$b\to ss{\bar d} $ decay in Randall-Sundrum models

The extremely small branching ratio of $b\to ss{\bar d}$ decay in the Standard Model makes it a suitable channel to explore new-physics signals. We study this $\Delta S=2$ process in Randall-Sundrum models, including the custodially protected and the bulk-Higgs Randall-Sundrum models. Exploring the experimentally favored parameter spaces of these models, it suggests a possible enhancement of the decay rate, compared to the Standard Model result, by at most two orders of magnitude.


Introduction
In studying flavor-changing neutral-current (FCNC) transitions in rare B decays for exploring new physics (NP), one major difficulty is, how to reliably subtract the Standard Model (SM) background. Theoretical uncertainties in FCNC transitions make it hard to conclude about definite new physics signals against SM predictions. For this reason, an alternative approach suggested in [1,2] is to consider processes which have tiny strengths in SM so that mere detection of such processes will indicate NP. One such process is the rare b → ssd decay, as reported in [1,2], which can serve the purpose of exposing NP.
The ∆S = 2 b → ssd process is box mediated in SM and is found to occur with a branching ratio of the order of 10 −12 . The authors of Ref. [1] suggested B − → K − K − π + as the most appropriate mode for experimental searches and many other studies of the b → ssd decay have been conducted in * lucd@ihep.ac.cn † faisalmunir@ihep.ac.cn ‡ qin@physik.uni-siegen.de various beyond SM scenarios [3,4,5]. The first search was reported in [6] and upper limits were given by both B factories [7,8,9], with the current upper limit reported by BABAR Collaboration to be B(B − → K − K − π + ) < 1.6 × 10 −7 . Moreover, two-body exclusive decays of B − [10] and B c [11], which are driven by the b → ssd transition, have also been studied in SM and in various extensions of it.
In this paper, we consider the inclusive b → ssd decay in Randall-Sundrum (RS) models [12,13]. We shall study two models known as the RS model with custodial protection (RS c ) [14,15,16,17,18] and the bulk-Higgs RS model [19,20], in both of which FCNC transitions occur at tree level. M KK is the KK scale always larger than 1 TeV) and the importance of Higgs FCNCs is limited with the most pronounced effects occurring in the case of the CP-violating parameter ǫ K , but even there they are typically smaller than the corrections due to KK-gluon exchanges [22]. Therefore, in view of the possible Higgs-boson effects to be insignificant in ∆F = 2 processes, we simply neglect them in our study of the with the Wilson coefficients corresponding to µ = O(M g (1) ) is given by

RS model with custodial protection
where and with i = 1, 2 and j = V LL, V RR, LR, RL. Note that, in the RS c model, compared to the analogous processes K 0 −K 0 and B 0 s −B 0 s mixings [23], the b → ssd decay receives additional contributions from the RL operators. [C j i (M g (1) )] G in Eq. (3) denote the contributions from the KK gluon to the Wilson coefficients that are calculated to be where, p UV parameterizes the influence of brane kinetic terms on the SU(3) c coupling. In our analysis we set p UV ≡ 1. Similarly, for the KK photon and (Z H , Z ′ ) contributions, we find the following corrections to the Wilson coefficients C j i (M g (1) ), [∆C LR 2 (M g (1) )] QED = 0, where the overlap integrals ∆ sb L,R (Z (1) ), ∆ sb L,R (Z where the three contributions in the bracket correspond to the KK gluon, the KK photon and combined (Z H , Z ′ ) exchange, respectively. The Wilson coefficients C LR 2 (M g (1) ) and C RL 2 (M g (1) ) receive only the KKgluon contributions. We see that the EW contributions, dominated by (Z H , Z ′ ) exchanges, give +87% and +150% corrections in the case of C V LL 1 (M g (1) ) and C V RR 1 (M g (1) ), respectively, while corrections of −174% are observed for C LR 1 (M g (1) ) and C RL 1 (M g (1) ). The Hamiltonian in Eq. (1) is valid at scales of O(M g (1) ) and has to be evolved to a low energy scale µ b = 4.6 GeV. For that, the anomalous dimension matrices for ∆F = 2 four-quark dimension-six operators have already been calculated at two loop level in [24,25]. As gluons are flavor blind and QCD preserves chirality so the anomalous dimension matrices of the operators in b → ssd are the same as for the case of B 0 d,s −B 0 d,s mixing operators. Therefore, the renormalization group running of the Wilson coefficients for the b → ssd decay is performed by using analytic formulae for the relevant QCD factors given in Section 3.1 and appendix C of [26]. Finally, the decay width for the b → ssd decay in the RS c model is given by

Bulk-Higgs RS model
The bulk-Higgs RS model is based on the 5D gauge group SU where A summation over color indices α and β is understood. TheÕ n operators are obtained from O n by L ↔ R exchange. Wilson coefficients at O(M KK ) are given by where Higgs and scalar field φ Z give opposite contributions to the Wilson coefficient C 2 , thus they cancel each other giving C 2 = 0. Similarly,C 2 = 0. The expressions of the mixing matrices ( and similarly in the lepton sector) in terms of the overlap integrals of boson and fermion profiles in the bulk-Higgs RS model, will be reported in [19]. For the present study, we restrict ourselves to the 3 × 3 submatrices governing the couplings of the SM fermion fields. In the zero mode approximation (ZMA), the required expressions are simplified considerably with (see also [27]) where Hence for the evolution of the Wilson coefficients we use the formulae of NLO QCD factors given in [28]. After that, the decay width in the bulk-Higgs RS model is given by

Phenomenological bounds on RS models
In this section we discuss the relevant constraints on the parameter spaces of the RS models coming from the EW precision tests and the latest measurements of the Higgs signal strengths at the LHC. In addition, we will also consider the constraints coming from K 0 −K 0 and B 0 s −B 0 s mixing in Section 5. First, considering the RS c model, the bounds induced from EW precision tests allow for KK masses in the few TeV range. A recent tree-level analysis of the S and T parameters yields M g (1) > 4.8 TeV at 95% confidence level (CL) for the mass of the lightest KK gluon and photon resonances [29]. While comparing the predictions of the signal rates for the various Higgs-boson decays with the latest data from the LHC, it is suggested in [30] that the most stringent bounds emerge from the signal rates for pp → h → ZZ * , W W * . In the RS c model, KK gluon masses lighter than 22.7 TeV × (y ⋆ /3) in the brane-Higgs case and 13.2 TeV × (y ⋆ /3) in the narrow bulk-Higgs scenario are excluded at 95% CL, where the y ⋆ = O(1) is a free parameter and is defined as the upper bound on the various entries of the Yukawa matrices that are taken to be complex random numbers such that |(Y f ) ij | ≤ y ⋆ . Thus, for y ⋆ = 3 the bounds derived from Higgs physics are much stronger than those stemming from EW precision measurements. In order to lower these bounds, smaller values of y ⋆ can be considered. For that it was also presented in Ref. [30] that for the lowest value of the lightest KK gluon mass M g (1) = 4.8 TeV implied by EW precision constraints, in the RS c model, the constraints at 95% CL on the values of the y ⋆ are given by y ⋆ < 0.3 for the brane-Higgs scenario, and y ⋆ < 1.1 for the narrow bulk-Higgs case.
However, realizing the fact that too small Yukawa couplings would give rise to enhanced corrections to ǫ K and hence they would reinforce the RS flavor problem, relatively loose bound on the values of the y ⋆ can be obtained for the lightest KK gluon mass of M g (1) = 10 TeV. For instance, in the RS c model, the constraints on the value of y ⋆ at 95% CL valid for M g (1) = 10 TeV are given by y ⋆ < 1.1 and y ⋆ < 2.25 for the brane-Higgs case and the narrow bulk-Higgs case, respectively [30].
Next, we consider the bulk-Higgs RS model. The constraints on the KK mass scale in the bulk-Higgs RS model implied by the analyses of EW precision data are given in [20]. Under a constrained fit (i.e.

Numerical analysis
In this section we present the results of the b → ssd decay rate in RS models. Before proceeding to analyze the NP, we first estimate the size of the leading order SM result.
Next, we explore the parameter space of the RS c model by the strategy outlined in [23]. It was pointed out in [23] that there exist regions in parameter space, without much fine-tuning in the 5D Yukawa couplings, which satisfy all existing ∆F = 2 and EW precision constraints for scales of masses of lightest KK gauge bosons M KK ≃ 3 TeV. However, as mentioned above that for the anarchic Yukawa couplings with y ⋆ = 3 in the RS c model with the a brane Higgs, the constraints on M g (1) emerging from Higgs physics, are much stronger than the EW precision constraints, so in our study of the RS c model, we generate two sets of fundamental 5D Yukawa matrices with y ⋆ = 1. previously. In order to determine the nine quark bulk-mass parameters c i Q,u,d , we take 0.4 ≤ c 3 Q ≤ 0.45 in our scan, allowing for consistency with EW precision data, so that the remaining bulk mass parameters are determined making use of the analytic formulae presented in section 3 of [23]. Finally, by diagonalising numerically the obtained effective 4D Yukawa coupling matrices, we keep only those parameter sets that in addition to the quark masses and CKM mixing angles also reproduce the proper value of the Jarlskog determinant, all within their respective 2σ ranges. The flavor transitions that would be involved in the b → ssd mode will commonly also give contributions to K 0 −K 0 and B 0 s −B 0 s mixings, so we consider ∆M K , ǫ K and ∆M Bs constraints on the parameter space in addition to EW precision constraints and the Higgs constraints mentioned above. Expressions of (M K 12 ) KK and (M s 12 ) KK relevant for K 0 −K 0 and B 0 s −B 0 s mixings constraints, calculated in the RS c model, are contained in Eqs. (4.32) and (4.33) of [23], respectively. Figure 1 shows the branching ratio of the RS c predictions for the b → ssd decay as a function of M g (1) with two different values of y ⋆ . Note that we have excluded the SM contribution to display the decoupling behavior of the NP contribution as M g (1) increases. The red and blue scatter points represent the cases of y ⋆ = 1.5 and 3, respectively. While imposing the experimental constraints for ∆M K , ∆M Bs and ǫ K in both cases, we set input parameters in table 2 to their central values and allow the resulting observables to deviate by ±50%, ±30% and ±30%, respectively. The predictions of the b → ssd decay rates for the parameter points with y ⋆ = 1.5 are generally larger than those with y ⋆ = 3, but it can be seen in figure 1 that after applying the ∆M K , ǫ K and ∆M Bs constraints simultaneously, the maximum possible y ⋆ = 1.5 prediction is reduced relatively close to that for the case of y ⋆ = 3.
However, after imposing the K 0 −K 0 and B 0 s −B 0 s mixings constraints, still for some parameter points with y ⋆ = 1.5 in the low M g (1) range, the branching ratio of b → ssd decay in the RS c model can be close to the order of 10 −10 , which is approximately two orders of magnitude larger compared to the SM result.
Considering the effects of the new heavy EW gauge bosons Z H and Z ′ in the RS c model, we found in agreement with [23] that while imposing the ∆M K and ǫ K constraints Z H and Z ′ give subleading contributions because the strong QCD renormalization group enhancement of the C LR For the bulk-Higgs RS model, following the directions given in [20,21], for a given value of β and M KK , we generate two sets of random and anarchic 5D Yukawa matrices, whose entries satisfy |(Y u,d ) ij | ≤ y ⋆ with y ⋆ = 1.5 and 3. These values of y ⋆ lie below the perturbativity bound, which is given by y ⋆ < y max with y max ∼ 8.3/ √ 1 + β [20]. Moreover, for values of y ⋆ < 1 it becomes increasingly difficult to fit the top-quark mass. Next, we require that the 5D Yukawa matrices with proper bulk-mass parameters c Q i < 1.5 and c q i < 1.5 reproduce the correct values for the SM quark masses evaluated at the scale µ = 1 TeV. In our analysis, we consider the two representative values β = 1 and β = 10 corresponding to broad Higgs profile and narrow Higgs profile, respectively. In figure 2, we show the NP predictions with β = 1 and 10, respectively, for the b → ssd decay rate as a function of M g (1) , after simultaneously imposing the ∆M K , ǫ K and ∆M Bs constraints. The red and blue scatter points again correspond to model points obtained using y ⋆ = 1.5 and 3, respectively. For the case of y ⋆ = 1.5, the branching ratios are generally larger because of less suppressed FCNCs compared to y ⋆ = 3 case, but as mentioned earlier the lower values of y ⋆ are subject to more stringent constraints from flavour physics, so after imposing the ∆M K , ǫ K and ∆M Bs constraints, the maximum possible branching ratio of the parameter points  with y ⋆ = 1.5 in the bulk-Higgs RS model lies close to the SM result as shown in figure 2(a). While for the case of y ⋆ = 3 in figure 2(a), subject to relatively less severe constraints from the K 0 −K 0 and B 0 s −B 0 s mixings compared with y ⋆ = 1.5 case, the maximum possible branching ratio for some of the parameter points, even with suppressed FCNCs, lies close to the order 10 −11 . Situation is similar in the β = 10 case, except that compared to the β = 1 scenario, an order of magnitude enhancement for the maximum possible branching ratio is observed for both cases of y ⋆ , as displayed in figure 2(b).

Conclusions
We studied the b → ssd decay in the RS c and the bulk-Higgs RS model. In both models, main contribution to the b → ssd decay comes from tree level exchanges of KK gluons, while in the RS c model the contributions from the new heavy EW gauge bosons Z H and Z ′ can compete with the KK-gluon contributions. We employed renormalization group runnings of the Wilson coefficients with NLO QCD factors in both models. Although this decay receives tree level contributions, the parameter space is severely constrained by K 0 −K 0 mixing and B 0 s −B 0 s mixing experiments such that for broad Higgs profile corresponding to β = 1 case no significant increase in the branching ratio is observed in the bulk-Higgs RS model compared to the SM result. Whereas, for the value β = 10, it is possible to achieve an order of magnitude enhancement of the branching ratio for some of the parameter points. While, the RS c model with additional contributions from the new heavy EW gauge bosons Z H and Z ′ enhances the branching ratio, compared to SM result, by at least one order of magnitude for some points in the parameter space with y ⋆ = 1.5, which leaves this decay free for search of new physics in future experiments.