Interpreting Hints for Lepton Flavor Universality Violation

TL;DR

This paper analyzes recent experimental hints of lepton flavor universality violation in rare B meson decays using a model-independent approach, identifying potential new physics regions and suggesting further measurements.

Contribution

It provides a model-independent analysis of new physics explanations for lepton flavor universality violation in B decays and proposes specific experimental directions.

Findings

Identifies parameter space consistent with experimental anomalies.

Shows compatibility with other B decay anomalies.

Suggests new physics phenomena below 100 TeV scale.

Abstract

We interpret the recent hints for lepton flavor universality violation in rare $B$ meson decays. Based on a model-independent effective Hamiltonian approach, we determine regions of new physics parameter space that give a good description of the experimental data on $R_K$ and $R_{K^*}$, which is in tension with Standard Model predictions. We suggest further measurements that can help narrowing down viable new physics explanations. We stress that the measured values of $R_K$ and $R_{K^*}$ are fully compatible with new physics explanations of other anomalies in rare $B$ meson decays based on the $b \to s \mu\mu$ transition. If the hints for lepton flavor universality violation are first signs of new physics, perturbative unitarity implies new phenomena below a scale of $\sim 100$ TeV.

Tables

Table 1. Table 1: Best-fit values and pulls for scenarios with NP in one individual Wilson coefficient, taking into account only LFU observables.

Coeff.	best fit	$1\sigma$	$2\sigma$	pull
$C_9^{\mu }$	$-1.56$	[$-2.12$, $-1.10$]	[$-2.87$, $-0.71$]	$4.1\sigma$
$C_{10}^{\mu }$	$+1.20$	[$+0.88$, $+1.57$]	[$+0.58$, $+2.00$]	$4.2\sigma$
$C_9^e$	$+1.54$	[$+1.13$, $+1.98$]	[$+0.76$, $+2.48$]	$4.3\sigma$
$C_{10}^e$	$-1.27$	[$-1.65$, $-0.92$]	[$-2.08$, $-0.61$]	$4.3\sigma$
$C_9^{\mu }=-C_{10}^{\mu }$	$-0.63$	[$-0.80$, $-0.47$]	[$-0.98$, $-0.32$]	$4.2\sigma$
$C_9^e=-C_{10}^e$	$+0.76$	[$+0.55$, $+1.00$]	[$+0.36$, $+1.27$]	$4.3\sigma$
$C_9^e=C_{10}^e$	$-1.91$	[$-2.30$, $-1.51$]	[$-2.71$, $-1.10$]	$3.9\sigma$
$C_9^{\prime \mu }$	$-0.05$	[$-0.31$, $+0.21$]	[$-0.57$, $+0.46$]	$0.2\sigma$
$C_{10}^{\prime \mu }$	$+0.03$	[$-0.21$, $+0.27$]	[$-0.44$, $+0.51$]	$0.1\sigma$
$C_9^{\prime e}$	$+0.07$	[$-0.21$, $+0.37$]	[$-0.49$, $+0.69$]	$0.2\sigma$
$C_{10}^{\prime e}$	$-0.04$	[$-0.30$, $+0.21$]	[$-0.57$, $+0.45$]	$0.2\sigma$

Equations

$$R_K=\frac{\mathcal{B}(B\to K{\mu }^{+}{\mu }^{-})}{\mathcal{B}(B\to K{e}^{+}{e}^{-})},R_{K^{\ast }}=\frac{\mathcal{B}(B\to K^{\ast }{\mu }^{+}{\mu }^{-})}{\mathcal{B}(B\to K^{\ast }{e}^{+}{e}^{-})}.$$

$$R_K^{[1,6]}=0.745_{-0.074}^{+0.090}\pm 0.036,$$

$$R_{K^{\ast }}^{[0.045,1.1]}$$

$$R_{K^{\ast }}^{[1.1,6]}$$

$${\mathcal{H}}_{\text{eff}}^{\text{NP}}=-\frac{4G_F}{\sqrt{2}}{V}_{tb}{V}_{ts}^{\ast }\frac{e^2}{16{\pi }^2}\sum _{i,\ell }(C_i^{\ell }{O}_i^{\ell }+C_i^{\prime \ell }{O}_i^{\prime \ell })+\text{h.c.},$$

$$O_9^{\ell }$$

$$O_{10}^{\ell }$$

$$C_9^{\mu }-C_9^e-C_{10}^{\mu }+C_{10}^e\simeq -1.4,$$

$$D_{P_4^{\prime }}$$

$$D_{P_5^{\prime }}$$

$${\Lambda }_i=\frac{4\pi }{e}\frac{1}{\sqrt{|V_{tb}{V}_{ts}^{\ast }|}}\frac{1}{\sqrt{|C_i|}}\frac{v}{\sqrt{2}}\simeq \frac{35\text{TeV}}{\sqrt{|C_i|}}.$$

Full text

Introduction.

The wealth of data on rare leptonic and semi-leptonic $b$ hadron decays that has been accumulated at the LHC so far allows the Standard Model (SM) Cabibbo-Kobayashi-Maskawa picture of flavor and CP violation to be tested with unprecedented sensitivity. Interestingly, current data on rare $b\to s\ell\ell$ decays show an intriguing pattern of deviations from the SM predictions both for branching ratios Aaij et al. (2014a, 2016a, 2015) and angular distributions Aaij et al. (2016b); Wehle et al. (2017). The latest global fits find that the data consistently points with high significance to a non-standard effect that can be described by a four fermion contact interaction $C_{9}\,(\bar{s}\gamma^{\nu}P_{L}b)(\bar{\mu}\gamma_{\nu}\mu)$ Altmannshofer et al. (2017a) (see also earlier studies Altmannshofer and Straub (2015); Descotes-Genon et al. (2016); Hurth et al. (2016)). Right now the main obstacle towards conclusively establishing a beyond-SM effect is our inability to exclude large hadronic effects as the origin of the apparent discrepancies (see e.g. Jäger and Martin Camalich (2016); Descotes-Genon et al. (2014); Lyon and Zwicky (2014); Ciuchini et al. (2016); Capdevila et al. (2017); Chobanova et al. (2017); Blake et al. (2017)).

In this respect, observables in $b\to s\ell\ell$ transitions that are practically free of hadronic uncertainties are of particular interest. Among them are lepton flavor universality (LFU) ratios, i.e. ratios of branching ratios involving different lepton flavors such as Hiller and Kruger (2004); Bobeth et al. (2007); Hiller and Schmaltz (2015)

[TABLE]

In the SM, the only sources of lepton flavor universality violation are the leptonic Yukawa couplings, which are responsible for both the charged lepton masses and their interactions with the Higgs.111Neutrino masses provide another source of lepton flavor non-universality, but the effects are negligible here. Higgs interactions do not lead to any observable effects in rare $b$ decays and lepton mass effects become relevant only for a very small di-lepton invariant mass squared close to the kinematic limit $q^{2}\sim 4m_{\ell}^{2}$. Over a very broad range of $q^{2}$ the SM accurately predicts $R_{K}=R_{K^{*}}=1$, with theoretical uncertainties of $O(1\%)$ Bordone et al. (2016). Deviations from the SM predictions can be expected in various models of new physics (NP), e.g. $Z^{\prime}$ models based on gauged $L_{\mu}-L_{\tau}$ Altmannshofer et al. (2014); Crivellin et al. (2015a); Altmannshofer and Yavin (2015); Crivellin et al. (2017a) or other gauged flavor symmetries Crivellin et al. (2015b); Celis et al. (2015); Falkowski et al. (2015); Bhatia et al. (2017); Ko et al. (2017), models with partial compositeness Niehoff et al. (2015); Carmona and Goertz (2016); Megias et al. (2016); Garcia Garcia (2017), and models with leptoquarks Hiller and Schmaltz (2014); Gripaios et al. (2015); de Medeiros Varzielas and Hiller (2015); Bauer and Neubert (2016); Bečirević et al. (2016a, b); Barbieri et al. (2017); Cox et al. (2017); Crivellin et al. (2017b).

A first measurement of $R_{K}$ by the LHCb collaboration Aaij et al. (2014b) in the di-lepton invariant mass region $1\,\text{GeV}^{2}<q^{2}<6\,\text{GeV}^{2}$,

[TABLE]

shows a $2.6\sigma$ deviation from the SM prediction. Very recently, LHCb presented first results for $R_{K^{*}}$ Bifani ; Bifani (2017); Aaij et al. (2017a),

[TABLE]

where the superscript indicates the di-lepton invariant mass bin in GeV2. These measurements are in tension with the SM at the level of $2.4$ and $2.5\sigma$, respectively. Intriguingly, they are in good agreement with the recent $R_{K^{*}}$ predictions in Altmannshofer et al. (2017a) that are based on global fits of $b\to s\mu\mu$ decay data, assuming $b\to see$ decays to be SM-like.

In this letter we interpret the $R_{K^{(*)}}$ measurements using a model-independent effective Hamiltonian approach (see Hurth et al. (2014); Glashow et al. (2015); Ghosh et al. (2014); Bhattacharya et al. (2015); Alonso et al. (2015); Greljo et al. (2015); Bečirević et al. (2015) for earlier model independent studies of $R_{K}$). We also include Belle measurements of LFU observables in the $B\to K^{*}\ell^{+}\ell^{-}$ angular distibutions Wehle et al. (2017). We do not consider early results on $R_{K^{(*)}}$ from BaBar Lees et al. (2012) and Belle Wei et al. (2009) which, due to their large uncertainties, have little impact. We identify the regions of NP parameter space that give a good description of the experimental data. We show how future measurements can lift flat directions in the NP parameter space and discuss the compatibility of the $R_{K^{(*)}}$ measurements with other anomalies in rare $B$ meson decays.

Model independent implications for new physics.

We assume that NP in the $b\to s\ell\ell$ transitions is sufficiently heavy such that it can be model-independently described by an effective Hamiltonian, $\mathcal{H}_{\text{eff}}=\mathcal{H}_{\text{eff}}^{\text{SM}}+\mathcal{H}_{\text{eff}}^{\text{NP}}$,

[TABLE]

with the following four-fermion contact interactions,

[TABLE]

and the corresponding Wilson coefficients $C_{i}^{\ell}$, with $\ell=e,\mu$. We do not consider other dimension-six operators that can contribute to $b\to s\ell\ell$ transitions. Dipole operators and four-quark operators Jäger et al. (2017) cannot lead to violation of LFU and are therefore irrelevant for this work. Four-fermion contact interactions containing scalar currents would be a natural source of LFU violation. However, they are strongly constrained by existing measurements of the $B_{s}\to\mu\mu$ and $B_{s}\to ee$ branching ratios Aaij et al. (2017b); Aaltonen et al. (2009). Imposing $SU(2)_{L}$ invariance, these bounds cannot be avoided Alonso et al. (2014). We have checked explicitly that $SU(2)_{L}$ invariant scalar operators cannot lead to any appreciable effects in $R_{K^{(*)}}$ (cf. Altmannshofer et al. (2017b)).

For the numerical analysis we use the open source code flavio Straub et al. (2017). Based on the experimental measurements and theory predictions for the LFU ratios $R_{K^{(*)}}$ and the LFU differences of $B\to K^{*}\ell^{+}\ell^{-}$ angular observables $D_{P_{4,5}^{\prime}}$ (see below), we construct a $\chi^{2}$ function that depends on the Wilson coefficients and that takes into account the correlations between theory uncertainties of different observables. We use the default theory uncertainties in flavio, in particular $B\to K^{*}$ form factors from a combined fit to light-cone sum rule and lattice results Bharucha et al. (2016). The experimental uncertainties are presently dominated by statistics, so their correlations can be neglected. For the SM we find $\chi^{2}_{\text{SM}}=24.4$ for 5 degrees of freedom.

Tab. 1 lists the best fit values and pulls, defined as the $\sqrt{\Delta\chi^{2}}$ between the best-fit point and the SM point for scenarios with NP in one individual Wilson coefficient. The plots in Fig. 1 show contours of constant $\Delta\chi^{2}\approx 2.3,6.2,11.8$ in the planes of two Wilson coefficients for the scenarios with NP in $C_{9}^{\mu}$ and $C_{10}^{\mu}$ (top), in $C_{9}^{\mu}$ and $C_{9}^{e}$ (center), or in $C_{9}^{\mu}$ and $C_{9}^{\prime\,\mu}$ (bottom), assuming the remaining coefficients to be SM-like.

The fit prefers NP in the Wilson coefficients corresponding to left-handed quark currents with high significance $\sim 4\sigma$. Negative $C_{9}^{\mu}$ and positive $C_{10}^{\mu}$ decrease both $\mathcal{B}(B\to K\mu^{+}\mu^{-})$ and $\mathcal{B}(B\to K^{*}\mu^{+}\mu^{-})$ while positive $C_{9}^{e}$ and negative $C_{10}^{e}$ increase both $\mathcal{B}(B\to Ke^{+}e^{-})$ and $\mathcal{B}(B\to K^{*}e^{+}e^{-})$, allowing a good description of the data in each case. Also along the direction $C_{9}^{\ell}=-C_{10}^{\ell}$ that corresponds to a left-handed lepton current, we find excellent fits to the data. As can be seen from the top plot in Fig. 1, right-handed muon currents ($C_{9}^{\mu}=C_{10}^{\mu}$) cannot describe the data. We find that right-handed electron currents can explain $R_{K}$ and $R_{K^{*}}$ if the corresponding Wilson coefficients are sizable ($C_{9}^{e}=C_{10}^{e}\sim-2$ or $+3$).

The primed Wilson coefficients, that correspond to right-handed quark currents, cannot improve the agreement with the data by themselves. As is well known Hiller and Schmaltz (2015), the primed coefficients imply $R_{K^{*}}>1$ given $R_{K}<1$ and vice versa. The complementary sensitivity of $R_{K^{*}}$ and $R_{K}$ to right-handed currents is illustrated in the bottom plot of Fig. 1 for the example of $C_{9}^{\mu}$ vs. $C_{9}^{\prime\,\mu}$. In combination with sizable un-primed coefficients, the primed coefficients can slightly improve the fit.

Among the un-primed Wilson coefficients, there are approximate flat directions. We find that a good description of the experimental results is given by

[TABLE]

unless some of the individual coefficients are much larger than $1$ in absolute value. The flat direction is clearly visible in the top and center plot of Fig. 1. In many NP models one has relations among these coefficients. In models with leptoquarks one finds $C_{9}^{\ell}=\pm C_{10}^{\ell}$ Hiller and Schmaltz (2014); Buras et al. (2015), models based on gauged $L_{\mu}-L_{\tau}$ predict $C_{9}^{e}=C_{10}^{\ell}=0$ Altmannshofer et al. (2014), while in some $Z^{\prime}$ models one finds $C_{9}^{\ell}=aC_{10}^{\ell}$, where $a$ is a constant of $\mathcal{O}(1)$ (see e.g. Buras and De Fazio (2016)).

We find that a non-standard $C_{10}^{\ell}$ ($C_{9}^{\ell}$) leads to slightly larger (smaller) effects in $R_{K^{*}}$ than in $R_{K}$. Therefore, $R_{K^{*}}\lesssim R_{K}<1$ is best described by a non-standard $C_{10}^{\ell}$. The opposite hierarchy, $R_{K}\lesssim R_{K^{*}}<1$, would lead to a slight preference for NP in $C_{9}^{\ell}$.

A more powerful way to distinguish NP in $C_{9}^{\ell}$ from NP in $C_{10}^{\ell}$ is through measurements of LFU differences of angular observables Altmannshofer and Yavin (2015); Capdevila et al. (2016); Serra et al. (2017). We find that the observables

[TABLE]

are particularly promising (for a definition of the observables $P_{4,5}^{\prime}$ see Descotes-Genon et al. (2013)). Predictions for the observables $D_{P_{4,5}^{\prime}}$ as functions of $q^{2}$ in the SM and various NP scenarios are shown in the plots of Fig. 2. The SM predictions are close to zero with very high accuracy across a wide $q^{2}$ range. In the presence of NP, $D_{P_{4,5}^{\prime}}$ show a non-trivial $q^{2}$ dependence. If the discrepancies in $R_{K^{(*)}}$ are explained by NP in $C_{9}^{\ell}$, we predict a negative $D_{P_{4}^{\prime}}\sim-0.1$ at low $q^{2}\lesssim 2.5$ GeV2 and a sizable positive $D_{P_{5}^{\prime}}\sim+0.5$. With NP in $C_{10}^{\ell}$ we predict instead a positive $D_{P_{4}^{\prime}}\sim+0.15$ and a small negative $D_{P_{5}^{\prime}}\sim-0.1$. We observe that $D_{P_{5}^{\prime}}$ has even the potential to distinguish between NP in $C_{9}^{e}$ and $C_{9}^{\mu}$. For $q^{2}\gtrsim 5$ GeV2, a negative $C_{9}^{\mu}$ leads to a sizable increase of $P_{5}^{\prime}(B\to K^{*}\mu^{+}\mu^{-})$, while a positive $C_{9}^{e}$ can decrease $P_{5}^{\prime}(B\to K^{*}e^{+}e^{-})$ only slightly, as the SM prediction for $P_{5}^{\prime}$ in this $q^{2}$ region is already close to its model-independent lower bound of $-1$. The recent measurements by Belle, $D_{P_{4}^{\prime}}^{[1,6]}=+0.498\pm 0.553$ and $D_{P_{5}^{\prime}}^{[1,6]}=+0.656\pm 0.496$ Wehle et al. (2017), have still sizable uncertainties and are compatible with NP both in $C_{9}^{\ell}$ and in $C_{10}^{\ell}$. They slightly favor NP in $C_{9}^{\ell}$. We note that, while the SM prediction for these observables has a tiny uncertainty, for fixed values of LFU violating Wilson coefficients, form factor and other hadronic uncertainties do play a role, as also shown in Fig. 2. However, these uncertainties are still so small that sufficient experimental precision could allow a clean identification of the underlying NP contact interaction.

We stress that the NP contact interactions in (5) lead also to a characteristic $q^{2}$ shape in the LFU ratios $R_{K^{(*)}}$. In Fig. 3 we show $R_{K^{(*)}}$ as functions of $q^{2}$ in the SM and in the same NP scenarios as in Fig. 2. In the SM, $R_{K^{(*)}}$ are to an excellent approximation $q^{2}$ independent. For very low $q^{2}\simeq 4m_{\mu}^{2}$ they drop to zero, due to phase space effects. NP contact interactions lead to an approximately constant shift in $R_{K}$. The ratio $R_{K^{*}}$, on the other hand, shows a non-trivial $q^{2}$ dependence in the presence of NP. In contrast to $B\to K\ell\ell$, the $B\to K^{*}\ell\ell$ decays at low $q^{2}$ are dominated by the photon pole, which gives a lepton flavor universal contribution. The effect of NP is therefore diluted at low $q^{2}$. Given the current experimental uncertainties, the measured $q^{2}$ shape of $R_{K^{*}}$ is compatible with NP in form of a contact interaction. Significant discrepancies from the shapes shown in Fig. 3 would imply the existence of light NP degrees of freedom around or below the scale set by $q^{2}$ and a breakdown of the effective Hamiltonian framework.

Assuming that the description in terms of contact interactions holds, we translate the best fit values of the Wilson coefficients into a generic NP scale. Reparameterizing the effective Hamiltonian (5) as $\mathcal{H}_{\text{eff}}^{\text{NP}}=-\sum_{i}\mathcal{O}_{i}/\Lambda_{i}^{2}$, one gets

[TABLE]

Based on perturbative unitarity we therefore predict the existence of NP degrees of freedom below a scale of $\Lambda_{\text{NP}}\sim\sqrt{4\pi}\times 35~{}\text{TeV}/\sqrt{|C_{i}|}\sim 100~{}\text{TeV}$.

Compatibility with other rare $B$ decay anomalies.

It is natural to connect the discrepancies in $R_{K^{(*)}}$ to the other existing anomalies in rare decays based on the $b\to s\mu\mu$ transition. In the plots of Fig. 1 we show in dotted gray the 1, 2, and 3$\sigma$ contours from our global $b\to s\mu\mu$ fit that does not take into account the measurements of the LFU observables $R_{K^{(*)}}$ and $D_{P_{4,5}^{\prime}}$ Altmannshofer et al. (2017a). We observe that the blue regions prefered by the LFU observables are fully compatible with the $b\to s\mu\mu$ fit. We have also performed a full fit, taking into account all the observables from the $b\to s\mu\mu$ fit, the branching ratio of $B_{s}\to\mu^{+}\mu^{-}$ (assuming it not to be affected by scalar NP contributions), and the BaBar measurement of the $B\to X_{s}e^{+}e^{-}$ branching ratio Lees et al. (2014). This fit, shown in red, points to a non-standard $C_{9}^{\mu}\simeq-1.2$ with very high significance. Wilson coefficients other than $C_{9}^{\mu}$ are constrained by the global fit.

Compared to the LFU observables, the global $b\to s\mu\mu$ fit depends more strongly on estimates of hadronic uncertainties in the $b\to s\ell\ell$ transitions. To illustrate the impact of a hypothetical, drastic underestimation of these uncertainties, we also show results of a global fit where uncertainties of non-factorisable hadronic contributions are inflated by a factor of 5 with respect to our nominal estimates. In this case, the global fit becomes dominated by the LFU observables, but the $b\to s\mu\mu$ observables still lead to relevant constraints. For instance, the best-fit value for $C_{10}^{\mu}$ in Tab. 1 would imply a 50% suppresion of the $B_{s}\to\mu^{+}\mu^{-}$ branching ratio, which is already in tension with current measurements Aaij et al. (2017b), barring cancellations with scalar NP contributions.

Conclusions.

The discrepancies between SM predictions and experimental results in the LFU ratios $R_{K}$ and $R_{K^{*}}$ can be explained by NP four-fermion contact interactions $(\bar{s}b)(\bar{\ell}\ell)$ with left-handed quark currents. Future measurements of LFU differences of $B\to K^{*}\ell^{+}\ell^{-}$ angular observables can help to identify the chirality structure of the lepton currents. If the hints for LFU violation in rare $B$ decays are first signs of NP, perturbative unitarity implies new degrees of freedom below a scale of $\Lambda_{\text{NP}}\sim 100$ TeV. These results are robust, i.e. they depend very mildly on assumptions about the size of hadronic uncertainties in the $B\to K^{(*)}\ell^{+}\ell^{-}$ decays.

Intriguingly, the measured values of $R_{K}$ and $R_{K^{*}}$ are fully compatible with NP explanations of various additional anomalies in rare $B$ meson decays based on the $b\to s\mu\mu$ transition. A combined fit singles out NP in the Wilson coefficient $C_{9}^{\mu}$ as a possible explanation.

Acknowledgments

WA acknowledges financial support by the University of Cincinnati. The work of PS and DS was supported by the DFG cluster of excellence “Origin and Structure of the Universe”.