#  Revisiting $B\to K^{(\ast)}\nu\bar{\nu}$ decays in the Standard Model and
beyond

Damir Bečirević<EMAIL_ADDRESS>IJCLab, Pôle Théorie (Bat.
210), CNRS/IN2P3 et Université, Paris-Saclay, 91405 Orsay, France  Gioacchino
Piazza<EMAIL_ADDRESS>IJCLab, Pôle Théorie (Bat. 210),
CNRS/IN2P3 et Université, Paris-Saclay, 91405 Orsay, France  Olcyr Sumensari
<EMAIL_ADDRESS>IJCLab, Pôle Théorie (Bat. 210), CNRS/IN2P3 et
Université, Paris-Saclay, 91405 Orsay, France

###### Abstract

In this letter we revisit the Standard Model predictions for $\mathcal{B}(B\to
K^{(\ast)}\nu\bar{\nu})$ and discuss the opportunities that open up when
combining its partial decay rate with that of $B\to K^{(\ast)}\ell\ell$. In
the Standard Model a suitable ratio of these two modes can be used to extract
$C_{9}^{\mathrm{eff}}$, which is essential for a reliable phenomenological
analysis of the $B\to K^{(\ast)}\ell\ell$ angular observables. The same ratio
also proves to be more sensitive to the presence of New Physics in many
plausible extensions of the Standard Model. We also suggest that the separate
measurement of $\mathcal{B}(B\to K\nu\bar{\nu})$ for high and for low
$q^{2}$’s can be helpful for testing the assumed shape of the vector form
factor, because the lattice QCD data are obtained at high $q^{2}$’s, whereas
the low $q^{2}$ region is obtained through an extrapolation.

##  I Introduction

Over the past decade a great effort in the high energy physics community has
been invested in studying the exclusive decays based on the $b\to sll$ mode,
with $l\in\\{e,\mu\\}$. Mediated by the flavor changing neutral current, these
decays can occur only through loops in the Standard Model (SM), and therefore
their measurement was expected to give us insight into the loop content, which
in turn could reveal a presence of physics beyond the SM (BSM). The LHC
experiments performed detailed studies of $B\to K^{\ast}(\to K\pi)\mu\mu$
LHCb:2015svh ; CMS:2020oqb  and $B\to K\mu\mu$ LHCb:2014auh ; CMS:2018qih .
The angular distribution of these decays offered access to a set of new
observables, free of the Cabibbo-Kobayashi-Maskawa (CKM) uncertainties, each
with an additional and often complementary way to test the presence of the BSM
physics Becirevic:2011bp ; Descotes-Genon:2013vna . It soon became clear,
however, that the main obstacles to such an endeavor are the hadronic
uncertainties. On the one hand and despite the improvement in controlling the
uncertainties of hadronic matrix elements of the local operators, an
$\mathcal{O}(10\%)$ uncertainty can be warrantied only in a few cases. On the
other hand, the matrix element of the non-local operators, arising from
couplings to the $c\bar{c}$-pairs, remains an open problem, see e.g. Ref.
Ciuchini:2015qxb . To get around the latter problem one tries to stay below
the region populated by the $c\bar{c}$-resonances. To evaluate the matrix
element of the non-local operator one then opts to either invoke the quark-
hadron duality which then means relying entirely on the perturbation theory
Greub:2008cy ; Asatryan:2001zw ; Asatrian:2019kbk , or to employ a (hadronic)
model calculation Khodjamirian:2010vf ; Khodjamirian:2012rm ;
Gubernari:2020eft ; Gubernari:2022hxn . The problems related to both kinds of
hadronic uncertainties appear to be almost entirely absent in the measurement
of the ratio $\smash{R_{K^{(\ast)}}=\mathcal{B}^{\prime}(B\to
K^{(\ast)}\mu\mu)/\mathcal{B}^{\prime}(B\to K^{(\ast)}ee)}$ Hiller:2003js ,
where $\mathcal{B}^{\prime}$ is used to indicate that the partial branching
fractions are measured in the interval $q^{2}\in[1.1,6]~{}\mathrm{GeV}^{2}$,
therefore below the first $c\bar{c}$ resonance,
$m_{J/\psi}^{2}=(3.097\,\mathrm{GeV})^{2}$. Indeed, the measurement of $R_{K}$
and $R_{K^{\ast}}$ LHCb:2022qnv , and of $\mathcal{B}(B_{s}\to\mu\mu)$
ATLAS:2018cur ; CMS:2020rox ; LHCb:2021awg  resulted in clean constraints on
the BSM couplings and on the models of physics BSM. Many BSM scenarios predict
a significant deviation of $\mathcal{B}(B\to K^{(\ast)}\nu\bar{\nu})$ with
respect to its SM prediction and therefore $\mathcal{B}(B\to
K^{(\ast)}\nu\bar{\nu})$ can provide us with either a test of validity of a
given model, or with a constraint when building an acceptable scenario of
physics BSM.

In this letter we will first evaluate $\mathcal{B}(B\to
K^{(\ast)}\nu\bar{\nu})$ in the SM by arguing that the most precise
information can be obtained if one splits the $B\to K\nu\bar{\nu}$ events to
those with high- and those with low-$q^{2}$’s. From the comparison of the
measured $\mathcal{B}(B\to K\nu\bar{\nu})_{\mathrm{high}-q^{2}}$ with the one
predicted in the SM, one can check for a consistency with the SM. Instead,
from the ratio $\mathcal{B}(B\to
K\nu\bar{\nu})_{\mathrm{low}-q^{2}}/\mathcal{B}(B\to
K\nu\bar{\nu})_{\mathrm{high}-q^{2}}$ one can check on the validity of the
shape of the sole form factor entering the expression for
$\mathrm{d}\mathcal{B}(B\to
K\nu\bar{\nu})/\mathrm{d}q^{2}\propto|f_{+}(q^{2})|^{2}$. We will then argue,
like in Ref. Bartsch:2009qp , that the ratio $\mathcal{B}^{\prime}(B\to
K\mu\mu)/\mathcal{B}^{\prime}(B\to K\nu\bar{\nu})$, in the low $q^{2}$-bin, is
essentially free of the form factor uncertainty and, if measured, it allows us
to extract the desired Wilson coefficient $C_{9}^{\mathrm{eff}}$, which is the
one plagued by uncertainties arising from the hadronic matrix element of the
non-local operator. A similar discussion, even if somewhat less accurate, can
then be extended to $B\to K^{\ast}\nu\bar{\nu}$. In that way one can also
assess the size of the non-factorizable contribution to the
$C_{9}^{\mathrm{eff}}$ which, according to Ref. Khodjamirian:2010vf , is
different in the case of $K$ from the case of $K^{\ast}$ in the final state.

On the basis of that information, the controversies in comparison of other
observables extracted from the experimental angular analysis of $B\to
K^{(\ast)}\mu\mu$ with their SM values would be removed, and the search of a
scenario of physics BSM consistent with many more experimental constraints
would become more compelling. We show that a study of the ratio
$\mathcal{B}^{\prime}(B\to K\mu\mu)/\mathcal{B}^{\prime}(B\to K\nu\bar{\nu})$
could provide us with a useful filter to select among the acceptable models of
physics BSM.

The remainder of this letter is organized as follows: In Sec. II we describe
the $B\to K^{(\ast)}\nu\bar{\nu}$ decays in the SM, with a focus on hadronic
uncertainties. In Sec. III, we propose alternative observables that are
potentially less sensitive to hadronic uncertainties, and we discuss their
sensitivity to physics beyond the SM in Sec. IV. Our findings are briefly
summarized in Sec. V.

##  II $B\to K^{(\ast)}\nu\bar{\nu}$ Decays in the SM

###  II.1 Effective theory description

Decays based on the $b\to s\nu\bar{\nu}$ transition are described by the
following effective Lagrangian,

$\displaystyle\mathcal{L}_{\mathrm{eff}}^{\mathrm{b\to
s\nu\nu}}=\dfrac{4G_{F}}{\sqrt{2}}\lambda_{t}\sum_{a}C_{a}\,\mathcal{O}_{a}+\mathrm{h.c.}\,,$
(1)

where $G_{F}$ is the Fermi constant, $\lambda_{t}=V_{tb}V_{ts}^{\ast}$ is a
suitable product of the CKM entries, and the only relevant operator in the SM
is given by 111The right-handed operator
$\mathcal{O}_{R}^{\nu_{i}\nu_{j}}=\frac{e^{2}}{(4\pi)^{2}}(\bar{s}_{R}\gamma_{\mu}b_{R})(\bar{\nu}_{i}\gamma^{\mu}(1-\gamma_{5})\nu_{j})$
is absent in the SM, but it can appear in some of the BSM scenarios (see Sec.
IV).

$\displaystyle\mathcal{O}_{L}^{\nu_{i}\nu_{j}}$
$\displaystyle=\dfrac{e^{2}}{(4\pi)^{2}}(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{\nu}_{i}\gamma^{\mu}(1-\gamma_{5})\nu_{j})\,.$
(2)

The SM effective Wilson coefficient
$\big{[}C_{L}^{\nu_{i}\nu_{j}}\big{]}_{\mathrm{SM}}\equiv\delta_{ij}\,C_{L}^{\mathrm{SM}}$
is known Buras:2014fpa ,

$\displaystyle C_{L}^{\mathrm{SM}}=-X_{t}/\sin^{2}\theta_{W}\,,\qquad
X_{t}=1.462(17)(2)\,,$ (3)

and it includes the NLO QCD corrections Buchalla:1993bv ; Buchalla:1998ba ;
Misiak:1999yg , as well as the two-loop electroweak contributions Brod:2010hi
. Using $\sin^{2}\theta_{W}=0.23141(4)$ ParticleDataGroup:2022pth , one
finally arrives at $C_{L}^{\mathrm{SM}}=-6.32(7)$, where the dominant source
of uncertainty comes from the higher order QCD crrections.

#### $B\to K\nu\bar{\nu}$

The SM differential decay rate of $B\to K\nu\bar{\nu}$ can be written as

$\displaystyle\dfrac{\mathrm{d}\mathcal{B}}{\mathrm{d}q^{2}}(B\to
K\nu\bar{\nu})=$
$\displaystyle\mathcal{N}_{K}(q^{2})\,|C_{L}^{\mathrm{SM}}|^{2}\,|\lambda_{t}|^{2}\left[f_{+}(q^{2})\right]^{2}\,,$
(4)

where $0<q^{2}\leq(m_{B}-m_{K})^{2}$ is the di-neutrino invariant mass,
$f_{+}(q^{2})$ is the $B\to K$ vector form factor which will be discussed in
Sec. II.2, and $\mathcal{N}_{K}(q^{2})$ denotes a known $q^{2}$-dependent
function,

$\displaystyle\mathcal{N}_{K}(q^{2})=\tau_{B}\dfrac{G_{F}^{2}\,\alpha_{\mathrm{em}}^{2}}{256\pi^{5}}\dfrac{\lambda^{3/2}_{K}}{m_{B}^{3}}\,,$
(5)

with $\lambda_{K}\equiv\lambda(q^{2},m_{B}^{2},m_{K}^{2})$ being the triangle
function
$\lambda(a^{2},b^{2},c^{2})\equiv\left(a^{2}-(b-c)^{2}\right)\left(a^{2}-(b+c)^{2}\right)$.
Note that in the above expressions we summed over the neutrino flavors.

#### $B\to K^{\ast}\nu\bar{\nu}$

Similarly to the previous case, the $B\to K^{\ast}\nu\bar{\nu}$ branching
fraction can be written as:

$\displaystyle\dfrac{\mathrm{d}\mathcal{B}}{\mathrm{d}q^{2}}(B\to
K^{\ast}\nu\bar{\nu})=$
$\displaystyle\mathcal{N}_{K^{\ast}}(q^{2})|C_{L}^{\mathrm{SM}}|^{2}|\lambda_{t}|^{2}\mathcal{F}(q^{2})\,,$
(6)

where the kinematical factor reads,

$\displaystyle\mathcal{N}_{K^{\ast}}(q^{2})=\tau_{B}\dfrac{G_{F}^{2}\,\alpha_{\mathrm{em}}^{2}}{128\pi^{5}}\dfrac{\lambda^{1/2}_{K^{\ast}}q^{2}}{m_{B}^{3}}\left(m_{B}+m_{K^{\ast}}\right)^{2}\,,$
(7)

with $\lambda_{K^{\ast}}\equiv\lambda(q^{2},m_{B}^{2},m_{K^{\ast}}^{2})$, and
$\mathcal{F}(q^{2})$ given by

$\displaystyle\begin{split}\mathcal{F}(q^{2})=[A_{1}(q^{2})]^{2}&+\dfrac{32\
m_{K^{\ast}}^{2}m_{B}^{2}}{q^{2}(m_{B}+m_{K^{\ast}})^{2}}[A_{12}(q^{2})]^{2}\\\
&+\dfrac{\lambda_{K^{\ast}}}{(m_{B}+m_{K^{\ast}})^{4}}[V(q^{2})]^{2}\,.\end{split}$
(8)

The $B\to K^{\ast}$ form factors $A_{1}(q^{2})$, $A_{12}(q^{2})$ and
$V(q^{2})$ will be defined shortly, in Sec. II.3.

Besides the small and controlled uncertainty in $C_{L}^{\mathrm{SM}}$, two
other sources of theoretical uncertainties in the above expressions come from:
(i) the $B\to K^{(\ast)}$ form factors that must be determined
nonperturbatively, see Sec. II.2 and II.3, and (ii) the product of the CKM
matrix elements $\lambda_{t}=V_{tb}V_{ts}^{\ast}$, which will be discussed in
Sec. II.4.

###  II.2 $B\to K$ form factors

The $B\to K$ hadronic matrix element is the main source of theoretical
uncertainty entering the $B\to K\nu\bar{\nu}$ branching fraction. It is
usually decomposed as

$\displaystyle\langle\bar{K}(k)|\bar{s}\gamma^{\mu}b|\bar{B}(p)\rangle$
$\displaystyle=\Big{[}(p+k)^{\mu}-\dfrac{m_{B}^{2}-m_{K}^{2}}{q^{2}}q^{\mu}\Big{]}f_{+}(q^{2})$
$\displaystyle+\dfrac{m_{B}^{2}-m_{K}^{2}}{q^{2}}q^{\mu}f_{0}(q^{2})\,,$ (9)

where $f_{+}$ ($f_{0}$) are the so-called vector (scalar) $B\to K$ form
factors, satisfying at $q^{2}=0$ the condition $f_{+}(0)=f_{0}(0)$. Note also
that the scalar form factor does not enter the theoretical expression for
$\mathcal{B}(B\to K\nu\bar{\nu})$, cf. Eq. (4), since the neutrino masses are
negligible.

While this letter was in writing an update of the lattice QCD result by the
HPQCD collaboration appeared Parrott:2022rgu . We used the information
provided in their paper, combined it with the lattice QCD results by the
FNAL/MILC collaboration Bailey:2015dka  and followed the same procedure as
FLAG Aoki:2021kgd  in order to provide the new average of the lattice QCD form
factors, cf. Appendix A. Our average therefore supersedes the one presented in
Ref. Aoki:2021kgd , in which now already obsolete HPQCD results from Ref.
Bouchard:2013eph  have been used. Since the lattice QCD results are obtained
for $q^{2}\gtrsim 16~{}\mathrm{GeV}^{2}$, an extrapolation is needed to cover
the entire $B\to K\nu\bar{\nu}$ physical region. This is provided by the
parametrization of the $q^{2}$ dependence of the form factors, which is
discussed in Ref. Aoki:2021kgd . In Fig. 2 we show the newly averaged form
factors and their shapes (solid curves), and compare them with the previous
ones (dashed) presented in Aoki:2021kgd . Clearly, the effect of the inclusion
of the new HPQCD results is that the form factors at low $q^{2}$’s are now
more accurate.

Figure 1: The results of our fit for $f_{+}(q^{2})$ and $f_{0}(q^{2})$ form
factors are depicted by the blue and red solid curves respectively. The dashed
lines correspond to the results reported by FLAG Aoki:2021kgd . The synthetic
data points by HPQCD (green) Parrott:2022rgu  and by FNAL/MILC (orange)
Bailey:2015dka  are also shown for comparison. $P_{+,0}(q^{2})$ are the
inverse pole terms defined in Eq. (58).

Figure 2: Rescaled differential branching fraction
$|\lambda_{t}|^{-2}\,\mathrm{d}\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})/\mathrm{d}q^{2}$ is plotted by using the form factor
$f_{+}(q^{2})$ from the FLAG review Aoki:2021kgd , and by using the one
discussed in this letter. The lower panel shows the relative uncertainty on
this quantity as a function of $q^{2}$ using the FLAG form factors.

One should keep in mind, however, that the low-$q^{2}$ region is a result of
an extrapolation which is a potential source of systematic uncertainty, not
accounted in the error budget. We propose a way to monitor such an uncertainty
by splitting the sample of $B\to K\nu\bar{\nu}$ events into two bins: low
$q^{2}$’s for $q^{2}/(m_{B}-m_{K})^{2}\in(0,1/2)$, and high $q^{2}$’s for
$q^{2}/(m_{B}-m_{K})^{2}\in(1/2,1)$. 222Here we propose to split the sample in
two bins, but obviously splitting the sample into more than two bins would be
even better when it comes to monitoring the shape of the form factor. From the
measured ratio:

$\displaystyle r_{\mathrm{lh}}={\mathcal{B}(B\to
K\nu\bar{\nu})_{\mathrm{low}-q^{2}}\over\mathcal{B}(B\to
K\nu\bar{\nu})_{\mathrm{high}-q^{2}}},$ (10)

one can check whether or not the result is consistent with prediction, which
is obtained by using lattice QCD results for $f_{+}(q^{2})$ at high $q^{2}$,
and the ones obtained through extrapolation to low $q^{2}$. Note that the
above ratio $r_{\mathrm{lh}}$ is independent on the CKM factor, and on the
Wilson coefficient regardless of the presence of physics BSM, provided we
consider the left-handed neutrinos. Using the new average for the form factor
provided in this paper we find,

$\displaystyle r_{\mathrm{lh}}=1.91(6)\,,$ (11)

which is obviously consistent with $r_{\mathrm{lh}}=1.92(20)$, obtained by
using solely the FNAL/MILC form factors Bailey:2015dka , as well as with
$r_{\mathrm{lh}}=1.85(9)$, that we obtain by using only the new HPQCD values
Parrott:2022rgu . Finally, and for future reference we note that from our
average $f_{+}(0)=0.336(10)$, which again is consistent with the FNAL/MILC
value, $f_{+}(0)=0.335(36)$, and with the new one obtained by HPQCD,
$f_{+}(0)=0.332(12)$. All these values are not far from $f_{+}(0)=0.304(42)$,
often used in the literature and obtained by using the so-called light cone
sum rules (LCSR) Ball:2004ye .

Regarding the total branching fraction, with our new average of the lattice
QCD form factor, we obtain:

$\displaystyle\mathcal{B}(B\to K$
$\displaystyle\nu\bar{\nu})^{\mathrm{SM}}/|\lambda_{t}|^{2}=\left\\{\begin{matrix}(1.33\pm
0.04)_{K_{S}}\times 10^{-3}\,,&\\\\[5.0pt] (2.87\pm 0.10)_{K^{+}}\times
10^{-3}\,,&\end{matrix}\right.$ (12)

where we factored out the CKM dependence and distinguished the charged from
the neutral kaon case. 333Note that the charged mode is also affected by the
non-negligible tree-level contribution proportional to $G_{F}^{4}$, as
discussed in Sec. II.5 of the present letter. The comparison of this
prediction with the ones available in the literature is provided in Table 1,
each corresponding to a different choice of the form factor $f_{+}(q^{2})$: in
Ref. Buras:2014fpa  the lattice results of Ref. Bouchard:2013eph  are combined
with the LCSR value; in Ref. Blake:2016olu  only the FNAL/MILC lattice results
Bailey:2015dka  have been used; in Ref. Parrott:2022dnu  only the new HPQCD
results are considered Parrott:2022rgu ; in obtaining our result for the
branching fraction we use the average of the form factor result obtained by
FNAL/MILC Bailey:2015dka  and the new HPQCD result Parrott:2022rgu . In Fig. 2
we show the impact of the new form factor on the differential branching
fraction. In the same plot we emphasize the consequence of using the new form
factor average on the error of the branching fraction in the low $q^{2}$
region. One should, however, keep in mind that the form factor at low
$q^{2}$’s is obtained through an extrapolation from large $q^{2}$’s where the
actual lattice QCD data are available. This is done by using a pole-like shape
of the form factor, multiplied by a suitable polynomial, as proposed in the
model of Ref. Bourrely:2008za . It is therefore of major importance to devise
strategies aiming at reducing the impact of hadronic uncertainties in the
observables that can be accessed experimentally, as we explore in Sec. III.

Ref. | Form factors | $\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})/|\lambda_{t}|^{2}$
---|---|---
Buras et al. Buras:2014fpa  | Ball:2004ye ; Bouchard:2013eph  | $(2.5\pm 0.3)\times 10^{-3}$
Blake et al. Blake:2016olu  | Bailey:2015dka  | $(2.8\pm 0.3)\times 10^{-3}$
Parrott et al. Parrott:2022dnu  | Parrott:2022rgu  | $(2.81\pm 0.15)\times 10^{-3}$
This work | Parrott:2022rgu ; Bailey:2015dka ; Aoki:2021kgd  | $(2.87\pm 0.10)\times 10^{-3}$
Table 1:  SM predictions for $\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})/|\lambda_{t}|^{2}$ and the corresponding form factors used
in the computation. Note that the non-negligible contribution proportional to
$G_{F}^{4}$ has not been included in the above results, cf. Sec. II.5.

Before continuing, we need to stress that the difference between
$\mathcal{B}(B^{0}\to K_{S}\nu\bar{\nu})$ and $\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})$ is related to the symmetry relation among the matrix
elements:

$\displaystyle\langle K_{S}|\bar{b}\gamma_{\mu}s|B^{0}\rangle$
$\displaystyle=-\langle K_{S}|\bar{s}\gamma_{\mu}b|\bar{B}^{0}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\langle
K^{+}|\bar{s}\gamma_{\mu}b|\bar{B}^{+}\rangle\,,$ (13)

where we accounted for the Clebsch-Gordan coefficient, so that in the end we
have

$\displaystyle\mathcal{B}(B^{0}\to K_{S}\,\nu\bar{\nu})$
$\displaystyle=\mathcal{B}(\overline{B}^{0}\to K_{S}\,\nu\bar{\nu})$
$\displaystyle=\frac{1}{2}\frac{\tau_{B^{0}}}{\tau_{B^{+}}}\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})\,,$ (14)

where, for illustration purposes, we neglect the tiny phase space difference.
In our numerics, however, we use the correct masses of the charged and of the
neutral kaons. Note in particular that it is important to properly account for
the $B$-meson lifetimes, because $\tau_{B^{+}}/\tau_{B^{0}}=1.076(4)$
ParticleDataGroup:2022pth .

###  II.3 $B\to K^{\ast}$ form factors

The hadronic matrix element entering the $B\to K^{\ast}\nu\bar{\nu}$ decay can
be parameterized as follows 444The convention used in Eq. (II.3) is
$\varepsilon_{0123}=+1$.

$\displaystyle\langle\bar{K}^{\ast}(k)|\bar{s}\gamma_{\mu}(1$
$\displaystyle-\gamma_{5})b|\bar{B}(p)\rangle=\varepsilon_{\mu\nu\rho\sigma}\varepsilon^{\ast\nu}p^{\rho}k^{\sigma}\dfrac{2V(q^{2})}{m_{B}+m_{K^{\ast}}}$
$\displaystyle-i\varepsilon_{\mu}^{\ast}(m_{B}+m_{K^{\ast}})A_{1}(q^{2})$
$\displaystyle+i(p+k)_{\mu}(\varepsilon^{\ast}\cdot
q)\dfrac{A_{2}(q^{2})}{m_{B}+m_{K^{\ast}}}$ (15)
$\displaystyle+iq_{\mu}(\varepsilon^{\ast}\cdot
q)\dfrac{2m_{K^{\ast}}}{q^{2}}\left[A_{3}(q^{2})-A_{0}(q^{2})\right]\,,$

where $\varepsilon_{\mu}$ is the polarization vector of $K^{\ast}$, while
$V(q^{2})$ and $A_{0,1,2,3}(q^{2})$ are the form factors. In the above
definition we use $A_{3}(q^{2})$, while in Eq. (8) we used $A_{12}(q^{2})$.
They are both related to $A_{1}(q^{2})$ and $A_{2}(q^{2})$ as:

$\displaystyle A_{3}(q^{2})=$
$\displaystyle\dfrac{m_{B}+m_{K^{\ast}}}{2m_{K^{\ast}}}A_{1}(q^{2})-\dfrac{m_{B}-m_{K^{\ast}}}{2m_{K^{\ast}}}A_{2}(q^{2}),$
$\displaystyle A_{12}(q^{2})=$
$\displaystyle\dfrac{(m_{B}+m_{K^{\ast}})(m_{B}^{2}-m_{K^{\ast}}^{2}-q^{2})}{16m_{B}m_{K^{\ast}}^{2}}A_{1}(q^{2})$
$\displaystyle-\dfrac{\lambda_{K^{\ast}}}{16m_{B}m_{K^{\ast}}^{2}(m_{B}+m_{K^{\ast}})}A_{2}(q^{2}),$
(16)

so that at $q^{2}=0$ they satisfy:
$8m_{B}m_{K^{\ast}}A_{12}(0)=(m_{B}^{2}-m_{K^{\ast}}^{2})A_{0}(0)$, and
$A_{3}(0)=A_{0}(0)$. Since the pseudoscalar form factor $A_{0}(q^{2})$ does
not contribute to the decay rate in the massless neutrino limit, three
independent form factors are needed to compute $B\to K^{\ast}\nu\bar{\nu}$,
namely $V$, $A_{1}$ and $A_{2}$.

The situation for the $B\to K^{\ast}$ transition is far more intricate than
for $B\to K$ because there are more form factors. Furthermore, the results of
only one lattice QCD study at nonzero recoil have been reported so far, with a
specific lattice setup Horgan:2013hoa . In this paper, we take the results of
Ref. Bharucha:2015bzk  in which the lattice QCD values from Ref.
Horgan:2013hoa  were combined with those obtained by using the LCSR. By
adopting the form factor parameterizations and inputs from Ref.
Bharucha:2015bzk , we obtain

$\displaystyle\mathcal{B}(B\to
K^{\ast}\nu\bar{\nu})^{\mathrm{SM}}/|\lambda_{t}|^{2}=\left\\{\begin{matrix}(5.9\pm
0.8)_{K^{\ast 0}}\times 10^{-3}\,,&\\\\[5.0pt] (6.4\pm 0.9)_{K^{\ast+}}\times
10^{-3}\,,&\end{matrix}\right.$ (17)

in good agreement with Ref. Blake:2016olu . 555Notice that slightly smaller
values are obtained in Ref. Buras:2022qip  by using the $B\to K^{\ast}$ form
factors provided in Ref. Gubernari:2018wyi . We should point out, however,
that this result is obviously less robust than the one for $\mathcal{B}(B\to
K\nu\bar{\nu})$. This is strengthening even more our motivation to look for
the options that allow one to reduce sensitivity to the form factor
uncertainties, see Sec. III.

###  II.4 CKM couplings

Figure 3: Predictions for $|\lambda_{t}|^{-2}\,\mathrm{d}\mathcal{B}(B^{0}\to
K^{\ast 0}\nu\bar{\nu})/\mathrm{d}q^{2}$ obtained by using the form factors
from Ref. Bharucha:2015bzk . The lower panel shows the relative uncertainty on
this quantity as a function of $q^{2}$.

The uncertainty on the CKM factor $\lambda_{t}=V_{tb}V_{ts}^{\ast}$ introduces
the largest parametric uncertainty in $\mathcal{B}(B\to
K^{(\ast)}\nu\bar{\nu})$. The usual procedure, often adopted in the
literature, is to determine $|V_{cb}|$ from the tree-level processes and then,
by virtue of the CKM unitarity, evaluate $|\lambda_{t}|$, cf. e.g. Ref.
Buras:2014fpa . In that way the loop-induced processes are used to probe the
effects of physics BSM. Unfortunately, that procedure is intrinsically
ambiguous too, because the CKM coupling $|V_{cb}^{\mathrm{incl}}|$ extracted
from the inclusive semileptonic decay, does not coincide with
$|V_{cb}^{\mathrm{excl}}|$ obtained from the exclusive modes. The latest HFLAV
average values of the inclusive $|V_{cb}|$ read:
$|V_{cb}^{\mathrm{incl}}|_{\mathrm{kin}}=(42.2\pm 0.8)\times 10^{-3}$ or
$|V_{cb}^{\mathrm{incl}}|_{\mathrm{1S}}=(42.0\pm 0.5)\times 10^{-3}$
HFLAV:2022pwe . Those values are larger than $|V_{cb}^{B\to D}|=(40.0\pm
1.0)\times 10^{-3}$ Aoki:2021kgd  obtained after combining the experimental
results on the exclusive $B\to Dl\nu$ decays HFLAV:2022pwe  (with $l=e,\mu$)
with the LQCD form factors from Refs. MILC:2015uhg ; Na:2015kha . This
discrepancy remains true if one compares the inclusive values with the one
derived from $B\to D^{\ast}l\nu$, namely $|V_{cb}^{B\to D^{\ast}}|=(38.5\pm
0.7)\times 10^{-3}$ HFLAV:2022pwe . 666The quoted $|V_{cb}^{B\to D^{\ast}}|$
value agrees with $|V_{cb}^{B\to D^{\ast}}|=(38.9\pm 0.9)\times 10^{-3}$
reported by FLAG Aoki:2021kgd . In other words, there is a discrepancy in
$|\lambda_{t}|$ depending on the particular input considered,

$|\lambda_{t}|\times 10^{3}=\left\\{\begin{matrix}42.2\pm
0.8\,,&\quad\;\,{\small(B\to X_{c}l\bar{\nu})}\\\\[5.0pt] 40.0\pm
1.0\,,&\quad{\small(B\to Dl\bar{\nu})}\\\\[5.0pt] 38.5\pm
0.7\,,&\quad\;\,{\small(B\to D^{\ast}l\bar{\nu})}\end{matrix}\right.$ (18)

where the inclusive value is about $1\sigma$ and $2\sigma$ larger the the ones
derived from $B\to Dl\bar{\nu}$ and $B\to D^{\ast}l\bar{\nu}$ decays,
respectively.

For future reference, we will use $|\lambda_{t}|_{\mathrm{excl}}=(40.0\pm
1.0)\times 10^{-3}$, and $|\lambda_{t}|_{\mathrm{incl}}$ as noted above.
Notice that by using $|\lambda_{t}|_{\mathrm{excl}}$ we obtain the branching
fractions in Eqs. (12,17) $1.5\sigma$ smaller than the values we get by using
$|\lambda_{t}|_{\mathrm{incl}}$, which again highlights the importance of
clarifying the issue of the determination of $V_{cb}$. 777Notice that this
discrepancy would be even stronger if instead of $|V_{cb}^{B\to D}|$ we used
the HFLAV value of $|V_{cb}^{B\to D^{\ast}}|$ HFLAV:2022pwe . Note also that a
larger $|V_{cb}^{B\to D^{\ast}}|$ has been recently advocated in Ref.
Martinelli:2021myh . Another possibility is to rely on the CKM unitarity and
extract the $V_{cb}$ from the global fit with data in which $b\to c\ell\nu$
experimental input is removed. This leads to
$|V_{cb}^{\mathrm{UTfit}}|=42.2(5)\times 10^{-3}$ UTfit:2022hsi , or
$|V_{cb}^{\mathrm{CKMfitter}}|=41.3(3)\times 10^{-3}$ Charles:2004jd , mostly
constrained by the $K^{0}-\overline{K}^{0}$ and $B_{s}-\overline{B}_{s}$
mixing. Clearly, one could extract $\lambda_{t}$ from $\Delta m_{B_{s}}$, as
advocated in Ref. Buras:2022qip . In all these alternative strategies, an
excellent control over the matrix element of the $B_{s}-\overline{B}_{s}$
mixing is required, usually referred to as $f_{B_{s}}\sqrt{\hat{B}_{B_{s}}}$,
which is currently not the case. In the FLAG review,
$f_{B_{s}}\sqrt{\hat{B}_{B_{s}}}=274(8)$ MeV ($N_{\mathrm{f}}=2+1$), and
$f_{B_{s}}\sqrt{\hat{B}_{B_{s}}}=256(6)$ MeV ($N_{\mathrm{f}}=2+1+1$), which
again renders the value of $\lambda_{t}$ ambiguous. One way to avoid the
ambiguity on the CKM couplings will be discussed in Sec. III.

###  II.5 Numerical Predictions

Before we give our final numerical results, we need to account for one
significant correction. It was first noted in Ref. Kamenik:2009kc  that there
is an important tree level contribution to the charged $B^{\pm}\to
K^{\pm}\nu\bar{\nu}$ mode arising from the weak annihilation mediated by the
on-shell $\tau$-lepton. We will follow Ref. Kamenik:2009kc  and call this new
contribution as “Tree”, while keeping in mind that its amplitude is $\propto
G_{F}^{2}$. To be more specific, we write:

$\displaystyle{d\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})_{\mathrm{Tree}}\over
dq^{2}dp_{\tau}^{2}}$
$\displaystyle={\tau_{B^{+}}G_{F}^{4}|V_{us}V_{ub}|^{2}f_{K}^{2}f_{B}^{2}\over
64\pi^{3}m_{B}^{3}}\,p_{\tau}^{4}$
$\displaystyle\times\frac{m_{B}^{2}(p_{\tau}^{2}-m_{K}^{2})-p_{\tau}^{2}(p_{\tau}^{2}+q^{2}-m_{K}^{2})}{(m_{\tau}^{2}-p_{\tau}^{2})^{2}+m_{\tau}^{2}\Gamma_{\tau}^{2}}\,,$
(19)

where $p_{\tau}^{2}$ and $\Gamma_{\tau}$ are the invariant mass and the full
width of the intermediate $\tau$-lepton. After integrating over the phase
space we have:

$\displaystyle\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})_{\mathrm{Tree}}$
$\displaystyle={\tau_{B^{+}}G_{F}^{4}|V_{us}V_{ub}|^{2}f_{K}^{2}f_{B}^{2}\over
128\pi^{2}m_{B}^{3}}\,$
$\displaystyle\times\frac{m_{\tau}}{\Gamma_{\tau}}(m_{\tau}^{2}-m_{K}^{2})^{2}(m_{B}^{2}-m_{\tau}^{2})^{2}\,,$
(20)

where we have used a narrow-width approximation. Similarly, for the $B^{+}\to
K^{\ast+}\nu\bar{\nu}$ mode we have,

$\displaystyle{d\mathcal{B}(B^{+}\to
K^{\ast+}\nu\bar{\nu})_{\mathrm{Tree}}\over dq^{2}dp_{\tau}^{2}}$
$\displaystyle={\tau_{B^{+}}G_{F}^{4}|V_{us}V_{ub}|^{2}f_{K^{\ast}}^{2}f_{B}^{2}\over
64\pi^{3}m_{B}^{3}}\,p_{\tau}^{4}$
$\displaystyle\times\frac{(m_{B}^{2}-p_{\tau}^{2})(p_{\tau}^{2}-m_{K^{\ast}}^{2})-q^{2}(p_{\tau}^{2}-2m_{K^{\ast}}^{2})}{(m_{\tau}^{2}-p_{\tau}^{2})^{2}+m_{\tau}^{2}\Gamma_{\tau}^{2}}\,,$
(21)

which leads to,

$\displaystyle\mathcal{B}(B^{+}\to
K^{\ast+}\nu\bar{\nu})_{\mathrm{Tree}}={\tau_{B^{+}}G_{F}^{4}|V_{us}V_{ub}|^{2}f_{K^{\ast}}^{2}f_{B}^{2}\over
128\pi^{2}m_{B}^{3}}\frac{m_{\tau}}{\Gamma_{\tau}}$
$\displaystyle\quad\,\,\,\,\times(m_{\tau}^{2}-m_{K^{\ast}}^{2})^{2}(m_{B}^{2}-m_{\tau}^{2})^{2}\left(1+\frac{2m_{K^{\ast}}^{2}}{m_{\tau}^{2}}\right)\,.$
(22)

To estimate the size of this contribution we take $f_{K}=155.7(3)$ MeV and
$f_{B}=190.0(1.3)$ MeV from the FLAG review Aoki:2021kgd , in addition to
$f_{K^{\ast}}=205(6)$ MeV, which we extracted from the measured
$\mathcal{B}(\tau\to K^{\ast}\nu)=1.20(7)\%$ ParticleDataGroup:2022pth , and
by using $|V_{us}|=0.2259(5)$ Becirevic:2020rzi , $|V_{ub}|=3.74(17)\times
10^{-3}$ Aoki:2021kgd . We finally obtain

$\displaystyle\begin{split}\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})_{\mathrm{Tree}}&=(6.28\pm 0.06)\times
10^{-7}\,,\\\\[3.50006pt] \mathcal{B}(B^{+}\to
K^{\ast+}\nu\bar{\nu})_{\mathrm{Tree}}&=(1.07\pm 0.10)\times
10^{-6}\,.\end{split}$ (23)

Despite being $\propto G_{F}^{4}$ this contribution is indeed numerically
significant. We find that it amounts to more than $10\%$ with respect to the
dominant effect $\propto G_{F}^{2}$, and therefore if we aim at the $10\%$
experimental precision or better at Belle-II, the “Tree” contribution must be
included Belle-II:2018jsg . We do so to obtain our final estimates for the
total branching fractions collected in Table. 2, which are just below the
current experimental limits Belle:2017oht . Note, once again, that in our
final results we used $|\lambda_{t}|=(4.00\pm 0.10)\times 10^{-2}$. For the
readers’ convenience, in Tables 3 and 4 we also provide the binned values for
the $B\to K^{(\ast)}\nu\bar{\nu}$ branching fractions.

Decay | Branching ratio
---|---
$B^{+}\to K^{+}\nu\bar{\nu}$ | $(5.22\pm 0.15\pm 0.28)\times 10^{-6}$
$B^{0}\to K_{S}\nu\bar{\nu}$ | $(2.12\pm 0.07\pm 0.13)\times 10^{-6}$
$B^{+}\to K^{\ast+}\nu\bar{\nu}$ | $(11.27\pm 1.38\pm 0.62)\times 10^{-6}$
$B^{0}\to K^{\ast 0}\nu\bar{\nu}$ | $(9.47\pm 1.28\pm 0.57)\times 10^{-6}$
Table 2:  Our final predictions for the branching fractions of the most relevant $b\to s\nu\bar{\nu}$ decay modes. The first uncertainty comes from the hadronic form factors and the second one is dominated by the uncertainty of $|\lambda_{t}|$.  $q^{2}$-bin $[\mathrm{GeV}^{2}]$ | $\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})\times 10^{6}$ | $\sigma_{\mathcal{B}_{K^{+}}}/\mathcal{B}_{K^{+}}$ | $\mathcal{B}(B^{0}\to K_{S}\nu\bar{\nu})\times 10^{6}$ | $\sigma_{\mathcal{B}_{K_{S}}}/\mathcal{B}_{K_{S}}$
---|---|---|---|---
$[0,4]$ | $(1.241\pm 0.058\pm 0.066)$ | 0.07 | $(0.506\pm 0.027\pm 0.030)$ | 0.08
$[4,8]$ | $(1.196\pm 0.041\pm 0.065)$ | 0.06 | $(0.493\pm 0.019\pm 0.030)$ | 0.07
$[8,12]$ | $(1.097\pm 0.028\pm 0.060)$ | 0.06 | $(0.455\pm 0.013\pm 0.027)$ | 0.07
$[12,16]$ | $(0.916\pm 0.021\pm 0.050)$ | 0.06 | $(0.378\pm 0.010\pm 0.023)$ | 0.07
$[16,q^{2}_{\mathrm{max}}]$ | $(0.765\pm 0.018\pm 0.040)$ | 0.06 | $(0.292\pm 0.008\pm 0.017)$ | 0.07
1-5 $[0,q^{2}_{\mathrm{max}}]$ | $(5.22\pm 0.15\pm 0.28)$ | 0.06 | $(2.12\pm 0.07\pm 0.13)$ | 0.07
Table 3:  SM predictions for the partially integrated $B\to K\nu\bar{\nu}$ branching fraction, in a given $q^{2}$-bin, obtained by using our fit to the lattice form factors from HQPCD Parrott:2022rgu  and FNAL/MILC Bailey:2015dka , and the CKM input $|\lambda_{t}|=4.00(10)\times 10^{-2}$. The first uncertainty comes from the hadronic form factors and the second one is dominated by the uncertainty on $|\lambda_{t}|^{2}$. The total relative uncertainty of each observable is shown in the last column.  $q^{2}$-bin $[\mathrm{GeV}^{2}]$ | $\mathcal{B}(B^{+}\to K^{\ast+}\nu\bar{\nu})\times 10^{6}$ | $\sigma_{\mathcal{B}_{K^{\ast+}}}/\mathcal{B}_{K^{\ast+}}$ | $\mathcal{B}(B^{0}\to K^{\ast 0}\nu\bar{\nu})\times 10^{6}$ | $\sigma_{\mathcal{B}_{K^{\ast 0}}}/\mathcal{B}_{K^{\ast 0}}$
---|---|---|---|---
$[0,4]$ | $(1.82\pm 0.20\pm 0.09)$ | 0.12 | $(1.44\pm 0.18\pm 0.09)$ | 0.14
$[4,8]$ | $(2.33\pm 0.24\pm 0.12)$ | 0.12 | $(1.93\pm 0.22\pm 0.12)$ | 0.13
$[8,12]$ | $(2.73\pm 0.32\pm 0.15)$ | 0.13 | $(2.32\pm 0.29\pm 0.14)$ | 0.14
$[12,16]$ | $(2.82\pm 0.41\pm 0.16)$ | 0.16 | $(2.42\pm 0.37\pm 0.14)$ | 0.17
$[16,q^{2}_{\mathrm{max}}]$ | $(1.57\pm 0.32\pm 0.09)$ | 0.21 | $(1.33\pm 0.29\pm 0.08)$ | 0.22
1-5 $[0,q^{2}_{\mathrm{max}}]$ | $(11.27\pm 1.38\pm 0.62)$ | 0.13 | $(9.47\pm 1.28\pm 0.57)$ | 0.15
Table 4:  SM predictions similar to those presented in Tab. 3 but for the case
of vector meson in the final state, $B\to K^{\ast}\nu\bar{\nu}$.

Before closing this section, we should stress that depending on the precision
the so-called “Tree” contribution can be important when discussing the ratio
$r_{\mathrm{lh}}$ (10). While the case of $B\to K_{S}\nu\bar{\nu}$ remains
unchanged, we note that for the charged mode:

$\displaystyle\frac{\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})_{\mathrm{Tree}}}{\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})}\approx\biggl{.}11\,\%\biggr{|}_{\mathrm{low}-q^{2}},\,\biggl{.}13\%\biggr{|}_{\mathrm{high}-q^{2}},$
(24)

where the denominator represents the sum of values obtained by using Eq. (4)
and Eq. II.5.

##  III Improved strategies

We now explore the strategies allowing us to reduce the impact of hadronic
uncertainties, while trying to keep the sensitivity to the BSM physics
pronounced. In the following discussion, we focus on $B\to
K_{S}\,\nu\bar{\nu}$ which is not impacted by the above mentioned “Tree”
contribution.

###  III.1 $B\to K\nu\bar{\nu}$ at high-$q^{2}$

For $B\to K\nu\bar{\nu}$ decays, the simplest strategy to reduce the
theoretical uncertainty is to focus on the high-$q^{2}$ region, which is where
$f_{+}(q^{2})$ is precisely determined in LQCD. This restriction to high
$q^{2}$’s does hamper the sensitivity of this quantity to New Physics which
would only rescale the entire $q^{2}$-spectrum for left-handed neutrinos (see
Sec. IV). 888See Ref. Felkl:2021uxi  for a discussion that includes right-
handed neutrinos too. The main disadvantage of this approach is that less
statistics would be available in the experimental measurement, which to our
view is outweighed by the advantage of having better theoretical control over
the SM prediction.

By considering the two intervals, $q^{2}\geq 12~{}\mathrm{GeV}^{2}$ (bin I)
and $q^{2}\geq 16~{}\mathrm{GeV}^{2}$ (bin II), we find that they comprise
about $30\,\%$ and $15\,\%$ of the full event sample, respectively. The
corresponding SM predictions in these bins are:

$\displaystyle\mathcal{B}(B\to
K_{S}\nu\bar{\nu})^{\mathrm{SM}}_{\text{bin~{}I}}$ $\displaystyle=(0.670\pm
0.018\pm 0.040)\times 10^{-6}\,,$ (25) $\displaystyle\mathcal{B}(B\to
K_{S}\nu\bar{\nu})^{\mathrm{SM}}_{\text{bin~{}II}}$ $\displaystyle=(0.292\pm
0.008\pm 0.017)\times 10^{-6}\,,$ (26)

where the form factor uncertainties become subdominant in comparison with the
one arising from the CKM matrix elements. Note also that the relative
uncertainties in these intervals are about a factor of $2$ smaller than the
one from the total branching fraction, see Table 2.

This strategy provides a clear way to avoid the uncontrolled form factor
uncertainties in the $B\to K$ transition, provided the binned information will
be made available by Belle-II. However, there are several limitations to this
approach. Firstly, the uncertainty associated with the CKM factor remains
important, in particular the one arising from discrepancies between different
determinations of $|V_{cb}|$, cf. Sec. II.4. Moreover, as of now, this idea
cannot be fully exploited for $B\to K^{\ast}$ decays since the LQCD results at
nonzero recoil have only been obtained by one collaboration and without a full
control over systematic uncertainties Horgan:2013hoa . These limitations call
for the alternative approaches to reduce the theoretical errors, as discussed
in the following.

###  III.2 $(\nu/\ell)$ ratios

In this Section we explore an alternative way to reduce the theoretical
uncertainty on both $B\to K\nu\bar{\nu}$ and $B\to K^{\ast}\nu\bar{\nu}$
decays by exploiting the similarity with the corresponding decays into charged
leptons. More precisely, we study the following ratio,

$\displaystyle\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}[q_{0}^{2},q_{1}^{2}]\equiv\dfrac{\mathcal{B}(B\to
K^{(\ast)}\nu\bar{\nu})}{\mathcal{B}(B\to
K^{(\ast)}ll)}\Bigg{|}_{[q_{0}^{2},q_{1}^{2}]}\,,$ (27)

where $l\in\\{e,\mu\\}$, and the branching fractions are integrated over the
same $q^{2}$ interval, $[q_{0}^{2},q_{1}^{2}]$, both in the numerator and in
the denominator. Since the considered lepton masses are negligible with
respect to the other mass scales in the process, we can expect a cancellation
not only of the CKM factors, but also of the form factors in Eq. (27),
provided the $q^{2}$-bin is chosen judiciously. Of course, the region around
the $c\bar{c}$ resonances must be avoided, as the resonances would completely
spoil the benefits of the ratio
$\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}[q_{0}^{2},q_{1}^{2}]$. Moreover, one
should consider a region where $C_{9}^{\mathrm{eff}}(q^{2})$ is under
reasonable theoretical control. The optimal choice turns out to be the
interval $q^{2}\in[1.1,6]~{}\mathrm{GeV}^{2}$, the one that is already
considered in the experimental tests of lepton flavor universality (LFU)
LHCb:2022qnv .

We briefly remind the reader that the processes based on $b\to s\ell\ell$ can
be described by the following effective Lagrangian Altmannshofer:2008dz :

$\displaystyle\mathcal{L}_{\mathrm{eff}}^{b\to
s\ell\ell}=\dfrac{4G_{F}}{\sqrt{2}}\lambda_{t}\sum_{i}\Big{(}C_{i}^{\ell}\,\mathcal{O}_{i}^{\ell}+C_{i^{\prime}}^{\ell}\,\mathcal{O}_{i^{\prime}}^{\ell}\Big{)}+\mathrm{h.c.}\,,$
(28)

where the effective coefficients $C_{i}^{\ell\ell}\equiv
C_{i}^{\ell\ell}(\mu)$ and operators
$\mathcal{O}_{i}^{\ell\ell}\equiv\mathcal{O}_{i}^{\ell\ell}(\mu)$ are defined
at the scale $\mu=m_{b}$. The relevant operators to our study are

$\displaystyle\mathcal{O}_{9}^{\ell\ell}$
$\displaystyle=\dfrac{e^{2}}{(4\pi)^{2}}\big{(}\bar{s}\gamma_{\mu}P_{L}b\big{)}\big{(}\bar{\ell}\gamma_{\mu}\ell\big{)}\,,$
(29) $\displaystyle\mathcal{O}_{10}^{\ell\ell}$
$\displaystyle=\dfrac{e^{2}}{(4\pi)^{2}}\big{(}\bar{s}\gamma_{\mu}P_{L}b\big{)}\big{(}\bar{\ell}\gamma_{\mu}\gamma_{5}\ell\big{)}\,,$
(30)

in addition to the chirality flipped ones,
$\mathcal{O}_{i^{\prime}}^{\ell\ell}$, obtained from
$\mathcal{O}_{i}^{\ell\ell}$ by replacing $P_{L}\leftrightarrow P_{R}$, and
the dipole operators $\mathcal{O}_{7,8}$. The contributions from the four-
quark operators $\mathcal{O}_{1-6}$ are included in the redefinition of the
coefficients $C_{7,9}$ Altmannshofer:2008dz ; Buras:1993xp ; Bobeth:1999mk .

By relying on the theoretical inputs described above, we obtain the following
predictions for the $B\to K^{(\ast)}ll$ partial branching fractions (with
$l=e,\mu$),

$\displaystyle\mathcal{B}(B^{0}\to
K_{S}ll)^{\mathrm{SM}}_{[1.1,6]}/|\lambda_{t}|^{2}$
$\displaystyle=0.507(24)\times 10^{-4}\,,$ (31)
$\displaystyle\mathcal{B}(B^{0}\to K^{\ast
0}ll)^{\mathrm{SM}}_{[1.1,6]}/|\lambda_{t}|^{2}$ $\displaystyle=1.46(21)\times
10^{-4}\,.$ (32)

The difference between the rates with muons and with electrons in the final
state is completely marginal with respect to the current theoretical
uncertainties. By combining these results with $\mathcal{B}(B\to
K^{(\ast)}\nu\bar{\nu})_{[1.1,6]}$, we find:

$\displaystyle\biggl{.}\mathcal{R}_{K}^{(\nu/l)}[1.1,6]\biggr{|}_{\mathrm{SM}}$
$\displaystyle=7.58\pm 0.04\,,$ (33)

and

$\displaystyle\biggl{.}\mathcal{R}_{K^{\ast}}^{(\nu/l)}[1.1,6]\biggr{|}_{\mathrm{SM}}$
$\displaystyle=8.6\pm 0.3\,,$ (34)

which are shown in Fig. 4. For the ratios based on $B\to K$ decays, we obtain
a relative uncertainty less than $1\%$, which is much smaller the $7.5\%$
error on $\mathcal{B}(B\to K\nu\bar{\nu})_{[1.1,6]}$. The cancellation of the
$f_{+}(q^{2})$ form factor in $\smash{\mathcal{R}_{K}^{(\nu/l)}}$ is indeed
efficient, since the contributions involving the scalar ($f_{0}$) and tensor
($f_{T}$) form factors are suppressed by $m_{l}^{2}/q^{2}$ and by
$|C_{7}|/|C_{9}|\approx 0.1$, respectively. The same argument holds true for
the pseudoscalar ($A_{0}$) and the tensor form factors ($T_{1,2,3}$) entering
the $B\to K^{\ast}$ observables. However, in the latter case, there are more
form factors that survive even in the limit in which the charged lepton mass
and the photon-dipole are neglected. Therefore the cancellation of the form
factors work to a lesser extent and the ratio
$\smash{\mathcal{R}_{K^{\ast}}^{(\nu/l)}}$ is predicted with a relative
uncertainty of about $5\%$ in the bin we chose,
$q^{2}\in[1.1,6]~{}\mathrm{GeV}^{2}$. This resulting uncertainty is, however,
almost a factor $3$ smaller than the one in $\mathcal{B}(B\to
K^{\ast}\nu\bar{\nu})_{[1.1,6]}$, proving that the proposal to study the ratio
could be advantegous also in this case.

Figure 4: The ratios $\mathcal{R}_{K}^{(\nu/\mu)}$ and
$\mathcal{R}_{K^{\ast}}^{(\nu/\mu)}$ are plotted in orange and blue
respectively, as functions of $q^{2}$ in the region well below the first
$c\bar{c}$ resonance. In the lower panel we display the relative error in
order to better appreciate the precision with respect to those shown in Figs.
2 and 3.

###  III.3 Standard Model Considerations

The caveat in the above discussion is related to the choice of
$C_{9}^{\mathrm{eff}}(q^{2})$ that one has to make in order to evaluate the
full and/or partial branching fractions, $\mathcal{B}(B\to K^{(\ast)}ll)$.
That issue attracted quite a lot of attention in the literature and we will
not dwell on it here. To obtain the results in Eqs. (33,34) we relied on the
quark-hadron duality and used the two-loop results from Ref. Asatryan:2001zw .
The inclusion of the non-factorizable nonperturbative $c\bar{c}$-contribution
to $C_{9}^{\mathrm{eff}}$ has been discussed in Ref. Khodjamirian:2012rm , and
recently improved in Refs. Gubernari:2020eft ; Gubernari:2022hxn . A reliable
estimate of that contribution remains a challenging task. In practice, for
example, the quark-hadron duality, mentioned above, in the interval
$q^{2}\in[1.1,6]~{}\mathrm{GeV}^{2}$, leads to $C_{9}^{\mathrm{eff}}\simeq
4.4$, also used in the popular flavio code Straub:2018kue . Instead, in Ref.
Ciuchini:2020gvn  the value $C_{9}^{\mathrm{eff}}\simeq 4.1$ is preferred.
This is actually the main source of disagreement among various collaborations
while performing the global analyses of the experimental data for the angular
observables in the exclusive $b\to sll$ decay modes Ciuchini:2020gvn ;
Alguero:2019ptt .

We can now turn that problem around and actually use our ratios
$\mathcal{R}_{K^{(\ast)}}^{(\nu/e)}[1.1,6]$ in order to extract the
$C_{9}^{\mathrm{eff}}$ from the data. If we stick to the SM, we obtain

$\displaystyle\biggl{.}\frac{1}{\mathcal{R}_{K}^{(\nu/l)}[1.1,6]}\biggr{|}_{\mathrm{SM}}$
$\displaystyle\simeq$ $\displaystyle\left[7.15-0.45\cdot
C_{9}^{\mathrm{eff}}+0.42\cdot\left(C_{9}^{\mathrm{eff}}\right)^{2}\right]\times
10^{-2}\,,$ (35)
$\displaystyle\biggl{.}\frac{1}{\mathcal{R}_{K^{\ast}}^{(\nu/l)}[1.1,6]}\biggr{|}_{\mathrm{SM}}$
$\displaystyle\simeq$ $\displaystyle\left[9.98-1.45\cdot
C_{9}^{\mathrm{eff}}+0.42\cdot\left(C_{9}^{\mathrm{eff}}\right)^{2}\right]\times
10^{-2}\,,$ (36)

where we neglect the $q^{2}$-variation of $C_{9}^{\mathrm{eff}}$, which is why
in Fig. 5 we employed $\langle C_{9}^{\mathrm{eff}}\rangle$ to emphasize that
it corresponds to the average of $C_{9}^{\mathrm{eff}}(q^{2})$ over the
interval of $q^{2}$’s in which the ratio $\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}$
has been measured. It is interesting to note that if
$\mathcal{R}_{K}^{(\nu/l)}[1.1,6]$ is measured with a precision of $10\%$
($20\%$), the error on the extracted $\langle C_{9}^{\mathrm{eff}}\rangle$
would be $\pm 0.4$ ($\pm 0.8$). Finally, we should stress that it is important
to consider both the pseudoscalar and the vector kaon in the final state,
since the non-factorizable contributions are expected to be smaller in the
case of $K$ than in the case of outgoing $K^{\ast}$ Khodjamirian:2012rm .

Figure 5: The ratios $\mathcal{R}_{K}^{(\nu/\mu)}$ and
$\mathcal{R}_{K^{\ast}}^{(\nu/\mu)}$ are plotted as a function of the average
value of $C_{9}^{\mathrm{eff}}$ in a fixed $q^{2}$ bin. The black points
correspond to the predictions obtained using the $C_{9}^{\mathrm{eff}}$ input
from Ref. Straub:2018kue . Note that the ratio $\mathcal{R}_{K}^{(\nu/\mu)}$
is practically constant in the low-$q^{2}$ region and, for this reason, is
only plotted in the $[1.1,6]~{}\mathrm{GeV}^{2}$ bin, see Fig. 4 .

##  IV BSM implications

In this Section we discuss the sensitivity of the observables discussed above
to the contributions arising from physics BSM. We will provide the most
general expressions for the ratios defined in Eq. (27) and briefly illustrate
their variation with respect to the SM in the case of a peculiar New Physics
scenario.

#### General expressions:

We factorize the SM contribution and, in a given $q^{2}$-bin, we write:

$\displaystyle\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}=\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}\Big{|}_{\mathrm{SM}}\left(1+\delta\mathcal{R}^{(\nu/l)}_{K^{(\ast)}}\right)\,,$
(37)

where $\smash{\delta\mathcal{R}^{(\nu/l)}_{K^{(\ast)}}}$ is the New Physics
contribution. Similarly, $\delta\mathcal{B}_{K^{(\ast)}}^{\nu\bar{\nu}}$ and
$\delta\mathcal{B}_{K^{(\ast)}}^{ll}$ are the respective shifts of
$\mathcal{B}(B\to K^{(\ast)}\nu\bar{\nu})$ and $\mathcal{B}(B\to
K^{(\ast)}ll)$ in a given $q^{2}$ bin, arising from BSM,

$\displaystyle\begin{split}\mathcal{B}(B\to{K^{(\ast)}}{\nu\nu})&=\mathcal{B}(B\to{K^{(\ast)}}{\nu\nu})\Big{|}_{\mathrm{SM}}\,\big{(}1+\delta\mathcal{B}_{K^{(\ast)}}^{\nu\nu}\big{)}\,,\\\\[3.15005pt]
\small\mathcal{B}(B\to{K^{(\ast)}}{ll})&=\mathcal{B}(B\to{K^{(\ast)}}{ll})\Big{|}_{\mathrm{SM}}\,\big{(}1+\delta\mathcal{B}_{K^{(\ast)}}^{ll}\big{)}\,.\end{split}$
(38)

In other words, we can write:

$\displaystyle\delta\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}=\displaystyle\dfrac{\delta\mathcal{B}_{K^{(\ast)}}^{\nu\bar{\nu}}-\delta\mathcal{B}_{K^{(\ast)}}^{ll}}{1+\delta\mathcal{B}_{K^{(\ast)}}^{ll}}\,.$
(39)

The $\delta\mathcal{B}_{K^{(\ast)}}^{\nu\bar{\nu}}$ can be easily expressed in
terms of the Wilson coefficients defined in the Lagrangian (1), namely,

$\displaystyle\begin{split}\delta\mathcal{B}_{K^{(\ast)}}^{\nu\bar{\nu}}&=\sum_{i}\dfrac{2\mathrm{Re}[C_{L}^{\mathrm{SM}}\,(\delta
C_{L}^{\nu_{i}\nu_{i}}+\delta
C_{R}^{\nu_{i}\nu_{i}})]}{3|C_{L}^{\mathrm{SM}}|^{2}}\\\
&+\sum_{i,j}\dfrac{|\delta C_{L}^{\nu_{i}\nu_{j}}+\delta
C_{R}^{\nu_{i}\nu_{j}}|^{2}}{3|C_{L}^{\mathrm{SM}}|^{2}}\\\
&-\eta_{K^{(\ast)}}\sum_{i,j}\dfrac{\mathrm{Re}[\delta
C_{R}^{\nu_{i}\nu_{j}}(C_{L}^{\mathrm{SM}}\delta_{ij}+\delta
C_{L}^{\nu_{i}\nu_{j}})]}{3|C_{L}^{\mathrm{SM}}|^{2}}\,,\end{split}$ (40)

Decays ($l=e,\mu$) | $a_{VV}$ | $a_{VA}$ | $a_{AV}$ | $a_{AA}$ | $b_{VV}$ | $b_{VA}$ | $b_{AV}$ | $b_{AA}$
---|---|---|---|---|---|---|---|---
$B\to Kll$ | $0.2430(1)$ | $-0.260(1)$ | $0$ | $0$ | $0.0316(2)$ | $0.0317(2)$ | $0$ | $0$
$B\to K^{\ast}ll$ | $0.0012(48)$ | $-0.038(8)$ | $-0.191(10)$ | $0.255(6)$ | $0.0048(10)$ | $0.0047(10)$ | $0.0312(7)$ | $0.0311(7)$
Table 5:  Numerical coefficients defined in Eq. (41) for $l=e,\mu$, which are
computed in the $[1.1,6]~{}\mathrm{GeV}^{2}$ bin by using the $B\to K$ and
$B\to K^{\ast}$ form factors from Ref. Aoki:2021kgd  and Ref. Bharucha:2015bzk
, respectively. The difference between electron and muon coefficients lies
within the quoted uncertainties.

where we sum over the neutrino flavors $i,j\in\\{1,2,3\\}$. Note that the last
term vanishes for $B\to K\nu\bar{\nu}$ ($\eta_{K}=0$), but it is nonzero for
$B\to K^{\ast}\nu\bar{\nu}$. Using the form factor described in Sec. II, we
find that $\eta_{K^{\ast}}=3.47(10)$ in the $[1.1,6]~{}\mathrm{GeV}^{2}$ bin.
Similarly for $\delta\mathcal{B}_{K^{(\ast)}}^{ll}$ we have,

$\displaystyle\delta\mathcal{B}_{K^{(\ast)}}^{ll}$
$\displaystyle=\sum_{i}a_{i}\,\mathrm{Re}\big{(}\delta
C_{i}^{ll}\big{)}+\sum_{i}b_{i}\,|\delta C_{i}^{ll}|^{2}$ (41)

where $i\in\\{{VV},{AV},{VA},{AA}\\}$, with

$\displaystyle\begin{split}\delta C_{VV}^{ll}&=\delta
C_{9^{\prime}}^{ll}+\delta C_{9}^{ll}\,,\qquad\delta C_{VA}^{ll}=\delta
C_{10^{\prime}}^{ll}+\delta C_{{10}}^{ll}\,,\\\\[3.99994pt] \delta
C_{AV}^{ll}&=\delta C_{9^{\prime}}^{ll}-\delta C_{9}^{ll}\,,\qquad\delta
C_{AA}^{ll}=\delta C_{10^{\prime}}^{ll}-\delta C_{{10}}^{ll}\,.\end{split}$
(42)

We computed the numerical coefficients $a_{i}$ and $b_{i}$, and the results
are collected in Table 5.

#### SMEFT:

Since we are interested in New Physics scenarios defined well above the
electroweak scale, the low-energy effective theory must be replaced by the SM
effective field theory (SMEFT), which is invariant under the full
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ gauge symmetry Buchmuller:1985jz .
The main interest of using this approach is that contributions to the $b\to
s\nu\bar{\nu}$ are partially correlated to $b\to s\mu\mu$ via $SU(2)_{L}$
gauge invariance Buras:2014fpa ; Bause:2021cna ; deGiorgi:2022vup ;
Rajeev:2021ntt .

The $d=6$ semileptonic operators relevant to our study are:

$\displaystyle\begin{split}\big{[}\mathcal{O}_{lq}^{(1)}\big{]}_{ijkl}&=\big{(}\overline{L}_{i}\gamma^{\mu}L_{j}\big{)}\big{(}\overline{Q}_{k}\gamma_{\mu}Q_{l}\big{)}\,,\\\\[3.99994pt]
\big{[}\mathcal{O}_{lq}^{(3)}\big{]}_{ijkl}&=\big{(}\overline{L}_{i}\gamma^{\mu}\tau^{I}L_{j}\big{)}\big{(}\overline{Q}_{k}\tau^{I}\gamma_{\mu}Q_{l}\big{)}\,,\\\\[3.99994pt]
\big{[}\mathcal{O}_{eq}\big{]}_{ijkl}&=\big{(}\overline{e}_{i}\gamma^{\mu}e_{j}\big{)}\big{(}\overline{Q}_{k}\gamma_{\mu}Q_{l}\big{)}\,,\\\\[3.99994pt]
\big{[}\mathcal{O}_{ld}\big{]}_{ijkl}&=\big{(}\overline{L}_{i}\gamma^{\mu}L_{j}\big{)}\big{(}\overline{d}_{k}\gamma_{\mu}d_{l}\big{)}\,,\\\\[3.99994pt]
\big{[}\mathcal{O}_{ed}\big{]}_{ijkl}&=\big{(}\overline{e}_{i}\gamma^{\mu}e_{j}\big{)}\big{(}\overline{d}_{k}\gamma_{\mu}d_{l}\big{)}\,,\end{split}$
(43)

where $Q,L$ denote the SM quark and lepton $SU(2)_{L}$ doublets, and $u,d,e$
are the quark and lepton weak singlets. Flavor indices are denoted by
$i,j,k,l$. In what follows, we work in the flavor basis defined with diagonal
down-quark Yukawa matrix, i.e. the CKM matrix appears in the upper component
of $Q_{i}=[(V^{\dagger}\,u)_{i}\,,\,d_{i}]^{T}$.

The matching of the SMEFT Lagrangian to Eq. (28) gives

$\displaystyle\delta C_{9}^{\ell_{i}\ell_{i}}-\delta
C_{10}^{\ell_{i}\ell_{i}}$
$\displaystyle=\dfrac{2\pi}{\alpha_{\mathrm{em}}\lambda_{t}}\dfrac{v^{2}}{\Lambda^{2}}\left\\{\big{[}\mathcal{C}_{lq}^{(1)}\big{]}_{ii23}+\big{[}\mathcal{C}_{lq}^{(3)}\big{]}_{ii23}\right\\}\,,$
$\displaystyle\delta C_{9}^{\ell_{i}\ell_{i}}+\delta
C_{10}^{\ell_{i}\ell_{i}}$
$\displaystyle=\dfrac{2\pi}{\alpha_{\mathrm{em}}\lambda_{t}}\dfrac{v^{2}}{\Lambda^{2}}\big{[}\mathcal{C}_{qe}\big{]}_{ii23}\,,$
$\displaystyle\delta C_{9^{\prime}}^{\ell_{i}\ell_{i}}-\delta
C_{10^{\prime}}^{\ell_{i}\ell_{i}}$
$\displaystyle=\dfrac{2\pi}{\alpha_{\mathrm{em}}\lambda_{t}}\dfrac{v^{2}}{\Lambda^{2}}\big{[}\mathcal{C}_{\ell
d}\big{]}_{ii23}\,,$ (44) $\displaystyle\delta
C_{9^{\prime}}^{\ell_{i}\ell_{i}}+\delta C_{10^{\prime}}^{\ell_{i}\ell_{i}}$
$\displaystyle=\dfrac{2\pi}{\alpha_{\mathrm{em}}\lambda_{t}}\dfrac{v^{2}}{\Lambda^{2}}\big{[}\mathcal{C}_{ed}\big{]}_{ii23}\,.$

The (pseudo)scalar Wilson coefficients are not explicitly written since their
contributions to $\mathcal{B}(B\to K^{(\ast)}ll)$ are suppressed by the light
lepton masses.

Similarly, for the $b\to s\nu\bar{\nu}$ effective Lagrangian defined in Eq.
(1) we get:

$\displaystyle\begin{split}\delta
C_{L}^{\nu_{i}\nu_{j}}&=\dfrac{\pi}{\alpha_{\mathrm{em}}\lambda_{t}}\dfrac{v^{2}}{\Lambda^{2}}\left\\{\big{[}\mathcal{C}_{lq}^{(1)}\big{]}_{ij23}-\big{[}\mathcal{C}_{lq}^{(3)}\big{]}_{ij23}\right\\}\,,\\\\[3.99994pt]
\delta
C_{R}^{\nu_{i}\nu_{j}}&=\dfrac{\pi}{\alpha_{\mathrm{em}}\lambda_{t}}\dfrac{v^{2}}{\Lambda^{2}}\big{[}\mathcal{C}_{\ell
d}\big{]}_{ij23}\,.\end{split}$ (45)

The above relations, together with the general expression given in Eq. (39),
allow us to predict $\smash{\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}}$ in any
extension of the SM.

Figure 6: The ratios $\mathcal{R}_{K^{(\ast)}}^{(\nu/e)}$ (bottom and top left
panels) and $\mathcal{R}_{K^{(\ast)}}^{(\nu/\mu)}$ (bottom and top right
panels) normalized to their SM values, in the region
$q^{2}\in[1.1,6]~{}\mathrm{GeV}^{2}$, and plotted against $\delta
C_{9}^{\mu\mu}=-\delta C_{10}^{\mu\mu}$ for selected New Physics scenarios
with left-handed couplings to muons, see Sec. IV.

#### Illustration:

We will now consider the simplest scenario of New Physics that can contribute
to the $b\to s\mu\mu$ decay modes, namely via left-handed effective operators
satisfying $\delta C_{9}^{\mu\mu}=-\delta C_{10}^{\mu\mu}$, and illustrate its
impact onto the ratios (27). In terms of the SMEFT operators, this scenario
can arise from any combination of $\smash{\mathcal{C}_{lq}^{(1)}}$ and
$\smash{\mathcal{C}_{lq}^{(3)}}$, with couplings to muons, and to $2\to 3$
quark-flavor indices. Several concrete models can induce these operators
through the exchange of new bosons Buras:2014fpa . Clearly, these operators
contribute not only to $b\to s\mu\mu$, but also to $b\to
s\nu_{\mu}\bar{\nu}_{\mu}$, i.e. to both the numerator and denominator of Eq.
(27). We classify them in terms of their SM quantum numbers
$(SU(3)_{c},SU(2)_{L},U(1)_{Y})$,

  * $\bullet$

$Z^{\prime}\sim(\mathbf{1},\mathbf{1},0)$ : One of the simplest scenarios is
to extend the SM with a $Z^{\prime}$ that couples exclusively to left-handed
quarks and leptons,

$\mathcal{L}_{Z^{\prime}}\supset\left[g_{Q}^{ij}\,\bar{Q}_{i}\gamma_{\mu}Q_{j}+g_{L}^{ij}\,\bar{L}_{i}\gamma_{\mu}L_{j}\right]\,Z^{\prime\,\mu}\,,$
(46)

where $g_{Q}$ and $g_{L}$ are Hermitian matrices. In this case, we find that

$\mathcal{C}_{lq}^{(1)}\neq 0\,,\qquad\mathcal{C}_{lq}^{(3)}=0\,.$ (47)

  * $\bullet$

$V\sim(\mathbf{1},\mathbf{3},0)$ : $Z^{\prime}$ could be a part of a weak
triplet $V$ with the following interaction Lagrangian Greljo:2015mma ,

$\qquad\mathcal{L}_{V}\supset\left[g_{Q}^{ij}\,\bar{Q}_{i}\tau^{a}\gamma_{\mu}Q_{j}+g_{L}^{ij}\,\bar{L}_{i}\tau^{a}\gamma_{\mu}L_{j}\right]\,V_{a}^{\mu}\,,$
(48)

where $g_{Q}$ and $g_{L}$ are again Hermitian matrices, and $\tau^{a}$ are the
Pauli matrices, with $a=1,2,3$. In this case,

$\mathcal{C}_{lq}^{(1)}=0\,,\qquad\mathcal{C}_{lq}^{(3)}\neq 0\,.$ (49)

  * $\bullet$

$S_{3}\sim(\bar{\mathbf{3}},\mathbf{3},1/3)$ : $S_{3}$ is the scalar
leptoquark, often used in the literature Buchmuller:1986zs ; Hiller:2014yaa .
Its Yukawa interaction is given by

$\mathcal{L}_{S_{3}}\supset
y_{L}^{ij}\,\overline{Q_{i}^{C}}i\tau_{2}\big{(}\vec{\tau}\cdot\vec{S}_{3}\big{)}L_{j}+\mathrm{h.c.}\,,$
(50)

which then imply at tree-level,

$\mathcal{C}_{lq}^{(1)}=3\,\mathcal{C}_{lq}^{(3)}\,.$ (51)

  * $\bullet$

$U_{1}\sim(\mathbf{3},\mathbf{1},2/3)$ : Another leptoquark often used in
literature is the vector leptoquark $U_{1}$ which interacts with the left-
handed fermions via Buchmuller:1986zs ,

$\mathcal{L}_{U_{1}}\supset
x_{L}^{ij}\,\overline{Q_{i}}\gamma_{\mu}L_{j}\,U_{1}^{\mu}+\mathrm{h.c.}\,,$
(52)

from which we have,

$\mathcal{C}_{lq}^{(1)}=\mathcal{C}_{lq}^{(3)}\,.$ (53)

This combination of Wilson coefficient is peculiar as it contributes to $b\to
s\ell\ell$ at tree-level, but not to $b\to s\nu\bar{\nu}$ due to a
cancellation in Eq. (45).

The concrete models listed above predict different patterns for the effective
coefficients $\mathcal{C}_{lq}^{(1)}$ and $\mathcal{C}_{lq}^{(3)}$, which
imply specific model-dependent correlations between the $b\to s\mu\mu$ and
$b\to s\nu_{\mu}\nu_{\mu}$ transitions.

In Fig. (6) we plot the New Physics contribution to the ratios
$R^{(\nu/\mu)}_{K}$ (left panel) and $R^{(\nu/\mu)}_{K^{\ast}}$ (right panel)
for each of the scenarios listed above. For illustration purposes we consider
the $[1.1,6]~{}\mathrm{GeV}^{2}$ interval. The SMEFT operators induce not only
a modification of the numerator, but also of the denominator, which actually
enhances the effect up to $\mathcal{O}(40\%)$ in some models. In other words,
besides the theoretical accuracy of the $\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}$
ratios, which is improved compared to the separate branching fractions, these
ratios also allow us to increase the sensitivity to the type of New Physics
effects considered here.

The above discussion is based on a minimalistic assumption that only the
muonic couplings in the $b\to s\ell\ell$ and $b\to s\nu\bar{\nu}$ transitions
are affected by New Physics, but obviously that assumption can be changed. For
example, one can test whether or not the New Physics couplings to $\tau$’s are
significant deGiorgi:2022vup ; Descotes-Genon:2020buf , such as the case in
some leptoquark scenarios that can accommodate the LFU discrepancies in the
$b\to c\tau\nu$ transition Angelescu:2021lln . Since these scenarios predict
an enhancement of $\mathcal{B}(B\to K\nu_{\tau}\bar{\nu}_{\tau})$
Becirevic:2022tsj ; Fuentes-Martin:2020hvc ; Gherardi:2020qhc , it is clear
that the effects depicted in Fig. 6 would only increase in this case.

##  V Summary

In this paper we revisited the SM estimate of $\mathcal{B}(B\to
K^{(\ast)}\nu\bar{\nu})$. This is particularly important for the case of the
pseudoscalar meson in the final state because the relevant form factor has
been extensively studied and computed by means of LQCD. Since the new such a
calculation appeared after the most recent release of the FLAG review, we
updated the FLAG average of all three form factors relevant to the $B\to K$
transitions. Since the lattice QCD results are obtained for large $q^{2}$’s
one should be careful when addressing the issue of systematic uncertainties.
For that reason we believe that the most reliable way to test the SM value for
$\mathcal{B}^{\prime}(B\to K^{(\ast)}\nu\bar{\nu})$ is in the region of large
$q^{2}$’s. Furthermore, the experimental information on
$r_{\mathrm{lh}}={\mathcal{B}(B\to
K\nu\bar{\nu})_{\mathrm{low}-q^{2}}/\mathcal{B}(B\to
K\nu\bar{\nu})_{\mathrm{high}-q^{2}}}$ would be helpful to test the validity
of the extrapolation of LQCD results (obtained at high-$q^{2}$) to low
$q^{2}$’s. Besides the hadronic uncertainties in $\mathcal{B}(B\to
K^{(\ast)}\nu\bar{\nu})$, it is important to further improve the
$\lambda_{t}=V_{tb}V_{ts}^{\ast}$ which, by virtue of the CKM unitarity, is
related to the problem of reconciling the value of $V_{cb}$ extracted from
exclusive and from inclusive semileptonic decays.

Most of the uncertainties mentioned above actually cancel out if one considers
the ratio of the partial decay rates of $\mathcal{B}^{\prime}(B\to
K^{(\ast)}\nu\bar{\nu})$ and of $\mathcal{B}(B\to K^{(\ast)}\ell\ell)$, which
we denote by $\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}$. The major hadronic
uncertainty in both rates comes from the form factor which cancels out in
$\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}$. The uncertainty from the multiplicative
CKM factor $\lambda_{t}$ cancels out as well. However, the price to pay is
that the Wilson coefficient $C_{9}$, entering $\mathcal{B}(B\to
K^{(\ast)}\ell\ell)$, becomes an obstacle because it is sensitive to the
contribution from the non-local operator arising from the vector current
couplings to $c\bar{c}$. In the literature that contribution is often
estimated by using the quark-hadron duality or by resorting to the model
calculations. If we stick to the SM, we show that from a measurement of the
ratio $\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}$ in a given interval of $q^{2}$’s
(preferably below the first $c\bar{c}$ resonance), one can extract the value
$\langle C_{9}^{\mathrm{eff}}\rangle$, and indeed check whether of not the
sizable non-factorizable contribution would result in $\langle
C_{9}^{\mathrm{eff}}\rangle_{K}\neq\langle
C_{9}^{\mathrm{eff}}\rangle_{K^{\ast}}$, as sometimes argued in the
literature. By $\langle C_{9}^{\mathrm{eff}}\rangle$ we denote the
$C_{9}^{\mathrm{eff}}(q^{2})$ averaged over the interval in which
$\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}$ is measured.

To further support the benefits of measuring
$\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}$, we also illustrate how it can be used to
look for the effects of physics BSM. In a scenario in which the New Physics
contributed at low energy scales through left-handed couplings, both to quarks
and to leptons, we find that $\mathcal{R}_{K^{(\ast)}}^{(\nu/l)}$ would be
more sensitive a test of presence of physics BSM than its numerator and/or its
denominator separately. We also provided an illustration of such a scenario in
several simple models.

$a_{0}^{+}$ | $a_{1}^{+}$ | $a_{2}^{+}$ | $a_{0}^{0}$ | $a_{1}^{0}$ | $a_{0}^{T}$ | $a_{1}^{T}$ | $a_{2}^{T}$
---|---|---|---|---|---|---|---
$0.4742(62)$ | $-0.894(51)$ | $-0.44(14)$ | $0.2939(36)$ | $0.277(40)$ | $0.4752(92)$ | $-0.921(82)$ | $-0.53(35)$
$1$ | $-0.2904$ | $-0.0347$ | $0.7480$ | $0.1844$ | $0.6558$ | $-0.2193$ | $-0.0751$
$\cdot$ | $1$ | $0.7757$ | $0.2291$ | $0.8527$ | $-0.2569$ | $0.5371$ | $0.2574$
$\cdot$ | $\cdot$ | $1$ | $0.1690$ | $0.8455$ | $-0.1029$ | $0.3700$ | $0.2653$
$\cdot$ | $\cdot$ | $\cdot$ | $1$ | $0.4568$ | $0.5232$ | $0.0314$ | $0.0257$
$\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $1$ | $0.0182$ | $0.4501$ | $0.2372$
$\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $1$ | $-0.0255$ | $-0.0535$
$\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $1$ | $0.6920$
$\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $1$
Table 6:  Values of the $z$-expansion coefficients and correlation matrix
obtained by our combined fit to the $B\to K$ form factors computed by HPQCD
Parrott:2022rgu  and FNAL/MILC Bailey:2015dka  (with
$\chi^{2}_{\mathrm{min}}/\mathrm{d.o.f}\simeq 9.2/10$). We consider the
parameterization from Eq. (55)–(57) with $N=3$, and we remove the $a_{2}^{0}$
coefficient by imposing the relation $f_{+}(0)=f_{0}(0)$. The covariance
matrix is provided with more digits in an ancillary file. See text for more
details.

## Acknowledgments

We thank F. Jaffredo and E. Lunghi for discussions regarding the fits to the
form factors computed in lattice QCD, M. Fedele and P. Stangl for discussion
regarding $C_{9}^{\mathrm{eff}}$, and F. Mescia for useful exchanges. This
project has received support from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement No
860881-HIDDeN.

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

####  $B\to K$ :

The definitions of the vector ($f_{+}$) and scalar ($f_{0}$) form factors are
given in Eq. II.2, while the tensor form factor $f_{T}$ is defined as follows

$\displaystyle\langle\bar{K}(k)|\bar{s}\sigma_{\mu\nu}b|\bar{B}(p)\rangle$
$\displaystyle=-i\left(p_{\mu}k_{\nu}-p_{\nu}k_{\mu}\right)\dfrac{2f_{T}(q^{2})}{m_{B}+m_{K}}\,.$
(54)

We consider the same parameterization for the $B\to K$ form factors as
provided by FLAG Aoki:2021kgd ,

$\displaystyle f_{+}(q^{2})$
$\displaystyle=\frac{1}{P_{+}(q^{2})}\sum_{n=0}^{N-1}a_{n}^{+}\,\left[z^{n}-(-1)^{n-N}\frac{n}{N}z^{N}\right]\,,$
(55) $\displaystyle f_{T}(q^{2})$
$\displaystyle=\frac{1}{P_{T}(q^{2})}\sum_{n=0}^{N-1}a_{n}^{T}\,\left[z^{n}-(-1)^{n-N}\frac{n}{N}z^{N}\right]\,,$
(56) $\displaystyle f_{0}(q^{2})$
$\displaystyle=\frac{1}{P_{0}(q^{2})}\sum_{n=0}^{N-1}a_{n}^{0}\,z^{n}\,,$ (57)

where $a_{n}^{i}$ (with $i\in\\{0,+,T\\}$) are numerical coefficients, the
pole factors are given by

$\displaystyle P_{i}(q^{2})=1-q^{2}/M_{i}^{2}\,,$ (58)

with $M_{+}=M_{T}=5.4154\ {\mathrm{GeV}}$ and $M_{0}=5.711\ {\mathrm{GeV}}$
Lang:2015hza , and $z\equiv z(q^{2})$ reads

$z(q^{2})=\frac{\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}}}\,,$
(59)

where $t_{+}=(m_{B}+m_{K})^{2}$ and we consider
$t_{0}=(m_{B}+m_{K})(\sqrt{m_{B}}-\sqrt{m_{K}})^{2}$. Since the scalar and
vector form factor satisfy $f_{+}(0)=f_{0}(0)$, it is possible to remove one
of the coefficients in Eq. (55) and (56), which we take to be $a_{2}^{0}$,

$a_{2}^{0}=\frac{f_{+}(0)-a_{0}^{0}-a_{1}^{0}\,z(q^{2}=0)}{z^{2}(q^{2}=0)}\,.$
(60)

Figure 7: The result of our fit for the $f_{T}(q^{2})$ form factors is
depicted by the blue region respectively, compared to the result reported by
FLAG Aoki:2021kgd  and the HPQCD (green points) Parrott:2022rgu  and FNAL/MILC
(orange points) Bailey:2015dka  results.

In this letter, we update the combined fit to the $B\to K$ form factors made
by FLAG Aoki:2021kgd , by including the latest HPQCD results Parrott:2022rgu
that are combined with the ones from FNAL/MILC Bailey:2015dka . To this
purpose, we follow the same procedure as FLAG which consists in generating
synthetic data points for $\\{f_{+}(q^{2}),f_{0}(q^{2}),f_{T}(q^{2})\\}$ for
both HPQCD and FNAL/MILC form factors at three $q^{2}$ values, namely
$q^{2}\in\\{18,20.5,23\\}~{}\mathrm{GeV}^{2}$. The central values and the
covariance matrix obtained for each collaboration are then fitted in a
combined $\chi^{2}$ assuming that the HPQCD and FNAL/MILC results are
uncorrelated. We consider the same parameterization of FLAG with $N=3$, as
specified above, and we remove the $a_{2}^{0}$ coefficient by exploiting the
scalar/vector form factor relations at $q^{2}=0$.

The results of our $\chi^{2}$ fit are given in Table (6), including the
correlation matrix between form factor coefficients. Our combined fit gives
$\chi^{2}_{\mathrm{min}}/\mathrm{d.o.f}\simeq 9.2/10$ and, differently from
FLAG Aoki:2021kgd , we opt for not rescaling the uncertainties of the fitted
parameters by $\sqrt{\chi^{2}_{\mathrm{min}}/\mathrm{d.o.f}}$. Our results for
$f_{+}(q^{2})$ are given in Fig. 2, where we see a good agreement between the
two calculations and our combined fit.

We have performed several cross-checks of our fitting procedure. In
particular, we are able to perfectly reproduce the FLAG results when combining
FNAL/MILC Bailey:2015dka  with the previous HPQCD results Bouchard:2013eph ,
provided we rescaled the uncertainties of the fitted parameters by
$\smash{\sqrt{\chi^{2}_{\mathrm{min}}/\mathrm{d.o.f}}}$. As already stated
above, we opt for not rescaling the uncertainties of our new combined fit.

####  $B\to K^{\ast}$:

The vector and axial form factors entering the $B\to K^{\ast}$ transition are
$\\{A_{0}$, $A_{1}$, $A_{2}$, $V\\}$, which are defined in Eq. II.3. The
tensor form factors $\\{T_{1}$, $T_{2}$, $T_{3}\\}$ are defined by

$\displaystyle\langle\bar{K}^{\ast}(k)$
$\displaystyle|\bar{s}\sigma_{\mu\nu}q^{\nu}(1-\gamma_{5})b|\bar{B}(p)\rangle=2i\varepsilon_{\mu\alpha\beta\gamma}\epsilon^{\ast\alpha}p^{\beta}k^{\gamma}T_{1}(q^{2})$
$\displaystyle+$
$\displaystyle\bigl{[}(m_{B}^{2}-m_{K^{\ast}}^{2})\epsilon^{\ast}_{\mu}-(\epsilon^{\ast}\cdot
q)(k+p)_{\mu}\bigr{]}\,T_{2}(q^{2})$ $\displaystyle+$
$\displaystyle(\epsilon^{\ast}\cdot
q)\left[q_{\mu}-\dfrac{q^{2}}{m_{B}^{2}-m_{K^{\ast}}^{2}}(k+p)_{\mu}\right]\,T_{3}(q^{2})\,,$
(61)

where $\varepsilon_{\mu}$ is the polarization vector of $K^{\ast}$. The $B\to
K^{\ast}$ form factor parameterization and input parameters used in our study
are taken from Ref. Bharucha:2015bzk , and we conservatively assume the fitted
parameters to be uncorrelated.

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