couplings relevant for the t → uZ decay areobtained from the above expressions by replacing subscripts 23 by 13 and m c by m u . Due tothe RS-GIM mechanism this decay is, however, strongly suppressed compared to t → cZ .Using the ZMA expressions (125) and (127), it is straightforward to derive that to leading53 (cid:3) (cid:2) CL CDFwithout Z (cid:3) bb (cid:2) CL ATLAS5 Σ ATLAS (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) M KK (cid:2) TeV (cid:3) (cid:2) (cid:4) t (cid:3) c Z (cid:5) (cid:2) (cid:2) CL CDFwith Z (cid:3) bb (cid:2) CL ATLAS5 Σ ATLAS (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) M KK (cid:2) TeV (cid:3) (cid:2) (cid:4) t (cid:3) c Z (cid:5) Figure 10: Branching ratio of the rare decay t → cZ as a function of M KK in theRS model. The red (dark gray) band is disfavored at 95% CL by the CDF searchfor t → u ( c ) Z . The dashed and dotted red (dark gray) lines indicate the expectedsensitivities of ATLAS for 100 fb − integrated luminosity. All scatter points reproducethe correct quark masses, mixing angles, and the CKM phase. In the left (right)plot points that violate the constraints from the Z → b ¯ b “pseudo observables” areincluded (excluded). The yellow (light gray) triangle and the orange (medium gray)stars represent the results for our RS reference points. See text for details.power in hierarchies the flavor-changing couplings inducing t → cZ decay take the form (cid:0) g uL (cid:1) → m Z M (cid:18) −
23 sin θ W (cid:19) F ( c Q ) F ( c Q )3 + 2 c Q ( Y u ) ( Y u ) (cid:18) L − c Q c Q ) (cid:19) − e − i ( φ − φ ) m c m t M F ( c u ) F ( c u ) (cid:20) − c u ( Y u ) ∗ ( M u ) ( Y u ) ∗ ( M u ) + 11 − c u ( Y u ) ∗ ( Y u ) ∗ (cid:21) , (cid:0) g uR (cid:1) → − e − i ( φ − φ ) m Z M
23 sin θ W F ( c u ) F ( c u )3 + 2 c u ( Y u ) ∗ ( Y u ) ∗ (cid:18) L − c u c u ) (cid:19) + m c m t M F ( c Q ) F ( c Q ) (cid:20) − c Q ( Y u ) ( M u ) ∗ ( Y u ) ( M u ) ∗ + 11 − c Q ( Y u ) ( Y u ) (cid:21) . (178)Here ( M u ) ij again denotes the minor of the up-type Yukawa matrix Y u , and the definitionsof the phase factors e iφ j can be found in (99).The RS predictions for B ( t → cZ ) as a function of M KK are shown in Figure 10. Each plotcontains 3000 randomly generated parameter points. The values of B ( t → cZ ) = 6 . · − and B ( t → cZ ) = 3 . · − corresponding to our RS reference point for M KK = 1 . M KK = 3 TeV are displayed by the triangle and stars. Notice that for M KK = 1 . Z → b ¯ b “pseudo54 (cid:3) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:3) (cid:2) (cid:2) (cid:2) (cid:2) t (cid:4) cZ (cid:3) P L R (cid:2) t (cid:4) c Z (cid:3) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:3) (cid:2) (cid:2) (cid:2) (cid:2) t (cid:4) cZ (cid:3) P L R (cid:2) t (cid:4) c Z (cid:3) Figure 11: Left-right polarization asymmetry of t → cZ as a function of the branchingratio of t → cZ . The results before and after removing the points that violate theconstraints from the Z → b ¯ b “pseudo observables” are shown in the left and rightpanel. Bins with the lowest density of points are colored light blue (light gray), whilethose with the highest density of points are colored yellow (light gray). The yellow(light gray) triangle and the orange (medium gray) stars represent the results for ourRS reference points. See text for details.observables”. The presently most precise experimental upper bound on FCNC t → u ( c ) Z decays stems from the CDF experiment and amounts to B ( t → u ( c ) Z ) < .
7% at 95% CL[99]. It is shown as a band. Notice that the recent CDF bound supersedes the 95% CL upperlimit B ( t → u ( c ) Z ) < .
7% [100] set by the L3 experiment from the non-observation ofFCNC single top-quark production. At the LHC, rare FCNC top-quark transitions can besearched for in top-quark production and decays. The best identification will be reached for t → u ( c ) Z → u ( c ) l + l − and t → u ( c ) γ . Simulation studies have been performed by both theATLAS [101] and the CMS [102] Collaboration. The minimum of B ( t → cZ ) allowing fora signal discovery with 5 σ significance with 100 fb − integrated luminosity is expected to be1 . · − at ATLAS. In the absence of a signal, the expected limit at 95% CL is 6 . · − .These sensitivities are indicated by the dashed and dotted lines in the panels.The distribution of scatter points in the plots of Figure 10 demonstrates that RS real-izations which lead to values of B ( t → cZ ) above 10 − are typically in conflict with theconstraints from R b , A b , and A ,b FB . This correlation arises from the presence of F ( c Q ) in(170) and (178). Making the profile F ( c Q ) bigger by locating ( t L , b L ) closer to the IR braneenhances B ( t → cZ ), but at the same time this leads to a positive shift in g bL , which worsensthe quality of the fit in the Z → b ¯ b sector. To elucidate this feature, we depict in Figure 11the left-right polarization asymmetry P LR ( t → cZ ) = Γ( t L → c L Z ) − Γ( t R → c R Z )Γ( t L → c L Z ) + Γ( t R → c R Z ) = | ( g uL ) | − | ( g uR ) | | ( g uL ) | + | ( g uR ) | (179)55s a function of B ( t → cZ ). The left and right panels represent the RS results before andafter removing the points that fail to reproduce R b , A b , and A ,b FB within 99% probability.The density of points in each bin is indicated by the color shading. The triangle and starsindicate the values P LR ( t → cZ ) = 0 .
44 and P LR ( t → cZ ) = 0 .
41 corresponding to our RSreference point with M KK = 1 . M KK = 3 TeV. The distribution of points, with thelargest concentration arising close to P LR ( t → cZ ) = ±
1, can be understood by recallingthat only the product | F ( c u ) F ( c Q ) | is constrained by the requirement to reproduce theobserved top-quark mass. From the structure of (178) and the fact that the contributionsproportional to m Z and m c m t are typical of the same size, it then follows that the magnitudesof ( g uL,R ) are strongly anti-correlated. Cases in which | ( g uL ) | ≈ | ( g uR ) | , corresponding to aleft-right polarization asymmetry close to zero, thus require a tuning of the elements of theup-type Yukawa matrix. Notice also that on average the coupling ( g uL ) tends to be larger inmagnitude than ( g uR ) due to the ratio of prefactors (1 / − / θ W ) / ( − / θ W ) ≈ − . P LR ( t → cZ ) = +1.The constraints from R b , A b , and A ,b FB restrict more strongly the magnitude of the left-handed than that of the right-handed coupling and so tend to exclude points close to P LR ( t → cZ ) = +1 and branching ratios above 10 − . This reduces the density of points correspond-ing to almost purely left-handed Z tc interactions, as can be seen from the right plot. Inconsequence, for the phenomenologically interesting range of B ( t → cZ ) above 10 − , theright-handed coupling more frequently gives the dominant contribution to the t → cZ decaywidth. The same conclusion has been reached in [42], although the correlation between the t → cZ and the Z → b ¯ b transition has not been studied there. The constraints from R b , A b ,and A ,b FB have also not been considered in [53], which finds that B ( t → cZ ) is dominated bythe left-handed coupling. The wide spread of points in our plots suggests however that the RSframework per se does not lead to a firm prediction of the chirality of the Z tc interactions.Since the RS predictions for B ( t → cZ ) are typically below the expected LHC sensitivity,the prospects of angular analyses [103] that would allow to determine the chirality of the Z tc interactions seem in any case quite challenging.Important constraints on the structure and size of flavor-changing top-quark couplingsalso arise from B physics. In particular, the constraints from B → X s γ and B → X s l + l − decay disfavor flavor-changing top-quark interactions at a level observable at the LHC dueto most of the operators containing left-handed up- or charm-quark fields [104]. It is easyto understand that in the minimal RS framework the correlation between the Z s ¯ b couplings( g dL,R ) affecting B → X s l + l − and ( g uL,R ) entering t → cZ provides the strongest constrainton the potential size of B ( t → cZ ). Corrections due to anomalous Z t ¯ t and W tb couplingsenter B → X s l + l − first at the loop level and are therefore subleading. Within experimentaland theoretical errors, the Z s ¯ b couplings ( g dL,R ) are constrained at the level of a few 10 − by B → X s l + l − decay. We have verified that the vast majority of points that satisfy the Z → b ¯ b It is not possible to attach a statistically rigorous meaning to the shown distributions, since they dependon the way in which the parameter space is sampled and the results are binned. In our analysis we neglect semileptonic four-fermion operators like (¯ bs )(¯ ll ) arising from integrating out theKK modes of the Z boson. This is justified, since these contributions are suppressed by the logarithm of thewarp factor with respect to the Z s ¯ b couplings. B ( B → X s l + l − ) that lie safely within the allowed range. Thisfeature is expected in models in which the modification of the flavor structure is connected tothe third generation [105]. Our analysis shows that values of the t → cZ branching ratio of upto 10 − are not in conflict with any currently available information on Z -penguin transitions.The branching ratio of t → uZ is typically suppressed by two orders of magnitude com-pared to the one of t → cZ . This suppression factor is readily understood from the scaling | F ( c Q ) | / | F ( c Q ) | ∼ λ of the fermion profiles and the smallness of the up-quark mass, whichrenders effects due to mixing of zero and KK fermion modes negligibly small. The structureof (178) then implies that generically | ( g uL ) | (cid:29) | ( g uR ) | , so that the chirality of the Z tu interactions is preferable left-handed. Unfortunately, in view of the smallness of B ( t → uZ ),it seems impossible to test this prediction at the LHC.We conclude the discussion of FCNC top-quark decays to Z bosons by noting that in RSmodels with custodial symmetry, the observed correlations between the t → c ( u ) Z , Z → b ¯ b ,and b → sZ transitions can be less pronounced. This is a consequence of the left-rightexchange symmetry, which allows to suppress tree-level corrections to the left-handed Z b ¯ b vertex, while at the same time leaving the t → c ( u ) Z and b → sZ transitions unprotected[34]. We thus expect that the RS predictions for B ( t → c ( u ) Z ) are typically larger in thesescenarios and captured by the plots on the left-hand side of Figures 10 and 11. As a result,the experimental prospects for observing the decays t → c ( u ) Z seem more favorable. Similarstatements apply to the case of the t → c ( u ) h processes discussed in the next section. The general form of the interactions of fermions with the Higgs boson has been given in (136).We will express the couplings ( g qh ) mn in this relation in units of quark masses divided by theHiggs vacuum expectation value. After adjusting the phase of the SM quark fields accordingto the standard CKM phase convention, we obtain for our default RS parameters( g uh ) ij = √ m u i m u j v . − . e − i . ◦ . e i . ◦ − . · − e i . ◦ .
000 0 . e i . ◦ − . · − e − i . ◦ − . e i . ◦ . ij , ( g dh ) ij = √ m d i m d j v . − . · − e i . ◦ . e i . ◦ − . · − e − i . ◦ .
000 0 . e i . ◦ . · − e − i . ◦ . · − e − i . ◦ . ij , (180)with m q i given in (157). The elements ( g qh ) i deviate notably from the SM expressions ( g qh ) = m q /v and ( g qh ) = ( g qh ) = 0. In particular, the ht ¯ t coupling can receive sizable correctionsof order − ht ¯ t coupling would lead to a reduction of the rate for the gluon-fusion process gg → h via a top-quark loop, resulting in a suppression of the Higgs-bosonproduction cross section. Using the di-photon or the W ± W ∓ , Z Z channels at the LHC, thetheoretical estimate of the Higgs-boson production cross section [106, 107] may be confrontedwith experiment. As both experiment and theory are limited to an accuracy of about 10%, it57 (cid:3) without Z (cid:2) bb (cid:3) CL LHC3 Σ LHC (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) M KK (cid:2) TeV (cid:3) (cid:2) (cid:4) t (cid:2) c h (cid:5) (cid:2) with Z (cid:2) bb (cid:3) CL LHC3 Σ LHC (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) M KK (cid:2) TeV (cid:3) (cid:2) (cid:4) t (cid:2) c h (cid:5) Figure 12: Branching ratio of t → ch as a function of M KK in the RS model. Thedashed and dotted red (dark gray) lines indicate the expected sensitivities of the LHC.In the left (right) plot points that violate the constraints from the Z → b ¯ b “pseudoobservables” are included (excluded). The yellow (light gray) triangle and the orange(medium gray) stars represent the results for our RS reference points. See text fordetails.will be challenging to detect the predicted suppression effects of the ht ¯ t coupling. A detailedstudy of the Higgs-boson production cross section in warped 5D scenarios, as performed in[108, 109], is beyond the scope of this article.When kinematically accessible, the couplings of (136) allow for the flavor-changing de-cay t → ch . Including terms up to first order in the charm-quark mass, the correspondingbranching ratio takes the form B ( t → ch ) = 2 (1 − r h ) r W (1 − r W ) (1 + 2 r W ) g (cid:26) | ( g uh ) | + | ( g uh ) | + 4 r c − r h Re (cid:2) ( g uh ) ( g uh ) (cid:3)(cid:27) , (181)where as before r i ≡ m pole i /m pole t , and g is the SU (2) L gauge coupling. In our numericalanalysis of the t → ch branching ratio we will use r h = 0 .
87, corresponding to a Higgs-bosonmass m h = 150 GeV.The RS predictions for B ( t → ch ) as a function of M KK are shown in Figure 12 for 3000randomly generated parameter points. The values of B ( t → ch ) = 2 . · − and B ( t → ch ) =1 . · − corresponding to our default RS point for M KK = 1 . M KK = 3 TeV aredisplayed by the triangle and stars. The LHC is expected to give 3 σ evidence for B ( t → ch )larger than 6 . · − or set a limit of 4 . · − with 95% CL if the decay is not observed [110].These limits are indicated by the dashed and dotted lines in the panels. Since B ( t → ch )barely reaches values of 10 − once the Z → b ¯ b constraints are enforced, a detection of aRS signal for the t → ch decay will be taxing at the LHC. The branching ratio of t → uh is suppressed relative to the one of t → ch by more than an order of magnitude, so that a58etection of t → uh at the LHC seems impossible in the minimal RS framework. Due to theweaker correlation between t → c ( u ) h and Z → b ¯ b in RS models with custodial symmetry,the prospects for observing the decays t → c ( u ) h are likely to be better in warped scenarioswith extended gauge symmetry. We have presented a detailed and comprehensive discussion of tree-level flavor-changing effectsin the RS model with gauge and matter fields in the bulk and the Higgs sector localized on theIR brane. We have derived exact expressions for the masses of the gauge bosons and fermionsand their KK excitations, as well as for the profiles of these fields along the extra dimension,by solving the bulk equations of motions with appropriate boundary conditions taking intoaccount the couplings to the Higgs sector. In all cases analytical expressions are obtained forthe bulk profiles of the fields in the effective 4D theory, including their normalization.For the gauge fields we have discussed the KK decomposition in a covariant R ξ gauge. Asingle gauge-fixing term in the 5D action suffices to define the theory completely. The resultingFaddeev-Popov ghost Lagrangian contains a ghost field of the usual form for each KK mode.We have derived compact expressions accounting for sums over the KK towers of gauge bosonsarising in tree-level diagrams at low energy. For fermions the KK decomposition is performedincluding the mixing between different generations sourced by the Yukawa couplings. Ourexact results allow us to study flavor mixing not only among the lowest-lying (SM) fermions,but also among their KK excitations. We have also commented on dimensional-analysis con-straints on the scale of the 5D Yukawa matrices, finding that these could in principle be largerthan commonly assumed in the literature. Potential phenomenological implications of thisobservation will be discussed elsewhere [55, 56].Our exact treatment of the field equations, which avoids a perturbative expansion in powersof v /M , extends previous analyses of flavor physics in the RS models in several ways. Animportant observation of our analysis is that the field equations in the fermion sector areincompatible with the naive orthonormality relations usually imposed on the bulk profiles. Theproperly generalized normalization conditions allow us to derive exact and compact expressionsfor flavor-changing effects arising from the mixing of SU (2) L doublet and singlet fermions viatheir KK excitations. These effects give rise to FCNC couplings of the Z and Higgs bosons,which so far have not been studied systematically in the literature.The hierarchies observed in the fermion spectrum and the CKM matrix can be explainednaturally in terms of anarchic 5D Yukawa matrices and wave-function overlap integrals. Wehave emphasized the invariance of the results for these masses and mixings under two typesof reparametrization transformations of the 5D bulk mass parameters and Yukawa matrices.We have also pointed out that the KK excitations of the SM fermions have generically largemixings between different generations as well as between SU (2) L singlets and doublets. Thereason is that without the Yukawa couplings the spectra of the KK towers of different fermionfields are nearly degenerate, so that even small Yukawa couplings can lead to large mixingeffects, which could be an important source of flavor-changing loop effects.The heart of our analysis is a comprehensive, quantitative study of the structure of gauge59nd Higgs boson interactions with SM fermions and their KK excitations. In particular,we have investigated in detail the flavor-changing couplings of the W ± and Z bosons toSM fermions. They give rise to the leading flavor-changing effects in ∆ F = 1 weak decayprocesses. The two main sources of non-standard flavor violations are the deviation of the Z -boson profiles along the extra dimension from a constant, and the SU (2) L singlet admixturein the wave function of the left-handed SM fermions. While the second effect is suppressed bytwo powers of light quark masses and has thus frequently been neglected in the literature, wehave shown that it is parametrically and numerically as important as the first one.Performing a careful analysis of electroweak precision observables, including the S and T parameters and the Z b ¯ b couplings, we have found that the simplest RS model containing onlySM particles and their KK excitations is consistent with all experimental bounds for reasonablylow KK scales if one allows for a heavy Higgs boson ( m h (cid:46) SU (2) R symmetry and an extended fermionspectrum. The reason is that the Higgs boson is naturally heavy in the RS framework, sinceradiative corrections to the Higgs-boson mass scale like the fourth power of the UV cutoff onthe IR brane. This gives rise to a large negative contribution to the T parameter, which canbe compensated in the RS model without custodial symmetry by a large positive tree-levelcontribution.We have concluded this work with a rather detailed study of tree-level flavor-changingeffects, which includes analyses of the non-unitarity of the quark mixing matrix, anomalousright-handed couplings of the W ± bosons, tree-level FCNC couplings of the Z and Higgsbosons, the rare decays t → c ( u ) Z and t → c ( u ) h , and the flavor mixing among KK fermions.The analytical and numerical results obtained in this paper form the basis for general calcula-tions of flavor-changing processes in the RS model and its extensions, including loop-mediateddecays. A comprehensive study of flavor effects in the B -, D -, and K -meson systems will bepresented in a companion paper [56]. Acknowledgments
We are grateful to Babis Anastasiou, Martin Bauer, Roberto Contino, Georgi Dvali, ChristopheGrojean, Leonhard Gr¨under, Jos´e Santiago, and Andreas Weiler for useful discussions, and toDonatello Dolce and Adam Falkowski for sharing unpublished notes and private communica-tions. One of us (MN) likes to thank the Kavli Institute for Theoretical Physics at UC SantaBarbara for hospitality and support during two visits while this paper was being written.60
Textures of Mixing Matrices
It is useful to have explicit expressions for the various mixing matrices at hand, which include the relevant combinations ofYukawa couplings. We work in the ZMA and use the relations given in (125), (127), and (134) along with the scaling relationsderived in Section 4.1. For simplicity, we evaluate polynomial factors involving the c i parameters by setting c i = − /
2, whichis a good approximation for all light fermions. It would be straightforward to reinstate the correct expressions if desired.To leading order in hierarchies, the matrices V L,R parametrizing the couplings of the W ± bosons to fermions in (132) aregiven by( V L ) ij (cid:39) (cid:104) ( M d ) ( M d ) − ( M u ) ( M u ) (cid:105) F Q F Q (cid:104) ( M u ) ( M u ) + ( Y d ) ( Y d ) − ( M u ) ( Y d ) ( M u ) ( Y d ) (cid:105) F Q F Q (cid:104) ( M u ) ∗ ( M u ) ∗ − ( M d ) ∗ ( M d ) ∗ (cid:105) F Q F Q (cid:104) ( Y d ) ( Y d ) − ( Y u ) ( Y u ) (cid:105) F Q F Q (cid:104) ( M d ) ∗ ( M d ) ∗ + ( Y u ) ∗ ( Y u ) ∗ − ( M d ) ∗ ( Y u ) ∗ ( M d ) ∗ ( Y u ) ∗ (cid:105) F Q F Q (cid:104) ( Y u ) ∗ ( Y u ) ∗ − ( Y d ) ∗ ( Y d ) ∗ (cid:105) F Q F Q ij , ( V R ) ij (cid:39) m u i m d j M F Q ( M d ) ( M d ) F Q F Q ( Y d ) ( Y d ) F Q F Q ( M u ) ∗ ( M u ) ∗ F Q F Q (cid:104) ( M u ) ∗ ( M d ) ∗ ( M u ) ∗ ( M d ) ∗ (cid:105) F Q (cid:104) ( Y d ) ( Y d ) + ( Y d ) ( M u ) ∗ ( Y d ) ( M u ) ∗ (cid:105) F Q F Q ( Y u ) ∗ ( Y u ) ∗ F Q F Q (cid:104) ( Y u ) ∗ ( Y u ) ∗ + ( Y u ) ∗ ( M d ) ( Y u ) ∗ ( M d ) (cid:105) F Q F Q (cid:104) ( Y u ) ∗ ( Y d ) ( Y u ) ∗ ( Y d ) + ( Y u ) ∗ ( Y d ) ( Y u ) ∗ ( Y d ) (cid:105) F Q − F Q ij . (A1)The corresponding expressions for the matrices ∆ ( (cid:48) ) A entering the Z -boson couplings to fermions in (121) read(∆ ( (cid:48) ) Q ) ij (cid:39) (cid:104) | ( M q ) | | ( M q ) | + | ( M q ) | | ( M q ) | (cid:105) F Q − (cid:104) ( M q ) ( M q ) + ( Y q ) ∗ ( M q ) ( Y q ) ∗ ( M q ) (cid:105) F Q F Q ( M q ) ( M q ) F Q F Q − (cid:104) ( M q ) ∗ ( M q ) ∗ + ( Y q ) ( M q ) ∗ ( Y q ) ( M q ) ∗ (cid:105) F Q F Q (cid:104) | ( Y q ) | | ( Y q ) | (cid:105) F Q − ( Y q ) ( Y q ) F Q F Q ( M q ) ∗ ( M q ) ∗ F Q F Q − ( Y q ) ∗ ( Y q ) ∗ F Q F Q F Q ij , (∆ ( (cid:48) ) q ) ij (cid:39) e − i ( φ i − φ j ) (cid:104) | ( M q ) | | ( M q ) | + | ( M q ) | | ( M q ) | (cid:105) F q − (cid:104) ( M q ) ∗ ( M q ) ∗ + ( Y q ) ( M q ) ∗ ( Y q ) ( M q ) ∗ (cid:105) F q F q ( M q ) ∗ ( M q ) ∗ F q F q − (cid:104) ( M q ) ( M q ) + ( Y q ) ∗ ( M q ) ( Y q ) ∗ ( M q ) (cid:105) F q F q (cid:104) | ( Y q ) | | ( Y q ) | (cid:105) F q − ( Y q ) ∗ ( Y q ) ∗ F q F q ( M q ) ( M q ) F q F q − ( Y q ) ( Y q ) F q F q F q ij . (A2) inally, the matrices δ A entering the same relations are given by( δ Q ) ij (cid:39) e − i ( φ i − φ j ) m q i m q j M F q ( M q ) ∗ ( M q ) ∗ F q F q ( Y q ) ∗ ( Y q ) ∗ F q F q ( M q ) ( M q ) F q F q (cid:104) | ( M q ) | | ( M q ) | (cid:105) F q (cid:104) ( Y q ) ∗ ( Y q ) ∗ + ( Y q ) ∗ ( M q ) ( Y q ) ∗ ( M q ) (cid:105) F q F q ( Y q ) ( Y q ) F q F q (cid:104) ( Y q ) ( Y q ) + ( Y q ) ( M q ) ∗ ( Y q ) ( M q ) ∗ (cid:105) F q F q (cid:104) | ( Y q ) | | ( Y q ) | + | ( Y q ) | | ( Y q ) | (cid:105) F q − F q ij , ( δ q ) ij (cid:39) m q i m q j M F Q ( M q ) ( M q ) F Q F Q ( Y q ) ( Y q ) F Q F Q ( M q ) ∗ ( M q ) ∗ F Q F Q (cid:104) | ( M q ) | | ( M q ) | (cid:105) F Q (cid:104) ( Y q ) ( Y q ) + ( Y q ) ( M q ) ∗ ( Y q ) ( M q ) ∗ (cid:105) F Q F Q ( Y q ) ∗ ( Y q ) ∗ F Q F Q (cid:104) ( Y q ) ∗ ( Y q ) ∗ + ( Y q ) ∗ ( M q ) ( Y q ) ∗ ( M q ) (cid:105) F Q F Q (cid:104) | ( Y q ) | | ( Y q ) | + | ( Y q ) | | ( Y q ) | (cid:105) F Q − F Q ij . (A3)In these formulae F Q i ≡ F ( c Q i ) and F q i ≡ F ( c q i ). The extra terms in the 33 entries of the matrices V R and δ Q,q should bekept in cases where F Q ,q = O (1). The phase factors e iφ j are defined in (99). B Reference Values for SM Parameters
The central values and errors of the quark masses used in our analysis are m u = (1 . ± .
0) MeV , m c = (550 ±
40) MeV , m t = (140 ±
5) GeV ,m d = (3 . ± .
0) MeV , m s = (50 ±
15) MeV , m b = (2 . ± .
1) GeV . (B1)They correspond to MS masses evaluated at the scale M KK = 1 . λ = 0 . ± . , A = 0 . ± . , ¯ ρ = 0 . +0 . − . , ¯ η = 0 . ± . . (B2)The central values and errors for the parameters entering our analysis of electroweak precision observables are [29, 79]∆ α (5)had ( m Z ) = 0 . ± . , m Z = (91 . ± . ,α s ( m Z ) = 0 . ± . , m t = (172 . ± .
4) GeV . (B3)We refer to the central values for these quantities as SM reference values. Unless noted otherwise, the reference value for theHiggs-boson mass is m h = 150 GeV. Mixing Matrices with Default Parameters
Here we collect results for some of the flavor-mixing matrices, which were not given in themain text. These results are obtained using the default parameters in (155) and (156), alongwith the KK scale M KK = 1 . W ± bosons to right-handed SM quark fields in (131), is V R = . · − . · − − . · − − . · − e − i . ◦ − . · − e − i . ◦ . · − − . · − e i . ◦ − . · − e − i . ◦ . · − . (C1)Potential effects of the largest coupling, ( V R ) , have been discussed in Section 6.3.1.The results for the matrices ∆ (cid:48) , which enter the Z -boson couplings to fermions in (119),read ∆ (cid:48) D = 10 − .
298 0 . e − i . ◦ − . e i . ◦ . e i . ◦ . − . e − i . ◦ − . e − i . ◦ − . e i . ◦ . , ∆ (cid:48) d = 10 − . . e − i . ◦ . e i . ◦ . e i . ◦ . . e i . ◦ . e − i . ◦ . e − i . ◦ . , ∆ (cid:48) U = 10 − .
782 1 . e − i . ◦ − . e i . ◦ . e i . ◦ . − . e − i . ◦ − . e − i . ◦ − . e i . ◦ . , ∆ (cid:48) u = 10 − . − . e − i . ◦ . e − i . ◦ − . e i . ◦ . − . e − i . ◦ . e i . ◦ − . e i . ◦ . . (C2)Based on the results (125) obtained in the ZMA, these matrices are expected to be approxi-mately equal to the corresponding matrices ∆ A . This expectation is supported by the com-parison with the results given in Section 6.3.2.63or completeness, we also list the mixing matrices (cid:15) ( (cid:48) ) A . They read (cid:15) D = 10 − .
117 0 . e i . ◦ − . e i . ◦ . e − i . ◦ . − . e i . ◦ − . e − i . ◦ − . e − i . ◦ . , (cid:15) d = 10 − . − . e − i . ◦ . e i . ◦ − . e i . ◦ . . e − i . ◦ . e − i . ◦ . e i . ◦ . , (cid:15) U = 10 − .
232 0 . e − i . ◦ − . e i . ◦ . e i . ◦ . − . e i . ◦ − . e − i . ◦ − . e − i . ◦ . , (cid:15) u = 10 − . − . e − i . ◦ . e i . ◦ − . e i . ◦ .
089 143 . e − i . ◦ . e − i . ◦ . e i . ◦ . , (C3)and (cid:15) (cid:48) D = 10 − . . e i . ◦ − . e i . ◦ . e − i . ◦ . − . e i . ◦ − . e − i . ◦ − . e − i . ◦ . , (cid:15) (cid:48) d = 10 − . − . e − i . ◦ . e i . ◦ − . e i . ◦ . . e − i . ◦ . e − i . ◦ . e i . ◦ . , (cid:15) (cid:48) U = 10 − .
168 0 . e − i . ◦ − . e i . ◦ . e i . ◦ . − . e i . ◦ − . e − i . ◦ − . e − i . ◦ . , (cid:15) (cid:48) u = 10 − . − . e − i . ◦ . e i . ◦ − . e i . ◦ .
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