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% Journal sections (see http://pkp.jinr.ru/index.php/PEPAN_LETTERS/about/editorialPolicies#focusAndScope)
%\issuearea{Physics of Elementary Particles and Atomic Nuclei. Theory}

\title{ $B \to K^{(*)} \nu \bar\nu$ decays: data and theoretical predictions\\ }

%\title{Распады $B \to K^{(*)} \nu \bar\nu$: экспериментальные данные и теоретические предсказания}

\maketitle
\authors{Aidos Issadykov\vspace{1mm} $^{a,b}$ \vspace{2mm}\footnote{E-mail: issadykov.a@gmail.com}}
%\authors{Айдос Исадыков$^{a,b}$\footnote{E-mail: issadykov.a@gmail.com (русский вариант)},}

\from{$^{a}$\,The Institute of Nuclear Physics, \\
Ministry of Energy of the Republic of Kazakhstan, 050032 Almaty,  {\it KAZAKHSTAN}} 
%\from{$^{a}$\,Институт ядерной физики Министерства Энергетики Республики Казахстан, 050032 Алматы, Казахстан

\from{$^{b}$\,Joint Institute for Nuclear Research, 141980 Dubna, \textit{RUSSIA}}
%\from{$^{b}$\,Объединенный институт ядерных исследований, 141980 Дубна, Россия 


\begin{abstract}
In this paper we compare data from Belle II and BaBar with available results of theoretical approaches for $B \to K^{(*)} \nu \bar\nu$ decays.
The theoretical predictions for $B^+ \to K^{+} \nu \bar{\nu}$ are lower than experimental data. This may indicate on the possibility of New physics effects in this decay.
\end{abstract}
\vspace*{6pt}

%\begin{abstract}
%В этой статье мы сравниваем данные от Belle II и BaBar с имеющимися результатами теоретических подходов для распадов $B \to K^{(*)} \nu \bar\nu$. Теоретические предсказания для $B^+ \to K^{+} \nu \bar{\nu}$ на порядок меньше экспериментальных данных. Это может указывать на возможность эффектов Новой физики в этом распаде.
%\end{abstract}

\noindent
12.40.$-$y; 14.40.$-$n

\label{sec:intro}
\section*{INTRODUCTION}
Decays involving flavor-changing neutral currents are of great interest for studying and uncovering new physical phenomena. In the decay process $B \to K^{(*)} \mu^+ \mu^{-}$, discrepancies have been observed between theoretical predictions and experimental data. Over the past several years, research studies have been carried out to detect the decay of B mesons to K mesons accompanied by a pair of neutrinos~\cite{Belle:2007vmd,BaBar:2008wiw,BaBar:2010oqg,Belle:2013tnz,BaBar:2013npw,Belle:2017oht,Belle-II:2021rof}.
The rare decays of B mesons to final states containing a pair of neutrinos are theoretically considered one of the cleanest processes involving neutral currents that facilitate flavor changes in quarks. The electrical neutrality of neutrinos enables an accurate separation of hadronic and lepton currents, unlike other B meson decays. Various theoretical frameworks have provided predictions for the branching fraction of this decay~\cite{Altmannshofer:2009ma,Buras:2021nns,Buras:2022wpw,Parrott:2022dnu,Kamenik:2009kc,Colangelo:1996ay,Melikhov:1997zu,Faessler:2002ut,Wang:2012ab,Bause:2021cna}.
The theoretical predictions for the $B^+ \to K^{+} \nu \bar{\nu}$ decay process obtained within the CCQM(Covariant Confined Quark Model) framework~\cite{Issadykov:2022iwp} are consistent with other theoretical methods, and were substantially lower than the experimental upper bounds known at the time. In late 2023, the Belle II collaboration reported the measured branching fraction for $B^+ \to K^{+} \nu \bar{\nu}$~\cite{Belle-II:2023esi}, providing data for the inclusive, hadronic, and combined decay modes.
\bea
\Br_{inc}(B^+\to K^+\nu\bar\nu) &=& \langle 2.7 \pm 0.5(stat) \pm 0.5(syst)\rangle 10^5
\nn
\Br_{hadr}(B^+\to K^+\nu\bar\nu) &=& \langle1.1^{+0.9}_{-0.8}(stat) ^{+0.5}_{-0.4}(syst)\rangle 10^5
\nn
\Br_{comb}(B^+\to K^+\nu\bar\nu) &=& \langle2.3 \pm 0.5(stat) ^{+0.5}_{-0.4}(syst)\rangle 10^5
\ena
In this paper, we reviewed theoretical predictions within the Standard Model and experimental data.


\label{sec:model}
\section*{Covariant confined quark model}
We will use the covariant confined quark model~\cite{Issadykov:2022iwp,Issadykov:2022imz,Soni:2021fky,Soni:2020bvu,Issadykov:2023pwl,Dubnicka:2020xoh,Issadykov:2018myx} in our calculations. This analysis is based on nonlocal interpolation currents that possess the relevant quantum numbers of hadrons. The effective Hamiltonian for $b\to s\nu\bar\nu$~transitions in the Standard Model can be expressed in the following form
%
\be
{\mathcal H}_{\rm eff}^{\rm SM} =
\frac {G_F}{\sqrt2}  V_{tb} \, V^*_{ts}
\Big[\frac {\alpha_{em}}{2\pi} \frac {X_t}{\sin^2\Theta_W}\Big]
(\bar s\, O_\mu\, b) (\bar\nu O^\mu \nu) + h.c.
\en
Here $G_F$  is the Fermi constant,  $V_{tb}$ and  $V^\ast_{ts}$
are the matrix elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix,
$\alpha_{em}$ is the fine structure constant calculated at the
electroweak scale of $Z$-boson mass,  $\theta_W$ is the Weinberg angle,
$O_\mu=\gamma_\mu(1-\gamma_5)$ is the weak Dirac matrix with the left
chirality. The Wilson coefficient $X_t$ is known with a high accuracy,
including NLO QCD corrections~\cite{Misiak:1999yg,Buchalla:1998ba} and
two-loop electroweak contributions~\cite{Brod:2010hi}, resulting in
%
\be
X_t=1.469 \pm 0.017.
\en
The matrix elements $\langle K^{(\ast)} | \bar s O^\mu b) | B \rangle$ can be expressed as scalar functions, known as form factors, which depend on the square of the transferred momentum
%
\bea
\langle K(p_2)\,|\,\bar s\, O^{\,\mu}\, b\,| B(p_1)\rangle
&=&
F_+(q^2)\, P^{\,\mu} + F_-(q^2)\, q^{\,\mu},
\label{eq:PP'}\\[1.5ex]
\langle K^\ast(p_2,\epsilon_2)\,|\,\bar s\, O^{\,\mu}\,b\, |\,B(p_1)\rangle
&=&
\frac{\epsilon^{\,\dagger}_{2\,\alpha}}{m_1+m_2}\,
\Big( - g^{\mu\alpha}\,Pq\,A_0(q^2) + P^{\,\mu}\,P^{\,\alpha}\,A_+(q^2)
\nn
&+& q^{\,\mu}\,P^{\,\alpha}\,A_-(q^2)
+ i\,\varepsilon^{\mu\alpha P q}\,V(q^2)\Big).
\label{eq:PV}
\ena
Here $P =p_1+p_2$, \,$q=p_1-p_2$, and $\epsilon_2$ is the polarization vector
of the $K^\ast$ meson, so $\epsilon_2^\dagger\cdot p_2 = 0$ .
The abbreviation
$\varepsilon^{\mu\alpha P q} = \varepsilon^{\mu\alpha \nu\delta} P_\nu q_\delta$
is adopted, where the absolutely antisymmetric Levi-Civita tensor is
defined as $\varepsilon^{0123} = -\varepsilon_{0123} = -1$.
The particles are on the mass shell, i.e.
$p_1^2=m_1^2=m_B^2$ and $p_2^2=m_2^2=m_{K^{(\ast)}}^2$.

The calculation of form factors necessitates approaches beyond the perturbation theory, such as quark models, QCD sum rules, and lattice QCD. Considering the aforementioned definitions of form factors, the differential branching fraction can be expressed as follows:
%
\begin{align}
\frac{ d \Br (B\to K^{(\ast)}+\nu\bar\nu)}{dq^2} &=
3\tau_{B} \frac{ (G_F \lambda_t \alpha_{em})^2 }{3(2\pi)^5}
\Big(\frac{X_t}{\sin^2\theta_W}\Big)^2
\nn
&\times \frac{|{\bf p_2}|}{4m_1^2}\,
\Big( {\tilde H}_+^2 + {\tilde H}_-^2 + {\tilde H}_0^2 \Big).
\label{eq:BR}
\end{align}
The factor 3 arises from the summation over the three neutrino flavors: $\nu_e$, $\nu_\mu$ and $\nu_\tau$. In the Standard Model,
they are assumed to be massless and therefore do not interfere with each other.
Additionally, $\tau_{B}$ is the B-meson lifetime,
$\lambda_t\equiv| V_{tb}V^\ast_{ts} |$,
$|{\bf p_2}|=\lambda^{1/2}(m_1^2,m_2^2,q^2)/2m_1 $ is the  daughter meson momentum
in the rest frame of the initial $B$~meson.
Scaled helicity amplitudes ${\tilde H}$
are related to the helicity amplitudes defined in \cite{Ivanov:2015tru}
as ${\tilde H} = \sqrt{q^2} H$. Consequently, we have found that\\[1.2ex]
%
\noindent
{\boldmath $B\to K$ \,\,\bf transition:}
\be
{\tilde H}_\pm = 0,
\qquad
{\tilde H}_0 = 2\,m_1\,|{\bf p_2}|\,F_+.
\label{eq:hel_PP}
\en
%
\noindent
{\boldmath$B\to K^\ast$ \,\,\bf transition:}
\bea
{\tilde H}_\pm &=&
\frac{\sqrt{q^2}}{m_1+m_2}
\Big(-Pq\, A_0 \pm 2\,m_1\,|{\bf p_2}|\, V \Big),
\nn[1.2ex]
{\tilde H}_0 &=&
\frac{1}{m_1+m_2}\frac{1}{2\,m_2}
\Big(-Pq\,(Pq - q^2)\, A_0 + 4\,m_1^2\,|{\bf p_2}|^2\, A_+\Big).
\label{eq:hel_PV}
\ena

The scaling is done in order to avoid uncertainties at the point $q^2 = 0$ in
numerical calculations. We note that in these decays the physical range of
the variable $q^2$ is $0\le q^2 \le q^2_{\rm max}=(m_1-m_2)^2$. The more information about model and parameters can be found in~\cite{Issadykov:2022iwp}.

\section*{Numerical results and conclusions}
\label{sec:results}
Table~\ref{tab:BR_tot} presents the results of the total branching fraction calculations for the $B \to K^{(\ast)}\nu\bar\nu$ process, as computed within the CCQM framework. For comparison, the table also includes the findings from experimental measurements and theoretical predictions derived using various approaches.

The theoretical predictions for the branching ratio of $B \to K\nu\bar\nu$ differs from the experimental measurement by approximately $2.4$ standard deviations.
%
%%%%%%%%%%%%%%%%%%
\begin{table}[ht]
\caption{Summary of experimental and theoretical results for the branching
  fractions $\mathcal{B}(B\to K^{(\ast)}\nu\bar{\nu})$. Experimental upper
  limits are given at 90\% confidence level.}
\label{tab:BR_tot}  
  \begin{center}
    \def\arraystretch{1.2}
\begin{tabular}{c|c|c}
  \hline
Ref &{$10^5\Br(B^+\to K^+\nu\bar\nu)$} &
{$10^5\Br(B\to K^{\ast}\nu\bar\nu)$} \\
\hline
BaBar~\cite{BaBar:2013npw} &$<1.6$  & $<6.4$  \\

Belle-II~\cite{Belle-II:2023esi,Belle:2017oht} &$2.3 \pm 0.5(stat) ^{+0.5}_{-0.4}(syst)$  & $<6.1$ \\
\hline
Our model~\cite{Issadykov:2022iwp} & 0.496(0.074) &  0.957(0.143) \\
\cite{Altmannshofer:2009ma} & 0.45(0.07) &  0.68$^{+0.10}_{-0.11}$  \\
 \cite{Buras:2021nns}&0.445(0.062) & 0.970(0.092) \\
  \cite{Buras:2022wpw}&0.465(0.062) & 1.013(0.092)  \\
 \cite{Kamenik:2009kc}& 0.51(0.08) &  0.84(0.14) \\
\cite{Colangelo:1996ay} &0.24(0.06)   & 0.51(0.08) \\
 \cite{Melikhov:1997zu}  &0.52(0.11) &  1.3(0.5)   \\
 \cite{Bause:2021cna} & 0.423(0.056) & 0.893(0.107) \\[1.2ex]
 \hline
\end{tabular}
\end{center}
\end{table}

\label{sec:funding}
\section*{Funding}

This research has been funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. BR24992891). 

\begin{thebibliography}{1}
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\cyrdash\hskip.2em\ignorespaces}

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%----------------- Xt ---------------------------------------------

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%------------- helicities --------------------

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