Relaxing Constraints from Lepton Flavor Violation in 5D Flavorful Theories
aa r X i v : . [ h e p - ph ] J u l September 18, 2018UMD-PP-09-030
Relaxing Constraints from Lepton Flavor Violation in D Flavorful Theories
Kaustubh Agashe Maryland Center for Fundamental Physics, Department of Physics, University of Maryland,College Park, MD 20742, USA
Abstract
We propose new mechanisms for ameliorating the constraints on the Kaluza-Klein (KK) mass scale fromcharged lepton flavor violation in the framework of the Standard Model (SM) fields propagating in a warpedextra dimension, especially in models accounting for neutrino data. These mechanisms utilize the extendedfive-dimensional (5 D ) electroweak gauge symmetry h SU (2) L × SU (2) R × U (1) X i which is already stronglymotivated in order to satisfy electroweak precision tests in this framework. We show that new choices ofrepresentations for leptons under this symmetry (naturally) can allow small mixing angles for left-handed(LH) charged leptons and simultaneously large mixing angles for their SU (2) L partners, i.e., the LH neutrinos,with the neutrino data being accounted for by the latter mixings. Enhancement of charged lepton flavorviolation by the large mixing angle observed in leptonic charged currents, which is present for the minimalchoice of representations where the LH charged lepton and neutrino mixing angles are similar, can thus beavoided in these models. This idea might also be useful for suppressing the contributions to B d, s mixing inthis framework and in order to suppress flavor violation from exchange of superpartners (instead of from KKmodes) in 5 D “flavorful supersymmetry” models. Additionally, the less minimal representations can providecustodial protection for shifts in couplings of fermions to Z and, in turn, further suppress charged leptonflavor-violation from tree-level Z exchange in the warped extra-dimensional scenario. As a result, ∼ O (3)TeV KK mass scale can be simultaneously consistent with charged lepton flavor violation and neutrino data,even without any particular structure in the 5 D flavor parameters in the framework of a warped extradimension. email: [email protected] Introduction
The framework of a warped extra dimension was proposed in order to provide a solution to thePlanck-weak hierarchy problem of the Standard Model (SM) [1]. With the SM gauge and fermionsfields propagating in the extra dimension [2, 3, 4], it can also account for the flavor hierarchy ofthe SM via extra-dimensional profiles for SM fermions [3, 4]. Inherent to this approach is flavorviolation from the resulting non-universal couplings of SM fermions to the Kaluza-Klein (KK)modes of the SM fields which are the new particles present in this framework [5]. In spite of ananalog of Glashow-Iliopoulos-Maiani (GIM) mechanism being built-in to this framework [4, 6, 7],the lower limits on the mass scale of the KK gauge bosons from the flavor violation in the quark sector can still be ∼ O (5 −
10) TeV , depending on the details of the model [10, 11, 12] (see also[13, 14]). Whereas, flavor violation in the charged lepton sector requires a gauge KK mass scaleat least as large as ∼ O (5) TeV without consideration of neutrino data [15]. It turns out that, in minimal models, enhancement of charged lepton flavor violation by the large mixing required inorder to account for neutrino data results in constraints being even stronger than ∼ O (5) TeV [16].Such a large gauge KK scale might imply a tension with a resolution of the Planck-weak hier-archy problem of the SM which requires a KK scale ∼ TeV. Also, signals from direct productionof these KK modes at the Large Hadron Collider (LHC) (including upgrades) are then extremelychallenging (if not unlikely). Therefore, it is very interesting to study mechanisms to ameliorateconstraints from this flavor violation, thus allowing for a lower gauge KK mass scale. Recently,five-dimensional (5 D ) flavor symmetries (for both quark and lepton sectors) have been suggestedfor this purpose [17, 13, 18, 16, 19] such that a gauge KK scale as low as ∼ O (3) TeV might beallowed.In this paper, we propose alternative mechanisms in order to suppress charged lepton flavorviolation. The idea is to use new (less minimal) representations for leptons under the extendedelectroweak (EW) 5 D gauge symmetry – such an extended symmetry is in any case stronglymotivated in order to suppress contributions to electroweak precision tests, in particular, the T parameter [21]. We demonstrate that A clarification about notation is in order here. The uncertainty in the bounds on KK mass scale from flavorviolation denoted by the symbol “ ∼ ... ” comes from effects of modifications to the minimal model such as brane-localized kinetic terms for bulk fields [8] or replacement of the endpoint of the extra dimension by a “soft wall” [9].Such variations are present even for limits on KK mass scale from electroweak precision tests. On the other hand,the symbol “ O ( ... )” refers to uncertainties in the bounds from flavor violation due to presence of O (1) factors in 5 D Yukawa (which is an inherent feature of 5 D flavor “anarchy”) and due to the presence (typically) of more than oneterm (of similar size, but uncorelated) in the flavor-violating amplitude. Contributions to electroweak precision testsare not very sensitive to the latter types of effects and hence the “ O ( ... )” factor is absent in that case. Very recently in reference [20], such non-minimal representations for leptons were studied in the context ofgauge-Higgs unification, but their relevance for suppression of charged lepton flavor violation was not discussed. certain such choices of representations can allow small and large mixing angles to naturally co-exist for left-handed (LH) charged leptons and neutrinos, respectively, in spite of thembeing SU (2) L partners.Thus it is possible (unlike in the minimal models) to avoid the large mixing angles required toexplain the neutrino data from exacerbating charged lepton flavor violation. As a result, evenafter including neutrino data, the constraint on the gauge KK scale can relax to the ∼ O (5) TeVvalue in the case without neutrino data. This new idea can suppress contributions to B d, s mixing(which, however, are not the dominant constraints from flavor violation in quark sector) as well byallowing LH down-type and up-type quark mixing angles to be parametrically different. Similarlyit can also be applied to other extra-dimensional models (which account for flavor hierarchy viaprofiles for SM fermions in the extra dimension) in order to suppress flavor violation (especially incharged lepton sector). Independently of the decoupling of large neutrino mixings from charged lepton sector, we showthat • such new representations can result in a custodial symmetry which protects shifts in couplingof SM fermions to Z in the framework of a warped extra dimension [24].Hence flavor-violating Z couplings to leptons and the resulting tree-level Z exchange contributionsto processes such as µ to e conversion in nuclei and µ → e can be suppressed . Combining theabove two ideas, we show that it is possible to reduce the lower limit on gauge KK scale fromcharged lepton flavor violation (including neutrino data) down to ∼ O (3) TeV from > O (5) TeV inthe minimal model.The outline of the rest of the paper is as follows. We begin with an overview of the frameworkof warped extra dimension (including both the quark and lepton sectors) and a qualitative outlineof the problem of charged lepton flavor violation and the solutions proposed in this paper. Thenwe present quantitative estimates for charged lepton flavor violation in section 3, including howcharged lepton flavor violation is enhanced by large neutrino mixing. The central observations ofthis paper are in the next two sections: in section 4, we show how to decouple mixings of theLH charged lepton and neutrino sectors, with many example representations for leptons under theextended 5 D EW gauge symmetry: the general idea is illustrated in Fig. 2. In section 5 we considera choice of profiles in order to obtain large neutrino mixing with mild tuning and then discuss thecustodial protection mechanism which is critical to suppressing flavor-violating couplings to Z with Recently [22], it was shown that with a profile for the SM Higgs in the extra dimension (but still peaked near theendpoint of the extra dimension) [23] (instead of a δ -function localized Higgs) and with SM neutrinos being Dirac, itis possible to achieve a similar “decoupling” of LH neutrino and charged lepton mixing angles. The role of such a custodial symmetry in suppressing flavor violation in the quark sector was discussed in [11, 25]. D “flavorful supersymmetry”(SUSY). A slice of anti-de Sitter (AdS) space in 5 D [1] provides a solution to the Planck-weak hierarchyproblem of the SM. Basically, the warped geometry implies that the UV cut-off of the effective 4 D theory depends on location in the extra dimension ( y ): M D eff. ∼ M D e − ky , where k is the AdScurvature scale, e − ky is called the warp factor and M D is the fundamental 5 D mass scale. The 4 D graviton (zero-mode of the 5 D gravitational field) is automatically localized near the y = 0 end ofthe extra dimension (hence called the Planck/UV brane). Suppose the Higgs sector is taken to belocalized near the other end of the extra dimension (called the TeV/IR brane): y = πR , where R is the proper size (or radius) of the extra dimension – such a localization happens automaticallyif Higgs is the 5 th component of a 5 D gauge field [26]. Then the Planck-weak hierarchy can beexplained by a mild hierarchy between the AdS curvature radius ( ∼ /k which is taken to be ∼ /M D ) and R : M Higgs or M weak ∼ M D e − kπR . Note that M D ∼ M P l ∼ GeV is requiredin order to reproduce the observed (4 D ) Planck scale due to warp factor being 1 at the locationof the 4 D graviton. In turn, such a mild hierarchy between the proper size of the extra dimensionand curvature radius: kπR ∼ log ( M P l / TeV) ∼
30 can be stabilized by suitable mechanisms [27].It is also that interesting that, based on the AdS/CFT correspondence [28], such a scenario isconjectured to be dual to SM Higgs being a composite of TeV-scale strong dynamics [29, 26].
Such a framework can also provide a solution to the flavor hierarchy of the SM if the SM fermionsarise as zero-modes of fermions propagating in the extra dimension [3, 4]. Namely, the profiles ofthe SM fermions in the extra dimension are then controlled by their 5 D masses. The crucial featureis that small variations in the 5 D masses enable the SM fermions to have profiles which are peakedeither near the Planck or TeV branes or are flat. This feature results in small/large/intermediatecouplings of the SM fermions to the SM Higgs (which is localized near the TeV brane), simplybased on overlaps of profiles in the extra dimension, i.e., without any hierarchy in the fundamental(5 D ) parameters (Yukawa couplings and 5 D fermion masses).SM gauge fields must also then originate as zero-modes of 5 D fields (“SM in the bulk”) [2, 4]4 it turns out that the SM gauge fields have a flat profile in the extra dimension. In additionto the zero-modes, the 5 D fields have other, non-trivial excitations in the extra dimension (calledKaluza-Klein or KK modes) which appear as heavier particles from the 4 D point of view. In thewarped case, these KK modes turn out to be automatically localized near the TeV brane and havemasses ∼ ke − kπR , i.e., at the ∼ TeV-scale. Thus all SM particles (except perhaps the SM Higgs)have KK modes in this scenario. So, contributions from these KK modes to precision tests of theSM can constrain this scenario. In particular, electroweak precision tests (EWPT) can be undercontrol, using custodial symmetries to protect contributions to the T parameter [21] and the shiftin Zb ¯ b coupling [24], even with KK masses of a few (or several) TeV [21, 30, 31]. More relevant to this paper, there is flavor violation from exchange of KK modes which necessarilyhave non-universal couplings to the SM fermions (given that the flavor hierarchy is accounted forby SM fermions’ non-universal profiles) [5]. However, there is an analog of the GIM mechanismof the SM which is automatic in this scenario since the non-universalities in the couplings of SMfermions to KK modes are of size of 4 D Yukawa couplings (due to KK’s having similar profile toHiggs) [4, 6, 7]. However, even in the presence of this RS-GIM mechanism, recently [10, 11] (seealso [13, 14]) it was shown that the constraint on the KK mass scale from tree-level contributionsof KK gluon to ǫ K is quite stringent. In particular, for the model with the SM Higgs (strictly)localized on the TeV brane, the limit on the KK gluon mass scale from ǫ K is ∼ O (10) TeV forthe smallest allowed 5 D QCD coupling obtained by loop -level matching to the 4 D coupling withnegligible tree-level brane kinetic terms. On the other hand, for larger brane kinetic terms suchthat the 5 D QCD coupling (in units of the AdS curvature scale, k ) is ∼ π , the lower limit onKK gluon mass scale increases to ∼ O (40) TeV. In addition, the constraint on the KK gluon massscale is weakened as the size of the 5 D Yukawa (in units of k ) is increased. However, this directionreduces the regime of validity of the 5 D effective field theory (EFT): the above limits on KK gluonmass scale are for the size of 5 D Yukawa such that about two KK modes are allowed in the 5 D EFT.Whereas, with a profile for the SM Higgs in the extra dimension (but which is still peaked nearTeV brane [23], called a “bulk Higgs”) and choosing the smallest allowed 5 D QCD coupling and twoKK modes in the 5 D EFT, it was demonstrated in reference [12] that ∼ O (3) TeV KK gluon massscale can be consistent with ǫ K . However, in the “two-site” model [32] (which is an economicalapproach to studying this framework by restricting to the SM fields and their first KK excitations), It was also argued in the same reference that a larger size of the 5 D QCD coupling might in fact conflict with5 D perturbativity.
5t was also shown in reference [12] that there is a “tension” between satisfying constraints form ǫ K and BR( b → sγ ) (the latter observable being sensitive to loop effects of KK modes). Thus, the limiton the mass scale for the new particles (assuming the heavy fermions and gauge bosons have samemass) must actually be a bit larger, namely, ∼ O (5) TeV to be consistent with this combination ofconstraints. Hence, it was also suggested reference [12] that the 5 D models with a bulk Higgs canallow a similar, i.e., ∼ O (5) TeV, gauge KK scale to be consistent with the entire body of data onflavor violation in the quark sector.Furthermore, 5 D flavor symmetries in the quark sector can add more structure to the 5 D model,for example, by relating the 5 D (or bulk) fermion masses to the 5 D Yukawa couplings and/or byenforcing degenerate bulk masses [17, 13, 19]. Such a reduction in the number of flavor parametersresults in suppressed quark sector flavor violation. Also, by lowering the UV-IR hierarchy, i.e., kπR ,it is possible to lower the gauge KK scale allowed by quark sector flavor violation [33], although inthis paper we will always assume Planck-weak hierarchy, i.e., kπR ∼
30. Further studies of flavorviolation, especially experimental signals, appear in references [34, 35].In this paper, we focus instead on flavor violation and hierarchy of masses in the lepton sector.With out consideration of neutrino data, it was shown in reference [15] that the constraint fromcharged lepton flavor violation on gauge KK mass scale is ∼ O (5) TeV – such a strong constraintis mainly due to a tension between the two processes µ to e conversion in nuclei (which occurs attree-level in this framework) and loop-induced µ → eγ . Note that this constraint was obtained forhierarchies in charged lepton masses being explained by the choice of hierarchical profiles near theTeV brane for both right-handed (RH) and LH charged leptons so that both RH and LH chargedlepton mixing angles (given by ratio of respective profiles at the TeV brane) were set to be small(roughly square root of ratio of charged lepton masses). However, including neutrino data, two new and distinct issues come up (see also related discussionin reference [16]):(i)
Enhancement due to large mixing angle : With the simplest representations under theextended bulk EW gauge symmetry, i.e., SU (2) L × SU (2) R × U (1) X (such an extension istypically required to satisfy EWPT) and a (strictly) TeV brane-localized Higgs, the chargedlepton and neutrino (Dirac) masses originate from the same LH lepton bulk profiles evaluatedat the TeV brane. Thus the mixing angles for LH charged leptons and neutrino are similarand, in turn, a combination these two mixing angles is what enters charged current leptoninteractions. So, this mixing is required to be large in order to explain neutrino oscillationdata. 6uch large LH charged lepton mixing results in an enhancement of charged lepton flavorviolation relative to without considerations of neutrino data as in [15] – as mentioned above,in reference [15] both
RH and LH charged lepton mixing was set to be small. Thus, the gaugeKK scale is constrained to be larger than ∼ O (5) TeV in order to be consistent with all thedata, i.e., charged lepton flavor violation and neutrino mixings.(ii) Flat profiles for mild tuning : For the case of a brane-localized Higgs, large LH neutrinomixing clearly requires non-hierarchical (i.e., with ∼ O (1) ratios) profiles for LH leptons nearthe TeV brane where the 4 D Yukawa coupling originates. However, if the LH lepton profilesare peaked near the Planck brane, i.e., exponentially suppressed near the TeV brane, then itis clear that we need to tune the bulk masses (which control the exponentials) to be (almost)degenerate in order for the profiles near the TeV brane to be non-hierarchical.
If we require no tuning of bulk masses , then we might be forced to choose close-to-flat profilesfor all generation LH leptons such that the profiles near the TeV brane can be non-hierarchicalwith only a mild tuning of bulk masses. However, such a choice results in a larger couplingof SM leptons to gauge KK modes (which are localized near IR brane) relative to the caseof without considerations of neutrino data, i.e., where lepton profiles – both LH and RH –are peaked near the Planck brane (motivated by smallness of charged lepton masses). Inturn, the larger couplings of leptons to KK modes enhance charged lepton flavor violation viatree-level Z exchange further, i.e., in addition to the effect of large LH charged lepton mixingangles mentioned in point (i) above.Invoking 5 D flavor symmetries is one way to solve the above problems [18, 16]. In particular,even if LH lepton profiles are peaked near the Planck brane, the (almost) degenerate LH lepton bulkmasses required to give non-hierarchical profiles near TeV brane (and hence large mixing) are thenenforced by a symmetry. Also, the resulting universal couplings of LH charged leptons to gaugeKK modes (GIM mechanism) suppress LH charged lepton flavor violation from zero-KK gaugeboson mixing: see top right-hand side of Fig. 1. Independently, such symmetries can relate bulkmasses to 5 D Yukawa couplings (just like for quarks discussed above) thus reducing the number offlavor parameters (i.e., adding structure). Hence LH charged lepton flavor violation from zero-KK fermion mixing (see top left-hand side of Fig. 1) and similarly µ → eγ are suppressed as well.Alternatively [22], for Higgs with a profile in the bulk (but still peaked near the TeV brane)[23] and with neutrinos being Dirac particles, neutrino masses of the observed size (i.e., requiredto account for neutrino oscillations) can arise from overlap near the Planck brane, whereas chargedlepton masses originate (as usual) from the overlap of profiles near the TeV brane. Then themuch smaller neutrino masses (relative to charged lepton) and large vs. small mixing in neutrino7nd charged lepton sectors arise naturally. The point is that the LH lepton profiles can be non-hierarchical and large near the Planck brane (giving large mixing for LH neutrinos and ultra-smallmasses due to small Higgs profile at the Planck brane), while simultaneously being small andhierarchical near the TeV brane (giving small mixing and small masses for LH charged leptons). D gauge representations for leptons In this paper, we propose an alternative to both the above ideas to suppress charged lepton flavorviolation while obtaining large neutrino mixings. We still consider neutrino masses originating fromnear the
TeV brane (say, Higgs is localized on the TeV brane or it leaks into the bulk, but notsufficiently for Dirac neutrino masses from overlap near the Planck brane to be of the observed size).The new idea is to use less minimal representations under the SU (2) R × U (1) X gauge symmetry.In particular, there are two new mechanisms as follows.(a) Decoupling large neutrino mixing from charged lepton masses : the idea is that LHlepton zero-mode for each generation can arise as a combination of zero-modes from 2 different5 D multiplets: see Fig. 2. Such a scenario allows LH mixing angles to be parametricallydifferent for charged leptons vs. neutrinos since the two mass matrices (and hence mixingangles) can originate from the two different components of the LH lepton zero-mode. Inparticular, mixing angles can then be small for charged vs. large for neutrinos. This novelpossibility prevents large mixing angles in leptonic charged currents from infiltrating bothtree-level Z exchange (giving µ to e conversion in nuclei) and loop-induced dipole operators(giving µ → eγ ).(b) Custodial protection : Independently, some choices of representations under the extendedbulk EW gauge symmetry can result in a custodial symmetry for the shift in the couplingsof leptons to Z (similar to one used to suppress shift in Zb ¯ b [24]). Such a symmetry canthen suppress the flavor-violating couplings of leptons to Z and hence charged lepton flavorviolation via tree-level Z exchange. Such a suppression is especially desirable if we choose(close-to-) flat profiles in order to generate large neutrino mixings without tuning (as men-tioned in point (ii) in section 2.3). In such a case, the enhanced coupling of charged leptonsto KK modes is still problematic for charged lepton flavor violation (as discussed in point (ii)in section 2.3), even if we obtain small charged lepton mixing angles using the idea in point(a) above.It is in fact possible to combine the above two features for some choice of representations of leptonsunder the 5 D EW gauge symmetry, resulting in ∼ O (3) TeV KK scale being consistent with charged8epton flavor violation and large neutrino mixings (without any particular structure in the 5 D flavorparameters). Various cases utilizing the above two ideas: (a) and/or (b) are listed in table 1. In this section, we collect formulae for charged lepton flavor violation valid for the general case andthen specialize to the models with neutrino masses. Since we are mainly concerned with parametriceffects and mechanisms, estimates of these effects (i.e., formulae valid up to O (1) factors) willsuffice for our purpose. For more detailed formulae, the reader is referred to previous studies (seereferences [7, 15] for example). In addition to the effects of KK modes summarized below, thereare also operators induced by physics at the cut-off of the 5 D theory. For simplicity, we assumehere that we have a bulk Higgs (but with a profile which is peaked near the TeV brane), wheresuch cut-off effects can be shown to be smaller than KK-induced ones (see references [7, 15, 16]).We first perform a KK decomposition for SM gauge and fermion fields setting the Higgs vevto zero. The 4 D Yukawa coupling, i.e., the coupling of SM Higgs to two zero-mode fermions (saycharged leptons), is given by: Y ( c e L , c e R ) ∼ Y √ kf ( c e L ) f ( c e R ) (1)where Y is 5 D Yukawa coupling of mass dimension − / and f ’s are ratio of zero-mode and KKprofiles near the TeV brane: f ( c ) ≈ q(cid:0) c − (cid:1) e kπR (1 − c ) for c > / q kπR for c = 1 / q(cid:0) − c (cid:1) for c < / c is the 5 D mass for the corresponding 5 D fermion in units of k . We can show that the KKYukawa coupling, i.e., the coupling of Higgs to two KK fermions, is given by: Y KK ∼ Y √ k (3)which (along with the above definition of f ’s) explains Eq. (1). Similarly, the coupling of Higgs toone zero-mode and one KK fermion is given by Y mixed ( c ) ∼ Y √ kf ( c ) (4)where c is that of the zero-mode fermion. Finally, the c -dependent part (which is the one relevantfor flavor-violation) of the coupling of two zero-mode fermions to gauge KK mode is given by g KK ( c ) ∼ g SM √ kπRf ( c ) , (5) due to SM Higgs being in the bulk. D and 5 D gauge couplings at the tree-level and without anybrane-localized kinetic terms for gauge fields. We can use the above forumlae to estimate chargedlepton flavor violation in this framework which is of two types: tree-level and loop processes whichwe now review in turn. The tree-level flavor-violation occurs dominantly via Z exchange with the following flavor-violating Z couplings to leptons (we focus on µ and e in this paper, but the formulae can be easily generalizedto the case of τ ’s): δg Zµ L e L ∼ h M Z M KK × kπR + (cid:16) Y √ k × v (cid:17) M KK ih f ( c µ L ) i ( U L ) (6)where 1st term originates from mixing between zero and KK gauge modes and 2nd term fromfermion zero-KK mode mixing, both effects being induced by the Higgs vev: see Fig. 1. Finally,( U L ) denotes the mixing angle of the transformation from weak to mass basis for the chargedleptons.In particular, the assumption of a structureless or anarchic Y (which we will make throughoutthis paper) implies that the mixing angles between charged leptons are given by ratio of profiles atthe TeV brane (i.e., f ’s) ( U L ) ij ∼ f ( c e L i ) f (cid:0) c e L j (cid:1) for i < j (7)Similar formulae apply for δg Zµ R e R and RH charged lepton mixing.In the special case of LH and RH charged lepton profiles being similar, i.e., hierarchies in chargedlepton masses being explained equally by ratios of RH and LH profiles at the TeV brane, we findthat both mixing angles are small and given by ( U L, R ) ∼ p m e /m µ (based on Eqs. (1) and (7)).We then obtain δg Zµ L e L , δg Zµ R e R ∼ h (cid:18) M Z M KK × kπR (cid:19) Y µ Y √ k + Y µ Y √ kv M KK ir m e m µ (8)Note that there are flavor- preserving shifts in couplings to Z which are given by similar formulae(except there are no mixing angles involved here). We assume small brane-localized kinetic terms for 5 D fields so that the KK fermion and KK gauge masses are(almost) the same. (0) vv v vvµ (0) L e (0) L f ( c µ L ) Y √ kf ( c e L ) Y √ k HY √ kµ (0) R e (0) L e ( n ) L i e ( n ) R j f ( c µ R ) Y √ k f ( c e L ) Y √ kµ (0) L e (0) L g Z √ kπRf ( c µ L ) ( U L ) g Z √ kπRZ ( n ) Z (0) µ (0) L e (0) L g Z / √ kπRf ¯ fZ ( n ) g Z √ kπR × f ( c µ L ) ( U L ) Figure 1:
Flavor violating couplings to Z generated by zero-KK mode fermion mixing (top left-hand side) and by zero-KK mode gauge mixing (top right-hand side), ∆ F = 1 4 -fermion operatorsgenerated by exchange of gauge KK modes (with out mixing with the zero-mode, bottom left-handside) and dipole operators generated by Higgs-KK fermion loops (bottom right-hand side). .1.1 Direct KK Z exchange The coefficient of the 4-fermion operator ∼ µ L γ µ e L ¯ f γ µ f (where f = quark, lepton) generated bydirect exchange of KK Z (i.e., without mixing with zero-mode Z ) is given by (see Fig. 1) A KK Z (cid:0) µ L → e L f ¯ f (cid:1) ∼ g Z M KK h f ( c µ L ) i ( U L ) (9)where we have used the result that the flavor- preserving coupling of KK Z is ∼ g SM / √ kπR andsimilarly for µ R → e R f ¯ f .Comparing this effect to the one from Z exchange (based on Eq. 6), we see that the direct KK Z exchange is suppressed by kπR ∼ log ( M P l / TeV). However, as mentioned earlier, we will invokecustodial symmetry to protect flavor violation from Z couplings, whereas direct KK Z exchange isnot suppressed by this mechanism and thus might become relevant in these cases. The coefficient of dipole operator: e F µν µ L, R σ µν e R, L induced by loops of KK fermions and Higgs(including longitudinal
W/Z ), as in Fig. 1 , is given by A ( µ R → e L γ ) ∼ (cid:16) Y √ k (cid:17) π m µ M KK ( U L ) (10)and similarly for µ L → e R γ .Again, in the case of LH and RH being similar, we find A ( µ R → e L γ ) , A ( µ L → e R γ ) ∼ (cid:16) Y √ k (cid:17) π m µ M KK r m e m µ (11)Note that there is some tension between tree-level and loop processes from the size of Y in thesense that the former (1st term in Eq. (8)) is enhanced for small Y while the latter (Eq. (11))is suppressed in this limit. Without considerations of neutrino data (in particular, not taking intoaccount the large charged current mixing which is a combination of LH charged lepton and neutrinomixing angles), we can assume LH and RH charged lepton profiles are similar, i.e., both sets ofprofiles are hierarchical at the TeV brane and mixing angles are small as in Eqs. (8) and (11). Thisis the case studied in reference [15] with the result that the least constrained scenario (i.e., lowest KK photon will also induce similar effects. And, in the models with extended EW gauge symmetry, there isan addition neutral gauge boson tower (denoted by Z ′ ), i.e., the combination of the U (1) subgroup of SU (2) R and U (1) X which is orthogonal to the hypercharge gauge symmetry, U (1) Y . However, flavor- preserving couplings of Z ′ to light SM fermions which are localized near the Planck brane are suppressed compared to the coupling to KK Z –roughly the former couplings are of size given by 4 D Yukawa couplings. It turns out that the loops with KK
W/Z or transverse SM
W/Z and KK fermions are approximately alignedwith 4 D Yukawa and hence do not contribute to µ → eγ [7, 15].
12K scale) is with Y √ k ∼ O (1) which still requires ∼ O (5) TeV gauge KK mass scale in orderto be consistent with charged lepton flavor violation data. It turns out that the flavor- preserving shifts in Z couplings to leptons are then quite safe. Having estimated that ∼ O (5) TeV gauge KK mass scale can be consistent with charged leptonflavor violation with out considering neutrino masses, we next discuss how incorporating neutrinodata affects these estimates. It is usually assumed that LH profile (at the TeV brane) governingcharged lepton mass is the same as that for neutrino mass (for each generation) because LH leptonzero-mode originates from a single 5 D multiplet, i.e., f ( c e L i ) = f ( c ν L i ) ≡ f ( c L i ). Clearly, alongwith the assumption of a anarchic Y , the mixing angles (appearing in the bi-unitary transformationto go from weak to mass basis) for LH charged leptons and neutrinos are then of the same order(but not exactly the same) in these minimal models. The reason for this feature is that the mixingangles are dictated by the ratios of profiles of the three L zero-modes near the TeV brane: see Eq.(7). In turn, the neutrino oscillation data (i.e., large mixing in leptonic charged currents which is acombination of LH charged lepton and neutrino mixing) then requires this LH lepton mixing angleto be large.Thus we make the following change compared to the case without neutrino masses consideredin reference [15]: ( U L ) ∼ p m e /m µ →∼ O (1), which must result from no hierarchies in LH leptonprofiles near the TeV brane, i.e., f ( c L ) ∼ f ( c L ) ∼ f ( c L ). Thus, once we include neutrinodata, it seems that LH and RH profiles cannot be chosen to be similar for charged leptons. Inturn, no hierarchies in LH charged lepton profiles at the TeV brane implies that the hierarchiesin charged lepton masses are then explained entirely by hierarchies in RH charged lepton profilesat the TeV brane, resulting in RH charged lepton mixing actually being smaller than in the caseassumed in reference [15]: ( U R ) ∼ p m e /m µ → m e /m µ (based on Eqs. (1) and (7)).In short, with the above changes for mixing angles in the estimates for charged lepton violationfrom sections 3.1 and 3.2, we find that the “best” case, i.e., with lowest KK mass scale, allowed bycharged lepton flavor violation and taking into account the constraints from flavor- preserving shifts δg Ze R i e R i , δg Ze L i e L i , δg Zν L i ν L i < ∼ a few 0 .
1% is the following: • M KK ∼ O (10) TeV for Y √ k ∼ O (0 . f ( c τ R ) ∼ O (1) and f ( c L i ) ∼ O (0 . Note that this is the limit on KK scale obtained by considering only one term at a time in the flavor-violatingamplitude (from among several uncorrelated terms of similar size), whereas some of the limits quoted in reference[15] were based on the combined effect of all terms in this amplitude (for a certain choice of relative phases betweenthe various terms). We assume these f ’s are not exactly equal since that would require a tuning of c ’s.
13o that c L i > /
2. Thus, the LH lepton profiles are peaked near the Planck brane so that we doneed to choose c L i ’s to be close to each other in order to achieve the (exponentially suppressed)profiles near the TeV brane being non-hierarchical: see Eq. (2) (we will return to this issue later).In more detail (this discussion is an elaboration of point (i) of section 2.3 and will be usefullater), there are more than one “count” of enhancement of charged lepton flavor violation once weinclude neutrino masses relative to the case without neutrino masses:I For µ R → e L γ , we have enhancement from ( U L ) ∼ p m e /m µ →∼ O (1), although µ L → e R γ is suppressed compared to the case without neutrino masses due to ( U R ) ∼ p m e /m µ →∼ m e /m µ . There is a similar enhancement and suppression for the two tree-level contributions,i.e., δg Zµ L e L and δg Zµ R e R , respectively.II (A) Another count of enhancement (relative to case without neutrino masses) for δg Zµ L e L comesdue to three f ( c L i )’s being similar, i.e., f ( c L ) in Eq. (6) is clearly dictated by m τ (insteadof depending only on m µ earlier) since it is now (roughly) similar to f ( c L ) and hence canbe larger.II (B) Moreover, we might try to choose smaller Y in order to to suppress µ R → e L γ (see Eq. (10)),keeping several TeV KK mass scale (in the light of point I above). Such a smaller Y impliesthat, in order to obtain correct m τ , f ( c L ) (and hence f ( c L ) also), in turn, might have tobe larger than in the case without neutrino masses in reference [15].Of course, we are free to choose RH and LH charged lepton profiles to be different (unlike the caseconsidered in reference [15]), in particular, we can increase f ( c τ R ) in order to make f ( c L ) smallerwhile keeping m τ fixed. Hence δg Zµ L e L can be smaller, avoiding the enhancements in point II (A)and (B) above. However, a too large f ( c τ R ) is constrained by δg Zτ R τ R < ∼ a few 0 . f ( c τ R ) and Y such that constraint on M KK is the same from threeobservables: δg Zτ R τ R , A ( µ R → e L γ ) and δ Zµ L e L (it can be checked that the other processes – bothflavor-violating and flavor-preserving – are more easily satisfied and so are not the bottlenecks).It is such an analysis which shows that the lowest allowed KK scale is O (10) TeV (as mentionedabove). Clearly, the “cornering” involving the various observables discussed above can be simply avoidedif the LH profiles (at the TeV brane) which govern the charged lepton and neutrino masses, and f ( c L ) – and hence f ( c L , ) (up to ∼ O (1) factor) – is then fixed by m τ . SU (2) L partners) butremarkably it is possible as follows! The central idea is that the SM SU (2) L doublet LH lepton( l (0) ) (for one generation) is actually a combination of zero-mode SU (2) L doublets from two 5 D multiplets with different profiles such that the charged lepton masses originate from one componentof this zero-mode, whereas obtaining neutrino masses requires using the other component. Althoughwe focus on leptons here, a similar argument can apply to quarks in order to obtain parametricallydifferent mixing angles for LH down-type vs. up-type quarks. Let us see how to implement this idea in detail. We will begin with a discussion of the general casewhich will enable us to see how to apply it to other extra-dimensional models and also to the quarksector. There are three main ingredients of this idea (which is summarized in Fig. 2):(1) Suppose the 5 D gauge symmetry is extended beyond the SM gauge symmetry and is reducedto the SM gauge symmetry by boundary conditions at the Planck brane (or equivalently by alarge scalar vev on the Planck brane). In other words, the gauge symmetry of the 4 D effectivetheory (at the level of zero-modes) is only the SM symmetry, but it is a subgroup of the 5 D gauge symmetry.(2) Consider two 5 D fermion multiplets, L e and L ν , which transform differently under the 5 D gauge symmetry (and hence cannot mix in the bulk/on the TeV brane). Moreover, thesetwo multiplets contain zero-modes (to begin with: see later) – denoted by l (0) e, ν , respectively –which transform like LH leptons (i.e., identically) under the SM EW gauge symmetry. Hencethese two zero-modes can mix on the Planck brane ( only ) since the Planck brane respectsonly the SM (and not the full 5 D ) gauge symmetry.Specifically, one combination of the two zero-modes is given a (Planck-scale) mass with afermion localized on the Planck brane, l ′ R i (effectively this combination of the two 5 D mul-tiplets has Dirichlet boundary condition on the Planck brane): L UV brane ∋ l ′ R i (cid:16) sin α i l (0) e i − cos α i l (0) ν i (cid:17) (12)The orthogonal combination of the two zero-modes is left over as the only massless mode andis then identified with the SM LH lepton: l (0) i = cos α i l (0) e i + sin α i l (0) ν i (13) In fact, two different 5
D SU (2) L multiplets have already been used in references [31, 10, 19] in order to obtainup and down-type quark masses, but the implication for decoupling the down-type quark mixing angles from theup-type was not specifically considered in these references. The gauge couplings ofthe SM fermion to leading order (i.e., couplings to the gauge zero-mode) are obviously notaffected by such a combination of the fermion zero-modes.(3) Moreover, the representations of the RH charged leptons and neutrinos under the 5 D gaugesymmetry are chosen to be such that their couplings to Higgs – localized near the TeV brane– must involve the two different components of the l (0) . The reason for these new “selectionrules” is that the Higgs couplings (in general, all bulk and TeV brane interactions) respectthe full 5 D gauge symmetry (which is, again, larger than the SM one). Therefore, chargedlepton and neutrino masses depend on the different profiles of the two components of l (0) ,giving different LH mixing angles.Note that it is the enlarged 5 D symmetry which forces this “decoupling” of LH profiles involvedin the charged lepton masses from those involved in the neutrino masses – obviously the SM/4 D symmetry would allow charged lepton and neutrino masses to proceed via the same component ofthe l (0) . Also, the Higgs couplings must satisfy the larger 5 D gauge symmetry even in the minimal case where we require (for simplicity) that the SM LH lepton originates as zero-mode of a single D field – it is just that in this case these selection rules then get translated into specific representationsfor the RH charged leptons and neutrinos under the 5 D gauge symmetry. SU (2) L × SU (2) R × U (1) X Specifically, consider SU (2) L × SU (2) R × U (1) X as the 5 D EW gauge symmetry with U (1) Y ,being a combination of U (1) X and U (1) subgroup of SU (2) R , i.e., Y = T R + X . As alreadymentioned, such an extension is motivated by satisfying EWPT, in particular, the constraint fromthe T parameter. (However, this idea can be generalized to other extended 5 D gauge symmetries,such as a grand-unified one.) The SM Higgs transforms as ( , ) , where 1st/2nd symbol in ( ... ) isthe representation under SU (2) L,R symmetry and the subscript denotes the charge under U (1) X .The two different 5 D multiplets which will constitute the l (0) transform as L e : ( , r L e ) X E and L ν : ( , r L ν ) X N , respectively. Note that, in general, r L e , ν = so that each multiplet can contain more than one SU (2) L doublet. We must choose the various charges such that Y = T R + X = − / Y for the SM LH lepton) for one SU (2) L doublet contained in each of the two multiplets, In any case, we can show that such effects not significant. For the mixing of l (0) ν components of different generationson the Planck brane, this conclusion is due to either to custodial protection for the resulting flavor-violating couplingsto Z or to the choice f ( c L ν ) ≪ l (0) e components which can be shown to be equivalent to mixing via Planck brane localized kinetic terms. Such mixingappears even in the minimal models (and even without consideration of neutrino masses) where LH lepton arisesfrom a single 5 D field. Flavor violation due to such kinetic terms (even if they are ∼ O (1), i.e., comparable to bulkcontributions) can also be shown to be small. x µ ✻ x µ ✲ yy = 0Planck/UV braneSM gaugesymmetry ❍❍❍❍❥ y = πR TeV/IR brane ✛ ✲ extended 5 D gauge symmetry l (0) ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ cos α ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ sin α L e E HY E ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ NL ν HY N PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP
Figure 2:
The mechanism for decoupling LH charged lepton and neutrino mixing angles. Thesingle lines denote mixing of the zero-modes from the bulk SU (2) L doublet multiplets, L e, ν onthe Planck brane (which respects only the D /SM gauge symmetry) and double lines denote theircouplings (which respect the enlarged D gauge symmetry) to Higgs, H and the bulk SU (2) L singletmultiplets, E and N . e and L ν . Moreover, we choose only these two components of the two multiplets to have zero-modes, i.e., to begin with, we choose Neumann boundary condition on both branes only for thecorresponding 5 D fields. Thus these two zero-modes correspond to the l (0) e, ν in the general discussionabove. Any extra “would-be” SU (2) L doublet zero-modes from the rest of the multiplets can beprojected out using Dirichlet boundary condition for the corresponding 5 D fields on the Planckbrane. Next, these two zero-modes mix as in Eqs. (12) and (13) since on the Planck braneonly U (1) Y is preserved so that finally we are left with only one SU (2) L massless doublet (pergeneration). Equivalently, ultimately only the combination of the two 5 D multiplets L e and L ν inEq. (13) has Neumann boundary condition on the Planck brane.The SM e R can arise as zero-mode of a single D multiplet (denoted by E ) and ν R as zero-mode of a different (single) 5 D multiplet (denoted by N ). As discussed above, we must chooserepresentations for the RH charged lepton and neutrino 5 D multiplets under the bulk EW gaugesymmetry, ( , r E ) X E and ( , r N ) X N (respectively), such that charged lepton and neutrino massesmust proceed via the l (0) e and l (0) ν components of l (0) . Schematically, we desiregeneral case : Y E : ( , r L e ) X E ( , r E ) X E ( , ) Y N : ( , r L ν ) X N ( , r N ) X N ( , ) (14)Thus, we require r L e × r E ∋ L ν × r N ∋ (15)so that E can couple to L e and Higgs (and similarly for N ), but X E = X N or r L e × r N ∋6 , r L ν × r E ∋6 (16)so that E cannot couple to L ν and Higgs (and similarly for N ). Note that these Higgs couplingsmust preserve the enlarged, i.e., SU (2) L × SU (2) R × U (1) X , bulk gauge symmetry, and not justthe SM symmetry. Hence, if Eq. (16) is satisfied, then charged lepton mass cannot proceed viathe l (0) ν component of l (0) and vice versa. Clearly, then the hierarchies in charged lepton andneutrino masses are set by the hierarchies in profiles (at the TeV brane) of ( l (0) e , e (0) R ) and ( l (0) ν , ν (0) R ), respectively. In particular, large (small) LH mixing desired for charged leptons (neutrinos) isachieved via small (large) hierarchies in the profiles at the TeV brane of the l (0) e ( l (0) ν ) componentsof the SM LH lepton. Clearly we cannot do such a splitting for e L and ν L due to SU (2) L symmetry being preserved at the zero-mode/4 D level (of course before Higgs acquires a vev). As for the case of L e, ν , in general, we have r E, N = so that there are extra components (other than the SM e R and ν R ) in the 5 D E and N multiplets. In fact, it is possible that the 5 D multiplet E has a component withquantum numbers of ν R and vice versa. “Unwanted” zero-modes for such components will have to be projected outusing Dirichlet boundary condition on the Planck brane. Again, only SM gauge symmetry is preserved on the Planckbrane so that such “splitting” of multiplets can be realized. .2.1 X = 1 / B − L )We will now discuss explicit examples. Begin with the minimal case: X = 1 / B − L ) and onlyone 5 D LH lepton multiplet, L : ( , ) − / (of course this case will have the constraints discussedin section 3.3). The RH charged leptons and neutrinos can be obtained from different ( , ) − / D multiplets (labeled with superscripts e and ν below) with the extra states in each multiplethaving no zero-modes due to Dirichlet boundary condition on the Planck brane. Charged leptonand Dirac neutrino masses then both arise fromCase (0) : Y E and Y N : ( , ) − / ( , ) e , ν − / ( , ) (17)giving similar (and hence large) mixing angle for LH charged leptons and neutrinos and resulting inthe KK mass limit ∼ O (10) TeV. Recall that this choice of parameters additionally requires tuningof c L i ’s in order to obtain large mixings (as mentioned earlier).Note, however, that, even with the choice X = 1 / B − L ), all that we require is T R =0 , − / , +1 / L , E and N , respectively, i.e., we are not forced to choose L to be singlet of SU (2) R or E , N to be doublets of SU (2) R – it suffices to choose integer and half-integer spinrepresentations of SU (2) R , respectively, for them. Thus we could instead choose L e : ( , ) − / , L ν : ( , ) − / , E : ( , ) − / and N : ( , ) − / in order to satisfy the 2nd condition in Eq. (16)for different mixing angles (even though X E = X N in this case). So, we haveCase (1) : Y E : ( , ) − / ( , ) − / ( , ) Y N : ( , ) − / ( , ) − / ( , ) (18) X = 1 / B − L )In general, X = 1 / B − L ) such that even X E = X N is possible. For example, L e : ( , ) − / , L ν : ( , ) , E : ( , ) − / and N : ( , ) so thatCase (2) : Y E : ( , ) − / ( , ) − / ( , ) Y N : ( , ) ( , ) ( , ) (19)As a final example of decoupling of LH charged and neutrino mixing angles, we choose L e : ( , ) − , L ν : ( , ) , E : ( , ) − and N : ( , ) so thatCase (3) : Y E : ( , ) − ( , ) − ( , ) Y N : ( , ) ( , ) ( , ) (20)The motivation for the above two cases will be discussed later. The above idea of decoupling large neutrino mixing from LH charged lepton sector resolves onlypart of the problem discussed in section 3.3, i.e., count (I) only, namely, the enhancing effect of19ixing angles. In particular, the EW structure of the charged lepton dipole operator is similar tothe masses so that the l (0) ν component of l (0) does not enter the amplitude for µ → eγ . Hence, theestimates for µ → eγ are similar to the case without neutrino masses.However, the flavor violation via tree-level Z exchange is still modified (relative to the casewithout neutrino masses) as follows. The point is that, although the l (0) ν component of l (0) doesnot determine the LH charged lepton mixing angles , it does contribute to the coupling of LHcharged leptons to KK Z which is (approximately) diagonal in generation space in the weak basisfor leptons . And, even though we have f ( c L ν ) ∼ f ( c L ν ) ∼ f ( c L ν ), we are not assuming strict universality of these f ’s (which would require tuning or a symmetry) so that couplings ofSM LH leptons to KK Z (see Eq. (5)) induced by their l (0) ν components do differ by ∼ O (1)factors. In turn, via KK Z -zero-mode Z mixing (see top right-hand side diagram in Fig. 1), thesecouplings to KK Z result in non -universal shifts (relative to the Z (0) coupling) in the coupling ofLH charged leptons to SM Z . In the weak basis, these shifts in the couplings to Z are still diagonalin generation space. Recall that these non-universal shifts in coupling to Z then get convertedinto flavor- violating coupling to Z up on going from weak to mass basis, as in 1st term of Eq. (6)(albeit with small mixing angle in this case). Therefore, depending on the size of f ( c L ν ), the l (0) ν component can still be important for tree-level charged lepton flavor violation via Z exchange.We can then distinguish two cases (this discussion is an elaboration of point (ii) of section2.3). If c L ν > /
2, then we have f ( c L ν ) ≪
1, i.e., the l (0) ν are peaked near the Planck brane. Inparticular, we can choose f ( c L ν ) < ∼ f ( c L e ), where the latter parameter determines m µ , so thatthe effect of l (0) ν component in tree-level charged lepton flavor violation (in zero-KK gauge modemixing) is smaller than that of the L e component . Then, the tree-level charged lepton flavorviolation is also same as in the case without neutrino masses, i.e., generically KK mass limit is O (5) TeV. However, obtaining non-hierarchical profiles near TeV brane for the l (0) ν component inorder to generate large LH neutrino mixings then requires (almost) degenerate bulk masses, i.e.,tuning, due to the profiles’ exponential sensitivity to the bulk masses: see Eq. (2). Specifically, weneed a splitting in c of ∼ / ( kπR ) ∼ .
03 if we require (at most) a factor of ∼ c > / c L ν < ∼ / f ’s have a milder (power-law instead of exponential)dependence on c L (see Eq. (2)), i.e., the l (0) ν have a flat/peaked near TeV brane profile. Thus thereis no need for any tuning of bulk masses in this case in order to obtain large LH neutrino mixing. Here, we are assuming that the off-diagonal couplings of leptons (in this basis) to KK Z which are induced viabrane-localized kinetic terms are small. Similarly, such off-diagonal effects generated by zero-KK fermion mixing (seetop left-hand side of Fig. 1, with Z (0) replaced by Z ( n ) ) are also suppressed, assuming Y √ k ∼ O (1). Similarly, the l (0) ν component of LH lepton also contributes to off-diagonal couplings to SM Z (already in the weakbasis for leptons) via zero-KK fermion mode mixing. This effect can also be suppressed by the choice of c L ν > / c L ν < / Z (see Eq. (5)), the flavor- preserving shift of the couplings of leptons to Z and 4-fermion operatorsinduced by direct KK gauge exchange become too large .As a compromise, we are then led to considering c L ν ∼ / • c = 0 . ↔ .
45 gives only a factor of ∼ c ∼ . specific models: see, for example, reference[13] for one possible fit of c ’s to quark masses. In this paper, we will accept this mild tuning.With this choice, the flavor- preserving shifts in Z couplings (see Eq. (6) without mixing angle)are marginal, i.e., ∼ a few 0 . f ( c L ν i ) ∼ / p log ( M P l / TeV) (seeEq. (2)). And 4-fermion operators induced by direct KK gauge exchange are quite safe for severalTeV KK mass scale, using couplings in Eq. (5). However, with this size of f ’s, one problem is thatthe flavor- violating coupling to Z , δg Zµ L e L in Eq. (6), is still larger than in the case without neutrinomasses (even with small mixing angle) – in the latter case, we get f ( c L ) ∼ p m µ /v ∼ /
40 forthe case of similar RH and LH charged lepton profiles and Y √ k ∼ O (1). Thus, including neutrinomasses is still dangerous on a count similar to (II A) in section 3.3 even though count (I) is avoided.Thus we get the KK mass limit > O (5) TeV from charged lepton flavor violation. Similarly, for theminimal case (0) considered earlier with the large LH charged lepton mixing angle, the KK masslimit will be even larger than that mentioned before, i.e., > O (10) TeV if we insist on no tuning,i.e., choose c L < ∼ / mass splittings for neutrinos (in additionto mixing angles). We can achieve this goal by choosing c N < ∼ /
2, i.e., mild or no tuning of bulkmasses for RH neutrinos giving non-hierarchical profiles near the TeV brane. Combined with thenon-hierarchical LH neutrino profiles near the TeV brane (as above), the resulting Dirac neutrinomasses will then be non-hierarchical, but too large since both RH and LH profiles generatingneutrino masses near TeV brane are larger than those of charged leptons. However, we can include(a Planck/GUT-scale) Majorana mass term for RH neutrino on Planck brane and thereby use thesee-saw mechanism [36, 16] to obtain very small neutrino masses (for other neutrino mass models As discussed below, we can invoke custodial symmetries to suppress shifts in couplings of fermions to Z in thiscase, but these can only protect either (not both) LH charged lepton or neutrino couplings to Z from being shifted. Explicitly, with dimensionless coefficient being O ` g Z ´ , such 4-fermion operators have to be suppressed by severalTeV mass scale.
21n warped extra dimension, see references [37, 20]).
Next, we discuss how a custodial symmetry for the shift in the coupling of fermions to Z canrelax the above tension for the choice of close-to-flat profiles for l (0) ν components of LH leptons. Inparticular, we • choose L ν : ( , ) which implies T L = T R for this component of the LH charged leptons.If we further choose the 5 D SU (2)
L, R couplings to be equal, then we realize the P LR custodialsymmetry [24] for this component of the LH charged leptons. Such a symmetry protects thecouplings of SM Z to leptons (in the weak basis) from receiving a non -universal shift via zero-KKgauge mixing (again on account of the l (0) ν component) as follows – note that these shifts are flavor- preserving . Recall that the couplings of LH charged leptons to KK Z and similarly to (KK) Z ′ induced by their l (0) ν components are not universal (although they have similar size). However, thissymmetry enforces a cancellation between the non-universal contributions of KK Z and KK Z ′ inthe mixing with Z (0) so that the net shift in the couplings of LH charged leptons (coming fromtheir l (0) ν components) to SM Z is (approximately) universal, i.e., δg Ze L i e L i is i -independent. Hencethe resulting flavor- violating SM Z coupling (after rotating from weak to mass basis for leptons) issuppressed as well . Note that we cannot simultaneously protect the (flavor-preserving) Zν L ν L coupling from being shifted due to the l (0) ν component (due to T L = +1 / − T R for ν L ).However, as mentioned above, in any case this effect is marginal (i.e., ∼ a few 0 . c L ν i ∼ /
2, i.e., f ( c L ν i ) ∼ / p log ( M P l / TeV) in Eq. (6) (without mixing angle). Also, notethat the δg Zµ L e L from l (0) e component is not protected (since L e must transform differently under SU (2) R than L ν , i.e., it must have T R = − /
2, in order to decouple LH neutrino and chargedlepton mixings), but anyway this effect is of similar size to the case without neutrino masses andsafe since we can choose f ( c L e ) < / p log ( M P l / TeV).We would like to emphasize that this mechanism to suppress flavor-violating couplings to Z does not require the profiles at the TeV brane ( f ( c L ν i )’s) and hence the couplings to KK Z (seeEq. (5)) to be universal at all , but rather relies up on cancellations between KK Z and KK Z ′ due to the couplings of leptons to KK Z , Z ′ being diagonal in this basis, as mentioned earlier. It is clear that even in presence of off-diagonal coupling of leptons (in the weak basis) to KK Z , Z ′ inducedvia brane-localized kinetic terms (or zero-KK fermion mixing due to Higgs vev), this custodial protection for flavor-violating lepton couplings to SM Z still works since it is the result of a cancellation between KK Z and Z ′ . Similarly,there is a cancellation between the contributions of various KK fermions to the shift in coupling to SM Z fromzero-KK fermion mode mixing (see top left-hand side of Fig. 1) so that this effect also enjoys custodial protection. There is also a shift in charged current lepton couplings (vs. those for quarks) due to the mixing of KK andzero-mode W (especially due to l (0) ν component of SM lepton), but again this effect is marginal. again, we are assuming these f ’s are non-hierarchical in order to obtain large neutrino mixing, but still differingby ∼ O (1) factors. non -universal) to the shifts in couplings of fermions to Z . In thissense this mechanism to suppress flavor-violating coupling to Z is quite distinct from the idea of 5 D flavor symmetries which set c ’s (and thus f ’s) to be degenerate. Hence, with 5 D flavor symmetries,couplings of leptons to KK Z (see Eq. (5)) and thus the contribution of KK Z to the shift in thecoupling of fermions to SM Z is by itself universal, giving flavor-preserving Z couplings after goingfrom weak to mass basis. Note that this result applies also to KK Z ′ contributions, i.e., it is valid separately for KK Z and KK Z ′ , unlike for the custodial symmetry case considered here. Direct KK Z , Z ′ exchange : a detailed analysis of this effect (including the effects of Z ′ whichhave not been calculated before ) is beyond the scope of this paper, but it suffices to note thatthis effect does not enjoy custodial protection (unlike the effect of mixing of KK Z with zero-mode Z ). Moreover, due to the l (0) ν component of LH lepton with c L ν ∼ /
2, i.e., f ( c L ν ) ∼ / √ kπR ,this effect can be enhanced (for the LH leptons only) compared to the case studied in reference[15] without neutrino masses. Specifically, with RH and LH charged lepton profiles being similar,we get f ( c L ) ∼ m µ /v ∼ / Y √ k ∼ O (1) in the latter model. So, ratio of direct KK Z exchange in the two models is ∼ sin α × / ( kπR ) based on Eq. (9), where sin α is the admixtureof l (0) ν in the SM LH lepton as in Eq. (13). As mentioned in section 3.1.1, direct KK Z exchangeis suppressed compared to Z exchange in the model without neutrino masses by ∼ kπR and latteris on the edge of data for M KK ∼ O (5) TeV. Thus we see that direct KK Z exchange in the modelunder consideration here is marginal if sin α ∼ M KK ∼ O (5) TeV. Choice of L e representations (responsible for charged lepton masses) : one possibilityis L e : ( , ) − / and E : ( , ) − / as in case (2) above. The KK mass can then be as small as O (5) TeV in case (2), even without any large tuning of c L ν in order to obtain large LH neutrinomixing angles. If we allow tuning of c L ν ’s, we already saw in the beginning of section 5 that KKmass limit is same as that without neutrino masses, i.e., ∼ O (5) TeV, as long as we decoupleLH charged lepton and neutrino mixings. Recall that case (1) also has small LH charged leptonmixing angle so that the KK mass limit can also be ∼ O (5) TeV, but this case does not have thecustodial protection. So in case (1), we need to choose c L ν > / δg Zµ L e L from l (0) ν component, implying that we need tuning for obtaining large LH neutrino mixing angle. Astronger limit on KK scale results if we instead choose c L ν ∼ / although the couplings of Z ′ to quarks, relevant for µ to e conversion in nuclei, are expected to be negligible. Note that, as mentioned earlier, we assume that the couplings of leptons (in weak basis) to KK Z , Z ′ are(approximately) diagonal (but non-universal) in generation space. So, the off -diagonal couplings to KK Z , Z ′ givingflavor violation arise only after rotating to mass basis: the charged lepton mixing angle entering this effect in thecases we are considering here is the same as in the case without neutrino masses (i.e., this angle is small). .1.1 Best case scenario So far, we have been able to obtain a similar level of charged lepton flavor violation (and hence ∼ O (5) TeV KK scale) as in the case without neutrino masses discussed in reference [15]. In fact,we can obtain more safety (relative to the case without neutrino masses in reference [15]) using case (3) above which has T L = T R = 0 for RH charged leptons. Such a choice of quantumnumbers results in custodial protection ( P C symmetry [24]) for non-universal shifts in couplings of Z to RH charged leptons (in weak basis). As for the P LR custodial symmetry discussed before,there is is a cancellation between the non-universal contributions of KK Z and (KK) Z ′ in themixing with zero-mode Z , but we do not need the 5 D SU (2)
L, R gauge couplings to be equal inthis case (unlike for P LR symmetry). The idea then is to • increase all f ( c E )’s by, say, 2 √ violating coupling to Z , i.e., δg Zµ R e R (after rotating from weak to mass basis), is also protectedby this custodial symmetry and • reduce all f ( c L e i ) by ∼ √ δg Zµ L e L resulting from L e component is not protectedsince we have T L = T R for this component of LH charged lepton.Hence, we can reduce Y by ∼
2, keeping charged lepton masses fixed. Then both the tree-level and loop amplitudes are reduced by ∼ ∼
2, i.e., O (2 .
5) TeV. If we keep increasing f ( c E )’seven more, then, eventually, RH charged lepton flavor violation from direct KK Z exchange (whichis not protected) becomes relevant. Thus, we conclude that ∼ O (3) TeV KK scale can be consistentin case (3) with both charged lepton flavor violation and neutrino masses and with at most mildtuning of c L ν ’s in order to obtain large neutrino mixing. no decoupling of mixing angles In order to illustrate the independence of the above two mechanisms, namely, decoupling of largeneutrino mixing from charged lepton sector (discussed in section 4) and custodial symmetry studiedin this section, we consider a final case with only one 5 D multiplet for LH leptons, but choose again, a similar effect occurs for zero-KK mode fermion mixing. Again, the LH contribution of Eq. (6) is suppressed only by ∼
2, but there is no RH contribution due to custodialprotection, giving another reduction by factor of 2. We chose hierarchies in LH charged lepton profiles (at the TeV brane) from l (0) e component to be similar to thoseof RH charged leptons, i.e., both RH and LH charged lepton 1 − ∼ p m e /m µ , as in reference [15]. Itis easy to check that such a choice minimizes the constraint from µ → eγ . Based on the previous discussion, it can be seen that a very mild tuning of sin α is required to make the effect ofdirect KK Z exchange, coming from l (0) ν component of LH leptons, marginal for ∼ O (3) TeV KK scale. : ( , ) for custodial protection for LH charged leptons. With E : ( , ) and N : ( , ) , weget Case (4) : Y E : ( , ) ( , ) ( , ) Y N : ( , ) ( , ) ( , ) (21)The large, i.e., O (1), mixing both for LH charged leptons and neutrinos due to the choice of one5 D L multiplet implies that we must choose Y smaller (i.e., ∼ O (1 / ∼ O (5) TeV KKmass scale can be allowed by µ → eγ (see Eq. (10)). In order to compensate the effect of smaller Y in τ Yukawa coupling, we can then increase f ( c E ) such that we are on the edge of the δg Zτ R τ R constraint for several TeV KK mass scale, i.e., f ( c E ) ∼ / √ kπR (recall that we do not havecustodial protection for RH charged lepton couplings to Z in this case since T R = − f ( c E ), we still need f ( c L ) ∼ / √ kπR , i.e., c L ∼ / m τ . Anyway, c L ∼ / We can check that the tree-level δg Zµe is quite safe, due to custodial protection for LH contributionand due to (very) small mixing angle for RH contribution (even for such large f ( c E )).Based on the previous discussion, it is clear that the flavor violation from direct KK Z exchangein this case violates the experimental constraint by ∼ O (10) due to mixing angle being larger thanin cases (2) and (3) discussed above. A mild tuning sin α ∼ O (0 .
1) can make this effect marginalfor M KK ∼ O (5) TeV in case (4). Note that we could also have invoked sin α ≪ Z exchange (instead of using custodial protection), but clearly we wouldhave needed significant tuning in this case.The various possibilities discussed in this paper are summarized in table 1. Clearly, otherpossibilities with low KK scale can be constructed from combinations of the cases presented in thistable. As we eagerly await the start of the LHC, where new physics at the TeV scale related to Planck-weakhierarchy of the SM might be discovered, it is interesting to study whether clues of flavor hierarchyof the SM could lie in this physics. In this paper, we considered one such possibility, namely, theframework of a warped extra dimension with the SM gauge and fermion fields propagating in thebulk. The flavor hierarchy of the SM can be accounted for in this framework using profiles forthe SM fermions in the bulk, but the flip side is the resulting flavor violation from KK modes.Even though there is an automatic GIM-type mechanism, the limit on KK mass scale from flavor Also, we cannot choose c L < / Z since wedo not simultaneously have such protection for ν L couplings. r L e and r L ν under the bulk gauge symmetry, SU (2) L × SU (2) R × U (1) X , for the two components of LH leptons, l (0) e and l (0) ν , respectively, discussed in the text. The“Y” and “N” in 3rd column convey whether charged lepton mixing angle is small or not. Similarlythey convey whether the case has custodial symmetry for the l (0) ν component of LH lepton with non-hierarchical profiles at the TeV brane (which give neutrino masses with large mixing) or not (4thcolumn) and finally custodial symmetry for RH charged lepton multiplet (5th column). The lastcolumn shows lower limit on M KK from charged lepton flavor violation. For the cases with smallLH charged lepton mixing, we are considering flavor-violating contributions from l (0) e componentof LH lepton (which give charged lepton masses) and RH charged lepton multiplets only, i.e.,contribution from the non-hierarchical l (0) ν profiles is assumed to be negligible in these cases. Thisassumption is justified due to either (i) the choice of these profiles peaked near Planck brane, whichrequires tuning of bulk masses in order to obtain large neutrino mixing or (ii) presence of custodialprotection (i.e., “Y” in 4th column) for the case of close-to-flat profiles, which does not requiretuning. r L e r L ν small mixing angle? L ν custodial? RH custodial? lower limit on M KK ( , ) − / ( , ) − / N N N ∼ O (10) TeV( , ) − / ( , ) − / Y N N ∼ O (5) TeV( , ) − / ( , ) Y Y N ∼ O (5) TeV( , ) − ( , ) Y Y Y ∼ O (3) TeV( , ) ( , ) N Y N ∼ O (5) TeVviolation in both the quark and charged lepton sector (with out considerations of neutrino data) isstill ∼ O (5) TeV.Moreover, if we include the neutrino data, then the charged lepton flavor violation tends to beenhanced by the large charged current mixing required to account for neutrino oscillations. Thepoint is that, in the minimal model, the mixings are similar for LH charged leptons and neutrinos,being dictated by LH profiles (at the TeV brane) which are same for the two sectors. Hence, thelimit on gauge KK mass scale from charged lepton flavor violation when combined with neutrinodata is larger than ∼ O (5) TeV, making any signals at the LHC from direct production of gaugeKK modes unlikely.In this paper, we presented new mechanisms which can suppress charged lepton flavor violationin this framework. The central point is to use less minimal representations for leptons under theextended 5 D gauge symmetry , allowing mixing angles to be (simultaneously) small and large inthe LH charged lepton and neutrino sectors, respectively. The trick is that the LH lepton zero-mode is actually a combination of two zero-modes with different profiles, one giving charged leptonmasses and the other neutrino masses. Furthermore, such representations can lead to custodial such an extension of 5 D gauge symmetry is motivated for satisfying EWPT. Z (ala Zb ¯ b ) and hence suppress chargedlepton flavor violation from tree-level exchange of Z .The bottom line is that ∼ O (3) TeV gauge KK mass scale might then allowed by chargedlepton flavor violation, including neutrino masses and without any particular structure in the 5 D flavor parameters. However, charged lepton flavor violation is still not “super-safe” (unlike in somemodels with 5 D flavor symmetries) so that the upcoming lepton flavor violation experiments (MEGat PSI [38], PRIME at JPARC [39] and the proposed mu2e experiment at Fermilab [40]) should seea signal. The situation is similar to reference [15] without considerations of neutrino masses sincethe two issues of charged lepton flavor violation and neutrino masses are now decoupled. Also, with ∼ O (3) TeV gauge KK scale, signals form direct production of these KK modes at the LHC arethen still viable [41]. We would like to emphasize that the mechanism discussed in this paper is quite general as wediscuss below with several examples.
Quark sector in warped extra dimensional framework : the mechanism for decoupling ofmixing angles of the LH charged leptons and neutrinos can be applied to quark SU (2) L doubletsas well for the warped extra-dimensional scenario. In particular, we can arrange for LH down-typequark mixing to be parametrically smaller than LH up-type quark mixing – the latter would thenhave to entirely account for the CKM mixing. Thus flavor-violating effects involving LH down-typequarks can be suppressed compared to the minimal models, where the LH down-type and up-typequark mixings are similar (just like for LH charged leptons and neutrinos) and hence of CKM-size.However, the dominant constraint on gauge KK scale from flavor violation in the quark sectorcomes from contributions to ǫ K involving both LH and RH down-type quarks. While LH down-typequark mixings can be suppressed using the trick used for leptons here, it is easy to see that thethe RH down-type quark mixings are enhanced compared to minimal models so that this mixedcontribution to ǫ K is not affected. However, the dominant contribution to B d, s mixing does comefrom operators with LH down-type quarks only and hence it can be suppressed using the mechanismof decoupling LH up and down-type quark mixing angles. Of course, the constraint on KK massscale from these systems is (generically) weaker than the one from ǫ K . In short, it seems difficultto fully ameliorate the constraints from quark sector flavor violation using this mechanism. Combining with other proposals within warped extra dimensional framework : wehave presented the new mechanisms for suppressing charged lepton flavor violation in the warped just like we found for charged leptons in section 3.3 that enhancement of LH mixing angles implies reduction inRH mixing angles. minimal choice of representations of the bulkgauge symmetry, in a framework where neutrinos are Dirac particles and with a bulk Higgs. Inthis framework, we choose non-degenerate c L > /
2, i.e., with LH lepton profiles being hierarchicalnear the TeV brane and non-hierarchical near the Planck brane. The point is that we can thenobtain small charged lepton mixing angles (as usual) since charged lepton masses are dominatedby overlap with Higgs near the TeV brane, whereas neutrino masses can be dominated by overlapof profiles near the Planck brane thus giving large neutrino mixings. Thus, even without using thenew SU (2) R representations (i.e., the two mechanisms of this paper), the lepton flavor violationconstraints in this framework for neutrino masses reduce to the case without neutrino massesstudied in [15], i.e., the gauge KK mass limit is ∼ O (5) TeV. In addition, the mild tuning of c L ’srequired in the models considered here in order to obtain large neutrino mixings is avoided in theidea of reference [22] .Interestingly, we can add the custodial protection for charged lepton couplings to Z to theabove idea. Namely, we move either LH or RH charged lepton profile closer to the TeV brane(relative to the choice of same RH and LH profiles), invoking custodial symmetry to protect tree-level flavor violation via Z exchange from this chirality. Simultaneously, we move the profile of theother chirality (which does not enjoy custodial protection) away from the TeV brane (thus reducingtree-level µ to e conversions) in such a way as to allow a reduction in 5 D Yukawa coupling andthus suppressing, in turn, loop-induced µ → eγ as well. We note that • such a strategy (along the lines discussed in section 5.1.1) can allow us to lower the limit(from lepton flavor violation) on the gauge KK scale in this framework from ∼ O (5) TeV(which is the value without custodial protection) down to < ∼ O (3) TeV.Similarly, these mechanisms can be suitably combined with 5 D flavor symmetries. We will leavethese directions for future work. Beyond applications to the warped extra dimensional framework : these mechanismsmight enable suppression of flavor violation (especially in charged lepton sector) in other extra-dimensional models which explain flavor hierarchy via profiles, as long as there is an extended gaugesymmetry to play the decoupling trick. In particular, another framework where the origins of flavorleave their imprint on physics at the TeV scale (i.e., within LHC reach) is the recently proposed 5 D flavorful SUSY [42]. This 5 D set-up can be quite similar to that considered in this paper, namelyHiggs localized on one brane in an extra dimension with light fermion profiles being peaked near28he other end of the extra dimension – the smallness of the fermion profiles near the Higgs branethen account for the lightness of these fermions.More importantly, SUSY breaking can occur on the Higgs brane in this framework such that thenon-universalities/mixing among squarks and sleptons are governed by the (s)fermion profiles nearHiggs brane. Thus the structure of squark and slepton masses is correlated with the SM Yukawacouplings, possibly suppressing SUSY contributions to flavor violation at least for the 1st/2ndgeneration. This effect for 5 D flavorful SUSY is the analog of the GIM-like mechanism for KKcontributions considered here – of course, the actual KK contributions could be much smaller for5 D flavorful SUSY due to higher compactification scale with the resulting hierarchy between thatscale and the weak scale being explained by SUSY. However, charged lepton flavor violation in 5 D flavorful SUSY could be enhanced due to large neutrino mixing just like discussed here. It will beinteresting to further study the mechanisms for suppressing flavor violation discussed in this paperin the context of 5 D flavorful SUSY. Acknowledgments
This work is supported by NSF grant No. PHY-0652363. The author would like to thank RobertoContino, Takemichi Okui, Gilad Perez and Raman Sundrum for discussions, Gilad Perez for com-ments on the manuscript and the Aspen Center for Physics for hospitality during part of thiswork.
References [1] L. Randall and R. Sundrum, Phys. Rev. Lett. , 3370 (1999) [arXiv:hep-ph/9905221] andPhys. Rev. Lett. , 4690 (1999) [arXiv:hep-th/9906064].[2] H. Davoudiasl, J. L. Hewett and T. G. Rizzo, Phys. Lett. B , 43 (2000)[arXiv:hep-ph/9911262]; A. Pomarol, Phys. Lett. B , 153 (2000) [arXiv:hep-ph/9911294];S. Chang, J. Hisano, H. Nakano, N. Okada and M. Yamaguchi, Phys. Rev. D , 084025 (2000)[arXiv:hep-ph/9912498].[3] Y. Grossman and M. Neubert, Phys. Lett. B , 361 (2000) [arXiv:hep-ph/9912408].[4] T. Gherghetta and A. Pomarol, Nucl. Phys. B , 141 (2000) [arXiv:hep-ph/0003129].[5] A. Delgado, A. Pomarol and M. Quiros, JHEP , 030 (2000) [arXiv:hep-ph/9911252].[6] S. J. Huber and Q. Shafi, Phys. Lett. B , 256 (2001) [arXiv:hep-ph/0010195].297] K. Agashe, G. Perez and A. Soni, Phys. Rev. D , 016002 (2005) [arXiv:hep-ph/0408134].[8] H. Davoudiasl, J. L. Hewett and T. G. Rizzo, Phys. Rev. D , 045002 (2003)[arXiv:hep-ph/0212279]; M. Carena, E. Ponton, T. M. P. Tait and C. E. M. Wagner, Phys.Rev. D , 096006 (2003) [arXiv:hep-ph/0212307]; M. S. Carena, A. Delgado, E. Ponton,T. M. P. Tait and C. E. M. Wagner, Phys. Rev. D , 035010 (2003) [arXiv:hep-ph/0305188];M. S. Carena, A. Delgado, E. Ponton, T. M. P. Tait and C. E. M. Wagner, Phys. Rev. D ,015010 (2005) [arXiv:hep-ph/0410344].[9] P. McGuirk, G. Shiu and K. M. Zurek, JHEP , 012 (2008) [arXiv:0712.2264 [hep-ph]];G. Shiu, B. Underwood, K. M. Zurek and D. G. E. Walker, Phys. Rev. Lett. , 031601(2008) [arXiv:0705.4097 [hep-ph]]; A. Falkowski and M. Perez-Victoria, arXiv:0806.1737 [hep-ph]; B. Batell, T. Gherghetta and D. Sword, arXiv:0808.3977 [hep-ph].[10] C. Csaki, A. Falkowski and A. Weiler, JHEP , 008 (2008) [arXiv:0804.1954 [hep-ph]].[11] M. Blanke, A. J. Buras, B. Duling, S. Gori and A. Weiler, arXiv:0809.1073 [hep-ph].[12] K. Agashe, A. Azatov and L. Zhu, arXiv:0810.1016 [hep-ph], accepted for publication in Phys.Rev. D.[13] A. L. Fitzpatrick, G. Perez and L. Randall, arXiv:0710.1869 [hep-ph].[14] S. Davidson, G. Isidori and S. Uhlig, Phys. Lett. B , 73 (2008) [arXiv:0711.3376 [hep-ph]].[15] K. Agashe, A. E. Blechman and F. Petriello, Phys. Rev. D , 053011 (2006)[arXiv:hep-ph/0606021].[16] G. Perez and L. Randall, JHEP , 077 (2009) [arXiv:0805.4652 [hep-ph]].[17] G. Cacciapaglia, C. Csaki, J. Galloway, G. Marandella, J. Terning and A. Weiler, JHEP ,006 (2008) [arXiv:0709.1714 [hep-ph]]; J. Santiago, JHEP , 046 (2008) [arXiv:0806.1230[hep-ph]]; C. Csaki, G. Perez, Z. Surujon and A. Weiler, arXiv:0907.0474 [hep-ph].[18] M. C. Chen and H. B. Yu, arXiv:0804.2503 [hep-ph]; C. Csaki, C. Delaunay, C. Grojean andY. Grossman, JHEP , 055 (2008) [arXiv:0806.0356 [hep-ph]].[19] C. Csaki, A. Falkowski and A. Weiler, arXiv:0806.3757 [hep-ph].[20] M. Carena, A. D. Medina, N. R. Shah and C. E. M. Wagner, arXiv:0901.0609 [hep-ph].[21] K. Agashe, A. Delgado, M. J. May and R. Sundrum, JHEP , 050 (2003)[arXiv:hep-ph/0308036]. 3022] K. Agashe, T. Okui and R. Sundrum, arXiv:0810.1277 [hep-ph], accepted for publication inPhys. Rev. Lett.[23] H. Davoudiasl, B. Lillie and T. G. Rizzo, JHEP , 042 (2006) [arXiv:hep-ph/0508279];G. Cacciapaglia, C. Csaki, G. Marandella and J. Terning, JHEP , 036 (2007)[arXiv:hep-ph/0611358]; M. Piai, arXiv:hep-ph/0608241; J. Hirn and V. Sanz, JHEP ,100 (2007) [arXiv:hep-ph/0612239]; C. D. Carone, J. Erlich and J. A. Tan, Phys. Rev. D ,075005 (2007) [arXiv:hep-ph/0612242].[24] K. Agashe, R. Contino, L. Da Rold and A. Pomarol, Phys. Lett. B (2006) 62[arXiv:hep-ph/0605341].[25] M. Blanke, A. J. Buras, B. Duling, K. Gemmler and S. Gori, arXiv:0812.3803 [hep-ph].[26] R. Contino, Y. Nomura and A. Pomarol, Nucl. Phys. B , 148 (2003)[arXiv:hep-ph/0306259].[27] W. D. Goldberger and M. B. Wise, Phys. Rev. Lett. , 4922 (1999) [arXiv:hep-ph/9907447];J. Garriga and A. Pomarol, Phys. Lett. B , 91 (2003) [arXiv:hep-th/0212227].[28] J. M. Maldacena, Adv. Theor. Math. Phys. , 231 (1998) [Int. J. Theor. Phys. , 1113(1999)] [arXiv:hep-th/9711200]; S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett.B , 105 (1998) [arXiv:hep-th/9802109]; E. Witten, Adv. Theor. Math. Phys. , 253 (1998)[arXiv:hep-th/9802150].[29] N. Arkani-Hamed, M. Porrati and L. Randall, JHEP , 017 (2001) [arXiv:hep-th/0012148];R. Rattazzi and A. Zaffaroni, JHEP , 021 (2001) [arXiv:hep-th/0012248].[30] K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B , 165 (2005) [arXiv:hep-ph/0412089];K. Agashe and R. Contino, Nucl. Phys. B , 59 (2006) [arXiv:hep-ph/0510164];M. Carena, E. Ponton, J. Santiago and C. E. M. Wagner, Nucl. Phys. B , 202(2006) [arXiv:hep-ph/0607106] and Phys. Rev. D , 035006 (2007) [arXiv:hep-ph/0701055];A. D. Medina, N. R. Shah and C. E. M. Wagner, Phys. Rev. D , 095010 (2007)[arXiv:0706.1281 [hep-ph]]; C. Bouchart and G. Moreau, Nucl. Phys. B , 66 (2009)[arXiv:0807.4461 [hep-ph]].[31] R. Contino, L. Da Rold and A. Pomarol, Phys. Rev. D , 055014 (2007)[arXiv:hep-ph/0612048].[32] R. Contino, T. Kramer, M. Son and R. Sundrum, JHEP , 074 (2007)[arXiv:hep-ph/0612180]. 3133] H. Davoudiasl, G. Perez and A. Soni, Phys. Lett. B , 67 (2008) [arXiv:0802.0203 [hep-ph]].[34] For studies with ∼
10 TeV KK masses, see S. J. Huber, Nucl. Phys. B , 269 (2003)[arXiv:hep-ph/0303183]; S. Khalil and R. Mohapatra, Nucl. Phys. B , 313 (2004)[arXiv:hep-ph/0402225].[35] G. Burdman, Phys. Lett. B , 86 (2004) [arXiv:hep-ph/0310144]; K. Agashe, G. Perezand A. Soni, Phys. Rev. Lett. , 201804 (2004) [arXiv:hep-ph/0406101]; G. Moreau andJ. I. Silva-Marcos, JHEP , 090 (2006) [arXiv:hep-ph/0602155]; K. Agashe, G. Perez andA. Soni, Phys. Rev. D , 015002 (2007) [arXiv:hep-ph/0606293]; S. Chang, C. S. Kim andJ. Song, JHEP , 087 (2007) [arXiv:hep-ph/0607313] and Phys. Rev. D , 075001 (2008)[arXiv:0712.0207 [hep-ph]]; P. M. Aquino, G. Burdman and O. J. P. Eboli, Phys. Rev. Lett. ,131601 (2007) [arXiv:hep-ph/0612055]; W. F. Chang, J. N. Ng and J. M. S. Wu, Phys. Rev.D , 096003 (2008) [arXiv:0806.0667 [hep-ph]] and arXiv:0809.1390 [hep-ph]; S. Casagrande,F. Goertz, U. Haisch, M. Neubert and T. Pfoh, JHEP , 094 (2008) [arXiv:0807.4937[hep-ph]]; M. Bauer, S. Casagrande, L. Gruender, U. Haisch and M. Neubert, arXiv:0811.3678[hep-ph].[36] S. J. Huber and Q. Shafi, Phys. Lett. B , 293 (2004) [arXiv:hep-ph/0309252].[37] S. J. Huber and Q. Shafi, Phys. Lett. B , 295 (2002) [arXiv:hep-ph/0205327];T. Gherghetta, Phys. Rev. Lett. , 161601 (2004) [arXiv:hep-ph/0312392]; M. C. Chen,Phys. Rev. D , 113010 (2005) [arXiv:hep-ph/0504158]; G. Moreau and J. I. Silva-Marcos,JHEP , 048 (2006) [arXiv:hep-ph/0507145].[38] G. Signorelli, J. Phys. G , 2027 (2003).[39] A. Sato, talk given at 7th International Workshop on Neutrino Factories and Superbeams(NuFact 05), Frascati, Italy, 21-26 June 2005.[40] R. Bernstein, talk given at 4th International Workshop On Nuclear And Particle Physics AtJ-PARC (NP08), Mito, Ibaraki, Japan, 5-7 March 2008.[41] K. Agashe, A. Belyaev, T. Krupovnickas, G. Perez and J. Virzi, Phys. Rev. D , 015003(2008) [arXiv:hep-ph/0612015]; B. Lillie, L. Randall and L. T. Wang, JHEP , 074 (2007)[arXiv:hep-ph/0701166]; B. Lillie, J. Shu and T. M. P. Tait, Phys. Rev. D , 115016 (2007)[arXiv:0706.3960 [hep-ph]]; A. Djouadi, G. Moreau and R. K. Singh, Nucl. Phys. B , 1(2008) [arXiv:0706.4191 [hep-ph]]; M. Guchait, F. Mahmoudi and K. Sridhar, Phys. Lett. B , 347 (2008) [arXiv:0710.2234 [hep-ph]]; U. Baur and L. H. Orr, Phys. Rev. D , 094012322007) [arXiv:0707.2066 [hep-ph]] and Phys. Rev. D , 114001 (2008) [arXiv:0803.1160 [hep-ph]]; K. Agashe et al. , Phys. Rev. D , 115015 (2007) [arXiv:0709.0007 [hep-ph]]; M. Carena,A. D. Medina, B. Panes, N. R. Shah and C. E. M. Wagner, Phys. Rev. D , 076003(2008) [arXiv:0712.0095 [hep-ph]]; Y. Bai and Z. Han, arXiv:0809.4487 [hep-ph]; K. Agashe,S. Gopalakrishna, T. Han, G. Y. Huang and A. Soni, arXiv:0810.1497 [hep-ph].[42] Y. Nomura, M. Papucci and D. Stolarski, Phys. Rev. D , 075006 (2008) [arXiv:0712.2074[hep-ph]]; Y. Nomura, M. Papucci and D. Stolarski, JHEP0807