Rare Λb → Λl+l− decay in the Bethe-Salpeter equation approach

We study the rare decays in the Bethe-Salpeter equation approach. We find that the branching ratio is in our model. This result agrees with the experimental data well. In the same parametric region, we find that the branching ratio is .


Introduction
Recently, some interesting experimental results have been obtained in the studies of rare decays of b baryons induced by the transition [1][2][3][4][5]. The rare decay was observed by the CDF [1] and LHCb [2] collaborations. The first observation of the baryonic flavor changing neutral current decay by the CDF collaboration [1] had a signal yield of events, corresponding to the absolute branching fraction (stat) (syst) . Following previous measurements, the LHCb collaboration [2] reported the branching fraction of (stat) (syst) (norm) based on the events and updated experimental data intergrating over GeV [3]. The first observation of the radiative decay was reported in [4], and the branching fraction was measured as based on the events with the significance of 5.6 . The analysis of the angular distribution of the decay was reported in [5], and the first analysis of the differential fraction and the angular distribution of was given in [3].
However, in most of these works with the FFs of being based on light-cone QCD sum rules and assumed to have the same shape, the results for the branching ratios of are different and do not agree with the existing experimental data. One important approach to search for new physics in b-physics is to analyze the rare B decay models that are induced by the flavor-changing neutral current (FCNC) transitions. The FCNC transitions are forbidden at the tree level in the standard model, and thus provide a good testing ground for new physics. To use rare decays in search of new physics, the transition matrix must be determined more exactly.
In general, there could be other types of diquarks contributing to and . However, considering that the bquark is very heavy, the diquark has good spin and isospin quantum numbers, which are both zero in the heavy quark limit; hence, only the composition is taken into account, which is dominant. Then, for the transition, only the composition in contributes.
This paper is organized as follows. In Section 2, we establish the BS equation for and . In Section 3 we derive the FFs for in the BS equation approach. In Section 4 the numerical results for the FFs and the decay branching ratios of are given. Finally, the summary and discussion are given in Section 5.
In our work, can be described as a system (the first and second subscripts correspond to the spin and the isospin of , respectively). The BS wave function of the system can be defined as follows [33][34][35][36][37][38][39][40]: where and are the field operators of the bquark and diquark, respectively, and P is the momentum of . We use to represent the masses of the , the b-quark and the diquark, respectively. We define the BS wave function in the momentum space, as follows: where is the coordinate of the center of mass, , , , and p is the relative momentum of the quark and the diquark. As is show in Fig. 1 in the momentum space, the BS equation for satisfies the homogeneous integral equation [33,34,[36][37][38][39][40] where the quark momentum and the diquark momentum , and are the propagators of the quark and the scalar diquark, respectively. and are the scalar confinement and onegluon-exchange terms, respectively [41].
is introduced to describe the structure of the scalar diquark [6,36,42]. By analyzing the proton electromagnetic FFs, it was found that GeV 2 can yield results that are consistent with the existing experimental data [6]. Motivated by the potential model, and have the following forms in the covariant instantaneous approximation ( ) [36,37,40,43]: where is the transverse projection of the relative momenta along the momentum P, defined as , ( ). The second term of is introduced to avoid infrared divergence at the point , is a small parameter to avoid the infrared divergence. The parameters and are related to the scalar confinement and the one-gluon-exchange diagram, respectively. b(ud) 00 In general, the system needs two scalar functions to describe the BS wave function [33,34,38] where are the Lorentz-scalar functions of , is the spinor of , is the transverse projection of the relative momenta along the momentum P, and . The quark and diquark propagators can be written as follows: where . are the projection operators that satisfy the relations, . On the order of [36], the quark propagator can be written as where is the binding energy. In general, is approximately GeV [38]; then, the diquark mass is approximately GeV, which is in a good agreement with [44]. Then, we obtain that is approximately GeV for [39]. Defining , the scalar BS wave functions satisfy the coupled integral equatioñ The discussion about the BSE of is similar. In Eq. (3), when , can be replaced by , and we find that the BSE for has the following form [36] Then, considering that , we find that . Therefore, the BS wave function of can be written as where is a scalar function.
In general, the BS wave function can be normalized under the condition of the covariant instantaneous approximation [43]: where and represent the color indices of the quark and the diquark, respectively, is the spin index of the baryon , is the inverse of the fourpoint propagator, written as follows: In this section, we derive the matrix element of in the BS equation approach. On the quark level, is described by the transition. The effective Hamiltonian describing this process can be given as the following [45] where and are the Fermi coupling constant and the electromagnetic coupling constant, respectively, q is the total momentum of the lepton pair, and are the Wilson coefficients. The amplitude of the decay is obtained by calculating the matrix element of the effective Hamiltonian for the transition between the initial and final states, . The matrix element can be parameterized in terms of the FFs, as follows: where is the momentum transfer, and , , , ) can be expressed as functions solely of , which is the energy of the baryon in the rest frame.
In the pole formulae for the extrapolation to in the decay one has (monopole) and (dipole) [6], while in [6] CLEO data from [31] were combined, to obtain , ignoring the mass of baryon. Lattice QCD (LQCD) gives in the leading order in the heavy quark effective theory [46]. In [47] it was assumed that . The QCD sum rules analysis yielded and at the point GeV. Therefore, we expect , considering the correction of . The ratio (stat) (syst) has been previously measured by the CLEO collaboration using experimental data for the semileptonic decay with the invariant mass in the range from to , assuming the same shape for and and ignoring the corrections [48]. In [12] was given at , and in [32] and were obtained. However, according to the pQCD scaling law, the FFs should have different shapes for large [42,49,50]; therefore, we expect , which agrees with the results in [13]. Using experimental data [48], we have estimated the value of and found it to be ranging from to approximately. Considering the results in [12], we let vary from to . Comparing Eq. (20) with Eq. (21), we obtain the following relations: where . The transition matrix for can be expressed in terms of the BS wave function of and . Then, we find the following relations with and The differential decay rate is obtained as follows: where the parameters , and , ( and ) are defined as In the physical region , the decay rate of is obtained as where , and is the lepton velocity. The decay amplitude is given as [23] where Chinese Physics C Vol. 44, No. 8 (2020) 083107 4 Numerical analysis To analyze the decay rate and the branching ratio, we use the following numerical values: for the Wilson coefficients, , , [51][52][53], for the masses of baryons, GeV, GeV [54], while for the masses of the quark, GeV and GeV [34,37,38]. The variable varies from to , and for , and , respectively.
Solving Eqs. (10) and (11) for with the above parameters, we obtain the numerical solutions of the BS wave functions. For we need to solve Eq. (16). In Table 1, we list the values of , for different binding energies and different for . In Table 2, we list the values of for different binding energies and different for . It can be seen from Tables 1 and 2 that the dependence of on the parameters and for is obviously stronger than that for .
In Figs  and (the value of R decreases with increasing , and with increasing the line thickens ( from to ) for the same color line). (color online) The differential decay width of with the binding energy GeV (the decay width increases with increasing from to GeV 3 for the same color line).
Chinese Physics C Vol. 44, No. 8 (2020) 083107 ). This range agrees with our result and with the result in [12]. Considering the experimental data for in [48] and that the value of decreases with increasing the values of or , we believe that the optimal range for our model parameters is GeV and from to GeV, because in this region and R varying from to agrees with our previous results. On the other hand, we find that LQCD also gives the value [46].
In Figs. 9-11, we show the -dependent differential decay width of for different parameters. In our optimal range of parameters and in the range GeV 3 , and GeV, we obtain the branching ratios, respectively, which are listed in Table 3. From this table, we see that our results differ from those of the heavy quark effective theory (HQET) and QCD sum rules, but our results are consistent with the most recent experimental data. With GeV 3 and GeV, we find , and in our optimal parameter range this value is . The values of in the above two ranges are and , respectively. When the parameters and vary in their regions, we find that the differential branching ratio of does not peak at approximately . In [2,3] when in the range (corresponding to in the range GeV 2 ), the experimental data exhibit a peak. Considering this difference, there could be new physics in this region.

Summary and discussion
Theoretical studies of the decay require knowing the matrix element . In the leading order in the HQET, this matrix element is given by two FFs. In the past few decades, in most of the published works the FFs were studied based on the QCD sum rules [12], and by fitting the available experimental data [31]. With experimental advances, the data pertaining to the rare decay have been updated.
In the present work, we have performed the first BS equation calculation of these FFs. In our work, is regarded as a bound state of a Q-quark and a scalar diquark. In this picture, we established the BS equations for , and derived the FFs for in the BS equation approach. After solving the BS equations of and , we calculated the ratio and the decay branching ratio for , and compared our results with those reported in other works. We found that the shape of the differential decay branching ratio for in our model is similar to the experimental data throughout most of the region, and in our work the shape of the decay differential branching ratio of agreed with the LQCD results [29,46]. The experimental data for the differential decay width of exhibited a peak with , but in most of the existing theoretical works such a pole does not appear. Therefore, new physics could exist in this region. More accurate experimental data should be obtained by repeated measurements in that region. Our result for is very close to the experimental data, and we also provide predictions for the decays , which need to be tested in future experiments. We find that for different parameters the ratio of FFs, changes from to in our approach. This result agrees with the experimental data and with [12], and agrees with the LQCD results at [46]. On the other hand, while using the BSE to investigate the octet and decuplet baryon properties in the ladder approximation, the quark exchanges generate the kernel when irreducible 3-quark interactions are neglected and separable 2-quark (diqaurk) correlations are assumed [56][57][58]. Then, both the scalar diquark current and axial-vector diquark current contributions are considered. We will also consider the axial-vector current contributions for the transition in future works.
In the HQET, the approximation is uncertain within . Considering the uncertainties from the parameters and , the maximal uncertainty is approximately 22% in our optimal data region. In the future, our model will also be used to study the forward-backward asymmetries, T violation, and angular distributions in the decays induced by , to further validate our FFs. Table 3. The values of the branching ratios for , and comparison with other models.