Towards excluding a light $Z^\prime$ explanation of $b\to s\ell^+\ell^-$

The discrepancies between $b\to s\ell^+\ell^-$ data and the corresponding Standard Model predictions constitute the most significant hints for new physics (at the TeV scale or below) currently available. In fact, many scenarios that can account for these anomalies have been proposed in the literature. However, only a single light new physics explanation, i.e. with a mass below the $B$ meson scale, is possible: a light $Z^\prime$ boson. In this article, we point out that improved limits on $B\to K^{(*)}\nu\nu$, including the experimental sensitivities required for a proper treatment of the necessarily sizable $Z^\prime$ width, together with the forthcoming Belle~II analyses of $e^+e^-\to\mu^+\mu^-+{\rm invisible}$, can rule out a $Z^\prime$ explanation of $b\to s\ell^+\ell^-$ data with a mass below $\approx4\,$GeV. Importantly, such a light $Z^\prime$ is the only viable single particle solution to the $b\to s\ell^+\ell^-$ anomalies predicting $R(K^{(*)})>0$ in high $q^2$ bins, therefore providing an essential consistency test of data.


I. INTRODUCTION
In recent years, multiple hints for the violation of lepton flavour universality (LFU), which is satisfied by the Standard Model (SM) gauge interactions, have been accumulated (see Refs. [1,2] for recent reviews). Among them, the discrepancies between b → s + − data and the corresponding SM predictions are statistically most significant (see Refs. [3][4][5] for an overview). Combining the current measurements of the LFU ratios R(K ( * ) ) one observes that several new physics (NP) scenarios are statistically preferred over the SM hypothesis with significances close to 5σ [6][7][8][9][10]. Furthermore, global analyses including also muon-specific observables, like the branching ratio of B s → φµ + µ − and angular observables in B → K * µ + µ − , like P 5 , show preferences compared to the SM hypothesis with pulls of up to even more than 7σ, depending on theoretical assumptions and data included in the fits [6][7][8][9][11][12][13].
Because the b → s + − anomalies constitute such tantalizing hints for NP, a plethora of SM extensions have been proposed in the literature, including lepto-quarks , models with loop effects of new scalars and fermions [43][44][45][46][47][48][49] and in particular models with new neutral gauge bosons, i.e. Z s . Because all these solutions, except the Z one, involve charged particles they must be realized at the electroweak scale, or even significantly above, due to the constraints from direct searches. However, a Z boson can be light and, in fact, such solutions to the b → s + − anomalies have been proposed and studied in the literature [108][109][110][111][112][113][114][115]. Importantly, this is the only NP model addressing the b → s + − anomalies which predicts R(K ( * ) ) > 0 in the high q 2 bins, i.e. above the charm resonances, where precise experimental data is still missing but expected in the near future.
While a light Z explanation of b → s + − data is experimentally well constrained, it still remains viable if it is assumed that the Z decays dominantly to invisible final states. This avoids direct searches such as e + e − → 4µ and provides the sizable width necessary for the Z to affect multiple q 2 bins in b → s + − observables and thus give a good fit to data. However, also processes with invisible final states are constrained experimentally, such as the di-muon invariant mass distribution in Drell-Yan production close to the Z mass [113] or e + e − → µ + µ − +invisible at Belle II [116].
In this letter, we analyze these processes together with B → K ( * ) νν [117] using a proper treatment of the Z contribution, including the effects of its large width and the mass of the invisible states it decays to. We assess the possibility that a light Z can in fact be responsible for the b → s + − anomalies and how this option can be tested in the future with the forthcoming BELLE II analyses.

II. SETUP AND OBSERVABLES
As outlined in the introduction, we supplement the SM by a light neutral gauge boson, i.e. a Z , with a mass below the B-meson mass scale (m Z 6 GeV). We will be agnostic about the origin of this new state and simply parametrize its couplings necessary to explain b → s + − data by the Lagrangian We solely include couplings to muons because the ones to electrons are not necessary to explain the b → s + − anomalies and are experimentally well constrained, in particular when they appear simultaneously with muon couplings. Furthermore, we do not consider couplings to muon neutrinos as they are very stringently constrained by neutrino trident production. Note that this is possible even for left-handed muon couplings because we assume our model to be realized below the EW symmetry breaking scale such that SU (2) L invariance is not necessarily obeyed by the Z couplings. Furthermore, in order to achieve the large width necessary to affect multiple q 2 = s bins in b → s + − observables such that a good fit to data is possible, we will assume that the Z has a large decay rate to invisible final states χ with m χ < m Z /2. As the couplings tosb andμµ turn out to be small, we will assume that the branching ratio to invisible final states is to a good approximation 100%. Furthermore, for specificity χ is taken to be a fermion with vectorial couplings to the Z . 1 Using the standard parametrization of semi-leptonic B decays, the effect of a light Z can be described by a q 2 dependent contribution to the effective Wilson coefficients defined at the B meson scale. Here Γ Z is the width of the light vector boson which we approximate here to be q 2 independent. For the phenomenological analysis, we implemented these contributions into flavio [118] to perform the global fit of our model to data. This includes e.g. the measurement of the LFU ratios R K [119], R K ( * ) [120], R K 0 S [121] and R K * + [121], as well as the branching ratio for B s → µ + µ − [122][123][124], the angular observables of B → K * µ + µ − [125] and the branching ratio and angular distribution of B s → φµ + µ − [126,127] which exhibit the most significant deviations from SM predictions.
The most important constraints on Z − b − s couplings, in case the Z decays dominantly to invisible final states, can be obtained from the processes B → K ( * ) νν measured most precisely at BaBar [128] and Belle [117]. However, only the latest Belle II analysis [129] with the bound provides the necessary q 2 dependence of the experimental efficiency necessary to easily recast it in terms of the decay B + → K + χχ. 2 In the case of a large Z width, the branching ratio B(B → K ( * ) χχ) can be approximated by . In these expressions we have kept the s-dependence of the Z width obtained from our fermionic model of the dark sector, gives the desired width Γ Z . The reason for keeping the s-dependence is that it can affect significantly the limits obtained from B → K ( * ) νν searches for large m χ .
With the SM predictions for the differential decay width dΓ(B + → K + νν)/dq 2 [131], the relevant form factors [132] and the experimental efficiency function reported by Belle II [133], we can translate Eq. (3) into a limit on our Z model, given the masses m Z and m χ as well as the width Γ Z . The experimental signal efficiency [133] is shown in Fig. 1 together with the form factor, the Breit-Wigner distribution of the Z and the squared matrix element of the amplitude (excluding the form factor). The resulting branching ratio is obtained by integrating the product, starting at s min . As the amplitude and the efficiency function increase at small s = q 2 , the bounds on g sb are stronger in case of a large width compared to a narrow one.
Finally, let us remark that B → Kχχ is only sensitive to the vector current g L sb + g R sb such that data from B → K * χχ would be required to probe the axial-vector coupling g R sb − g L sb . However, the former process is sufficient to constraint the NP scenarios needed to explain the b → s + − anomalies as right-handed bs couplings are bounded by the fits to data.

C. Bs −Bs mixing
Tree-level exchange of the Z contributes to B s −B s mixing. For light Z masses one can set up an operator product expansion in m Z /m b to calculate this new physics contribution to the mixing amplitude and obtain bounds on g sb [130]. However, these limits turn out the be much weaker than the ones from B → K ( * ) νν. While in principle for higher Z masses there could be a (close to) resonant enhancement, it is not clear how to calculate these effects reliably and we will therefore not use B s −B s mixing as a constraint in our analysis.
The anomalous magnetic moment of the muon receives 1-loop corrections from the Z . With the results given e.g. in Ref. [134] we find This expression has to be compared with the experimental value [135,136] and the SM prediction [137] 3 , resulting in ∆a µ = a exp µ − a SM µ = 251(59) × 10 −11 .
Ref. [113] pointed out that Drell-Yan (DY) searches for muon pairs at the LHC place relevant limits on the parameter space. The Z can be radiated from the final state muons and significantly modify the di-muon invariant mass distribution close the the Z pole. It is found that for a Z mass between 1 − 5 GeV the muon coupling should be smaller than ≈ 0.1 in case of a dominant branching ratio to invisible.
The Belle II experiment released a search of invisible Z decays in the process e + e − → µ + µ − +invisible [116] using the commissioning run data. Although limited by the size of the data sample analyzed (276 pb −1 ), 90% confidence level limits on the coupling g V µµ of the order of 10 −2 − 10 −1 were obtained. Belle II has also provided sensitivity projections for this model for integrated luminosities up to 50 fb −1 [163]; in addition, we obtain projections of the sensitivity up to 5 ab −1 by accounting for a scaling factor equivalent to L 1/4 . While Ref. [116] gives bounds on the vectorial coupling, the cross section for e + e − → µ + µ − + Z scales as (g V µµ ) 2 + (g A µµ ) 2 and thus can be easily adjusted to the case of other chiralities. Note that the analysis of Ref. [116] was done for a Z with a narrow width. We therefore recasted the analysis such that it applies to our case with a sizable Z width by recalculating the expected signal yield in each bin of the original analysis, assuming a Breit-Wigner with Γ Z = 0.1M Z (Fig.3 left) and Γ Z = 0.15M Z (Fig.3 right) convoluted with a Gaussian resolution function for the signal. We then set up a binned likelihood fit and used the profile likelihood ratio method to extract the 90% C.L. intervals.

III. PHENOMENOLOGY
First of all, as already noted in Ref. [108], a sizable width for the Z is necessary such that it does not only affect a single bin of P 5 , R(K ( * ) ) etc. This can be achieved by assuming that the Z decays dominantly into invisible final states χ which at the same time avoids constraints from searches like e + e − → 4µ. Recasting the B + → K + νν analysis of Belle II the limits on g L sb for a 100% branching ratio to undetected final states 4 are shown in Fig. 2. In this plot we see that a large m χ ≤ m Z /2 weakens the bound on g L sb such that for 2m 2 χ 15GeV 2 no limit can be obtained because the experimental sensitivity vanishes.
Let us now turn to the couplings of the Z to leptons. For purely vectorial couplings, the bound from (g − 2) µ would be so strong that it would exclude a Z explanation of b → s + − . However, the effect vanishes for g V = − √ 5g A . As this scenario (i.e. C eff 9 = − √ 5C eff 10 ) gives a good fit to b → s + − data (as any scenario between the limiting cases C 9 and C 9 = −C 10 ) we will use it as a benchmark scenario here. Note that if we choose g V slightly bigger, we could account for the tension in (g − 2) µ while leaving the b → s + − fit unchanged to a very good approximation.
Performing the b → s + − fit under these assumptions, we have three free parameters, m Z , Γ Z and g L sb × g V µµ (with g V µµ = − √ 5g A µµ ). First of all, note that a width ≈ 15% gives the best fit to data with ∆χ 2 = χ 2 − χ 2 SM ≈ 40 which is however still smaller than what can be achieved with heavy NP that give a q 2 independent effect in the same scenario. Furthermore, ∆χ 2 does not change significantly for 0.1m Z < Γ Z < 0.2m Z .
In order to minimize the effect in direct searches for the Z (i.e. DY and e + e − → µ + µ − invisible), given that it provides an explanation to b → s + − data, we can assume that g L sb takes its maximal value allowed by B + → K + νν. The resulting regions preferred by b → s + − data in the m Z and g V µµ = − √ 5g A µµ plane are shown in Fig. 3. From there, one can see that the constraints from DY searches at the LHC and e + e − → µ + µ − +invisible cannot exclude a light Z explanation of the b → s + − anomalies, yet. However, the forthcoming Belle II analysis of e + e − → µ + µ − +invisible has the potential of excluding a mass below 4 GeV depending on m χ and the width of the Z . Alternatively, we can use the e + e − → µ + µ − +invisible to derive an upper limit on g V µµ = − √ 5g A µµ and show the exclusions from B + → K + νν in the m Z -g L sb as depicted in Fig. 4, were the 50 fb −1 prospects of Belle II have been used. Note, that for a width of 15% a Z with 4 GeV < m Z < 4.5 GeV gives a good fit to data and cannot be excluded due to the vanishing experimental sensitivity in B + → K + νν for a DM mass close to one half of m Z . However, a Z with such a mass would not lead to R(K ( * ) ) > 1 in the high q 2 bins above the J/Ψ resonances.

IV. CONCLUSIONS AND OUTLOOK
In this letter we pointed out that a light Z explanation (with a mass below 4 GeV) of the b → s + − anomalies can be confirmed or disproved by combining the forthcoming Belle II searches for e + e − → µ + µ + +invisible and B → K ( * ) νν. Concerning the latter, it is imperative to properly take into account the sizable Z width and the experimental efficiencies. This endeavour is very important to limit the number of viable models addressing b → s + − , in particular in the absence of a signal in direct searches. Furthermore, a light Z is the only remaining viable NP explanation of b → s + − for which the high q 2 bin (above the charm resonances) in e.g. R(K ( * ) ) could lie above unity (assuming that the situation in the low q 2 bins remains unchanged). While a light Z with a mass between ≈ 4 − 6 GeV, that enhances the SM amplitude at high q 2 , cannot be excluded for m χ ≈ m Z /2 due to the limited experimental sensitivity of the B → K ( * ) νν analysis to low energetic K ( * ) , this gap could be closed in the future, e.g. with a reliable calculation of B s −B s mixing for such Z masses. Belle II 50 fb −1 proj.
3: Preferred regions in the m Z − g V µµ plane from b → s + − (whole data-set, green) and the fit to the LFU ratios R(K) and Bs → µ + µ − (red) at the 1σ , 2σ and 3σ level for g V µµ = − √ 5 g A µµ assuming that g L sb takes it maximally allowed value from B → Kνν for different Z widths and χ masses. The regions above the solid lines are excluded by the current DY (cyan) and e + e − → µ + µ − invisible searches (blue) while the dashed lines indicate future sensitivities. Note that a smaller width and a larger χ mass lead to weaker constraints on the model. assuming that g V µµ = − √ 5 g A µµ takes its maximally allowed value allowed by the 50fb −1 sensitivity of Belle II for e + e − → µ + µ − invisible. The regions above the lines, depending on the width and mχ, can be excluded by future B → Kνν bounds.

Acknowledgments
We thank Luc Schnell for checking the heavy NP fit for the g V µµ = − √ 5 g A µµ scenario. Support by the Swiss National Science Foundation of the Project B decays and lepton flavour universality violation (PP00P21 76884) is gratefully acknowledged by A. Crivellin  For the decay width of a B meson into K ( * ) and a vector of invariant mass s, the operator product expansion analysis of Refs. [132,164] gives