$R_K$ and $R_{K^{\ast}}$ beyond the Standard Model

TL;DR

Recent measurements of $R_K$ and $R_{K^*}$ suggest lepton universality violation, indicating potential new physics beyond the Standard Model, with leptoquarks as a plausible explanation testable at the LHC.

Contribution

This paper analyzes the implications of $R_K$ and $R_{K^*}$ anomalies, proposing leptoquark models as explanations and predicting testable signatures at the LHC.

Findings

Leptoquark models can explain $R_K$ and $R_{K^*}$ anomalies at tree level.

The anomalies favor SM-like chiral new physics contributions.

Predicted leptoquark masses are within reach of current collider experiments.

Abstract

Measurements of the ratio of $B \to K^* \mu \mu $ to $B \to K^* e e $ branching fractions, $R_{K^*}$, by the LHCb collaboration strengthen the hints from previous studies with pseudoscalar kaons, $R_K$, for the breakdown of lepton universality, and therefore the Standard Model (SM), to $\sim 3.5 \sigma$. Complementarity between $R_K$ and $R_{K^*}$ allows to pin down the Dirac structure of the new contributions to be predominantly SM-like chiral, with possible admixture of chirality-flipped contributions of up to ${\cal{O}}( \mbox{few} 10 \%)$. Scalar and vector leptoquark representations $(S_3,V_1,V_3)$ plus possible ($\tilde S_2,V_2$) admixture can explain $R_{K,K^*}$ via tree level exchange. Flavor models naturally predict leptoquark masses not exceeding a few TeV, with couplings to third generation quarks at $O(0.1)$, implying that this scenario can be directly tested at the LHC.

Equations

$$R_H=\frac{\mathcal{B}(B\to H{\mu }^{+}{\mu }^{-})}{B(B\to H{e}^{+}{e}^{-})},H=K,K^{\ast },X_s,\dots$$

$$R_K^{\text{LHCb}}$$

$$R_{K^{\ast }}^{\text{LHCb}}$$

$${\mathcal{H}}_{\text{eff}}=-\frac{4G_F{\lambda }_t}{\sqrt{2}}\frac{\alpha }{4\pi }\sum _i{C}_i^{\ell }{\mathcal{O}}_i^{\ell }+\text{h.c.},$$

$$\begin{array}{rl}{\mathcal{O}}_9^{\ell }& =(\bar{s}{\gamma }^{\mu }{P}_{L}b)(\bar{\ell }{\gamma }_{\mu }\ell ),{\mathcal{O}}_9^{\prime \ell }=(\bar{s}{\gamma }^{\mu }{P}_{R}b)(\bar{\ell }{\gamma }_{\mu }\ell ),\\ {\mathcal{O}}_{10}^{\ell }& =(\bar{s}{\gamma }^{\mu }{P}_{L}b)(\bar{\ell }{\gamma }_{\mu }{\gamma }_5\ell ),{\mathcal{O}}_{10}^{\prime \ell }=(\bar{s}{\gamma }^{\mu }{P}_{R}b)(\bar{\ell }{\gamma }_{\mu }{\gamma }_5\ell ),\end{array}$$

$${\mathcal{O}}_{AB}^{\ell }=(\bar{s}{\gamma }^{\mu }{P}_{A}b)(\bar{\ell }{\gamma }_{\mu }{P}_B\ell ),A,B=L,R,$$

$$\begin{array}{rl}{C}_9^{\ell }& =\frac{1}{2}(C_{LL}^{\ell }+C_{LR}^{\ell }),C_{10}^{\ell }=\frac{1}{2}(C_{LR}^{\ell }-C_{LL}^{\ell }),\\ C_9^{\prime \ell }& =\frac{1}{2}(C_{RL}^{\ell }+C_{RR}^{\ell }),C_{10}^{\prime \ell }=\frac{1}{2}(C_{RR}^{\ell }-C_{RL}^{\ell }).\end{array}$$

$$\begin{array}{rl}{R}_K& =1+{\Delta }_{+}+{\Sigma }_{+},\\ R_{K^{\ast }}& =1+{\Delta }_{+}+{\Sigma }_{+}+p({\Sigma }_{-}-{\Sigma }_{+}+{\Delta }_{-}-{\Delta }_{+}),\end{array}$$

$$\begin{split}\Delta_{\pm}&=2\Re\bigg{(}\frac{C^{\rm NP\,\mu}_{LL}\pm C^{\mu}_{RL}}{C^{\rm SM}_{LL}}-(\mu\rightarrow e)\bigg{)}\,,\\ \Sigma_{\pm}&=\frac{|C^{\rm NP\,\mu}_{LL}\pm C^{\mu}_{RL}|^{2}+|C^{\mu}_{LR}\pm C^{\mu}_{RR}|^{2}}{|C^{\rm SM}_{LL}|^{2}}-(\mu\rightarrow e).\end{split}$$

$$X_{K^{\ast }}=R_{K^{\ast }}/R_K$$

$$R_{K^{\ast }}+R_K-2$$

$$\mathrm{Re}[C_9^{\mathrm{NP}\mu }-C_{10}^{\mathrm{NP}\mu }-(\mu \to e)]$$

$$\mathrm{Re}[C_9^{\prime \mu }-C_{10}^{\prime \mu }-(\mu \to e)]$$

$$C_{LL}^{\text{NP}\ell }=$$

$$C_{RL}^{\ell }=$$

$$\frac{Y_{b\mu }{Y}_{s\mu }^{\ast }-Y_{be}{Y}_{se}^{\ast }}{M^2}\simeq \frac{1.1}{(35\text{TeV}{)}^2},(S_3)$$

$$\frac{Y_{b\mu }{Y}_{s\mu }^{\ast }-Y_{be}{Y}_{se}^{\ast }}{M^2}\simeq \frac{-0.1}{(24\text{TeV}{)}^2}.({\tilde{S}}_2)$$

$$Y_{q_3\ell }\sim c_l,Y_{q_2\ell }\sim c_l{\lambda }^2,q_3=b,t,\text{leavevmode}{q}_2=s,c,$$

$$M\lesssim 40\text{mbox}TeV,45\text{mbox}TeV,20\text{mbox}TeV\text{mbox}for\text{leavevmode}{S}_3,V_1,V_3.$$

$${\mathcal{H}}_{\text{eff}}^{\Delta B=2}=(C_1^{SM}+C_1^{LQ})(\bar{b}{\gamma }_{\mu }(1-{\gamma }_5)s)(\bar{b}{\gamma }_{\mu }(1-{\gamma }_5)s)+\text{h.c.}$$

$$C_1^{LQ}=\frac{p_{LQ}(Y{Y}^{\ast }{)}^2}{128{\pi }^2{M}^2},\text{leavevmode}{p}_{LQ}=5,4,20\text{mbox}for{S}_3,V_1,V_3,$$

$$\Delta m_{B_s}^{\text{LQ}}/\Delta m_{B_s}^{SM}=\frac{p_{LQ}(Y{Y}^{\ast }{)}^2}{8M^2{G}_F^2{m}_W^2{\lambda }_t^2{S}_0(x_t)},$$

$${\mathcal{L}}_{QQ}=Y_{\kappa }{\bar{Q}}_L^{C\alpha }(i{\sigma }^2{)}^{\alpha \beta }(S_3^{†}{)}^{\beta \gamma }{Q}_L^{\gamma }+\text{h.c.},$$

$$R_{X_s}\sim 0.73\pm 0.07,$$

Full text

††preprint: DO-TH 17/04

Introduction.  Gauge interactions of the leptons within the Standard Model (SM) exhibit exact universality. The only known source of lepton non-universality (LNU) are the Yukawa couplings of the leptons to the Higgs. Tests of lepton universality are provided by rare (semi)leptonic $|\Delta B|=|\Delta S|=1$ transitions, which are induced in the SM at one loop and additionally suppressed by the Glashow-Iliopoulos-Maiani mechanism, therefore allowing to probe physics from scales significantly higher than the weak scale. Useful observables are the ratios of branching fractions of $B$ meson decays into strange hadrons $H$ and muon pairs over electron pairs Hiller:2003js

[TABLE]

in which (lepton universal) hadronic effects cancel. The ratios are therefore predicted within the SM to be very close to one and provide a clean test of the SM Hiller:2003js .

The LHCb collaboration measured $R_{K}$ in the dilepton invariant mass squared ($q^{2}$) bin $1\,\text{GeV}^{2}\leq q^{2}\leq 6\,\text{GeV}^{2}$ using the $1\,\text{fb}^{-1}$ data set Aaij:2014ora

[TABLE]

and, very recently, $R_{K^{\ast}}$ within $1.1\,\text{GeV}^{2}\leq q^{2}\leq 6\,\text{GeV}^{2}$ LHCb2017

[TABLE]

with deviation from $R=1$ by $2.6\,\sigma$ each. (Here and in the following we add statistical and systematic uncertainties in quadrature.) Corrections to $R=1+O(m_{\mu}^{2}/m_{B}^{2})$ Hiller:2003js arise from electromagnetic interactions Huber:2005ig ; Bobeth:2007dw ; Huber:2007vv ; Huber:2015sra . This affects the SM prediction at low $q^{2}$ at percent level Bordone:2016gaq , not qualitatively altering the fact that the data, (2) and (3) constitute a challenge to universality, and the SM.

Moreover, the importance of the measurement of $R_{K^{*}}$ in addition to $R_{K}$ is in its diagnosing power regarding different beyond the SM (BSM) contributions Hiller:2014ula . Left-handed and right-handed $b\to s$ currents enter $B\to K\ell\ell$ and $B\to K^{*}\ell\ell$ in almost orthogonal combinations in both regions of $q^{2}$ sensitive to LNU. Comparison of $R_{K}$ with $R_{K^{*}}$, for instance through a double ratio $X_{K^{\ast}}=R_{K^{\ast}}/R_{K}$ Hiller:2014ula , probes directly right-handed LNU currents. The aim of this paper is to exploit this model-independently and pursue interpretations within leptoquark extensions of the SM.

Model-independent interpretation.  We employ the usual effective Hamiltonian for $b\to s\ell\ell$, $\ell=e,\mu,\tau$ transitions

[TABLE]

where $C^{\ell}_{i},\mathcal{O}^{\ell}_{i}$ denote lepton-specific Wilson coefficients and dimension-six operators, respectively, renormalized at the scale $\mu\sim m_{b}$. Furthermore, $G_{F}$, $\alpha$, and $\lambda_{t}=V_{tb}V^{*}_{ts}$ stand for Fermi’s constant, the finestructure constant and the product of relevant Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, respectively. The semileptonic operators read

[TABLE]

with chiral projectors $P_{L,R}=1/2(1\mp\gamma_{5})$. The operators with chiral lepton currents,

[TABLE]

are related to the ${\cal{O}}_{9,10}^{(\prime)\ell}$ as

[TABLE]

Within the SM the (lepton universal) Wilson coefficients are $C^{\text{SM}}_{9}=4.07$, $C^{\text{SM}}_{10}\simeq-4.31$ global fits , thus $C^{\rm SM}_{LL}=C^{\text{SM}}_{9}-C^{\text{SM}}_{10}\simeq 8.4$, while scalar or tensor Wilson coefficients are negligible. We define $C^{\ell}_{LL}=C^{\rm SM}_{LL}+C^{\text{NP}\ell}_{LL}$ Hiller:2014ula and drop the label ”NP” (new physics) for Wilson coefficients negligible within the SM.

In the $B\to K^{(\ast)}\ell\ell$ branching fractions contributions from photon exchange enter, notably from charm loops and dipole operators. These contributions are numerically small at high and low $q^{2}$, sufficiently away from the photon pole, and lepton universal. Within current accuracy of $R_{K,K^{*}}$ these contributions can be safely neglected. In this limit Hiller:2014ula

[TABLE]

where

[TABLE]

Since BSM contributions in $|\Delta B|=|\Delta S|=1$ transitions are smaller than the SM ones global fits the dominant BSM effect is captured by the linear (interference) terms $\Delta_{\pm}$.

The coefficient $p$ in Eq. (8) denotes the fraction of transverse parallel and longitudinal contributions to the $B\to K^{\ast}\ell\ell$ branching ratio Hiller:2014ula . Due to helicity arguments $p\sim 1$ both at low recoil (high $q^{2}$) and at low $q^{2}$. Consequently, ${\cal{B}}(B\to K^{\ast}\ell\ell)$ is dominated by contributions proportional to $|C-C^{\prime}|^{2}$. Since ${\cal{B}}(B\to K\ell\ell)\propto|C+C^{\prime}|^{2}$ due to parity invariance of the strong interaction both modes are complementary and deviations of $R_{K}$ from $R_{K^{*}}$ probe primed operators Hiller:2014ula .

Using (2),(3) one obtains

[TABLE]

which gives, at $1\sigma$

[TABLE]

As anticipated, $|C^{\rm NP}|\ll|C^{\rm SM}|$. Therefore, the linear approximation, that is, neglecting the $\Sigma_{\pm}$-terms, is meaningful within the current experimental precision. Dropping quadratic terms greatly simplifies the interpretation of the data: Only BSM in $O_{LL}^{\ell}$ or $O_{RL}^{\ell}$ is able to explain $R_{K,K^{*}}$.

In Fig. 1 a $\chi^{2}$ fit for the left- and right-handed Wilson coefficients is shown. The discrepancy with the SM is about $\sim 3.5\sigma$, where we allowed for a few percent deviations from $R=1$ Bordone:2016gaq .

Interestingly, solutions with $C_{9}^{{\rm NP}\mu}\sim-1$ are also favored by a global fit global fits to $b\to s\mu\mu$ observables. Taking this into account suggests an explanation of $R_{K,K^{*}}$ anomalies with BSM predominantly residing in the muons.

Leptoquark explanations.  We consider leptoquark extensions of the SM with tree level couplings to down-type quarks and leptons. Representations under $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ with relevant Wilson coefficients are given in Table 1 for scalar $S_{i}$111In the literature the scalar leptoquarks $S_{2}$ and $\tilde{S}_{2}$ are also denoted by $R_{2}$ and $\tilde{R}_{2}$, respectively. and in Table 2 for vector leptoquarks $V_{i}$, respectively. The index $i=1,2,3$ refers to the dimension of the $SU(2)_{L}$ multiplet, see e.g. Kosnik:2012dj ; Hiller:2016kry ; Dorsner:2016wpm for overviews.

The scalar leptoquarks $S_{2}$ and $\tilde{S}_{1}$ generate only $C_{LR}$ and $C_{RR}$, respectively, which do not interfere with the SM contribution, see Eq. (9), and lead to $R_{K},R_{K^{\ast}}$ near 1. We therefore discard these two possibilities as explanations of the $R_{K,K^{\ast}}$ anomalies.

In view of the experimental constraints shown in Fig. 1 we focus on leptoquarks that can give a sizable $C^{\text{NP}\ell}_{LL}=2C_{9}^{\text{NP}\ell}=-2C_{10}^{\text{NP}\ell}$. This singles out the scalar triplet $S_{3}$, the vector singlet $V_{1}$ and the vector triplet $V_{3}$. This scalar and the vectors have been considered as a possible explanation of $R_{K}$ (2) in  Hiller:2014yaa ; Gripaios:2014tna ; Hiller:2016kry ; Barbieri:2015yvd ; Varzielas:2015iva and in Barbieri:2015yvd ; Fajfer:2015ycq ; Becirevic:2016oho ; Alonso:2015sja ; Calibbi:2015kma ; Hiller:2016kry , respectively. Subdominant contributions from right-handed currents can be provided by additional leptoquarks $\tilde{S}_{2}$ or $V_{2}$, which induce $C_{RL}^{\ell}=2C_{9}^{\prime\ell}=-2C_{10}^{\prime\ell}$. In these models Dorsner:2016wpm ; Hiller:2016kry

[TABLE]

Here, $M$ ($Y$) denotes the leptoquark mass (coupling).

Model-independent and leptoquark specific predictions for $R_{K}$ versus $R_{K^{*}}$ are shown in Fig. 2. The green and blue band denote the $1\sigma$ band of $R_{K}$ (2) and $R_{K^{*}}$ (3), respectively. Also shown are BSM scenarios which can (red solid and dashed lines) or cannot (blue dotted and gray dashed lines) simultaneously explain the data. Concretely, leptoquark $\tilde{S}_{2}$, corresponding to the blue dotted curve, and which has been considered in the context of $R_{K}$ Hiller:2014yaa ; Sahoo:2015wya ; Becirevic:2015asa ; Cox:2016epl , is disfavoured as the sole source of LNU by the measurement of $R_{K^{*}}$. The numerics are based on the full expressions for the decay rates, for $\ell=\mu$. Recall, however, to linear approximation only non-universality matters.

We find for the dominant, SM-like chiral contribution $S_{3}$

[TABLE]

and similarly for $V_{1}$ or $V_{3}$. To accommodate an admixure of right-handed currents we need contributions from another leptoquark, such as $\tilde{S}_{2}$

[TABLE]

Understanding the mass range is linked to flavor. The leptoquark coupling matrix $Y$ is a $3\times 3$ matrix in generation space, with rows corresponding to quark flavor and columns corresponding to lepton flavor. The presence of both kinds of fermions in one vertex is beneficial; it allows to probe flavor in new ways beyond SM fermion masses and mixings. Viable models are those employing a Froggatt-Nielsen $U(1)_{FN}$ to generate mass hierarchies for quarks and charged leptons together with a discrete, non-abelian group such as $A_{4}$, which allows to accommodate neutrino properties Altarelli:2010gt ; Varzielas:2010mp . Applied to leptoquark models this allows to select lepton species – for instance having only couplings to one lepton species, muons, or electrons Varzielas:2015iva . Corrections to lepton isolation arise from rotations to the mass basis and at higher order in the spurion expansion, and induce lepton flavor violation Glashow:2014iga ; Becirevic:2016zri ; Varzielas:2015iva ; Hiller:2016kry ; Duraisamy:2016gsd ; Crivellin:2017zlb such as $B\to K\mu\tau$, which can be probed with $B$-physics experiments but also $\mu-e$-converison, rare $K$ and $\ell\to\ell^{\prime}$ decays. Together with $B\to K^{(*)}\nu\bar{\nu}$ modes the latter constitute the leading constraints on flavor models and LNU anomalies, and improved experimental study is promising.

A generic prediction for $S_{3},V_{1},V_{3}$ – all of them couple quark doublets to lepton doublets– is obtained from simple flavor patterns such as $\ell$-isolation, $\ell=e,\mu$, Varzielas:2015iva ; Hiller:2016kry

[TABLE]

where $c_{l}\sim\lambda\sim 0.2$. Note that the FN-mechanism is only able to explain parametric suppressions in specific powers of the parameter $\lambda$ up to numbers of order one. Irrespective of the concrete flavor symmetry, each coupling $Y$ to lepton doublets brings in a non-abelian spurion insertion suppression, the factor $c_{l}$, which is unavoidable as lepon doublets are necessarily charged under the non-abelian group to obtain a viable PMNS-matrix. The suppression of the additional couplings to right-handed leptons in $V_{1,2}$ can be achieved using flavor symmetries Alonso:2015sja ; Hiller:2016kry .

Putting lepton and neutrino properties aside, a minimal prediction is $Y_{s\ell}/Y_{b\ell}\sim m_{s}/m_{b}$, hence $Y_{b\ell}Y_{s\ell}^{*}\sim\lambda^{2}\simeq\mbox{ few}\times 0.01$. Eq. (16) implies $M\sim 5-10$ TeV, accessible at the LHC at least partly with single production.

Eq. (18) points to lower values of leptoquark masses, see Fig. 3. Also shown are constraints from $B_{s}-\bar{B}_{s}$ mixing, induced at one loop through box diagrams and which constrains the square of $YY^{*}$ over $M^{2}$ Hiller:2014yaa . A data-driven upper limit, irrespective of flavor, is obtained as

[TABLE]

We assume vector leptoquarks to be gauge-like and employ the usual Hamiltonian

[TABLE]

where

[TABLE]

see, e.g Davidson:1993qk . In general, $(YY^{*})^{2}\to\sum_{\ell_{i},\ell_{j}}(Y_{b\ell_{i}}Y^{\ast}_{s\ell_{i}})(Y_{b\ell_{j}}Y^{\ast}_{s\ell_{j}})$.

It follows that

[TABLE]

where $S_{0}$ is an Inami-Lim function, $x_{t}=m_{t}^{2}/m_{W}^{2}$. We use $\Delta m_{B_{s}}^{exp}/\Delta m_{B_{s}}^{SM}=1.02\pm 0.10$ Olive:2016xmw ; Becirevic:2016zri .

Direct limits for scalar leptoquarks decaying 100 % into a muon (electron) and a jet are are $M>1050$ GeV Aaboud:2016qeg ($M>1755$ GeV Khachatryan:2015qda ). For vector leptoquarks, the limits are model-dependent and read $M>1200-1720$ GeV ($M>1150-1660$ GeV) for 100 % decays to muon (electron) plus jet Khachatryan:2015vaa . The bounds weaken if decays into neutrinos are taken into account.

UV considerations.  The main challenge for embedding light scalar leptoquarks into (complete) short-distance models is proton decay. From Table 1 only $S_{2},\tilde{S}_{2}$ do not couple to quark bilinears $(\bar{q}q)$ and, thus, do not induce proton decay at tree-level. In addition, dangerous couplings to the Higgs doublet should be suppressed Arnold:2012sd ; Dorsner:2016wpm .

SM gauge-invariance allows $S_{3}$ to couple to quark billinears

[TABLE]

with isospin indices $\alpha,\beta,\gamma$. The Yukawa coupling $Y_{\kappa}$ is anti-symmetric in flavor space Vecchi:2011ab ; Dorsner:2012nq , thus $S_{3}$ does not introduce proton decay at tree level, however, couplings to $ut(c)$ can induce the process via higher orders diagrams Dorsner:2012nq .

Within flavor models, the dangerous terms in (23) receive suppressions. For $U(1)_{FN}\times A_{4}\times Z_{3}$, and assuming that the quarks transform trivially under $A_{4}$, we find that this requires at least 2 spurion fields $\xi$ and $\xi^{\prime}$, see Varzielas:2015iva for details. Including the FN-suppression for the up-quark this amounts to $\lambda^{4}\kappa\kappa^{\prime}\lesssim 10^{-4}$. Viable patterns for $R_{K}$ are obtained with second generation quarks in non-trivial representations of $A_{4}$. This way, however, the $ut$ coupling cannot be suppressed further.

If the evidence for LNU in $C^{\text{NP}}_{LL}$ strengthens, it would be important to understand the origin of the leptoquark $S_{3}$ which provides an explicit high-energy realization. One possibility was suggested in Ref. Gripaios:2014tna . The $S_{3}$ appears, along with the Higgs doublet, as a pseudo-Goldstone boson of the strong dynamics around TeV scale, while the proton decay is avoided with a discrete symmetry.

An alternative possibility is to trace the origin of $S_{3}$ to a Grand Unified Theory (GUT). The $S_{3}$ is contained in the $\overline{126}_{H}$ scalar multiplet of $SO(10)$ Slansky:1981yr ; Dorsner:2016wpm . The dangerous couplings to quark billinears are forbidden by $SO(10)$-invariance - the corresponding Yukawa coupling to fermion multiplets is $y_{ij}\,16_{i}\,16_{j}\,\overline{126}_{H}$ which embeds only the couplings to leptons and quarks, but not to quark billinears. The $16_{i}$ denotes the spinor representation of $SO(10)$ that unifies all SM fermions of a single generation and a right-handed neutrino, and $i=1,2,3$ is a flavor index. The $S_{3}$ might play a role in correcting the phenomenologically unsuccessful prediction of the relation between the mass matrices of down-type quarks and charged leptons in the minimal $SU(5)$ Georgi:1979df ; Georgi:1974sy .

Vector leptoquarks appear as super-heavy gauge bosons in a GUT, with masses near the unification scale. For example, the state with quantum numbers of $V_{1}$ is a gauge boson in models of quark-lepton unification, e.g. the original Pati-Salam model or variants thereof, see Perez:2013osa . $V_{1},V_{3}$ do not couple to quark billinears and are safe with regards to proton decay. If $V_{1}$ is a gauge boson, the corresponding left- and right-handed couplings are unitary. It is then more difficult to suppress the unwanted (right-handed) couplings and simultaneously avoid the constraints from the first generation fermions. The embedding of $V_{3}$ into a UV complete model is challenging Biggio:2016wyy .

The low scale non-gauge spin-1 leptoquarks might be obtained as composite states from strongly coupled dynamics, in which case they are accompanied by other composite states.

Conclusions.  The recent measurement of $R_{K^{*}}$ (3) by the LHCb collaboration challenges lepton universality, an inbuilt feature of the SM and many of its extensions, further: combined with $R_{K}$ (2) the discrepancy with the SM is $\sim 3.5\sigma$. The LNU contributions to $|\Delta B|=|\Delta S|=1$ FCNCs are predominantly SM-like chiral, with possible admixture of right-handed contributions up to the order of few $10\%$, see Fig. 1. Since $R_{K}$ and $R_{K^{*}}$ suffice to determine the chiral structure, measurements of further LNU ratios $R_{H}$ into different final states and angular distributions Hiller:2014ula ; Wehle:2016yoi provide consistency checks. Using $R_{K,K^{*}}$ data we predict the ratio of inclusive $B\to X_{s}\ell\ell$ branching fractions

[TABLE]

consistent with earlier findings by Belle $R_{X_{s}}=0.42\pm 0.25$ iijimaLP09 and BaBar $R_{X_{s}}=0.58\pm 0.19$ Lees:2013nxa 222The BaBar collaboration finds an excess of electrons relative to the SM prediction, especially in the lowest $q^{2}$-bin. .

Leptoquarks naturally induce LNU in semileptonic decays at tree level. The scalar $S_{3}$ and the vector $V_{1,3}$ representations can account for the dominant, SM-like chiral contribution (12). Their masses are limited to not exceed the multi-10 TeV range in order to comply with data, see (19) for details. Leptoquark explanations of $R_{K,K^{*}}$ within flavor models, which simultaneously address the masses and mixings of SM fermions require leptoquark masses in the few TeV-region, which can be explored at the LHC, see Fig. 3. The dominant decay modes of the triplets $S_{3}$ and $V_{3}$ are $b\mu,t\mu,b\nu$ and $t\nu$, whereas the $SU(2)_{L}$-singlet $V_{1}$ decays predominantly to $b\mu,t\nu$. The respective Yukawa couplings are at the level $O(0.1)$. Ignoring the pull from the global fit to $b\to s\mu\mu$ LNU can also stem from sizable BSM contributions to $b\to see$. In this case modes into final state electrons (and corresponding neutrinos) are dominant.

Acknowledgements.  This project is supported in part by the Bundesministerium für Bildung und Forschung (BMBF). We thank Ivo de Medeiros Varzielas for useful discussions.