# Emerging patterns of New Physics with and without Lepton Flavour Universal contributions

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## Abstract

We perform a model-independent global fit to \(b\rightarrow s\ell ^+\ell ^-\) observables to confirm existing New Physics (NP) patterns (or scenarios) and to identify new ones emerging from the inclusion of the updated LHCb and Belle measurements of \(R_K\) and \(R_{K^*}\), respectively. Our analysis, updating Refs. Capdevila et al. (J Virto JHEP 1801:093, 2018) and Algueró et al. (J Matias Phys Rev D 99(7):075017, 2019) and including these new data, suggests the presence of right-handed couplings encoded in the Wilson coefficients \({{{\mathcal {C}}}}_{9'\mu }\) and \({{{\mathcal {C}}}}_{10'\mu }\). It also strengthens our earlier observation that a lepton flavour universality violating (LFUV) left-handed lepton coupling (\({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}=-\,{{{\mathcal {C}}}}_{10\mu }^{\mathrm{V}}\)), often preferred from the model building point of view, accommodates the data better if lepton-flavour universal (LFU) NP is allowed, in particular in \({{{\mathcal {C}}}}_{9}^{\mathrm{U}}\). Furthermore, this scenario with LFU NP provides a simple and model-independent connection to the \(b\rightarrow c\tau \nu \) anomalies, showing a preference of \(\approx 7\,\sigma \) with respect to the SM. It may also explain why fits to the whole set of \(b\rightarrow s\ell ^+\ell ^-\) data or to the subset of LFUV data exhibit stronger preferences for different NP scenarios. Finally, motivated by \(Z^\prime \) models with vector-like quarks, we propose four new scenarios with LFU and LFUV NP contributions that give a very good fit to data.

## 1 Introduction

Most prominent 1D patterns of NP in \(b\rightarrow s\mu ^+\mu ^-\). \(\hbox {Pull}_{\mathrm{SM}}\) is quoted in units of standard deviation

1D Hyp. | All | LFUV | ||||||
---|---|---|---|---|---|---|---|---|

Best fit | 1\(\sigma \)/2\(\sigma \) | \(\hbox {Pull}_{\mathrm{SM}}\) | p-value | Best fit | 1 \(\sigma \)/ 2 \(\sigma \) | \(\hbox {Pull}_{\mathrm{SM}}\) | p-value | |

\({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}\) | \({-\,0.98}\) | \([-\,1.15,-\,0.81]\) | 5.6 | 65.4% | \({-\,0.89}\) | \([-\,1.23,-\,0.59]\) | 3.3 | 52.2% |

\([-\,1.31,-\,0.64]\) | \([-\,1.60,-\,0.32]\) | |||||||

\({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}=-\,{{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}}\) | \({-\,0.46}\) | \([-\,0.56,-\,0.37]\) | 5.2 | 55.6% | \({-\,0.40}\) | \([-\,0.53,-\,0.29]\) | 4.0 | 74.0% |

\([-\,0.66,-\,0.28]\) | \([-\,0.63,-\,0.18]\) | |||||||

\({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}=-\,{{{\mathcal {C}}}}_{9'\mu }\) | \({-\,0.99}\) | \([-\,1.15,-\,0.82]\) | 5.5 | 62.9% | \({-\,1.61}\) | \([-\,2.13,-\,0.96]\) | 3.0 | 42.5% |

\([-\,1.31,-\,0.64]\) | \([-\,2.54,-\,0.41]\) | |||||||

\({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}=-\,3 {{{\mathcal {C}}}}_{9e}^{\mathrm{NP}}\) | \({-\,0.87}\) | \([-\,1.03,-\,0.71]\) | 5.5 | 61.9% | \({-\,0.66}\) | \([-\,0.90,-\,0.44]\) | 3.3 | 52.2% |

\([-\,1.19,-\,0.55]\) | \([-\,1.17,-\,0.24]\) |

Most prominent 2D patterns of NP in \(b\rightarrow s\mu ^+\mu ^-\). The last five rows correspond to Hypothesis 1: \(({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}=-\,{{{\mathcal {C}}}}_{9^\prime \mu } , {{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}}={{{\mathcal {C}}}}_{10^\prime \mu })\), 2: \(({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}=-\,{{{\mathcal {C}}}}_{9^\prime \mu } , {{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}}=-\,{{{\mathcal {C}}}}_{10^\prime \mu })\), 3: \(({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}=-\,{{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}} , {{{\mathcal {C}}}}_{9^\prime \mu }={{{\mathcal {C}}}}_{10^\prime \mu }\)), 4: \(({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}=-\,{{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}} , {{{\mathcal {C}}}}_{9^\prime \mu }=-\,{{{\mathcal {C}}}}_{10^\prime \mu })\) and 5: \(({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}} , {{{\mathcal {C}}}}_{9^\prime \mu }=-\,{{{\mathcal {C}}}}_{10^\prime \mu })\)

2D Hyp. | All | LFUV | ||||
---|---|---|---|---|---|---|

Best fit | \(\hbox {Pull}_{\mathrm{SM}}\) | p-value | Best fit | \(\hbox {Pull}_{\mathrm{SM}}\) | p-value | |

\(({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}},{{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}})\) | (\(-\,\)0.91, 0.18) | 5.4 | 68.7% | (\(-\,\)0.16, 0.56) | 3.4 | 76.9% |

\(({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}},{{{\mathcal {C}}}}_{7^{\prime }})\) | (\(-\,\)1.00, 0.02) | 5.4 | 67.9% | (\(-\,\)0.90, \(-\,\)0.04) | 2.9 | 55.1% |

\(({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}},{{{\mathcal {C}}}}_{9^\prime \mu })\) | (\(-\,\)1.10, 0.55) | 5.7 | 75.1% | (\(-\,\)1.79, 1.14) | 3.4 | 76.1% |

\(({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}},{{{\mathcal {C}}}}_{10^\prime \mu })\) | (\(-\,\)1.14, \(-\,\)0.35) | 5.9 | 78.6% | (\(-\,\)1.88, \(-\,\)0.62) | 3.8 | 91.3% |

\(({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}, {{{\mathcal {C}}}}_{9e}^{\mathrm{NP}})\) | (\(-\,\)1.05, \(-\,\)0.23) | 5.3 | 66.2% | (\(-\,\)0.73, 0.16) | 2.8 | 52.3% |

Hyp. 1 | (\(-\,\)1.06, 0.26) | 5.7 | 75.7% | (\(-\,\)1.62, 0.29) | 3.4 | 77.6% |

Hyp. 2 | (\(-\,\)0.97, 0.09) | 5.3 | 65.2% | (\(-\,\)1.95, 0.25) | 3.2 | 66.6% |

Hyp. 3 | (\(-\,\)0.47, 0.06) | 4.8 | 55.7% | (\(-\,\)0.39, \(-\,\)0.13) | 3.4 | 76.2% |

Hyp. 4 | (\(-\,\)0.49, 0.12) | 5.0 | 59.3% | (\(-\,\)0.48, 0.17) | 3.6 | 84.3% |

Hyp. 5 | (\(-\,\)1.14, 0.24) | 5.9 | 78.7% | (\(-\,\)2.07, 0.52) | 3.9 | 92.5% |

We have also updated our average for \({{{\mathcal {B}}}}(B_s \rightarrow \mu ^+\mu ^-)\) including the latest measurement from the ATLAS collaboration [10] and taking into account the most recent lattice update of \(f_{B_s}\) for \(N_f=2+1+1\) simulations collected in Ref. [11].

1 and 2 \(\sigma \) confidence intervals for the NP contributions to Wilson coefficients in the 6D hypothesis allowing for NP in \(b\rightarrow s\mu ^+\mu ^-\) operators dominant in the SM and their chirally-flipped counterparts, for the fit “All”

\({{{\mathcal {C}}}}_{7}^{\mathrm{NP}}\) | \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}\) | \({{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}}\) | \({{{\mathcal {C}}}}_{7^\prime }\) | \({{{\mathcal {C}}}}_{9^\prime \mu }\) | \({{{\mathcal {C}}}}_{10^\prime \mu }\) | |
---|---|---|---|---|---|---|

Best fit | + 0.01 | \(-\,\)1.10 | + 0.15 | + 0.02 | + 0.36 | \(-\,\)0.16 |

1 \(\sigma \) | \([-\,0.01,+ 0.05]\) | \([-\,1.28,-\,0.90]\) | \([-\,0.00,+ 0.36]\) | \([-\,0.00,+ 0.05]\) | \([-\,0.14,+ 0.87]\) | \([-\,0.39,+ 0.13]\) |

2 \(\sigma \) | \([-\,0.03,+ 0.06]\) | \([-\,1.44,-\,0.68]\) | \([-\,0.12,+ 0.56]\) | \([-\,0.02,+ 0.06]\) | \([-\,0.49,+ 1.23]\) | \([-\,0.58,+ 0.33]\) |

In addition to updating the experimental inputs, our analysis explores new emerging directions in the parameter space spanned by the effective operators driven by data within two different frameworks. First, following Ref. [1] we assume in Sect. 2 that NP affects only muons and is thus purely Lepton-Flavour Universality Violating (LFUV). In Sect. 3 we follow the complementary approach discussed in Ref. [2], where we consider the consequences of removing the frequently made hypothesis that NP is purely LFUV. We then explore the implications of allowing both LFU and LFUV NP contributions to the Wilson coefficients \({{{\mathcal {C}}}}_{9^{(\prime )}}\) and \({{{\mathcal {C}}}}_{10^{(\prime )}}\).

Coefficients for the polynomial parameterisation of the numerator and denominator of \(R_K^{[1.1,6]}\) in the vicinity of the SM point

\(\alpha _{0\mu }\) | \(\alpha _{1\mu }\) | \(\alpha _{2\mu }\) | \(\alpha _{3\mu }\) | \(\alpha _{4\mu }\) | \(\alpha _{5\mu }\) | \(\alpha _{6\mu }\) | \(\alpha _{7\mu }\) | \(\alpha _{8\mu }\) | \(\alpha _{9\mu }\) | \(\alpha _{10\mu }\) |
---|---|---|---|---|---|---|---|---|---|---|

4.00 | 0.92 | 0.12 | 0.92 | 0.12 | 0.24 | \(-\,\)1.06 | 0.12 | \(-\,\)1.06 | 0.12 | 0.25 |

\(\alpha _{0e}\) | \(\alpha _{1e}\) | \(\alpha _{2e}\) | \(\alpha _{3e}\) | \(\alpha _{4e}\) | \(\alpha _{5e}\) | \(\alpha _{6e}\) | \(\alpha _{7e}\) | \(\alpha _{8e}\) | \(\alpha _{9e}\) | \(\alpha _{10e}\) |
---|---|---|---|---|---|---|---|---|---|---|

3.99 | 0.92 | 0.12 | 0.92 | 0.12 | 0.24 | \(-\,\)1.05 | 0.12 | \(-\,\)1.05 | 0.12 | 0.24 |

## 2 Global fits in presence of LFUV NP

- 1.
The scenario \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}=-\,{{{\mathcal {C}}}}_{9'\mu }\), which favours a SM-like value of \(R_K^{[1.1,6]}\) [2, 15], has an increased significance in the “All” fit compared to our earlier analysis.

- 2.
The scenario \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}\) has the largest

*p*-value in the “All” fit while \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}=-\,{{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}}\) has the largest*p*-value in the LFUV fit, a difference which can be solved through the introduction of LFU NP (see Ref. [2] and next section). - 3.
The best-fit point for the scenario \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}\) coincides now in the “All” and LFUV fits.

- 4.
The scenario with only \({{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}}\) has a significance in the “All” fit of only 4.0\(\sigma \) level and 3.9\(\sigma \) for the LFUV fit, which explains its absence from Table 1 as happens in Ref. [1].

^{1}Indeed, these RHC contributions tend to increase the value of \(R_K^{[1.1,6]}\) while \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}<0\) tend to decrease it as can be seen from the explicit expression of \(R_K^{[1.1,6]}=A_{\mu }/A_e\) where the numerator and the denominator can be given by an approximate polynomial parameterisation near the SM point

*SU*(2) singlet vector-like quark can still be decoupled [18].

Most prominent patterns for LFU and LFUV NP contributions from Fit “All”

Scenario | Best-fit point | 1 \(\sigma \) | 2 \(\sigma \) | \(\hbox {Pull}_{\mathrm{SM}}\) | p-value | |
---|---|---|---|---|---|---|

Scenario 5 | \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}\) | \(-\,0.36\) | \([-\,0.86,+0.10]\) | \([-\,1.41,+0.52]\) | 5.2 | 71.2% |

\({{{\mathcal {C}}}}_{10\mu }^{\mathrm{V}}\) | \(+\,0.67\) | \([+0.24,+1.03]\) | \([-\,1.73,+1.36]\) | |||

\({{{\mathcal {C}}}}_{9}^{\mathrm{U}}={{{\mathcal {C}}}}_{10}^{\mathrm{U}}\) | \(-\,0.59\) | \([-\,0.90,-\,0.12]\) | \([-\,1.13,+0.68]\) | |||

Scenario 6 | \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}=-\,{{{\mathcal {C}}}}_{10\mu }^{\mathrm{V}}\) | \(-\,0.50\) | \([-\,0.61,-\,0.38]\) | \([-\,0.72,-\,0.28]\) | 5.5 | 71.0% |

\({{{\mathcal {C}}}}_{9}^{\mathrm{U}}={{{\mathcal {C}}}}_{10}^{\mathrm{U}}\) | \(-\,0.38\) | \([-\,0.52,-\,0.22]\) | \([-\,0.64,-\,0.06]\) | |||

Scenario 7 | \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}\) | \(-\,0.78\) | \([-\,1.11,-\,0.47]\) | \([-\,1.45,-\,0.18]\) | 5.3 | 66.2% |

\({{{\mathcal {C}}}}_{9}^{\mathrm{U}}\) | \(-\,0.20\) | \([-\,0.57,+0.18]\) | \([-\,0.92,+0.55]\) | |||

Scenario 8 | \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}=-\,{{{\mathcal {C}}}}_{10\mu }^{\mathrm{V}}\) | \(-\,0.30\) | \([-\,0.42,-\,0.20]\) | \([-\,0.53,-\,0.10]\) | 5.7 | 75.2% |

\({{{\mathcal {C}}}}_{9}^{\mathrm{U}}\) | \(-\,0.74\) | \([-\,0.96,-\,0.51]\) | \([-\,1.15,-\,0.25]\) | |||

Scenario 9 | \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}=-\,{{{\mathcal {C}}}}_{10\mu }^{\mathrm{V}}\) | \(-\,0.57\) | \([-\,0.73,-\,0.41]\) | \([-\,0.87,-\,0.28]\) | 5.0 | 60.2 % |

\({{{\mathcal {C}}}}_{10}^{\mathrm{U}}\) | \(-\,0.34\) | \([-\,0.60,-\,0.07]\) | \([-\,0.84,+0.18]\) | |||

Scenario 10 | \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}\) | \(-\,0.95\) | \([-\,1.13,-\,0.76]\) | \([-\,1.30,-\,0.57]\) | 5.5 | 69.5 % |

\({{{\mathcal {C}}}}_{10}^{\mathrm{U}}\) | \(+\,0.27\) | [0.08, 0.47] | \([-\,0.09,0.66]\) | |||

Scenario 11 | \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}\) | \(-\,1.03\) | \([-\,1.22,-\,0.84]\) | \([-\,1.38,-\,0.65]\) | 5.6 | 73.6 % |

\({{{\mathcal {C}}}}_{10'}^{\mathrm{U}}\) | \(-\,0.29\) | \([-\,0.47,-\,0.12]\) | \([-\,0.63,0.05]\) | |||

Scenario 12 | \({{{\mathcal {C}}}}_{9'\mu }^{\mathrm{V}}\) | \(-\,0.03\) | \([-\,0.22,0.15]\) | \([-\,0.40,0.32]\) | 1.6 | 15.7% |

\({{{\mathcal {C}}}}_{10}^{\mathrm{U}}\) | \(+0.41\) | [0.21, 0.63] | [0.02, 0.83] | |||

Scenario 13 | \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}\) | \(-\,1.11\) | \([-\,1.28,-\,0.91]\) | \([-\,1.41,-\,0.71]\) | 5.4 | 78.7% |

\({{{\mathcal {C}}}}_{9'\mu }^{\mathrm{V}}\) | \(+0.53\) | [0.24, 0.83] | \([-\,0.10,1.11]\) | |||

\({{{\mathcal {C}}}}_{10}^{\mathrm{U}}\) | \(+0.24\) | [0.01, 0.48] | \([-\,0.21,0.69]\) | |||

\({{{\mathcal {C}}}}_{10'}^{\mathrm{U}}\) | \(-\,0.04\) | \([-\,0.28,0.20]\) | \([-\,0.48,0.42]\) |

## 3 Global fits in presence of LFUV and LFU NP

We update some of the scenarios considered in Ref. [2] in Table 5. Concerning new directions in parameter space we allow for RHC, motivated by the results of the previous section, and focus on scenarios that could be fairly easily obtained in simple NP models.

With the updated experimental inputs, we confirm our earlier result [2] that a LFUV left-handed lepton coupling structure (corresponding to \({{{\mathcal {C}}}}_{9}^{\mathrm{V}}=-\,{{{\mathcal {C}}}}_{10}^{\mathrm{V}}\) and preferred from a model-building point of view) yields a better description of data with the addition of LFU-NP in the coefficients \({{{\mathcal {C}}}}_{9,10}\), as shown by the scenarios 6, 8 in Table 5 with *p*-values larger than \(70\%\).

We observe a very slight decrease in significance for the scenarios 5–7, with the exception of scenario 8 which exhibits one of the most significant pulls with respect to the SM.

Scenario 8 of Ref. [2] can actually be realized via off-shell photon penguins [19] in a leptoquark model explaining also \(b\rightarrow c\tau \nu \) data (we will return to this point in the following section).

Updated plots of the 2D LFU-LFUV scenarios discussed in Ref. [2] are shown in Fig. 3.

*Z*couplings (to a good approximation \({{{\mathcal {C}}}}_{9(')}^{\mathrm{U}}\) can be neglected). The pattern of scenario 9 occurs in Two-Higgs-Doublet models where this flavour universal effect can be supplemented by a \({{{\mathcal {C}}}}_{9}^{\mathrm{V}}=-\,{{{\mathcal {C}}}}_{10}^{\mathrm{V}}\) effect [20].

In case of scenarios 11–13, one can invoke models with vector-like quarks where modified *Z* couplings are even induced at tree level. The LFU effect in \({{{\mathcal {C}}}}_{10(')}^{\mathrm{U}}\) can be accompanied by a \({{{\mathcal {C}}}}_{9,10(')}^{\mathrm{V}}\) effect from \(Z^\prime \) exchange [21]. Vector-like quarks with the quantum numbers of right-handed down quarks (left-handed quarks doublets) generate effect in \({{{\mathcal {C}}}}_{10}^{\mathrm{U}}\) and \({{{\mathcal {C}}}}_{9'}^{\mathrm{V}}\) (\({{{\mathcal {C}}}}_{10(')}^{\mathrm{U}}\) and \({{{\mathcal {C}}}}_{9}^{\mathrm{V}}\)) for a \(Z^\prime \) boson with vector couplings to muons [21].

## 4 Model-independent connection to \(b\rightarrow c\ell \nu \)

In complement with the above EFT analysis, we focus now on the NP interpretation of scenario 8. Indeed, this scenario allows for a model-independent connection between the anomalies in \(b\rightarrow s\ell ^+\ell ^-\) and those in \(b\rightarrow c\tau \nu \), which are now at the \(3.1\sigma \) level [22].

Such a correlation arises in the SMEFT scenario where \({{{\mathcal {C}}}}_{}^{(1)}={{{\mathcal {C}}}}_{}^{(3)}\) expressed in terms of gauge-invariant dimension-6 operators [23, 24]. This scenario stems naturally from models with an *SU*(2) singlet vector leptoquark [25, 26, 27]. The operator involving-third generation leptons explains \(R_{D^{(*)}}\) and the one involving the second generation gives a LFUV effect in \(b\rightarrow s\mu ^+\mu ^-\) processes. The constraint from \(b\rightarrow c\tau \nu \) and \(SU(2)_L\) invariance leads generally to large contributions to the operator \({\bar{s}} \gamma ^\mu P_Lb {{\bar{\tau }}} \gamma _\mu P_L \tau \), which enhances \(b\rightarrow s\tau ^+\tau ^-\) processes [24], but also mixes into \({{{\mathcal {O}}}}_9\) and generates \({{{\mathcal {C}}}}_{9}^{\mathrm{U}}\) at \(\mu =m_b\) [19]. Note that not all models addressing the charged and neutral current anomalies simultaneously have an anarchic flavour structure. In fact, in the case of alignment in the down-sector [29, 30] one does not find large effects in \(b\rightarrow s\tau ^+\tau ^-\) or \({{{\mathcal {C}}}}_9^{\mathrm{U}}\).

## 5 Conclusions

In summary, including recent updates (\(R_K\), \(R_{K^*}\) and \({{{\mathcal {B}}}}(B_s \rightarrow \mu ^+\mu ^-)\)) our global model-independent analysis yields a very similar picture to the one previously found in Refs. [1, 2] for the various NP scenarios of interest with some important peculiarities. In presence of LFUV NP contributions only, the 1D fits to “All” observables remain basically unchanged showing the preference for \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}\) scenario over \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}=-\,{{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}}\). If only LFUV observables are considered the situation is reversed, as already found in Ref. [1], but now with an increased gap between the significances. This difference between the preferred hypotheses, depending on the data set used, can be solved introducing LFU NP contributions [2].

The main differences arise for the 2D scenarios: the cases including RHC, (\({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}, {{{\mathcal {C}}}}_{10^\prime \mu }\)), (\({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}, {{{\mathcal {C}}}}_{9^\prime \mu }\)) or (\({{{\mathcal {C}}}}_{9\mu }^{\mathrm{NP}}, {{{\mathcal {C}}}}_{9^\prime \mu }=-\,{{{\mathcal {C}}}}_{10^\prime \mu }\)), can accommodate better the recent updates, which enhances the significance of these scenarios compared to Ref. [1], pointing to new patterns including RHC. A more precise experimental measurement of the observable \(P_1\) [34, 35] would be very useful to confirm or not the presence of RHC NP encoded in \({{{\mathcal {C}}}}_{9^\prime \mu }\) and \({{{\mathcal {C}}}}_{10^\prime \mu }\).

We also observe interesting changes in the 2D fits in the presence of LFU NP, where new scenarios (not considered in Ref. [2]) give a good fit to data with \({{{\mathcal {C}}}}_{10^{(\prime )}}^{\mathrm{U}}\) and additional LFUV contributions. For example scenario 11 (\({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}, {{{\mathcal {C}}}}_{10^\prime \mu }\)) can accommodate \(b\rightarrow s\ell ^+\ell ^-\) data very well, at the same level as scenario 8. Scenarios including LFU NP in left-handed currents (discussed in Ref. [2]) stay practically unchanged but with some preference for scenarios 6 and 8, which have a \((V-A)\) structure for the LFUV-NP and a *V* or \((V+A)\) structure for the LFU-NP. Furthermore, we have included additional scenarios 9 and 10 that exhibit a significance of 5.0\(\sigma \) and 5.5\(\sigma \) respectively.

We note that the amount of LFU NP is sensitive to the structure of the LFUV component. For instance, in scenario 7 (\({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}\) and \({{{\mathcal {C}}}}_{9}^{\mathrm{U}}\)) the LFU component is negligible at its best fit point. On the contrary, if the LFUV-NP has a \((V-A)\) structure, the LFU-NP component (\({{{\mathcal {C}}}}_{9}^{\mathrm{U}}\)) is large, as illustrated by scenarios 6, 8 and 9. Scenarios with NP in RHC (either LFU or LFUV) prefer such contributions at the \(2\sigma \) level (see scenarios 11 and 13) with the exception of scenario 12 with negligible \({{{\mathcal {C}}}}_{9'\mu }^{\mathrm{V}}\). The new values of \(R_K\) and \(R_{K^*}\) seem thus to open a window for RHC contributions while the new \({{{\mathcal {B}}}}(B_s \rightarrow \mu \mu )\) update (theory and experiment) helps only marginally scenarios with \({{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}}\).

Finally, we showed that scenario 8, which allows for a model-independent connection between the \(b\rightarrow c\tau \nu \) anomalies and the ones in \(b\rightarrow s\ell ^+\ell ^-\), can explain all data consistently and is preferred over the SM by \(7\,\sigma \).

Figure 5 illustrates the impact on the largest anomaly (\(P_5^\prime \)) of some of the most significant scenarios. Interestingly, several of the scenarios currently favoured cluster around the same values for the bins showing deviations with respect to the SM.

We have thus identified a number of NP scenarios with similarly good *p*-values and pulls with respect to the SM, which are able to reproduce the \(b\rightarrow s\ell ^+\ell ^-\) data very well. Hierarchies among these scenarios can be identified, but additional data and reduced uncertainties are required to come to a final conclusion. The full exploitation of LHC run-2 data by the LHCb experiment (as well as by ATLAS and CMS) and the forthcoming results from the Belle and Belle II collaborations are expected to improve the situation very significantly in the forthcoming years, helping us to pin down the actual NP pattern hinted at by the \(b\rightarrow s\ell ^+\ell ^-\) anomalies currently observed and to build accurate phenomenological models to be confirmed through other experimental probes such as direct production experiments.

*Note added* After the completion of this work, several global analyses have been performed to assess NP scenarios affecting \(b\rightarrow s\ell ^+\ell ^-\) processes [14, 28, 36, 37]. They agree well with our findings, with small differences stemming mainly from slightly different theoretical approaches as well as theoretical and experimental inputs. The improvement brought by RHC has been observed in Refs. [14, 36], whereas the interest of LFU NP contributions is also identified in Refs. [14, 28, 38]. Most of the analyses observe that the slight deviation from \({{{\mathcal {B}}}}(B_s\rightarrow \mu ^+\mu ^-)\) plays no specific role in the global fit [36, 37], apart from Ref. [28]. In the latter analysis, the significance of a scenario with only \({{{\mathcal {C}}}}_{10\mu }^{\mathrm{NP}}\) is much more important than in our case, and the hierarchies between the significances of 2D scenarios is different. After discussion with the authors of Ref. [28], this difference comes from their inclusion of \(B_s\)-\(\bar{B}_s\) mixing and the assumption that \(\Delta F=2\) observables are purely governed by the SM, which helps them sharpening the prediction for \({{{\mathcal {B}}}}(B_s\rightarrow \mu ^+\mu ^-)\) and increase the weight of this observable in the fit. Our present analysis does not rely on this strong hypothesis, which should be contrasted with the fact that most models invoked to explain \(b\rightarrow s\ell ^+\ell ^-\) anomalies typically affect also \(\Delta F=2\) observables.

## Footnotes

## Notes

### Acknowledgements

This work received financial support from European Regional Development Funds under the Spanish Ministry of Science, Innovation and Universities (Projects FPA2014-55613-P and FPA2017-86989-P) and from the Agency for Management of University and Research Grants of the Government of Catalonia (project SGR 1069) [MA, BC, PM, JM] and from European Commission (Grant Agreements 690575, 674896 and 69219) [SDG]. The work of PM is supported by the Beatriu de Pinos postdoctoral program co-funded by the Agency for Management of University and Research Grants of the Government of Catalonia and by the COFUND program of the Marie Sklodowska-Curie actions under the framework program Horizon 2020 of the European Commission. JM gratefully acknowledges the financial support by ICREA under the ICREA Academia programme. JV acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement no 700525, ‘NIOBE’ and support from SEJI/2018/033 (Generalitat Valenciana). The work of AC is supported by a Professorship Grant (PP00P2_176884) of the Swiss National Science Foundation.

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