Polarized forward-backward asymmetries of lepton pair in $B\rightarrow K_{1}\ell^{+}\ell^{-}$ decay in the presence of New physics

Double polarized forward-backward asymmetries in $B\rightarrow K_{1}(1270,1400)\ell^{+}\ell^{-}$ with $\ell=\mu , \tau$ decays are studied, using most general non-standard local four-fermi interactions, where the mass eigenstates $K_{1}(1270)$ and $K_{1}(1400)$ are the mixture of $^{1}{P}_{1}$ and $^{3}{P}_{1}$ states with the mixing angle $\theta_{K}$. We have calculated the expressions of nine doubly polarized forward-backward asymmetries and it is presented that the polarized lepton pair forward-backward asymmetries are greatly influenced by the new physics. Therefore, these asymmetries are interesting tool to explore the status of new physics in near future, specially at LHC.


I. INTRODUCTION
Rare B decays mediated through the flavor changing neutral current (FCNC) b → s(d) + − transitions not only provide a testing ground for the gauge structure of standard model (SM) but are also an effective way to look for the physics beyond the SM. As we know that in SM the Wilson coefficients C 7 , C 9 and C 10 of the operators O 7 , O 9 and O 10 at µ = m b are used to describe the b → s + − transition. Therefore, in these transitions, NP effects can be incorporated in two different ways: one is through new contributions to Wilson coefficients and the other is via the introduction of new operators in effective Hamiltonian which are absent in the SM.
Though the decay distribution of inclusive decays such as B → X s,d + − is theoretically better understood but hard to be measured experimentally. In opposite, the exclusive decays such as B → (K, K * , K 1 , ρ) + − are easy to detect experimentally but are tough to calculate theoretically as the difficulty lies in describing the hadronic structure, which are the main source of uncertainties in the predictions of exclusive rare decays.
The exploration of physics beyond the SM through various inclusive B meson decays such as B → X s,d + − and their corresponding exclusive processes, B → M + − with M = K, K * , K 1 , ρ etc., have already been studied [1][2][3][4][5]. These studies showed that the above mentioned inclusive and exclusive decays of B meson are very sensitive to the flavor structure of the SM and provide an effective way to explore NP effects.
It has been mentioned in [18] that measurement of many additional observables, would be possible by studying the simultaneous polarizations of both leptons in the final state, which in turn would be useful in testing the SM and highlighting new physics beyond the SM. It should be mentioned here that double lepton polariztion asymmetries in B → K * τ + τ − [19], B → K + − [20], B → ρ + − [21] and B → K 1 + − [22,23] have already been studied. Along with other observables, forward backward asymmetry is also an efficient observable to explore NP beyond the SM. In this regard, double lepton polarization forward-backward asymmertries in B → K * + − [24,25], B → K + − [26], B → ρ + − [27] and in B s → γ + − [28] have already been explored. We would like to emphasise here that the situation which makes B → K 1 + − decay more interesting than B → K * + − is the mixing of axial vector states K 1A and K 1B which are the 3 P 1 and 1 P 1 states respectively. Therefore, it is also interesting to see that how polarized forward-backward asymmetries of B → K 1 + − are influenced in the presence of new physics. So in the present work polarized forward-backward asymmetry in the exclusive decay B → K 1 + − are addressed using most general effective Hamiltonian, including all forms of possible interactions, similar to the case of B → K * + − [24] decay. The physical states K 1 (1270) and K 1 (1400) are superposition of the P-wave states in the following way If we define, y = sin θ K then above Eqs. become where the magnitude of the mixing angle θ K has been estimated [29] to be 34 • ≤ |θ K | ≤ 58 • and the study of B → K 1 (1270)γ impose the limit [30] on the mixing angle as where minus sign of θ K is related to the chosen phase of K 1A and K 1B [30]. The manuscript is presented as follows. In sec. II, we devise our required theoretical framework which is followed by two Subsections. II A and II B, relating to mixing of K 1 (1270) and K 1 (1400), form factors and constraints on the coefficients of NP operators used in this study. Sec. III, is devoted to analytical calculations and the explicit expressions of doubly polarized forward-backward asymmetries. In Sec. IV, we give the numerical analysis with discussion about the observables underconsiderations. We end our work by giving concluding remarks in Sec. V.

II. THEORETICAL FORMALISM
At the quark level B → K 1 (1270, 1400) + − decays are induced by the transition b → s + − , which in the SM, is described by the following effective Hamiltonian [31] H where R, L = (1 ± γ 5 ) /2 are the projector operators and q 2 is the square of momentum transfer while C s are Wilson coefficients. The effective Wilson coefficient C ef f SM 9 (µ), can be decomposed into the following three parts [3,5] C ef f SM where the parameters z andŝ are defined as z = m c /m b ,ŝ = q 2 /m 2 b . It is important to mention here that in our numerical calculations of asymmetries and their average values, we do not include Y LD (z,ŝ), otherwise the asymmetries would be largely effected by the contributions of J/ψ and ψ(2s) resonance around s = 10GeV 2 and s = 14GeV 2 respectively. The explicit expressions for short-distance contributions Y SD (z,ŝ) and long distance contributions Y LD (z,ŝ) are given in [3,5].
New physics effects are explored for B → K 1 l + l − channel by considering the most general local four-fermi interactions. In this regard the total effective Hamiltonian is given by where while H SM ef f is given in Eq. (4) and C X are the coefficients of the four-Fermi interactions. Defining the combinations where R A , R V , R A , R V , R S , R P , R S , R P , C T and C T E represents the NP couplings. Using the expression of the effective Hamiltonian Eq. (5) the decay amplitude for B → K 1 l + l − is given by Note: One can also consider the new physics contribution coming from the operator O 7 = C 7s σ µν bLF µν . However, in the present study we do not include these effects. The hadronic matrix elements of quark operators appearing in Eq. (7) over the meson states, for the exclusive B → K 1 (1270, 1400) + − decays can be parameterized in terms of the form factors as: where p B (p k1 ) are the momenta of the B(K 1 ) mesons and ε µ correspond to the polarization of the final state axial vector K 1 meson. In Eq.(8) we have with with F 1 (0) = 2F 2 (0). Where Eqs. (12a,12b) are obtained by contracting Eq. (11) with q ν . Moreover, the matrix element K 1 (p k1 , ε)|s(1 ± γ 5 )b|B(p B ) can be calculated by contracting Eq. (8) with q µ and by making use of the equation of motions along with Eq. (10), we have where the mass of strange quark has been neglected. As the physical states K 1 (1270) and K 1 (1400) are mixed states of the K 1A and K 1B with mixing angle θ K as defined in Eqs. (1)(2). The B → K 1 form factors can be parameterized as [22] where the mixing matrix M is So the form factors A K1 , V K1 0,1,2 and F K1 0,1,2 satisfy the following relations where we have supposed that p µ K1(1270),K1(1400) p µ K 1A ,K 1B . Using the above matrix elements, the decay amplitude for B → K 1 l + l − can be written as The auxiliary functions appearing in (24) can be written as follows:

B. Phenomenological bounds on NP couplings
In the present paper, we use the constraints on the NP couplings parameters from A. Kumar et al [32]. However, for self consistency these bounds are given below: In the absence of R V,A the bounds are however these bounds are weakened when we include R V,A On the other hand the constraints on tensor coupling entirely come from B(B → X s µ + µ − ) which are The limits on scalar and pseudo scalar couplings are extracted from B(B 0 and from B(B → X s µ + µ − ) [33,34] are

III. ANALYTICAL CALCULATIONS OF DOUBLY POLARIZED FORWARD BACKWARD ASYMMETRIES
Now we have all the ingredients to calculate the physical observables. The double differential decay rate is given as [32] By using the expression of the decay amplitude given in Eq. (24) one can get the expression of the dilepton invariant mass spectrum as where we first define the six orthogonal vectors belonging to the polarizations of l − and l + which we denote here by S i and W i respectively where i =L, N and T corresponding to the longitudinally, Normally and transversally polarized lepton l ± respectively. [18,35,36] S µ L ≡ (0, e L ) = 0, where p + , p − and p K1 denote the three momenta vectors of the final particles l + , l − and K 1 respectively. These polarization vectors S µ i (W µ i ) in Eqs. (32) and (33) are defined in the rest frame of l − (l + ). When we apply lorentz boost to bring these polarization vectors from rest frame of l − (l + ) to the centre of mass frame of l + and l − , only the longitudinal polarization four vector get boosted while the other two polarization vectors remain unchanged. After this operation the longitudinal four vector read as To achieve the polarization asymmetries one can use the spin projector 1 2 (1 + γ 5 / S) for l − and for the l + spin projector is 1 2 (1 + γ 5 / W ). Normalized, unpolarized differential forward-backward asymmetry is defined as When the spins of both leptons are taken into account, the A F B will be a function of the spins of final leptons, and is defined as Using these definitions for the double polarized FB asymmetries, we have found the expressions of numerators as follows: Note: It is worthful to mention here we have included short distance part, Y SD (z,ŝ), of C ef f SM 9 in our numerical calculation which contains also the imaginary part, therefore, in A N T F B (A T N F B ) and A LN F B (A N L F B ) only those terms contribute which contain auxiliary functions A, B 1 and B 2 .

IV. NUMERICAL RESULTS AND DISCUSSION
In this section we examine the effects of different new physics operators on polarized lepton pair forward-backward asymmetries. For this purpose, we analyze the behaviour of polarized FB asymmetries and their average values in the presence of constraints on NP couplings that are given in section II B. Regarding this, different scenarios for NP Lorentz structure are displayed in Table IV-VI. Numerical values of different input parameters are given in Table I, while the SM Wilson coefficients at µ = m b are given in Table II. In addition to calculate the numerical values of observables under consideration, we have used the light-cone QCD sum rules form factors [38], summarized in Table  III. The momentum dependence dipole parametrization for these form factors is:  where T denotes the A, V or F form factors and the subscript i can take the value 0, 1, 2 or 3. The superscript X belongs to K 1A or K 1B state. Before proceeding to analyze the NP, first we would like to mention here that the authors of ref [30,39] concluded that all observables such as branching ratio, forward backward and single lepton polarization asymmetries, etc for B → K 1 (1430)µ + µ − are sensitive to mixing angle θ K . In this context, it is interesting to see the dependence of the values of double lepton polarizations forward-backward asymmetries on mixing angle θ K . In this study, we have found that A LL F B , A LT F B and A T L F B are sensitive to θ K for the decay B → K 1 (1430)µ + µ − as shown in fig 11(a-c) but not much sensitive for B → K 1 (1270)µ + µ − . Therefore, besides te other observables, the precise measurements of these asymmetries (for former decay channel) at LHC may also provide help to put some stringent constraint on the mixing angle θ K in near future. However, as it is mentioned in ref. [40] that the branching ratio for K 1 (1430) is two order suppressed i.e. Br(B → K 1 (1270)µ + µ − (τ + τ − )) are of the order of 10 −6 (10 −8 ) while Br(B → K 1 (1430)µ + µ − (τ + τ − )) are of the order of 10 −8 (10 −10 ). For this reason we are not interested in the results of Now to see the behaviour of double polarized FB asymmetries under the influence of new physics couplings, we have drawn the s-dependence of these asymmetries in figs. 1-10. In all these graphs the grey shaded band corresponds to the region of the SM values of these asymmetries due to uncertainties in mixing angle θ K while dashed line corresponds to the SM value when the central values of the form factors are taken. In fig. 1 (6 fig. 1a depicts scenario S1(see Table IV fig. 2c shows the case of S9 when both tensor couplings C T and C T E are present with opposite polarity. It is important to mention here that only those scenarios of all NP couplings are shown in figures for which the zero position of A LL F B is shifted distinctly in comparison to that of the zero position in SM. In contrast to B → K 1 (1270)µ + µ − , A LL F B does not have zero crossing for B → K 1 (1270)τ + τ − . figs. 6(a-c) depict scenarios S1, S4, S6 and fig. 7(a-c) depict scenarios S7, S8 and S9 which show, respectively, the possible effects  [38], where a and b are the parameters of the form factors in dipole parametrization.
when only vector and tensor type couplings are present in A LL F B for B → K 1 (1270)τ + τ − . In all these scenarios the value of A LL F B remains positive in high s region as predicted by SM value except S7. fig. 7a shows that when tensor coupling C T is present only (Scenario S7), A LL F B can get the negative values in opposite to SM prediction. Therefore, if negative values of A LL F B are measured in future experiments for B → K 1 (1270)τ + τ − , these results will be unambiguous indication of existence of new physics beyond the SM (i.e. existence of tensor type interactions). It is emphasized here that in our analysis only A LL F B , A LT F B and A T L F B are observed to be considerably effected by NP couplings of different types. Therefore the other remaining polarized lepton pair forward-backward asymmetries are not discussed.
Moreover, we eliminate the dependence of forward-backward polarized asymmetries on s by performing integration over s and find the average values of above mentioned asymmetries which are also experimentally useful tools to explore the new physics. We calculate the averaged double lepton polarization forward-backward asymmetries by As mentioned in Sec. II that in the calculation of average values we do not include long distance contribution, Y LD (z,ŝ). Now we discuss the effects of NP on A ij F B , in the following sections.

A. Tensor type interactions present only
In this section, we discuss the explicit dependence of tensor type couplings on the average values of different polarized forward-backward asymmetries. For this purpose 12e and 12f show the effects of NP tensor and axial tensor operators, respectively, on A ij F B for the case of muons. fig.12e depicts the scenario S7 (see Table-VI. i.e. when C T present only), in which A LL F B significantly varies from its SM value. The value of A LL F B increases and reaches to a maximum value of ≈ 0.21 and then again decreases within the allowed range (−1.14 ≤ C T ≤ 1.14). It is also clear that When only R V and R A couplings are present, for the case of muons, figs. 12a and 12b represent scenarios S1 and S2 respectively. In fig. 12a, When the value of R A = −1.10 is fixed and R V is varied in allowed range from -6.5 to 1, A LL F B is drastically changed from its SM value, while A LT F B and A T L F B are also modified appreciably from their SM values. The value of A LL F B remains negative for the values of R V from -6 to -3 and it acquires positive values for (−3 ≤ R V ≤ 1), where the maximum value A LL F B = 0.24 is observed at R V = 1. It is also clear from this plot that A LT F B and A T L F B follow the opposite pattern, such that A LT F B ( A T L F B ) remains positive (negative) for (−6 ≤ R V ≤ −1.2) and negative (positive) from -1.2 to 1. Similarly when S2 is considered ( fig. 12b) In conclusion, we calculate double polarized FB asymmetries using most general model independent form of the effective Hamiltonian including all possible non-standard local four-fermi interactions. Our analysis shows that similar to the other observables, polarized FB asymmetries are also sensitive to the mixing angle θ K . While considering the different NP scenarios our analysis exhibit that the averaged double lepton polarization forward-backward asymmetries are very sensitive to NP couplings. The key points are as under.
When vector axial-vector couplings are considered for the decay B → K 1 (1270)µ + µ − , only averaged polarized forward-backward asymmetries, A LL Additionally, the dependence of polarized lepton pair forward-backward asymmetries A LL F B , A LT F B and A T L F B on s for the decay B → K 1 (1270)µ + µ − depict the left and right-side shifting of zero crossing positions of these forwardbackward polarized asymmetries from their corresponding SM values, when vector axial-vector and tensor type NP operators are considered. Moreover, signs of some of these polarized FB asymmetries are also flipped for few allowed values of different NP couplings. Similar conclusion is drawn for the case of tauons as final state leptons.