# Understanding the first measurement of $\mathcal{B}(B\to K\nu\bar{\nu})$

L. Allwicher<EMAIL_ADDRESS>Physik-Institut, Universität Zürich,
CH-8057 Zürich, Switzerland  D. Bečirević<EMAIL_ADDRESS>IJCLab, Pôle Théorie (Bat. 210), CNRS/IN2P3 et Université, Paris-Saclay, 91405
Orsay, France  G. Piazza<EMAIL_ADDRESS>IJCLab, Pôle
Théorie (Bat. 210), CNRS/IN2P3 et Université, Paris-Saclay, 91405 Orsay,
France  S. Rosauro-Alcaraz<EMAIL_ADDRESS>IJCLab, Pôle
Théorie (Bat. 210), CNRS/IN2P3 et Université, Paris-Saclay, 91405 Orsay,
France  O. Sumensari<EMAIL_ADDRESS>IJCLab, Pôle Théorie
(Bat. 210), CNRS/IN2P3 et Université, Paris-Saclay, 91405 Orsay, France

###### Abstract

Recently, Belle II reported on the first measurement of
$\mathcal{B}(B^{\pm}\to K^{\pm}\nu\bar{\nu})$ which appears to be almost
$3\sigma$ larger than predicted in the Standard Model. We point out the
important correlation with $\mathcal{B}(B\to K^{\ast}\nu\bar{\nu})$ so that
the measurement of that decay mode could help restraining the possible options
for building the model of New Physics. We then try to interpret this new
experimental result in terms of physics beyond the Standard Model by using
SMEFT and find that a scenario with coupling only to $\tau$ can accommodate
the current experimental constraints but fails in getting a desired
$R_{D^{(\ast)}}^{\mathrm{exp}}/R_{D^{(\ast)}}^{\mathrm{SM}}$, unless one turns
the other SMEFT operators that are not related to $b\to s\ell\ell$ or/and
$b\to s\nu\nu$.

##  I Introduction

Experimental studies of the angular distribution of $B\to K^{\ast}(\to
K\pi)\mu\mu$ LHCb:2015svh ; CMS:2020oqb  and $B\to K\mu\mu$ LHCb:2014auh ;
CMS:2018qih  offered a number of observables, the study of which could help
unveiling the effects of presence of New Physics (NP), i.e. physics beyond the
Standard Model (BSM). It appeared, however, that interpreting several apparent
deviations with respect to the Standard Model (SM) predictions required a very
good control over the hadronic uncertainties, and in particular those related
to the low-energy operators coupling to $\bar{c}\gamma_{\mu}c$. Those latter
contributions are particularly problematic in the regions of
$q^{2}=(p_{\mu^{+}}+p_{\mu^{-}})^{2}$ populated by the $c\bar{c}$ resonances.
Clearly, in order to resolve the effects of BSM physics from those related to
the SM weak interaction in these regions, one needs to evaluate the relevant
hadronic matrix element of a non-local operator, which cannot be done yet in
lattice QCD. Instead, one resorts to using various assumptions such as the
quark-hadron duality to treat the problem perturbatively (even though the
energy window is narrow) Greub:2008cy ; Asatryan:2001zw ; Asatrian:2019kbk ,
or a specific hadronic model Khodjamirian:2010vf ; Khodjamirian:2012rm ;
Gubernari:2020eft ; Gubernari:2022hxn . That problem is commonly circumvented
when measuring $\smash{R_{K^{(\ast)}}=\mathcal{B}^{\prime}(B\to
K^{(\ast)}\mu\mu)/\mathcal{B}^{\prime}(B\to K^{(\ast)}ee)}$ Hiller:2003js ,
where $\mathcal{B}^{\prime}$ is used to indicate that the partial branching
fractions are measured in the interval $q^{2}\in[1.1,6]~{}\mathrm{GeV}^{2}$,
well below the first $c\bar{c}$ resonance,
$m_{J/\psi}^{2}=(3.097\,\mathrm{GeV})^{2}$. First measurements of $R_{K}$ and
$R_{K^{\ast}}$, as well as of $\mathcal{B}(B_{s}\to\mu\mu)$ ATLAS:2018cur ;
CMS:2020rox ; LHCb:2021awg  indicated an important departure from the SM
predictions. Many models used to accommodate these deviations were used to
constrain the BSM couplings, most of which also implied a significant
deviation of $\mathcal{B}(B\to K^{(\ast)}\nu\bar{\nu})$ from its SM value.
Very recently, however, LHCb reported $R_{K}=0.998(90)$,
$R_{K^{\ast}}=0.930(97)$ LHCb:2022qnv , thus fully consistent with
$R_{K^{(\ast)}}^{\mathrm{SM}}=1.00(1)$ Bordone:2016gaq .

Even though it is experimentally much more challenging, $B\to
K^{(\ast)}\nu\bar{\nu}$ is theoretically cleaner than the equivalent mode with
charged leptons instead of neutrinos in the final state. This is so because
the coupling to the problematic operators involving $c\bar{c}$ resonances is
absent. The remaining non-perturbative QCD obstacle, that both decay modes
share, is a reliable theoretical estimate of the hadronic matrix elements of
the local operators in the entire physical region, $0\lesssim
q^{2}\leq(m_{B}-m_{K^{(\ast)}})^{2}$. That task is also very challenging for
lattice QCD because the available $q^{2}$ range is too large. This is why the
lattice QCD results are used to constrain the parameters entering a model
$q^{2}$-dependence of the relevant form factor, necessary for the SM
prediction of $\mathcal{B}(B\to K\nu\bar{\nu})$. Interestingly, in the case of
$B\to K\nu\bar{\nu}$ the $q^{2}$ shape of the hadronic form factor can be
checked experimentally by measuring the partial branching fractions
$\mathcal{B}^{\prime}(B\to K\nu\bar{\nu})$, as discussed in Ref.
Becirevic:2023aov .

In the following we will often use the ratio
$R_{\nu\nu}^{K^{(\ast)}}=\mathcal{B}(B\to
K^{(\ast)}\nu\bar{\nu})/\mathcal{B}(B\to
K^{(\ast)}\nu\bar{\nu})^{\mathrm{SM}}$, for which the experimental upper
bounds exist Belle:2017oht :

$\displaystyle R^{K}_{\nu\nu}<3.6\quad(90\%\ \text{C.L.})\,,$ (1)
$\displaystyle R^{K^{*}}_{\nu\nu}<2.7\quad(90\%\ \text{C.L.})\,.$ (2)

Very recently, the first of these bounds was superseded by the first
measurement of this decay mode by Belle II, $\mathcal{B}(B^{\pm}\to
K^{\pm}\nu\bar{\nu})=2.40(67)\times 10^{-5}$ Glazov , which appears to be
$2.9\sigma$ larger than its SM estimate. 111Note that for the SM estimate we
take $\mathcal{B}(B^{\pm}\to K^{\pm}\nu\bar{\nu})=4.44(14)(27)\times
10^{-5}$Becirevic:2023aov  i.e. the value which does not include the tree
level contribution Kamenik:2009kc  which was also subtracted away during the
experimental data analysis Glazov . This then leads to

$\displaystyle R^{K}_{\nu\nu}=5.4\pm 1.5\,.$ (3)

In this letter we discuss how this departure from the SM prediction can be
interpreted in terms of generic BSM scenarios in the SMEFT framework, which
then requires one to remain consistent with the stringent bounds on NP in
$b\to s$ transitions arising from the measured $R_{K^{(\ast)}}$ mentioned
above, as well as from the measured
$\mathcal{B}(B_{s}\to\mu\mu)=3.35(27)\times 10^{-8}$ CMS:2022mgd , which is
consistent with the SM estimate
$\mathcal{B}(B_{s}\to\mu\mu)^{\mathrm{SM}}=3.66(4)\times 10^{-8}$
Beneke:2019slt .

##  II Effective Theory considerations

The low energy effective theory relevant to the $b\to s\nu\nu$ decay is
described by the effective Hamiltonian,

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

with

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

in a standard notation with $\lambda_{t}=V_{tb}V_{ts}^{\ast}$, the suitable
combination of the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements. In the SM
we know that $C_{R}^{\mathrm{SM}}=0$ and
$\big{[}C_{L}^{\nu_{i}\nu_{j}}\big{]}_{\mathrm{SM}}\equiv\delta_{ij}\,C_{L}^{\mathrm{SM}}$,
with $C_{L}^{\mathrm{SM}}=-6.32(7)$ Buchalla:1993bv ; Buchalla:1998ba ;
Misiak:1999yg ; Brod:2010hi ; Buras:2014fpa . A detailed discussion of
hadronic uncertainties and those arising from the choice of $\lambda_{t}$ can
be found in Ref. Becirevic:2023aov . Here we will just mention that in the SM
the values of decay rates of the charged $B$-meson read:

$\displaystyle\mathcal{B}(B^{\pm}\to K^{\pm}\nu\nu)$ $\displaystyle=(4.44\pm
0.30)\times 10^{-6}\,,$ $\displaystyle\mathcal{B}(B^{\pm}\to
K^{\pm\ast}\nu\nu)$ $\displaystyle=(9.8\pm 1.4)\times 10^{-6}\,,$ (6)

and they do not comprise the triply Cabibbo-suppressed tree-level contribution
(also subtracted away in the experimental data analysis). It is convenient to
factor out the SM contribution to the branching fractions and write

$\displaystyle\mathcal{B}(B\to{K^{(\ast)}}{\nu\nu})$
$\displaystyle=\mathcal{B}(B\to{K^{(\ast)}}{\nu\nu})\Big{|}_{\mathrm{SM}}\,\big{(}1+\delta\mathcal{B}_{K^{(\ast)}}^{\nu\nu}\big{)}\,,$
(7)

so that the NP piece, $\delta\mathcal{B}_{K^{(\ast)}}^{\nu\bar{\nu}}$, can be
expressed in terms of $\delta C_{L,R}^{\nu_{i}\nu_{j}}$, the BSM contribution
to the left- or right-handed operator given in Eq. (II). By writing,
$C_{L,R}^{\nu_{i}\nu_{j}}=\delta_{ij}C_{L,R}^{\mathrm{SM}}+\delta
C_{L,R}^{\nu_{i}\nu_{j}}$, we have Buras:2014fpa ; Becirevic:2023aov :

$\displaystyle\begin{split}\delta\mathcal{B}_{K^{(\ast)}}^{\nu\bar{\nu}}&=\sum_{i}\dfrac{2\mathrm{Re}[C_{L}^{\mathrm{SM}}\,(\delta
C_{L}^{\nu_{i}\nu_{i}}+\delta
C_{R}^{\nu_{i}\nu_{i}})]}{3|C_{L}^{\mathrm{SM}}|^{2}}\\\
&+\sum_{i,j}\dfrac{|\delta C_{L}^{\nu_{i}\nu_{j}}+\delta
C_{R}^{\nu_{i}\nu_{j}}|^{2}}{3|C_{L}^{\mathrm{SM}}|^{2}}\\\
&-\eta_{K^{(\ast)}}\sum_{i,j}\dfrac{\mathrm{Re}[\delta
C_{R}^{\nu_{i}\nu_{j}}(C_{L}^{\mathrm{SM}}\delta_{ij}+\delta
C_{L}^{\nu_{i}\nu_{j}})]}{3|C_{L}^{\mathrm{SM}}|^{2}}\,,\end{split}$ (8)

where the sum over neutrino flavor indices is understood, $i,j\in\\{1,2,3\\}$.
In the above expression, $\eta_{K}=0$, and $\eta_{K^{\ast}}=3.33(7)$. In that
way, one can easily check the response of
$\mathcal{B}(B\to{K^{\ast}}{\nu\nu})$ to the new experimentally established
$\mathcal{B}(B\to{K}{\nu\nu})$, which is shown in Fig. 1.

Figure 1: The correlation between $\mathcal{B}(B\to{K^{\ast}}{\nu\nu})$ and
$\mathcal{B}(B\to K{\nu\nu})$ decays with respect to the variation of $\delta
C_{L}$ or $\delta C_{R}$. The shaded gray area correspond to $1\sigma$ and
$2\sigma$ of the recent Belle II result for $\mathcal{B}(B\to K{\nu\nu})$. The
red point corresponds to the SM predictions for these observables. We also
show the region of experimentally excluded
$\mathcal{B}(B\to{K^{\ast}}{\nu\nu})$ values.

By switching on either $\delta C_{L}$ or $\delta C_{R}$ and by considering all
three diagonal couplings to neutrino flavors, we obtain that accommodating the
measured $\mathcal{B}(B\to K{\nu\nu})$ to $2\sigma$ results in:

$\displaystyle\delta C_{L,R}\in[-12.0,-3.5]\cup[16.1,24.7]\,.$ (9)

These solutions become strongly restricted after imposing the experimental
bound $\mathcal{B}(B\to{K^{\ast}}{\nu\nu})^{\mathrm{exp}}<2.7\times 10^{-5}$
Belle:2017oht . In particular, the large positive values of $\delta C_{R}$ are
discarded, and the above ranges become:

$\displaystyle\delta C_{L}\neq 0:$ $\displaystyle\ \delta
C_{L}\in[-4.2,-3.5]\cup[16.9,17.3],\quad$ $\displaystyle
R_{\nu\nu}^{K^{\ast}}\in(2.4,2.7)\,,$ $\displaystyle\delta C_{R}\neq 0:$
$\displaystyle\ \delta C_{R}\in[-12.0,-3.5],\quad$ $\displaystyle
R_{\nu\nu}^{K^{\ast}}\in(0.6,2.1)\,,$ (10)

where we also give the resulting $\mathcal{B}(B\to{K^{\ast}}{\nu\nu})$
compatible with $\mathcal{B}(B\to K{\nu\nu})^{\mathrm{exp}}$ to $2\sigma$ and
the experimental bound on
$\mathcal{B}(B\to{K^{\ast}}{\nu\nu})^{\mathrm{exp}}$. At this stage we
consider the BSM contributions to be neutrino flavor universal, which is a
relevant information when interpreting the values for $\delta C_{L,R}$.
However, the correlation shown in Fig. 1 and the ranges of
$R_{\nu\nu}^{K^{\ast}}$ given in Eq. (II) remain as such regardless of whether
we assume the NP to be lepton flavor universal or not.

One can go a step further and predict the behavior of a fraction of the
$B\to{K^{\ast}}{\nu\nu}$ corresponding to a specific polarization state of the
outgoing $K^{\ast}$. For example, one can check how $F_{L}$, the fraction of
decay rate corresponding to the longitudinally polarized $K^{\ast}$ (cf. Refs.
Buras:2014fpa ; Das:2017ebx ), responds to a non-zero $\delta C_{L}$ or
$\delta C_{R}$. $\mathcal{R}_{F_{L}}=F_{L}/F_{L}^{\mathrm{SM}}$ is presented
in Fig. 2 to better emphasize the fact that $F_{L}$ remains insensitive to
$\delta C_{L}\neq 0$, whereas it becomes drastically depleted with respect to
the SM value, $F_{L}^{\mathrm{SM}}=0.49(7)$, when $\delta C_{R}\neq 0$ is
chosen such that it is consistent with both $\mathcal{B}(B\to
K{\nu\nu})^{\mathrm{exp}}$ and the experimental bound on $\mathcal{B}(B\to
K^{\ast}{\nu\nu})^{\mathrm{exp}}$. In fact, we obtain that with $\delta
C_{R}\neq 0$, the value of $F_{L}$ gets more than $50\%$ suppressed with
respect to its SM value. From the plot in Fig. 2 we read off:

$0\leq\mathcal{R}_{F_{L}}\leq 0.44\quad\Rightarrow\quad F_{L}\in[0,0.21]\,.$
(11)

This is a clear prediction that could be tested experimentally. Note, in
particular, that $F_{L}$ has a much smaller theoretical uncertainty in the SM
than the $\mathcal{B}(B\to K^{\ast}\nu\bar{\nu})$, since most of the form
factor uncertainties cancel out in the ratio.

Figure 2: Similar to Fig. 1 we show how the fraction of
$B\to{K^{\ast}}{\nu\nu}$ with longitudinally polarized $K^{\ast}$ responds to
the variation of $\delta C_{L}$ or $\delta C_{R}$. More highlighted are the
parts of the curves consistent with the constraints plotted in Fig. 1.

##  III SMEFT considerations

When discussing the BSM scenarios in which the new degrees of freedom enter
the stage well above the electroweak scale, it is convenient to work in the SM
effective field theory (SMEFT), invariant under the full $SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}$ gauge symmetry Buchmuller:1985jz . This allows one
to relate the $b\to s\nu\bar{\nu}$ and $b\to s\ell\ell$ via $SU(2)_{L}$ gauge
symmetry Buras:2014fpa ; Aebischer:2015fzz ; Bause:2021cna ; deGiorgi:2022vup
; Rajeev:2021ntt . Of all the $d=6$ operators in

$\mathcal{L}^{(6)}_{\mathrm{SMEFT}}\supset\sum_{i}\frac{\mathcal{C}_{i}}{\Lambda^{2}}{\mathcal{O}}_{i}\,,$
(12)

we select those relevant to our study, namely,

$\displaystyle\begin{split}\big{[}\mathcal{O}_{Hq}^{(1)}\big{]}_{kl}&=\big{(}\overline{Q}_{k}\gamma^{\mu}Q_{l}\big{)}\big{(}H^{\dagger}{\overleftrightarrow{D}}_{\mu}H\big{)}\,,\\\\[3.99994pt]
\big{[}\mathcal{O}_{Hq}^{(3)}\big{]}_{kl}&=\big{(}\overline{Q}_{k}\tau^{I}\gamma_{\mu}Q_{l}\big{)}\big{(}H^{\dagger}{\overleftrightarrow{D}}_{\mu}\tau^{I}H\big{)}\,,\\\\[3.99994pt]
\big{[}\mathcal{O}_{Hd}\big{]}_{kl}&=\big{(}\overline{d}_{kR}\gamma_{\mu}d_{lR}\big{)}\big{(}H^{\dagger}{\overleftrightarrow{D}}_{\mu}H\big{)}\,,\\\\[8.99994pt]
\end{split}$ (13)
$\displaystyle\begin{split}\big{[}\mathcal{O}_{lq}^{(1)}\big{]}_{ijkl}&=\big{(}\overline{L}_{i}\gamma^{\mu}L_{j}\big{)}\big{(}\overline{Q}_{k}\gamma_{\mu}Q_{l}\big{)}\,,\\\\[3.99994pt]
\big{[}\mathcal{O}_{lq}^{(3)}\big{]}_{ijkl}&=\big{(}\overline{L}_{i}\gamma^{\mu}\tau^{I}L_{j}\big{)}\big{(}\overline{Q}_{k}\tau^{I}\gamma_{\mu}Q_{l}\big{)}\,,\\\\[3.99994pt]
\big{[}\mathcal{O}_{ld}\big{]}_{ijkl}&=\big{(}\overline{L}_{i}\gamma^{\mu}L_{j}\big{)}\big{(}\overline{d}_{kR}\gamma_{\mu}d_{lR}\big{)}\,,\\\\[3.99994pt]
\end{split}$ (14)

where $Q$ and $L$ denote the quark and lepton $SU(2)_{L}$ doublet,
respectively, while $u,d,e$ stand for the quark and lepton weak singlets. In
what follows, we will work in the flavor basis defined with diagonal down-type
quark Yukawa matrix, i.e. with the CKM matrix element in the upper component
of $Q_{i}=[(V^{\dagger}\,u)_{i}\,,\,d_{i}]^{T}$. Since we are focusing on the
$b\to s$ processes we will fix $k=2$ and $l=3$, and discuss the two subsets of
operators (13,14) separately.

###  III.1 Quark bilinears and Higgs

Figure 3: Tree-level contributions of four-fermion (left panel) and Higgs-
fermion (right panel) operators to the $b\to s\nu\bar{\nu}$ transition in the
SMEFT.

Firstly, we consider the operators with quark-bilinears and the Higgs current,
as defined in Eq. (13). These contribute to $B\to K\nu\bar{\nu}$ via a tree-
level induced $Z$-coupling to the $(\bar{s}\gamma^{\mu}b)$ current, as
depicted in the right panel of Fig. 3. Clearly, these contributions would
impact not only the $b\to s\nu_{\ell}\bar{\nu}_{\ell}$ transition, but also
$b\to s\ell\ell$, with lepton-flavor universal contributions Buras:2014fpa ,
e.g. the precisely determined $B_{s}\to\mu\mu$ branching ratio. Moreover, a
double insertion of these coefficients will also have an effect in
$B_{s}-\bar{B}_{s}$ mixing. Writing the $Z$-boson interaction Lagrangian with
down-type quarks as

$\displaystyle\mathcal{L}_{\mathrm{eff}}^{Z}$
$\displaystyle=-\dfrac{g}{\cos\theta_{W}}\sum_{i,j}\bar{d}_{i}\gamma^{\mu}\left(g_{d_{L}}^{Z\,ij}\,P_{L}+g_{d_{R}}^{Z\,ij}\,P_{R}\right)d_{j}Z_{\mu}\,,$
(15)

where we find

$\displaystyle\delta g_{d_{L}}^{Z\,ij}$
$\displaystyle=-\dfrac{v^{2}}{2\Lambda^{2}}\left\\{\big{[}C_{Hq}^{(1)}\big{]}_{ij}+\big{[}C_{Hq}^{(3)}\big{]}_{ij}\right\\}\,,$
(16) $\displaystyle\delta g_{d_{R}}^{Z\,ij}$
$\displaystyle=-\dfrac{v^{2}}{2\Lambda^{2}}\big{[}C_{Hd}\big{]}_{ij}\,,$ (17)

with $g_{\psi}^{Z\,ij}=\delta_{ij}\,g_{\psi}^{Z}+\delta g_{\psi}^{Z\,ij}$,
$g_{d_{L}}^{Z}=-1/2+1/3\sin^{2}\theta_{W}$ and
$g_{d_{R}}^{Z}=+1/3\sin^{2}\theta_{W}$, and $\theta_{W}$ the Weinberg angle.
These relations can be used to match onto the relevant low-energy effective
four-fermion operators, after integration of the SM vector bosons
Jenkins:2017jig . Using the constraints from $\mathcal{B}(B_{s}\to\mu\mu)$
CMS:2022mgd  and $\Delta m_{B_{s}}$ Jenkins:2017dyc ; Aebischer:2020dsw , we
determine the 2$\sigma$ confidence intervals for the coefficients
$[\mathcal{C}_{Hq}^{(1)}]_{23}$, $[\mathcal{C}_{Hq}^{(3)}]_{23}$, and
$[\mathcal{C}_{Hd}]_{23}$, switching on one at a time at $\Lambda=1$ TeV and
accounting from the renormalization group evolution from $\Lambda$ down to
$\mu=m_{b}$ Jenkins:2013zja . The allowed intervals for the coefficients can
then be translated onto intervals for the $B\to K\nu\bar{\nu}$ and $B\to
K^{*}\nu\bar{\nu}$ branching fractions. We find that the
$\mathcal{B}(B^{\pm}\to K^{\pm}\nu\bar{\nu})$ value can only be enhanced by
$\approx 20\%$, which is largely insufficient to accommodate the deviation
shown in Belle-II data.

###  III.2 Four-fermion operators

We turn now our attention to the four-fermion operators defined in Eq. (14).
Let us first explicitly rewrite $\mathcal{L}^{(6)}_{\mathrm{SMEFT}}$ using the
operators given in Eq. (14). We have:

$\displaystyle\mathcal{L}^{(6)}_{\mathrm{SMEFT}}\supset\frac{1}{\Lambda^{2}}\biggl{\\{}\left(\mathcal{C}_{lq}^{(1)}+\mathcal{C}_{lq}^{(3)}\right)_{ij}\,(\overline{s}_{L}\gamma^{\mu}b_{L})(\overline{e}_{Li}\gamma_{\mu}e_{Lj})\biggr{.}$
$\displaystyle\qquad+\left(\mathcal{C}_{lq}^{(1)}-\mathcal{C}_{lq}^{(3)}\right)_{ij}\,(\overline{s}_{L}\gamma^{\mu}b_{L})(\overline{\nu}_{Li}\gamma_{\mu}\nu_{Lj})$
$\displaystyle\qquad+2\,V_{cs}\,\left[\mathcal{C}_{lq}^{(3)}\right]_{ij}\,(\overline{c}_{L}\gamma^{\mu}b_{L})(\overline{e}_{Li}\gamma_{\mu}\nu_{Lj})$
$\displaystyle\biggl{.}+\left[\mathcal{C}_{ld}\right]_{ij}\,(\overline{s}_{R}\gamma^{\mu}b_{R})\left[(\overline{\nu}_{Li}\gamma_{\mu}\nu_{Lj})+(\overline{e}_{Li}\gamma_{\mu}e_{Lj})\right]+\mathrm{h.c.}\biggr{\\}}.$
(18)

By comparing the above Lagrangian with the one given in Eq. (4) it is easy to
identify:

$\displaystyle\begin{split}\delta
C_{L}^{\nu_{i}\nu_{j}}&=\dfrac{\pi}{\alpha_{\mathrm{em}}\lambda_{t}}\dfrac{v^{2}}{\Lambda^{2}}\left\\{\big{[}\mathcal{C}_{lq}^{(1)}\big{]}_{ij}-\big{[}\mathcal{C}_{lq}^{(3)}\big{]}_{ij}\right\\}\,,\\\\[3.99994pt]
\delta
C_{R}^{\nu_{i}\nu_{j}}&=\dfrac{\pi}{\alpha_{\mathrm{em}}\lambda_{t}}\dfrac{v^{2}}{\Lambda^{2}}\big{[}\mathcal{C}_{ld}\big{]}_{ij}\,.\end{split}$
(19)

As it was discussed in Ref. Becirevic:2023aov , several simple scenarios can
be distinguished if we switch one of the above SMEFT operators at the time.
For example, by allowing only $\mathcal{C}_{lq}^{(1)}\neq 0$ we get the
simplest scenario with a $Z^{\prime}$ boson coupled only to the left-handed
fermions. If, instead, the $Z^{\prime}$ is considered to be member of the weak
triplet then one has only $\mathcal{C}_{lq}^{(3)}\neq 0$. Quite peculiar is
the case with $\mathcal{C}_{lq}^{(3)}=\mathcal{C}_{lq}^{(1)}\neq 0$, which
corresponds to the BSM scenario often encountered in the literature and
involves a singlet vector leptoquark state of hypercharge $Y=2/3$. Similarly,
a choice $\mathcal{C}_{lq}^{(1)}=3\mathcal{C}_{lq}^{(3)}$ corresponds to the
BSM model with a triplet of scalar leptoquarks with hypercharge $Y=1/3$
Buchmuller:1986zs .

##  IV Phenomenology

So far we did not touch on the issue of lepton flavor. In this Section we
discuss simple BSM scenarios and check whether or not one can build a simple
scenario compatible with experimental data, and most importantly with
$\mathcal{B}(B\to K{\nu\nu})^{\mathrm{exp}}$. In doing so we will separately
treat the lepton flavor conserving case in which the couplings are diagonal
($i=j$), from the lepton flavor violating, i.e. with non-diagonal couplings
($i\neq j$) being non-zero.

###  IV.1 New Physics coupling to muons only

Figure 4: Turning on the couplings to $\mu$’s only, i.e.
$\left[\mathcal{C}_{lq}^{(3)}\right]_{22}\equiv\mathcal{C}_{lq}^{(3)}\neq 0$
or $\left[\mathcal{C}_{ld}\right]_{22}\equiv\mathcal{C}_{ld}\neq 0$, leads to
the same result (purple curve in the plot) which cannot be reconciled with
$\mathcal{B}(B\to K{\nu\nu})^{\mathrm{exp}}$. Instead, by only allowing
$\mathcal{C}_{lq}^{(1)}\neq 0$ (orange curve) one can get its small fraction
to be consistent with both $\mathcal{B}(B\to K{\nu\nu})^{\mathrm{exp}}$ and
$\mathcal{B}(B_{s}\to\mu\mu)^{\mathrm{exp}}$ to $2\sigma$, which is also
highlighted in orange in the plot. The red point corresponds to the SM values
(i.e. $\mathcal{C}_{lq}^{(1)}=\mathcal{C}_{lq}^{(3)}=\mathcal{C}_{ld}=0$).

We first attempt attributing the discrepancy between the measured
$\mathcal{B}(B\to K{\nu\nu})$ and $\mathcal{B}(B\to K{\nu\nu})^{\mathrm{SM}}$
to the couplings to muon only, i.e. $i=j=2$. The best suited quantity to check
the validity of such a solution is $\mathcal{B}(B_{s}\to{\mu\mu})$ for which
we need to include the modification of the Wilson coefficient
$C_{10}=C_{10}^{\mathrm{SM}}+\delta C_{10}$, that in terms of
$\mathcal{C}_{lq}^{(1,3)}$ and $\mathcal{C}_{ld}$ writes:

$\displaystyle\delta C_{10}^{\ell_{i}\ell_{i}}$
$\displaystyle=\dfrac{\pi}{\alpha_{\mathrm{em}}\lambda_{t}}\dfrac{v^{2}}{\Lambda^{2}}\left\\{\big{[}\mathcal{C}_{ld}\big{]}_{ii}-\big{[}\mathcal{C}_{lq}^{(1)}\big{]}_{ii}-\big{[}\mathcal{C}_{lq}^{(3)}\big{]}_{ii}\right\\}\,,$
(20)

In Fig. 4 we show how $\mathcal{B}(B_{s}\to{\mu\mu})$ varies when switching
$\mathcal{C}_{lq}^{(1,3)}$ or $\mathcal{C}_{ld}$ one at the time. Clearly, any
$\mathcal{C}_{lq}^{(3)}$ or $\mathcal{C}_{ld}$ cannot alone enhance
$\mathcal{B}(B\to K{\nu\nu})^{\mathrm{SM}}$ in order to be consistent with
experiment. The situation improves when $\mathcal{C}_{lq}^{(1)}\neq 0$ in
which case one can reach the $2\sigma$ edge of $\mathcal{B}(B\to
K{\nu\nu})^{\mathrm{exp}}$, while remaining consistent with
$\mathcal{B}(B_{s}\to{\mu\mu})^{\mathrm{exp}}$ to $2\sigma$ as well. If we fix
$\Lambda=10$ TeV we get that the resulting allowed
$\mathcal{C}_{lq}^{(1)}\in[1.29,1.31]$, which translates to very stringent
bound: $2.16\leq\mathcal{B}(B\to{K^{\ast}}{\nu\nu})\times 10^{5}\leq 2.19$.
This scenario, however, seems quite unlikely even though it would be
consistent with the recent LHCb results regarding the lepton flavor
universality $R_{K^{(\ast)}}$. In fact, consistency with $R_{K^{(\ast)}}$
would require $\mathcal{C}_{lq}^{(1)}\in[1.29,1.34]$.

###  IV.2 New Physics coupling to two or three lepton species

With respect to the previous case, the situation considerably improves if we
turn on the couplings to both electrons and muons, in such a way that
$\left[\mathcal{C}_{lq}^{(1)}\right]_{11}=\left[\mathcal{C}_{lq}^{(1)}\right]_{22}\equiv\mathcal{C}$.
Quite obviously, in that case $R_{K^{(\ast)}}$ remains at its SM value and
thus consistent with recent experimental analyses at LHCb. Compatibility with
$\mathcal{B}(B\to K{\nu\nu})^{\mathrm{exp}}$ is so improved that even a small
$1\sigma$ overlap with $\mathcal{B}(B\to K{\nu\nu})^{\mathrm{exp}}$ can be
reached.

Figure 5: Similar to Fig. 4, except that in this case we turn on all of the
lepton species with
$\left[\mathcal{C}\right]_{11}=\left[\mathcal{C}\right]_{22}=\left[\mathcal{C}\right]_{33}$
where $\mathcal{C}$ stands for either $\mathcal{C}_{lq}^{(1)}$,
$\mathcal{C}_{lq}^{(3)}$ or $\mathcal{C}_{ld}$.

To go deep into the $1\sigma$ region of both $\mathcal{B}(B\to
K{\nu\nu})^{\mathrm{exp}}$ and $\mathcal{B}(B_{s}\to\mu\mu)$ one can take the
flavor universal situation, and assume
$\left[\mathcal{C}_{lq}^{(1)}\right]_{11}=\left[\mathcal{C}_{lq}^{(1)}\right]_{22}=\left[\mathcal{C}_{lq}^{(1)}\right]_{33}\equiv\mathcal{C}$,
which is what we plot in Fig. 5. Like before, $R_{K^{(\ast)}}$ remains
unchanged with respect to its SM value, and overall consistency with the $b\to
s$ data is achieved. However, as a result of the severe constraint arising
from $\mathcal{B}(B_{s}\to{\mu\mu})^{\mathrm{exp}}$, in both cases we obtain
$\mathcal{C}\in[1.2,1.3]$, which then results in either
$R^{K^{\ast}}_{\nu\nu}\in(3.6,3.9)_{e,\mu}$ or
$R^{K^{\ast}}_{\nu\nu}\in(4.8,5.3)_{e,\mu,\tau}$. Since both these intervals
are larger than $R^{K^{\ast}\,\mathrm{(exp)}}_{\nu\nu}<2.7$, all three
scenarios discussed so far cannot meet the experimental constraints.

Another important shortcoming of the two scenarios discussed here, is that the
contribution to ratios $R_{D}^{(*)}=\mathcal{B}(B\to
D^{(\ast)}\tau\bar{\nu})/\mathcal{B}(B\to D^{(\ast)}l\bar{\nu})$, where
$l\in(e,\mu)$, is absent, since this needs to go through the triplet operator
$\mathcal{C}_{lq}^{(3)}$ (cf. Eq. (III.2)). We remind the reader that the most
recent averages of experimental values for $R_{D^{(\ast)}}$
HeavyFlavorAveragingGroup:2022wzx ,

$R_{D}=0.257(29),\quad R_{D^{\ast}}=0.284(12)\,$ (21)

are larger than predicted in the SM FlavourLatticeAveragingGroupFLAG:2021npn ;
MILC:2015uhg ; FermilabLattice:2021cdg , leading to
$R_{D}/R_{D}^{\mathrm{SM}}=1.19(10)$ and
$R_{D^{\ast}}/R_{D^{\ast}}^{\mathrm{SM}}=1.15(5)$, which can be averaged for
our purpose to:

$R_{D^{(\ast)}}^{\mathrm{exp}}/R_{D^{\ast}}^{\mathrm{SM}}=1.16(5)\,.$ (22)

From Figure 4 one can clearly see that the constraints from $B_{s}\to\mu\mu$
cannot be respected in the case
$\left[\mathcal{C}_{lq}^{(3)}\right]_{11}=\left[\mathcal{C}_{lq}^{(3)}\right]_{22}=\left[\mathcal{C}_{lq}^{(3)}\right]_{33}$.
Moreover, this lepton-flavor universal scenario predicts
$R_{D^{(\ast)}}=R_{D^{(\ast)}}^{\mathrm{SM}}$ which disagrees with Eq. (22).
Also the case
$\left[\mathcal{C}_{lq}^{(3)}\right]_{11}=\left[\mathcal{C}_{lq}^{(3)}\right]_{22}$is
in disagreement with data, since it predicts
$R_{D^{(\ast)}}<R_{D^{(\ast)}}^{\mathrm{SM}}$ in correlation with an enhanced
$\mathcal{B}(B\to K{\nu\nu})$.

One concludes that modification of the couplings to $\tau$’s is what is
desired in order to be consistent with experimental data. More specifically
one needs $\left[\mathcal{C}_{lq}^{(3)}\right]_{33}\neq 0$.

###  IV.3 New Physics coupling to taus only

Finding
$\mathcal{C}_{lq}^{(3)}\equiv\left[\mathcal{C}_{lq}^{(3)}\right]_{33}\neq 0$
such that $\mathcal{B}(B\to K{\nu\nu})$ and $R_{D^{(\ast)}}$ are
simultaneously consistent with their experimental values is possible as we
avoid the constraint coming from
$\mathcal{B}(B_{s}\to{\mu\mu})^{\mathrm{exp}}$. We note that in this situation

$\frac{R_{D^{(\ast)}}}{R_{D^{(\ast)}}^{\mathrm{SM}}}=\left(1-\frac{v^{2}}{\Lambda^{2}}\,\frac{V_{cs}}{V_{cb}}\,\mathcal{C}_{lq}^{(3)}\right)^{2}\,,$
(23)

from which we see that $\mathcal{C}_{lq}^{(3)}\neq 0$ is significantly
enhanced by $V_{cs}/V_{cb}=24(1)$. The solution is shown in Fig. 6.

Figure 6: The curve shows $R_{D^{(\ast)}}/R_{D^{\ast}}^{\mathrm{SM}}$ as a
function of $\mathcal{B}(B\to K{\nu\nu})$ varying
$\bigl{[}\mathcal{C}_{lq}^{(3)}\bigr{]}_{33}$ (purple) or
$\bigl{[}\mathcal{C}_{ld}\bigr{]}_{33}$ (blue). The highlighted purple region
correspond to $\bigl{[}\mathcal{C}_{lq}^{(3)}\bigr{]}_{33}$ allowed by
$\mathcal{B}(B\to K{\nu\nu})^{\mathrm{exp}}$ which becomes restricted to the
hatched region once the condition $R^{K^{\ast}\,\mathrm{(exp)}}_{\nu\nu}<2.7$
is imposed. That range is smaller than
$R_{D^{(\ast)}}^{\mathrm{exp}}/R_{D^{\ast}}^{\mathrm{SM}}$ to $2\sigma$
(horizontal gray area). The red point corresponds to the SM.

Again, with $\Lambda=10$ TeV, we find that the acceptable
$\mathcal{C}^{(3)}_{lq}$ falls in:

$\mathcal{C}^{(3)}_{lq}\in[-3.9,-1.9].$ (24)

As can be seen from (23), and in view of Eq. (22), only negative
$\mathcal{C}^{(3)}_{lq}$ values will lead to
$R_{D^{(\ast)}}^{\mathrm{exp}}/R_{D^{\ast}}^{\mathrm{SM}}>1$, cf. Fig. 6, and
we obtain $R_{D^{(\ast)}}/R_{D^{\ast}}^{\mathrm{SM}}\in[1.06,1.12]$. However,
the $\mathcal{C}^{(3)}_{lq}$ range allowed by
$R^{K^{\ast}\,\mathrm{(exp)}}_{\nu\nu}<2.7$ and $\mathcal{B}(B\to
K{\nu\nu})^{\mathrm{exp}}$ is
$\mathcal{C}^{(3)}_{lq}\in[-1.5,-1.3]\cup[3.3,3.6]$, incompatible with Eq.
(24).

If that last constraint turns out to be experimentally invalidated, this model
would result in an increase of the decay rates to the pair of $\tau$-leptons
in the final state Capdevila:2017iqn . Using Eq. (24) we would obtain:

$\displaystyle\frac{\mathcal{B}(B_{s}\to\tau\tau)}{\mathcal{B}(B_{s}\to\tau\tau)^{\mathrm{SM}}}$
$\displaystyle\in[44,157],$ $\displaystyle\frac{\mathcal{B}(B\to
K\tau\tau)}{\mathcal{B}(B\to K\tau\tau)^{\mathrm{SM}}}$
$\displaystyle=\frac{\mathcal{B}(B\to K^{\ast}\tau\tau)}{\mathcal{B}(B\to
K^{\ast}\tau\tau)^{\mathrm{SM}}}\in[16,48]\,.$ (25)

###  IV.4 Lepton flavor violating case

We can now assume that the difference between $\mathcal{B}(B\to
K{\nu\nu})^{\mathrm{exp}}$ and its SM prediction can be described by turning
on the off-diagonal couplings to lepton flavors ($i\neq j$). This can be done
with a number of various assumptions. Here, as an example, we turn either
$\bigl{[}\mathcal{C}_{lq}^{(3)}\bigr{]}_{ij}$, or
$\bigl{[}\mathcal{C}_{lq}^{(1)}\bigr{]}_{ij}$, or
$\bigl{[}\mathcal{C}_{ld}\bigr{]}_{ij}$ at the time and assume the couplings
to $\mu$ and $\tau$ to be non-zero by taking
$\mathcal{C}_{32}=\mathcal{C}_{23}$. As a result we get that the $2\sigma$
compatibility with $\mathcal{B}(B\to K{\nu\nu})^{\mathrm{exp}}$ is achieved
for $\mathcal{C}_{32}=\mathcal{C}_{23}\in[-3.4,-1.5]\cup[1.5,3.4]$, which,
after imposing $R^{K^{\ast}\,\mathrm{(exp)}}_{\nu\nu}<2.7$, reduces to
$\mathcal{C}_{32}=\mathcal{C}_{23}\in[-1.7,-1.5]\cup[1.5,1.7]$, for
$\Lambda=10$ TeV, as before. As a consequence the decay rates of the
corresponding lepton flavor violating modes will be significant. By using the
expressions derived in Ref. Becirevic:2016zri  we obtain:

$\displaystyle\mathcal{B}(B\to K\mu\tau)\in[2,3]\times 10^{-6},$ (26)

not far from but consistent with the experimental bound LHCb:2022wrs ;
ParticleDataGroup:2022pth .

###  IV.5 Summary

In this letter we discuss the possible consequences on $\mathcal{B}(B\to
K^{\ast}{\nu\nu})$ due to the recent Belle II measurement of $\mathcal{B}(B\to
K{\nu\nu})$ which was found to be nearly $3\sigma$ larger than predicted. We
find that the values $\mathcal{B}(B\to K^{\ast}{\nu\nu})\lesssim
1.8\times\mathcal{B}(B\to K^{\ast}{\nu\nu})^{\mathrm{SM}}$ could only occur if
the coupling to the right-handed operator $\bar{b}_{R}\gamma_{\mu}s_{R}$ is
non-zero, while larger branching fractions would be consistent with couplings
to either left-handed or right-handed operators (or a combination of both).

By relying on SMEFT we show that the increase in $\mathcal{B}(B\to K{\nu\nu})$
cannot be described by switching on the couplings to one or more lepton
flavors if that choice comprises the coupling to muons. The reason is that
$\mathcal{B}(B_{s}\to\mu\mu)^{\mathrm{exp}}$ provides a very stringent
constraint on the values of the Wilson coefficient $\mathcal{C}_{lq}^{(1)}$,
which is the only one that can provide the desired enhancement to
$\mathcal{B}(B\to K^{(\ast)}{\nu\nu})$. In the SMEFT Lagrangian the Wilson
coefficients relevant to $b\to s\ell\ell$ and $b\to s\nu\nu$ decay modes are
also related to the semileptonic $b\to c\ell\nu$, so that one should make sure
that the resulting $R_{D^{(\ast)}}>R_{D^{\ast}}^{\mathrm{SM}}$, as suggested
by experiment. For that it is necessary that $\mathcal{C}_{lq}^{(3)}\neq 0$ or
to have the scalar and tensor operators that are not related to $b\to
s\ell\ell$ and $b\to s\nu\nu$ Angelescu:2018tyl . We find that a significant
increase in $R_{D^{(\ast)}}/R_{D^{\ast}}^{\mathrm{SM}}$ can be achieved if we
allow only the coupling to $\tau$ and not to other species. In this way one
can find the agreement with all of the above-mentioned experimental
constraints, including $R_{K^{(\ast)}}$, except that
$R_{D^{(\ast)}}/R_{D^{\ast}}^{\mathrm{SM}}\approx 1.05$.

At this stage we do not evoke a concrete model Angelescu:2018tyl ;
Greljo:2015mma ; Buchmuller:1986zs . It is worth mentioning that a BSM model
with a weak triplet of scalar leptoquarks $S_{3}$ cannot be made consistent
with all the data discussed here. The reason is well known, namely it cannot
give a significant increase to $R_{D^{(\ast)}}$. Neither a $U_{1}$ vector
leptoquark, in a simplified model approach (involving couplings to the left-
handed operators only), can be made consistent with data. This is so because
at tree-level $\mathcal{C}_{lq}^{(1)}=\mathcal{C}_{lq}^{(3)}$, or $\delta
C_{L}=0$, which is excluded by the new Belle II result for $\mathcal{B}(B\to
K{\nu\nu})^{\mathrm{exp}}$. It is important to stress, however, that the
$U_{1}$ being a massive vector necessarily calls for a UV completion, which
typically requires also the inclusion of additional fermionic degrees of
freedom. The full theory of the $U_{1}$ has been extensively studied, and in
particular it has been shown that the relation
$\mathcal{C}_{lq}^{(1)}=\mathcal{C}_{lq}^{(3)}$ is broken at one-loop, leading
to an increase of $\mathcal{B}(B\to K{\nu\nu})$ of up to $\approx$ 50% with
respect to the Standard Model, given the current values of $R_{D^{(*)}}$
Fuentes-Martin:2019ign ; Fuentes-Martin:2020luw . Another possibility is to
take the couplings to $\tau$’s verifying
$\mathcal{C}_{lq}^{(1)}=-\mathcal{C}_{lq}^{(3)}$, which is valid in a scenario
with the so called $S_{1}$ leptoquark. In that case one gets
$R_{D^{(\ast)}}/R_{D^{\ast}}^{\mathrm{SM}}\approx 1.02$, thus much smaller
than in the case discussed in Sec. IV.3. The way out would then be to turn on
the right handed couplings Angelescu:2018tyl . Similarly, in the case of the
so called $\widetilde{R}_{2}$ leptoquarks one needs to turn on the right-
handed couplings to quarks (and left-handed to $\tau$), which in the SMEFT
language means that $\mathcal{C}_{ld}\neq 0$ Becirevic:2016yqi . It thus can
explain the new experimental result $\mathcal{B}(B\to
K{\nu\nu})^{\mathrm{exp}}$. Moreover, by combining the constraints
$\mathcal{B}(B\to K{\nu\nu})^{\mathrm{exp}}$ and $[\Delta
m_{s}]^{\mathrm{exp}}/[\Delta m_{s}]^{\mathrm{SM}}$ we get the bound
$m_{\widetilde{R}_{2}}\lesssim 3$ TeV and $m_{S_{1}}\lesssim 4.5$ TeV. This
scenario, however, fails to explain
$R_{D^{(\ast)}}^{\mathrm{exp}}/R_{D^{\ast}}^{\mathrm{SM}}$. Another similar
model is the one with a $Z^{\prime}$ boson coupling to left-handed $\tau$ and
$\bar{s}\gamma_{\mu}P_{R}b$, again giving rise to $\mathcal{C}_{ld}\neq 0$
Becirevic:2015asa . While it can accommodate all of the $b\to s$ constraints,
it does not contribute to $b\to c\tau\nu$.

We also considered a possibility with the off-diagonal couplings to lepton
species. Accommodating $\mathcal{B}(B\to K{\nu\nu})^{\mathrm{exp}}$ in this
setting implies a large $\mathcal{B}(B\to K\mu\tau)$, not too far from the
current experimental bound.

We need to emphasize that all the couplings discussed in this paper are well
within the ranges allowed by the experimental data collected in the region of
high-$p_{T}$ tails of the differential distribution of $pp\to\ell\ell$ (+ soft
jets), cf. Ref. Allwicher:2022gkm .

One final comment regards the experimental value used in our analysis, which
is the new measurement of $\mathcal{B}(B\to K\nu\nu)$ only. In Glazov , also
an average with the previous bound from Belle-II was provided Belle:2017oht ,
giving $\mathcal{B}(B\to K\nu\nu)^{\exp}=1.4(4)\times 10^{-5}$. How
considering this average affects our conclusions can be mostly read off the
same plots we presented, the main effect being a higher compatibility with the
bound on $R_{\nu}^{K^{*}}$ in the left-handed scenario, and an increased
(although still below the SM) range for $F_{L}$. Similarly, the $\tau$-only
coupling scenario would call for a larger right-handed contribution in order
to enhance the contributions to $R_{D}^{(*)}$.

## Acknowledgments

This project has received funding /support from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 860881-HIDDeN and No 101086085-ASYMMETRY. The work of LA is
supported by the Swiss National Science Foundation (SNF) under contract
200020-204428. LA is also grateful to the Mainz Institute for Theoretical
Physics (MITP) of the Cluster of Excellence PRISMA+ (Project ID 39083149), for
its hospitality and its partial support during the completion of this work.

## Note added

While this paper was in writing the results of studies similar to our’s were
released in Refs. Bause:2023mfe ; Athron:2023hmz . In particular the authors
of Ref. Bause:2023mfe  interpret the new Belle II results in terms of the
SMEFT operators reaching the conclusions which agree with ours for the most
part.

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