# Lepton flavor non-universality in $b\to s\ell^{+}\ell^{-}$ processes

Damir Bečirević<EMAIL_ADDRESS>Laboratoire de Physique
Théorique, CNRS/Université Paris-Sud 11 (UMR 8627), 91405 Orsay, France
Svjetlana Fajfer<EMAIL_ADDRESS>Department of Physics, University of
Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia  J. Stefan Institute, Jamova
39, P. O. Box 3000, 1001 Ljubljana, Slovenia  Nejc Košnik<EMAIL_ADDRESS>Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana,
Slovenia  J. Stefan Institute, Jamova 39, P. O. Box 3000, 1001 Ljubljana,
Slovenia

(August 27, 2024)

###### Abstract

We explore a scenario of New Physics entering the description of $B\to
K^{(\ast)}\mu\mu$ decay through couplings to the operators ${\cal
O}_{9,10}^{\prime}$, satisfying $C_{9}^{\prime}=-C_{10}^{\prime}$. From the
current data on ${\cal B}(B_{s}\to\mu\mu)$ and ${\cal B}(B\to
K\mu\mu)_{[15,22]{\rm GeV}^{2}}$, we obtain constraints on
$\mathrm{Re}C_{10}^{\prime}$ and $\mathrm{Im}C_{10}^{\prime}$ which we then
assume to be lepton specific, and find $R_{K}={\cal B}(B\to K\mu\mu)/{\cal
B}(B\to Kee)_{[1,6]{\rm GeV}^{2}}=0.88(8)$, consistent with recent value
measured at LHCb. A specific realization of this scenario is the one with a
scalar leptoquark state $\Delta$, in which $C_{10}^{\prime}$ is related to the
mass of $\Delta$ and its Yukawa couplings. We then show that this scenario
does not make any significant impact on $B_{s}-\overline{B}_{s}$ mixing
amplitude nor to ${\cal B}(B\to K\nu\bar{\nu})$. Instead, it can modify
$R_{K^{\ast}}={\cal B}(B\to K^{\ast}\mu\mu)/{\cal B}(B\to
K^{\ast}ee)_{[1,6]{\rm GeV}^{2}}$, which will soon be experimentally measured
and we find it to be $R_{K^{\ast}}=1.11(8)$, while
$R_{K^{\ast}}/R_{K}=1.27(19)$. A similar ratio of forward-backward asymmetries
also becomes lower than in the Standard Model.

###### pacs:

13.20.He,12.60.-i,14.80.Sv

††preprint: LPT-Orsay-15-25

##  I Introduction

The $b\to s$ transitions were in the focus of many theoretical and
experimental studies during the last two decades due to the possibility to
constrain potential New Physics (NP) contributions at low energies. With LHC7
and LHC8 runs direct searches for NP became available. This gives us an
excellent opportunity to question the appearance of physics beyond Standard
Model (SM). At low energies $B$-factories and the LHCb experiment provided
flavor physics community with a lot of rather precise results on $b\to s$
transitions. The LHCb experiment has observed slight discrepancies between the
SM predictions and the experimental results for the angular observables in
$B\to K^{\ast}\mu^{+}\mu^{-}$ decay. This effect has been attributed to NP,
although the tension might be a result of the SM QCD effects. Recently,
another anomaly in $b\to s\ell^{+}\ell^{-}$ transition has been found in the
ratio of the branching fractions,

$R_{K}=\frac{\mathcal{B}(B\to
K\mu^{+}\mu^{-})_{q^{2}\in[1,6]\mathrm{\leavevmode\nobreak\
GeV^{2}}}}{\mathcal{B}(B\to
Ke^{+}e^{-})_{q^{2}\in[1,6]\mathrm{\leavevmode\nobreak\ GeV^{2}}}}\,.$ (1)

LHCb Collaboration measured this ratio for the square of dilepton invariant
mass in the bin $1$ GeV${}^{2}\leq q^{2}\leq 6$ GeV2, and found Aaij et al.
(2014a),

$R_{K}^{\mathrm{LHCb}}=0.745\pm^{0.090}_{0.074}\pm 0.036\,,$ (2)

lower than the SM prediction, $R_{K}^{SM}=1.0003\pm 0.0001$, in which next-to-
next-to-leading QCD corrections have been included Hiller and Kruger (2004).
In other words, the LHCb result points towards a $2.6\ \sigma$ effect of the
lepton flavor universality violation.

Furthermore, the combined data analysis of the $B_{s}\to\mu^{+}\mu^{-}$ events
gathered at LHCb and CMS resulted in
$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})=(2.8^{+0.7}_{-0.6})\times 10^{-9}$
Khachatryan et al. (2015), in good agreement with the SM prediction
$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})=(3.65\pm 0.23)\times 10^{-9}$ Bobeth et
al. (2014). This offers an excellent probe of $b\to s\mu^{+}\mu^{-}$
transition in the light of SM and gives rather tight constraints on parameter
space of many models of NP. The $R_{K}$ anomaly has been approached in the
literature in different ways: either by using the effective Lagrangian
approach or in a specific model of NP. For example the effective Lagrangian
approach used in references Alonso et al. (2014); Hiller and Schmaltz (2014);
Glashow et al. (2015); Bhattacharya et al. (2015) indicated that in order to
understand the measured value of $R_{K}$ one must include the effects of NP,
and that the effects of non-perturbative QCD alone could not explain such a
large deviation of $R_{K}$ from unity Hiller and Schmaltz (2014); Kosnik
(2012); Ghosh et al. (2014); Biswas et al. (2015); Hurth et al. (2014);
Glashow et al. (2015); Altmannshofer and Straub (2014); Sahoo and Mohanta
(2015); Varzielas and Hiller (2015); Bhattacharya et al. (2015); Gripaios et
al. (2014); Jäger and Martin Camalich (2013); Descotes-Genon et al. (2013a);
Altmannshofer and Straub (2013); Crivellin et al. (2015); Aristizabal et al.
(2015). In particular, it was found that the NP contribution most likely
affects $C_{9}$, $C_{10}$ or $C_{9}^{\prime}$, $C_{10}^{\prime}$ effective
Wilson coefficients, and that some kind of lepton flavor universality
violation is needed, e.g. $C_{9}^{\mu}\neq C_{9}^{e}$ Hiller and Schmaltz
(2015); Ghosh et al. (2014). In order to determine whether $R_{K}$ anomaly is
due to NP in electron or/and muon couplings through a combined analysis of
several decay modes, it is very important to have a high precision knowledge
of hadronic form factors Gripaios et al. (2014); Jäger and Martin Camalich
(2013); Descotes-Genon et al. (2013a), which can be computed in the region of
large $q^{2}$’s by means of numerical simulations of QCD on the lattice
Becirevic et al. (2012); Bouchard et al. (2013); Horgan et al. (2014).

In this study we first use a model independent approach, assuming that NP
contributes at low energies to an operator that is a product of a right-handed
quark and a left-handed muon current. In the language of $b\to s\mu\mu$
effective Hamiltonian such a situation corresponds to a combination of Wilson
coefficients $C_{9}^{\prime}$ and $C_{10}^{\prime}$, and that they obey
$C_{9}^{\prime}=-C_{10}^{\prime}$. Decays to the final states with electron-
positron pair are instead governed by the SM only. This assumption is
motivated by the fact that measured quantities of $b\to se^{+}e^{-}$ processes
agree with the SM predictions better than they do for the $b\to
s\mu^{+}\mu^{-}$ processes Hurth et al. (2014), which are also more precisely
measured than the electronic modes. We consider simultaneously the constraints
posed by ${\cal B}(B\to K\mu^{+}\mu^{-})$ and ${\cal
B}(B_{s}\to\mu^{+}\mu^{-})$ on such a scenario, and then predict the $R_{K}$
as well as $R_{K^{\ast}}$. We discuss other observables which might serve as
additional probes of the observed lepton-flavor universality violation.

A specific realization of the scenario we discuss in this paper is a model
with a light scalar leptoquark $\Delta$ with quantum numbers of
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ being $(3,2,1/6)$. It indeed
verifies the relation, $C_{9}^{\prime}=-C_{10}^{\prime}$ Kosnik (2012), and
leads to a consistency with the measured value of $R_{K}$. The features of
this leptoquark state have been already described in the literature Dorsner et
al. (2014). While there is no theoretical motivation to forbid leptoquark
contributing to $b\to see$ decays, simultaneous presence of both muonic and
electronic couplings could be problematic because they would, together, induce
lepton flavor violation in $B_{s}\to e\mu$ and $\mu\to e\gamma$ decays. It is
interesting that the flavor physics constraints at low energies agree and are
complementary with the constraints obtained from the direct experimental
searches at LHC Aad et al. (2012); CMS (2013). Furthermore, the atomic parity
violation experiments provided a strong constraint on the interaction of the
down-quark–electron interaction with the leptoquark state Dorsner et al.
(2014); Gresham et al. (2012), while the couplings to muons appear to be less
constrained via ${\cal B}(K_{L}\to\mu^{\pm}e^{\mp})<4.7\times 10^{-12}$ Olive
et al. (2014); Dorsner et al. (2014). We therefore assume in our analysis that
in the $b\to s\ell^{+}\ell^{-}$ processes only the muons can interact with the
leptoquark state. A few other leptoquark states have been discussed in the
literature Hiller and Schmaltz (2014); Sahoo and Mohanta (2015); Kosnik
(2012); Gripaios et al. (2014) as possible candidates to contribute to the
$R_{K}$ anomaly. However, the leptoquark with quantum numbers $(3,2,1/6)$ has
a desired feature that it can be light without destabilizing the proton
Weinberg (1980); Buchmuller and Wyler (1986); Dorsner et al. (2012). Notice
also that another light leptoquark scalar state, not mediating the proton
decay, is $(3,2,7/6)$ and it leads to the relation $C_{9}=C_{10}$. That latter
scenario, however, cannot explain the $R_{K}$ anomaly as discussed in Refs.
Hiller and Schmaltz (2014); Sahoo and Mohanta (2015).

In Sec. II we remind the reader of the main definitions and give basic
expressions for ${\cal B}(B_{s}\to\mu^{+}\mu^{-})$ and ${\cal B}(B\to
K\mu^{+}\mu^{-})$, which are then used, together with the experimental data in
Sec. III, to constraint $C_{10}^{\prime}=-C_{9}^{\prime}$ and show the
consistency of our value for $R_{K}$ with the measured one at LHCb.
Furthermore, we make a prediction of the similar ratio in the case of $B\to
K^{\ast}\ell^{+}\ell^{-}$ decays and discuss other observables that might be
of interest for testing the lepton flavor universality violation. In Sec. IV
we discuss a model with scalar leptoquark in which the relation
$C_{10}^{\prime}=-C_{9}^{\prime}$ holds exactly, and is connected to other
similar processes involving the $b\to s$ transitions which we also discuss. We
finally summarize our findings in Sec. V.

##  II Effective Hamiltonian and basic formulas

The processes with flavor structure $(\bar{s}b)\,(\bar{\mu}\mu)$ at scale
$\mu=\mu_{b}=4.8$ GeV are governed by dimension-6 effective Hamiltonian
Grinstein et al. (1989); Misiak (1993); Buras and Munz (1995):

$\mathcal{H}_{\mathrm{eff}}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\left[\sum_{i=1}^{6}C_{i}(\mu)\mathcal{O}_{i}(\mu)+\sum_{i=7,\ldots,10}\left(C_{i}(\mu)\mathcal{O}_{i}(\mu)+C^{\prime}_{i}(\mu)\mathcal{O}^{\prime}_{i}(\mu)\right)\right]\,.$
(3)

The contributions of the charged-current operators $\mathcal{O}_{1,2}$, QCD
penguins $\mathcal{O}_{3,\ldots,6}$, and the electromagnetic (chromomagnetic)
dipole operators $\mathcal{O}_{7}\,(\mathcal{O}_{8})$ will be assumed to be
saturated by the SM. On the other hand, operators involving a quark and a
lepton current will contain the SM and potential NP contributions. The basis
of operators may be further extended to account for possible (pseudo)scalar or
tensor operators Becirevic et al. (2012), whereas for the purposes of this
work the following operators will suffice:

$\begin{split}\mathcal{O}_{7}&=\frac{e}{g^{2}}m_{b}(\bar{s}\sigma_{\mu\nu}P_{R}b)F^{\mu\nu}\,,\qquad\mathcal{O}_{8}=\frac{1}{g}m_{b}(\bar{s}\sigma_{\mu\nu}G^{\mu\nu}P_{R}b)\,,\\\
\mathcal{O}_{9}&=\frac{e^{2}}{g^{2}}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}\gamma^{\mu}\ell)\,,\qquad\quad\\!\mathcal{O}_{10}=\frac{e^{2}}{g^{2}}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}\gamma^{\mu}\gamma_{5}\ell)\,.\end{split}$
(4)

Here $P_{L/R}=(1\mp\gamma_{5})/2$, while $e$ is the electromagnetic and $g$
the color gauge coupling. $F^{\mu\nu}$ and $G^{\mu\nu}$ are the
electromagnetic and color field strength tensors, respectively. The basis is
further extended by the wrong-chirality operators,
$\mathcal{O}^{\prime}_{9,10}$, which are related to $\mathcal{O}_{9,10}$ by
replacing $P_{L}\leftrightarrow P_{R}$ in the quark current.

###  II.1 $B\to K\mu^{+}\mu^{-}$

In calculating the amplitude for the $B\to K\mu^{+}\mu^{-}$ decay it is
convenient to group the combinations of Wilson coefficients multiplying the
same hadronic matrix element. Namely, the operators $\mathcal{O}_{1-6}$ mix at
leading order into $\mathcal{O}_{7,8,9}$ and it is customary to define
effective Wilson coefficients as Buras et al. (1994):

$\begin{split}C_{7}^{\rm
eff}(\mu_{b})&=\frac{4\pi}{\alpha_{s}}\,C_{7}-\frac{1}{3}\,C_{3}-\frac{4}{9}\,C_{4}-\frac{20}{3}\,C_{5}\,-\frac{80}{9}\,C_{6}\,,\\\
C_{9}^{\rm eff}(\mu_{b})&=\frac{4\pi}{\alpha_{s}}\,C_{9}+Y(q^{2})\,,\\\
C_{10}^{\rm eff}(\mu_{b})&=\frac{4\pi}{\alpha_{s}}\,C_{10}\,,\qquad
C_{7,8,9,10}^{\prime\mathrm{eff}}(\mu_{b})=\frac{4\pi}{\alpha_{s}}\,C^{\prime}_{7,8,9,10}\,,\end{split}$
(5)

where the function $Y(q^{2})$ at NLL can be found in Ref. Altmannshofer et al.
(2009a). We also incorporate the NNLL mixing of $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$ into $\mathcal{O}_{7}$ and $\mathcal{O}_{9}$ as calculated
in Ref. Greub et al. (2008). The Wilson coefficients on the right-hand sides
are evaluated at $\mu=\mu_{b}$. For the sake of readability we will from here
on discuss only the effective Wilson coefficients that will be addressed
simply as “Wilson coefficients” and denoted without the “eff” label. The
values of the SM Wilson coefficients at scale $\mu_{b}$ are $C_{7}=-0.304$,
$C_{9}=4.211$, and $C_{10}=-4.103$ Buras et al. (1994); Bobeth et al. (2000);
Altmannshofer et al. (2009a).

The decay spectrum as a function of the invariant mass of the muon pair is
given by

$\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}(B\to
K\mu^{+}\mu^{-})=2a_{\mu}(q^{2})+\frac{2}{3}c_{\mu}(q^{2})\,,$ (6)

where $q^{2}=(p_{\mu^{-}}+p_{\mu^{+}})^{2}$, while functions $a_{\mu}(q^{2})$,
$c_{\mu}(q^{2})$ are combinations of Wilson coefficients and hadronic form
factors and their explicit expressions can be found in Ref. Becirevic et al.
(2012) and in the Appendix of the present paper in the limit of $m_{\ell}\to
0$. The rate depends on the sums of the Wilson coefficients of opposite
chiralities, $C_{7}+C_{7}^{\prime}$, $C_{9}+C_{9}^{\prime}$,
$C_{10}+C_{10}^{\prime}$, from what follows that even in principle we cannot
determine the chirality of the quark-current in $B\to K\mu^{+}\mu^{-}$.

Definitions of the hadronic form factors are relegated to the Appendix. We
employ the form factors calculated in the unquenched lattice simulation using
non-relativistic formulation of the $b$ quark and staggered fermion
formulation for the light quarks Bouchard et al. (2013). We use the
$z$-expansion to parameterize the form factors and take into account the
statistical errors given by the covariance matrix of the parameters, both
given in Bouchard et al. (2013). However, we neglect additional systematic
errors that should come on top of the ones contained in the covariance matrix.
The correlations between form factor parameters are propagated onto
observables of interest, namely we can construct $\chi^{2}$ statistic for
$\mathcal{B}(B\to K\mu^{+}\mu^{-})$ and $R_{K}$, that are functions of the
form factor parameters, as well as the Wilson coefficients. Nonlocal
contributions to the decay amplitude due to operators $\mathcal{O}_{1,2}$ are
taken into account by leading order in operator product expansion together
with next-to-next-to-leading logarithmic QCD corrections Greub et al. (2008).
Higher orders in operator product expansion have been shown to have small
effect in the large $q^{2}$ region Grinstein and Pirjol (2004). Since the
partial branching ratio that we are interested in corresponds to an integral
over a large region of $q^{2}$ we rely on the semi-local quark-hadron duality
Beylich et al. (2011). In the SM limit the prediction of the branching ratio
in the high-$q^{2}$ bin is

$\mathcal{B}(B^{+}\to
K^{+}\mu^{+}\mu^{-})|^{\mathrm{SM}}_{q^{2}\in[15,22]\mathrm{GeV}^{2}}=(10.2\pm
0.5)\times 10^{-8}\,.$ (7)

The LHCb collaboration measured partial branching fractions below and above
the region of charmonium resonances. For the
$q^{2}>15\mathrm{\leavevmode\nobreak\ GeV}^{2}$ region we can predict the
partial branching ratio using form factors determined on the lattice that are
largely free from extrapolation errors and parameterization dependence. Thus
we will use Aaij et al. (2014b),

$\mathcal{B}(B^{+}\to
K^{+}\mu^{+}\mu^{-})|_{q^{2}\in[15,22]\mathrm{GeV}^{2}}=(8.5\pm 0.3\pm
0.4)\times 10^{-8}\,,$ (8)

as an experimental constraint, where the errors quoted are statistical and
systematic, respectively. In our analysis we will sum the two and treat the
observable with a Gaussian $\chi^{2}$.

###  II.2 $B_{s}\to\mu^{+}\mu^{-}$

This decay receives contributions from operators with axial, scalar, and
pseudoscalar lepton currents, and, owing to the pseudoscalar nature of the
$B_{s}$ meson, the wrong-chirality Wilson coefficients will affect the decay
with opposite sign. In the absence of (pseudo)scalar operators, the amplitude
is proportional to the difference $C_{10}-C_{10}^{\prime}$:

$\displaystyle P=\frac{2m_{\mu}}{m_{B_{s}}}(C_{10}-C_{10}^{\prime})\,,$ (9)

and the “theoretical” branching ratio is expressed as

$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})^{\mathrm{th}}=\mathcal{B}_{0}|P|^{2}\,,\qquad\mathcal{B}_{0}=\frac{f_{B_{s}}^{2}m_{B_{s}}^{3}}{\Gamma_{s}}\frac{G_{F}^{2}\alpha^{2}|V_{tb}V_{ts}|^{2}}{(4\pi)^{3}}\,\sqrt{1-\frac{4m_{\mu}^{2}}{m_{B_{s}}^{2}}}\,.$
(10)

For the decay constant of the $B_{s}$ meson we take $f_{B_{s}}=(228\pm
8)\mathrm{\leavevmode\nobreak\ MeV}$, consistent with the average made by FLAG
Aoki et al. (2014). Due to $B_{s}-\bar{B}_{s}$ oscillations and relatively
large $y_{s}=\Delta\Gamma_{s}/(2\Gamma_{s})$ in the $B_{s}$ sector, the
measured branching fraction actually corresponds to a time-integrated rate of
the oscillating $B_{s}$ system to $\mu^{+}\mu^{-}$ De Bruyn et al. (2012). In
effect, the value reported by the experimentalists is different from
$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})^{\mathrm{th}}$:

$\displaystyle\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})^{\mathrm{exp}}$
$\displaystyle=\frac{\mathcal{B}_{0}}{1-y_{s}^{2}}\left[|P|^{2}+y_{s}\mathrm{Re}(P^{2})\right]\,.$
(11)

Latest average of the LHCb and CMS measurements of $B_{s}\to\mu^{+}\mu^{-}$
branching fraction is Khachatryan et al. (2015)

$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})^{\mathrm{exp}}=(2.8_{-0.6}^{+0.7})\times
10^{-9}\,.$ (12)

The relative decay width difference $y_{s}=0.061\pm 0.009$ has been determined
from LHCb simultaneous measurement of total width $\Gamma_{s}$ and width
difference $\Delta\Gamma_{s}$ in decay channels $B_{s}\to J/\psi P^{+}P^{-}$
Aaij et al. (2015). The above determined value agrees very well with the HFAG
and PDG averages Amhis et al. (2014); Olive et al. (2014). In the fits we use
the values for $\Gamma_{s}$ and $\Delta\Gamma_{s}$ reported by LHCb with
summed statistical and systematic errors

$\Delta\Gamma_{s}=(0.0805\pm 0.0123)\mathrm{\leavevmode\nobreak\
ps}^{-1}\,,\qquad\Gamma_{s}=(0.6603\pm 0.0042)\mathrm{\leavevmode\nobreak\
ps}^{-1}\,,$ (13)

with correlation coefficient $-0.3$ Aaij et al. (2015).

##  III New Physics in $C_{9}^{\prime}=-C_{10}^{\prime}$ and prediction for
$R_{K}$

We focus now on the SM extensions that affect the effective Hamiltonian solely
by a single operator that is a product of right-handed quark current with a
left-handed lepton current. In our operator basis it corresponds to a linear
combination $\mathcal{O}_{9}^{\prime}-\mathcal{O}_{10}^{\prime}$ implying

$C_{9}^{\prime}(\Lambda)=-C_{10}^{\prime}(\Lambda)\,,$ (14)

where $\Lambda$ is a scale where NP degrees of freedom are integrated out. An
explicit example of such a scenario can be made in a leptoquark model that
will be discussed in Section IV. If Eq. (14) holds at scale $\Lambda$ it is
neccessary to run the Wilson coefficients down to the low scale $\mu_{b}$
using the renormalization group equations. Under QCD renormalization group the
two operators do not run, keeping the constraint (14) intact Bobeth et al.
(2004). 111Eq. (14) is broken only by tiny effects from QED renormalization.
Thus we have, at low energies, a SM modification that satisfies

$C_{9}^{\prime}=-C_{10}^{\prime}\,,$ (15)

where $C_{9,10}^{\prime}$ are scale invariant, modulo small QED corrections.

In Fig. 1 we show in gray the $1\sigma$ region in the $C_{10}^{\prime}$ plane
as obtained from the fit to the partial branching fraction of $B^{+}\to
K^{+}\mu^{+}\mu^{-}$, cf. Eq. (8). The $1\sigma$ region is defined here as
$\chi^{2}<2.30$. The width of the “donut” reflects both experimental and form
factor uncertainties. The SM point in the parameter space is marked with a dot
and exhibits a tension with the measurement with $\chi^{2}=3.9$. In Fig. 1 the
$1\sigma$ region (defined as before) of fit to the
$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})$ according to Eq. (12) is depicted in
blue. In this case the SM point is in comfortable agreement with the
observable ($\chi^{2}=0.7$). Then we perform combined fit to all of the above
quantities and find the best value to be $\chi^{2}_{\mathrm{min}}=2.26$, which
is substantially better than the SM point with $\chi^{2}_{\mathrm{SM}}=4.6$.
The green patch is defined by $\chi^{2}<\chi^{2}_{\mathrm{min}}+1=3.26$ with
$39\,\%$ C.L. and corresponds to the $1\sigma$ region of predicted $R_{K}$
given below.

Figure 1: Regions in the complex $C_{10}^{\prime}$ plane that are in $1\sigma$
agreement with $B_{s}\to\mu^{+}\mu^{-}$ (blue), $B\to K\mu^{+}\mu^{-}$ (gray).
Green area corresponds to the $1\sigma$ coverage of $R_{K}$ from fit to both
observables. Black dot is the SM.

Assuming that the effective Hamiltonian (3), tailored for $b\to se^{+}e^{-}$,
receives only SM contributions, unlike $b\to s\mu^{+}\mu^{-}$ that also
receives NP contributions from $C_{9,10}^{\prime}$, we can now predict the
value of $R_{K}$. In $R_{K}$ the uncertainties of the hadronic form factors
cancel out to a large extent in the ratio and the formula boils down to:

$R_{K}(C_{10}^{\prime})=1.001(1)-0.46\ \mathrm{Re}[C_{10}^{\prime}]-0.094(3)\
\mathrm{Im}[C_{10}^{\prime}]+0.057(1)|C_{10}^{\prime}|^{2}\,.$ (16)

Remaining uncertainties are indicated by the numbers in parentheses. In Fig. 2
we show contours of constant $R_{K}$ in the $C_{10}^{\prime}$ plane using the
formula (16) with central values for the coefficients. By dark gray we
indicate the region corresponding to the measured value of $R_{K}$. In the
same figure we plot again the $1\sigma$ prediction of $C_{10}^{\prime}$, also
shown in Fig. 1. We see an appreciable overlap with the measured $R_{K}$.
Mapping the fitted region (green) to $R_{K}$ we obtain the prediction

$R_{K}^{\mathrm{pred.}}=0.88\pm 0.08\,,$ (17)

which is indeed in good agreement with
$R_{K}^{\mathrm{LHCb}}=0.745\pm^{0.090}_{0.074}\pm 0.036$ Aaij et al. (2014a).

Figure 2: Contours of constant $R_{K}$ are indicated by dashed lines. Gray
region represents the $1\sigma$ measured range of $R_{K}$ projected onto the
$C_{10}^{\prime}$ plane, whereas green contour denotes the region allowed by
$B_{s}\to\mu^{+}\mu^{-}$ and $B\to K\mu^{+}\mu^{-}$. Black dot is the SM.

###  III.1 Impact on $B\to K^{\ast}\ell^{+}\ell^{-}$

$B\to K^{\ast}\ell^{+}\ell^{-}$ is particularly interesting for the NP
searches because of the observables that one can construct from the
$q^{2}$-dependent coefficients $I_{1\dots 9}$ which appear in the angular
distribution,

$\displaystyle{d^{4}\Gamma(\bar{B}^{0}\to\bar{K}^{\ast
0}\ell^{+}\ell^{-})\over dq^{2}\ d\cos\theta_{\ell}\ d\cos\theta_{K}\ d\phi}$
$\displaystyle={9\over
32\pi}\Biggl{[}I_{1}^{s}\sin^{2}\theta_{K}+I_{1}^{c}\cos^{2}\theta_{K}+(I_{2}^{s}\sin^{2}\theta_{K}+I_{2}^{c}\cos^{2}\theta_{K})\cos
2\theta_{\ell}\Biggr{.}$
$\displaystyle+I_{3}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\cos
2\phi+I_{4}\sin 2\theta_{K}\sin 2\theta_{\ell}\cos\phi+I_{5}\sin
2\theta_{K}\sin\theta_{\ell}\cos\phi$
$\displaystyle+(I_{6}^{s}\sin^{2}\theta_{K}+I_{6}^{c}\cos^{2}\theta_{K})\cos\theta_{\ell}+I_{7}\sin
2\theta_{K}\sin\theta_{\ell}\sin\phi$ (18) $\displaystyle\Biggl{.}+I_{8}\sin
2\theta_{K}\sin
2\theta_{\ell}\sin\phi+I_{9}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin
2\phi\,\Biggr{]}.$

Differential decay rate is then simply
$d\Gamma/dq^{2}=(3I_{1}^{c}+6I_{1}^{s}-I_{2}^{c}-2I_{2}^{s})/4$, and the
similar expressions can be written for the transverse/longitudinal part of the
decay rate, for the forward-backward asymmetry, $A_{\rm
fb}(q^{2})=3I_{6}^{s}/(4d\Gamma/dq^{2})$, CP-asymmetry, and several other
observables. Each of the coefficient functions, $I_{i}\equiv I_{i}(q^{2})$,
can be written in terms of transversity amplitudes,
$A_{\perp,\parallel,0}^{L,R}(q^{2})$, which are related to the respective spin
states of the on-shell $K^{\ast}$-meson, and the amplitude
$A_{t}^{L,R}(q^{2})$ which is related to the off-shell virtual gauge boson
decaying into the lepton pair. The superscripts $L,R$ indicate the chirality
of the lepton. Detailed expressions can be found, for example, in Refs.
Altmannshofer et al. (2009a); Bobeth et al. (2008); Descotes-Genon et al.
(2013b); Becirevic and Schneider (2012).

The strategy of looking for the NP effects through a detailed analysis of the
angular distribution of $B\to K^{\ast}\ell^{+}\ell^{-}$ is somewhat plagued by
hadronic uncertainties. The observables built up of
$A_{\perp,\parallel}^{L,R}(q^{2})$ turn out to be less sensitive to hadronic
uncertainties because they involve the (combinations of) hadronic form factors
which appear to be under a rather good theoretical control, especially in the
region of small $q^{2}$’s Charles et al. (1999); Beneke et al. (2001, 2005)
(see also discussion in Ref. Becirevic and Schneider (2012)). On the other
hand, the observables made of $A_{0,t}^{L,R}(q^{2})$ entail the hadronic form
factors that are less well understood. Moreover, the latter observables are
subject to another kind of hadronic uncertainty, i.e. the one arising from
misidentification of the $K\pi$ pairs coming from $B\to K^{\ast}(\to
K\pi)\ell^{+}\ell^{-}$ with those emerging from $B\to K_{0}^{\ast}(\to
K\pi)\ell^{+}\ell^{-}$, where $K_{0}^{\ast}$ stands for a broad scalar state
Becirevic and Tayduganov (2013); Das et al. (2014); Blake et al. (2013);
Matias (2012); Döring et al. (2013). Finally, and to avoid problems of the
$c\bar{c}$-resonances in the $q^{2}$-spectrum of the decay, a standard
strategy is to either work at low $q^{2}<m_{J/\psi}^{2}$ or large
$q^{2}\gtrsim 15{\rm GeV}^{2}$, in which the impact of the
$c\bar{c}$-resonances is expected to be small. To be more specific, we fully
rely on quark-hadron duality since we avoid the region in which the prominent
narrow $c\bar{c}$-resonances appear, and integrate over a window $\gtrsim
5\leavevmode\nobreak\ {\rm GeV}^{2}$. 222For a recent attempt to more
realistically model the effects of such resonances see Ref. Lyon and Zwicky
(2014) or those discussed previously in Refs. Ali et al. (1991); Kruger and
Sehgal (1997).

With the information obtained in the previous section of this paper, i.e. with
$C_{10}^{\prime}=-C_{9}^{\prime}$ extracted from the comparison of the
measured $\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})$ and $\mathcal{B}(B\to
K\mu^{+}\mu^{-})_{q^{2}>15\ {\rm GeV}^{2}}$ with the corresponding theoretical
expressions, we already showed that we were able to verify the consistency of
our result for $R_{K}$ with the one measured at LHCb. With our approach, in
which only the decay to muon-pair is modified, we can also predict
$R_{K^{\ast}}$, defined as

$\displaystyle R_{K^{\ast}}={\Gamma(B\to
K^{\ast}\mu^{+}\mu^{-})_{q^{2}\in[1,6]\ {\rm GeV}^{2}}\over\Gamma(B\to
K^{\ast}e^{+}e^{-})_{q^{2}\in[1,6]\ {\rm GeV}^{2}}},$ (19)

as well as the ratio of the two Hiller and Schmaltz (2014, 2015), namely,

$\displaystyle X_{K}={R_{K^{\ast}}\over R_{K}}-1\,.$ (20)

In Ref. Altmannshofer and Straub (2013) it was shown that the ratio of
forward-backward asymmetries integrated between
$q^{2}\in[4,6]\leavevmode\nobreak\ {\rm GeV}^{2}$ can also be sensitive to
lepton flavor universality violation. After defining,

$\displaystyle A_{\rm fb[4-6]}^{\ell}=\frac{3}{4}\,{\displaystyle{\int_{4\
{\rm GeV}^{2}}^{6\ {\rm GeV}^{2}}I_{6}^{s}(q^{2})\
dq^{2}}\over\quad\Gamma(B\to K^{\ast}\ell^{+}\ell^{-})_{q^{2}\in[4,6]\ {\rm
GeV}^{2}}}\,,$ (21)

the ratio of forward-backward asymmetries is then simply,

$\displaystyle R_{\rm fb}={A_{\rm fb[4-6]}^{\mu}\over A_{\rm fb[4-6]}^{e}}\,.$
(22)

To compute the above-mentioned quantities we use the standard values of the
Wilson coefficients Bobeth et al. (2000), and include the effect of quark
loops in the coefficients $C_{7,9}$ arising from the operators $O_{1,2}$, as
calculated in Ref. Greub et al. (2008). We neglect the soft gluon corrections
to the charm quark loop at low $q^{2}$, which according to Ref. Khodjamirian
et al. (2010) is reasonable. At low $q^{2}$ the hard scattering contributions
are neglected. For the form factors we use the values computed by means of QCD
sum rules on the light cone Ball and Zwicky (2005). In Fig. 3 we show our
results for $R_{K^{\ast}}$, $X_{K}$ and $R_{\rm fb}$ as functions of
Re[$C_{10}^{\prime}$]. For an easier comparison, in the same plot we also show
$R_{K}$. The range $0.075\leq{\rm Re}[C_{10}^{\prime}]\leq 0.41$ has been
obtained in the previous section of this paper, where we showed that for a
given value of $0.075\leq{\rm Re}[C_{10}^{\prime}]\leq 0.41$ there is a region
of allowed Im[$C_{10}^{\prime}$], and therefore instead of curves in Fig. 3 we
actually have the corresponding regions of values determined by ${\rm
Im}[C_{10}^{\prime}]$. We should emphasize again that the uncertainties
related to form factors cancel to a large extent in the ratios.

Figure 3: $R_{K}$, $R_{K^{\ast}}$, $X_{K}$ and $R_{\rm fb}$, defined in Eq.
(1,19,20,22) respectively, are plotted as functions of ${\rm
Re}[C_{10}^{\prime}]$, in the range allowed by the measured values of ${\rm
B}(B_{s}\to\mu^{+}\mu^{-})$ and $\mathcal{B}(B\to K\mu^{+}\mu^{-})_{q^{2}>15\
{\rm GeV}^{2}}$. Instead of a curve for each quantity we actually have a
region of values, reflecting the fact that for each ${\rm
Re}[C_{10}^{\prime}]$ there is a range of allowed values of ${\rm
Im}[C_{10}^{\prime}]$, as shown in Fig. 1.

As for the results, we first see that in the scenario with
$C_{10}^{\prime}=-C_{9}^{\prime}\neq 0$, allowing coupling to muons only, and
explicitly realized in the model with a $(3,2,1/6)$ leptoquark state, we get

$\begin{split}R_{K}&=0.88\pm 0.08\,,\qquad R_{K^{\ast}}=1.11\pm 0.08\,,\\\
X_{K}&=0.27\pm 0.19\,,\qquad R_{\rm fb}=0.84\pm 0.12\,,\end{split}$ (23)

which are obviously different from the values obtained in the SM, $R_{K}^{\rm
SM}=1.00$, $R_{K^{\ast}}^{\rm SM}=0.996(5)\approx 1$, $R_{\rm
fb}=0.995(4)\approx 1$, and $X_{K}=-0.004(5)\approx 0$. Notice, however, that
while our value for $R_{K}$ is lower than the one in the SM, our prediction
for $R_{K^{\ast}}$ is larger than that obtained in the SM. The measurement of
$R_{K^{\ast}}$ at LHCb will therefore help to either confirm or discard our
model as a viable description of the lepton flavor universality violation. The
errors in Eq. (23) are completely dominated by the range of
$\mathrm{Re}[C_{10}^{\prime}]$ and $\mathrm{Im}[C_{10}^{\prime}]$, while those
arising from form factors are reduced in the ratios and induce an uncertainty
negligible in comparison with that coming from the variation of
$C_{10}^{\prime}$.

Besides the above quantities, one can also check on the asymmetries
$A_{T}^{(2)}$ and $A_{T}^{(\mathrm{Im})}$, defined in the Appendix, which are
experimentally more difficult to study but which could be very useful to
compare with predictions as their values can considerably change if
$C_{9,10}^{\prime}\neq 0$. To exemplify that feature we consider the bin
$q^{2}\in[2,6]\leavevmode\nobreak\ {\rm GeV}^{2}$ and in the SM we have
$\langle A_{T}^{(2)}(q^{2})\rangle_{q^{2}\in[2,6]\ {\rm
GeV}^{2}}^{\mathrm{SM}}=-0.05(2)$, and $\langle A_{T}^{({\rm
Im})}(q^{2})\rangle_{q^{2}\in[2,6]\ {\rm GeV}^{2}}^{\mathrm{SM}}\simeq 0$, for
either electrons or muons in the final state. If, instead, the coefficients
$C_{10}^{\prime}=-C_{9}^{\prime}$ become non-zero and take values within the
green region shown in Fig. 1, then in the case of $B\to
K^{\ast}\mu^{+}\mu^{-}$, the above values change to

$\displaystyle\langle A_{T}^{(2)}(q^{2})\rangle_{q^{2}\in[2,6]\ {\rm
GeV}^{2}}^{\mu}\simeq\biggl{.}-0.09\biggr{|}_{\mathrm{for}\
\mathrm{Re}[C_{10}^{\prime}]=0.08},-0.24\ \biggr{|}_{\mathrm{for}\
\mathrm{Re}[C_{10}^{\prime}]=0.4},\qquad|\langle A_{T}^{({\rm
Im})}(q^{2})\rangle_{q^{2}\in[2,6]\ {\rm GeV}^{2}}^{\mu}|\lesssim 0.27\,.$
(24)

Notice also that $\langle A_{T}^{({\rm Im})}(q^{2})\rangle$ has not yet been
measured, and that the current errors on $\langle A_{T}^{(2)}(q^{2})\rangle$
are still too large for making a meaningful quantitative comparison with our
results Aaij et al. (2013a).

Finally, before closing this part of our paper, we need to comment on
$P_{5}^{\prime}(q^{2})$, an observable constructed from coefficients of the
angular distribution of the $B\to K^{\ast}\ell^{+}\ell^{-}$ decay Descotes-
Genon et al. (2013c),
$P_{5}^{\prime}(q^{2})=I_{5}/\sqrt{-4I_{2}^{c}I_{2}^{s}}$, which has been
measured at LHCb, and turned out to be $4\sigma$ away from the value predicted
in the SM when integrated over an interval
$q^{2}\in[4.3,8.68]\leavevmode\nobreak\ {\rm GeV}^{2}$ Aaij et al. (2013b).
More specifically, the SM value is $\langle
P_{5}^{\prime}\rangle_{[4.3-8.68]}^{\rm SM}=-0.90(5)$, while the measured one
is $\langle P_{5}^{\prime}\rangle_{[4.3-8.68]}^{\rm LHCb}=-0.19(16)$ Aaij et
al. (2013b), which can be compactly written as, $\langle
P_{5}^{\prime}\rangle_{[4.3-8.68]}^{\rm LHCb}/\langle
P_{5}^{\prime}\rangle_{[4.3-8.68]}^{\rm SM}=0.22(18)$. While the
interpretation of this discrepancy is somewhat controversial Descotes-Genon et
al. (2013a); Altmannshofer and Straub (2013); Jager and Martin Camalich
(2014), it is nevertheless interesting to check whether or not the leptoquark
model used in this paper (and discussed in more details in the following
Section) can describe the manifest disagreement between theory and experiment.
With the values of $C_{10}^{\prime}=-C_{9}^{\prime}$ discussed above we indeed
see that $\langle P_{5}^{\prime}\rangle_{[4.3-8.68]}^{\rm LQ}/\langle
P_{5}^{\prime}\rangle_{[4.3-8.68]}^{\rm SM}<1$, but with the leptoquark model
discussed here we cannot reach very low values. We instead obtain
$0.78\leq\langle P_{5}^{\prime}\rangle_{[4.3-8.68]}^{\rm LQ}/\langle
P_{5}^{\prime}\rangle_{[4.3-8.68]}^{\rm SM}\leq 0.98$. A similar tendency is
observed for other bins, and in particular the one corresponding to
$q^{2}\in[1,6]\leavevmode\nobreak\ {\rm GeV}^{2}$.

##  IV Model with a Scalar Leptoquark

In this Section we discuss a specific model in which the scenario discussed
above, i.e. $C_{9}^{\prime}=-C_{10}^{\prime}$, is explicitly realized and
involves the presence of a light scalar leptoquark state $\Delta$. More
specifically, we choose the leptoquark $\Delta$ to carry the quantum numbers
$(3,2,1/6)$ of the SM gauge group. Its couplings to fermions are described by
a renormalizable Lagrangian

$\begin{split}\mathcal{L}&=Y_{ij}\overline{L}_{i}\,i\tau^{2}\Delta^{*}d_{Rj}+\mathrm{h.c.}\\\
&=Y_{ij}\left(-\bar{\ell}_{Li}d_{Rj}\Delta^{(2/3)*}+\bar{\nu}_{Lk}(V^{\mathrm{PMNS}})^{\dagger}_{ki}d_{Rj}\Delta^{(-1/3)*}\right)+\mathrm{h.c.}\,,\end{split}$
(25)

where $Y$ is a $3\times 3$ complex matrix, $L_{i}$ and $d_{Rj}$ are the lepton
doublet and down-quark singlet. Charge eigenstates of the leptoquark doublet
are denoted with $\Delta^{(2/3)}$ and $\Delta^{(-1/3)}$ and we will assume
that they are degenerate. The second line in the above Lagrangian is written
in the fermion mass basis, and a relative PMNS rotation in lepton doublet
components has been assigned to the neutrino sector.

Clearly, the lepton flavor universality is explicitly broken by the terms
presented in Eq. (25). This might appear questionable because in a similar
situation in which the coupling of leptoquark to $\mu c$ would be allowed, the
ratio of the electronic and muonic widths of the decay of $J/\psi$ and its
radial excitations have been accurately measured, and shows no violation of
the lepton flavor coupling universality. In particular, the measured
$\Gamma(J/\psi\to\mu^{+}\mu^{-})/\Gamma(J/\psi\to e^{+}e^{-})=1.0016\pm
0.0031$ Olive et al. (2014) is in excellent agreement with its SM value,
$1.00001$. 333By explicitly including the lepton mass in the calculation of
phase space we obtain,
$\displaystyle\Gamma(J/\psi\to\ell^{+}\ell^{-})={16\pi\alpha^{2}\over
27m_{J/\psi}}\left(1+{2m_{\ell}^{2}\over
m_{J/\psi}^{2}}\right)\sqrt{1-{4m_{\ell}^{2}\over m_{J/\psi}^{2}}}\
f_{J/\psi}^{2}$ and the effect on the ratio of the electronic and muonic
widths is extremely small. That situation is, however, much different from the
examples discussed in this paper, because the amplitude for
$J/\psi\to\ell^{+}\ell^{-}$ is dominated by the tree-level electromagnetic
interaction diagram which is much larger than the weak interaction one,
suppressed by $1/m_{Z}^{2}$ with respect to the dominant one, and therefore
completely negligible. Our leptoquark state is $m_{\Delta}\gg m_{Z}$, and its
contribution to $J/\psi\to\ell^{+}\ell^{-}$ is even smaller than the weak
interaction diagram and cannot make an impact on the decay of charmonia at the
present level of accuracy.

Instead, the weak $b\to s\mu^{+}\mu^{-}$ decays in the SM are loop-induced so
that the tree level contribution involving couplings to the leptoquark state
may become comparable in size to the SM amplitude, which is why the $b\to
s\mu^{+}\mu^{-}$ is likely to be more sensitive to the presence of the term
described by the lagrangian (25). The relevant leptoquark coupling for the
$b\to s\mu^{+}\mu^{-}$ is the product $Y_{\mu b}Y_{\mu s}$, which enters the
Wilson coefficients divided by $m_{\Delta}^{2}$. The scalar particle exchange
generates scalar operators in the Fierzed basis and those appear as
(pseudo)vector currents in the ordinary operator basis Kosnik (2012):

$C_{10}^{\prime}=-C_{9}^{\prime}=\frac{\pi}{2\sqrt{2}G_{F}V_{tb}V_{ts}^{*}\alpha}\,\frac{Y_{\mu
b}Y_{\mu s}^{*}}{m_{\Delta}^{2}}\,.$ (26)

We assume other elements of Yukawa matrix $Y$ to vanish. The same state will
also contribute at loop level to electro- and chromo-magnetic operators
$C_{7}^{\prime}(m_{\Delta})$ and $C_{8}^{\prime}(m_{\Delta})$ where these
coefficients will be suppressed by electromagnetic $\alpha(m_{\Delta})/(4\pi)$
and strong $\alpha_{S}(m_{\Delta})/(4\pi)$ couplings at high scale
$m_{\Delta}$, respectively. We have explicitly checked that these
modifications result in negligibly small value of $C_{7}^{\prime}$ when
compared to the $C_{7}$ of SM, cf. Eq. (5). In the remainder of this Section
we will analyze additional observables that constrain this leptoquark
scenario.

The considered leptoquark state $\Delta$ couples to the neutrinos with the
same couplings as to the charged leptons, only modified by a PMNS rotation
matrix. Namely, the charge $-1/3$ state will generate
$(\bar{s}b)(\bar{\nu}\nu)$ operators while the box diagrams will lead to
$B_{s}-\bar{B}_{s}$ mixing.

###  IV.1 Contribution of $(3,2,1/6)$ leptoquark in $B_{s}-\bar{B}_{s}$
oscillation frequency

The state $(3,2,1/6)$ will induce $\Delta B=2$ box diagrams with $\mu$ and
$\Delta^{(2/3)}$ or $\nu$ and $\Delta^{(-1/3)}$ running in the box. The two
contributions of boxes with $\mu$ and $\nu$ are equal in the $m_{\mu}=0$ limit
and in sum they amount to

$C_{6}^{\mathrm{LQ}}(m_{\Delta})=-\frac{Y_{\mu b}^{*2}Y_{\mu
s}^{2}}{64\pi^{2}m_{\Delta}^{2}}\,.$ (27)

The effective $\Delta B=2$ Hamiltonian is defined as

${\cal H}_{\mathrm{eff}}=C_{1}^{\rm
SM}(\bar{b}\gamma_{\mu}P_{L}s)\,(\bar{b}\gamma^{\mu}P_{L}s)+C_{6}^{\rm
LQ}(\bar{b}\gamma_{\mu}P_{R}s)\,(\bar{b}\gamma^{\mu}P_{R}s)+\mathrm{h.c.}\,,$
(28)

where $P_{L/R}=(1\mp\gamma_{5})/2$. The coefficient in Eq. (27) is subject to
QCD renormalization and has to be evaluated at scale $\mu_{b}$. The anomalous
dimensions of $C_{6}^{\rm LQ}$ is however equal to the one of $C_{1}^{\rm
SM}$. Therefore the two Wilson coefficients renormalize with the same
multiplicative factor between scales $\mu=m_{t}$, where SM is matched onto
effective Hamiltonian (28), and $\mu_{b}$, where the hadronic matrix elements
are computed. Remaining $C_{6}^{\mathrm{LQ}}$ running from $m_{\Delta}$ down
to $m_{t}$ is already in the asymptotic regime of QCD and can be safely
neglected. The mass difference of the $B_{s}-\bar{B}_{s}$ system is then

$\displaystyle\Delta
m_{B_{s}}=\frac{2}{2m_{B_{s}}}\left|\frac{G_{F}^{2}m_{W}^{2}}{16\pi^{2}}(V_{tb}^{\ast}V_{ts})^{2}\eta_{B}S_{0}(x_{t})+\frac{\eta_{B}}{4}C_{6}^{\rm
LQ}(m_{\Delta})\right|\langle\bar{B}_{s}^{0}|\bar{b}\gamma_{\mu}(1-\gamma_{5})s\,\bar{b}\gamma^{\mu}(1-\gamma_{5})s|B_{s}^{0}\rangle\,.$
(29)

By using Eq. (26) we can write

$\displaystyle C_{6}^{\mathrm{LQ}}(m_{\Delta})=-{G_{F}^{2}\over
8\pi^{4}}(V_{tb}^{\ast}V_{ts})^{2}\alpha^{2}m_{\Delta}^{2}(C_{10}^{\prime\ast})^{2}\,,$
(30)

which, together with
$\langle\bar{B}_{s}^{0}|\bar{b}\gamma_{\mu}(1-\gamma_{5})s\,\bar{b}\gamma^{\mu}(1-\gamma_{5})s|B_{s}^{0}\rangle=(8/3)f_{B_{s}}^{2}m_{B_{s}}^{2}B_{B_{s}}$,
gives

$\displaystyle\Delta m_{B_{s}}=\underbrace{{G_{F}^{2}m_{W}^{2}\over
6\pi^{2}}|V_{tb}^{\ast}V_{ts}|^{2}f_{B_{s}}^{2}m_{B_{s}}B_{B_{s}}\eta_{B}S_{0}(x_{t})}_{\Delta
m_{B_{s}}^{\rm SM}}\left|1-\frac{1}{2\pi^{2}}{\alpha^{2}\over
S_{0}(x_{t})}(C_{10}^{\prime\ast})^{2}\frac{m_{\Delta}^{2}}{m_{W}^{2}}\right|\,.$
(31)

With the current values for $f_{B_{s}}=228(8)$ MeV and $B_{B_{s}}=1.33(6)$, as
obtained in numerical simulations of QCD on the lattice Aoki et al. (2014),
and $m_{t}^{\overline{\rm MS}}(m_{t})=160^{+5}_{-4}$ GeV Olive et al. (2014),
we get 444To evaluate $\Delta m_{B_{s}}^{\rm SM}$ we also used
$\eta_{B}=0.55(1)$ Buras et al. (1990), and $S_{0}(x_{t})=2.25^{+11}_{-09}$,
the Inami-Lim function at $x_{t}=m_{t}^{2}/m_{W}^{2}$.

$\displaystyle\Delta m_{B_{s}}^{\rm SM}=17.3\pm 1.7\ {\rm ps}^{-1},$ (32)

which is in excellent agreement with the measured $\Delta
m_{B_{s}}=17.7(2){\rm ps}^{-1}$ Olive et al. (2014). With the values of
$C_{10}^{\prime}$ determined in the previous Section, we see that Eq. (31)
leads to a very loose upper bound for $m_{\Delta}$. For example, for
$\mathrm{Re}[C_{10}^{\prime}]\in[0.15,0.35]$, we get the upper bound of the
order $100$ TeV.

###  IV.2 Impact of $(3,2,1/6)$ leptoquark on $B\to K\nu\bar{\nu}$

In the presence of leptoquark $\Delta$ the pair of neutrinos in the final
state of $B\to K\nu\bar{\nu}$ may be in any flavor combination. In order to
encompass such a possibility we must extend the effective Hamiltonian of Ref.
Altmannshofer et al. (2009b) to account for the disparity in neutrino flavors:

${\cal
H}_{\mathrm{eff}}=-\frac{4G_{F}}{{\sqrt{2}}}V_{tb}V_{ts}^{*}(C_{L}^{ij}{\cal
O}^{ij}_{L}+C_{R}^{ij}{\cal O}^{ij}_{R})\,.$ (33)

The operators are defined as ${\cal
O}^{ij}_{L,R}=\frac{e^{2}}{16\pi^{2}}(\bar{s}\gamma_{\mu}P_{L,R}b)(\bar{\nu}_{i}\gamma^{\mu}(1-\gamma_{5})\nu_{j})$.
The authors of Altmannshofer et al. (2009b) found that in the SM the Wilson
coefficient at next-to-leading order in QCD is

$C_{L}^{\mathrm{SM}}\equiv C_{L}^{ii}=-6.38\pm 0.06\,,\qquad\textrm{(no sum
over $i$ implied)}.$ (34)

If the leptoquark state $(3,2,1/6)$ is present then it will manifest itself in
$B\to K\nu\bar{\nu}$ through right-handed operators:

$C_{R}^{ij}=-\frac{1}{N}\,\frac{(VY)_{ib}(VY)^{*}_{js}}{4m_{\Delta}^{2}}\,,\qquad
N\equiv\frac{G_{F}V_{tb}V_{ts}^{*}\alpha}{\sqrt{2}\pi}\,.$ (35)

Here $V$ denotes the PMNS matrix. The experimentally accessible decay width of
$B\to K\nu\bar{\nu}$ is a sum of partial widths of $B\to
K\nu^{i}\bar{\nu}^{j}$. The amplitudes are proportional to the sum of the SM
and leptoquark contribution and the two will interfere in the $B\to
K\nu\bar{\nu}$ decays width as

$\begin{split}\Gamma(B\to
K\nu\bar{\nu})&\sim\sum_{i,j=1}^{3}\left|\delta_{ij}C_{L}^{\mathrm{SM}}+C_{R}^{ij}\right|^{2}\\\
&=3|C_{L}^{\mathrm{SM}}|^{2}+|C_{10}^{\prime}|^{2}-2\mathrm{Re}[C_{L}^{\mathrm{SM}*}C_{10}^{\prime}]\,.\end{split}$
(36)

$C_{10}^{\prime}$ is the Wilson coefficient of $b\to s\mu^{+}\mu^{-}$ that we
obtained from the fits to experimental data in the previous Section. Last line
of Eq. (36) was obtained by applying the unitarity of matrix $V$, and assuming
that $Y_{\mu b}$ and $Y_{\mu s}$ are the only non-zero elements of the matrix
$Y$. Finally, the $q^{2}$-spectrum of this decay reads,

$\frac{\\!\mathrm{d}\Gamma}{\mathrm{d}q^{2}}(B\to
K\nu\bar{\nu})=\frac{|N|^{2}}{384\pi^{3}m_{B}^{3}}\,f_{+}(q^{2})\left[\lambda(m_{B}^{2},m_{K}^{2},q^{2})\right]^{3/2}\biggl{(}3|C_{L}^{\mathrm{SM}}|^{2}+|C_{10}^{\prime}|^{2}-2\mathrm{Re}[C_{L}^{\mathrm{SM}*}C_{10}^{\prime}]\biggr{)}\,,$
(37)

where $q^{2}$ in this case stands for the invariant mass of the neutrino pair.
Notice that the above expression, for $C_{10}^{\prime}=0$, confirms Eq. (2.14)
of Ref. Altmannshofer et al. (2009b). The expression (37) can be recast into a
product of the SM $q^{2}$-spectrum and a correction factor,

$1.01<\left[1+\frac{1}{3}\left|C_{10}^{\prime}/C_{L}^{\mathrm{SM}}\right|^{2}-\frac{2}{3}\
\mathrm{Re}[C_{10}^{\prime}/C_{L}^{\mathrm{SM}}]\right]<1.05\ ,$ (38)

where its lower and upper bounds have been derived from the $1\sigma$ region
of $C_{10}^{\prime}$, obtained in the previous Section. We learn that the
$\mathcal{B}(B\to K\bar{\nu}\nu)$ will increase by at most $5\,\%$ if
leptoquark $\Delta$ is present.

##  V Summary and conclusion

In this paper we discussed a possibility of constraining a scenario of New
Physics affecting the $b\to s\mu^{+}\mu^{-}$ decays through coupling with the
operators ${\cal O}_{9,10}^{\prime}$. Such a scenario is explicitly verified
in a model with a light scalar leptoquark state, $\Delta$, carrying the
quantum numbers $(3,2,1/6)$ of the Standard Model gauge group. In this
scenario, $C_{9}^{\prime}=-C_{10}^{\prime}$ is specific for the muons in the
final state. In the leptoquark model discussed in this paper,
$C_{10}^{\prime}$ is related to ${Y_{\mu b}Y_{\mu s}^{*}}/m_{\Delta}^{2}$.
From the currently available experimental data on ${\cal
B}(B_{s}\to\mu^{+}\mu^{-})$ and ${\cal B}(B\to K\mu^{+}\mu^{-})_{[15,22]{\rm
GeV}^{2}}$ we were able to constrain $\mathrm{Re}[C_{10}^{\prime}]$ and
$\mathrm{Im}[C_{10}^{\prime}]$, which are then used to compute $R_{K}={\cal
B}(B\to K\mu^{+}\mu^{-})/{\cal B}(B\to Ke^{+}e^{-})_{[1,6]{\rm
GeV}^{2}}=0.88(8)$. We find a good agreement with a recent experimental
result, $R_{K}^{\rm LHCb}=0.75(14)$.

After having passed this test we focused on $B\to K^{\ast}\ell^{+}\ell^{-}$
decays. Within our scenario, and the range of $C_{10}^{\prime}$ obtained from
${\cal B}(B_{s}\to\mu^{+}\mu^{-})$ and ${\cal B}(B\to
K\mu^{+}\mu^{-})_{[15,22]{\rm GeV}^{2}}$, we predict $R_{K^{\ast}}={\cal
B}(B\to K^{\ast}\mu^{+}\mu^{-})/{\cal B}(B\to K^{\ast}e^{+}e^{-})_{[1,6]{\rm
GeV}^{2}}=1.11(8)$, and $R_{K^{\ast}}/R_{K}=1.27(10)$. Similarly, in this
scenario the ratio of forward-backward asymmetries becomes different from
unity. In particular, we find $R_{\rm fb}=({A_{\rm fb}^{\mu}/A_{\rm
fb}^{e}})_{[4-6]\ {\rm GeV}^{2}}=0.8(1)$. Furthermore, we checked that a
combination of coefficients of the angular distribution of $B\to
K^{\ast}\mu^{+}\mu^{-}$, known as $P_{5}^{\prime}$, and weighed over a
specific bin of $q^{2}$’s, indeed becomes smaller than its value predicted in
the Standard Model. However, it cannot explain a very low value of $|\langle
P_{5}^{\prime}\rangle|_{[4.3-8.68]}$ measured at LHCb, for which the Standard
Model prediction is still a subject to controversies mainly related to the
issue of treatment of the charm quark loops.

Finally, in the leptoquark model our constraints on the Wilson coefficient
$C_{10}^{\prime}$ can have impact on other physical processes. We checked, in
particular, that the contribution to the frequency of oscillation in the
$B_{s}-\overline{B}_{s}$ system is insignificant, and that only up to five
percent enhancement of ${\cal B}(B\to K\nu\bar{\nu})$ can be obtained.

###### Acknowledgements.

S.F. and N.K. acknowledge support of the Slovenian Research Agency.

##  Appendix A $B\to K^{(\ast)}$ form factors

For completeness we remind the reader of the standard parameterization of the
$B\to K^{(\ast)}\ell^{+}\ell^{-}$ hadronic matrix elements in terms of the
relevant form factors,

$\displaystyle\langle K(k)|\bar{s}\gamma_{\mu}b|B(p)\rangle$
$\displaystyle=\left[(p+k)_{\mu}-{m_{B}^{2}-m_{K}^{2}\over
q^{2}}q_{\mu}\right]f_{+}(q^{2})+{m_{B}^{2}-m_{K}^{2}\over
q^{2}}q_{\mu}f_{0}(q^{2})\,,$ $\displaystyle\langle
K(k)|\bar{s}\sigma_{\mu\nu}b|B(p)\rangle$
$\displaystyle=-i\left(p_{\mu}k_{\nu}-p_{\nu}k_{\mu}\right)\frac{2f_{T}(q^{2})}{m_{B}+m_{K}}\,,$
(39)
$\displaystyle\langle\bar{K}^{\ast}(k)|\bar{s}\gamma_{\mu}(1-\gamma_{5})b|\bar{B}(p)\rangle$
$\displaystyle=$
$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\varepsilon^{*\nu}p^{\rho}k^{\sigma}\,\frac{2V(q^{2})}{m_{B}+m_{K^{\ast}}}-i\varepsilon^{\ast}_{\mu}(m_{B}+m_{K^{\ast}})A_{1}(q^{2})$
(40) $\displaystyle+i(p+k)_{\mu}(\varepsilon^{\ast}\cdot
q)\,\frac{A_{2}(q^{2})}{m_{B}+m_{K^{\ast}}}+iq_{\mu}(\varepsilon^{\ast}\cdot
q)\,\frac{2m_{K^{\ast}}}{q^{2}}\,\left[A_{3}(q^{2})-A_{0}(q^{2})\right],$
$\displaystyle\langle\bar{K}^{\ast}(k)|\bar{s}\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})b|\bar{B}(p)\rangle$
$\displaystyle=$ $\displaystyle
2i\varepsilon_{\mu\nu\rho\sigma}\varepsilon^{*\nu}p^{\rho}k^{\sigma}\,T_{1}(q^{2})$
(41)
$\displaystyle+\left[\varepsilon^{\ast}_{\mu}(m_{B}^{2}-m_{K^{\ast}}^{2})-(\varepsilon^{\ast}\cdot
q)\,(2p-q)_{\mu}\right]T_{2}(q^{2})$ $\displaystyle+(\varepsilon^{\ast}\cdot
q)\left[q_{\mu}-\frac{q^{2}}{m_{B}^{2}-m_{K^{\ast}}^{2}}\,(p+k)_{\mu}\right]T_{3}(q^{2}),$

where
$2m_{K^{\ast}}A_{3}(q^{2})=(m_{B}+m_{K^{\ast}})A_{1}(q^{2})-(m_{B}-m_{K^{\ast}})A_{2}(q^{2})$,
and $T_{1}(0)=T_{2}(0)$. In the limit of massless lepton, the
$q^{2}$-dependent functions entering eqs. (6,III.1), relevant for the present
study, read

$\displaystyle a_{\mu}(q^{2})=\frac{3}{2}{\cal F}\,,\qquad
I_{2}^{s}=\frac{1}{4}\left(|A_{\perp}^{L,R}|^{2}+|A_{\parallel}^{L,R}|^{2}\right),$
$\displaystyle I_{2}^{c}=-|A_{0}^{L,R}|^{2},\qquad
I_{6}^{s}=2\mathrm{Re}\left(A_{\perp}^{L}A_{\parallel}^{L\
\ast}-A_{\perp}^{R}A_{\parallel}^{R\ \ast}\right)\,,$ (42)

where

$\displaystyle{\cal F}(q^{2})$ $\displaystyle=$ $\displaystyle
N\lambda(q^{2})\left[\biggl{|}(C_{9}+C_{9}^{\prime})f_{+}(q^{2})+{2m_{b}\over
m_{B}+m_{K}}(C_{7}+C_{7}^{\prime})f_{T}(q^{2})\biggr{|}^{2}+\biggl{|}(C_{10}+C_{10}^{\prime})f_{+}(q^{2})\biggr{|}^{2}\right],$
$\displaystyle A_{\perp}^{L,R}(q^{2})$ $\displaystyle=$
$\displaystyle\sqrt{Nq^{2}\lambda(q^{2})}\bigg{[}\frac{2m_{b}}{q^{2}}(C_{7}+C_{7}^{\prime})\
T_{1}(q^{2})+\left[(C_{9}+C_{9}^{\prime})\mp(C_{10}+C_{10}^{\prime})\right]{V(q^{2})\over
m_{B}+m_{K^{\ast}}}\bigg{]},$ $\displaystyle A_{\parallel}^{L,R}(q^{2})$
$\displaystyle=$
$\displaystyle-\sqrt{Nq^{2}}(m_{B}^{2}-m_{K^{\ast}}^{2})\bigg{[}\frac{2m_{b}}{q^{2}}(C_{7}-C_{7}^{\prime})\
T_{2}(q^{2})+\left[(C_{9}-C_{9}^{\prime})\mp(C_{10}-C_{10}^{\prime})\right]{A_{1}(q^{2})\over
m_{B}-m_{K^{\ast}}}\bigg{]}\,,$ $\displaystyle A_{0}^{L,R}(q^{2})$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{N}}{2m_{K^{\ast}}}\biggl{\\{}\left[\left(C_{9}-C^{\prime}_{9}\right)\mp\left(C_{10}-C^{\prime}_{10}\right)\right]\times\biggr{.}$
$\displaystyle\qquad\left[\left(m_{B}^{2}-m_{K^{\ast}}^{2}-q^{2}\left)\right(m_{B}+m_{K^{\ast}}\right)A_{1}(q^{2})-\lambda(q^{2})\frac{A_{2}(q^{2})}{m_{B}+m_{K^{\ast}}}\right]$
$\displaystyle\qquad\left.+2m_{b}\left(C_{7}-C^{\prime}_{7}\right)\left[\left(m_{B}^{2}+3m_{K^{\ast}}^{2}-q^{2}\right)T_{2}(q_{2})-\frac{\lambda(q^{2})}{m_{B}^{2}-m_{K^{\ast}}^{2}}T_{3}(q^{2})\right]\right\\}\,,$

and $N=|V_{tb}V_{ts}^{\ast}|^{2}\ \lambda^{1/2}(q^{2})\
G_{F}^{2}\alpha^{2}/(3072\pi^{5}m_{B}^{3})$,
$\lambda(q^{2})=[q^{2}-(m_{B}+m_{K^{\ast}})^{2}]\
[q^{2}-(m_{B}-m_{K^{\ast}})^{2}]$. In the same massless lepton limit,
$a_{\mu}=-c_{\mu}$, $I_{1}^{s}=3I_{2}^{s}$ and $I_{1}^{c}=-I_{2}^{c}$, so that
$d\Gamma(B\to K\mu\mu)/dq^{2}=4a_{\mu}/3$, and $d\Gamma(B\to
K^{\ast}\mu\mu)/dq^{2}=4I_{2}^{s}-I_{2}^{c}=|A_{\perp}^{L,R}|^{2}+|A_{\parallel}^{L,R}|^{2}+|A_{0}^{L,R}|^{2}$.
The two transverse asymmetries discussed in the text are defined as
$A_{T}^{(2)}(q^{2})=I_{3}/(2I_{2}^{s})=(|A_{\perp}|^{2}-|A_{\parallel}|^{2})/(|A_{\perp}|^{2}+|A_{\parallel}|^{2})$,
and
$A_{T}^{(\mathrm{Im})}(q^{2})=I_{9}/(2I_{2}^{s})=-2\mathrm{Im}(A_{\parallel}A_{\perp}^{\ast})/(|A_{\perp}|^{2}+|A_{\parallel}|^{2})$.
In the computation of decay amplitudes we rely on the full QCD form factors
computed on the lattice (in case of $B\to K\ell\ell$) or by means of the QCD
sum rules near the light cone (in case of $B\to K^{\ast}\ell\ell$) and to the
standard (universal) Operator Product Expansion. We did not rely on the
effective theory approaches such as QCD factorization because we do not know
how to reliably compute the relevant form factors and include all power
corrections. At the level of present accuracy however, both approaches lead to
compatible results.

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