# Recent \(\varvec{B}\) physics anomalies: a first hint for compositeness?

## Abstract

We scrutinize the recently further strengthened hints for new physics in semileptonic *B*-meson decays, focusing on the ‘clean’ ratios of branching fractions \(R_K\) and \(R_{K^*}\) and examining to which pattern of new effects they point to. We explore in particular the hardly considered, yet fully viable, option of new physics in the right-handed electron sector and demonstrate how a recently proposed framework of leptons in composite Higgs setups naturally solves both the \(R_K\) and \(R_{K^*}\) anomalies via a peculiar structure of new physics effects, predicted by minimality of the model and the scale of neutrino masses. Finally, we also take into account further observables, such as \(\mathcal{B}(B_s \rightarrow \mu ^+\mu ^-)\), \(\varDelta M_{B_s}\), and angular observables in \(B \rightarrow K^{*} \mu ^+ \mu ^-\) decays, to arrive at a comprehensive picture of the model concerning (semileptonic) *B* decays. We conclude that – since it is in good agreement with the experimental situation in flavor physics and also allows to avoid ultra-light top partners – the model furnishes a very promising scenarios of Higgs compositeness in the light of LHC data.

## 1 Introduction

Decays of *B* mesons offer a promising place to search for new physics (NP), since in the Standard Model (SM) of Particle Physics flavor changing neutral processes are strongly suppressed and thus effects of NP might be sizable in (flavor-changing) *B* decays. The case is strengthened by the fact that the bottom quark is the heaviest down-type quark and resides in the same weak doublet as the \(t_L\), which due to its large mass is thought to furnish a major link to the completion of the SM at smallest distances.

In fact, several anomalies have been found in \(b \rightarrow s \ell ^+ \ell ^-\) transitions, such as the long-standing anomaly in the angular analysis of the \(B \rightarrow K^* \mu ^+ \mu ^-\) decay [1, 2, 3, 4, 5], as well as deficits in the branching fractions \(B \rightarrow K \mu ^+ \mu ^-\) [6] and \(B_s \rightarrow \phi \mu ^+ \mu ^-\) [7]. While these anomalies might be interpreted as a sign of new physics,^{1} some caution is in order because potentially sizable hadronic uncertainties are challenging to control.

^{2}in the \(q^2 \equiv (p_{\ell ^-}+p_{\ell ^+})^2 \in [1,6]\,\mathrm{GeV}^2\) bin, i.e.,

^{3}

In this article, we will focus on these clean observables, and study first the structure of NP required to explain the found deviations in an effective field theory (EFT) approach. We will in particular stress a solution with sizable effects in the right-handed electron sector, complementary to the common solution which is linked to the left-handed muon sector. We will then show how a recently proposed composite Higgs model, incorporating a non-trivial, yet minimal, implementation of the lepton sector [21, 22] can explain the \(R_K\) and \(R_{K^*}\) anomalies *simultaneously*, due to the peculiar chirality structure of the involved currents. The setup contains less degrees of freedom than standard realizations and is very predictive, leading in general to a non-negligible violation of LFU, while allowing at the same time for a strong suppression of flavor changing neutral currents (FCNCs).^{4}

We will also discuss predictions of the setup for less clean observables in \( b \rightarrow s \ell ^+ \ell ^-\) decays and take into account further flavor constrains on the model. We will in particular focus on the angular analysis of the \(B \rightarrow K^*\mu ^+ \mu ^-\) decay, where pseudo-observables have been defined that also allow to cancel leading hadronic uncertainties [23, 24, 25, 26, 27, 28, 29, 30, 31], and where results are available from *all* 3 LHC *pp* experiments as well as from Belle, which are again pointing to a \(\sim 4\,\sigma \) deviation from the SM. It turns out that non-negligible effects are also predicted in this decay, allowing a significant improvement with respect to the SM, while still addressing the \(R_{K^{(*)}}\) anomalies and meeting the most stringent flavor bounds.

The remainder of this article is organized as follows. In the next section, we will provide an analysis of the NP required to address the anomalies in LFU violating decays in terms of \(D=6\) operators, parametrizing heavy physics beyond the SM in a model-independent way. We will then examine the structure of operators generated by the composite lepton model and present numerical predictions for \(R_{K^*}\), scrutinizing the correlation with \(R_K\) as well as \(B_s - {\bar{B}}_s\) mixing and taking into account constraints from \(B_s \rightarrow \mu ^+\mu ^-\). Finally, we will give our predictions for the angular observable \(P_5^\prime \), confronting them with the experimental results, before ending with our conclusions.

## 2 Pattern of the \(R_K, R_{K^*}\) anomalies

^{5}In fact, as the relevant energy scale is much below \(m_W\), also the SM contributions are best expressed in terms of contributions to the operators (6), \(C=C^\mathrm{SM}+C^\mathrm{NP}\) , where (at \(\mu =4.8\) GeV) [38]

^{6}

^{7}Neglecting also the strongly suppressed interference with \(C_{ bs_{L} \ell _{R}}^\mathrm{SM}\), we arrive at

^{8}In particular, it captures the

*leading*effects of

*all*NP Wilson coefficients considered.

We directly observe that \(R_K<1\) can be realized in two ways. It could origin from a destructive (constructive) interference of the combined left-handed muon (electron) contributions with the leading SM piece or, more generally, a negative sign in the difference of muon and electron contributions \(C_{ bs_{L+R} (\mu -e)_{L}}^{\mathrm{NP}} \equiv C_{ bs_{L} \mu _{L}}^{\mathrm{NP}}+C_{ bs_{R} \mu _{L}}^{\mathrm{NP}}-C_{ bs_{L} e_{L}}^{\mathrm{NP}}-C_{ bs_{R} e_{L}}^{\mathrm{NP}}\). On the other hand, it could stem from couplings to right-handed lepton currents. In that case, as discussed, the quadratic NP contribution dominates in general. From (14) it then follows directly that \(R_K<1\) requires the effect to come from the electron sector. Of course, a combination is possible, such that right-handed muon currents are allowed, however, while for the case of electron currents, *any* operator alone could accommodate \(R_K<1\), for the muon case, right handed contributions *alone* are not feasible, no matter what is the quark chirality.

*B*-physics anomalies can be obtained by considering in addition the ratio \(R_{K^*}\), which tests different combinations of Wilson coefficients, to which we will turn now. The theoretical prediction in this case reads (\(p\approx 0.86\))

^{9}

*both*left- and right-handed quark currents appear in LFU violating NP contributions.

The findings above are visualized in Fig. 1, where we show the correlations between \(R_K\) and \(R_{K^*}\), employing Eqs. (13) and (18). The colored lines correspond to the effects of the various NP Wilson coefficients. We consider all the coefficients entering these expressions, including combinations, such as to allow simultaneously for NP in the muon and electron sectors and in different chirality combinations. The most important dependencies of \(R_K\) and \(R_{K^*}\) are on the *difference* of purely left-handed contributions involving muons and electrons \(C_{ bs_{L} (\mu -e)_{L}}^{\mathrm{NP}} \equiv C_{ bs_{L} \mu _{L}}^{\mathrm{NP}} - C_{ bs_{L} e_{L}}^{\mathrm{NP}}\), on the corresponding quantity involving right-handed quark currents \(C_{ bs_{R} (\mu -e)_{L}}^{\mathrm{NP}} \equiv C_{ bs_{R} \mu _{L}}^{\mathrm{NP}} - C_{ bs_{R} e_{L}}^{\mathrm{NP}}\), where the direction of positive values is indicated by an arrow, and on the four coefficients involving right handed lepton currents, entering at quadratic order in NP. Note that for the latter case, a *simultaneous* presence of left- and right-handed quark currents is necessary, in order to break the degeneracy \(R_K=R_{K^*}\), while in case only one coefficient is turned on, the effect of either of them is indistinguishable in the \(R_K\) vs. \(R_{K^*}\) plane. We thus consider the distinct individual contributions \(C_{ bs_{X} \mu _{R}}^{\mathrm{NP}}\) and \(C_{ bs_{X} e_{R}}^{\mathrm{NP}}\) (being equal for \(X=L,R\)) as well as the simultaneous presence of left- and right-handed quark currents via \(C_{ bs_{L} \ell _{R}}^{\mathrm{NP}}=C_{ bs_{R} \ell _{R}}^{\mathrm{NP}}\equiv C_{ bs_{(L=R)} \ell _{R}}^{\mathrm{NP}}\), to capture the most relevant different scenarios. The effect of further combining different contributions to \(R_K\) and \(R_{K^*}\) can be easily inferred by considering the analytic Eqs. (13) and (18) in addition to the figure. The size of the coefficients corresponding to a certain point in the plane is visualized via the shape of the lines – solid lines correspond to \(0< |C_{ bs_{X} \ell _{Y}}^{\mathrm{NP}}| < 1\), dashed lines to \(1< |C_{ bs_{X} \ell _{Y}}^{\mathrm{NP}}| < 2.5\), while dotted lines feature \(2.5< |C_{ bs_{X} \ell _{Y}}^{\mathrm{NP}}| < 5\). Since the interference of NP effects in the right-handed lepton sector with SM contributions is suppressed, generically larger coefficients are required here in order to obtain sizable effects.

*electron*sector via \(C_{ bs_{X} e_{R}}^{\mathrm{NP}}\), as the preferred solution to the \(R_K^{(*)}\) anomalies.

^{10}This is a very interesting finding with respect to the model considered in the remainder of the paper. In fact, while a number of models accommodate the former option of dominating left-handed effects (including leptonic vector currents), see [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73] for \(Z^\prime \) realizations, the latter solution hardly exists in the form of explicit models in the literature, however just emerges in the composite model presented in Sect. 3.

We finally stress again that for the predictions of the model discussed in the next section, we employ full results including higher order corrections and the (small) effects of \(C_7^\mathrm{SM}\), performing a quadratic fit in the NP Wilson coefficients to the NLO results.^{11} We also display, in the right panel of Fig. 1, the results employing these full expressions, visualized via faint colored lines. Note that now, in the case of right-left contributions, the prediction does not just depend on the difference of muon and electron contributions any more and varying \(C_{ bs_{R} \mu _{L}}^{\mathrm{NP}}\) and \(C_{ bs_{R} e_{L}}^{\mathrm{NP}}\) independently leads to a (very modest) spread of the predictions in the \(R_K\) vs. \(R_{K^*}\) plane, depicted by the blue shaded region. Generally, the approximate results describe the relevant physics quite accurately.

In summary, consistently explaining the \(R_K\) and \(R_{K^*}\) anomalies requires both quark FCNCs involving the *b* quark and LFU violation in the \(\mu \) vs. *e* system (with effects either in left-handed currents in both sectors, or with a non-negligible *right-handed electron* contribution). The model that we will discuss now naturally leads to both effects, via exchange of composite vector resonance, whose couplings are not aligned with the SM couplings and the biggest contributions are actually expected in quark transitions involving the third generation and LFU violation involving light SM leptons. Since larger corrections are predicted for electrons, the setup matches nicely with the fact that this sector is less constrained, and, as we will see, sizable effects are in fact possible without problems with, e.g., \(B_s \rightarrow \ell ^+ \ell ^-\) decays.

## 3 Predictions in composite framework and further observables

Composite Higgs models offer a priori a compelling framework to explain the neutral flavor anomalies. The presence of a rich spectrum of bound states at the TeV scale, including heavy vector resonances (of \(Z^{\prime }\) type) with sizable couplings to some of the SM fermions, make these scenarios natural candidates to address the tension between data and SM predictions. Moreover, and contrary to most of the solutions to these anomalies that one can find in the literature, they also offer an interplay with electroweak symmetry breaking (EWSB) and some rationale to solve the hierarchy problem. If one considers that fermion masses are generated via the mechanism of *partial compositeness*, a sizable violation of lepton flavor universality, like suggested by \(R_K\) and \(R_{K^{*}}\), necessarily requires the charged leptons to feature a sizable degree of compositeness in their left-handed (LH) and/or right-handed (RH) chirality, \(\epsilon _{\ell _R}\), \(\epsilon _{\ell _L}\). Since charged lepton masses scale in general as \( \sim g_{*} v\epsilon _{\ell _L} \epsilon _{\ell R}\), where \(g_{*}\) is the characteristic coupling within the strong sector, both chiralities can not be composite at the same time. Therefore, these models will either predict effects in \(\mathcal {O}_{bs_{L,R} \ell _L}\) or in \(\mathcal {O}_{bs_{L,R}\ell _R}\) scaling like \(\sim g_{*}^2/m_{*}^2 V_{ts}\epsilon _{b_X}^2 \epsilon _{\ell _Y}^2\), where *X* and *Y* denote the possible chiralities involved in the quark- and lepton-sector, respectively, and \(m_{*}\) is the typical mass scale of the first vector resonances.

The model under consideration falls into the second category and was presented in detail in [21, 22]. One of its most interesting features is that charged leptons partially substitute the role of the top quark as a trigger of EWSB, and a link between violation of lepton flavor universality and the absence of top partners at the LHC is established. Indeed, if the composite operators interacting with the RH charged leptons transform in sufficiently large irreducible representations of the global group within the strong sector, the leading charged lepton contribution to the Higgs quartic coupling will appear at order \(\sim |\epsilon _{\ell _R}|^2\) instead of the usual \(\sim |\epsilon _{\ell _R}|^4\). Since the leading top contribution can be expected to appear at order \(|\epsilon _{t_L}|^4\), the contribution arising from the lepton sector can be comparable to the top one, even with a smaller degree of compositeness. Moreover, if all the three lepton generations are partially composite, the lepton contribution will be enhanced by a factor \(N_\mathrm{gen}\sim 3\), compensating the color factor \(N_c = 3\) present in the top case and allowing to lift the problematic top partners via destructive interference between the two sectors in the Higgs potential.^{12} One of the important findings of [21, 22] is that the very same representations making this possible also provide the required quantum numbers for a minimal implementation of a type-III seesaw mechanism for neutrino masses, which can in fact motivate RH charged-lepton compositeness, as we will see now.^{13} If one follows this very minimal avenue, considering each generation of RH leptons to interact with a *single* composite operator, all RH charged leptons in fact inherit the degree of compositeness of their RH neutrino counterparts, and the latter is required to be sizable to allow for large enough neutrino masses (see [21, 22, 77, 78, 79, 80] for more details in both type-I and type-III seesaw models). Then, the different scaling of the neutrino and charged lepton mass matrices requires \(0\ll \epsilon _{\tau _R}\ll \epsilon _{\mu _R} \ll \epsilon _{e_R}\), in order to have simultaneously hierarchical charged lepton masses and a non-hierarchical neutrino mass matrix [21]. An immediate consequence of the above chirality structure is that mostly the operators \(\mathcal{O}_{ bs_{X} e_{R}}\) will be generated (as well as subdominantly \(\mathcal{O}_{ bs_{X} \mu _{R}}\)). Moreover, possible modifications of *Z* couplings which are extremely constrained by electroweak precision data are avoided due to custodial symmetry, contrary to what happens for the case of composite LH leptons, where it is not possible to protect the coupling to both fields in the SM doublet at the same time.

To be concrete, we are considering a strongly interacting sector featuring the Higgs as a pseudo Nambu–Goldstone boson (pNGB) arising from the symmetry breaking \(SO(5)\rightarrow SO(4)\), known as the Minimal Composite Higgs model (MCHM) [81, 82]. The quark fields are embedded in \(\mathbf{5}^u_L,\mathbf{1}^u_R,\mathbf{5}^d_L,\mathbf{1}^d_R\) representations of *SO*(5), while *all* lepton fields are embedded in only *two* representations, \(\mathbf{5}_L^\ell , \mathbf{14}_R^\ell \), per generation. As mentioned before, we can explain the tiny neutrino masses via a type-III seesaw mechanism, since the symmetric representation \(\mathbf{14}_R^\ell \cong \mathbf {(1,1) + (2,2) + (3,3)}\) of \(SO(5) \cong SU(2)_L \times SU(2)_R\) can host both an electrically charged \(SU(2)_L\) singlet lepton (\(\ell _R\)) and a heavy fermionic seesaw triplet of \(SU(2)_L\). This *unification* of right handed leptons comes along with a more minimal quark representation than in known models, because the enhanced leptonic contribution to the Higgs potential, originating from the symmetric *SO*(5) representation, allows for a viable electroweak symmetry breaking with all right handed SM-quarks inert under *SO*(5) and the left-handed ones in the fundamental (see [21, 22] for details). Thus, the model features less degrees of freedom than standard incarnations, such as the MCHM\(_5\).

One of the main challenges of all these scenarios featuring composite leptons is the generation of dangerous FCNCs through the exchange of the very same vector resonances producing the violation of LFU. Since, in general, they will also contribute to extremely well measured lepton flavor violating processes like \(\mu \rightarrow e\gamma \), \(\tau \rightarrow \mu \gamma \), \(\mu \rightarrow eee\), and \(\mu -e\) conversion, they typically require the addition of some non-trivial flavor symmetry. In the model at hand, it turns out that the reduced number of composite operators mixing with the light leptons naturally allows for a very strong flavor protection along the lines of minimal flavor violation, since a single spurion allows to break the flavor symmetry [21]. Regarding the quark sector, in the model at hand the left handed current \({\bar{s}}_L \gamma ^\mu b_L\) (i.e., \(X=L\)) will in general dominate, with the compositeness of \(b_L\) following from the large top mass, but also non-negligible contributions from right-handed quarks are possible.

*f*to be relatively high, and that one needs to overcome the absence of interference with the SM contribution, agreement with the experimental values of \(R_K\) and \(R_{K^{*}}\) requires points with rather sizable FCNCs, which can be obtained in the considered parameter space.

^{14}

For the composite Higgs model considered, effects are coming from non-vanishing \(C_{ bs_{L-R} \mu _{R}}^{\mathrm{NP}}, \) but are modest in general, as can be seen from Fig. 4 where we show the ratio \(\mathcal{B}(B_s \rightarrow \mu ^+ \mu ^-)/\mathcal{B}(B_s \rightarrow \mu ^+ \mu ^-)_\mathrm{SM}\), together with the experimental \(1\,\sigma \) range, the latter depicted by the red dashed lines. Most of the points lie within the corresponding range (while basically all meet the \(2\,\sigma \) constraints). The impact of the \(1\,\sigma \) bound on the composite solution to the \(R_{K^{(*)}}\) anomalies is visualized in the right panel of Fig. 3, where we reject the parameter-space points that do not meet this constraint. As can be seen, the effect is very modest.

*only*for the case studied here of NP mostly in RH electrons, one can have an effect in both \(\bar{b}_R \gamma ^{\mu } s_R\) and \(\bar{b}_L \gamma ^{\mu } s_L\) and still get agreement with

*both*\(R_K\) and \(R_{K^{*}}\) (as discussed in the previous section). This means in particular that, if the prediction for \(\varDelta M_{B_s}\) quoted by [88] is taken at face value, accepting some accidental cancellations, the scenario considered here would provide the only allowed \(Z^{\prime }\) model accommodating both anomalies.

^{15}

*B*decays are the angular dependencies of the \(B \rightarrow K^*\mu ^+ \mu ^-\) decay rate. Of particular interest is the coefficient \(P_5^\prime \), which belongs to a class of observables constructed such as to cancel hadronic uncertainties [23, 24, 25, 26, 27, 28, 29, 30, 31], which shows an interesting deviation with respect to the SM prediction, see e.g., [8, 16]. Our results for \(P_5^\prime \) in the \(q^2\in [4.3,8.68]\,\mathrm{GeV}^2\) bin are given in Fig. 6, where we plot the correlation between \(P_5^\prime \) and \(R_K\), visualizing again the size of the corrections to \(\varDelta M_{B_s}\) as different shades of blue. The SM prediction \({P_5^\prime }_{[4.3,8.68]}\approx -\ 0.8\) and the experimentally preferred value \({P_5^\prime }_{[4.3,8.68]}\approx - \ 0.2\) [8, 16] are given as red and green dashed lines, respectively. From this final plot it is evident that the proposed composite model can address the \(R_{K}\) and \(R_{K^{*}}\) anomalies, being in agreement with \(B_s - {\bar{B}}_s\) mixing and \(\mathcal{B}(B_s \rightarrow \ell ^+ \ell ^-)\), while also the fit to the \(P_5^\prime \) results can be improved (via the light-blue points approaching \({P_5^\prime }_{[4.3,8.68]}\approx -\ 0.2\) and featuring \(R_K \sim \) (0.7–0.8)). It thus furnishes an interesting setup both with respect to the gauge hierarchy problem – avoiding light top partners – and concerning the current pattern of experimental results in flavor physics.

Before concluding, let us mention further ways to test the proposed composite Higgs scenario. First, finding non-vanishing coefficients of four-lepton operators at future lepton colliders (in particular of type \(({\bar{e}}_R \gamma ^\mu e_R) ({\bar{e}}_R \gamma _\mu e_R)\) [21, 80]) could be interpreted as a hint for lepton compositeness. Moreover, along the line of ‘conventional’ light top partners, the model at hand contains *leptonic* resonances \(\ell ^\prime \) somewhat below the typical resonance scale, \(m_{\ell ^\prime } < m_{Z^\prime }\) [22], which could be searched for directly. Finally, due to the very fact that these particles can be lighter than the \(Z^\prime \), new decay channels such as \(Z^\prime \rightarrow \ell ^\prime \ell \) open up, which have been investigated in [90], and would be a smoking gun for the setup.

## 4 Conclusions

We have scrutinized hints for lepton flavor universality violation in *B*-meson decays, focusing first on the general properties of the anomalies in an EFT approach. Here, we emphasized a simultaneous solution to the \(R_K\) and \(R_{K^*}\) anomalies via effects in right-handed lepton currents, not worked out before. We stressed that this solution requires the dominant contribution originating from electron (and not muon) currents and presented a composite Higgs scenario, where this pattern emerges in a natural way. In fact, the model provides one of the very few scenarios, that features all ingredients to consistently resolve the anomalies, without being actually constructed for that purpose: it *predicts* LFU violation and sizable FCNCs involving the third quark generation, features a strong protection from FCNCs in the lepton sector, and allows for the absence of ultra-light top partners at the LHC. We also discussed the impact of operators addressing the \(R_{K^{(*)}}\) measurements on other flavor observables, such as \(\mathcal{B}(B_s \rightarrow \ell ^+\ell ^-)\), \(\varDelta M_{B_s}\), and \(P_5^\prime \). For the explicit model at hand, we found that all constraints are met, while it is possible to simultaneously resolve the (more controversial) \(P_5^\prime \) anomaly.

## Footnotes

- 1.
- 2.
This is valid above the muon threshold, \(q^2 \sim 4 m_\mu ^2\).

- 3.
Note that in the case of \(R_K\) this holds also in the presence of NP, while for \(R_{K^*}\) it holds within the SM, still allowing for a clean test of the latter.

- 4.
Beyond that, in these models the leptonic contribution to the Higgs mass is parametrically enhanced relative to the quark contribution by (inverse) powers of

*SO*(5) breaking spurions, such that a light Higgs does not necessarily lead to light top partners, resolving tensions with LHC searches. - 5.
The scalar and pseudo-scalar operators \({\mathcal{O}^\ell _{S,P}}^{(\prime )}\) are already considerably constrained from \(B_s \rightarrow \mu ^+ \mu ^-\) and play no important role in the following discussions [34, 35] (for the electron channel, sizable effects are still possible in principle [36], however the operators play no role for our analysis of Sect. 3). This is also true for tensor operators, which can not be generated from operators invariant under the SM gauge group to leading order [37].

- 6.The individual branching fractions are given by [13, 17, 39, 41] where \(m_{A\pm B}^2 \equiv m_A^2\pm m_B^2\), and we neglected lepton masses, CP violation, and higher order corrections (which are however included in our numerical analysis, employing CP averaged quantities). Here, \(f_+\) and \(f_T\) are the QCD form factors (see [42, 43]) and \(h_K(q^2)\) parametrizes non-factorizable contributions from the weak hamiltonian [13]. Neglecting the strongly suppressed \(C_7^{(\prime )}\) contributions (which could only become relevant approaching the photon pole at \(q^2 = 0\) and are in any case constrained to be pretty SM like [44]) and the non-factorizable \(h_K(q^2)\), the QCD form factor \(f_+(q^2)\) drops out in the ratio \(R_K\) (13).
- 7.
If quadratic terms in \(C_{ bs_{X} \ell _{R}}\) are kept, a \(\sim 30\%\) effect in \(R_K\) is per se in agreement with a convergence of the expansion in NP contributions, no matter from which operator it is induced (see also [45]).

- 8.If the fourth order in \(C_{ bs_{X} \ell _{R}}\) is included, which corresponds to adding it holds even at the \(\mathcal{O}(1\%)\) level, which becomes negligible compared to other uncertainties. Still, for the numerical results presented in Sect. 3, we will use the exact expressions, including in addition higher order QCD corrections [46] as well as the effect of \(C_7^\mathrm{SM}\).
- 9.Neglecting terms suppressed by \(m_\ell ^2/q^2\), NLO, and non-factorizable corrections (as well as CP violation), the individual branching fractions can be expressed in terms of six transversity amplitudes \(A_{0,\perp ,\parallel }^{L,R}\) as [47] (see also [39, 41, 48, 49])where the form factors \(\xi _{\perp ,\parallel }\) are given, e.g., in Appendix E of [47]. We directly dropped electromagnetic dipole contributions, becoming important only for \(q^2 \rightarrow 0\), which would appear in the three square brackets above as \(2 m_b m_B/q^2 \{C_7+C_7^\prime ,\,C_7-C_7^\prime ,\,q^2/m_B^2 (C_7-C_7^\prime )\}\). Defining the integrated form factorswe can write \(R_{K^*}\) as two combinations of Wilson coefficients, weighted by the polarization fraction \(p \equiv \frac{g_0+g_\parallel }{g_0+g_\parallel +g_\perp }\), where \(p \approx 0.86\) to good approximation for the \(q^2\) range considered here (with some per cent deviation for the low \(q^2\) bin) [39, 47].$$\begin{aligned}&g_{\perp ,\parallel ,0}^{[q_\mathrm{min}^2,q_\mathrm{max}^2]}= \int _{q_\mathrm{min}^2}^{q_\mathrm{max}^2} dq^2 ([m_{B-K^*}^2]^2-2m_{B+K^*}^2q^2 +q^4)^{\frac{1}{2}}\\&\quad \times \frac{2(q^3 - m_B^2 q)^2}{m_B^2}\ \left\{ |\xi _\perp |^2,\,|\xi _\perp |^2,\, {\frac{(m_B^2-q^2)^2}{8q^2 m_{K^*}^2}}|\xi _\parallel |^2 \right\} , \end{aligned}$$
- 10.
It becomes also evident that a negative NP contribution to \(C_9^\mu =(C_{ bs_{L} \mu _{R}}+C_{ bs_{L} \mu _{L}})/2\), as advocated as a solution to the \(B \rightarrow K^*\mu ^+ \mu ^-\) anomaly (see, e.g., [8, 16]), allows for a good fit of the \(R_K^{(*)}\) anomalies, basically because (for moderate values of the coefficients) the effect of \(C_{ bs_{L} \mu _{L}}^{\mathrm{NP}}\) dominates via the SM interference. A positive \(C_9^e\), on the other hand, also allows for a straightforward solution.

- 11.
We used the code Flavio (v0.21.1) [46] for the numerical prediction.

- 12.
See also [74] for a (different) composite explanation of LFU violation.

- 13.
- 14.
Here, we neglect scalar and pseudoscalar operators, not generated in the model at hand to good approximation.

- 15.
Note that solutions to the \(R_K\) anomaly are also subject to competitive constraints from searches for tails in the high \(p_T\) di-lepton spectrum [89]. However, due to the suppressed couplings to light quarks, our setup avoids these bounds. Moreover, di-lepton searches will be less effective, since one typically expects the neutral vector mediators to decay via a lepton partner, leading to different decay topologies [90], see also below.

## Notes

### Acknowledgements

We are grateful to Marco Nardecchia, David Straub and Jorge Martin Camalich, for useful comments and discussions. FG thanks the CERN Theory Department, where part of this work was performed, for its hospitality. The research of AC was supported by a Marie Skłodowska-Curie Individual Fellowship of the European Community’s Horizon 2020 Framework Programme for Research and Innovation under contract number 659239 (NP4theLHC14) and by the Cluster of Excellence *Precision Physics, Fundamental Interactions and Structure of Matter* (PRISMA—EXC 1098) and Grant 05H12UME of the German Federal Ministry for Education and Research (BMBF).

## References

- 1.R. Aaij et al., Phys. Rev. Lett.
**111**, 191801 (2013). https://doi.org/10.1103/PhysRevLett.111.191801 ADSCrossRefGoogle Scholar - 2.R. Aaij et al., JHEP
**02**, 104 (2016). https://doi.org/10.1007/JHEP02(2016)104 ADSCrossRefGoogle Scholar - 3.S. Wehle et al., Phys. Rev. Lett.
**118**(11), 111801 (2017). https://doi.org/10.1103/PhysRevLett.118.111801 ADSCrossRefGoogle Scholar - 4.CMS Collaboration, Measurement of the \(P_1\) and \(P_5^{\prime }\) angular parameters of the decay \(\rm B\mathit{^0 \rightarrow \rm K}^{*0} \mu ^+ \mu ^-\) in proton-proton collisions at \(\sqrt{s}=8~\rm TeV\). CMS-PAS-BPH-15-008 (2017)Google Scholar
- 5.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. ATLAS-CONF-2017-023 (2017)Google Scholar
- 6.R. Aaij et al., JHEP
**06**, 133 (2014). https://doi.org/10.1007/JHEP06(2014)133 ADSCrossRefGoogle Scholar - 7.R. Aaij et al., JHEP
**09**, 179 (2015). https://doi.org/10.1007/JHEP09(2015)179 ADSCrossRefGoogle Scholar - 8.S. Descotes-Genon, J. Matias, J. Virto, Phys. Rev. D
**88**, 074002 (2013). https://doi.org/10.1103/PhysRevD.88.074002 ADSCrossRefGoogle Scholar - 9.W. Altmannshofer, D.M. Straub, Eur. Phys. J. C
**73**, 2646 (2013). https://doi.org/10.1140/epjc/s10052-013-2646-9 ADSCrossRefGoogle Scholar - 10.F. Beaujean, C. Bobeth, D. van Dyk, Eur. Phys. J. C
**74**, 2897 (2014). https://doi.org/10.1140/epjc/s10052-014-2897-0 [Erratum: Eur. Phys. J. C**74**, 3179 (2014). https://doi.org/10.1140/epjc/s10052-014-3179-6] - 11.T. Hurth, F. Mahmoudi, JHEP
**04**, 097 (2014). https://doi.org/10.1007/JHEP04(2014)097 ADSCrossRefGoogle Scholar - 12.R. Gauld, F. Goertz, U. Haisch, Phys. Rev. D
**89**, 015005 (2014). https://doi.org/10.1103/PhysRevD.89.015005 ADSCrossRefGoogle Scholar - 13.W. Altmannshofer, D.M. Straub, Eur. Phys. J. C
**75**(8), 382 (2015). https://doi.org/10.1140/epjc/s10052-015-3602-7 ADSCrossRefGoogle Scholar - 14.S. Descotes-Genon, L. Hofer, J. Matias, J. Virto, JHEP
**06**, 092 (2016). https://doi.org/10.1007/JHEP06(2016)092 ADSCrossRefGoogle Scholar - 15.T. Hurth, F. Mahmoudi, S. Neshatpour, Nucl. Phys. B
**909**, 737 (2016). https://doi.org/10.1016/j.nuclphysb.2016.05.022 ADSCrossRefGoogle Scholar - 16.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 ADSCrossRefGoogle Scholar - 17.G. Hiller, F. Kruger, Phys. Rev. D
**69**, 074020 (2004). https://doi.org/10.1103/PhysRevD.69.074020 ADSCrossRefGoogle Scholar - 18.R. Aaij et al., Phys. Rev. Lett.
**113**, 151601 (2014). https://doi.org/10.1103/PhysRevLett.113.151601 ADSCrossRefGoogle Scholar - 19.M. Bordone, G. Isidori, A. Pattori, Eur. Phys. J. C
**76**(8), 440 (2016). https://doi.org/10.1140/epjc/s10052-016-4274-7 ADSCrossRefGoogle Scholar - 20.R. Aaij et al., JHEP
**08**, 055 (2017). https://doi.org/10.1007/JHEP08(2017)055 ADSCrossRefGoogle Scholar - 21.A. Carmona, F. Goertz, Phys. Rev. Lett.
**116**(25), 251801 (2016). https://doi.org/10.1103/PhysRevLett.116.251801 ADSCrossRefGoogle Scholar - 22.A. Carmona, F. Goertz, JHEP
**05**, 002 (2015). https://doi.org/10.1007/JHEP05(2015)002 ADSCrossRefGoogle Scholar - 23.J. Matias, F. Mescia, M. Ramon, J. Virto, JHEP
**04**, 104 (2012). https://doi.org/10.1007/JHEP04(2012)104 ADSCrossRefGoogle Scholar - 24.S. Jäger, J. Martin Camalich, JHEP
**05**, 043 (2013). https://doi.org/10.1007/JHEP05(2013)043 ADSCrossRefGoogle Scholar - 25.S. Descotes-Genon, T. Hurth, J. Matias, J. Virto, JHEP
**05**, 137 (2013). https://doi.org/10.1007/JHEP05(2013)137 ADSCrossRefGoogle Scholar - 26.J. Lyon, R. Zwicky, Resonances gone topsy turvy—the charm of QCD or new physics in \(b \rightarrow s \ell ^+ \ell ^-\)? Report no. EDINBURGH-14-10, CP3-ORIGINS-2014-021-DNRF90, DIAS-2014-21 (2014). arXiv:1406.0566
- 27.S. Descotes-Genon, L. Hofer, J. Matias, J. Virto, JHEP
**12**, 125 (2014). https://doi.org/10.1007/JHEP12(2014)125 ADSCrossRefGoogle Scholar - 28.S. Jäger, J. Martin Camalich, Phys. Rev. D
**93**(1), 014028 (2016). https://doi.org/10.1103/PhysRevD.93.014028 ADSCrossRefGoogle Scholar - 29.M. Ciuchini, M. Fedele, E. Franco, S. Mishima, A. Paul, L. Silvestrini, M. Valli, JHEP
**06**, 116 (2016). https://doi.org/10.1007/JHEP06(2016)116 ADSCrossRefGoogle Scholar - 30.B. Capdevila, S. Descotes-Genon, L. Hofer, J. Matias, JHEP
**04**, 016 (2017). https://doi.org/10.1007/JHEP04(2017)016 ADSCrossRefGoogle Scholar - 31.V.G. Chobanova, T. Hurth, F. Mahmoudi, D. Martinez Santos, S. Neshatpour, JHEP
**07**, 025 (2017). https://doi.org/10.1007/JHEP07(2017)025 ADSCrossRefGoogle Scholar - 32.B. Grinstein, M.J. Savage, M.B. Wise, Nucl. Phys. B
**319**, 271 (1989). https://doi.org/10.1016/0550-3213(89)90078-3 ADSCrossRefGoogle Scholar - 33.G. Buchalla, A.J. Buras, M.E. Lautenbacher, Rev. Mod. Phys.
**68**, 1125 (1996). https://doi.org/10.1103/RevModPhys.68.1125 ADSCrossRefGoogle Scholar - 34.G. Hiller, M. Schmaltz, Phys. Rev. D
**90**, 054014 (2014). https://doi.org/10.1103/PhysRevD.90.054014 ADSCrossRefGoogle Scholar - 35.F. Beaujean, C. Bobeth, S. Jahn, Eur. Phys. J. C
**75**(9), 456 (2015). https://doi.org/10.1140/epjc/s10052-015-3676-2 ADSCrossRefGoogle Scholar - 36.R. Fleischer, R. Jaarsma, G. Tetlalmatzi-Xolocotzi, JHEP
**05**, 156 (2017). https://doi.org/10.1007/JHEP05(2017)156 ADSCrossRefGoogle Scholar - 37.R. Alonso, B. Grinstein, J. Martin Camalich, Phys. Rev. Lett.
**113**, 241802 (2014). https://doi.org/10.1103/PhysRevLett.113.241802 ADSCrossRefGoogle Scholar - 38.B. Capdevila, S. Descotes-Genon, L. Hofer, J. Matias, J. Virto, PoS
**LHCP2016**, 073 (2016)Google Scholar - 39.G. Hiller, M. Schmaltz, JHEP
**02**, 055 (2015). https://doi.org/10.1007/JHEP02(2015)055 ADSCrossRefGoogle Scholar - 40.G. D’Amico, M. Nardecchia, P. Panci, F. Sannino, A. Strumia, R. Torre, A. Urbano, JHEP
**09**, 010 (2017). https://doi.org/10.1007/JHEP09(2017)010 ADSCrossRefGoogle Scholar - 41.A. Ali, P. Ball, L.T. Handoko, G. Hiller, Phys. Rev. D
**61**, 074024 (2000). https://doi.org/10.1103/PhysRevD.61.074024 ADSCrossRefGoogle Scholar - 42.C. Bouchard, G.P. Lepage, C. Monahan, H. Na, J. Shigemitsu, Phys. Rev. D
**88**(5), 054509 (2013). https://doi.org/10.1103/PhysRevD.88.079901 [Erratum: Phys. Rev. D**88**(7), 079901 (2013). https://doi.org/10.1103/PhysRevD.88.054509] - 43.A.J. Buras, J. Girrbach-Noe, C. Niehoff, D.M. Straub, JHEP
**02**, 184 (2015). https://doi.org/10.1007/JHEP02(2015)184 ADSCrossRefGoogle Scholar - 44.A. Paul, D.M. Straub, JHEP
**04**, 027 (2017). https://doi.org/10.1007/JHEP04(2017)027 ADSCrossRefGoogle Scholar - 45.R. Contino, A. Falkowski, F. Goertz, C. Grojean, F. Riva, JHEP
**07**, 144 (2016). https://doi.org/10.1007/JHEP07(2016)144 ADSCrossRefGoogle Scholar - 46.D. Straub, P. Stangl, C. Niehoff, E. Gurler, W. Zeren, J. Kumar, S. Reicher, F. Beaujean, flav-io/flavio v0.21.1 (2017). https://doi.org/10.5281/zenodo.569011
- 47.C. Bobeth, G. Hiller, G. Piranishvili, JHEP
**07**, 106 (2008). https://doi.org/10.1088/1126-6708/2008/07/106 ADSCrossRefGoogle Scholar - 48.C. Hambrock, G. Hiller, S. Schacht, R. Zwicky, Phys. Rev. D
**89**(7), 074014 (2014). https://doi.org/10.1103/PhysRevD.89.074014 ADSCrossRefGoogle Scholar - 49.A. Bharucha, D.M. Straub, R. Zwicky, JHEP
**08**, 098 (2016). https://doi.org/10.1007/JHEP08(2016)098 ADSCrossRefGoogle Scholar - 50.A. Crivellin, G. D’Ambrosio, J. Heeck, Phys. Rev. Lett.
**114**, 151801 (2015). https://doi.org/10.1103/PhysRevLett.114.151801 ADSCrossRefGoogle Scholar - 51.A. Crivellin, G. D’Ambrosio, J. Heeck, Phys. Rev. D
**91**(7), 075006 (2015). https://doi.org/10.1103/PhysRevD.91.075006 ADSCrossRefGoogle Scholar - 52.C. Niehoff, P. Stangl, D.M. Straub, Phys. Lett. B
**747**, 182 (2015). https://doi.org/10.1016/j.physletb.2015.05.063 ADSCrossRefGoogle Scholar - 53.D. Aristizabal Sierra, F. Staub, A. Vicente, Phys. Rev. D
**92**(1), 015001 (2015). https://doi.org/10.1103/PhysRevD.92.015001 ADSCrossRefGoogle Scholar - 54.A. Celis, J. Fuentes-Martin, M. Jung, H. Serodio, Phys. Rev. D
**92**(1), 015007 (2015). https://doi.org/10.1103/PhysRevD.92.015007 ADSCrossRefGoogle Scholar - 55.A. Greljo, G. Isidori, D. Marzocca, JHEP
**07**, 142 (2015). https://doi.org/10.1007/JHEP07(2015)142 ADSCrossRefGoogle Scholar - 56.A. Falkowski, M. Nardecchia, R. Ziegler, JHEP
**11**, 173 (2015). https://doi.org/10.1007/JHEP11(2015)173 ADSCrossRefGoogle Scholar - 57.B. Allanach, F.S. Queiroz, A. Strumia, S. Sun, Phys. Rev. D
**93**(5), 055045 (2016). https://doi.org/10.1103/PhysRevD.93.055045 [Erratum: Phys. Rev. D**95**(11), 119902 (2017). https://doi.org/10.1103/PhysRevD.95.119902] - 58.C.W. Chiang, X.G. He, G. Valencia, Phys. Rev. D
**93**(7), 074003 (2016). https://doi.org/10.1103/PhysRevD.93.074003 ADSCrossRefGoogle Scholar - 59.S.M. Boucenna, A. Celis, J. Fuentes-Martin, A. Vicente, J. Virto, Phys. Lett. B
**760**, 214 (2016). https://doi.org/10.1016/j.physletb.2016.06.067 ADSCrossRefGoogle Scholar - 60.E. Megias, G. Panico, O. Pujolas, M. Quiros, JHEP
**09**, 118 (2016). https://doi.org/10.1007/JHEP09(2016)118 ADSCrossRefGoogle Scholar - 61.W. Altmannshofer, S. Gori, S. Profumo, F.S. Queiroz, JHEP
**12**, 106 (2016). https://doi.org/10.1007/JHEP12(2016)106 ADSCrossRefGoogle Scholar - 62.A. Crivellin, J. Fuentes-Martin, A. Greljo, G. Isidori, Phys. Lett. B
**766**, 77 (2017). https://doi.org/10.1016/j.physletb.2016.12.057 ADSCrossRefGoogle Scholar - 63.I. Garcia Garcia, JHEP
**03**, 040 (2017). https://doi.org/10.1007/JHEP03(2017)040 ADSCrossRefGoogle Scholar - 64.D. Bhatia, S. Chakraborty, A. Dighe, JHEP
**03**, 117 (2017). https://doi.org/10.1007/JHEP03(2017)117 ADSCrossRefGoogle Scholar - 65.C. Bonilla, T. Modak, R. Srivastava, J.W.R. Valle, \(U(1)_{B_3-3L_\mu }\) gauge symmetry as a simple description of \(b\rightarrow s\) anomalies. Phys. Rev. D
**98**(9), 095002 (2018). https://doi.org/10.1103/PhysRevD.98.095002 ADSCrossRefGoogle Scholar - 66.R. Alonso, P. Cox, C. Han, T.T. Yanagida, Phys. Lett. B
**774**, 643 (2017). https://doi.org/10.1016/j.physletb.2017.10.027 ADSCrossRefGoogle Scholar - 67.J. Ellis, M. Fairbairn, P. Tunney, Anomaly-free models for flavour anomalies. Eur. Phys. J. C
**78**(3), 238 (2018). https://doi.org/10.1140/epjc/s10052-018-5725-0. arXiv:1705.03447 ADSCrossRefGoogle Scholar - 68.Y. Tang, Y.-L. Wu, Flavor non-universal gauge interactions and anomalies in B-meson decays. Chin. Phys. C
**42**(3), 033104 (2018). https://doi.org/10.1088/1674-1137/42/3/033104. arXiv:1705.05643 ADSCrossRefGoogle Scholar - 69.C.-W. Chiang, X.-G. He, J. Tandean, X.-B. Yuan, \(R_{K^{(*)}}\) and related \(b\rightarrow s\ell \bar{\ell }\) anomalies in minimal flavor violation framework with \(Z^{\prime }\) boson. Phys. Rev.
**D96**(11), 115022. https://doi.org/10.1103/PhysRevD.96.115022. arXiv:1706.02696 - 70.
- 71.D. Buttazzo, A. Greljo, G. Isidori, D. Marzocca, JHEP
**11**, 044 (2017). https://doi.org/10.1007/JHEP11(2017)044 ADSCrossRefGoogle Scholar - 72.E. Megias, M. Quiros, L. Salas, Phys. Rev. D
**96**(7), 075030 (2017). https://doi.org/10.1103/PhysRevD.96.075030 ADSCrossRefGoogle Scholar - 73.J.M. Cline, J. Martin Camalich, Phys. Rev. D
**96**(5), 055036 (2017). https://doi.org/10.1103/PhysRevD.96.055036 ADSCrossRefGoogle Scholar - 74.J.M. Cline, Phys. Rev. D
**97**(1), 015013 (2018). https://doi.org/10.1103/PhysRevD.97.015013 ADSCrossRefGoogle Scholar - 75.G. Ballesteros, A. Carmona, M. Chala, Eur. Phys. J. C
**77**(7), 468 (2017). https://doi.org/10.1140/epjc/s10052-017-5040-1 ADSCrossRefGoogle Scholar - 76.
- 77.M. Carena, A.D. Medina, N.R. Shah, C.E.M. Wagner, Phys. Rev. D
**79**, 096010 (2009). https://doi.org/10.1103/PhysRevD.79.096010 ADSCrossRefGoogle Scholar - 78.F. del Aguila, A. Carmona, J. Santiago, JHEP
**08**, 127 (2010). https://doi.org/10.1007/JHEP08(2010)127 ADSCrossRefGoogle Scholar - 79.C. Hagedorn, M. Serone, JHEP
**02**, 077 (2012). https://doi.org/10.1007/JHEP02(2012)077 ADSCrossRefGoogle Scholar - 80.A. Carmona, F. Goertz, Nucl. Part. Phys. Proc.
**285–286**, 93 (2017). https://doi.org/10.1016/j.nuclphysbps.2017.03.017 CrossRefGoogle Scholar - 81.R. Contino, Y. Nomura, A. Pomarol, Nucl. Phys. B
**671**, 148 (2003). https://doi.org/10.1016/j.nuclphysb.2003.08.027 ADSCrossRefGoogle Scholar - 82.K. Agashe, R. Contino, A. Pomarol, Nucl. Phys. B
**719**, 165 (2005). https://doi.org/10.1016/j.nuclphysb.2005.04.035 ADSCrossRefGoogle Scholar - 83.K. De Bruyn, R. Fleischer, R. Knegjens, P. Koppenburg, M. Merk, A. Pellegrino, N. Tuning, Phys. Rev. Lett.
**109**, 041801 (2012). https://doi.org/10.1103/PhysRevLett.109.041801 ADSCrossRefGoogle Scholar - 84.C. Bobeth, M. Gorbahn, T. Hermann, M. Misiak, E. Stamou, M. Steinhauser, Phys. Rev. Lett.
**112**, 101801 (2014). https://doi.org/10.1103/PhysRevLett.112.101801 ADSCrossRefGoogle Scholar - 85.R. Aaij et al., Phys. Rev. Lett.
**118**(19), 191801 (2017). https://doi.org/10.1103/PhysRevLett.118.191801 ADSCrossRefGoogle Scholar - 86.T. Aaltonen et al., Phys. Rev. Lett.
**102**, 201801 (2009). https://doi.org/10.1103/PhysRevLett.102.201801 ADSCrossRefGoogle Scholar - 87.A.J. Buras, in
*Probing the standard model of particle interactions. Proceedings, Summer School in Theoretical Physics, NATO Advanced Study Institute, 68th Session, Les Houches, France, July 28–September 5, 1997. Pt. 1, 2*(1998), pp. 281–539Google Scholar - 88.L. Di Luzio, M. Kirk, A. Lenz, Phys. Rev. D
**97**, 095035 (2018). https://doi.org/10.1103/PhysRevD.97.095035 ADSCrossRefGoogle Scholar - 89.A. Greljo, D. Marzocca, Eur. Phys. J. C
**77**(8), 548 (2017). https://doi.org/10.1140/epjc/s10052-017-5119-8 ADSCrossRefGoogle Scholar - 90.M. Chala, M. Spannowsky, Phys. Rev. D
**98**(3), 035010 (2018). https://doi.org/10.1103/PhysRevD.98.035010 ADSCrossRefGoogle Scholar

## 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}