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

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].

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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 \({{{\mathcal {C}}}}_{10^{(\prime )}}\).

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.

The new scenarios 9–13 are characterized by a \({{{\mathcal {C}}}}_{10(')}^{\mathrm{U}}\) contribution. This arises naturally in models with modified 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].

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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].

The comparison of scenarios 10 and 12 illustrates that \({{{\mathcal {C}}}}_{9\mu }^{\mathrm{V}}\) plays an important role in LFU NP scenarios and cannot be swapped for its chirally-flipped counterpart without consequences. Finally, the allowed regions for the new LFU scenarios are displayed in Fig. 4.

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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 \({{{\mathcal {C}}}}_9^{\mathrm{U}}\).

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.