Inspection of new physics in $\rm B_s^0 \to K^+K^-$ decay mode

We scrutinize the penguin dominated $ B_s \to K^+ K^-$ decay mode involving $b \to s$ quark level transition in family non-universal $Z^ \prime$ and vector-like down quark model. There is discrepancy in the standard model branching ratio value of this mode with the experimental results reported by Belle, CDF and LHCb Collaborations. Additionally, the measured values of CP-violating asymmetries $ C_{K^+K^-}$ and $ S_{K^+K^-}$ deviate from the SM results. We constrain the new parameter space by using the existing experimental limits on leptonic $ B_s\to \ell \ell$ ($\ell = e, \mu, \tau$) processes. We then check the effects of new physics on the branching ratio and CP violating parameters of the $ B_s \to K^+ K^-$ process.


I. INTRODUCTION
The in-depth search for physics beyond the standard model (SM) plays an important role in the area of particle physics. It is known that the CP asymmetry, the symmetry violation of combination of charge conjugation (C) and parity (P), is the main source for matterantimatter asymmetry that is observed in our present universe. In the sector of quarks, the Cabbibo-Kobayasi-Maskawa(CKM) matrix indicates a message for an insight to the gate way of CP violation particularly in B and K meson decays in the SM. However, it is not sufficient to understand the observed baryon asymmetry. Recently, various experimental hunts are going on to probe the physics beyond the SM. In this regard, B meson system provides an important role to study prominent observables like branching ratio, and CP violating parameters such as direct and mixing induced CP asymmetry to probe new physics.
We would like to study the b → s penguin dominated B 0 s → K + K − decay mode which appears to have discrepancies in standard model values of CP-averaged branching ratio and CP violating parameters with the corresponding observed values. The theoretical result for the observables are given in the TABLE-I. Additionally, the TABLE-II shows the results from Belle, CDF, LHCb Collaborations along with world averages. Thus these discrepancies between observed and predicted values could lead to probe the physics beyond the SM.
In addition to this, the leptonic decays of pseudo-scalar meson B 0 s sector plays a vital role and enthusiastically makes more attention to explore the physics beyond the SM. In particular, we study B 0 s → µ + µ − decay mode because of careful observation of decay constant of neutral B 0 s meson from lattice. On the other side, the study of B 0 s → (where = e, τ ) put a less mark on the board as they have upper bounds. The former one has branching ratio with an upper limit of 2.8 × 10 −7 (90% C.L) [1], reported by LHCb where as < 6.8 × 10 −3 (95% C.L) reported by CDF Collaboration [2] for later decay mode. The SM values of branching ratio of B 0 s → τ + τ − and B 0 s → e + e − decays have O(10 −7 ) and O(10 −14 ) respectively where as for B 0 s → µ + µ − , it is of the order of 10 −9 [3]. Thus there is a possibility of contribution to both decays along with B 0 s → µ + µ − mode in the new physics scenario. Inspired by these discrepancies of the B 0 s → K + K − decay mode, we would like to investigate, in QCD factorisation approach, the NP effect on CP-averaged branching ratio as well as the CP violation parameters arising due to Z model where an extra U (1) gause boson Z takes part in the play. Several studies [4][5][6][7] have been done in the scenario of FCNC   [7][8][9][10] where an extra SU (2) L singlet down type quark has been added to SM quark sector which include a CP and flavor violating FCNC mediated by Z boson at tree level. The associated with Z (VLDQ) model can be constrained by using the experimental limit for all leptonic modes, and using the allowed parameter space, we examine the new physics impact on B 0 s → K + K − decay mode observables. The layout of this paper is structured as follows. In section II, we discuss the effective Hamiltonian responsible for the non-leptonic b → sqq processes. We have also presented the framework for B s → K + K − observables such as branching ratio and CP-violating parameters in the standard model. We constrain the new parameter space of Z model from the branching ratios of leptonic B s modes in section III, and address the footprint of this model on B s → K + K − process by using the new couplings. In section IV, we draw an attention to the interactions of the VLDQ model and check the impact on the aforementioned observables for B s → K + K − decay mode. In the end, our results are summarized in section V.
In the standard model, the penguin dominated B s → K + K − decay mode receives contribution from quark level transition b → s where the weak effective Hamiltonian describing the decay b → sqq is given as [22]  Using the framework of QCD factorization approach [23], the decay amplitude can be written in the form as where K + K − |O i |B 0 s fact represents the hadronic factorized matrix element, the second and third terms in the square bracket are higher order corrections. α s is the strong coupling constant, Λ QCD = 0.225 GeV is the QCD scale.
In the heavy quark limit, the amplitude of this decay mode can be represented as [23], where which includes the form factor F Bs→K 0 (0) at zero recoil momentum, and decay constants f Bs and f K . The expressions of coefficients α i and β i (b i ) are given in detail in ref. [23], which include factorisable along with non factorisable contributions to the above decay amplitude.
The CP-averaged branching ratio can be obtained using the following formula, where τ Bs (m Bs ) are the life time (mass) of B s meson and the center of mass (c.o.m) momentum in the rest frame of B meson is given as The time dependent CP asymmetry of B 0 s meson decaying to final CP eigenstate K + K − can be written as [24] where C KK = |λ| 2 −1 1+|λ| 2 and S KK = 2 Im(λ) 1+|λ| 2 are the direct and the mixing-induced CP asymmetries, respectively. The parameter λ is given as where, q and p are mixing parameters which are connected to the standard model CKM elements as Symbolically, the amplitude of the B s → K + K − decay mode can be written as where Ac | , γ is the weak phase of CKM element V ub , and δ 1 is the relative weak phase between A u and A c . From the amplitude given in 10, the parameters BR, C KK and S KK can be obtained respectively as For numerical computation of these observables in the SM, the CKM matrix elements along with the weak angle γ, all particle masses and the life time of B s meson are taken from [21]. We use the value of β s angle from [25]. The calculated Wilson coefficients in naive dimensional regularization scheme at NLO are taken from [22] at m b scale. The value of form factor F Bs (0) at zero recoil momentum and the decay constant f Bs are taken from [26].
In addition to this the decay constant f K is taken from [27]. Using these input values, the predicted SM values of the B s → K + K − observables are given as Here, the theoretical uncertainties for the above observables mainly arise from form factor, decay constants [26,27], CKM matrix elements [21].

III. Z MODEL
In this section we discuss the effects of new physics associated with Z model on the observables of B s → K + K − decay process. We constrain the Z new couplings by using the experimental limits on B s → (where is any charged leptons), mediated by the FCNC transitions b → s . These are the theoretically cleanest B decays as the only nonperturbative quantity involved in the description of these processes is the B s meson decay constant.
In the SM, the effective Hamiltonian for quark level transitions b → s + − is given by [28,29] where , λ (q) k = V kb V * kq and C i 's are the Wilson coefficients. Using effective Hamiltonian 17 , the transition amplitude for this process is given as where α is the fine structure constant. Here, we have used the vacuum insertion method to define the decay constant in the matrix element as where p µ B = p µ + + p µ − . In general, from equation 19 , the associated branching ratio is given as [30] BR Using the B s decay constant from [26] and remaining input parameters from PDG [21], the predicted SM branching ratio values are presented below. The errors in the SM results are coming mainly from decay constants and CKM matrix elements. Here we also show the corresponding experimental limits for all leptonic decay modes [21] .
Though B s → + − decays occur only at one-loop level in the SM, these processes can occur at tree level in the presence of new Z gauge boson arising due to the U (1) gauge extension of the SM. The effective Hamiltonian corresponding to the transition b → s + − process is given by [31,32] where g 1 (g ) is the coupling constant of Z(Z ) boson. According to the SM effective Hamiltonian 17 , the Hamiltonian in Z can be written as where the new Wilson coefficients are given as with φ s is the associated weak phase of U bs . We consider g g 1 ∼ 1 with the assumption that both the U (1) groups have same origin from some grand unified theory. For a TeV-scale Z boson, their ratio of masses M Z /M Z will be ∼ 10 −1 . In this analysis, the coupling of Z  for B s → K + K − decay mode is given as [5] H we obtain where C 9 , C 7 are the new Wilson coefficients arising due to Z gauge boson. Many studies have been done in [5,[33][34][35][36][37] with the manifestation of electroweak contribution assuming dd . Thus, Now for convenience, these coefficients can be written in the following parametric form as where the assumption of U L(R) qq ∼ 1 has been taken out from experimental data of B s meson [4]. The decay amplitude in presence of additional Z boson can be written as where termsα andβ arise due to new physics contributions. We can represent the above transition amplitude in the parametrized form as In addition to ℘ and a, given in the previous section, the new parameters in the above amplitude are defined as b = | ζt ζc |, ℘ = | A NP Ac |, and δ 2 is the relative strong phase.

IV. VECTOR LIKE DOWN QUARK (VLDQ) MODEL
Here we study the minimal extension of SM where the quark sector is expanded by an extra vector-like down quark. Because of this, we obtain a 4 × 4 matrix V iα ( i = u, c, t, t and α = d, s, b, b ) from which the interaction of this extra down-type quark with the SM quarks could be obtainable and scrutinize the deviations of the unitarity relation of the CKM matrix. This mixing provides a remarkable study of flavor changing neutral current (FCNC) interaction where Z particle is mediated through tree level contribution. In general, this model include the following Lagrangian [10].
where L denote the left handed chiral particles, i and α, β denote the generation indices for up-type and down-type quarks respectively. The second term in the above Lagrangian corre- sponds to the mixing in the down-type quark sector and the matrix Q αβ can be represented as Here V is not unitary as an extra down-type vector like quark of charge (− 1 3 ) has been added to the SM. It provides a new signal to probe the physics beyond the SM and modify the CP asymmetries and branching ratio predictions. We constrain the new parameters from the Br(B s → + − ), to be presented in the subsequent section.
A. B s → + − ( = e, µ, τ ) processes Though B s → + − process are suppressed in the SM, but can be significant in the presence of extra vector like down quark particle where Z is mediated at tree level whose contribution provides the physics beyond the SM. The branching ratio of B s → + − in Z mediated VLDQ model is given by [38] where The effective Hamiltonian corresponding to new interaction describing b → sqq can be represented as where the new Wilson coefficients at the M Z scale are given as [9,39] Here Q bs = |Q bs |e iφs and sin 2 θ W = 0.231. As the new couplings are in M Z scale, so these can be evolved to m b scale employing renormalization group equation [22]. By using RGE, these three couplings can be generated and we consider the above values from the ref. [40]. Now using the unitarity condition from equation 39 , we get To this relation, we can express the decay amplitude along with the new physics contributions as Here,α p andβ p provides the dominant contributions to NP amplitude which contain all the above three new couplings as given in the equation 40 . Symbolically, the full amplitude can be written as where Here, γ is the weak phase of V ub , φ s is the weak phase of Q bs and δ 1 (δ 2 ) is the relative strong phase between A u and A c (A N P and A c ). From the parametrized amplitude, the CP-averaged branching ratio can be written as On the other hand, the direct CP asymmetry can be written as One can obtain the mixing induced CP asymmetry parameter as where G = 1 + ( a ) 2 + ( b ) 2 and M = sin 2β s + 2ra cos δ 1 sin(2β s + γ) − 2r b cos δ 2 sin(2β s − φ s ) + (ra) 2 sin(2β s + 2γ) (51) Qbs=8.51×10 to the weak phase φ s for three benchmark Q bs values. Here, green,blue and orange colors represent the predictions obtained by using Q bs = 5 × 10 −4 , 7 × 10 −4 and 8.51 × 10 −4 , respectively. For φ below 50 • (above 290 • ) and all the discussed benchmark points of Q bs , the predicted branching of B s → KK are accommodating within 1σ range of experimental central value. We also notice that, the CP violating parameters are also explained within  Table III .

V. CONCLUSION
We have investigated the observables of B 0 s → K + K − , a penguin induced decay mode occurring at quark level transition b → s, both in standard model as well as beyond the SM scenarios. In the new physics scenario, we consider both the Z and vector-like down quark