#  $R^{\nu}_{K^{(*)}}$ and non-standard neutrino interactions

Xiao Gang He  and German Valencia

###### Abstract

We discuss the modes $B\to K^{(*)}\nu\bar{\nu}$ in the context of non-standard
neutrino interactions that add incoherently to the SM rates. We consider two
scenarios: an additional light neutrino; and neutrino lepton flavour
violation. We find that an additional light neutrino that interacts with SM
fields via a non-universal $Z^{\prime}$ can increase $R^{\nu}_{K^{(*)}}$ by up
to a factor of two without conflicting with $B_{s}-\bar{B}_{s}$ mixing. This
model then predicts rates for $B_{s}\to\tau^{+}\tau^{-}$ up to six times
larger than the SM. In the context of neutrino lepton flavour violation
mediated by leptoquarks we find that the current experimental upper bounds on
$R^{\nu}_{K^{(*)}}$ are already more constraining than direct bounds from
$B_{s}\to\tau\ell$ and $B\to K^{(*)}\tau\ell$ modes for $\ell=e,\mu$.

##  1 Introduction

Rare $B$ decays play an important role in understanding the dynamics of the
standard model (SM) as well as being a fertile ground for the search for new
physics. The $B\to K^{(*)}\nu\bar{\nu}$ decays are amongst the cleanest modes
to search for new physics due to their well controlled theoretical
uncertainty. Experiments at Belle and Babar have already published upper
limits on these modes at 2-3 times the SM rate and further improvement is
expected from Belle-II, which can reach a sensitivity on the branching ratios
of about 10% with 50 ab-1 [1]. Interesting constraints for certain BSM physics
can be obtained already with current bounds.

We consider two types of models that add incoherently to the SM rates. First
we entertain the possibility of a fourth light neutrino that couples to SM
fields through a non-universal $Z^{\prime}$. We find that existing constraints
on the model allow enhancements of the $B\to K^{(*)}\nu\bar{\nu}$ rates by up
to factors of two and that these are correlated with the
$B_{s}\to\tau^{+}\tau^{-}$ mode which could reach a rate up to six times
larger than in the SM.

We then consider possible contributions from neutrino flavour violating final
states in the context of scalar and vector leptoquarks. These contributions
are correlated to the charged lepton flavour violating (CLFV) modes
$B_{s}\to\ell\ell^{\prime}$ and $B\to K^{(*)}\ell\ell^{\prime}$ and we find
that the current limits on $B\to K^{(*)}\nu\bar{\nu}$ are more restrictive for
modes with tau-leptons. The leptoquark scenario also correlates $B\to
K^{(*)}\nu\bar{\nu}$ to the neutral and charged B anomalies, as has been
extensively discussed in the literature, and we comment on this.

Within the SM the effective Hamiltonian responsible for the $B\to
K^{(*)}\nu\bar{\nu}$ transitions originates at lowest order from box and
penguin diagrams and is usually written as [2]

$\displaystyle{\cal H}_{SM}$
$\displaystyle=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V^{\star}_{ts}C_{L}^{SM}\sum_{i}{\cal
O}^{ii}_{L}+{\rm~{}h.c.}$ $\displaystyle{\cal O}^{ii}_{L}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\nu}_{i}\gamma^{\mu}(1-\gamma_{5})\nu_{i}),$
(1)

with an accurately known Wilson coefficient that is independent of the
neutrino flavour and that including NLO QCD corrections [3] and two-loop
electroweak corrections [4] is given by

$\displaystyle C_{L}^{SM}=-\frac{X(x_{t})}{s^{2}_{W}},\quad X(x_{t})=1.469\pm
0.017.$ (2)

Typical SM predictions obtained with flavio are111These numbers agree within
errors with published numbers as in [5, 1]. Neglecting isospin breaking, the
neutral and charged modes have the same rates so we choose to present the two
modes with the strongest experimental limits.

$\displaystyle\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})_{SM}=(4.4\pm 0.7)\times
10^{-6},\quad\mathcal{B}(B^{0}\to K^{*0}\nu\bar{\nu})_{SM}=(9.5\pm 1.0)\times
10^{-6}.$ (3)

We list the best current experimental constraints on these modes in Table 1.

Mode | 90% c.l upper limit | Reference
---|---|---
$\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})$ | $1.6\times 10^{-5}$ | Babar [6]
$\mathcal{B}(B^{+}\to K^{*+}\nu\bar{\nu})$ | $4.0\times 10^{-5}$ | Belle [7]
$\mathcal{B}(B^{0}\to K^{0}\nu\bar{\nu})$ | $2.6\times 10^{-5}$ | Belle [8]
$\mathcal{B}(B^{0}\to K^{*0}\nu\bar{\nu})$ | $1.8\times 10^{-5}$ | Belle [8]
Table 1: Current experimental upper bounds on the modes considered.

Belle-II is expected to improve these limits, and has produced a preliminary
result $\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})\leq 4.1\times 10^{-5}$ at the
90% confidence level [9]. They have averaged this result with the previous
ones to arrive at $\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})=(1.1\pm 0.4)\times
10^{-5}$ [9]. For the $K^{*}$ channel, Belle has also combined the charged and
neutral modes to obtain the limit $\mathcal{B}(B\to K^{*}\nu\bar{\nu})\leq
2.7\times 10^{-5}$ [8] but here we will use the limit in Table 1. These
results are usually presented as ratios, for which we obtain

$\displaystyle R^{\nu}_{K}=\frac{\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})}{\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})_{SM}}=2.5\pm
1.0,\quad R^{\nu}_{K^{*}}=\frac{\mathcal{B}(B\to
K^{*0}\nu\bar{\nu})}{\mathcal{B}(B\to K^{*0}\nu\bar{\nu})_{SM}}\leq 1.9.$ (4)

The second number is simply the ratio of the limit in Table 1 and the central
value in Eq. 3 and somewhat lower than what is used in [10].

##  2 Effective Hamiltonian at the $b$ scale

We can parameterise any new physics relevant for these decays through an
effective Hamiltonian at the $b$ mass scale. The effective theory originates
in extensions of the SM containing new particles at or above the electroweak
scale that have been integrated out. In general, this results in additional
contributions to $C_{L}$ in Eq. 2 as well as in new operators. Because our
discussion is tied to two types of models, we only need to consider the
following

$\displaystyle{\cal
H}_{NP}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V^{\star}_{ts}\frac{e^{2}}{16\pi^{2}}$
$\displaystyle\sum_{ij}\left(C_{L}^{ij}{\cal O}_{L}^{ij}+C_{R}^{ij}{\cal
O}_{R}^{ij}+C_{L}^{\prime~{}ij}{\cal
O}_{L}^{\prime~{}ij}+C_{R}^{\prime~{}ij}{\cal O}_{R}^{\prime~{}ij}\right.$
$\displaystyle+\left.C_{9}^{ij}{\cal O}_{9}^{ij}+C_{10}^{ij}{\cal
O}_{10}^{ij}+C_{9^{\prime}}^{ij}{\cal
O}_{9^{\prime}}^{ij}+C_{10^{\prime}}^{ij}{\cal
O}_{10^{\prime}}^{ij}\right)+{\rm~{}h.c.}$ (5)

where the operators are

$\displaystyle{\cal O}_{L}^{ij}$
$\displaystyle=(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{\nu}_{i}\gamma^{\mu}(1-\gamma_{5})\nu_{j})$
$\displaystyle{\cal O}_{R}^{ij}$
$\displaystyle=(\bar{s}_{R}\gamma_{\mu}b_{R})(\bar{\nu}_{i}\gamma^{\mu}(1-\gamma_{5})\nu_{j})$
$\displaystyle{\cal O}_{L}^{\prime~{}ij}$
$\displaystyle=(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{\nu}_{i}\gamma^{\mu}(1+\gamma_{5})\nu_{j})$
$\displaystyle{\cal O}_{R}^{\prime~{}ij}$
$\displaystyle=(\bar{s}_{R}\gamma_{\mu}b_{R})(\bar{\nu}_{i}\gamma^{\mu}(1+\gamma_{5})\nu_{j})$
$\displaystyle{\cal O}_{9}^{ij}$
$\displaystyle=(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{\ell}_{i}\gamma^{\mu}\ell_{j})$
$\displaystyle{\cal O}_{10}^{ij}$
$\displaystyle=(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{\ell}_{i}\gamma^{\mu}\gamma_{5}\ell_{j})$
$\displaystyle{\cal O}_{9^{\prime}}^{ij}$
$\displaystyle=(\bar{s}_{R}\gamma_{\mu}b_{R})(\bar{\ell}_{i}\gamma^{\mu}\ell_{j})$
$\displaystyle{\cal O}_{10^{\prime}}^{ij}$
$\displaystyle=(\bar{s}_{R}\gamma_{\mu}b_{R})(\bar{\ell}_{i}\gamma^{\mu}\gamma_{5}\ell_{j})$
(6)

The Wilson coefficients are defined so that they only contain NP contributions
and the SM is counted separately through Eq. 2. The list in Eq. 6 includes
${\cal O}^{(\prime)}_{L}$, ${\cal O}^{(\prime)}_{R}$, which contribute to
$B^{(*)}\to K^{(*)}\nu\bar{\nu}$. It excludes operators with scalar and tensor
neutrino bi-linears that have been considered in [10] because they do not
appear in the models we discuss. The operators with charged leptons appear in
the models we discuss with coefficients that are related to
$C_{L,R}^{(\prime)ij}$. The flavour diagonal operators ${\cal
O}^{(\prime)\mu\mu}_{9,10}$ affect $b\to s\mu\bar{\mu}$ decays including the
$B\to K^{(*)}\mu\bar{\mu}$ anomalies and have been studied extensively in that
context.

We can classify the contributions to $B\to K^{(*)}\nu\bar{\nu}$ from these
operators into two types: those that interfere with the SM, ${\cal
O}_{L,R}^{ii}$; and those that do not, ${\cal O}_{L,R}^{i\neq j}$ and ${\cal
O}^{\prime~{}ij}_{L,R}$. In $B\to K\nu\bar{\nu}$ only the vector current
enters the hadronic matrix element so that the contributions to the rate from
${\cal O}_{L}^{ij}$ and ${\cal O}_{R}^{ij}$ are the same. Similarly for those
from ${\cal O}_{L}^{\prime~{}ij}$ and ${\cal O}_{R}^{\prime~{}ij}$. At the
same time, the different neutrino chirality eliminates interference between
the primed and un-primed-operators for massless neutrinos. The only operators
that interfere with the SM are thus the diagonal ones (in neutrino flavour)
${\cal O}_{L}^{ii}$ and ${\cal O}_{R}^{ii}$. The rates can be evaluated
numerically using $\tt flavio$ [11], and the central value (uncertainty will
be shown in the figures) is given approximately by

$\displaystyle\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})\times 10^{6}$
$\displaystyle\approx 4.39-0.457~{}{\rm
Re}\sum_{i}\left(C^{ii}_{L}+C^{ii}_{R}\right)$
$\displaystyle+0.0357~{}\sum_{ij}\left(\left|C^{ij}_{L}+C^{ij}_{R}\right|^{2}+\left|C_{L}^{\prime~{}ij}+C_{R}^{\prime~{}ij}\right|^{2}\right)$
(7)

In $B\to K^{*}\nu\bar{\nu}$ both the vector and axial-vector currents enter
the hadronic matrix element resulting in different contributions for ${\cal
O}_{L}^{ij}$ and ${\cal O}_{R}^{ij}$ as well as for ${\cal
O}_{L}^{\prime~{}ij}$ and ${\cal O}_{R}^{\prime~{}ij}$. Numerically, the rate
is approximately given by

$\displaystyle\mathcal{B}(B^{0}\to K^{\star 0}\nu\bar{\nu})\times 10^{6}$
$\displaystyle\approx 9.53+{\rm
Re}\sum_{i}\left(-0.993~{}C^{ii}_{L}+0.661~{}C^{ii}_{R}\right)$
$\displaystyle+\sum_{ij}\left(0.0775\left({C^{ij}_{L}}^{2}+{C^{ij}_{R}}^{2}+{C_{L}^{\prime~{}ij}}^{2}+{C_{R}^{\prime~{}ij}}^{2}\right)-0.103\left(C^{ij}_{L}C^{ij}_{R}+C_{L}^{\prime~{}ij}C_{R}^{\prime~{}ij}\right)\right)$
(8)

The parametric uncertainty in these predictions, as estimated by flavio is
illustrated in Fig. 1222The uncertainty in these predictions is around 15% and
is mostly due to the form factors for the $B\to K$ and $B\to K^{*}$ hadronic
transitions, which are responsible for about 10%, while the value of $V_{cb}$
contributes an additional 5%.. We show $\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})$ as a function of $C_{L}^{33}$ (the figure is identical for
$C_{L,R}^{ii}$) and as a function of $C_{L}^{23}$ (the figure is identical for
$C_{L}^{i\neq j}$ and $C_{L,R}^{\prime~{}ij}$). The red band marks the
experimental combination $\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})=(1.1\pm
0.4)\times 10^{-5}$ [9] and the green band the SM in Eq. 3 at 1$\sigma$. For
$\mathcal{B}(B^{0}\to K^{\star 0}\nu\bar{\nu})$ we show the dependence on
$C_{L}^{33}$ (same for all $C_{L}^{ii}$), $C_{R}^{33}$ (same for all
$C_{R}^{ii}$) and $C_{L}^{23}$ (same for all $C_{L}^{i\neq j}$ and
$C_{L,R}^{\prime~{}ij}$). In this case the red line shows the 90% c.l.
experimental upper bound from Table 1 and the green band the SM at 1$\sigma$.

Figure 1: Top row: $\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})$ as a function of
$C_{L}^{33}$ (left panel) and $C_{L}^{23}$ (right panel). Bottom row:
$\mathcal{B}(B^{0}\to K^{\star 0}\nu\bar{\nu})$ as a function of $C_{L}^{33}$
(left panel), $C_{R}^{33}$ (centre panel) and $C_{L}^{23}$ (right panel).

For example, new physics contributions allowed by the value of
$\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})$ in Eq. 7 at the 1$\sigma$ level and
taking only one non-zero parameter at a time are shown in Table 2 and can also
be read off Fig. 1.

Mode | $C^{ii}_{L}$ | $C^{ii}_{R}$ |  $|C^{i\neq j}_{L,R}|$ or $|C^{\prime~{}ij}_{L,R}|$
---|---|---|---
$1.5\leq R^{\nu}_{K}\leq 3.5$ |  $[-12,-4]$ or $[17,25]$ |  $[-12,-4]$ or $[17,25]$ | $[8.5,17]$
$R^{\nu}_{K^{*}}\leq 2.2$ | $[-6,18.5]$ | $[-15.5,7]$ | $[0,10.5]$
Table 2: Limits for the coefficients $C_{L,R}^{ij(\prime)}$ taken one at a
time implied by Eq. 4.

##  3 An additional light neutrino

The first type of new physics we consider that can increase the SM value of
$B\to K^{(*)}\nu\bar{\nu}$ consists of new light neutrinos. In fact, modes
with neutrino pairs in the final state count the number of light neutrinos in
the SM due to lepton universality. The existence of new light neutrinos is
severely constrained by measurements of the invisible $Z$ width and by
cosmological considerations. Assuming lepton universality, the former implies
that $N_{\nu}=2.9840\pm 0.0082$ [12]. Cosmological constraints depend on other
parameters and, for example, $\Delta N_{eff}<0.77$ for a Hubble constant
$H_{0}=71.3^{+1.9}_{-2.2}$ km/s/Mpc [13].

It is possible to avoid these limits with a light sterile neutrino that
interacts with the SM through a $Z^{\prime}$. The contribution of this
neutrino to the $Z$ width neutrino count is proportional to the square of the
$Z-Z^{\prime}$ mixing parameter and can thus be negligibly small. In addition,
if the $Z^{\prime}$ is non-universal and couples predominantly to the third
generation SM fermions, the new neutrino reaches thermal equilibrium with SM
particles at a temperature near the $\tau$-lepton mass. However, at the time
of big-bang nucleosynthesis the temperature is about 1 MeV and this difference
results in a suppression of the contribution of this neutrino to $\Delta
N_{eff}$ to a safe level, $\Delta N_{eff}<0.1$ [14].

We have previously constructed a detailed example of a model with these
properties [15, 16] so we do not repeat the details here. The $Z^{\prime}$ is
responsible for two new operators that contribute to $B\to
K^{(*)}\nu\bar{\nu}$ and to $B_{s}\to\tau^{+}\tau^{-}$ [17]:

$\displaystyle{\cal H}_{T}$
$\displaystyle=-\frac{G_{F}}{\sqrt{2}}2s^{2}_{W}\frac{M_{Z}^{2}}{M_{Z^{\prime}}^{2}}\cot^{2}\theta_{R}V^{d*}_{Rbs}V^{d}_{Rbb}\bar{s}_{R}\gamma_{\mu}b_{R}\
\left(\bar{\nu}_{R3}\gamma^{\mu}\nu_{R3}-\bar{\tau}_{R}\gamma^{\mu}\tau_{R}\right)$
$\displaystyle{\cal H}_{L}$
$\displaystyle=-\frac{G_{F}}{\sqrt{2}}\frac{\alpha}{\pi}\frac{M_{Z}^{2}}{M_{Z^{\prime}}^{2}}\cot^{2}\theta_{R}V^{*}_{ts}V_{tb}I(\lambda_{t},\lambda_{H})\bar{s}_{L}\gamma_{\mu}b_{L}\
\left(\bar{\nu}_{R3}\gamma^{\mu}\nu_{R3}-\bar{\tau}_{R}\gamma^{\mu}\tau_{R}\right)$
(9)

In the notation of Eq. 5, the Wilson coefficients that result in this model
are thus:

$\displaystyle C_{L}^{\prime~{}\tau\tau}=-C_{9}^{\tau\tau}=-C_{10}^{\tau\tau}$
$\displaystyle=2\left(\frac{m_{Z}^{2}}{m_{Z^{\prime}}^{2}}\right)\cot^{2}\theta_{R}I(\lambda_{t},\lambda_{H})$
$\displaystyle
C_{R}^{\prime~{}\tau\tau}=-C_{9^{\prime}}^{\tau\tau}=-C_{10^{\prime}}^{\tau\tau}$
$\displaystyle=4\left(\frac{V^{d*}_{Rbs}V^{d}_{Rbb}}{V_{tb}V^{\star}_{ts}}\right)\left(\frac{m_{Z}^{2}}{m_{Z^{\prime}}^{2}}\right)\cot^{2}\theta_{R}\frac{\pi
s^{2}_{W}}{\alpha}.$ (10)

The first operator in Eq. 9 originates in a flavour changing tree-level
exchange of the $Z^{\prime}$. The parameters that appear in this result are:
the $Z^{\prime}$ mass; a ratio parameterising the strength of the new
interaction relative to the weak interaction, $\cot\theta_{R}$; and two
elements of the matrix that rotates the down-type quarks between the weak and
the mass bases. The second operator in Eq. 9 arises from a new penguin diagram
and depends on details of the scalar sector through the Inami-Lim function
$I(\lambda_{t},\lambda_{H})$ [18]. The existing constraints on these
parameters can be summarised as:

  * •

A combination of perturbative unitarity [15] and LHC non-production of
$Z^{\prime}$ from $b\bar{b}$ annihilation [19] restrict the overall strength
of the new interaction to

$\displaystyle\left(\frac{m_{Z}^{2}}{m_{Z^{\prime}}^{2}}\right)\cot^{2}\theta_{R}\lesssim
0.15.$ (11)

  * •

$B_{s}$ mixing constrains $|V^{d}_{Rbs}V^{d*}_{Rbb}|$ and
$I(\lambda_{t},\lambda_{H})$ [17]. Both the SM calculation and the
experimental situation regarding $\Delta M_{B_{s}}$ have changed significantly
so we repeat that analysis here.

In terms of the parameters of interest, the effective Hamiltonian below the
$Z^{\prime}$ scale is:

$\displaystyle{\cal H}=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\left(\frac{m_{Z}^{2}}{m_{Z^{\prime}}^{2}}\right)\cot^{2}\theta_{R}\left(\left(\frac{\alpha}{2\pi
s_{W}}V^{*}_{tb}V_{ts}\right)^{2}I(\lambda_{t},\lambda_{H})^{2}~{}{\cal
O}_{LL}\right.$
$\displaystyle+\left.s_{W}^{2}(V^{d}_{Rbs}V^{d*}_{Rbb})^{2}~{}{\cal
O}_{RR}+2s_{W}\left(\frac{\alpha}{2\pi
s_{W}}V^{*}_{tb}V_{ts}\right)I(\lambda_{t},\lambda_{H})(V^{d}_{Rbs}V^{d*}_{Rbb})~{}{\cal
O}_{LR}\right)$ (12)

with the usual $\Delta B=2$ operators:

$\displaystyle{\cal
O}_{LL,RR}=(\bar{s}_{L,R}\gamma_{\mu}b_{L,R})(\bar{s}_{L,R}\gamma_{\mu}b_{L,R}),\quad{\cal
O}_{LR}=(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{s}_{R}\gamma_{\mu}b_{R}).$ (13)

QCD renormalisation group running modifies the Wilson coefficients and
introduces one more operator at the $b$ scale, ${\cal
O}_{SLR}=(\bar{s}_{R}b_{L})(\bar{s}_{L}b_{R})$. Making use of flavio once
more, we find

$\frac{\Delta M_{B_{s}}}{(\Delta M_{B_{s}})_{SM}}\approx 1+1.5\times
10^{-4}I(\lambda_{t},\lambda_{H})^{2}-8.2~{}I(\lambda_{t},\lambda_{H})|V^{d}_{Rbs}V^{d*}_{Rbb}|+3282.~{}|V^{d}_{Rbs}V^{d*}_{Rbb}|^{2}.$
(14)

The current experimental average $\Delta M_{B_{s}}=(17.741\pm
0.020){\rm~{}ps}^{-1}$ [20] combined with a recent SM prediction $(\Delta
M_{B_{s}})_{SM}=(18.4^{+0.7}_{-1.2}){\rm~{}ps}^{-1}$ [21] results in $\Delta
M_{B_{s}}/(\Delta M_{B_{s}})_{SM}=0.96^{+0.4}_{-0.6}$ and we compare this
ratio to the prediction of Eq. 14 in Fig. 2 for $I(\lambda_{t},\lambda_{H})=0$
and $I(\lambda_{t},\lambda_{H})=3$. These two values were chosen because a
scan over the parameters in the model [18] suggests $-2\lesssim
I(\lambda_{t},\lambda_{H})\lesssim 3$ as a range for the Inami-Lim function.
The allowed range for $C_{L,R}^{\prime\tau\tau}$, assuming that
$V^{d}_{Rbs}V^{d*}_{Rbb}$ is real, is then showed in the right panel of Fig.
2. The key point is that the tree level $Z^{\prime}$ exchange tends to
increase $\Delta M_{B_{s}}$ over its SM value and this is severely constrained
by current data. It is the new penguin contribution to ${\cal O}_{LR}$ and
${\cal O}_{SLR}$ that allows $\Delta M_{B_{s}}$ to drift below its SM value.
As can be seen from Eq. 14, allowing $V^{d}_{Rbs}V^{d*}_{Rbb}$ to have a phase
can augment the allowed parameter range but a complete phenomenological study
of this general case is beyond the scope of the present work.

Figure 2: Left panel: $\Delta M_{B_{s}}/(\Delta M_{B_{s}})_{SM}$ as a function
of $|V^{d}_{Rbs}V^{d*}_{Rbb}|$ for $I(\lambda_{t},\lambda_{H})=0,3$ showing
parametric uncertainty. Right panel: values taken by
$C_{L,R}^{\prime\tau\tau}$ with parameters that satisfy $0.9\leq\Delta
M_{B_{s}}/(\Delta M_{B_{s}})_{SM}\leq 1.0$ assuming that
$V^{d}_{Rbs}V^{d*}_{Rbb}$ is real.

The allowed region in $C_{L,R}^{\prime\tau\tau}$ shown in Fig. 2 results in an
increase of the $B\to K^{(*)}\nu\bar{\nu}$ rates over their SM value and this
is shown in the left panel of Fig. 3. The figure indicates a near perfect
correlation between the two neutrino modes and the largest values, near two,
are obtained for the upper-right corner of the region in Fig. 2.
Interestingly, the same model with the constraint of Eq. 11 can no longer
enhance $K\to\pi\nu\bar{\nu}$ modes by more than a few percent.333In the
notation of [22] it predicts $\tilde{X}\leq 0.1$. The main difference between
these two cases is the strong constraint on $V^{d*}_{Rbd}$ (from $B_{d}$
mixing) that enters $K\to\pi\nu\bar{\nu}$ in place of $V^{d*}_{Rbb}$ which can
be close to one.

The model predicts through Eq. 10 a correlation between $B\to
K^{(*)}\nu\bar{\nu}$ modes and $B_{s}\to\tau^{+}\tau^{-}$. The latter
currently only has a weak experimental limit from LHCb
$\mathcal{B}(B_{s}\to\tau^{+}\tau^{-})\leq 6.8\times 10^{-3}$ at 95%c.l. [23].
The prediction can be written in a very simple form because only the hadronic
axial vector current contributes,

$\displaystyle\frac{\mathcal{B}(B_{s}\to\tau^{+}\tau^{-})}{\mathcal{B}(B_{s}\to\tau^{+}\tau^{-})_{SM}}=\left(1+\frac{(C_{10}^{\tau\tau}-C_{10^{\prime}}^{\tau\tau})}{C_{10}^{\tau\tau
SM}}\right)^{2}.$ (15)

The allowed parameter region seen in Fig. 2, combined with $C_{10}^{\tau\tau
SM}=-Y(x_{t})/s_{W}^{2}\sim-4.3$ [3], implies that the model allows

$\frac{\mathcal{B}(B_{s}\to\tau^{+}\tau^{-})}{\mathcal{B}(B_{s}\to\tau^{+}\tau^{-})_{SM}}\lesssim
6.$ (16)

This can be read off Fig. 3 which illustrates the correlation with
$R^{\nu}_{K}$. On the other hand, the model also allows for
$(C_{10^{\prime}}^{\tau\tau}-C_{10}^{\tau\tau})\sim-4$ where the
$\mathcal{B}(B_{s}\to\tau^{+}\tau^{-})$ is significantly suppressed with
respect to its SM value.

Figure 3: Correlation between $R^{\nu}_{K}$ and $R^{\nu}_{K^{*}}$ (left panel)
and between $\mathcal{B}(B_{s}\to\tau^{+}\tau^{-})$ and $R^{\nu}_{K}$ (right
panel) for the parameter space allowed by $B_{s}$ mixing as shown in Fig. 2.

The corresponding Wilson coefficients affecting the modes $b\to
s\mu^{+}\mu^{-}$ exhibiting anomalies,
$C_{9,10,9^{\prime},10^{\prime}}^{\mu\mu}$ are suppressed with respect
to$C_{9,10,9^{\prime},10^{\prime}}^{\tau\tau}$ by factors
$|V^{\ell}_{R3\mu}|^{2}$ which can be very small [24]. For this reason, this
model yields predictions for $b\to s\mu^{+}\mu^{-}$ processes that are very
similar to the SM. Correlations between these dimuon modes and $b\to
s\nu\bar{\nu}$ modes have also been explored in other models [25, 26].

##  4 Models with leptoquarks

In this section we consider models that can increase $R^{\nu}_{K^{(*)}}$ by
producing final states with neutrino pairs of different lepton flavour, but
with only the three SM neutrinos. The starting point is then scalar $S$ and
vector $V$ leptoquarks with couplings to SM fermions which include a left-
handed neutrino $\nu_{L}$ of any flavour. They are [27, 28] ,

$\displaystyle{\cal L}_{S}=$
$\displaystyle\lambda_{LS_{0}}\bar{q}^{c}_{L}i\tau_{2}\ell_{L}S_{0}^{\dagger}+\lambda_{L\tilde{S}_{1/2}}\bar{d}_{R}\ell_{L}\tilde{S}^{\dagger}_{1/2}+\lambda_{LS_{1}}\bar{q}^{c}_{L}i\tau_{2}\vec{\tau}\cdot\vec{S}^{\dagger}_{1}\ell_{L}+{\rm~{}h.~{}c}.\;,$
$\displaystyle{\cal L}_{V}=$
$\displaystyle\lambda_{LV_{1/2}}\bar{d}_{R}^{c}\gamma_{\mu}\ell_{L}V^{\dagger\mu}_{1/2}+\lambda_{LV_{1}}\bar{q}_{L}\gamma_{\mu}\vec{\tau}\cdot\vec{V}^{\dagger\mu}_{1}\ell_{L}+{\rm~{}h.~{}c.}\;,$
(17)

where the leptoquark fields and their transformation properties under the SM
group are given by

$\displaystyle S^{\dagger}_{0}$ $\displaystyle=S_{0}^{1/3}:(\bar{3},1,1/3)$
$\displaystyle\tilde{S}_{1/2}^{\dagger}$
$\displaystyle=\left(\tilde{S}_{1/2}^{-1/3},\tilde{S}_{1/2}^{2/3}\right):(3,2,1/6)$
$\displaystyle\vec{\tau}\cdot\vec{S}_{1}^{\dagger}$
$\displaystyle=\left(\begin{array}[]{cc}S^{1/3}_{1}&\sqrt{2}S^{4/3}_{1}\\\
\sqrt{2}S^{-2/3}_{1}&-S^{1/3}_{1}\end{array}\right):(\bar{3},3,1/3)$ (20)
$\displaystyle V_{1/2}^{\dagger}$
$\displaystyle=\left(V_{1/2}^{1/3},V_{1/2}^{4/3}\right):(\bar{3},2,5/6)$
$\displaystyle\vec{\tau}\cdot\vec{V}_{1}^{\dagger}$
$\displaystyle=\left(\begin{array}[]{cc}V^{2/3}_{1}&\sqrt{2}V^{5/3}_{1}\\\
\sqrt{2}V^{-1/3}_{1}&-V^{2/3}_{1}\end{array}\right):(3,3,2/3)$ (23)

Exchange of these particles at tree-level, assuming leptoquark multiplets that
are degenerate in mass, generates the following effective Lagrangian

$\displaystyle{\cal L}_{eff}$
$\displaystyle=\frac{\lambda^{ij}_{LS_{0}}\lambda^{*kl}_{LS_{0}}}{2m_{S_{0}}^{2}}\left(\bar{d}_{Lk}\gamma_{\mu}d_{Li}\bar{\nu}_{L_{l}}\gamma^{\mu}\nu_{Lj}+\bar{u}_{Lk}\gamma_{\mu}u_{Li}\bar{e}_{L_{l}}\gamma^{\mu}e_{Lj}-\bar{u}_{Lk}\gamma_{\mu}d_{Li}\bar{e}_{L_{l}}\gamma^{\mu}\nu_{Lj}-\bar{d}_{Lk}\gamma_{\mu}u_{Li}\bar{\nu}_{L_{l}}\gamma^{\mu}e_{Lj}\right)$
$\displaystyle+\frac{\lambda^{ij}_{LS_{1}}\lambda^{*kl}_{LS_{1}}}{2m_{S_{1}}^{2}}\left(\bar{d}_{Lk}\gamma_{\mu}d_{Li}\bar{\nu}_{L_{l}}\gamma^{\mu}\nu_{Lj}+2\bar{d}_{Lk}\gamma_{\mu}d_{Li}\bar{e}_{L_{l}}\gamma^{\mu}e_{Lj}+2\bar{u}_{Lk}\gamma_{\mu}u_{Li}\bar{\nu}_{L_{l}}\gamma^{\mu}\nu_{Lj}+\bar{u}_{Lk}\gamma_{\mu}u_{Li}\bar{e}_{L_{l}}\gamma^{\mu}e_{Lj}\right)$
$\displaystyle-\frac{\lambda^{kj}_{LV_{1}}\lambda^{*il}_{LV_{1}}}{m_{V_{1}}^{2}}\left(2\bar{d}_{Lk}\gamma_{\mu}d_{Li}\bar{\nu}_{L_{l}}\gamma^{\mu}\nu_{Lj}+\bar{d}_{Lk}\gamma_{\mu}d_{Li}\bar{e}_{L_{l}}\gamma^{\mu}e_{Lj}+\bar{u}_{Lk}\gamma_{\mu}u_{Li}\bar{\nu}_{L_{l}}\gamma^{\mu}\nu_{Lj}+2\bar{u}_{Lk}\gamma_{\mu}u_{Li}\bar{e}_{L_{l}}\gamma^{\mu}e_{Lj}\right)$
$\displaystyle-\frac{\lambda^{kj}_{L\tilde{S}_{1/2}}\lambda^{*il}_{L\tilde{S}_{1/2}}}{2m_{S_{1/2}}^{2}}\left(\bar{d}_{Rk}\gamma_{\mu}d_{Ri}\bar{\nu}_{L_{l}}\gamma^{\mu}\nu_{Lj}+\bar{d}_{Rk}\gamma_{\mu}d_{Ri}\bar{e}_{L_{l}}\gamma^{\mu}e_{Lj}\right)$
$\displaystyle+\frac{\lambda^{ij}_{LV_{1/2}}\lambda^{*kl}_{LV_{1/2}}}{m_{V_{1/2}}^{2}}\left(\bar{d}_{Rk}\gamma_{\mu}d_{Ri}\bar{\nu}_{L_{l}}\gamma^{\mu}\nu_{Lj}+\bar{d}_{Rk}\gamma_{\mu}d_{Ri}\bar{e}_{L_{l}}\gamma^{\mu}e_{Lj}\right).$
(24)

If the fermions in Eq. 24 are in their weak eigenstate basis, rotation to the
mass eigenstate basis will introduce mixing angles. Here we will work with
$\lambda^{ij}_{I}$ defined in a basis in which the down-type fermions are
already mass eigenstates [29]. The $\nu$ and up-type quarks need to be further
rotated by $u_{Lk}=(V^{*}_{KM})_{km}u_{Lm}$, and
$\nu_{Lk}=(V_{PMNS})_{km}\nu_{Lm}$ respectively. However, since the neutrino
flavour is not measured, working in either their weak or mass basis yields the
same results. Collecting the Wilson coefficients for Eq. 6 gives,

$\displaystyle C_{L}^{ij}$ $\displaystyle=\frac{\pi}{\sqrt{2}\alpha
G_{F}V_{tb}V_{ts}^{*}}\left(\frac{\lambda^{3j}_{LS_{0}}\lambda^{*2i}_{LS_{0}}}{2m_{S_{0}}^{2}}+\frac{\lambda^{3j}_{LS_{1}}\lambda^{*2i}_{LS_{1}}}{2m_{S_{1}}^{2}}-2\frac{\lambda^{2j}_{LV_{1}}\lambda^{*3i}_{LV_{1}}}{m_{V_{1}}^{2}}\right),$
$\displaystyle C_{R}^{ij}$
$\displaystyle=C_{9^{\prime}}^{ij}=-C_{10^{\prime}}^{ij}=\frac{\pi}{\sqrt{2}\alpha
G_{F}V_{tb}V_{ts}^{*}}\left(-\frac{\lambda^{2j}_{L\tilde{S}_{1/2}}\lambda^{*3i}_{L\tilde{S}_{1/2}}}{2m_{S_{1/2}}^{2}}+\frac{\lambda^{3j}_{LV_{1/2}}\lambda^{*2i}_{LV_{1/2}}}{m_{V_{1/2}}^{2}}\right),$
$\displaystyle C_{9}^{ij}$
$\displaystyle=-C_{10}^{ij}=\frac{\pi}{\sqrt{2}\alpha
G_{F}V_{tb}V_{ts}^{*}}\left(\frac{\lambda^{3j}_{LS_{1}}\lambda^{*2i}_{LS_{1}}}{m_{S_{1}}^{2}}-\frac{\lambda^{2j}_{LV_{1}}\lambda^{*3i}_{LV_{1}}}{m_{V_{1}}^{2}}\right).$
(25)

All of these leptoquarks contribute to $R^{\nu}_{K^{(*)}}$ but their
contributions are correlated with different modes [30, 31, 32, 33]. We begin
with the lepton flavour number violating case which adds incoherently to the
SM values for $R^{\nu}_{K^{(*)}}$. There are several CLFV modes with existing
experimental upper bounds and we list them in Table 3. The corresponding
predictions using Eq. 25 are

$\displaystyle\mathcal{B}(B_{s}\to e^{\pm}\mu^{\mp})$
$\displaystyle\approx.98\left(C_{10}^{\mu
e~{}2}+C_{10^{\prime}}^{e\mu~{}2}+C_{10^{\prime}}^{\mu
e~{}2}+C_{10}^{e\mu~{}2}\right)\times 10^{-10}$
$\displaystyle\mathcal{B}(B_{s}\to\mu^{\pm}\tau^{\mp})$
$\displaystyle\approx.22\left(C_{10}^{\mu\tau~{}2}+C_{10^{\prime}}^{\tau\mu~{}2}+C_{10^{\prime}}^{\mu\tau~{}2}+C_{10}^{\tau\mu~{}2}\right)\times
10^{-7}$ $\displaystyle\mathcal{B}(B^{+}\to K^{+}e^{-}\mu^{+})$
$\displaystyle\approx 0.18\left((C_{10}^{\mu e}+C_{10^{\prime}}^{\mu
e})^{2}+(C_{9}^{\mu e}+C_{9^{\prime}}^{\mu e})^{2}\right)\times 10^{-7}$
$\displaystyle\mathcal{B}(B^{+}\to K^{+}\tau^{+}e^{-})$ $\displaystyle\approx
0.114\left((C_{10}^{e\tau}+C_{10^{\prime}}^{e\tau})^{2}+(C_{9}^{e\tau}+C_{9^{\prime}}^{e\tau})^{2}\right)\times
10^{-7}$ $\displaystyle\mathcal{B}(B^{+}\to K^{+}\tau^{+}\mu^{-})$
$\displaystyle\approx\left(0.117(C_{10}^{\mu\tau}+C_{10^{\prime}}^{\mu\tau})^{2}+0.111(C_{9}^{\mu\tau}+C_{9^{\prime}}^{\mu\tau})^{2}\right)\times
10^{-7}$ $\displaystyle\mathcal{B}(B^{+}\to K^{*+}e^{-}\mu^{+})$
$\displaystyle\approx 0.18\left((C_{10}^{\mu e}+C_{10^{\prime}}^{\mu
e})^{2}+(C_{9}^{\mu e}+C_{9^{\prime}}^{\mu e})^{2}\right)\times 10^{-7}$
$\displaystyle\mathcal{B}(B^{0}\to K^{*0}e^{-}\mu^{+})$ $\displaystyle\approx
0.18\left((C_{10}^{\mu e}+C_{10^{\prime}}^{\mu e})^{2}+(C_{9}^{\mu
e}+C_{9^{\prime}}^{\mu e})^{2}\right)\times 10^{-7}$ (26)

The best current experimental bounds on these modes as given in [20] are
listed in Table 3 along with the constraints they impose on the Wilson
coefficients taken one non-zero at a time.

Mode | 90% c.l | one $|C_{i}^{\ell\ell^{\prime}}|\neq 0$ |  $C_{9^{\prime}}^{\ell\ell^{\prime}}=-C_{10^{\prime}}^{\ell\ell^{\prime}}$ or
---|---|---|---
|  | at a time | $C_{9}^{\ell\ell^{\prime}}=-C_{10}^{\ell\ell^{\prime}}$
$\mathcal{B}(B_{s}\to e^{\pm}\mu^{\mp})$ | $5.4\times 10^{-9}$ | 7.4 | 7.4
$\mathcal{B}(B_{s}\to\mu^{\pm}\tau^{\mp})$ | $4.2\times 10^{-5}$ | 44 | 44
$\mathcal{B}(B^{+}\to K^{+}e^{-}\mu^{+})$ | $6.4\times 10^{-9}$ | 0.6 | 0.4
$\mathcal{B}(B^{+}\to K^{+}e^{-}\tau^{+})$ | $1.5\times 10^{-5}$ | 36 | 25
$\mathcal{B}(B^{+}\to K^{+}\mu^{-}\tau^{+})$ | $2.8\times 10^{-5}$ | 49 | 35
$\mathcal{B}(B^{+}\to K^{*+}e^{-}\mu^{+})$ | $9.9\times 10^{-7}$ | 7.4 | 5.2
$\mathcal{B}(B^{0}\to K^{*0}e^{-}\mu^{+})$ | $1.2\times 10^{-7}$ | 2.6 | 1.8
Table 3: Current experimental upper bounds on lepton flavour changing modes
and the limits they imply for the coefficients $C_{i}^{\ell\ell^{\prime}}$ of
Eq. 26 taken one non-zero at a time for the corresponding lepton flavour
indices. The last column shows the upper bound on
$|C_{i}^{\ell\ell^{\prime}}|$ assuming that
$C_{9^{\prime}}^{ij}=-C_{10^{\prime}}^{ij}\neq 0$, $C_{9}^{ij}=-C_{10}^{ij}=0$
or $C_{9^{\prime}}^{ij}=-C_{10^{\prime}}^{ij}=0$, $C_{9}^{ij}=-C_{10}^{ij}\neq
0$ as per Eq. 25.

The minimal set of Wilson coefficients consistent with the leptoquark origin
of Eq. 25 implies more than one non-zero Wilson coefficient at a time, either
$C_{9^{\prime}}^{ij}=-C_{10^{\prime}}^{ij}\neq 0$, $C_{9}^{ij}=-C_{10}^{ij}=0$
or $C_{9^{\prime}}^{ij}=-C_{10^{\prime}}^{ij}=0$, $C_{9}^{ij}=-C_{10}^{ij}\neq
0$. Both situations result in the same bound due to the symmetry between
primed and unprimed coefficients in Eq. 26. Without additional assumptions on
the leptoquark couplings, in particular allowing $C_{i}^{\ell\ell^{\prime}}$
to differ from $C_{i}^{\ell^{\prime}\ell}$ the tightest bounds that follow in
this case are shown in the last column of Table 3.

To explore the connection with $R^{\nu}_{K^{(*)}}$ it is useful to consider
Eq. 25 for each leptoquark multiplet separately. We see that $S_{0}$ only
produces $C_{L}^{ij}$ and is therefore not correlated with CLFV modes. $S_{1}$
and $V_{1}$ generate $C_{L}^{ij}$ and $C_{9,10}^{ij}$ whereas $S_{1/2}$ and
$V_{1/2}$ induce $C_{R}^{ij}$ and $C_{9^{\prime},10^{\prime}}^{ij}$ resulting
in all cases in $R_{K}^{\nu}=R_{K^{*}}^{\nu}$. To study the numerical
implications of these predictions we consider one lepton flavour pair at a
time and present the results in Table 4. These numbers indicate that the CLFV
are currently less restrictive on these leptoquark couplings than
$R_{K^{(*)}}^{\nu}$, Eq. 4, except for the $e\mu$ modes.

LQ | upper bound on $C_{L,R}^{ij}$ | $R^{\nu}_{K}=R^{\nu}_{K^{*}}$
---|---|---
|  | $\mu e$ | $e\tau$ | $\mu\tau$
$S^{0}$ | $|C_{L}^{\mu e}|\lesssim 0.4$ | $|C_{L}^{e\tau}|\lesssim 26$ | $|C_{L}^{\mu\tau}|\lesssim 35$ | 1.001 | 6.4 | 11
$\tilde{S}_{1/2}$ | $|C_{R}^{\mu e}|\lesssim 0.4$ | $|C_{R}^{e\tau}|\lesssim 26$ | $|C_{R}^{\mu\tau}|\lesssim 35$ | 1.001 | 6.4 | 11
${S}_{1}$ | $|C_{L}^{\mu e}|\lesssim 0.2$ | $|C_{L}^{e\tau}|\lesssim 13$ | $|C_{L}^{\mu\tau}|\lesssim 18$ | 1.0003 | 2.4 | 3.5
$V_{1/2}$ | $|C_{R}^{\mu e}|\lesssim 0.4$ | $|C_{R}^{e\tau}|\lesssim 26$ | $|C_{R}^{\mu\tau}|\lesssim 35$ | 1.001 | 6.4 | 11
$V_{1}^{\dagger}$ | $|C_{L}^{\mu e}|\lesssim 0.8$ | $|C_{L}^{e\tau}|\lesssim 52$ | $|C_{L}^{\mu\tau}|\lesssim 70$ | 1.005 | 23 | 40
Table 4: Implications of limits on charged lepton flavour violation for
$R_{K^{(*)}}^{\nu}$ for different leptoquarks shown separately for each lepton
flavour combination.

###  4.1 The B anomalies

Leptoquark models have been studied extensively in the context of the B
anomalies and we comment on that here. The neutral B anomalies are observed in
$b\to s\mu\bar{\mu}$ modes, and Eq. 24 shows that, with the exception of
$S_{0}$, these leptoquarks correlate $\bar{s}b\bar{\nu}^{\mu}\nu^{\mu}$ with
$\bar{s}b\bar{\mu}\mu$ operators. In particular

$\displaystyle S_{1}$ $\displaystyle\implies
C_{9}^{\mu\mu}=-C_{10}^{\mu\mu}=2C_{L}^{\mu\mu}$ $\displaystyle V_{1}$
$\displaystyle\implies
C_{9}^{\mu\mu}=-C_{10}^{\mu\mu}=\frac{1}{2}C_{L}^{\mu\mu}$ $\displaystyle
S_{1/2}$ $\displaystyle\implies
C_{9^{\prime}}^{\mu\mu}=-C_{10^{\prime}}^{\mu\mu}=2C_{R}^{\mu\mu}$
$\displaystyle V_{1/2}$ $\displaystyle\implies
C_{9^{\prime}}^{\mu\mu}=-C_{10^{\prime}}^{\mu\mu}=2C_{R}^{\mu\mu}$ (27)

Extensive fits to data from modes induced by $b\to s\mu\bar{\mu}$ indicate
that $C_{9}^{\mu\mu}=-C_{10}^{\mu\mu}=-0.46$ [34]444The precise number varies
depending on the fit, a more recent one gives -0.41 instead [35]. is a
possible solution whereas $C_{9^{\prime}}^{\mu\mu}=-C_{10^{\prime}}^{\mu\mu}$
is not [36]. This implies that

$R^{\nu}_{K^{(*)}}=\left\\{\begin{array}[]{c c}1.02&{\rm~{}for~{}}S_{1}\\\
1.1&{\rm~{}for~{}}V_{1}\end{array}\right.$ (28)

and $R^{\nu}_{K^{(*)}}=1$ for $S_{1/2},~{}V_{1/2}$.

In a similar manner $S_{0}$ correlates $\bar{s}b\bar{\nu}^{\tau}\nu^{i}$ with
$\bar{c}b\bar{\tau}\nu^{i}$ and therefore relates $R^{\nu}_{K^{(*)}}$ to the
so-called charged B-anomalies, $R(D^{(*)})$. We use the latest experimental
and theoretical averages from [37] (adding errors in quadrature and using
their arithmetic average of theoretical results) in terms of the ratios

$r_{D}=\frac{R(D)}{R(D)_{SM}}=1.14\pm 0.10,\quad
r_{D^{*}}=\frac{R(D)}{R(D^{*})_{SM}}=1.14\pm 0.06$ (29)

Depending on the neutrino lepton flavour, the operator will interfere or not
with the SM and both cases were considered in [38, 39]. The results are

$\displaystyle r_{D^{(*)}}$
$\displaystyle=|\Delta^{3,2}|^{2}+|\Delta^{3,2}|^{2}+|1+\Delta^{3,3}|^{2}$
$\displaystyle\Delta^{3,j}$
$\displaystyle=-\frac{\sqrt{2}}{4G_{F}V_{cb}}\sum_{i}V_{ci}\frac{\lambda^{i3*}_{LS_{0}}\lambda^{3j}_{LS_{0}}}{2m^{2}_{S_{0}}}$
(30)

The correlations simplify at leading order in CKM angles, where the term with
$V_{cs}$ dominates resulting in,

$C_{L}^{3j}=-\frac{2\pi}{\alpha}\Delta^{3,2}_{j}=\frac{\pi}{\sqrt{2}\alpha
G_{F}V_{cb}}\frac{\lambda^{23*}_{LS_{0}}\lambda^{3j}_{LS_{0}}}{2m^{2}_{S_{0}}}$
(31)

Taking one non-zero parameter at a time for this case, and assuming that the
leptoquark contribution results in the central value of Eq. 29 requires
$C_{L}^{33}\sim 54$ or $C_{L}^{31},C_{L}^{32}\sim 300$, both much larger than
allowed by $R^{\nu}_{K^{(*)}}$ as quantified in Table 2. Equivalently, the
most favourable scenario from Table 2, $C_{L}^{33}\sim 25$ would result in

$r_{D^{(*)}}\sim 1.06$ (32)

More complex leptoquark scenarios have been invoked in the study of the B
anomalies where it is possible to avoid a conflict with $R^{\nu}_{K^{(*)}}$
[40, 41].

##  5 Summary

We have studied the modes $B\to K^{(*)}\nu\bar{\nu}$ in the context of non-
standard neutrino interactions. We first considered a model with an additional
light neutrino that couples to a non-universal $Z^{\prime}$ and found that it
can result in $R^{\nu}_{K^{(*)}}$ close to two. The same model can also
enhance $B_{s}\to\tau^{+}\tau^{-}$ by up to a factor six over the SM within
the parameter range allowed by $B_{s}$ mixing and non-production of the
$Z^{\prime}$ at LHC.

Next we considered augmenting $R^{\nu}_{K^{(*)}}$ through neutrino lepton
flavour violating modes. We parameterised this possibility through scalar and
vector leptoquark exchange. This type of model correlates $R^{\nu}_{K^{(*)}}$
with CLFV modes $B_{s}\to\ell\ell^{\prime}$ and $B\to
K^{(*)}\ell\ell^{\prime}$ and we found that the former is currently more
restrictive than $e\tau$ and $\mu\tau$ CLFV modes.

Finally we briefly commented on the correlation with the B anomalies. In this
case we saw that global fits to $b\to s\mu\mu$ modes constrain
$C_{L}^{\mu\mu}$ for $S_{1}$ and $V_{1}$ leptoquarks so that, by itself, it
cannot add more than 10% to $R^{\nu}_{K^{(*)}}$. Similarly, current
measurements of $R^{\nu}_{K^{(*)}}$ constrain the parameters of $S_{0}$
couplings so that $R(D^{(*)})$ can be at most 1.06.

## Acknowledgments

This work was supported in part by the Australian Government through the
Australian Research Council. XGH was supported in part by Key Laboratory for
Particle Physics, Astrophysics and Cosmology, Ministry of Education, Shanghai
Key Laboratory for Particle Physics and Cosmology (Grant No. 15DZ2272100), in
part by the NSFC (Grant Nos. 11735010, 11975149, and 12090064), and also
supported in part by the MOST (Grant No. MOST 106-2112-M-002-003-MY3 ).

## References

  * [1] Belle-II collaboration, W. Altmannshofer et al., _The Belle II Physics Book_ , _PTEP_ 2019 (2019) 123C01, [1808.10567].
  * [2] G. Buchalla, A. J. Buras and M. E. Lautenbacher, _Weak decays beyond leading logarithms_ , _Rev. Mod. Phys._ 68 (1996) 1125–1144, [hep-ph/9512380].
  * [3] G. Buchalla and A. J. Buras, _The rare decays $K\to\pi\nu\bar{\nu}$, $B\to X\nu\bar{\nu}$ and $B\to l^{+}l^{-}$: An Update_, _Nucl. Phys._ B548 (1999) 309–327, [hep-ph/9901288].
  * [4] J. Brod, M. Gorbahn and E. Stamou, _Two-Loop Electroweak Corrections for the $K\to\pi\nu\bar{\nu}$ Decays_, _Phys. Rev._ D83 (2011) 034030, [1009.0947].
  * [5] T. Blake, G. Lanfranchi and D. M. Straub, _Rare $B$ Decays as Tests of the Standard Model_, _Prog. Part. Nucl. Phys._ 92 (2017) 50–91, [1606.00916].
  * [6] BaBar collaboration, J. P. Lees et al., _Search for $B\to K^{(*)}\nu\overline{\nu}$ and invisible quarkonium decays_, _Phys. Rev. D_ 87 (2013) 112005, [1303.7465].
  * [7] Belle collaboration, O. Lutz et al., _Search for $B\to h^{(*)}\nu\bar{\nu}$ with the full Belle $\Upsilon(4S)$ data sample_, _Phys. Rev. D_ 87 (2013) 111103, [1303.3719].
  * [8] Belle collaboration, J. Grygier et al., _Search for $\boldsymbol{B\to h\nu\bar{\nu}}$ decays with semileptonic tagging at Belle_, _Phys. Rev. D_ 96 (2017) 091101, [1702.03224].
  * [9] Belle-II collaboration, F. Dattola, _Search for $B^{+}\to K^{+}\nu\bar{\nu}$ decays with an inclusive tagging method at the Belle II experiment_, in _55th Rencontres de Moriond on Electroweak Interactions and Unified Theories_ , 5, 2021.  2105.05754.
  * [10] T. E. Browder, N. G. Deshpande, R. Mandal and R. Sinha, _Impact of $B\to K\nu\bar{\nu}$ measurements on beyond the Standard Model theories_, 2107.01080.
  * [11] D. M. Straub, _flavio: a Python package for flavour and precision phenomenology in the Standard Model and beyond_ , 1810.08132.
  * [12] ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavour Group collaboration, S. Schael et al., _Precision electroweak measurements on the $Z$ resonance_, _Phys. Rept._ 427 (2006) 257–454, [hep-ex/0509008].
  * [13] J. L. Bernal, L. Verde and A. G. Riess, _The trouble with $H_{0}$_, _JCAP_ 1610 (2016) 019, [1607.05617].
  * [14] A. D. Dolgov, _Neutrinos in cosmology_ , _Phys. Rept._ 370 (2002) 333–535, [hep-ph/0202122].
  * [15] X.-G. He and G. Valencia, _The $Z\to b\bar{b}$ decay asymmetry and left-right models_, _Phys. Rev._ D66 (2002) 013004, [hep-ph/0203036].
  * [16] X.-G. He and G. Valencia, _Lepton universality violation and right-handed currents in $b\to c\tau\nu$_, _Phys. Lett._ B779 (2018) 52–57, [1711.09525].
  * [17] X.-G. He and G. Valencia, _B(s) - anti-B(s) Mixing constraints on FCNC and a non-universal Z-prime_ , _Phys. Rev._ D74 (2006) 013011, [hep-ph/0605202].
  * [18] X.-G. He and G. Valencia, _$K^{+}\to\pi^{+}\nu\bar{\nu}$ and FCNC from non-universal $Z^{\prime}$ bosons_, _Phys. Rev._ D70 (2004) 053003, [hep-ph/0404229].
  * [19] A. Hayreter, X.-G. He and G. Valencia, _LHC constraints on $W^{\prime},~{}Z^{\prime}$ that couple mainly to third generation fermions_, _Eur. Phys. J. C_ 80 (2020) 912, [1912.06344].
  * [20] Particle Data Group collaboration, P. A. Zyla et al., _Review of Particle Physics_ , _PTEP_ 2020 (2020) 083C01.
  * [21] L. Di Luzio, M. Kirk, A. Lenz and T. Rauh, _$\Delta M_{s}$ theory precision confronts flavour anomalies_, _JHEP_ 12 (2019) 009, [1909.11087].
  * [22] X.-G. He, G. Valencia and K. Wong, _Constraints on new physics from $K\rightarrow\pi\nu{\bar{\nu}}$_, _Eur. Phys. J._ C78 (2018) 472, [1804.07449].
  * [23] LHCb collaboration, R. Aaij et al., _Search for the decays $B_{s}^{0}\to\tau^{+}\tau^{-}$ and $B^{0}\to\tau^{+}\tau^{-}$_, _Phys. Rev. Lett._ 118 (2017) 251802, [1703.02508].
  * [24] X.-G. He and G. Valencia, _$B$ decays with $\tau$ leptons in nonuniversal left-right models_, _Phys. Rev._ D87 (2013) 014014, [1211.0348].
  * [25] W. Altmannshofer, A. J. Buras, D. M. Straub and M. Wick, _New strategies for New Physics search in $B\to K^{*}\nu\bar{\nu}$, $B\to K\nu\bar{\nu}$ and $B\to X_{s}\nu\bar{\nu}$ decays_, _JHEP_ 04 (2009) 022, [0902.0160].
  * [26] S. Descotes-Genon, S. Fajfer, J. F. Kamenik and M. Novoa-Brunet, _Implications of $b\to s\mu\mu$ anomalies for future measurements of $B\to K^{(*)}\nu\bar{\nu}$ and $K\to\pi\nu\bar{\nu}$_, _Phys. Lett. B_ 809 (2020) 135769, [2005.03734].
  * [27] A. J. Davies and X.-G. He, _Tree Level Scalar Fermion Interactions Consistent With the Symmetries of the Standard Model_ , _Phys. Rev. D_ 43 (1991) 225–235.
  * [28] S. Davidson, D. C. Bailey and B. A. Campbell, _Model independent constraints on leptoquarks from rare processes_ , _Z. Phys. C_ 61 (1994) 613–644, [hep-ph/9309310].
  * [29] N. G. Deshpande and A. Menon, _Hints of R-parity violation in B decays into $\tau\nu$_, _JHEP_ 01 (2013) 025, [1208.4134].
  * [30] X.-G. He, J. Tandean and G. Valencia, _Charged-lepton-flavor violation in $|\Delta S|=1$ hyperon decays_, _JHEP_ 07 (2019) 022, [1903.01242].
  * [31] X.-G. He, J. Tandean and G. Valencia, _Lepton-flavor-violating semileptonic $\tau$ decay and $K\to\pi\nu\bar{\nu}$_, _Phys. Lett._ B797 (2019) 134842, [1904.04043].
  * [32] J.-Y. Su and J. Tandean, _Exploring leptoquark effects in hyperon and kaon decays with missing energy_ , _Phys. Rev. D_ 102 (2020) 075032, [1912.13507].
  * [33] R. Mandal and A. Pich, _Constraints on scalar leptoquarks from lepton and kaon physics_ , _JHEP_ 12 (2019) 089, [1908.11155].
  * [34] M. Algueró, B. Capdevila, A. Crivellin, S. Descotes-Genon, P. Masjuan, J. Matias et al., _Emerging patterns of New Physics with and without Lepton Flavour Universal contributions_ , _Eur. Phys. J. C_ 79 (2019) 714, [1903.09578].
  * [35] W. Altmannshofer and P. Stangl, _New Physics in Rare B Decays after Moriond 2021_ , 2103.13370.
  * [36] S. Descotes-Genon, L. Hofer, J. Matias and J. Virto, _Global analysis of $b\to s\ell\ell$ anomalies_, _JHEP_ 06 (2016) 092, [1510.04239].
  * [37] HFLAV collaboration, Y. S. Amhis et al., _Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of 2018_, _Eur. Phys. J._ C81 (2021) 226, [1909.12524].
  * [38] N. G. Deshpande and X.-G. He, _Consequences of R-parity violating interactions for anomalies in $\bar{B}\to D^{(*)}\tau\bar{\nu}$ and $b\to s\mu^{+}\mu^{-}$_, _Eur. Phys. J. C_ 77 (2017) 134, [1608.04817].
  * [39] P. S. Bhupal Dev, A. Soni and F. Xu, _Hints of Natural Supersymmetry in Flavor Anomalies?_ , 2106.15647.
  * [40] M. Bauer and M. Neubert, _Minimal Leptoquark Explanation for the $R_{D^{(*)}}$ , $R_{K}$ , and $(g-2)_{\mu}$ Anomalies_, _Phys. Rev. Lett._ 116 (2016) 141802, [1511.01900].
  * [41] Y. Cai, J. Gargalionis, M. A. Schmidt and R. R. Volkas, _Reconsidering the One Leptoquark solution: flavor anomalies and neutrino mass_ , _JHEP_ 10 (2017) 047, [1704.05849].