Leptoquark effects on $b \to s \nu \bar{\nu}$ and $B \to K l^+ l^-$ decay processes

We study the rare semileptonic decays of $B$ mesons induced by $b \to s \nu \bar{\nu}$ as well as $b \to s l^+ l^-$ transitions in the scalar leptoquark model where the leptoquarks transform as $(3,2,7/6)$ and $(3,2,1/6)$ under the standard model gauge group. The leptoquark parameter space is constrained using the most recent experimental results on ${\rm Br}(B_s \rightarrow \mu^+ \mu^-)$ and ${\rm Br}(B_d \to X_s \mu^+ \mu^-)$ processes. Considering only the baryon number conserving leptoquark interactions, we estimate the branching ratios for the exclusive $\bar{B} \rightarrow \bar{K}^{(*)} \nu \bar{\nu}$ and inclusive $B \rightarrow X_s \nu \bar{\nu}$ decay processes by using the constraint parameters. We also obtain the low recoil (large lepton invariant mass, i.e., $q^2 \sim m^2_b$) predictions for the angular distribution of $\bar B \rightarrow \bar{K} l^+ l^-$ process and several other observables including the flat term and lepton flavour non-universality factor in this model.


I. INTRODUCTION
It is well-known that the study of B physics plays an important role to critically test the standard model (SM) predictions and to look for possible signature of new physics beyond it. In particular, the rare decays of B mesons which are mediated by flavour changing neutral current (FCNC) transitions are well-suited for searching the effects of possible new interactions beyond the SM. This is due to the fact that the FCNC transitions b → s, d are highly suppressed in the SM as they occur only at one-loop level and hence, they are very sensitive to new physics. Recently the decay modes B → K ( * ) l + l − , which are mediated by the quark level transition b → sl + l − have attracted a lot of attention, as several anomalies at the level of few sigma are observed in the LHCb experiment [1][2][3]. Furthermore, the deviation in the ratio of rates of B → Kµµ over B → Kee (R K ) is a hint of violation of lepton universality [4]. This in turn requires the careful analyses of the angular observables for these processes both in the low and high q 2 regime.
Recently various B physics experiments such as BaBar, Belle, CDF and LHCb have provided data on the angular distributions of B → K * l + l − and B → Kl + l − decay processes both in the low and the large recoil region except the intermediate region around q 2 ∼ m 2 J/ψ and m 2 ψ ′ . The intermediate region is dominated by the pronounced charmonium resonance background induced by the decays B → K(cc) → Kl + l − , wherecc = J/ψ, ψ ′ . Using QCD factorization method the physical observables in the high recoil region can be calculated and the angular distribution ofB →Kl + l − at low recoil can be computed using simultaneous heavy quark effective theory and operator product expansions in 1/Q, with Q = (m b , q 2 ) i.e. q 2 is of the order of the b-quark mass [5,6]. In this work, we are interested to study the decay process B → Kll in the region of low hadronic recoil i.e. above the ψ ′ peak in the scalar leptoquark (LQ) model. We have studied the B → Kµ + µ − in the large recoil limit in Ref. [7] and found that the various anomalies associated with the isospin asymmetry parameter and the lepton flavour non-universality factor (R K ) for this process can be explained in this model.
Similarly the rare semileptonic decays of B mesons with νν pair in the final state, i.e., B → K ( * ) νν are also significantly suppressed in the SM and their long distance contributions are generally subleading. These decays are theoretically very clean due to the absence of photonic penguin contributions and strong suppression of light quarks. The experimental measurement of the inclusive decay rate probably be un-achievable due to the missing neutrinos, however, the exclusive channels like B → K * νν and B → Kνν are more promising as far as the measurement of branching ratios and other related observables are concerned.
Theoretically, study of these decays requires calculation of relevant form factors by nonperturbative methods.
In recent times, there are many interesting papers which are contemplated to explain the anomalies associated with the b → sl + l − processes, observed at LHCb experiment [1][2][3][4], both in the context of various new physics models as well as in model independent ways [8][9][10][11].
In this paper, we intend to study the effect of scalar leptoquarks transform as (3, 2, 7/6) and (3, 2, 1/6) under the standard model gauge group, on the branching ratio as well as on other asymmetry parameters in the low-recoil region of B → Kl + l − process. We also consider the processes B → K ( * ) νν and B → X s νν involving the quark level transitions b → sνν in the full physical regime. It is well-known that leptoquarks are scalar or vector color triplet bosonic particles which make leptons couple directly to quarks and vice versa and carry both lepton as well as baryon quantum numbers and fractional electric charge. Leptoquarks can be included in the low energy theory as a relic of a more fundamental theory at some high energy scale in the extended SM [12], such as grand unified theories [12,13], Pati-Salam models, models of extended technicolor [14] and composite models [15]. Leptoquarks are classified by their fermion number (F = 3B + L), spin and charge. Usually they have a mass near the unification scale to avoid rapid proton decay, even so leptoquarks may exist at a mass accessible to present collider, if baryon and lepton numbers would conserve separately.
The plan of the paper is follows. In section II we present the effective Hamiltonian responsible for b → sl + l − processes. We also discuss the new physics contributions due to the exchange of scalar leptoquarks. In section III we discuss the constraints on leptoquark parameter space by using the recently measured branching ratios of the rare decay modes The branching ratio, the flat term and the lepton nonuniversality factor (R K ) for the decay modeB →Kl + l − , where l = e, µ, τ at low recoil limit are computed in section IV. In section V we work out the branching ratio ofB →Kνν process in the full kinematically accessible physical region. The branching ratio, polarization and other asymmetries inB →K * νν process have been computed in section VI. The inclusive decay process B → X s νν is discussed in section VII and section VIII contains the summary and conclusion.

II. THE EFFECTIVE HAMILTONIAN FOR
The effective Hamiltonian describing the processes induced by the FCNC b → sl + l − transitions is given by [21] H It should be noted that the primed operators are absent in the SM. The values of Wilson coefficients C i=1,··· ,10 , which are evaluated in the next-to-next leading order at the renormalization scale µ = m b are taken from [22]. Here V qq ′ denotes the CKM matrix element, G F is the Fermi constant, α is the fine-structure constant and P L,R = (1 ∓ γ 5 )/2 are the chiral projectors. Due to the negligible contribution of the CKM-suppressed factor V ub V * us , there is no CP violation in the decay amplitude in the SM. These processes will receive additional contributions due to the exchange of scalar leptoquarks. In particular there will be new contributions to the electroweak penguin operators O 9 and O 10 as well their righthanded counterparts O ′ 9 and O ′ 10 . In the following subsection we will present these additional contributions to the SM effective Hamiltonian due to the exchange of such leptoquarks.
A. Scalar LQ Contributions to b → sl + l − effective Hamiltonian There are ten different types of leptoquarks under the SU(3) ×SU(2) ×U(1) gauge group [23], half of them have scalar nature and other halves have vector nature under the Lorentz transformation. The scalar leptoquarks have spin zero and could potentially contribute to the quark level transition b → sl + l − . Here we would like to consider the minimal renormalizable scalar leptoquark model [17], containing one single additional representation of SU(3) × SU(2) × U(1), which does not allow proton decay. There are only two such models with representations under the SM gauge group as ∆ (7/6) ≡ (3, 2, 7/6) and ∆ (1/6) ≡ (3, 2, 1/6) [17], which have sizeable Yukawa couplings to matter fields. These scalar leptoquarks do not have baryon number violation in the perturbation theory and could be light enough to be accessible in accelerator searches. The interaction Lagrangian of the scalar leptoquark ∆ (7/6) with the fermion bilinear is given as [18] where Q L is the left handed quark doublet and l R is the right-handed charged lepton singlet.
After performing the Fierz transformation and comparing with the SM effective Hamiltonian (1), one can obtain the new Wilson coefficients as discussed in Ref. [18] Similarly, the Lagrangian for the coupling of scalar leptoquark ∆ (1/6) to the SM fermions is given by where τ 2 is the Pauli matrix and consists of operators with right-handed quark currents.
Proceeding like the previous case one can obtain the new Wilson coefficients as which are associated with the right-handed semileptonic electroweak penguin operators O ′ 9 and O ′ 10 .

III. CONSTRAINT ON THE LQ PARAMETERS
After having the idea of possible scalar leptoquark contributions to the b → sll processes we now proceed to constraint the LQ couplings using the theoretical [24] and experimental branching ratio [25][26][27] of B s → µ + µ − process. This process is mediated by b → sµµ transition and hence well-suited for constraining the LQ parameter space. In the SM the branching ratio for this process depends only on the Wilson coefficient C 10 . However, in the scalar LQ model there will be additional contributions due to the leptoquark exchange which are characterized by the new Wilson coefficients C N P 10 and C ′ N P 10 depending on the nature of the LQs. Thus, in this model the branching ratio has the form [7,19] Br which can be expressed as where Br SM is the SM branching ratio and we define the parameters r and φ N P as Now comparing the SM theoretical prediction of Br(B s → µµ) [24] Br with the corresponding experimental value one can obtain the constraint on the new physics parameters r and φ N P . The constraint on the leptoquark parameter space has been extracted in [7,19] from this process, therefore, here we will simply quote the results. The allowed parameter space in r − φ N P plane which is compatible with the 1σ range of the experimental data is 0 ≤ r ≤ 0.1 for the entire range However, in this analysis we will use relatively mild constraint, consistent with both mea- It should be noted that the use of this limited range of CP phase, i.e., (π/2 ≤ φ N P ≤ 3π/2) is an assumption to have a relatively larger value of r. These bounds can be translated to obtain the bounds for the leptoquark couplings as After obtaining the bounds on leptoquark couplings, we now proceed to study the decay processes B → Kll and B → K ( * ) (X s )νν and the associated observables in the following sections.
The transition amplitude for the B → Kl + l − decay process can be obtained using the effective Hamiltonian presented in Eq. (1). The matrix elements of the various hadronic currents between the initial B meson and the final K meson can be parameterized in terms of the form factors f 0 , f T and f + as [28] K where p, k are the four-momentum of the B-meson and Kaon respectively and q = p − k is the four-momentum transferred to the dilepton system. Furthermore, using the QCD operator identity [5,29,30], an improved Isgur-Wise relation between f T and f + can be obtained as where strange quark mass has been neglected. Thus, one can obtain the amplitude for thē B →Kl + l − process in low recoil limit [28,31], after applying form factor relation (18) as where In Eqn. (20), and C tot 10 = C SM 10 + C N P 10 − C ′ N P 10 , where C (′) N P 9 and C (′) N P 10 are the new contributions to the Wilson coefficients arising due to the exchange of leptoquarks and the effective Wilson coefficients C ef f 7,9 are given in Ref. [32]. The corresponding differential decay distributions is given by where θ l is the angle between the directions ofB meson and the l − , in the dilepton rest frame. The expressions for the q 2 dependent parameters a l , c l are presented in Appendix A.
Thus, the decay rate for the processB →Kl + l − can be written as Another useful observable known as the flat term is defined as where the hadronic uncertainties are reduced due to cancellation between the numerator and denominator. It should be noted that the lepton mass suppression of (a l + c l ) follows as (F l H ) SM ∝ m 2 l , hence, it vanishes in the limit m l → 0. After obtaining the expressions for branching ratio and the observable F l H , we now proceed for numerical estimation for B → Kl + l − process in the low recoil region. In our analysis we use the following parametrization for the q 2 dependence of form factors f i (i = +, T, 0) as [28,33] where we have used the notation q 2 ≡ s. The z(s) functions are given as The values of f i (0) and b i 1 are taken from [28]. For numerical evaluation, we have used the particle masses and the lifetimes of B meson from [34]. For the CKM matrix elements, we have used the Wolfenstein parametrization with values A = 0.814 +0.023 −0.024 , λ = 0.22537 ± 0.00061,ρ = 0.117 ± 0.021 andη = 0.353 ± 0.013 and the fine structure coupling constant α = 1/137. With these input parameters, the differential branching ratios forB 0 d →K 0 e + e − (left panel),B 0 d →K 0 µ + µ − (right panel) andB 0 d →K 0 τ + τ − (lower panel) processes with respect to high q 2 , both in the SM and in the leptoquark model are shown in Fig. 1 for ∆ (7/6) leptoquark and in Fig. 2 for ∆ (1/6) .
The grey bands in these plots correspond to the uncertainties arising in the SM due to the uncertainties associated with the CKM matrix elements and the hadronic form factors. The green bands correspond to the LQ contributions. For B → Kµµ process, we vary the values of the leptoquark couplings as given in Eq. (14) and for B → Kee and B → Kτ τ processes we use the limits on the LQ couplings extracted from B d → X s e + e − and B s → τ + τ − processes [7] as and Since the leptoquark couplings are more tightly constrained in b → sµµ transitions, the deviations of the branching ratios in the LQ model from the corresponding SM values are found to be small. For B → Kee and B → Kτ τ these deviations are found to be significantly large. The bin-wise experimental values are shown in black in B → Kµµ process. From these figures it can be seen that the observed experimental data can be explained in the scalar LQ model but the deviation from the SM branching ratios are more in the ∆ (1/6) model. For the other observables in B → Kll processes we will show the results only for ∆ (7/6) leptoquark model. In Fig. 3, we have shown the lepton non-universality factors R µe K (left panel) (i.e. the ratio of branching ratios ofB →Kµ + µ − andB →Ke + e − ), R τ e K (right panel) and R τ µ K (lower panel) variation with high q 2 . From the figure one can see that there is significant deviations in the lepton-flavour non universality factor from their corresponding SM values in all the above three cases. The flat term for theB 0 d →K 0 µ + µ − (left panel) andB 0 d →K 0 τ + τ − (right panel) decay processes in the low recoil region are presented in Fig. 4 for ∆ (7/6) . In this case there is practically no deviation in B → Kµµ whereas there is significant deviation in B → Kτ τ process. The integrated branching ratios, flat terms and the lepton flavour non-universality factors for the B → Kll processes over the range 14.18, 22.84] are given in Table I. The flat term for B → Ke + e − process has been found to be negligibly small (F e H ∼ O(10 −7 )) due to tiny electron mass. In the low recoil region, the process having tau lepton in the final state has significant deviation from the SM.
The integrated branching ratio for B 0 → Kµµ process in the range q 2 ∈ [15, 22] GeV 2 has been measured by the LHCb Collaboration [1] and is given as Our predicted value in this range of q 2 is found to be The predicted values of the branching ratios are slightly higher than the central measured value but consistent with its 1-σ range.

V. B → Kνν PROCESS
The B → Kνν process is mediated by the quark level transition b → sνν and the effective Hamiltonian describing such transition is given as [35] H where (30) are the dimension-six operators and C ν L,R are their corresponding Wilson coefficients. The coefficient C ν R has negligible value within the standard model while C ν L can be calculated by using the loop function and is given by The necessary loop functions are presented in Appendix B. The decay distribution with respect to the di-neutrino invariant mass can be expressed as [36] dΓ The decay rate has been multiplied with an extra factor 3 due to the sum over all neutrino flavours. It should be noted that in Eq. (32) C ν R is the new Wilson coefficient arises due to the exchange of the leptoquark ∆ (1/6) . In order to find out its value, we consider the new contribution to the effective Hamiltonian due to the exchange of such leptoquark which is given as

Oservables
SM predictions Values in ∆ (7/6)   Comparing Eqs. (29) and (33), one can obtain the new Wilson coefficient as For numerical estimation, we use the B → K form factor f K + evaluated in the light cone sum rule (LCSR) approach [37] as which is valid in the full physical region. Furthermore, in contrast to B → Kl + l − process, which has dominant charmonium resonance background from there are no such analogous long-distance QCD contributions in this case as there are no intermediate states which can decay into two neutrinos. For the b → sνν LQ couplings we use the values as we used for b → sµµ as these two processes are related by SU(2) L symmetry. The variation of branching ratio with respect to s B in the full physical regime 0 s B (1 −m K ) 2 is shown in Fig. 5 and the predicted branching ratio is given in Table   II, which is well below the present upper limit Br(B 0 d → Kνν) < 4.9 × 10 −5 [34]. The study of B → K * νν is also quite important as this process is related to B → K * µµ process by SU(2) L and therefore, the recent LHCb anomalies in B → K * µµ would in principle also show up in B → K * νν. The experimental information about this exclusive decay process can be described by the double differential decay distribution. In order to compute the decay rate, we must have the idea about the matrix element of the effective Hamiltonian (29) between the initial B meson and the final particles. Due to the nondetection of the two neutrinos, experimentally we can't distinguish between the transverse polarization, so the decay rate will be the addition of both longitudinal and transverse polarizations. The double differential decay rate with respect to s B and cos θ is given by [35,38] where the longitudinal and transverse decay rate are The transversality amplitudes A ⊥, ,0 in terms of the form factors and Wilson coefficients are listed in Appendix C.
The fractions of K * longitudinal and transverse polarizations are given as and the K * polarization factor is The transverse asymmetry parameters are given as [39,40] A (1) However, one can't extract A T from the full angular distribution of B → K * νν, as it is not invariant under the symmetry of the distribution function and requires measurement of the neutrino polarization. So it can't be measured experimentally at B factories or in LHCb.
The transverse asymmetry A (2) T is theoretically clean and could be measurable in Belle-II. For numerical evaluation we use the q 2 dependence of the B → K * form factors V (q 2 ), A 1 (q 2 ), A 2 (q 2 ) from [41,42]. The variation of the branching ratio of B → K * νν with respect to the neutrino invariant mass, s B is shown in Fig. 6. Fig. 7 contains the longitudinal and transverse polarizations of K * verses s B . The polarization factor and the transverse asymmetry variation with respect to s B in the full region are shown in Fig. 8.
Although there is certain deviation found between the SM and LQ model predictions for the branching fraction, but no such noticeable deviations found between the SM and LQ predictions for the longitudinal/transverse polarizations, transverse asymmetry parameters A (2) T . The integrated values of branching ratio over the range s B ∈ [0, 0.69] are presented in Table II, which are well below the the present upper limit Br(B 0 d → K * νν) < 5.5 × 10 −5 [34].
wherem i = m i /m b and κ(0) = 0.83 is the QCD correction to the b → sνν matrix element [43]. The full kinematically accessible physical region is 0 ≤ s b ≤ (1 −m s ) 2 . In Fig. 9, we have shown the variation of the branching ratio with respect to s b and the integrated branching ratio values over the range s b ∈ [0, 0.96] both in the SM and in the leptoquark model are presented in Table II.
We define the ratio of branching ratios as [36], and and the variation of R K and R K * with respect to s B in the full kinematically allowed region is shown in Fig. 10. In this case also no deviation found between the SM and leptoquark predictions.

VIII. CONCLUSION
In this paper we have studied the effect of scalar leptoquarks on the rare semileptonic decays of B meson. In particular, we focus on the decay processes B → Kl + l − in low recoil  The predicted branching ratios for B → (K, K * , X s )νν processes and R K,K * for B → X s νν in their respective full physical ranges.
limit and the di-neutrino decay channels B → K ( * ) (X s )νν. The leptoquark parameter space is constrained by considering the recently measured branching ratios of B s → µ + µ − and B d → X s µ + µ − processes. Using the allowed parameter space we predicted the branching ratio, lepton non-universality factors and the flat terms for the B → Kl + l − process in the low recoil region. We found that the measured branching ratio can be accommodated in the scalar leptoquark model. We have also calculated the branching ratios of B → K ( * ) νν and B → X s νν processes. The predicted branching ratios for B → K ( * ) νν processes are well below the present upper limits. The polarization of K * and transverse asymmetry for B → K * νν are also computed using the constraint leptoquark parameters. However, we found no deviation between the SM prediction and the LQ results for different polarization variables and the transverse asymmetry parameter.

Acknowledgments
We would like to thank Science and Engineering Research Board (SERB), Government of India for financial support through grant No. SB/S2/HEP-017/2015. The a l and c l parameters in the decay distribution of the B → Kl + l − processes (21) can be expressed as and 3 · 2 10 π 5 s B λ 1/2 (1,m 2 K * , s B ) The various form factors V (s B ), A 1 (s B ), A 2 (s B ) associated with B → K * transition in Eqs.
(C1-C3) are defined as where q = p l + + p l − and ǫ µ is the polarization vector of K * .