Combining Direct & Indirect Kaon CP Violation to Constrain the Warped KK Scale
CCombining Direct & Indirect Kaon CP Violationto Constrain the Warped KK Scale
Oram Gedalia a , Gino Isidori b and Gilad Perez a a Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel b INFN, Laboratori Nazionali di Frascati, Via E. Fermi 40 I-00044 Frascati, Italy
Abstract
The Randall-Sundrum (RS) framework has a built in protection against flavour violation,but still generically suffers from little CP problems. The most stringent bound on flavourviolation is due to (cid:15) K , which is inversely proportional to the fundamental Yukawa scale.Hence the RS (cid:15) K problem can be ameliorated by effectively increasing the Yukawa scale witha bulk Higgs, as was recently observed in arXiv:0810.1016. We point out that incorporatingthe constraint from (cid:15) (cid:48) /(cid:15) K , which is proportional to the Yukawa scale, raises the lower boundon the KK scale compared to previous analyses. The bound is conservatively estimated tobe 5.5 TeV, choosing the most favorable Higgs profile, and 7.5 TeV in the two-site limit.Relaxing this bound might require some form of RS flavour alignment. As a by-product ofour analysis, we also provide the leading order flavour structure of the theory with a bulkHiggs. In generic RS models of a warped extra dimension with bulk fields, the flavour puzzle is solvedby the split fermion mechanism, where the localization of fermions is determined based on theirmasses and mixing angles [1, 2]. Within the RS, this yields extra protection against excess offlavour changing neutral current (FCNC) processes in the form of RS-GIM [3]. A residual littleCP problem is, however, still found in the form of too large contributions to the neutron electricdipole moment [3] and sizable contributions to (cid:15) K [4, 5, 6, 7] (see also [8] for some related recentRS flavour studies). Given an IR-localized Higgs field, a lower bound of O (20) TeV on the KKscale at leading order is obtained [6, 7].Recently in [9] it was pointed out, based on matching the full RS set-up onto a two site model(originally suggested in [10]), that if the Higgs is in the bulk and one-loop matching of the gaugecoupling is included, the KK scale can be lowered down to O (5) TeV. An important ingredient inthat paper’s analysis is the ability to raise the overall size of the 5D down-type Yukawa coupling, y d .The resulting weaker bound is actually controlled by simultaneously minimizing the contributionto (cid:15) K , which effectively falls like 1 / ( y d ) , with the contribution to b → sγ , which grows like ( y d ) .In this paper we point that a contribution to (cid:15) (cid:48) /(cid:15) K , similar in structure to b → sγ , actually yieldsa much stronger constraint on the 5D Yukawa size, which implies a strict conservative bound onthe KK scale of 7.5 TeV in the two site case. The bound is weakened to 5.5 TeV if one allows theHiggs profile to saturate the AdS stability bound [11]. This is still beyond the LHC reach [12], and1 a r X i v : . [ h e p - ph ] M a y mplies a rather severe little hierarchy problem. We also show that UV-sensitive operators raisethe bound significantly, for instance in case the Higgs is localized on the IR brane. In [9] it was pointed out that when the Higgs is in the bulk, the light fermions can be madeless composite while still keeping their masses constant, and also the overall Yukawa scale can beincreased without violating the corresponding perturbative bound. Both effects allow to amelioratethe RS (cid:15) K problem. In our analysis below we carefully analyze the flavour structure of the theory,allowing a rather general bulk Higgs profile. In most of the past studies, the flavour structure ofRS was analyzed via the approximation that the Higgs and any relevant KK states are localized onthe IR brane, where a transparent spurion structure can be formulated [3]. Here we consider thecouplings by calculating full overlap integrals of wavefunctions, and parametrize these correctionsby appropriate functions of the form: r = wavefunctions overlapapproximate coupling on the IR . (1)This can be understood from some relevant sample terms in the 4D effective Lagrangian [3]: L D ⊃ (cid:88) i,j Y dij H (cid:34) ψ † Q i f Q i ψ d j f d j r H ( β, c Q i , c d j ) + √ (cid:88) n ψ † Q i f Q i ψ nd j r H n ( β, c Q i , c d j )+ √ (cid:88) n ψ n † Q i ψ d j f d j r Hn ( β, c Q i , c d j ) + 2 (cid:88) n,m ψ n † Q i ψ md j r Hnm ( β, c Q i , c d j ) (cid:35) + g s ∗ (cid:88) i G ψ † i ψ i (cid:18) − kπR + f i r g ( c i ) (cid:19) . (2)The term with square brackets is the coupling of the Higgs, H , to quarks of various zero/KK levels, ψ ,n , respectively. The other term is the coupling of zero-mode quarks to the first KK gluon, G and i, j are flavour indices. For simplicity we only present the down type quark couplings, where Q ( d )stands for an SU(2) doublet (singlet) quark. The f ’s parametrize the values of SM quarks’ profileson the IR brane (note that in the convention we follow, the value of KK fermions’ wavefunctionon the IR brane is √
2) and the c ’s are their bulk masses in units of k , f ( c ) = (cid:115) − c − ( z v /z h ) c − . (3)The coupling Y dij in Eq. (2) is the 5D anarchic down-type Yukawa matrix. We use y d to denotea generic entry in Y d (in units of √ k ). Note that in comparison to the notation of [9], Y d ∗ =2 y d r H ( β = 1 , c = 0 . , c = 0 . H ( x, z ) = ˜ v ( β, z ) + (cid:80) n H ( n ) ( x ) φ n ( z ) [9], where ˜ v ( β, z )is the Higgs VEV profile, which is very close to the physical Higgs profile when m h (cid:28) M KK (here2 = (cid:112) µ , with µ being the bulk Higgs mass in units of k ). This profile can be chosen to peaknear the IR brane: ˜ v ( β, z ) = vz v (cid:115) β ) z h (1 − ( z h /z v ) β ) (cid:18) zz v (cid:19) β . (4)For the purposes of the following discussion, the only important parameter affecting the overlapcorrections is β . The β = 0 case describes a Higgs maximally-spread into the bulk (saturating thestability bound), while β = 1 corresponds to the two-site model considered in [9]. For a concretecomparison we take β = 1, and add the case of the weakest expected bound on the KK scale,which is obtained for β = 0.Note that the case of an IR Higgs corresponds to setting the r H ’s to unity. Full definitionsand discussion of the correction factors are presented in appendix A. (cid:15) K and b → sγ We start by considering the bound from (cid:15) K . In this case the largest contribution is generated byleft-right effective operators, and in particular by Q K = ¯ d αR s αL ¯ d βL s βR . (5)In the RS framework the leading contribution to C K (the effective coupling of Q K ) is generatedby a tree-level KK-gluon exchange. Up to O (1) complex factors, this leads to C K (cid:39) g s ∗ M KK f Q f Q f d f d r g ( c Q ) r g ( c d ) ≈ g s ∗ M KK λ d λ s ( y d ) r g ( c Q ) r g ( c d ) r H ( β, c Q , c d ) r H ( β, c Q , c d ) . (6)Here M KK is the scale of the first KK state, g s ∗ is the dimensionless 5D coupling of the gluon,and λ i is the SM Yukawa coupling of the quark i ( λ i = m q i /v , v ≈
174 GeV). SM and 5D Yukawacouplings are connected by the relation λ i ≈ y d f Q i f d i r H ( β, c Q i , c d i ). Eq. (6) uses the fact that themixing angles of the rotation matrices from the interaction basis to the mass basis are f Q i /f Q j and f d i /f d j ( i ≤ j ) for the quark doublets and singlets, respectively. We have verified numericallythat the correction to these relations due to the presence of a bulk Higgs in the relevant range ofparameters is a subleading effect. In principle, terms proportional to r g ( c Q ,d ) also contribute,with the same f (cid:48) x s structure; however, since r g ( c Q ,d ) < r g ( c Q ,d ), the contribution shown inEq. (6) is the dominant one.The result in (6) is similar to the one given in [9]. Taking into account the chirally-enhanced (cid:104) K | Q K | ¯ K (cid:105) matrix element, assuming an O (1) CP violating phase for C K , requiring that the NPcontribution to | (cid:15) K | is 60% of the experimental value [13] and evaluating the resulting suppressionscale [4] and the quark masses [14] at 5 TeV leads to M KK (cid:38) g s ∗ y d TeV , (7)for β = 1. The sources of difference from the result of [9] are a correction to the overlap of thequarks with the KK gluon and the 60% saturation requirement. This bound can be ameliorated Note that here and below we assume a single maximal CP violating phase. Given the fact that each of theobservables discussed by us actually receives contributions from multiple independent terms, this is a conservativeassumption. A more reasonable approach might be to estimate the sum of the different contributions via a “randomwalk” approach, which will increase the amplitude by factor of roughly (cid:112) N/
2, where N is the number of independentterms. b → sγ and (cid:15) (cid:48) /(cid:15) K from Yukawa interactions.by taking the Higgs to be maximally spread into the bulk ( β = 0), which enhances its couplingto the quarks (and raises the value of the mass corrections r H included in the calculation above).The result in this case is M KK (cid:38) . g s ∗ y d TeV . (8)Next we consider b → sγ . Here the largest contribution is generated by the effective operator(we follow the conventions of [9]) Q (cid:48) = em b π ¯ bσ µν F µν (1 + γ ) s . (9)The effective coupling of Q (cid:48) is generated in RS by a loop diagram with a Higgs propagating in theloop [3], as shown in Figure 1. The corresponding Wilson coefficient, evaluated in appendix B, is C (cid:48) ≈ λ b M KK f Q ( y d ) f d ˜ r (cid:48) bs ≈ M KK ( y d ) λ s λ b V ts ˜ r (cid:48) bs r H ( β, c Q , c d ) , (10)where in the last equation we have used the relation f Q /f Q ≈ V ts between the left-handed profiles( f Q i ) and the CKM matrix elements ( V ij ) . In Eq. (10) we grouped together the overlap correctionsto the loop diagram under ˜ r (cid:48) bs , which contains contributions from different quarks running in theloop. Under the assumption that the Yukawa is anarchical, so that in the bulk interaction basisthe bulk masses are diagonal and the c (cid:48) s are well defined [3], ˜ r (cid:48) bs is:˜ r (cid:48) bs ≈ (cid:88) i,j r H n ( β, c Q , c d i ) (cid:20) r Hn − m − ( β, c Q j , c d i ) − r Hnm ( β, c Q j , c d i ) (cid:21) † r Hm ( β, c Q j , c d ) , (11)where i, j are flavour indices and m, n are the KK levels of the fermions in the loop, and weonly consider the first KK state, since at one-loop the above contribution is finite (or at mostlog-divergent) [3, 15, 16]. Following the analysis in [9] (allowing 20% departure from the SM valueof B ( B → X s γ )) and using the values of the quark Yukawa couplings at 5 TeV [14], we obtain for β = 1 by requiring C (cid:48) (5 TeV) /C SM7 ( M W ) < . M KK (cid:38) . y d TeV . (12) We do not distinguish here between gluon and quark KK masses, which only slightly differ in the relevant rangeof parameters. lower than the corresponding bound reported in [9], because of the overlap correctionsconsidered here. Contrary to (cid:15) K , the case of β = 0 yields the same bound as for β = 1. As discussedin [3, 5, 9], the b → sγ operator with opposite chirality leads to a weaker condition.In [9] the constraints from (cid:15) K and b → sγ in (7) and (12) are combined to evaluate the valueof Y d ∗ that minimizes the lower bound on M KK , and the coupling g s ∗ is matched to the measured4D coupling at one-loop, resulting in g s ∗ ≈ g s ∗ ≈ y d is above the perturbativity bound(4 π/ √ N KK ) [9]. Taking y d equal to this bound with the minimal conceivable value N KK = 2 givesfor β = 1: M KK (cid:38) . (13) (cid:15) (cid:48) /(cid:15) K As anticipated in the introduction, here we show that the bound in (12) becomes substantiallystronger after we include an additional constraint from Re( (cid:15) (cid:48) /(cid:15) ), the direct CP violating observableof the K → π system. As pointed out in [5], the constraint following from the contribution of thechromomagnetic operator to Re( (cid:15) (cid:48) /(cid:15) ) is similar in structure to the b → sγ one, but is numericallymore stringent.Before analysing the extra contribution to Re( (cid:15) (cid:48) /(cid:15) ) generated in the RS framework, it is worthto briefly recall the experimental status of this observable and its prediction within the SM: • After a series of measurements by the KTeV and the NA48 collaborations, the present ex-perimental world average is Re( (cid:15) (cid:48) /(cid:15) ) exp = (1 . ± . × − [17]. • Re( (cid:15) (cid:48) /(cid:15) ) SM is dominated by the contributions of two operators: the electroweak penguin,contributing to Im( A ) = Im[ A ( K → (2 π ) I =2 )], and the QCD penguin ( Q in the standardnotations), contributing to Im( A ). The destructive interference of these two contributionsis one of the reasons why it is difficult to obtain a precise estimate of Re( (cid:15) (cid:48) /(cid:15) ) SM . Thenegative contribution to Re( (cid:15) (cid:48) /(cid:15) ) SM generated by the electroweak penguins is estimated with20%–30% errors, both on the lattice and using analytic methods [18]. On the other hand,the chiral structure of Q and the sizable final-state interactions in the (2 π ) I =0 channelprevent, at present, a reliable estimate of this matrix element on the lattice [19]. Recentestimates based on analytic methods [20, 21] lead to values of Re( (cid:15) (cid:48) /(cid:15) ) SM in good agreementwith Re( (cid:15) (cid:48) /(cid:15) ) exp , with errors ranging from 30% to 50%. As a conservative approach, in thefollowing we assume a 100% error, or 0 < Re( (cid:15) (cid:48) /(cid:15) ) SM < . × − , consistently with theconservative range suggested in [22].The potentially large new contribution to Re( (cid:15) (cid:48) /(cid:15) ) in the RS framework is induced by the twoeffective chromomagnetic operators Q G = g s H † ¯ s R σ µν T a G aµν d L , Q (cid:48) G = g s H ¯ s L σ µν T a G aµν d R . (14) The difficulty in estimating (cid:104) π | Q | K (cid:105) on the lattice is confirmed by the difficulty of reproducing the experi-mental value of Re( A ) on the lattice. The latter is affected by similar problems (in particular the large final-stateinteractions), but it is free from new-physics contaminations. In particular, lattice estimates tend to underestimateRe( A ): this provide a qualitative understanding of why lattice estimates of Re( (cid:15) (cid:48) /(cid:15) ) SM are typically smaller (oreven negative) compared to the analytic ones. b → sγ case, these are generated by the Higgs-mediated one-loop amplitude inFig. 1. The coefficients, evaluated in appendix B, are C G ≈ π M KK f Q ( y d ) f d ˜ r sd ≈ π M KK ( y d ) λ s V us ˜ r sd r H ( β, c Q , c d ) ,C (cid:48) G ≈ π M KK f Q ( y d ) f d ˜ r (cid:48) sd ≈ π M KK ( y d ) λ d V us ˜ r (cid:48) sd r H ( β, c Q , c d ) , (15)where ˜ r sd and ˜ r (cid:48) sd group the overlap corrections (see Eq. (48)). Defining δ (cid:15) (cid:48) = Re( (cid:15) (cid:48) /(cid:15) ) RS − Re( (cid:15) (cid:48) /(cid:15) ) SM Re( (cid:15) (cid:48) /(cid:15) ) exp , (16)we obtain δ (cid:15) (cid:48) = ω (cid:104) (2 π ) I =0 | λ s Q G | K (cid:105)√ A Re( (cid:15) (cid:48) /(cid:15) ) exp | (cid:15) | exp (cid:20) Im( C G − C (cid:48) G ) λ s (cid:21) ≈ (58 TeV) B G (cid:20) Im( C G − C (cid:48) G ) λ s (cid:21) , (17)where B G is the hadronic bag-parameter defined by [23] (cid:104) π I =0 | λ s Q G | K (cid:105) = (cid:114)
32 114 m π m K F π B G . (18)The value B G = 1 corresponds to the estimate of this hadronic matrix element in the chiral quarkmodel and to the first order in the chiral expansion [24]. A similar numerical value is also obtainedusing different hadronization techniques [25]. The hadronic matrix element of the chromomagneticoperator is affected by the same difficulties appearing in (cid:104) π | Q | K (cid:105) : beyond the lowest-order inthe chiral expansion we expect large positive corrections from final-state interactions. Moreover,as pointed out in [23], higher order chiral corrections should remove the accidental m π suppressionin Eq. (18). Therefore the estimate of δ (cid:15) (cid:48) obtained with B G = 1 can be considered as a conservativelower bound. The leading QCD corrections in running down the Wilson coefficients from the highscale ( ∼ ∼ C ( (cid:48) ) G /λ s and thematrix element in (18) are, to a good approximation, scale-independent quantities.Assuming O (1) CP violating phases for C G and C (cid:48) G , barring accidental cancellations amongthese two terms and imposing | δ (cid:15) (cid:48) | <
1, leads to M KK (cid:38) . y d TeV (19)for β = 1, and M KK (cid:38) . y d TeV (20)for β = 0.The constraint in Eq. (19) is substantially stronger with respect to the one in Eq. (12) (notealso that the former only depends on the down-type Yukawa, while the b → sγ amplitude, which Here we adopt the notation of [23], where F π = 131 MeV and the K → (2 π ) I amplitudes are normalisedsuch that Re( A ) = 3 . × − MeV (note that the analog of Eq. (18) reported in [5] has a missing factor 1/2).Additional numerical inputs are ω = | A /A | = 0 . | (cid:15) | exp = 2 . × − . a) (b) Figure 2: The combination of the bounds on M KK , as a function of y d , for (a) β = 1, whichcorresponds to the two site model; (b) β = 0, which corresponds to the weakest bound.is dominated by a charged Higgs contribution, implicitly depends on the up-type Yukawa, too).When combined with Eq. (7), the overall bound is obtained for y d ≈ . g s ∗ ≈
3) and is M KK (cid:38) . . (21)The lowest possible bound comes from combining Eq. (8) and Eq. (20): M KK (cid:38) . . (22)The combination of the bounds is shown in Fig. 2.The reason why the (cid:15) (cid:48) /(cid:15) K constraint is substantially stronger than b → sγ one (more thana factor of 10 at the amplitude level) can be understood by comparing the parametric depen-dence from quark masses and CKM factors of the LR (chromo-)magnetic operators in the RSframework [5]: A ( b L → s R γ ( g )) KK A ( b R → s L γ ( g )) SM ∝ m s m b | V ts | ∼ | V us | vs . A ( s R → d L γ ( g )) KK A ( s R → d L γ ( g )) SM ∝ | V us || V ∗ ts V td | ∼ | V us | . (23)In principle, the large enhancement of the s R → d L magnetic transitions in the RS frameworkoccurs also in the short-distance component of the s → dγ amplitude. In most K decays thisamplitude is unmeasurable, being obscured by long-distance contributions. The only case were itcould be detected is the rare decay K L → π e + e − [23, 26]. However, present experimental data onthis decay mode are still far from the SM level [17], and the corresponding bound on M KK is notcompetitive even with Eq. (12) (see [27] for a study of rare decys in the context of RS). As already mentioned above, there are cases where the leading contributions are from UV sensitiveoperators. This is expected since the 5D theory is non-renormalizable with negative mass dimensionoperators. For instance, when the Higgs is localized on the IR brane, the above one-loop isdivergent, and a counter-term in the form of a higher dimensional IR operator is included [3, 15, 16].We can derive a bound on the corresponding cutoff of the theory required to satisfy the constraints7rom b → sγ and (cid:15) (cid:48) /(cid:15) K (similarly to the µ → eγ case [16]). The effective operator on the IR canbe written as ( O IRx ) bs,sd = g x Q d (cid:16) Λ bs,sdIR (cid:17) Y d H ¯ Q i σ µν d j K µν , (24)where for the case of b → sγ ( (cid:15) (cid:48) /(cid:15) K ), x = 1 ( x = 3), i, j = 3 , i, j = 2 , i, j = 1 , Q d = 1 / Q d = 1) and K µν = F µν ( K µν = G µν ), the electromagnetic (gluon) field strength. The aboveexpression is simplified when we switch from 5D fields to canonically normalized 4D ones. Thiscutoff operator can be rewritten (in terms of the zero-modes and 4D couplings) on the IR braneas [16] O IRbs ∼ e (cid:0) Λ bsIR (cid:1) m s V ts F µν ¯ b σ µν (1 + γ ) s , O IRsd ∼ g (cid:0) Λ sdIR (cid:1) m d V us G µν ¯ s σ µν d R , (25)where we have replaced the Higgs by its vev and, for simplicity, we only consider the case where i, j = 2 , (cid:15) (cid:48) /(cid:15) K . Repeating the above analysis, we find the following bounds for Λ bs,sdIR Λ bsIR (cid:38) , Λ sdIR (cid:38)
20 TeV . (26) Generic flavour models within the RS framework provide an elegant explanation of the fermionmass hierarchy; however, the resulting suppression of FCNC processes might not be enough. Inthis paper we have shown that the constraints stemming from (cid:15) (cid:48) /(cid:15) K yield a lower bound of at least ∼ . • the bag parameter of the chromo-magnetic operator is likely to exceed the reference valuewe have adopted ( B G = 1), and we have allowed the RS contribution to saturate the exper-imental measurement of (cid:15) (cid:48) /(cid:15) K ; • we have only considered the first KK level of the quarks and the zero-mode of the Higgs; • since the value of y d is close to the perturbativity bound, the contribution from higher loopsis probably not negligible and, a piori, does not need not to be suppressed by r H − − ; • the final bound is obtained with an “ideal” Higgs profile.A more realistic evaluation should result in a stronger bound.The bound thus obtained is stronger than the one derived from electroweak precision tests,and induces a rather severe little hierarchy problem. If taken a face value, it also implies thata direct LHC discovery of the relevant degrees of freedom is unlikely. This motivates the searchfor alternative solutions of the residual RS flavour problem, such as the inclusion of some form offlavour alignment of the fundamental down-type sector [16, 28].8 cknowledgements We are grateful to Kaustubh Agashe for very valuable discussions and comments on the manuscript.We also thank Uli Haisch and Marek Karliner for useful discussions, and Yossi Nir for commentson the manuscript. The work of GP is supported by the Peter and Patricia Gruber Award. Thework of GI is supported by the EU under contract MTRN-CT-2006-035482 (
Flavianet ). A Overlap Corrections
A common approach is to use the values of fermion zero-modes on the IR brane, parameterizedby f ’s, to evaluate their coupling to the Higgs. This is exact in case the Higgs is localized on theIR brane, but it is only an approximation for a bulk Higgs. A similar approximation is used inthe coupling of fermions to the KK gluon, which is concentrated near the IR brane. Since we aretrying to constrain the RS flavour model, a more careful treatment is required. In this appendixwe estimate the corrections to these approximations by calculating full overlap integrals of thewavefunctions. For consistency, we follow the conventions of [9], and use the definitions given inappendix E of that paper.The correction factor for the overlap of the Higgs zero-mode with two zero-mode fermions isgiven by r H ( β, c L , c R ) ≡ (cid:82) z v z h dz ( z h /z ) ˜ v ( β, z ) χ ( c L , z ) χ ( c R , z ) / ( v √ k ) χ ( c L , z v ) χ ( c R , z v ) z h /z v == (1 − e − (2+ β − c L − c R ) kπR ) (cid:112) β )2 + β − c L − c R (cid:39) (cid:112) β )2 + β − c L − c R , (27)where ˜ v ( β, z ) is the Higgs zero-mode wavefunction defined in Eq. (4) and χ ( c, z ) is the fermionzero-mode wavefunction (see also Eq. (3)): χ ( c, z ) = f ( c ) √ z h (cid:18) z h z v (cid:19) / − c (cid:18) zz h (cid:19) − c . (28)The last approximate equality in Eq. (27) is valid to a very good accuracy for 2 + β > c L + c R (which is related to the “switching” behavior discussed in [29]). This result can be convenientlyfactorized to some approximation by r H ≈ √ β β − c L − c R , (29)which is valid for β (cid:46) c ’s is close to 0.Similarly, we define the correction factors for the overlap of the Higgs with a zero-mode fermionand a KK fermion, and with two KK fermions, respectively r H ( β, c , c ) ≡ (cid:82) z v z h dz ( z h /z ) ˜ v ( β, z ) χ ( c , z ) χ ( c , z ) / ( v √ k ) χ ( c , z v ) χ ( c , z v ) z h /z v ,r H ( β, c , c ) ≡ (cid:82) z v z h dz ( z h /z ) ˜ v ( β, z ) χ ( c , z ) χ ( c , z ) / ( v √ k ) χ ( c , z v ) χ ( c , z v ) z h /z v . (30)9or simplicity, we focus only on the first KK state of the fermions χ ( c, z ), defined by χ ( c, z ) = 1 N √ πR (cid:18) zz h (cid:19) / [ J α ( m z ) + b α ( m ) Y α ( m z )] , (31)with − b α ( m ) = J α − ( m z h ) Y α − ( m z h ) = J α − ( m z v ) Y α − ( m z v ) ,N = 12 πR (cid:8) z v [ J α ( m z v ) + b α ( m ) Y α ( m z v )] − z h [ J α ( m z h ) + b α ( m ) Y α ( m z h )] (cid:9) , (32)and α ≡ | c + 1 / | . r H ( β, c , c ) can be computed analytically, but the result is very complicated,while for r H ( β, c , c ) we could not find an analytic solution. Reasonable polynomial fits are givenby r H ( β, c , c ) ≈ .
41 + 0 . c − . βc + 0 . c c + 0 . c − . c ,r H ( β, c , c ) ≈ .
32 + 0 . β − . β + 0 . c + c ) + 0 . c c − . c c ( c + c ) , (33)The first one is valid to an accuracy of about 10% for β (cid:46)
10 (and breaks down in the regionwhere β ∼ | c − c | (cid:38)
1) and the second for β (cid:46) { ++ } boundary conditions. There is also a KK fermionwith opposite chirality ( {−−} boundary conditions) ˜ χ ( c, z ), defined in the same way as in Eq. (31),but with − ˜ b α ( m ) = J α ( m z h ) Y α ( m z h ) = J α ( m z v ) Y α ( m z v ) , ˜ N = 12 πR (cid:26) z v (cid:104) J α − ( m z v ) + ˜ b α ( m ) Y α − ( m z v ) (cid:105) − z h (cid:104) J α − ( m z h ) + ˜ b α ( m ) Y α − ( m z h ) (cid:105) (cid:27) , (34)and the replacement c → − c . Regarding the overlap of two KK fermions with the Higgs, weactually mostly use the opposite chirality states r H − − ( β, c , c ) ≡ (cid:82) z v z h dz ( z h /z ) ˜ v ( β, z ) ˜ χ ( c , z ) ˜ χ ( c , z ) / ( v √ k )˜ χ ( c , z v ) ˜ χ ( c , z v ) z h /z v , (35)rather than r H ( β, c , c ). A polynomial fit to r H − − is given by r H − − ( β, c , c ) ≈ . − . β + 0 . β + 0 . c + c ) − . c + c ) , (36)valid to β (cid:46) r g ( c ) ≡ (cid:82) z v z h dz ( z h /z ) ( f ( z ) − f ( z h )) χ ( c, z ) / √ kχ ( c, z v ) z h /z v , (37)where f ( z ) is the wavefunction of the first KK gluon, and we subtract its value on the UV branebecause it represents the flavour-universal part. This formula has a useful approximation, obtainedby neglecting the Y-type Bessel function in the KK gluon wavefunction [6, 30]: r g ( c ) = √ J ( x ) (cid:90) x − c J ( x x ) dx ≈ √ J ( x ) 0 . − c (cid:0) e c/ (cid:1) , (38)10ith x ≈ . J ( x ) = 0.In order to calculate realistic correction factors for the operators considered above, we employthe following procedure. First we choose values for three basic parameters: β , y d and c Q (the bulkmass parameter of the third generation left-handed quark doublet). The masses of the other left-handed doublets are obtained by the relation f Q i /f Q j ∼ V ij , and the masses of the right-handedquarks are extracted from λ i (cid:39) y d f Q i f d i r H ( β, c Q i , c d i ). Finally, the relevant correction factors foreach operator are computed together (note that r H appears only when the mass relation is used).Using this procedure, it was found that the corrections are actually quite insensitive to thevalue of c Q in the range of 0 − . y d around the value that minimizes the overall bound,which makes this analysis robust (although for a third generation quark running in the loop thecorrection is a bit more sensitive to c Q ). The only important parameter is β , as can be expected(e.g. the overlap of two light fermions with a bulk Higgs is quite different than with an IR brane-localized Higgs).The main result is that for the operators responsible for b → sγ and (cid:15) (cid:48) /(cid:15) K , the coefficientsare reduced by about an order of magnitude (for β = 0 , r H − − and the mass correction r H ( r H − − is always smallerthan r H , so the inclusion of the former in the dipole operators reduces their contribution relativeto what might be naively expected). The (cid:15) K contribution is actually raised for β = 1, as a resultof the KK gluon overlap correction r g .An important comment is in order. The bulk Higgs zero-mode wavefunction is usually obtainedby adding a bulk mass for the Higgs and kinetic terms on both branes (otherwise the zero-modevanishes by boundary conditions). Hence, There is no smooth limit (e.g. β → ∞ ) in which thebulk Higgs zero-mode wavefunction corresponds to an IR-localized Higgs. B One-Loop Coefficients of the Dipole Operators
B.1 The One-Loop Integral
Up to the overall coupling dictated by the flavour structure and the wave-function overlaps, theamplitude for the diagram in Fig. 1 with an external gluon line (including only the contributionsfrom the down-type flavour sector) is iA ( s → d g ) = (cid:90) d k (2 π ) u ( p (cid:48) ) (ˆ /p (cid:48) + M ( i ) KK )ˆ p (cid:48) − M ( i ) , KK ( g s γ µ t a G aµ ) i (ˆ /p + M ( i ) KK )ˆ p − M ( i ) , KK (1 ± γ ) u ( p ) · k − m H = (cid:90) d k (2 π ) u ( p (cid:48) ) (cid:34) g s t a G aµ M ( i ) KK ˆ /p (cid:48) γ µ + γ µ ˆ /p (ˆ p (cid:48) − M ( i ) , KK )(ˆ p − M ( i ) , KK )( k − m H ) (cid:35) (1 ± γ ) u ( p ) , (39)where ˆ p ( (cid:48) ) = p ( (cid:48) ) + k . Neglecting the Higgs mass, this leads to A ( s → d g ) == g s t a G aµ M KK (cid:90) d l (2 π ) (cid:90) dx (cid:90) − x dyu ( p (cid:48) ) − x ( p µ + iσ µν p ν ) − y ( p (cid:48) µ − iσ µν p (cid:48) ν ) (cid:104) l − M ( i ) , KK ( x + y ) (cid:105) (1 ± γ ) u ( p )= g s t a G aµ π ) M KK u ( p (cid:48) ) σ µν q ν (1 ± γ ) u ( p ) , (40)11here q ≡ p (cid:48) − p , we have used the equations of motion on the external spinors, and we have setthe KK mass equal to the value of the first KK state. B.2 Diagonalization of the Quark Mass Matrix
In order to compute the overall coupling of the loop amplitude, we need to address the diagonal-ization of the quark mass matrix. In general, a zero-mode quark in the interaction basis mixeswith the KK states. Restricting the discussion to the first KK level, there are two different states:one with the “right” chirality ( { ++ } boundary conditions, similar to the zero-mode), the otherwith “wrong” chirality ( {−−} boundary conditions, projecting out the zero-mode). The actualcontribution to a measurable quantity is then calculated after the mixing matrix is diagonalizedto the mass basis (see appendix B in [9] and [31] for similar analyses).For simplicity, we consider a one generation case, so the mass matrix M q is given by (cid:16) ¯ Q (0) ¯ d (1) L ¯ Q (1) L (cid:17) M q d (0) Q (1) R d (1) R , M q = M KK xf Q f d r H √ xf Q r H xr H − − √ xf d r H xr H , (41)with x ≡ vy d /M KK . M q is diagonalized to first order in x by a bi-unitary transformation: M mass q = O † L M q O R = M KK × diag( xf Q f d r H , x ( r H + r H − − ) , − x ( r H + r H − − )) , (42)where O L = 1 √ √ √ xf Q r H − xf Q − xf Q r H x ( r H − − − r H ) − x ( r H − − − r H )0 1 − x ( r H − − − r H ) 1 + x ( r H − − − r H ) ,O R = 1 √ √ √ xf d r H xf d − xf d r H − . (43)The interaction matrix of the quarks with the Higgs in the interaction basis is λ = y d f Q f d r H √ f Q r H r H − − √ f d r H r H , (44)and in the quark-mass basis it is simply λ mass = O † L λO R .The process that couples two opposite chirality zero-mode quarks is carried out through theinteraction of one quark with a heavy mass eigenstate, a propagator that couples it to the oppositechirality mass eigenstate and a coupling to the other light quark, summing over the two heavyeigenstates. Specifically, the effective coupling of the dipole amplitude is A ∝ λ mass21 λ mass12 / ( M mass q ) + λ mass31 λ mass13 / ( M mass q ) , (45)which results into the overall coupling12 v ( y d ) f Q f d r H r H r H − − M KK . (46) Here M mass q should actually be divided by M KK , to avoid double-counting of the propagator with the calculationof the previous appendix, and only consider the flavour structure that the propagator introduces. .3 Coefficients of the Effective Operators We are now ready to complete the calculation of the one-loop dipole amplitudes and derive thecoefficients of the corresponding effective operators by a matching procedure.In the case of the s → d g amplitude, the complete result is obtained multiplying Eq. (40) andEq. (46): A ( s → d g ) = g s t a G aµ π ) M KK u ( p (cid:48) ) σ µν q ν (1 ± γ ) u ( p ) 12 v ( y d ) f Q f d r H r H r H − − M KK . (47)Inserting the appropriate projection operator, u ( p (cid:48) ) σ µν G µ q ν (1 − γ ) u ( p ) can be identified with¯ s R σ µν G µν d L . Hence the coefficient C G of the operator in Eq. (14) is given by C G = 316 π M KK f Q ( y d ) f d r H r H r H − − , (48)and a similar expression is obtained for the opposite chirality coefficient C (cid:48) G .For b → sγ , the entire calculation follows in the same way. The coupling of a down-typeKK quark to the photon adds a factor of 1 /
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