Impact of scalar leptoquarks on heavy baryonic decays

We present a study on the impact of scalar leptoquarks on the semileptonic decays of $ \Lambda_b$, $\Sigma_b $ and $\Xi_b $. To this end, we calculate the differential branching ratio and lepton forward-backward asymmetry defining the processes $ \Lambda_b \rightarrow \Lambda \ell^+ \ell^-$, $\Sigma_b \rightarrow \Sigma \ell^+ \ell^-$ and $\Xi_b \rightarrow \Xi \ell^+ \ell^-$, with $\ell$ being $\mu$ or $\tau$, using the form factors calculated via light cone QCD in full theory. In calculations, the errors of form factors are taken into account. We compare the results obtained in leptoquark model with those of the standard model as well as the existing lattice QCD predictions and experimental data.


I. INTRODUCTION
The physics of transitions based on b → sℓ + ℓ − at quark level constitutes one of the main directions of the research in high energy and particle physics both theoretically and experimentally as new physics effects can contribute to such decay channels. The flavor changing neutral current (FCNC) transitions of Λ b → Λℓ + ℓ − , Σ b → Σℓ + ℓ − and Ξ b → Ξℓ + ℓ − are among important baryonic decay channels that can be used as sensitive probes to indirectly search for new physics contributions. Especially, the rare Λ b → Λℓ + ℓ − decay channel has been in the focus of much attention in recent years both theoretically and experimentally. The first measurement on the Λ b → Λµ + µ − process has been reported by the CDF Collaboration [1] with 24 signal events and a statistical significance of 5.8 Gaussian standard deviations. Using the pp collisions data samples corresponding to 6.8f b −1 and √ s = 1.96 TeV collected by the CDF II detector, the differential branching ratio for the Λ b → Λµ + µ − decay channel has been measured to be dBr(Λ 0 b → Λµ + µ − )/dq 2 = [1.73 ± 0.42(stat) ± 0.55(syst)] × 10 −6 [1]. The differential branching fraction of Λ 0 b → Λµ + µ − decay channel has also been measured as dBr(Λ 0 b → Λµ + µ − )/dq 2 = (1.18 + 0.09 − 0.08 ±0.03±0.27)×10 −7 GeV 2 /c 4 at 15 GeV 2 /c 4 ≤ q 2 ≤ 20 GeV 2 /c 4 region by the LHCb Collaboration [2]. The LHCb Collaboration has also measured the lepton forward-backward asymmetries associated to this transition as A µ F B = −0.05 ± 0.09(stat) ± 0.03(syst) at 15 GeV 2 /c 4 ≤ q 2 ≤ 20 GeV 2 /c 4 region [2]. The orders of branching fractions in Λ b → Λe + e − , Λ b → Λτ + τ − as well as those of Σ b → Σℓ + ℓ − and Ξ b → Ξℓ + ℓ − with all leptons indicate that (see Refs. [3][4][5][6][7]) these channels are accessible at LHC. We hope with the RUN II data at the center of mass energy 13 TeV it will be possible to measure different physical quantities related to these FCNC loop level rare transitions in near future.
The LHC RUN II may provide opportunities to search for various new physics scenarios. One of the important new physics models that has been proposed to overcome the problems of some inconsistencies between the SM predictions and experimental data, is the lep-toquark (LQ) model. As an example for the LHC constraints and prospects for scalar leptoquarks explaining the B → D ( * ) τ ν anomaly see [8]. LQs are hypothetical and color triplet boson particles that are formed from a lepton and a quark [9]. LQs carry both baryon (B) and lepton (L) quantum numbers with color and electric charge. The spin number of a leptoquark state can be 0 or 1, corresponding to a scalar leptoquark or vector leptoquark. If the leptoquarks violate both the baryon and lepton numbers, they are generally considered to be heavy particles at the level of O(10 15 ) GeV in order to prevent the proton decay. For more detailed information about leptoquark models and the recent experimental and theoretical progresses, see .
In the light of progresses about LQs, we calculate the differential branching ratio and lepton forward-backward asymmetry corresponding to the Λ b → Λℓ + ℓ − , Σ b → Σℓ + ℓ − and Ξ b → Ξℓ + ℓ − processes in a scalar LQ model. In the calculations, we use the form factors as the main inputs calculated from the light cone QCD sum rules in full theory without any approximation. We also encounter the errors of the form factors to the calculations. We compare the regions swept by the LQ model with those of the SM and search for deviations of the LQ model predictions with those of the SM. We also compare the results with the available experimental data.
The outline of this article is as follow. In next section, we present the effective Hamiltonian responsible for the transitions under consideration both in the SM and LQ models. In section III, we present the transition amplitude and matrix elements defining the above transitions. In section IV, we calculate the differential decay rate and the lepton forward-backward asymmetry in the baryonic Λ b → Λℓ + ℓ − , Σ b → Σℓ + ℓ − and Ξ b → Ξℓ + ℓ − channels and numerically analyze the results obtained. We compare the LQ predictions with those of the SM and existing experimental data also in this section.

II. THE EFFECTIVE HAMILTONIAN AND WILSON COEFFICIENTS
At the quark level the effective Hamiltonian, defining the above mentioned b → sℓ + ℓ − based transitions, in terms of Wilson coefficients and different operators in SM is defined as [32,33] where G F is the Fermi weak coupling constant, α em is the fine structure constant at Z mass scale, V tb and V * ts are elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, the C ef f 9 , C 10 and C ef f 7 are the SM Wilson coefficients and q 2 is the transferred momentum squared. We collect the explicit expressions of the Wilson coefficients in SM in the Appendix: A. Considering the additional contributions arising from the exchange of scalar leptoquarks, the effective Hamiltonian for b → sℓ + ℓ − transition in the LQ model can be written as where C ef f,LQ

III. TRANSITION AMPLITUDE AND MATRIX ELEMENTS
Generally, the amplitude of the transition responsible for the baryonic decays is provided with sandwiching the effective Hamiltonian between the initial and final baryonic states, where B represents Λ, Σ and Ξ baryons and Q corresponds to b quark. To get the transition amplitude, we need to consider the following transition matrix elements parametrized in terms of twelve form factors in full QCD: and where the u BQ and u B represent spinors of the initial and final states, respectively. The f (T ) i and g (T ) i (i running from 1 to 3) are transition form factors . In our calculations, we use these form factors calculated via light cone sum rules in full QCD. The values of these form factors corresponding to Λ b → Λℓ + ℓ − , Σ b → Σℓ + ℓ − and Ξ b → Ξℓ + ℓ − transitions are taken from [3], [4] and [5], respectively.
Using the above transition matrix elements in terms of form factors, we get the amplitude of the transitions and where R = (1 + γ 5 )/2 and L = (1 − γ 5 )/2 and the calligraphic coefficients are collected in Appendix: B.

IV. PHYSICAL OBSERVABLES
In this section we would like to calculate some physical observables such as the differential decay width, the differential branching ratio and the lepton forwardbackward asymmetry for the considered decay channels.
A. The differential decay width Using the decay amplitudes and transition matrix elements in terms of form factors, we find the differential decay rate defining the transitions under consideration in the LQ model as where v = 1 −

B. The differential branching ratio
Using the expression of the differential decay width, in this subsection, we numerically analyze the differential branching ratio in terms of q 2 for the decay channels under consideration. For this aim, we present the values of some input parameters and the quark masses in M S scheme used in the numerical analysis in Tables 1 and 2 [9]. As we previously said, we also use the values of form factors calculated via light cone QCD sum rules in full theory [3][4][5] in numerical analysis. The differential branching ratios of decay channels under consideration on q 2 , in the SM and LQ models, at µ and τ lepton channels are plotted in Figures 1-6. Note that, in these figures, the form factors are encountered with their uncertainties in SM that leads to some bands in this model. In LQ model we use the central values of the form factors. The bands in LQ model are due to the constrained regions of some parameters presented in Eq. (6). We do not present the results for e channel in the figures, because the predictions of µ channel are very close to those of the e channel. In figure 1, we also show the experimental data provided by LHCb [2]. From these figures it is clear that, • the differential branching ratios in terms of q 2 obtained in SM for all baryonic processes at both lepton channels include the bands of the LQ model except for Σ b → Σµ + µ − and Ξ b → Ξτ + τ − transitions. For latter we see some discrepancies between two models predictions at higher values of q 2 , while   in the case of Σ b → Σµ + µ − although bands of two models have some intersections, the LQ band shows considerable discrepancies from that of the SM at whole physical region of q 2 .
• The band of SM for Λ b → Λµ + µ − channel encompasses the experimental data provided by the LHCb Collaboration in the intervals 15 GeV 2 /c 4 ≤ q 2 ≤ 18 GeV 2 /c 4 , but these data cannot be described by LQ model. However, the predictions of SM and LQ models coincide with the experimental data in the region 18 GeV 2 /c 4 ≤ q 2 ≤ 20 GeV 2 /c 4 . The experimental data at 0 GeV 2 /c 4 ≤ q 2 ≤ 15 GeV 2 /c 4 interval remain out of the regions swept by the two models predictions.

C. The lepton forward-backward asymmetry
In this subsection, we present the results of the lepton forward-backward asymmetry (A F B ) which is one of useful observables to search for NP effects. This quantity is defined as  In order to see how predictions of LQ scenario deviate from those of the SM, we plot the dependence of the lepton forward-backward asymmetry on q 2 for the channels under discussion in Figures 7-12. In figure 7, we also present the measured values of the leptonic forward backward-asymmetries by the LHCb Collaboration [2] in the Λ b → Λµ + µ − decay channel. From these figures, we read that • in all decay channels the LQ model predictions demonstrate considerable discrepancies from the SM predictions except for Σ b → Σµ + µ − and Σ b → Στ + τ − transitions. In these two cases the SM bands include the areas swept by the LQ model   • Ignoring from the small intersection of the SM narrow bands with errors of the experimental data at very low and high values of q 2 , the LQ model, against the SM, can describe all data available in Λ b → Λµ + µ − channel.

V. CONCLUSION
In the present work, we have performed a comprehensive analysis of the semileptonic Λ b → Λℓ + ℓ − , Σ b → Σℓ + ℓ − and Ξ b → Ξℓ + ℓ − rare processes in the SM as well as the scalar leptoquark model. Using the parametrization of the matrix elements in terms of form factors calculated via light cone QCD sum rules in the full theory, we calculated the differential decay width and numerically analyzed the differential branching fraction and the lepton forward-backward asymmetry in terms of q 2 in different heavy baryonic decay channels for both the µ and τ leptons in both scenarios. We compared the predictions of the LQ model on the considered physical observables with those of the SM and the existing experimental data in Λ b → Λµ + µ − channel. We observed that the predictions of the LQ model on the differential decay width remain inside the bands of the SM except for Σ b → Σµ + µ − transition that we saw some deviations of the LQ model results from those of the SM. Both models describe some experimental data, available in Λ b → Λµ + µ − channel, at higher values of q 2 . The data on the differential branching fraction of this channel can not be explained by the two models at lower values of q 2 . In the case of lepton forward-backward asymmetry, the LQ model's predictions, overall, demonstrate considerable deviations from the SM results. The experimental data existing in Λ b → Λµ + µ − channel , overall, are described by the LQ model but remain outside of the SM band.
More experimental data in Λ b → Λτ + τ − as well as Σ b → Σℓ + ℓ − and Ξ b → Ξℓ + ℓ − with both leptons are needed to compare with the theoretical predictions. We hope, with the RUN II data, it will be possible to measure different physical quantities related to such FCNC transitions at LHCb in near future. Comparison of the experimental data with the theoretical predictions on different physical quantities in different decay channels can help us better explain some anomalies between the SM predictions and the experimental data in some channels and more important in the course of indirectly search for the new physics effects like leptoquarks.
Note Added: When providing this work we noticed that a part of our work, namely the Λ b → Λℓ + ℓ − channel has been investigated in [38,39] within the same framework. In these studies the authors use the form factors, as the main inputs, calculated in heavy quark effective theory while we use the form factors calculated via light cone QCD sum rules in full theory without using any approximation.

Appendix: A
The Wilson coefficient C ef f 7 in leading logarithm approximation in the SM is written by [34][35][36][37] are given as The parameter η in Eq.(A.1) is defined as where α s (m Z ) = 0.118 and β 0 = 23 3 . The coefficients h i and a i in Eq.(A.1) are also written by [35,36]  in SM is given by [35,36] in the naive dimensional regularization (NDR) scheme is written as where P N DR 0 = 2.60±0.25, sin 2 θ W = 0.23, Y = 0.98 and Z = 0.679 [35][36][37]. The last term in Eq.(A.9) is ignored due the negligible value of P E . In Eq.(A.8), the η(ŝ ′ ) is given as . (A.11) The function h(y,ŝ ′ ) is written as h(y,ŝ ′ ) = − 8 9 ln m b µ b − 8 9 ln y + 8 27 The coefficients C j (j=1,...6) at µ b = 5 GeV scale are also written as [37] C j = 8 i=1 k ji η ai (j = 1, ...6), (A.14) where the k ji are given as k 1i = ( 0, 0, The Wilson coefficient C 10 in the SM is given as: The calligraphic coefficients used in the transition amplitudes of the considered processes both in the SM and LQ models are find as (1 → 3) , (ŝ) in the differential decay width are given as