# Some results on lepton flavour universality violation

## Abstract

Motivated by recent experimental measurements on flavour physics, in tension with Standard Model predictions, we perform an updated analysis of new physics violating lepton flavour universality, by using the effective Lagrangian approach and in the \(Z^{'}\) and \(S_3\) leptoquark models. We explicitly analyze the impact of considering complex Wilson coefficients in the analysis of *B*-anomalies, by performing a global fit of \(R_{K}\) and \(R_{K^{*0}}\) observables, together with \(\varDelta M_s\) and \(A_{CP}^{\mathrm{mix}}\). The inclusion of complex couplings provides a slightly improved global fit, and a marginally improved \(\varDelta M_s\) prediction.

## 1 Introduction

*B*meson decay, \(B^0\rightarrow K^{0} \mu ^+ \mu ^-\) [19, 20, 21, 22].

A great theoretical effort has been devoted to the understanding of these deviations, see for example [15, 17, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42] and references therein. From the theoretical side, the ratios \(R_{K}\) and \(R_{K^{*0}}\) are very clean observables; essentially free of hadronic uncertainties that cancel in the ratios [15]. The experimental data has been used to constrain new physics (NP) models. One useful way to analyze the effects of NP in these observables and to quantify the possible deviations from the SM predictions is through the effective Hamiltonian approach, allowing us a model-independent analysis of new physics effects. In addition, one can compute this effective Hamiltonian in the context of specific NP models. It has been shown that \(Z^{'}\) and leptoquark models could explain the \(R_{K}, R_{K^{*0}}\) deviations.

*CP*asymmetries. The mixing-induced

*CP*asymmetry in the

*B*-sector can be measured through \( A_{CP}^{\mathrm{mix}}\equiv A_{CP}^{\mathrm{mix}}(B_s\rightarrow J/\psi \phi )\equiv \sin (\phi _s^{c\bar{c}s})\), experimentally it is measured to be [48]:

Reference [34] performed fits for the *B*-decay physics observables using complex Wilson coefficients, in the model independent and model dependent approaches. The analysis of Ref. [34] performs fits for the *B*-decay observables using complex couplings, without including the \(\varDelta M_s\) or \(A_{CP}^{\mathrm{mix}}\) observables, then Ref. [34] proceeds to provide predictions to *CP*-violation observables. Reference [34] only includes \(\varDelta M_s\) and \(A_{CP}^{\mathrm{mix}}\) in the \(Z'\)-model fit. Our results agree with the ones of Ref. [34] wherever comparable.

The aim of the present work is to investigate the effects of complex Wilson coefficients in the global analyses of NP in *B*-meson anomalies. We assume a model independent effective Hamiltonian approach and we study the region of NP parameter space compatible with the experimental data, by considering the dependence of the results on the assumptions of imaginary and/or complex Wilson coefficients. We compare our results with the case of considering only real Wilson coefficients. A brief summary of the NP contributions to the effective Lagrangian relevant for \(b \rightarrow s \ell \ell \) transitions and \(B_s\)-mixing is presented in Sect. 2, where we also recall the need to consider complex Wilson coefficients in the analysis. In Sect. 3 we discuss the effects of having imaginary or complex Wilson coefficients on \(R_{K}\) observables. The impact of these complex Wilson coefficients in the analysis of *B*-meson anomalies in two specific models, \(Z^{'}\) and leptoquarks, is included in Sect. 4. We consider a global fit of \(R_{K}\) and \(R_{K^{*0}}\) observables, together with \(\varDelta M_s\) and *CP*-violation observable \(A_{CP}^{\mathrm{mix}}\) in this analysis. Finally, conclusions are given in Sect. 5.

## 2 Effective Hamiltonians and new physics models

*CP*-asymmetry \(A_{CP}^{\mathrm{mix}}\) is given by[47, 49, 50]

Since \(\varDelta M_s^{\mathrm{exp}}<{\varDelta M_s^{\mathrm{SM}}}\) (2), Eq. (8) tells us that to obtain a prediction of \(\varDelta M_s\) closer to \(\varDelta M_s^{\mathrm{exp}}\) the NP Wilson coefficient \(C_{bs}^{LL}\) (7) must be negative (\(C_{bs}^{LL}<0\)). In a generic effective Hamiltonian approach, each Wilson coefficient is independent, and setting \(C_{bs}^{LL}<0\) has no effect on \(C_9^{\mathrm{NP}\, \mu }\), \(C_{10}^{\mathrm{NP}\, \mu }\), etc. However, explicit NP models give predictions on the Wilson coefficients which introduce correlations among them. We will concentrate on two specific models that have been proposed to solve the semi-leptonic \(B_s\)-decay anomalies: \(Z'\) and leptoquarks.

*CP*-asymmetry \(A_{CP}^{\mathrm{mix}}\) (9), so imaginary couplings might provide a way of improving the predictions on \(\varDelta M_s\) without introducing unwanted

*CP*-asymmetries.

## 3 Imaginary Wilson coefficients and \(R_{K}\) observables

Figure 1 shows the values of the ratios \(R_{K}\) and \(R_{K^{*0}}\), in their respective \(q^2\) ranges, when both Wilson coefficients \(C_9^{\mathrm{NP}\, \mu }\) and \(C_{10}^{\mathrm{NP}\, \mu }\) are imaginary (Fig. 1a) and when they are real (Fig. 1b), by assuming that \(C_9^{\mathrm{NP}\, \mu }= -C_{10}^{\mathrm{NP}\, \mu }\). If these two coefficients are imaginary, in all cases the minimum value for the ratio is obtained at the corresponding SM point \(C_9^{\mathrm{NP}\, \mu }= -C_{10}^{\mathrm{NP}\, \mu }=0\). The addition of non-zero imaginary Wilson coefficients results in larger values of \(R_{K}\) and \(R_{K^{*0}}\), at odds with the experimental values \(R_{K^{(*0)}}^{\mathrm {exp}} < R_{K^{(*0)}}^{\mathrm {SM}}\). This behaviour was already pointed out in Ref. [26], where it is shown that the interference of purely imaginary Wilson with the SM vanishes, and therefore they can not provide negative contributions to \(R_{K}\), \(R_{K^{*0}}\) (see also below). In contrast, as shown in the right panel, values of \(R_{K^{(*0)}} \sim 0.7\) (as in the experimental measurements) are possible when the Wilson coefficients are real.

We have done a global fit by including the ratios \(R_{K}\) and \(R_{K^{*0}}\), and the angular observables \(P_4'\) and \(P_5'\) [6, 19, 21, 22].^{1} Results are shown in Fig. 2. The allowed regions for imaginary values of \(C_9^{\mathrm{NP}\, \mu }\) and \(C_{10}^{\mathrm{NP}\, \mu }\) when fitting to measurements of a series of \(b\rightarrow s\mu ^+\mu ^-\) observables are presented in Fig. 2a, by assuming all other Wilson coefficients to be SM-like. The numerical analysis has been done by using the open source code *flavio* 0.28 [55], which computes the \(\chi ^2\) function with each \((C_9^{\mathrm{NP}\, \mu }, C_{10}^{\mathrm{NP}\, \mu })\) pair. The \(\chi ^2\) difference is evaluated with respect to the SM point, \(\varDelta \chi ^2_{\mathrm {SM}}= \chi ^2_{\mathrm {SM}} - \chi ^2_{\mathrm {min}}\). Then, the pull in \(\sigma \) is defined as \(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}\), in the case of only one Wilson coefficient, and for the two-dimensional case it can be evaluated by using the inverse cumulative distribution function of a \(\chi ^2\) distribution having two degrees of freedom; for instance, \(\varDelta \chi ^2 =2.29\) for \(1\,\sigma \). The darker red shaded regions in Fig. 2 correspond to the points with \(\varDelta \chi ^2=\chi ^2-\chi ^2_{\mathrm {min}}\le 2.29\), that is, they are less than \(1\,\sigma \) away from the best fit point, whereas the lighter red shaded regions correspond to \(\varDelta \chi ^2\le 6.18\) (\(\equiv 2\,\sigma \)). The crosses mark the position of the best fit points. In Fig. 2a the \(\chi ^2\) function has a broad flat region centered around the origin, with two nearly symmetric minima found at (\(C_9^{\mathrm{NP}\, \mu }= 0.72\,i\), \(C_{10}^{\mathrm{NP}\, \mu }= 0.74\;i\)) and (\(C_9^{\mathrm{NP}\, \mu }=-0.75\,i\), \(C_{10}^{\mathrm{NP}\, \mu }=-0.74\,i\)). The pull of the SM, defined as the probability that the SM scenario can describe the best fit assuming that \(\varDelta \chi ^2_{\mathrm {SM}}\) follows a \(\chi ^2\) distribution with 2 degrees of freedom, is of just \(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}=1.42\) (\(\equiv 0.91\,\sigma \)) and \(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}=1.38\) (\(\equiv 0.87\,\sigma \)) respectively, and both of them have the same \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}=2.25\), that is, purely complex couplings do not provide a good description of the data. For completeness, the fit to real values of the Wilson coefficients are included in Fig. 2b. Now the confidence regions are much tighter and do not include the SM point. In fact, the best fit point (\(C_9^{\mathrm{NP}\, \mu }= -1.09\), \(C_{10}^{\mathrm{NP}\, \mu }= 0.481\)) improves the SM by \(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}=6.28\) (\(\equiv 5.95\,\sigma \)), and a much lower \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}=1.24\).

*flavio*-computed value of \(R_{K^{*0}}\) to better than \(4\%\) in a large region of the parameter space. Now, if we assume that NP does not affect the electron channel (\(C_9^{\mathrm{NP}\, e}=C_{10}^{\mathrm{NP}\, e}=0\)), it is clear that to obtain \(R_{K^{*0}}<R_{K^{*0}}^{\mathrm {SM}}\) one needs to introduce \(C_9^{\mathrm{NP}\, \mu }\) and \(C_{10}^{\mathrm{NP}\, \mu }\) with a non-zero real part: the only possible negative contributions come from the \(\mathrm {Re}\,C_9^{\mathrm{NP}\, \mu }\), \(\mathrm {Re}\,C_{10}^{\mathrm{NP}\, \mu }\) terms, whereas the \(|C_9^{\mathrm{NP}\, \mu }|^2\), \(|C_{10}^{\mathrm{NP}\, \mu }|^2\) terms have a positive-defined sign, and can not reduce the value of \(R_{K^{*0}}\). Thus, purely imaginary values of \(C_{9,10}^{\mathrm{NP}\, \mu }\) contribute only to the modulus (positive-definite) and not to the real part, and can not bring the prediction of \(R_{K^{*0}}\) closer to the experimental value. In addition, this expression tells us that the better option to reduce the prediction of \(R_{K^{*0}}\) is using a real negative \(C_9^{\mathrm{NP}\, \mu }\), and a real positive \(C_{10}^{\mathrm{NP}\, \mu }\). This is actually the result that we have obtained in our numerical analysis. Figure 1b shows that, for real Wilson coefficients, the lowest prediction for \(R_{K^{*0}}\) is obtained for \(C_9^{\mathrm{NP}\, \mu }=-C_{10}^{\mathrm{NP}\, \mu }<0\), and Fig. 2b shows that the best fit is obtained for negative \(C_9^{\mathrm{NP}\, \mu }\) and positive \(C_{10}^{\mathrm{NP}\, \mu }\). Figure 1a shows that, in general, imaginary Wilson coefficients give positive contributions to \(R_{K}\), \(R_{K^{*0}}\), in accordance with Eq. (17). Of course, the full expression is richer than Eq. (17), and we expect some deviations, Fig. 2a shows that the best fit point is not the SM (\(C_9^{\mathrm{NP}\, \mu }=C_{10}^{\mathrm{NP}\, \mu }=0\)), but the best fit regions are centered around it, and the SM pull with respect the best fit points is small.

Best fit Wilson coefficients complex values to semi-leptonic decay observables \(R_{K}, R_{K^{*0}}, P_4'\) and \(P_5'\), allowing only one free coefficient at a time. Shown are also the corresponding pulls, and \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}\)

Best fit(s) | Pull (\(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}\)) | Pull (\(\sigma \)) | \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}\) | |
---|---|---|---|---|

\(C_9^{\mathrm{NP}\, \mu }\) | \(-1.11-0.02\;i\) | 5.94 | 5.60 \(\sigma \) | 1.35 |

\(C_{10}^{\mathrm{NP}\, \mu }\) | \(1.66+1.99\;i\) | 5.02 | 4.65 \(\sigma \) | 1.62 |

\(1.67-2.01\;i\) | ||||

\(C_9^{\mathrm{NP}\, \mu }= -C_{10}^{\mathrm{NP}\, \mu }\) | \(-1.16+1.14\;i\) | 6.06 | 5.72 \(\sigma \) | 1.31 |

\(-1.18-1.18\;i\) | ||||

\(C_9^{'\mathrm{NP}\, \mu }\) | \(-0.24-0.003\;i\) | 1.07 | 0.57 \(\sigma \) | 2.27 |

\(C_{10}^{'\mathrm{NP}\, \mu }\) | \(0.33-0.014\;i\) | 2.22 | 1.72 \(\sigma \) | 2.17 |

\(C_9^{\mathrm{NP}\, e}\) | \(-3.29+5.02\;i\) | 4.85 | 4.47 \(\sigma \) | 1.67 |

\(-3.35-5.04\;i\) | ||||

\(C_{10}^{\mathrm{NP}\, e}\) | \(-0.27+3.48\;i\) | 4.72 | 4.34 \(\sigma \) | 1.70 |

\(-0.27-3.48\;i\) | ||||

\(C_9^{\mathrm{NP}\, e}= -C_{10}^{\mathrm{NP}\, e}\) | \(-3.29+4.58\;i\) | 4.85 | 4.47 \(\sigma \) | 1.67 |

\(-3.35-4.59\;i\) | ||||

\(C_9^{'\mathrm{NP}\, e}\) | \(-0.59+3.89\;i\) | 4.81 | 4.43 \(\sigma \) | 1.68 |

\(-0.59-3.89\;i\) | ||||

\(C_{10}^{'\mathrm{NP}\, e}\) | \(0.52+3.88\;i\) | 4.81 | 4.43 \(\sigma \) | 1.68 |

\(0.53-3.88\;i\) |

\(R_{K}\), \(R_{K^{*0}}\) predictions with \(1\,\sigma \) uncertainties corresponding to the best fit Wilson coefficients of Table 1

\(R_{K}\) | \(R_{K^{*0}}^{[0.045,1.1]}\) | \(R_{K^{*0}}^{[1.1,6]}\) | |
---|---|---|---|

\(C_9^{\mathrm{NP}\, \mu }\) | \( 0.77 \pm 0.03 \) | \( 0.887 \pm 0.009 \) | \( 0.82\pm 0.04 \) |

\(C_{10}^{\mathrm{NP}\, \mu }\) | \( 0.78 \pm 0.05 \) | \( 0.87 \pm 0.03 \) | \( 0.80 \pm 0.10 \) |

\(C_9^{\mathrm{NP}\, \mu }= -C_{10}^{\mathrm{NP}\, \mu }\) | \(0.59 \pm 0.08\) | \(0.83 \pm 0.03\) | \(0.63 \pm 0.09\) |

\(C_9^{'\mathrm{NP}\, \mu }\) | \(0.95 \pm 0.05\) | \( 0.96\pm 0.03\) | \(1.09 \pm 0.09\) |

\(C_{10}^{'\mathrm{NP}\, \mu }\) | \( 0.92\pm 0.07\) | \(0.95 \pm 0.03\) | \(1.07 \pm 0.09\) |

\(C_9^{\mathrm{NP}\, e}\) | \(0.76 \pm 0.09\) | \(0.69 \pm 0.12\) | \(0.52 \pm 0.17\) |

\(C_{10}^{\mathrm{NP}\, e}\) | \(0.69 \pm 0.06\) | \(0.77 \pm 0.06\) | \(0.59 \pm 0.13\) |

\(C_9^{\mathrm{NP}\, e}= -C_{10}^{\mathrm{NP}\, e}\) | \(0.76 \pm 0.09\) | \(0.70 \pm 0.10\) | \(0.52 \pm 0.17\) |

\(C_9^{'\mathrm{NP}\, e}\) | \(0.75 \pm 0.09\) | \(0.71 \pm 0.10\) | \(0.52 \pm 0.18\) |

\(C_{10}^{'\mathrm{NP}\, e}\) | \(0.75 \pm 0.09\) | \(0.80 \pm 0.09\) | \(0.66 \pm 0.14\) |

We conclude that, actually, a NP explanation for \(R_{K}\), \(R_{K^{*0}}\) requires that \(C_9^{\mathrm{NP}\, \mu }\), \(C_{10}^{\mathrm{NP}\, \mu }\) have a non-zero real part, whereas we saw above that NP explanation for \(\varDelta M_s\) requires that \(C_9^{\mathrm{NP}\, \mu }\), \(C_{10}^{\mathrm{NP}\, \mu }\) have a non-zero imaginary part. Then, to have a NP explanation for both observables \(C_9^{\mathrm{NP}\, \mu }\), \(C_{10}^{\mathrm{NP}\, \mu }\) should be general complex numbers. Following this reasoning we have performed a global fit to the semi-leptonic decay observables \(R_{K}, R_{K^{*0}}, P_4'\) and \(P_5'\) using generic complex Wilson coefficients allowing only one free Wilson coefficient at a time. Table 1 shows the best fit values, pulls (defined as \(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}\)) and \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}\), for scenarios with NP in one individual complex Wilson coefficient, and Table 2 shows the prediction for \(R_{K}\), \(R_{K^{*0}}\) for the corresponding central values of each fit, together with the \(1\,\sigma \) uncertainties. The primed Wilson coefficients are also included. We found that the best fit of \(R_{K}\) and \(R_{K^{*0}}\) and the angular distributions is obtained for \(C_9^{\mathrm{NP}\, \mu }= -1.11-0.02 \;i\), for \(C_{10}^{\mathrm{NP}\, \mu }\) we find two points with similar minimum value for \(\chi ^2\) with opposite signs of the imaginary part, \(C_{10}^{\mathrm{NP}\, \mu }=1.66+1.99\;i\) and \(C_{10}^{\mathrm{NP}\, \mu }=1.65-2.10\;i\). Assuming \(C_9^{\mathrm{NP}\, \mu }=-C_{10}^{\mathrm{NP}\, \mu }\) we also obtain a double minimum \(C_9^{\mathrm{NP}\, \mu }=-C_{10}^{\mathrm{NP}\, \mu }=-1.16+1.14\;i\) and \(C_9^{\mathrm{NP}\, \mu }=-C_{10}^{\mathrm{NP}\, \mu }=-1.18-1.18\;i\) with a pull of \(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}=6.06\) (\(\equiv 5.72\,\sigma \)) and a \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}=1.31\). By looking at \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}\) we see that the scenarios with only \(C_9^{\mathrm{NP}\, \mu }\) or \(C_9^{\mathrm{NP}\, \mu }=-C_{10}^{\mathrm{NP}\, \mu }\) provide the best description of experimental data, whereas the scenarios with \(C_9^{'\mathrm{NP}\, \mu }\) and \(C_{10}^{'\mathrm{NP}\, \mu }\) provide the worst description. If only real Wilson coefficients are chosen the best fit of \(R_{K}\) and \(R_{K^{*0}}\) yields \(C_9^{\mathrm{NP}\, \mu }= -1.59\), \(C_{10}^{\mathrm{NP}\, \mu }=1.23\) or \(C_9^{\mathrm{NP}\, \mu }= -C_{10}^{\mathrm{NP}\, \mu }=-0.64\), with a pull around \(4.2 \,\sigma \) [36].

Reference [34] also provides fits for complex generic Wilson coefficients. Their *scenario I* corresponds to our first line in Table 1, our best fit value agrees with their result (\(C_9^{\mathrm{NP}\, \mu }= (-1.1\pm 0.2) + (0\pm 0.9\,i)\)), within the large uncertainties they give for the imaginary part, but we obtain larger pulls (\(5.6\,\sigma \) vs. \(4.2\,\sigma \) of Ref. [34]). Their *scenario II* corresponds to our third line in Table 1 (\(C_9^{\mathrm{NP}\, \mu }=-C_{10}^{\mathrm{NP}\, \mu }\)), we agree with the main features of their fit, for the real part they obtain \(\mathrm {Re}(C_9^{\mathrm{NP}\, \mu })=\mathrm {Re}(C_{10}^{\mathrm{NP}\, \mu })=-0.8\pm 0.3\), we obtain a slightly smaller real part, but they agree within uncertainties, both of us obtain a double minimum for the imaginary part \(\sim \pm (1.1-1.2)\,i\), again, we obtain a slightly larger pull (\(5.72\,\sigma \) vs. 4.0, \(4.2\,\sigma \) of Ref. [34]).

Choosing complex Wilson coefficients also implies additional constraints from *CP*-violating observables. This fact has not been considered in the previous analysis. In the next section we study the consequences of having these coefficients in the analysis of *B*-meson anomalies on some NP models and we consider a global fit of both the ratios \(R_{K}\) and \(R_{K^{*0}}\) and the angular observables \(P_4'\) and \(P_5'\), and also the *CP*-mixing asymmetry.

## 4 \(B_s\)-mixing and NP models

Several NP models that are able to explain the lepton flavour universality violation effects are constrained by other flavour observables like \(B_s\)-mixing. In particular the parameter space of \(Z^{'}\) and leptoquark models are severely constrained by the present experimental results of \(\varDelta M_s\) [47]. Besides, as already mentioned, additional constraints emerge from *CP*-violating observables when considering complex couplings. Reference [47] argues that nearly imaginary Wilson coefficients could explain the discrepancies with the \(\varDelta M_s\) experimental measurement, but a global fit of \(R_{K}\) and \(R_{K^{*0}}\) observables, together with \(\varDelta M_s\) and *CP*-violation observable \(A_{CP}^{\mathrm{mix}}\) in \(B_{s}\rightarrow J/\psi \phi \) decays should be performed. In the next subsections we investigate these issues for the case of \(Z^{'}\) and leptoquark models.

### 4.1 \(Z^{'}\) fit

*CP*-violation observable \(A_{CP}^{\mathrm{mix}}\) is included in our analysis.

*B*-meson decays, i.e. \(b\rightarrow s\mu ^+\mu ^-\) as in Fig. 2 plus the branching ratios \(\mathrm {BR}(B_s\rightarrow \mu ^+ \mu ^-)\) and \(\mathrm {BR}(B^0 \rightarrow \mu ^+ \mu ^-)\). The best fit region is the one between the curves; dotted lines: \(\varDelta \chi ^2= 1\), dash-dotted lines: \(\varDelta \chi ^2= 4\). Blue lines (solid, dashed) correspond to the fit to \(B_s\)-mixing observables \(\varDelta M_s\) and \(A_{CP}^{\mathrm{mix}}\). The best fit region is the one between the lines; solid lines \(\varDelta \chi ^2= 1\), dashed lines \(\varDelta \chi ^2= 4\), there are two regions with \(\varDelta \chi ^2<1\), but between them \(\varDelta \chi ^2\) is always smaller than 4. The green regions are the combined global fit: dark region \(\varDelta \chi ^2\le 1\), medium \(\varDelta \chi ^2\le 4\) and light \(\varDelta \chi ^2\le 9\).

Best fits, and corresponding pulls, to \(R_{K}, R_{K^{*0}}, \varDelta M_s\) and \(A_{CP}^{\mathrm{mix}}\); considering real, imaginary and complex Wilson coefficients on the \(Z^{'}\) model. Shown are also the corresponding pulls, \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}\), and the predictions for semi-leptonic decay observables \(R_{K}\), \(R_{K^{*0}}\); \(\varDelta M_s\) and \(A_{CP}^{\mathrm{mix}}\) with \(1\,\sigma \) uncertainties

Best fits | Real | Imaginary | Complex |
---|---|---|---|

\(\lambda _{23}^{Q}\) | \(-\,0.002 \) | \(\pm 0.047\;i\) | \(-0.0020-0.0021\;i\) |

\(M_{Z'}\) | 1.31 TeV | 12 TeV | 1.08 TeV |

Pull (\(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}\)) | 5.70 | 1.61 | 6.05 |

Pull (\(\sigma \)) | 5.39 \(\sigma \) | 1.09 \(\sigma \) | 5.43 \(\sigma \) |

\(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}\) | 1.41 | 2.12 | 1.34 |

\(R_{K}\) | \(0.66 \pm 0.05\) | \( 1.00 \pm 0.01 \) | \(0.65 \pm 0.07 \) |

\(R_{K^{*0}}^{[0.045,1.1]}\) | \(0.849 \pm 0.013\) | \( 0.93 \pm 0.02\) | \( 0.84\pm 0.02\) |

\(R_{K^{*0}}^{[1.1, 6]}\) | \(0.68 \pm 0.05\) | \(1.00 \pm 0.01\) | \( 0.68\pm 0.07\) |

\(\varDelta M_s\) | \( 20.41 \pm 1.26 \) ps\({}^{-1}\) | \(18.0 \pm 1.7 \) ps\({}^{-1}\) | \( 19.95\pm 1.27\) ps\({}^{-1}\) |

\(A_{CP}^{\mathrm{mix}}\) | \(-0.0369 \pm 0.0002\) | \( -0.041 \pm 0.002 \) | \(-0.035 \pm 0.003\) |

Figure 4 shows the best fit regions in the complex \(\lambda _{23}^Q\) plane for the best fit mass value \(M_{Z'}=1.08\,\mathrm{TeV}\) (Table 3). The red region shows the 2-dimensional 1 and 2-\(\sigma \) allowed values (\(\varDelta \chi ^2=2.29\), 6.18) including only the \(b\rightarrow s\mu ^+\mu ^-\) observables, the blue region shows the 1 and 2-\(\sigma \) allowed values including only \(\varDelta M_s\), and the green region show the 1 and 2-\(\sigma \) allowed values including only \(A_{CP}^{\mathrm{mix}}\), the violet region shows the combined fit. Here we see the tension between the \(b\rightarrow s\mu ^+\mu ^-\) and \(\varDelta M_s\) fits. \(b\rightarrow s\mu ^+\mu ^-\) selects a region around the real axis of the coupling, whereas \(\varDelta M_s\) selects regions away from it. There are two small intersection regions for the 1-\(\sigma \) allowed values of both fits. The \(A_{CP}^{\mathrm{mix}}\) fit selects one of these regions, and breaks the degeneracy. Actually, the \(b\rightarrow s\mu ^+\mu ^-\) fit selects fixed values of \(C_9^{\mathrm{NP}\, \mu }=-C_{10}^{\mathrm{NP}\, \mu }\), Eq. (11), since \(C_9^{\mathrm{NP}\, \mu }=-C_{10}^{\mathrm{NP}\, \mu }\) scale as \(\sim \lambda _{23}^Q/M_{Z'}^2\), for fixed \(C_9^{\mathrm{NP}\, \mu }=-C_{10}^{\mathrm{NP}\, \mu }\) the allowed values of \(\lambda _{23}^Q\) (red region in Fig. 4) around the real axis will grow as \(M_{Z'}^2\), but, at the same time, the allowed region will move away from the imaginary axis as \(M_{Z'}^2\). On the other hand, the fit on \(\varDelta M_s\) selects fixed values of \(C_{bs}^{LL}\), Eq. (12), since \(C_{bs}^{LL}\sim (\lambda _{23}^Q)^2/M_{Z'}^2\), for fixed \(C_{bs}^{LL}\) the 1-\(\sigma \) unfavored region around the origin (light blue region in Fig. 4) will grow as \(\lambda _{23}^Q\sim M_{Z'}\). As \(M_{Z'}\) grows, the red region moves away from the origin as \(M_{Z'}^2\), but the blue region expands only as \(M_{Z'}\), so that at some \(M_{Z'}\) value their 1-\(\sigma \) regions do not longer intersect. This is the reason why we obtain a relatively low \(M_{Z'}\) in the fits of Table 3.

Reference [34] provides also a fit for the \(Z'\) model, using a fixed \(M_{Z'}=1\,\mathrm{TeV}\), this value is close to our best fit value of Table 3. For \(\lambda _{22}^L=1\) they obtain the best fit coupling \(\lambda _{23}^Q= (-0.8\pm 0.3)\times 10^{-3} + (-0.4 \pm 3.1)\times 10^{-3}\,i\) with a pull of \(4.0\,\sigma \). Our best fit values agree with them within uncertainties. Note that we do not provide uncertainties for the best fit values, the reason being that the parameters are not independent, the 2-dimensional best fit regions in Fig. 4 are not ellipses, and the best fit points are not on the center of the figures, so that giving a central value with 1-dimensional uncertainties overestimates the uncertainty and leads to confusion about the meaning and position of the best fit point.

Best fits, and corresponding pulls, to \(R_{K}, R_{K^{*0}}, \varDelta M_s\) and \(A_{CP}^{\mathrm{mix}}\); considering real, imaginary and complex Wilson coefficients on the \(S_3\) leptoquark. Shown are also the corresponding pulls, \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}\) and the predictions for semi-leptonic decay observables \(R_{K}\), \(R_{K^{*0}}\); \(\varDelta M_s\) and \(A_{CP}^{\mathrm{mix}}\) with \(1\,\sigma \) uncertainties

Best fits | Real | Imaginary | Complex |
---|---|---|---|

\(y^{QL}_{32} y^{QL*}_{22}\) | 0.04 | \( -1.67\;i\) | \(0.033+0.034\;i\) |

\(M_{S_3}\) | 5.19 TeV | 50 TeV | 4.10 TeV |

Pull (\(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}\)) | 5.82 | 1.10 | 5.90 |

Pull (\(\sigma \)) | 5.47 \(\sigma \) | 0.60 \(\sigma \) | 5.27 \(\sigma \) |

\(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}\) | 1.38 | 2.16 | 1.39 |

\(R_{K}\) | \(0.64 \pm 0.06\) | \(1.00 \pm 0.01\) | \(0.62 \pm 0.14\) |

\(R_{K^{*0}}^{[0.045,1.1]}\) | \(0.835 \pm 0.015\) | \( 0.93\pm 0.02 \) | \(0.84 \pm 0.04\) |

\(R_{K^{*0}}^{[1.1, 6]}\) | \( 0.66 \pm 0.06\) | \(1.00 \pm 0.01\) | \(0.66 \pm 0.14\) |

\(\varDelta M_s\) | \(20.07 \pm 1.27\) ps\({}^{-1}\) | \(18.8 \pm 1.7\) ps\({}^{-1}\) | \(20.0 \pm 1.2\) ps\({}^{-1}\) |

\(A_{CP}^{\mathrm{mix}}\) | \(-\,0.0374 \pm 0.0006\) | \( -\,0.039 \pm 0.002\) | \(-\,0.032 \pm 0.003\) |

### 4.2 Leptoquark fit

*B*-meson decays, i.e. \(b\rightarrow s\mu ^+\mu ^-\) plus the branching ratios \(\mathrm {BR}(B_s\rightarrow \mu ^+ \mu ^-)\) and \(\mathrm {BR}(B^0 \rightarrow \mu ^+ \mu ^-)\), the best fit region is the one between the curves; dotted lines: \(\varDelta \chi ^2= 1\), dash-dotted lines: \(\varDelta \chi ^2= 4\). Blue lines (solid, dashed) correspond to the fit to \(B_s\)-mixing observables \(\varDelta M_s\) and \(A_{CP}^{\mathrm{mix}}\). The best fit region is the one between the lines; solid lines \(\varDelta \chi ^2= 1\), dashed lines \(\varDelta \chi ^2= 4\), there are two regions with \(\varDelta \chi ^2<1\), but between them \(\varDelta \chi ^2\) is always smaller than 4. The green regions are the combined global fit: dark region \(\varDelta \chi ^2\le 1\), medium \(\varDelta \chi ^2\le 4\) and light \(\varDelta \chi ^2\le 9\). In the \(b\rightarrow s\mu ^+\mu ^-\) fit the best fit parameters for imaginary couplings is \(y_{32}^{QL}y_{22}^{QL*}=-0.2\,i\), \(M_{S_3}=40.8\,\mathrm{TeV}\). The leptoquark fit to \(B_s\)-mixing observables has a double minimum, located at \(M_{S_3}= 44.9\,\mathrm{TeV}\), \(y_{32}^{QL} y_{22}^{QL*} = \pm 2\;i\), with a SM pull of \(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}= 1.74\) (\(\equiv 1.22\,\sigma \)) and \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}=0.51\). These points correspond to a value for the Wilson coefficient of \(C_{bs}^{LL}=-1.39\times 10^{-4}\). The global fit, including all observables, and considering only imaginary \(y_{32}^{QL} y_{22}^{QL*}\) couplings, is located at \(M_{S_3} = 50\,\mathrm{TeV}\), \(y_{32}^{QL} y_{22}^{QL*} = -1.67\;i\); with a SM pull of only \(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}= 1.1 (\equiv 0.6\,\sigma )\) and a large \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}=2.16\). Larger \(M_{S_3}\) masses provide similar values for the best fit couplings, and observable predictions, and the pulls improve slowly. The situation is similar than in the \(Z'\) case: by allowing larger \(M_{S_3}\) masses the best fit coupling reaches an asymptotic straight line, where the contribution to \(\varDelta M_s\) is constant (15), whereas the contribution to \(|C_{9,10}^{\mathrm{NP}\, \mu }|\) (14) decreases as \(M_{S_3}^{-1}\), the best fit coupling behaves as \(y_{32}^{QL} y_{22}^{QL*} \simeq i\,(4.43\times 10^{-2}\times M_{S_3}/\,\mathrm{TeV})\). Table 4 shows the best fit parameters for the leptoquark model considered in this work, corresponding pulls, predictions to the observables \(R_{K}\), \(R_{K^{*0}}\), \(\varDelta M_s\) and \(A_{CP}^{\mathrm{mix}}\) and \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}\), considering real, imaginary and complex Wilson coefficients. Table 4 shows that only imaginary couplings do not improve the results, they cannot explain the \(R_{K^{(*)}}\) anomaly. However, when complex couplings are considered, we found a better global fit of \(R_{K}, R_{K^{*0}}\) observables, the best global fit parameters emerge at \(M_{S_3} = 4.1\,\mathrm{TeV}\) and \(y_{32}^{QL} y_{22}^{QL*} = 0.033+0.034\;i\), with \(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}= 5.90\) (\(\equiv 5.27\, \sigma \)). The best fit point \(M_{S_3}\) and the coupling real part are similar to the real couplings case. The imaginary part of the coupling is similar to the real part. The pull with respect the SM is marginally better in the case of complex couplings (\(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}=5.9\) versus 5.82), but it actually worsens in units of \(\sigma \), since the complex coupling fit has one more free parameter. The \(\chi ^2_{\mathrm {min}}/\mathrm {d.o.f.}\) is similar in both scenarios. The predictions for the

*B*-meson physics observables are similar than in the real couplings case.

Figure 6 shows the best fit regions in the complex \(y_{32}^{QL} y_{22}^{QL*}\) plane, for the best fit mass parameter \(M_{S_3}=4.1\,\mathrm{TeV}\), Table 4. The meaning of each region is as in Fig. 4. In this model there is no intersection between the 1-\(\sigma \) best fit regions of the \(b\rightarrow s\mu ^+\mu ^-\) and the \(\varDelta M_s\) fits. Here we also find the tension between the \(b\rightarrow s\mu ^+\mu ^-\) and \(\varDelta M_s\) observables, and the different evolution of the best fit regions with the leptoquark mass \(M_{S_3}\). The \(\varDelta M_s\) fit moves the best fit point away from the real axis, and the \(A_{CP}^{\mathrm{mix}}\) fit selects of the of the signs for the imaginary part, however the global best fit region lies outside the 1-\(\sigma \) region for \(\varDelta M_s\), and the \(\varDelta M_s\) prediction does not improve with respect the SM.

Reference [34] also provides a fit for the leptoquark scenario, our model corresponds to their \(\mathbf {\varDelta }_{1/3}[S3]\) model. Reference [34] performs a fit fixing the leptoquark mass to \(M_{S_3}=1\,\mathrm{TeV}\), and they obtain a two nearly degenerate minimums with positive and negative imaginary parts. The reason for that is that they do not include the \(A_{CP}^{\mathrm{mix}}\) observable in the fit. Since the \(C_9^{\mathrm{NP}\, \mu }=-C_{10}^{\mathrm{NP}\, \mu }\) Wilson coefficient scales like \(\sim y_{32}^{QL} y_{22}^{QL*} /M_{S_3}^2\) (14) we can compare both results by scaling the best fit coupling with the mass squared, by taking their central value for the positive imaginary part, we obtain \(y_{32}^{QL} y_{22}^{QL*}=(1.4+1.7\,i)\times 10^{-3} \times (4.1)^2=0.023+0.029\,i\), which is similar to our third column in Table 4, and is inside the best fit region of Fig. 6. Again, we obtain a larger pull (\(\sqrt{\varDelta \chi ^2_{\mathrm {SM}}}=5.9\) versus 4.0).

If one relaxes the condition \(y_{33}^{QL} y_{23}^{QL*}\simeq y_{31}^{QL} y_{21}^{QL*} \simeq 0\) then the leptoquark contributions to \(\varDelta M_s\) (15) and \(C_{9,10}^{\mathrm{NP}\, \mu }\) (14) are no longer correlated, it would be possible to choose: a purely real coupling to muons, such that it fulfils the first column of Table 4; a vanishing coupling for electrons, such that it does not contribute to \(R_{K}\), \(R_{K^{*0}}\); and a complex coupling for taus, such that \(y_{33}^{QL} y_{23}^{QL*}+y_{32}^{QL} y_{22}^{QL*}\) is purely imaginary, and provides a good prediction for \(\varDelta M_s\) like in the second column of Table 4. Of course, this would be a quite strange arrangement for leptoquark couplings! Another option would be to take an specific model construction for the relations among the leptoquark couplings, and make a global fit on these parameters. This analysis is beyond the scope of the present work.

## 5 Conclusions

In this work, we have updated the analysis of New Physics violating lepton flavour universality, by using the effective Lagrangian approach and also in the \(Z^{'}\) and leptoquark models. By considering generic complex Wilson coefficients we found that purely imaginary coefficients do not improve significantly *B*-meson physics observable predictions, whereas complex coefficients (Table 1) do improve the predictions, with a slightly improved pull than using only real coefficients [36]. We have analyzed the impact of considering complex Wilson coefficients in the analysis of *B*-meson anomalies in two specific models: \(Z^{'}\) and leptoquarks, and we have presented a global fit of \(R_{K}\) and \(R_{K^{*0}}\) observables, together with \(\varDelta M_s\) and *CP*-violation observable \(A_{CP}^{\mathrm{mix}}\) when these complex couplings are included in the analysis. We confirm that real Wilson coefficients cannot explain the \(B_s\)-mixing anomaly; but also only imaginary Wilson coefficients cannot explain the \(R_{K}\), \(R_{K^{*0}}\) anomaly. Contrary, complex couplings offer a slightly better global fit. For complex couplings the predictions for \(R_{K}\), \(R_{K^{*0}}\) and \(\varDelta M_s\) are similar than for real couplings (Tables 3, 4). For \(Z'\) models the best fit in both cases is obtained for \(M_{Z'}\simeq \) 1–1.3 \(\,\mathrm{TeV}\), a negative real part of the coupling \(\mathrm{Re}(\lambda _{23}^Q)\simeq -0.002\), with possibly a similar imaginary coupling part \(\mathrm{Im}(\lambda _{23}^Q)\simeq -\,0.0021\). For leptoquark models the situation is similar, with a best fit mass of \(M_{S_3}=\) 4–5 \(\,\mathrm{TeV}\) and a coupling with a positive real part \(y_{32}^{QL} y_{22}^{QL*} \simeq \) 0.03–0.04, the presence of a similar imaginary part does not improve significantly the fit. One can obtain better fits in the leptoquark models by relaxing the assumption on the leptoquark couplings, or providing specific models for leptoquark couplings, this analysis is beyond the scope of the present work. In summary, new physics \(Z'\) or leptoquark models with complex couplings provide a slightly improved global fit to *B*-meson physics observables as compared with models with real couplings.

## 6 Note added

## Footnotes

- 1.
For the \(P_4'\), \(P_5'\) observables we include all \(q^2\) bins, except the ones around to the charm resonances \(q^2\in [8.7,14]\,\mathrm{GeV}^2\), where the theoretical computation is not reliable. In total we include 15 measurements for \(P_4'\) [6, 19, 22] and 21 measurements for \(P_5'\) [6, 19, 21, 22].

## Notes

### Acknowledgements

J.G. is thankful to F. Mescia for useful discussions. The work of J.A. and S.P. is partially supported by Spanish MINECO/FEDER grant FPA2015-65745-P and DGA-FSE grant 2015-E24/2. S.P. is also supported by CPAN (CSD2007-00042) and FPA2016-81784-REDT. J.A. is also supported by the *Departamento de Innovación, Investigación y Universidad* of Aragon government (DIIU-DGA/European Social Fund). J.G. has been supported by MCOC (Spain) (FPA2016-76005-C2-2-P), MDM-2014-0369 of ICCUB (Unidad de Excelencia ‘María de Maeztu’), AGAUR (2017SGR754) and CPAN (CSD2007-00042). J.G. thanks the warm hospitality of the Universidad de Zaragoza during the completion of this work.

## References

- 1.R. Aaij et al. [LHCb Collaboration], JHEP
**1406**133 (2014). https://doi.org/10.1007/JHEP06(2014)133. arXiv:1403.8044 [hep-ex] - 2.R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett.
**113**151601 (2014). https://doi.org/10.1103/PhysRevLett.113.151601. arXiv:1406.6482 [hep-ex] - 3.V. Khachatryan et al. [CMS and LHCb Collaborations], Nature
**522**68 (2015). https://doi.org/10.1038/nature14474. arXiv:1411.4413 [hep-ex] - 4.R. Aaij
*et al.*[LHCb Collaboration], Phys. Rev. Lett.**115**(11), 111803 (2015). https://doi.org/10.1103/PhysRevLett.115.159901. https://doi.org/10.1103/PhysRevLett.115.111803. arXiv:1506.08614 [hep-ex] [Erratum: Phys. Rev. Lett.**115**(15), 159901 (2015)] - 5.R. Aaij et al. [LHCb Collaboration], JHEP
**1509**179 (2015). https://doi.org/10.1007/JHEP09(2015)179. arXiv:1506.08777 [hep-ex] - 6.R. Aaij et al. [LHCb Collaboration], JHEP
**1602**104 (2016). https://doi.org/10.1007/JHEP02(2016)104. arXiv:1512.04442 [hep-ex] - 7.R. Aaij et al. [LHCb Collaboration], JHEP
**1611**047 (2016). https://doi.org/10.1007/JHEP11(2016)047. https://doi.org/10.1007/JHEP04(2017)142. arXiv:1606.04731 [hep-ex] [Erratum: JHEP**1704**(2017) 142] - 8.R. Aaij et al. [LHCb Collaboration], Eur. Phys. J. C
**77**(3), 161 (2017). https://doi.org/10.1140/epjc/s10052-017-4703-2. arXiv:1612.06764 [hep-ex] - 9.R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett.
**118**(25), 251802 (2017). https://doi.org/10.1103/PhysRevLett.118.251802. arXiv:1703.02508 [hep-ex] - 10.R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett.
**118**(19), 191801 (2017). https://doi.org/10.1103/PhysRevLett.118.191801. arXiv:1703.05747 [hep-ex] - 11.R. Aaij et al. [LHCb Collaboration], JHEP
**1708**, 055 (2017). https://doi.org/10.1007/JHEP08(2017)055. arXiv:1705.05802 [hep-ex] - 12.R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett.
**120**(17), 171802 (2018). https://doi.org/10.1103/PhysRevLett.120.171802. arXiv:1708.08856 [hep-ex] - 13.Talk by Simone Bifani for the LHCb collaboration, Search for new physics with \(b \rightarrow s l^+ l^-\) decays at LHCb, CERN, 18/4/2017. https://indico.cern.ch/event/580620/
- 14.R. Aaij et al. [LHCb Collaboration], JHEP
**1807**, 020 (2018). https://doi.org/10.1007/JHEP07(2018)020. arXiv:1804.07167 [hep-ex] - 15.G. Hiller, F. Krüger, Phys. Rev. D
**69**, 074020 (2004). https://doi.org/10.1103/PhysRevD.69.074020. arXiv:hep-ph/0310219 ADSCrossRefGoogle Scholar - 16.M. Bordone, G. Isidori, A. Pattori, Eur. Phys. J. C
**76**(8), 440 (2016). https://doi.org/10.1140/epjc/s10052-016-4274-7. arXiv:1605.07633 [hep-ph]ADSCrossRefGoogle Scholar - 17.W. Altmannshofer, C. Niehoff, P. Stangl, D.M. Straub, Eur. Phys. J. C
**77**(6), 377 (2017). https://doi.org/10.1140/epjc/s10052-017-4952-0. arXiv:1703.09189 [hep-ph]ADSCrossRefGoogle Scholar - 18.S. Wehle et al. [Belle Collaboration], Phys. Rev. Lett.
**118**(11), 111801 (2017). https://doi.org/10.1103/PhysRevLett.118.111801. arXiv:1612.05014 [hep-ex] - 19.The ATLAS Collaboration, Angular analysis of \(B^0_d\rightarrow K^* \mu ^+\mu ^-\) decays in \(pp\) collisions at \(\sqrt{s}=8\) TeV with the ATLAS detector, talk at the 52nd Rencontres de Moriond on Electroweak Interactions and Unified Theories, La Thuile, Italy, 18–25 Mar 2017, ATLAS-CONF-2017-023. https://cds.cern.ch/record/2258146
- 20.I. Carli [ATLAS Collaboration], PoS FPCP
**2017**, 043 (2017). https://doi.org/10.22323/1.304.0043 - 21.M. Aaboud et al. [ATLAS Collaboration], JHEP
**1810**, 047 (2018) https://doi.org/10.1007/JHEP10(2018)047. arXiv:1805.04000 [hep-ex] - 22.A.M. Sirunyan et al. [CMS Collaboration], Phys. Lett. B
**781**, 517 (2018) https://doi.org/10.1016/j.physletb.2018.04.030. arXiv:1710.02846 [hep-ex]ADSCrossRefGoogle Scholar - 23.W. Altmannshofer, D.M. Straub, Eur. Phys. J. C
**73**, 2646 (2013). https://doi.org/10.1140/epjc/s10052-013-2646-9. arXiv:1308.1501 [hep-ph]ADSCrossRefGoogle Scholar - 24.S. Descotes-Genon, L. Hofer, J. Matias, J. Virto, JHEP
**1412**, 125 (2014). https://doi.org/10.1007/JHEP12(2014)125. arXiv:1407.8526 [hep-ph]ADSCrossRefGoogle Scholar - 25.G. Hiller, M. Schmaltz, Phys. Rev. D
**90**, 054014 (2014). https://doi.org/10.1103/PhysRevD.90.054014. arXiv:1408.1627 [hep-ph]ADSCrossRefGoogle Scholar - 26.G. Hiller, M. Schmaltz, JHEP
**1502**, 055 (2015). https://doi.org/10.1007/JHEP02(2015)055. arXiv:1411.4773 [hep-ph]ADSCrossRefGoogle Scholar - 27.A. Crivellin, G. D’Ambrosio, J. Heeck, Phys. Rev. D
**91**(7), 075006 (2015). https://doi.org/10.1103/PhysRevD.91.075006. arXiv:1503.03477 [hep-ph]ADSCrossRefGoogle Scholar - 28.A. Crivellin, L. Hofer, J. Matias, U. Nierste, S. Pokorski, J. Rosiek, Phys. Rev. D
**92**(5), 054013 (2015). https://doi.org/10.1103/PhysRevD.92.054013. arXiv:1504.07928 [hep-ph]ADSCrossRefGoogle Scholar - 29.T. Hurth, F. Mahmoudi, S. Neshatpour, Nucl. Phys. B
**909**, 737 (2016). https://doi.org/10.1016/j.nuclphysb.2016.05.022. arXiv:1603.00865 [hep-ph]ADSCrossRefGoogle Scholar - 30.B. Capdevila, S. Descotes-Genon, L. Hofer, J. Matias, JHEP
**1704**, 016 (2017). https://doi.org/10.1007/JHEP04(2017)016. arXiv:1701.08672 [hep-ph]ADSCrossRefGoogle Scholar - 31.V.G. Chobanova, T. Hurth, F. Mahmoudi, D. Martinez Santos, S. Neshatpour, JHEP
**1707**, 025 (2017). https://doi.org/10.1007/JHEP07(2017)025. arXiv:1702.02234 [hep-ph]ADSCrossRefGoogle Scholar - 32.W. Altmannshofer, C. Niehoff, D.M. Straub, JHEP
**1705**, 076 (2017). https://doi.org/10.1007/JHEP05(2017)076. arXiv:1702.05498 [hep-ph]ADSCrossRefGoogle Scholar - 33.A. Crivellin, D. Müller, T. Ota, JHEP
**1709**, 040 (2017). https://doi.org/10.1007/JHEP09(2017)040. arXiv:1703.09226 [hep-ph]ADSCrossRefGoogle Scholar - 34.A.K. Alok, B. Bhattacharya, D. Kumar, J. Kumar, D. London, S.U. Sankar, Phys. Rev. D
**96**(1), 015034 (2017). https://doi.org/10.1103/PhysRevD.96.015034. arXiv:1703.09247 [hep-ph]ADSCrossRefGoogle Scholar - 35.B. Capdevila, A. Crivellin, S. Descotes-Genon, J. Matias, J. Virto, JHEP
**1801**, 093 (2018). https://doi.org/10.1007/JHEP01(2018)093. arXiv:1704.05340 [hep-ph]ADSCrossRefGoogle Scholar - 36.W. Altmannshofer, P. Stangl, D.M. Straub, Phys. Rev. D
**96**(5), 055008 (2017). https://doi.org/10.1103/PhysRevD.96.055008. arXiv:1704.05435 [hep-ph]ADSCrossRefGoogle Scholar - 37.G. D’Amico, M. Nardecchia, P. Panci, F. Sannino, A. Strumia, R. Torre, A. Urbano, JHEP
**1709**, 010 (2017). https://doi.org/10.1007/JHEP09(2017)010. arXiv:1704.05438 [hep-ph]ADSCrossRefGoogle Scholar - 38.G. Hiller, I. Nisandzic, Phys. Rev. D
**96**(3), 035003 (2017). https://doi.org/10.1103/PhysRevD.96.035003. arXiv:1704.05444 [hep-ph]ADSCrossRefGoogle Scholar - 39.L.S. Geng, B. Grinstein, S. Jäger, J. Martin Camalich, X.L. Ren, R.X. Shi, Phys. Rev. D
**96**(9), 093006 (2017). https://doi.org/10.1103/PhysRevD.96.093006. arXiv:1704.05446 [hep-ph]ADSCrossRefGoogle Scholar - 40.M. Ciuchini, A.M. Coutinho, M. Fedele, E. Franco, A. Paul, L. Silvestrini, M. Valli, Eur. Phys. J. C
**77**(10), 688 (2017). https://doi.org/10.1140/epjc/s10052-017-5270-2. arXiv:1704.05447 [hep-ph]CrossRefGoogle Scholar - 41.A.K. Alok, D. Kumar, J. Kumar, R. Sharma, Lepton flavor non-universality in the B-sector: a global analyses of various new physics models. arXiv:1704.07347 [hep-ph]
- 42.A.K. Alok, B. Bhattacharya, A. Datta, D. Kumar, J. Kumar, D. London, Phys. Rev. D
**96**(9), 095009 (2017). https://doi.org/10.1103/PhysRevD.96.095009. arXiv:1704.07397 [hep-ph]CrossRefGoogle Scholar - 43.A. Bazavov et al. [Fermilab Lattice and MILC Collaborations], Phys. Rev. D
**93**(11), 113016 (2016). https://doi.org/10.1103/PhysRevD.93.113016. arXiv:1602.03560 [hep-lat] - 44.T. Jubb, M. Kirk, A. Lenz, G. Tetlalmatzi-Xolocotzi, Nucl. Phys. B
**915**, 431 (2017). https://doi.org/10.1016/j.nuclphysb.2016.12.020. arXiv:1603.07770 [hep-ph]ADSCrossRefGoogle Scholar - 45.A.J. Buras, F. De Fazio, JHEP
**1608**, 115 (2016). https://doi.org/10.1007/JHEP08(2016)115. arXiv:1604.02344 [hep-ph]ADSCrossRefGoogle Scholar - 46.M. Kirk, A. Lenz, T. Rauh, JHEP
**1712**, 068 (2017). https://doi.org/10.1007/JHEP12(2017)068. arXiv:1711.02100 [hep-ph]ADSCrossRefGoogle Scholar - 47.L. Di Luzio, M. Kirk, A. Lenz, Phys. Rev. D
**97**, 095035 (2018). https://doi.org/10.1103/PhysRevD.97.095035. arXiv:1712.06572 [hep-ph]ADSCrossRefGoogle Scholar - 48.Y. Amhis et al. [HFLAV Collaboration], Eur. Phys. J. C
**77**(12), 895 (2017). https://doi.org/10.1140/epjc/s10052-017-5058-4. arXiv:1612.07233 [hep-ex]. Updated result for \(\phi _s^{c\bar{c}s}\) as of 2018: http://www.slac.stanford.edu/xorg/hflav/osc/PDG_2018/#BETAS - 49.M. Artuso, G. Borissov, A. Lenz, Rev. Mod. Phys.
**88**(4), 045002 (2016). https://doi.org/10.1103/RevModPhys.88.045002. arXiv:1511.09466 [hep-ph]ADSCrossRefGoogle Scholar - 50.A. Lenz, U. Nierste, JHEP
**0706**, 072 (2007). https://doi.org/10.1088/1126-6708/2007/06/072. arXiv:hep-ph/0612167 ADSCrossRefGoogle Scholar - 51.J. Charles et al. [CKMfitter Group], Eur. Phys. J. C
**41**(1), 1 (2005). https://doi.org/10.1140/epjc/s2005-02169-1. arXiv:hep-ph/0406184. Updated result as of summer 2016: http://ckmfitter.in2p3.fr/www/results/plots_ichep16/num/ckmEval_results_ichep16.html ADSCrossRefGoogle Scholar - 52.G. Buchalla, A.J. Buras, M.E. Lautenbacher, Rev. Mod. Phys.
**68**, 1125 (1996). https://doi.org/10.1103/RevModPhys.68.1125. arXiv:hep-ph/9512380 ADSCrossRefGoogle Scholar - 53.C. Bobeth, A.J. Buras, JHEP
**1802**, 101 (2018). https://doi.org/10.1007/JHEP02(2018)101. arXiv:1712.01295 [hep-ph]ADSCrossRefGoogle Scholar - 54.C. Patrignani et al. [Particle Data Group], Chin. Phys. C
**40**(10), 100001 (2016). https://doi.org/10.1088/1674-1137/40/10/100001 CrossRefGoogle Scholar - 55.D.M. Straub, flavio: a Python package for flavour and precision phenomenology in the Standard Model and beyond. arXiv:1810.08132 [hep-ph]. https://flav-io.github.io/. https://doi.org/10.5281/zenodo.59955
- 56.L. Di Luzio, M. Kirk, A. Lenz, \(B_s-\bar{B}_s\) mixing interplay with \(B\) anomalies, talk given at the 10th International Workshop on the CKM Unitarity Triangle (CKM 2018), Heidelberg University, 17–21 September 2018. arXiv:1811.12884 [hep-ph]

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}