Abstract
We study the exclusive semileptonic rare decay in the framework of light-cone quark model. The transition form factors and are evaluated in the timelike region using the analytic continuation method in frame. The analytic solutions of these form factors are compared with the results obtained from the double pole parametric form. The branching ratio for decay is calculated and compared with the other theoretical model predictions. The predicted results in this model can be tested at the LHCb experiments in near future which will help in testing the unitarity of CKM quark mixing matrix, thus providing an insight into the phenomenon of CP violation.
1. Introduction
In the past few years, great progress has been made in understanding the semileptonic decays in the sector as these are among the cleanest probes of the flavor sector of the Standard Model (SM) which not only provide valuable information to explore the SM but also are powerful means for probing different new physics (NP) scenarios beyond the SM (BSM) [1–3]. Due to the Glashow-Iliopoulos-Maiani (GIM) mechanism [4], flavor changing neutral current (FCNC) induced semileptonic decays are rare in the SM because these decays are forbidden at tree level and can proceed at the lowest order only via electroweak penguin and box diagrams [5, 6]. Therefore, these decay processes provide sensitive probes to look into physics BSM [7]. They also play a significant role in providing a new framework to study the mixing between different generations of quarks by extracting the most accurate values of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements which help us to test the charge-parity (CP) violation in the SM and to dig out the status of NP [8, 9].
The theoretical analysis of CP violating effects in rare semileptonic decays requires knowledge of the transition form factors that are model dependent quantities and are scalar functions of the square of momentum transfer [10]. These form factors also interrelate to the decay rates and branching ratios of all the observed decay modes of mesons and their calculation requires a nonperturbative treatment. Various theoretical approaches, such as relativistic constituent quark model [11–15], QCD sum rules [16–20], lattice QCD calculations [21–23], chiral perturbation theory [24, 25], and the light-front quark model (LFQM) [26–34], have been applied to the calculations of hadronic form factors for rare semileptonic decays. Experimentally, a significant effort has been made for the advancement of our knowledge of the flavor structure of the SM through the studies of inclusive [35] as well as exclusive [36] rare decays. The violation of CP symmetry in meson decays was first observed in 2001 (other than in neutral meson decays) by two experiments: the Belle experiment at KEK and the Babar experiment at SLAC [37]. Both these experiments were constructed and operated on similar time scales and were able to take flavor physics into a new realm of discovery [38]. The Babar and Belle experiments completed taking data in 2008 and 2010, respectively. Recently, numerous measurements of decays have been performed by the LHC experiments at CERN; in particular, the dedicated physics experiment LHCb makes a valuable contribution in the understanding of CP violation through the precise determination of the flavor parameters of the SM [39–41].
In particular, there has been an enormous interest in studying the decay properties of the meson due to its outstanding properties [42]. Unlike the symmetric heavy quark bound states (bottomonium) and (charmonium), meson is the lowest bound state of two heavy quarks ( and ) with different flavors and charge. Due to the explicit flavor numbers, mesons can decay only through weak interaction and are stable against strong and electromagnetic interactions, thereby providing us an opportunity to test the unitarity of CKM quark mixing matrix. The study of an exclusive semileptonic rare decay is prominent among all the meson decay modes as it plays a significant role for precision tests of the flavor sector in the SM and its possible NP extensions. At quark level, the decay proceeds via FCNC transition with the intermediate , , and quarks and most of the contribution comes from the intermediate quark. Also, due to the neutral and massless final states (), it provides an unique opportunity to study the penguin effects [10]. As a theoretical input, hadronic matrix elements of quark currents will be required to calculate the transition form factors [43] in order to study the decay rates and branching ratios of the above-mentioned decay.
The semileptonic rare decay has been studied by various theoretical approaches such as constituent quark model (CQM) [44] and QCD sum rules [45]. In this work, we choose the framework of light-cone quark model (LCQM) [46] for the analysis of this decay process. LCQM deals with the wave function defined on the four-dimensional space-time plane given by the equation and includes the important relativistic effects that are neglected in the traditional CQM [47, 48]. The kinematic subgroup of the light-cone formalism has the maximum number of interaction-free generators in comparison with the point form and instant form [49]. The most phenomenal feature of this formalism is the apparent simplicity of the light-cone vaccum, because the vaccum state of the free Hamiltonian is an exact eigen state of the total light-cone Hamiltonian [50]. The light-cone Fock space expansion constructed on this vacuum state provides a complete relativistic many-particle basis for a hadron [51]. The light-cone wave functions providing a description about the hadron in terms of their fundamental quark and gluon degrees of freedom are independent of the hadron momentum making them explicitly Lorentz invariant [52].
The paper is organized as follows. In Section 2, we discuss the formalism of light-cone framework and calculate the transition form factors for decay process in frame. In Section 3, we present our numerical results for the form factors and branching ratios and compare them with other theoretical results. Finally, we conclude in Section 4.
2. Light-Cone Framework
In the light-cone framework, we can write the bound state of a meson consisting of a quark and an antiquark with total momentum and spin as [53]where and denote the on-mass shell light-front momenta of the constituent quarks. The four-momentum is defined asThe momenta and in terms of light-cone variables arewhere represent the light-cone momentum fractions satisfying and is the relative transverse momentum of the constituent.
The momentum-space light-cone wave function in (1) can be expressed aswhere describes the momentum distribution of the constituents in the bound state and constructs a state of definite spin () out of the light-cone helicity () eigenstates. For convenience, we use the covariant form of for pseudoscalar mesons which is given bywhereThe meson state can be normalized asso thatWe choose the Gaussian-type wave function to describe the radial wave function :where is a scale parameter and denotes the internal momentum of meson. The longitudinal component is defined as
2.1. Form Factors for the Semileptonic Decay in LCQM
The form factors and can be obtained in frame with the “good” component of current, that is, , from the hadronic matrix elements given by [53]It is more convenient to express the matrix element defined by (11) in terms of and aswithHere and and .
Using the parameters of and quarks, the form factors and can be, respectively, expressed in the quark explicit forms as follows [46]:where represents the final state transverse momentum, , and . The term denotes the Jacobian of the variable transformation .
The LCQM calculations of form factors have been performed in the frame [54, 55], where (spacelike region). The calculations are analytically continued to the (timelike) region by replacing to in the form factors. To obtain the numerical results of the form factors, we use the change of variables as follows:The detailed procedure of analytic solutions for the weak form factors in timelike region has been discussed in literature [56].
For the sake of completeness and to compare our analytic solutions, we use a double pole parametric form of form factors expressed as follows [44]:where , denotes any of the form factors, and denotes the form factors at . Here , are the parameters to be fitted from (17). While performing calculations, we first compute the values of and from (15) in , followed by extraction of the parameters and using the values of and , and then finally fit the data in terms of parametric form.
2.2. Decay Rate and Branching Ratio for Decay
At the quark level, the rare semileptonic decay is described by the FCNC transition. As mentioned earlier, these kinds of transitions are forbidden at the tree level in the SM and occur only through loop diagrams as shown in the Figure 1. They receive contributions from the penguin and box diagrams [44]. Theoretical investigation of these rare transitions usually depends on the effective Hamiltonian density. The effective interacting Hamiltonian density responsible for transition is given by [57]where is the Fermi constant, is the electromagnetic fine structure constant, is the Weinberg angle, are the CKM matrix elements, and .
The function denotes the top quark loop function, which is given by
The differential decay rate for can be expressed in terms of the form factors as [46]where with and .
The differential branching ratio () can be obtained by dividing the differential decay rate () by the total width () of the meson and then, by integrating the differential branching ratio over , we can obtain the branching ratio (BR) for decay.
3. Numerical Results
Before obtaining the numerical results of the form factors for the semileptonic decay, we first specify the parameters appearing in the wave functions of the hadrons. We have used the constituent quark masses as [53, 58] The parameter that describes the momenta distribution of constituent quarks can be fixed by the meson decay constants and , respectively. The parameters that we have used in our work are given as [44] Using the above parameters, we present the analytic solutions of the form factors and (thick solid curve) for in Figures 2 and 3, respectively. We have also shown the results obtained from the parametric formula (dashed curve) given by (17). We would like to mention here that the point represents the maximum recoil point and the point represents the zero recoil point where the produced meson is at rest. As we can see from Figures 2 and 3, the form factors and increase and decrease exponentially with respect to . The analytic solutions of form factors given by (15) are well approximated by the parametric form in the physical decay region . For a deeper understanding of the results, we have listed the numerical results for the form factors and at and the parameters and of the double pole form in Table 1. For the sake of comparison, we have also presented the results of other theoretical models. It can be seen from the table that the values of form factors and at in our model agree quite well with the CQM. The difference in the values with respect to other models might be due to the different assumptions of the models or different choices of parameters.
To estimate the numerical value of the branching ratio for decay (defined in (20)), the various input parameters used are [46] , , GeV, GeV, and . The lifetime of ( ps) is taken from the Particle Data Group [59]. Our results for the differential branching ratio as a function of is shown in Figure 4.
Our prediction for the decay branching ratio of decay is listed in Table 2 and compared with the other theoretical predictions. As we can see from Table 2, the result predicted by LCQM approximately agrees with the prediction given by QCD sum rules whereas it is slightly larger when compared with the results of CQM. At present, we do not have any deep understanding of these values; however they do indicate that these results may be important even in a more rigorous model. The measurements can perhaps be substantiated by measurement of the decay width of mesons. Several experiments at LHCb are contemplating the possibility of searching for more meson decays.
4. Conclusions
We have studied the exclusive semileptonic rare decay within the framework of LCQM. In our analysis, we have evaluated the transition form factors and in the frame and then extended them from the spacelike region () to the timelike region () through the method of analytical continuation using the constituent quark masses (, , and ) and the parameters describing the momentum distribution of the constituent quarks ( and ), respectively. The numerical values of and have been fixed from the meson decay constants and , respectively. We have also compared the analytic solutions of transition form factors with the results obtained for the form factors using the double pole parametric form. Using the numerical results of transition form factors, we have calculated the decay branching ratio and compared our result with the other theoretical model predictions. The LCQM result for the decay branching ratio of decay comes out to be which approximately agrees with the prediction given by QCD sum rules [45]. This result can also be tested at the LHCb experiments in near future.
To conclude, new experiments aimed at measuring the decay branching ratios are not only needed for the profound understanding of decays but also to restrict the model parameters for getting better knowledge on testing the unitarity of CKM quark mixing matrix. This will provide us with a useful insight into the phenomenon of CP violation.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to acknowledge Chueng-Ryong Ji (North Carolina State University, Raleigh, NC) for the helpful discussions and Department of Science and Technology (Ref no. SB/S2/HEP-004/2013), Government of India, for financial support.