# Probing new physics in semileptonic \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) decays

## Abstract

Recently, several observed anomalies in semileptonic B meson decays have implied hints of lepton flavor universal violation. Motivated by these inspiring results, we study the baryon decays \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) which are mediated by \(b\rightarrow u(c)\tau ^{-}\bar{\nu }_{\tau }\) transitions at quark level in the Standard Model and different New Physics scenarios. In the framework of the extended Standard Model on assuming a general effective theory, we constrain the Wilson coefficients of the NP operators using the experimental measurement results for the \(Br(B_{c}^+\rightarrow \tau ^+ \nu _{\tau })\), \(R^{l}_{\pi }\), \(R_{D^{(*)}}\), \(R_{J/\psi }\) and \(F_{L}^{D^{*}}\) anomalies and investigate their New Physics effects on several observables relative to the \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) decays. We mention the differential branching fraction \({\text {d}}Br/{\text {d}}q^2\), the ratio of branching fractions \(R(q^2)\), the lepton-side forward–backward asymmetry \(A_{FB}(q^2)\), the longitudinal polarization \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) of the daughter baryons \(\Lambda (\Xi _{c})\) and \(P_{L}^{\tau }(q^2)\) of the \(\tau \) lepton, and the convexity parameter \(C_{F}(q^2)\).

## 1 Introduction

*B*meson decays. The Belle [1, 2, 3], BaBar [4, 5] and LHCb [6, 7, 8] collaborations have reported \(R_{D^{(*)}}\) and \(R_{J/\psi }\) anomalies in various semileptonic

*B*meson decays mediated by \(b\rightarrow c\) charged current interactions. The experimental measurement results for these anomalies show large deviations with their corresponding SM predictions, which may imply the LFUV and the existence of New Physics. The ratios are

It is really interesting and important to investigate the semileptonic baryon decays \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) mediated by the \(b\rightarrow u(c)\tau ^{-}\bar{\nu }_{\tau }\) transitions at the quark level. Studying them not only can provide an independent determination of the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements \(|V_{ub}|\) and \(|V_{cb}|\), but also it may confirm the LFUV again in the ratios of branching fractions \(R_{\Lambda (\Xi _{c})}\) having a similar formalism to the \(R_{D^{(*)}}\). On the basis of these advantages, many researchers have studied the decays \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) in the SM and NP scenarios [17, 18, 19, 20, 21, 22, 23, 24]. In this paper, we set out to explore the NP effects of these B meson decay anomalies in the \(\Xi _{b}\rightarrow \Lambda (\Xi _{c}) \tau ^{-}\bar{\nu }_{\tau }\) transitions. Using the Wilson coefficients of the NP operators constrained from the experimental values for \(Br(B_{c}^{+}\rightarrow \tau ^{+}\nu _{\tau })\), the \(R^{l}_{\pi }\), \(R_{D^{(*)}}\), \(R_{J/\psi }\) and \( F_{L}^{D^{*}}\) anomalies and the form factors for the \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\) transitions calculated by [24], we investigate the NP effects of these anomalies on the differential branching fraction \({\text {d}}Br/{\text {d}}q^2\), the ratio of branching fractions \(R(q^2)\), the lepton-side forward–backward asymmetry \(A_{FB}(q^2)\), the longitudinal polarization \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) of the daughter baryons \(\Lambda (\Xi _{c})\) and \(P_{L}^{\tau }(q^2)\) of the \(\tau \) lepton, and the convexity parameter \(C_{F}^{l}(q^2)\) relative to these baryon decays. It is worthwhile to note that our study is different from [23] whose authors also investigate the decay \(\Xi _{b}\rightarrow \Xi _{c}\tau ^{-}\bar{\nu }_{\tau }\) using a model-independent approach. We make the Wilson coefficients complex, though they are real in [23], and also we consider the constraints on the NP Wilson coefficients coming from \(R_{J/\psi }\) and \( F_{L}^{D^{*}}\) anomalies, which are not mentioned in [23]. In [13], the authors also consider the constraints on the complex NP Wilson coefficients, but they treat the \(b\rightarrow u\) and \(b\rightarrow c\) transitions separately. Therefore, the constraints on the NP Wilson coefficients are different from ours as is appropriate for both the \(b\rightarrow u\) and the \(b\rightarrow c\) transitions. In addition, we also consider the interplay of vector and scalar interactions scenarios, which are not mentioned in [13, 23]. In this paper, we consider the \(b\rightarrow u\) and \(b\rightarrow c\) transitions together in the same framework of effective theory and perform a combined analysis using the constraints on the Wilson coefficients coming from the experimental results for the \(Br(B_{c}^{+}\rightarrow \tau ^{+}\nu _{\tau })\), \(R^{l}_{\pi }\), \(R_{D^{(*)}}\), \(R_{J/\psi }\) and \( F_{L}^{D^{*}}\) anomalies in the end in determining the NP contributions to the observables aforementioned.

Our paper is organized as follows. In Sect. 2, we introduce the effective theory describing the \(b\rightarrow u(c)l^{-}\bar{\nu }_{\tau }\) transitions as well as the form factors, the helicity amplitudes and the observables of the \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})l^{-}\bar{\nu }_{\tau }\) decays. In Sect. 3, the Wilson coefficients of the NP operators are constrained from the experimental values for \(Br(B_{c}^{+}\rightarrow \tau ^{+}\nu _{\tau })\), \(R^{l}_{\pi }\), \(R_{D^{(*)}}\), \(R_{J/\psi }\) and \( F_{L}^{D^{*}}\) anomalies. In Sect. 4, the numerical results for the observables \({\text {d}}Br/{\text {d}}q^2\), \(R(q^2)\), \(A_{FB}(q^2)\), \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\), \(P_{L}^{\tau }(q^2)\) and \(C_{F}^{l}(q^2)\) in the SM and different NP scenarios are presented. Finally, we briefly conclude in Sect. 5.

## 2 Effective Hamiltonian, form factors, helicity amplitude and observables for \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})l^{-}\bar{\nu }_{\tau }\) decays

### 2.1 Effective Hamiltonian

Form factors for the weak \(\Xi _{b}\rightarrow \Lambda \) and \(\Xi _{b}\rightarrow \Xi _{c} \) transitions

\(\Xi _{b}\rightarrow \Lambda \) | \(f_1(q^2)\) | \(f_2(q^2)\) | \(f_3(q^2)\) | \(g_1(q^2)\) | \(g_2(q^2)\) | \( g_3(q^2)\) |
---|---|---|---|---|---|---|

| 0.092 | 0.029 | \(-\) 0.002 | 0.077 | 0.007 | \(-\) 0.041 |

\(f(q^2_{\mathrm {max}})/g(q^2_{\mathrm {max}}) \) | 0.609 | 0.745 | 0.290 | 0.369 | \(-\) 0.528 | \(-\) 1.36 |

\(a_0 \) | 0.139 | 0.170 | 0.098 | 0.122 | \(-\) 0.175 | \(-\) 0.292 |

\(a_1 \) | 0.136 | \(-\) 0.368 | \(-\) 0.323 | 0.016 | 0.865 | 0.554 |

\(a_2 \) | \(-\) 0.845 | \(-\) 0.180 | 0.059 | \(-\) 0.470 | \(-\) 0.947 | 0.630 |

\(M_{\mathrm {pole}} \) | 5.325 | 5.325 | 5.749 | 5.723 | 5.723 | 5.280 |

\(\Xi _{b}\rightarrow \Xi _{c} \) | \(f_1(q^2)\) | \(f_2(q^2)\) | \(f_3(q^2)\) | \(g_1(q^2)\) | \(g_2(q^2)\) | \( g_3(q^2)\) |
---|---|---|---|---|---|---|

| 0.474 | 0.150 | 0.081 | 0.449 | \(-\) 0.030 | \(-\) 0.285 |

\(f(q^2_{\mathrm {max}})/g(q^2_{\mathrm {max}}) \) | 0.945 | 0.426 | 0.161 | 0.962 | \(-\) 0.104 | \(-\) 0.752 |

\(a_0 \) | 0.684 | 0.308 | 0.121 | 0.729 | \(-\) 0.078 | \(-\) 0.541 |

\(a_1 \) | \(-\) 5.16 | \(-\) 4.18 | \(-\) 0.315 | \(-\) 7.11 | 0.775 | 6.93 |

\(a_2 \) | 28 | 25.9 | \(-\) 5.81 | 41.5 | 0.372 | \(-\) 44.9 |

\(M_{\mathrm {pole}} \) | 6.333 | 6.333 | 6.699 | 6.743 | 6.743 | 6.275 |

### 2.2 Form factors

### 2.3 Helicity amplitude and observables

## 3 Constraints on the NP Wilson coefficients

For the \(B_{c}\rightarrow J/\psi l^+\nu _{l}\) transitions, the hadronic helicity amplitudes and the form factors can easily be found in [35, 36].

Based on these functions, we calculate the predictions of the \(Br(B_{c}^{+}\rightarrow \tau ^{+}\nu _{\tau })\), \(R^{l}_{\pi }\), \(R_{D^{(*)}}\), \(R_{J/\psi }\) and \( F_{L}^{D^{*}}\) anomalies including only one new vector (scalar) coupling or both the vector and the scalar couplings at a time and compare the results with their corresponding \(2\sigma \) range of the observed experimental values, are in Sect. 1. First of all, we display the constrained range of NP Wilson coefficients \(V_L\), \(V_R\), \(S_L\) and \(S_R\) with real and imaginary parts in Fig. 1a–d, respectively. One can easily observe the constrained range of \(V_L\) and \(V_R\) in Fig. 1a, b. We show the constrained range of \(S_L\) and \(S_R\) more vividly in Fig. 1c, d and one can more easily notice that \(S_L\) and \(S_R\) are ruled out. The constraints coming from the \( F_{L}^{D^{*}}\) anomaly are not shown in Fig. 1c, d since they cannot be parameterized regularly but still have a common range with the other anomalies. This conclusion has been confirmed using the parameter scanning by us. The constraints on \(S_L\) coming from \(R_{D}\) and \(R_{D^{*}}\) as well as the constraints on \(S_R\) coming from \(R^{l}_{\pi }\) and \(R_{D^{(*)}}\) have no area in common on panels c and d, respectively. Secondly, we show the constrained range of \(V_L\) and \(S_L\), \(V_L\) and \(S_R\), \(V_R\) and \(S_L\), and \(V_R\) and \(S_R\) in Fig. 2a1–a4, b1–b4, c1–c4 and d1–d4, respectively. Taking the constrained range of \(V_L\) and \(S_L\) as an example, we calculate the predictions of the \(Br(B_{c}^{+}\rightarrow \tau ^{+}\nu _{\tau })\), \(R^{l}_{\pi }\), \(R_{D^{(*)}}\) and \( F_{L}^{D^{*}}\) anomalies including the contributions of both the NP couplings \(V_L\) and \(S_L\), and we then compare the predictions with their corresponding \(2\sigma \) range of the experimental measurement values. We show the range (yellow area) of different combinations of two parameters such as \(ReV_L\) and \(ReS_L\), \(ReV_L\) and \(ImS_L\), \(ImV_L\) and \(ReS_L\), and \(ImV_L\) and \(ImS_L\), contained in the couplings \(V_L\) and \(S_L\) in Fig. 2a1–a4, respectively. From Fig. 2a1–a4, one can see that the values of \(ReS_L\) and \(ImS_L\) are in a small range compared with the values of \(ReV_L\) and \(ImV_L\), since the NP coupling \(S_L\) is ruled out when it is considered separately.

## 4 Numerical results and discussions

In this section, we give the numerical results of the observables \({\text {d}}Br/{\text {d}}q^2\), \(R(q^2)\), \(A_{FB}(q^2)\), \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\), \(P_{L}^{\tau }(q^2)\) and \(C_{F}^{l}(q^2)\) relative to the \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) transitions including the contributions of only NP vector type couplings and both vector and scalar type couplings except the ones of only scalar type.

Before we present the numerical results, we have to clarify the values of the SM input parameters used for the numerical calculations in Table 2.

Input parameters in the SM used for our numerical analysis [38]

\(G_{F}=1.166378\times 10^{-5}~{\mathrm{GeV}}^{-2}\) | \(m_{\mu }=0.10565~{\mathrm{GeV}}\) |

\(m_{\tau }=1.77682~{\mathrm{GeV}}\) | \(m_{\Xi _{b}}=5.7945~{\mathrm{GeV}}\) |

\(m_{\Xi _{c}}=2.47087~{\mathrm{GeV}} \) | \(m_{\Lambda }=1.1156~{\mathrm{GeV}}\) |

\(|V_{cb}|=0.0422\pm 0.0008\) | \(|V_{ub}|= 0.00394\pm 0.00036\) |

\(R_{\Lambda (\Xi _{c})}(q^2)\) in the high kinematic region may be more useful for further confirming the NP effects of the B meson decay anomalies. We only show the SM predictions for \(A_{FB}(q^2)\), \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\), \(P_{L}^{\tau }(q^2)\) and \(C_{F}^{l}(q^2)\) of the two decays in Fig. 3. There is no difference between the NP predictions including contributions of \(V_L\) and the SM predictions of the \(A_{FB}(q^2)\), \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\), \(P_{L}^{\tau }(q^2)\) and \(C_{F}^{l}(q^2)\). Because the NP operators have the same Lorentz structure as the SM ones and the SM decay rates of the two transitions are modified by the factor \(|1+V_L|^2\). The factor \(|1+V_L|^2\) appears both in the numerator and in the denominator of the expressions describing \(A_{FB}(q^2)\), \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\), \(P_{L}^{\tau }(q^2)\) and \(C_{F}^{l}(q^2)\) simultaneously.

Secondly, using the constrained range within \(2\sigma \) allowed for real and imaginary parts of the new coefficient \(V_R\) in Fig. 1b, the NP predictions including the contributions of \(V_R\) and the SM predictions for the \({\text {d}}Br/{\text {d}}q^2\), \(R(q^2)\), \(A_{FB}(q^2)\), \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\), \(P_{L}^{\tau }(q^2)\) and \(C_{F}^{l}(q^2)\) relative to the \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) decays are displayed in Fig. 4. The red bands represent the SM predictions and the pink bands represent the NP predictions including the contributions of \(V_R\). The NP contributions of \(V_R\) to \({\text {d}}Br/{\text {d}}q^2\) and \(R(q^2)\) in Fig. 4a1, a2, b1, b2 have large deviations from their corresponding SM predictions in the same way as in the \(V_L\) scenario. Both \({\text {d}}Br/{\text {d}}q^2\) and \(R(q^2)\) are enhanced in the whole reasonable kinematic region. More interesting, the NP contributions of \(V_R\) to the observables \(A_{FB}(q^2)\) and \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) in Fig. 4a3, a4, b3, b4 similarly have large deviations from their SM predictions, which are really different from the scenario for \(V_L\). The \(A_{FB}(q^2)\) including NP contributions of \(V_R\) are decreased but \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) are enhanced in the whole kinematic region. The extent of the decrement and enhancement vary with the increasing of \(q^2\). In the lowest kinematic region, the enhancement of \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) is most prominent. Therefore, it is crucial to measure \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) in the low kinematic region. Additionally, one can see that the \(P_{L}^{\tau }(q^2)\) and \(C_{F}^{l}(q^2)\) of these two decays including NP contributions of \(V_R\) and their SM predictions in Fig. 4a5, a6, b5, b6 are almost the same, which indicates that they are both insensitive to the NP effects of the coupling \(V_R\).

Finally, we show the NP contributions of four combinations of vector and scalar type couplings such as \(V_L\) and \(S_L\), \(V_L\) and \(S_R\), \(V_R\) and \(S_L\), and \(V_R\) and \(S_R\) to \({\text {d}}Br/{\text {d}}q^2\), \(R(q^2)\), \(A_{FB}(q^2)\), \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\), \(P_{L}^{\tau }(q^2)\) and \(C_{F}^{l}(q^2)\) relative to the \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) decays in Figs. 5, 6, 7 and 8. In these four figures, it is still easily to see that the NP effects of the different combinations of vector or scalar type couplings on \({\text {d}}Br/{\text {d}}q^2\) and \(R(q^2)\) relative to the \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) decays are all similarly significant. In Fig. 5, different from the scenario only considering the NP couplings \(V_L\), the observables \(P_{L}^{\tau }(q^2)\) relative to the two decays are also sensitive to \(V_L\) and \(S_L\). In the highest kinematic region, the deviations between the NP predictions of \(P_{L}^{\tau }(q^2)\) including the contributions of \(V_L\) and \(S_L\) and the SM predictions are largest. Therefore, the observable \(P_{L}^{\tau }(q^2)\) is important to distinguish the NP originating from the vector type or from both vector and scalar type couplings. In Fig. 6, except that the \({\text {d}}Br/{\text {d}}q^2\) and \(R(q^2)\) observables are sensitive to \(V_L\) and \(S_R\), the observables \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) are also both enhanced significantly with the NP contributions of \(V_L\) and \(S_R\); nevertheless they are not sensitive to \(V_L\) and \(S_L\). Once the observable \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) is measured, they can be used to distinguish the NP contributions provided by \(V_L\) and \(S_L\) or \(V_L\) and \(S_R\). The NP predictions of the same observables in Figs. 7 and 8 show a similar variation tendency to the increasing of \(q^2\) and have similar deviations to their corresponding SM predictions. The contributions of \(V_R\) and \(S_L\), and \(V_R\) and \(S_R\) to the observables \({\text {d}}Br/{\text {d}}q^2\) and \(R(q^2)\) are larger than the contributions of only \(V_R\) shown in Fig. 4. In addition, the \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) are also more sensitive to \(V_R\) and \(S_L\), and \(V_R\) and \(S_R\) than \(V_R\), but the contributions of \(V_R\) and \(S_L\) on \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) are a little higher than those of \(V_R\) and \(S_R\).

## 5 Summary and conclusion

Several anomalies \(R_{D^{(*)}}\) and \(R_{J/\psi }\) observed in the semileptonic B meson decays have indicated the hints of LFUV and attracted many researchers’ attention. Much work has been done as regards the NP effects of these anomalies on the transitions mediated by the \(b\rightarrow c\) charged current, such as the baryon decays \(\Lambda _{b}\rightarrow \Lambda _{c}l^{-}\bar{\nu }_{l}\). It is important to investigate the semileptonic \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) baryon decays, which are mediated by \(b\rightarrow u(c)\tau ^{-}\bar{\nu }_{\tau }\) transitions at quark level. These decays not only can provide an independent determination of the CKM matrix elements \(|V_{ub}|\) and \(|V_{cb}|\) but also may be further confirmation of the hints of LFUV, which is helpful in exploring NP. Therefore, we investigate the \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) decays in the SM and various NP scenarios.

We consider the NP effects of new vector type couplings separately and the combinations of vector and scalar couplings on the observables \({\text {d}}Br/{\text {d}}q^2\), \(R(q^2)\), \(A_{FB}(q^2)\), \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\), \(P_{L}^{\tau }(q^2)\) and \(C_{F}^{l}(q^2)\) relative to the \(\Xi _{b}\rightarrow \Lambda (\Xi _{c})\tau ^{-}\bar{\nu }_{\tau }\) transitions. The results show that \({\text {d}}Br/{\text {d}}q^2\) and \(R(q^2)\) including any kind of NP couplings are all enhanced largely and have significant deviations comparing to their SM predictions. \(A_{FB}(q^2)\), \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\), \(P_{L}^{\tau }(q^2)\) and \(C_{F}^{l}(q^2)\) are the same as their corresponding SM predictions in \(V_L\) scenario, since the common coefficient \(|1+V_L|^2\) appears in the numerator and the denominator of the expressions describing these observables simultaneously. Nevertheless, the contributions of \(V_R\) to the \(A_{FB}(q^2)\) and \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) are also significant. In the lowest kinematic region, the enhancement of \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) is most prominent. Measuring \(R(q^2)\) in the large \(q^2\) region and \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) in the low \(q^2\) region are more important and useful to further confirm the NP effects of the B meson decay anomalies and can be used to distinguish the new left- and right-handed vector interactions. It is worth noting that the \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) are sensitive to \(V_L\) and \(S_R\) but not \(V_L\) and \(S_L\), so detecting \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) is important to confirm the origin of NP effects. For the \(V_R\) and \(S_L\) and the \(V_R\) and \(S_R\) scenarios, the characters of the NP effects on \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) are similar and both are more significant than in the scenario with only \(V_R\). In addition, the \(P_{L}^{\tau }(q^2)\) of these two decays are both sensitive to \(V_R\) and \(S_L\) and to \(V_R\) and \(S_R\), but not to only \(V_R\). In this case, the observables \(P_{L}^{\Lambda (\Xi _{c})}(q^2)\) and \(P_{L}^{\tau }(q^2)\) could be an effective target for differentiating the \(V_R\) from \(V_R\) and \(S_L\), and from \(V_R\) and \(S_R\). We think this investigation will be helpful in exploring and distinguishing different NP scenarios.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grants nos. 11547266 and 11905096; the Natural Science Foundation of Henan Province Grant nos. 16A140045, 16A140012 and 172102210404; the program for Youth Scholar teachers Supporting Plan in Universities of Henan province Grant no. 2018GGJS158.

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