Footprints of New Physics in Transitions
In this work, we perform a combined model independent analysis of the , and anomalies in the framework of the four–fermion effective field theory, by paying special attention to the employ of the hadronic form factors. For the transition form factors, we use the HQET parametrization that include the higher order corrections of and determined recently from a fit to lattice QCD and light-cone sum rule results in complementary kinematical regions of the momentum transfer. For the transitions, we use the form factors calculated in the covariant light-front quark model, which are found to be well consistent with the preliminary lattice results. From our analysis, the two classes of vector operators are shown to be the most favored single new physics (NP) operators by the current experimental constraints within and the LEP1 data on as well as the minimum fit, while the tensor operator is also allowed but severely constrained, and the scalar ones are excluded. Using the favored ranges and fitted values of the Wilson coefficients of the single NP operators, we also give prognosis for the physical observables such as the ratios of decay rates () and other polarized observables as well as the distributions.
pacs:12.38.Lg, 12.39.Mk, 14.40.Rt
The imprints of new physics (NP) beyond the standard model (SM) can be examined via both the direct approach (probing for NP signals) and the indirect approach (precisely testing the SM). For the indirect approach to explore NP, semi-leptonic decays via both the charged and neutral current processes play pivotal roles, especially given that several anomalies in these decays have been observed in recent years, which indicate deviations from the measurements to the SM predictions and therefore have attracted a lot of interest (for reviews, See Li and Lu (2018); Graverini (2018); Langenbruch (2018)). One of the involved physical observables, for the charged current process , is defined as
Unlike the branching fractions of these decay modes which are largely affected by the uncertainties that originates from the Cabibo–Kobayashi–Maskwa (CKM) matrix and the hadronic transition form factors, the reliance of and on the CKM matrix exactly cancels out, and the uncertainties due to the form factors can be largely reduced in these ratios. Hence digression of their values from the SM results would indicate the signature of NP. The combined results of and measured by BABAR Lees et al. (2012, 2013), BELLE Huschle et al. (2015); Sato et al. (2016); Hirose et al. (2017) and LHCb Aaij et al. (2015, 2018a, 2018b) are and , which clearly indicates the deviation from the SM predictions by and respectively Amhis et al. (2017).
Most recently LHCb reported the ratio of branching fractions
This result deviates away from the SM predictions, which lie in the range 0.23–0.29 Watanabe (2018); Tran et al. (2018); Bhattacharya et al. (2018). In addition to the ratio of decay rates, the longitudinal polarization has also been measured by BELLE Collaboration for the transition Hirose et al. (2017) with value . All these measurements clearly stipulate the deviation from the SM predictions. To address these anomalies and chalk out the status of NP, various approaches Watanabe (2018); Tanaka and Watanabe (2013); Sakaki et al. (2013, 2015); Iguro and Tobe (2017); Chauhan and Kindra (2017); Dutta (2017); Alok et al. (2017); Descotes-Genon et al. (2017); He and Valencia (2018); Choudhury et al. (2018); Capdevila et al. (2018); Wei et al. (2018); Tran et al. (2018); Yang et al. (2018); Abdullah et al. (2018); Azatov et al. (2018); Martinez et al. (2018); Fraser et al. (2018); Bhattacharya et al. (2018); Rui et al. (2018); Kumar et al. (2018); Crivellin et al. (2018); Cohen et al. (2018); Sannino et al. (2018); Albrecht et al. (2017); Bardhan et al. (2017); Li et al. (2017, 2018); Celis et al. (2013); Hu et al. (2018); Wang et al. (2017a) are under consideration.
In view of the new measurement done by LHCb for , we scrutinize and in a model independent framework, i.e. we consider the effects of each single NP operator in the general effective four-fermion lagrangian. To make a reliable analysis, we carefully consider the employ of the hadronic form factors for both the and transitions. For the formers, the heavy quark field theory (HQET) parametrization of the form factors are used in most of the existing experimental and theoretical analyses due to the lack of the experimental data to precisely determine them. Although HQET is expected to well describe the non-perturbative effects in these single heavy quark systems, deviations of its predictions for the form factors from those obtained by using lattice QCD in regions of small hadronic recoil have been observed Bigi and Gambino (2016); Bigi et al. (2017a), suggesting the importance of a reconsideration of the HQET form factors. In the study of transitions Jung and Straub (2018), the authors included the and (part of) the corrections to the HQET form factors, and performed a global fit of the HQET parametrization to the lattice Harrison et al. (2018); Aoki et al. (2017) and light-cone sum rule (LCSR) Faller et al. (2009) pseudo data points respectively taken in complementary kinematical regions of hadronic recoil, taking into account the strong unitarity constraints Bigi and Gambino (2016); Bigi et al. (2017b).
For the transitions, the theoretical determination of the form factors rely on other approaches and have been done in perturbative QCD (PQCD) Wang et al. (2013), QCD sum rules (QCDSR) Kiselev (2002), light-cone QCD sum rules (LCSR) Fu et al. (2018); Zhong et al. (2018), the covariant light-front quark model (CLFQM) Wang et al. (2009), the nonrelativistic quark model (NRQM) Hernandez et al. (2006), the relativistic quark model (RQM) Ebert et al. (2003), the covariant confined quark model (CCQM) Tran et al. (2018) etc.. Besides, the HPQCD Collaboration have also released the preliminary lattice QCD results on some of these form factors. Detailed comparison of the and form factors calculated using different approaches can be found in Wang et al. (2013, 2009); Tran et al. (2018). Among all these results, the CLFQM form factors computed by one of us and other collaborators have been found to be well consistent with the lattice results at all available points. Therefore, in this work we shall use the CLFQM form factors as our numerical inputs.
By using the above hadronic form factors, we study the constraints on the Wilson coefficients of the single NP operators from the measurement of , and within and . We also consider the limit on the branching fraction obtained from the LEP1 data Akeroyd and Chen (2017) as additional constraints on NP, which is much more stringent than the constraints from the lifetime Patrignani et al. (2016); Alonso et al. (2017) also considered in some other literatures. In addition, we also perform the minimum fit of the Wilson coefficients to the experimental data on obtained by LHCb Aaij et al. (2015, 2018a, 2018b), Belle Huschle et al. (2015); Sato et al. (2016); Hirose et al. (2017) and Babar Lees et al. (2012, 2013), and the longitudinal polarization obtained by Belle Hirose et al. (2017) and obtained by LHCb Aaij et al. (2018c). Using the obtained favored ranges and fitted results for the Wilson coefficients, we give predictions for the physical observables including the ratio of decay rates, longitudinal polarization, final state vector meson polarization, and the forward-backward asymmetry as well as the corresponding distributions.
This work will be organised as follows: In Section II, we shall introduce the general formalism of the effective field theory for the transitions. Then in Section II and IV we shall present detailed description of the and form factors respectively. The numerical analysis will be performed in Section V for the experimental constraints on the Wilson coefficients as well as the minimum fit, and in Section VI for the predictions on the physical observables. Finally, in Section VII we shall give the summary and conclusions.
Ii Effective Four-Fermion Interactions and Operator Basis
The semileptonic decays of mesons via in the SM can be described by the left handed four–fermion interaction as an effective theory. In the presence of NP the effective Lagrangian can get modified by incorporating new operator basis that includes all possible four fermion interactions. If the neutrinos are assumed to be left-handed and their flavors are not differentiated, the effective Lagrangian can be expressed as
where the four-fermion operator basis can be defined as
and (, , , , and ) are the corresponding Wilson coefficients with in the SM. By using the effective Lagrangian given in Eq. (3) one can compute the following hadronic matrix elements for the decays and Tanaka and Watanabe (2013); Sakaki et al. (2013):
where and represent the meson and virtual particle helicities with the values for pseudoscalar and vector meson respectively and for the virtual particle. The amplitudes given in Eq. (5) can be expressed in terms of transition form factors for the above mentioned decays and further used to calculate physical observables such as the unpolarized and polarized decay rates. Formulas for the physical observables in terms of the hadronic matrix elements are given in Tanaka and Watanabe (2013); Sakaki et al. (2013).
In order to compute the observables involved in semileptonic decays and make reliable conclusions on the possible NP effects, it is worthwhile to take care of the form factors which result in major theoretical uncertainties. In the next section we shall elaborate on the and form factors used in our analysis.
Iii form factors
For the form factors, we shall use as numerical inputs the results fitted in Jung and Straub (2018), of which the authors follow to use the HQET parametrization in Bernlochner et al. (2017) that include the corrections and part of the contributions. The HQET parameters such as the sub-leading Isgur-Wise functions are determined by a global fit to the lattice results Harrison et al. (2018); Aoki et al. (2017) at small hadronic recoil points and the LCSR results Faller et al. (2009) in the region of large hadronic recoil, with the the strong unitarity constraints Bigi and Gambino (2016); Bigi et al. (2017b) also being imposed. The HQET form factors for the transitions are defined through
where and are respectively the four velocities of the and mesons, and the dimensionless kinematic variable is used instead of the momentum transfer . For the transitions the following definitions are used:
where , , , and w are defined in the same way as for the transitions, and denotes the polarization vector of .
In the heavy quark limit, only the single leading Isgur-Wise function is needed for expressing the form factors. With inclusion of the contributions, these form factors are written as Bernlochner et al. (2017)111Here we follow the notations in Bernlochner et al. (2017) but the coefficients of the contributions should not be mixed up with the Wilson coefficients in Eq. (3).
where , with in which , are the coefficients of the terms calculated in the perturbation theory, and the functions can be expressed in terms of the sub-leading Isgur-Wise functions through:
With implied by the Luke’s theorem, up to the subleading Isgur-Wise functions can be approximated as follows:
The expressions for obtained by matching QCD and HQET at (corresponding to ) can be found in Bernlochner et al. (2017), which are lengthy and not presented in this work. To ensure the cancellation of the leading renormalon associated with the pole mass, the mass scheme has also been used, namely in the terms not multiplied by the sub-leading Isgur-Wise functions, the pole mass is treated as where is half of the mass, while in the other terms is imposed Bernlochner et al. (2017).
In the global fit performed in Jung and Straub (2018), contributions to , and of which the contributions vanish at zero hadronic recoil, have also been included, and the leading Isgur-Wise function is parameterized as in the expansion where . The fitted values of the parameters in the form factors along with the other inputs in this work are listed in Appendix A.
Iv and form factors
As mentioned in the introduction, the CLFQM form factors Wang et al. (2009)222Very recent application of the CLFQM form factors can be found in Wang et al. (2017b, c). for and transitions are well consistent with the preliminary lattice results Colquhoun et al. (2016); Lytle et al. (2016) obtained by the HPQCD Collaboration, therefore we use them as our numerical inputs in this work. The form factors of the vector and axial-vector operators are defined through the following matrix elements:
where the form factors are parametrized as with in the full kinematical range of , of which the results computed in the covariant light-front quark model Wang et al. (2009) are listed in Table 1. These results are consistent with the preliminary lattice results for , , and at all available values obtained by the HPQCD Collaboration Lytle et al. (2016); Colquhoun et al. (2016) , which can be clearly seen in Figure 1.
The form factors of the tensor operators are defined through
where can be related to and through the quark level equations of motion:
where we use the quark masses in the renormalization scheme at the scale .
V experimental constraints on the Wilson Coefficients
In this section, we shall perform our numerical analysis of the experimental constraints on the Wilson coefficients for single NP operators in the effective lagrangian given in Eq. (3). In order to make a general model-independent analysis, we first perform a minimum fit of the Wilson coefficients to the experimental data of different observables such as the ratios and and the polarization for each NP scenario. Other than the fit, we shall try to obtain the allowed region of the Wilson Coefficients by the current experimental data on and within and and limit of obtained from LEP1 data. Final conclusions will be made based on both the results of the fit and the favored regions by the experimental constraints.
In our methodology of minimum fit, the as a function of the Wilson coefficient is defined as
where are the theoretical predictions for , , etc., and are the corresponding experimental measurements, which are listed in Table 2. and are respectively the experimental and theoretical covariance matrices, which are calculated by taking the correlations listed in Table 2 and Table 3.
|BABARLees et al. (2012, 2013)|
|BelleHuschle et al. (2015)|
|Belle Sato et al. (2016)|
|BelleHirose et al. (2017)|
|LHCb Aaij et al. (2015)|
|LHCb Aaij et al. (2018a, b)|
|LHCbAaij et al. (2018c)|
The fitted Wilson coefficients in each NP scenario are listed in Table 4. Using these values, we predict all the observables in the SM and the NP scenarios, which are respectively listed in Table 5 and 7 in the next section.
Fig. 2 depicts the the correlations between , and in the presence of each single NP operator. The horizontal and vertical bands represent the experimental constraints at confidence level (C.L.). One can see that the , and operators can explain the experimental values of and within , but when is taken into account, all single-operator scenarios can no longer accommodate the experimental constraints.
At the C.L. of , typical allowed regions for the Wilson coefficients are obtained, as depicted in Fig. 3. In addition to the measurement of the ratios and , there have been other experimental constraints on the NP effects in transitions, namely the measurement of the life time Patrignani et al. (2016); Alonso et al. (2017) and the branching fraction of Akeroyd and Chen (2017). Here we consider the constraint from the latter, which is more restrictive than that from the life time. The LEP1 Data taken at the Z peak has given an upper limit on , which we consider in our analysis. The explicit expression for this constraint is Akeroyd and Chen (2017)
where and are respectively the lifetime and decay constant and their values used in this work are listed in Appendix A.
In Fig. 3 one can see that the scenario is excluded merely taking into account the constraints from and within . For the scenario, although it can simultaneously accommodate and and has the best value (as can be seen in Table 4), it is excluded by the constraint from . The remaining NP scenarios, namely the , and scenarios, are able to explain the current experimental data at . Among these scenarios, the ones corresponding to the and operators have distinctive allowed regions by the experimental measurements of and as well as small values in the fit. However the scenario is severely constrained and has a larger value. Therefore, in our analysis, the and scenarios are observed to be the most favored single-operator NP scenarios by the current experimental data, and has even a better value than in the fit.
The favored regions of different NP scenarios obtained in this work can be compared with those in Watanabe (2018), where the HQET form factors for the transitions are also used but only part of the