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\title{$\eta^{(\prime)}\to\pi^{0}\gamma\gamma$ decays: Test of meson exchange models and searches of leptophobic $B$ bosons}
\author{R.~Escribano\from{ins:uab}\from{ins:ifae},
S.~Gonz\`{a}lez-Sol\'{i}\from{ins:lanl}\from{ins:ub}\from{ins:icc}\thanks{Speaker.}
\atque
E.~Royo\from{ins:uab}\from{ins:ifae}}
\instlist{\inst{ins:uab} Grup de F\'{i}sica Te\`{o}rica, Departament de F\'{i}sica,
Universitat Aut\`{o}noma de Barcelona, 08193 Bellaterra (Barcelona), Spain
\inst{ins:ifae} Institut de F\'{i}sica d'Altes Energies (IFAE) and
The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain
\inst{ins:lanl} Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
\inst{ins:ub} Departament de F\'{i}sica Qu\`{a}ntica i Astrof\'{i}sica (FQA), Universitat de Barcelona (UB), c. Mart\'{i} i Franqu\'{e}s, 1, 08028 Barcelona, Spain
\inst{ins:icc} Institut de Ci\`{e}ncies del Cosmos (ICCUB), Universitat de Barcelona (UB), c. Mart\'{i} i Franqu\'{e}s, 1, 08028 Barcelona, Spain.
}
%% When only one author is present, please do not use the command \from{} near the author name.
\begin{document}
\maketitle
\begin{abstract}
We analyze the vector and scalar meson exchange contributions to the doubly radiative decays $\eta^{(\prime)}\to\pi^{0}\gamma\gamma$ and $\eta^{\prime}\to\eta\gamma\gamma$, and study the sensitivity of these decays to a leptophobic $B$ boson in the sub-GeV mass range.
%The SM predictions for the diphoton invariant mass spectra and the branching ratios are given and compared with experimental data, and the current constraints on the $B$-boson mass $m_B$ and coupling $\alpha_B$ are improved by adding an explicit $B$-boson resonance exchange to the dominant SM contribution.
Our results are relevant for studies of these decays at existing (A2, BESIII, KLOE-2) and forthcoming $\eta/\eta^{\prime}$-factories, such as the JEF and REDTOP experiments.
\end{abstract}
\section{Introduction}
The rare doubly radiative decays $\eta/\eta^{\prime}\to\pi^{0}\gamma\gamma$ and $\eta^{\prime}\to\eta\gamma\gamma$ have attracted a lot of attention recently\footnote{See, {\it{e.g.}} talks at ECT$^{*}$ workshop, ``{\it{Precision test of fundamental physics with light mesons}}", June 12-16, 2023~\cite{Trento23}.}, both from the experimental and theoretical sides, since the preliminary experimental measurement of the $\eta\to\pi^{0}\gamma\gamma$ decay by the KLOE-2 Collaboration\footnote{See talk by G.~Mandaglio at this workshop~\cite{KLOEhadron23}.}.
While our theoretical predictions of the branching ratios (${\rm{BR}}$) for the decays of the $\eta^{\prime}$ meson, ${\rm{BR}}(\eta^{\prime}\to\pi^{0}\gamma\gamma)=2.91(21)\times10^{-3}$ and ${\rm{BR}}(\eta^{\prime}\to\eta\gamma\gamma)=1.17(8)\times10^{-4}$~\cite{Escribano:2018cwg}, are compatible with the experimental results from the BESIII collaboration, ${\rm{BR}}(\eta^{\prime}\to\pi^{0}\gamma\gamma)=3.20(7)(23)\times10^{-3}$~\cite{BESIII:2016oet} and ${\rm{BR}}(\eta^{\prime}\to\eta\gamma\gamma)=8.25(3.41)(0.72)\times10^{-5}$~\cite{BESIII:2019ofm}, the situation for $\eta\to\pi^{0}\gamma\gamma$ is presently inconclusive.
For this decay, the new measurement from KLOE-2, ${\rm{BR}}(\eta\to\pi^{0}\gamma\gamma)=0.99(11)(24)\times10^{-4}$~\cite{KLOEhadron23}, is in agreement with our calculation, ${\rm{BR}}(\eta\to\pi^{0}\gamma\gamma)=1.35(8)\times10^{-4}$~\cite{Escribano:2018cwg}, but in tension with the PDG average, ${\rm{BR}}(\eta\to\pi^{0}\gamma\gamma)=2.55(22)\times10^{-4}$~\cite{Workman:2022ynf}; the main input being ${\rm{BR}}(\eta\to\pi^{0}\gamma\gamma)=2.54(27)\times10^{-4}$ from the A2 Collaboration at MAMI~\cite{A2atMAMI:2014zdf}.
On the other hand, these decays have been put forward as powerful probes to search for MeV-GeV signals of a new hypothetical gauge boson, named $B$ boson, that emerges from a new $U(1)_{B}$ gauge symmetry and couples predominantly to quarks over leptons~\cite{Tulin:2014tya}.
The aim of this contribution is to highlight the results for the three decays that we have obtained in Ref.~\cite{Escribano:2018cwg} based on meson exchange ideas and show the updated constraints on the $B$ boson parameters, mass $m_{B}$ and coupling to Standard Model particles $\alpha_{B}$, that we have recently placed in Ref.~\cite{Escribano:2022njt}.
The theoretical framework is detailed in Sec.~\ref{sec:ThFramework} and our results presented in Sec.~\ref{Sec:results}.
We close with an outlook in Sec.~\ref{Sec:Outlook}.
\section{Theoretical framework}\label{sec:ThFramework}
\subsection{Standard Model: Vector and scalar meson exchange contributions}\label{subsectionSM}
To calculate the vector meson exchange contributions we use VMD.
%The corresponding VMD amplitude represents not only the dominant contribution to the process $\eta\to\pi^{0}\gamma\gamma$, as shown long ago in~\cite{Ametller:1991dp}, but also to the decays $\eta^{\prime}\to\pi^{0}\gamma\gamma$ and $\eta^{\prime}\to\eta\gamma\gamma$~\cite{Escribano:2018cwg}.
In this framework, the $\eta\to\pi^{0}\gamma\gamma$ proceeds through the transition $\eta\to V\gamma$ followed by $V\to\pi^{0}\gamma$, resulting in a total of six diagrams contributing to the amplitude, which correspond to the exchange of the three neutral vector mesons $V=\rho^{0},\omega$ and $\phi$ in the $t$ and $u$ channels.
Combining the participating $V\eta\gamma$ and $V\pi^{0}\gamma$ interacting terms from the effective $VP\gamma$ Lagrangian from Ref.~\cite{Bramon:1997va} with the propagator of the corresponding vector meson, we find the vector meson contributions to $\eta\to\pi^{0}\gamma\gamma$~\cite{Escribano:2018cwg}:
\begin{eqnarray}
\label{AVMDetapi0}
\quad {\cal A}^{\mathrm{VMD}}_{\eta\to\pi^0\gamma\gamma}=
\sum_{V=\rho^0, \omega, \phi}g_{V\!\eta\gamma}g_{V\!\pi^0\gamma}\left[\frac{(P\cdot q_2-m_\eta^2)\{a\}-\{b\}}{D_V(t)}+
\bigg\{
\begin{array}{c}
q_2\leftrightarrow q_1\\
t\leftrightarrow u
\end{array}
\bigg\}\right]\ ,
\end{eqnarray}
where
$t,u=(P-q_{2,1})^2=m_\eta^2-2P\cdot q_{2,1}$ are the Mandelstam variables,
$\{a\}$ and $\{b\}$ are the Lorentz structures, which are defined as
\begin{equation}
\label{varGamma}
\begin{aligned}
\{a\}&=(\epsilon_1\cdot\epsilon_2)(q_1\cdot q_2)-(\epsilon_1\cdot q_2)(\epsilon_2\cdot q_1) \ , \\
\{b\}&=(\epsilon_1\cdot q_2)(\epsilon_2\cdot P)(P\cdot q_1)+(\epsilon_2\cdot q_1)(\epsilon_1\cdot P)(P\cdot q_2)\\
&-(\epsilon_1\cdot\epsilon_2)(P\cdot q_1)(P\cdot q_2)-(\epsilon_1\cdot P)(\epsilon_2\cdot P)(q_1\cdot q_2)\,,
\end{aligned}
\end{equation}
where $P$ is the four-momentum of the $\eta$ meson, and
$\epsilon_{1,2}$ and $q_{1,2}$ are, respectively, the polarisation and four-momentum
vectors of the photons.
The denominator $D_V(t)=m_V^2-t-i\,m_V\Gamma_V$ is the vector meson propagator; for the $\rho^0$ propagator, we use an energy-dependent decay width $\Gamma_{\rho^0}(t)=\Gamma_{\rho^0}\times[(t-4m_\pi^2)/(m_{\rho^0}^2-4m_\pi^2)]^{3/2}\times\theta(t-4m_\pi^2)$.
%\begin{equation}
%\label{varGamma}
%\Gamma_{\rho^0}(t)=\Gamma_{\rho^0}\times[(t-4m_\pi^2)/(m_{\rho^0}^2-4m_\pi^2)]^{3/2}\times\theta(t-4m_\pi^2) \ .
%\end{equation}
The amplitudes for the partner reactions $\eta^\prime\to\pi^0\gamma\gamma$ and
$\eta^\prime\to\eta\gamma\gamma$ have a similar structure to that of Eq.~(\ref{AVMDetapi0}), with the replacements $m_\eta^2\to m_{\eta^\prime}^2$, and $g_{V\eta\gamma}g_{V\pi^0\gamma}\to
g_{V\eta^\prime\gamma}g_{V\pi^0\gamma}$ for the $\eta^\prime\to\pi^0\gamma\gamma$ and
$g_{V\eta\gamma}g_{V\pi^0\gamma}\to g_{V\eta^\prime\gamma}g_{V\eta\gamma}$
for the $\eta^\prime\to\eta\gamma\gamma$.
For our analysis, we fix the $g_{VP\gamma}$ couplings in Eq.~(\ref{AVMDetapi0})
from the comparison of the calculated decay widths for the radiative transitions $V\to P\gamma$ and $P\to V\gamma$
with their empirical values from the PDG~\cite{Workman:2022ynf}.
We use the Linear Sigma Model to calculate the scalar meson exchange contributions to the amplitude.
These are small and are given in Ref.~\cite{Escribano:2018cwg}.
\subsection{Beyond the Standard Model: Leptophobic $B$-boson contribution}\label{subsectionBSM}
In analogy to the VMD contributions,
we next define the framework to include intermediate $B$-boson exchanges to the decay amplitude.
This contribution proceeds via the transition $\eta\to B\gamma\to\pi^{0}\gamma\gamma$ and can be assessed from the conventional VMD $VVP$ and $V\gamma$
Lagrangians~\cite{Bramon:1992kr}
supplemented by an effective Lagrangian that describes the $VB$ interaction.
The latter is formally
identical to the $V\gamma$ Lagrangian
with the substitutions $A^{\mu}\to B^{\mu}$, $e\to g_{B}$
and $Q\to{\rm{diag}}\{1/3,1/3,1/3\}$.
From the $VVP$ and $VB$ Lagrangians
along with the corresponding $V$-meson propagators,
it is straightforward to obtain expressions for the $g_{BP\gamma}$ couplings
in terms of the generic $B$-boson coupling $g_{B}$.
The $g_{BP\gamma}$ couplings are energy dependent
and read~\cite{Escribano:2022njt}:
\begin{equation}
\label{Eq:gBPgammaCouplings}
\begin{aligned}
g_{B\pi^{0}\gamma}(q^2)&=\frac{eg_{B}}{4\pi^{2}f_{\pi}}F_{\omega}(q^2)\ ,\quad
g_{B\eta\gamma}(q^2)=\frac{eg_{B}}{12\pi^{2}f_{\pi}}
\left[\cos\varphi_{P}F_{\omega}(q^2)+\sqrt{2}\sin\varphi_{P}F_{\phi}(q^2)\right]\ ,\\[1ex]
g_{B\eta^{\prime}\gamma}(q^2)&=\frac{eg_{B}}{12\pi^{2}f_{\pi}}
\left[\sin\varphi_{P} F_{\omega}(q^2)-\sqrt{2}\cos\varphi_{P}F_{\phi}(q^2)\right]\ ,
\end{aligned}
\end{equation}
where $\varphi_{P}$ is the $\eta$-$\eta^{\prime}$ mixing angle
in the quark-flavour basis~\cite{Bramon:1997va}.
%and the abbreviations $c\varphi_{P}\equiv\cos\varphi_{P}$ and $s\varphi_{P}\equiv\sin\varphi_{P}$ have been employed.
The functions $F_{V}(q^2)$ in the previous equations are form factors that account for the
$\omega$ and $\phi$ propagation, and are given by
$F_{V}(q^2)=m_{V}^{2}/(m_{V}^{2}-q^2-im_{V}\Gamma_{V})$.
Combining the couplings from Eq.~(\ref{Eq:gBPgammaCouplings}) with the propagator of the $B$ boson,
we get the $B$-boson exchange contribution to the amplitude of the
$\eta^{(\prime)}\to\pi^{0}\gamma\gamma$ decays:
%\begin{eqnarray}
\begin{equation}
\label{Eq:BbosonAmplitude}
{\cal A}^{B\,\mathrm{boson}}_{\eta^{(\prime)}\to\pi^0\gamma\gamma}=
g_{B\eta^{(\prime)}\gamma}(t)g_{B\pi^0\gamma}(t)
\left[\frac{(P\cdot q_2-m_{\eta^{(\prime)}}^2)\{a\}-\{b\}}{D_B(t)}+
\bigg\{
\begin{array}{c}
q_2\leftrightarrow q_1\\
t\leftrightarrow u
\end{array}
\bigg\}\right]\ ,
\end{equation}
%\end{eqnarray}
where $\mathcal{D}_{B}(q^2)=m_{B}^{2}-q^2-i\sqrt{q^2}\,\Gamma_{B}(q^2)$ is the $B$-boson propagator,
with $\Gamma_{B}(q^2)=\sum_{i}\Gamma_{B}^{i}(q^2)$ the energy-dependent width of the $B$ boson, with the sum running over the partial widths of the various decay channels the $B$ boson can decay into.
For our study, we include the partial widths of the channels
$B\to\pi^{0}\gamma$, $e^{+}e^{-}$, $\mu^{+}\mu^{-}$, $\pi^{+}\pi^{-}$, and $\pi^{0}\pi^{+}\pi^{-}$~\cite{Escribano:2022njt}.
%\begin{figure}[h]
%\centering\rotatebox{90}{\includegraphics[scale=0.25]{EtaToBbis.eps}}
%\caption{Schematic diagram of the $B$-boson exchange mechanism for the decay $\eta\to\pi^{0}\gamma\gamma$.}
%\label{Fig:BbosonExchange}
%\end{figure}
\section{Standard Model predictions and limits on the $B$ boson parameters}\label{Sec:results}
In Fig.~\ref{Fig:SMdistributions} we present our Standard Model predictions for the diphoton invariant mass distribution for $\eta\to\pi^0\gamma\gamma$ (left) and $\eta^{\prime}\to\pi^0\gamma\gamma$ (right) as compared to experimental data, while in Table~\ref{table:BRpredictions} we show the resulting branching ratios; the uncertainties in our predictions come from the errors of the $g_{VP\gamma}$ couplings.
%the shape of the measured spectra of $\eta\to\pi^0\gamma\gamma$ and $\eta^{\prime}\to\pi^0\gamma\gamma$ is well captured by our predictions.
For the decay $\eta\to\pi^0\gamma\gamma$, whereas our theoretical treatment shows a good agreement with the preliminary measurements of the spectrum by KLOE-2~\cite{KLOEhadron23}, it appears to present a normalization offset with respect to the data from the A2~\cite{A2atMAMI:2014zdf} and Crystal Ball~\cite{Prakhov:2008zz} collaborations, with a $\rm{BR}$ that is found to be approximately half of the averaged PDG one (see Table~\ref{table:BRpredictions}).
%As seen, the experimental situation for $\eta\to\pi^0\gamma\gamma$ is inconclusive.
%There is a growing interest in resolving this discrepancy.
%On the theory side, Ref.~\cite{Lu:2020qeo} suggests that contributions from the $a_{2}(1320)$ tensor resonance could be relevant at low diphoton invariant mass, while on the experimental side future analysis with improved statistics, {\it{e.g.}} from the A2 or JEF collaborations, would help clarify the experimental situation.
On the other hand, using the exact same treatment our predictions for the decays of the $\eta^{\prime}$ meson are compatible with the experimental results from BESIII~\cite{BESIII:2016oet,BESIII:2019ofm}.
%The exact same treatment for the spectrum $\eta^{\prime}\to\pi^0\gamma\gamma$ is in agreement with measured by BESIII~\cite{BESIII:2016oet,BESIII:2016oet}
%
%While our theoretical treatment shows a very good agreement with
%the $\eta^{\prime}\to\pi^0\gamma\gamma$ spectra measured by BESIII~\cite{BESIII:2016oet}, the exact same treatment for the spectrum of the decay $\eta\to\pi^0\gamma\gamma$ is
%appears to present a normalization offset with respect to the experimental measurements by the A2~\cite{A2atMAMI:2014zdf} and Crystal Ball~\cite{Prakhov:2008zz} collaborations, and the $\rm{BR}$ is found to be approximately half of the averaged PDG value (see Table~\ref{table:BRpredictions}).
%For this decay, however, our prediction for this channel is in good agreement with the preliminary results from KLOE-2~\cite{KLOEhadron23}.
%As seen, the experimental situation needs to be clarified.
%Finally, our $\rm{BR}$ prediction for $\eta^{\prime}\to\eta\gamma\gamma$ is consistent with the BESIII experimental value~\cite{BESIII:2019ofm}.
\begin{figure}[h]
\centering\includegraphics[scale=0.295]{DistributionSMerrorCouplingsBig.pdf}\includegraphics[scale=0.295]{EtaPIndividualSMpredictionBig.pdf}
\caption{Experimental diphoton energy spectra for $\eta\to\pi^0\gamma\gamma$ (left) and $\eta^{\prime}\to\pi^0\gamma\gamma$ (right) compared to our theoretical predictions from Ref.~\cite{Escribano:2018cwg} APS copyright.
The data is taken from Ref.~\cite{A2atMAMI:2014zdf}
(A2), Ref.~\cite{Prakhov:2008zz} (Crystal Ball), Ref.~\cite{KLOEhadron23} (KLOE-2, preliminary) and Ref.~\cite{BESIII:2016oet} (BESIII).}
\label{Fig:SMdistributions}
\end{figure}
{\centering
\begin{table}
\caption{\label{table:BRpredictions}Our predictions for the $\rm{BR}$ from Ref.~\cite{Escribano:2018cwg} compared to the experimental measurements.}
\begin{tabular}{|lll|}
\hline
Decay&
BR$_{\rm th}$~\cite{Escribano:2018cwg}&
BR$_{\rm exp}$\\
\hline
\multirow{2}{*}{$\eta\to\pi^0\gamma\gamma$} & \multirow{2}{*}{$1.35(8)\times 10^{-4}$} & $0.99(11)(24)\times10^{-4}$ (KLOE-2~\cite{KLOEhadron23})\\
& & $2.56(22)\times 10^{-4}$ (PDG~\cite{Workman:2022ynf})\\
\hline
$\eta^{\prime}\to\pi^0\gamma\gamma$& $2.91(21)\times 10^{-3}$ & $3.20(7)(23)\times10^{-3}$ (BESIII~\cite{BESIII:2016oet})\\
\hline
$\eta^{\prime}\to\eta\gamma\gamma$ & $1.17(8)\times10^{-4}$ & $8.25(3.41)(0.72)\times 10^{-5}$ (BESIII~\cite{BESIII:2019ofm})\\
\hline
\end{tabular}
\end{table}
}
Next, we calculate the constraints on the $B$-boson parameters, coupling $\alpha_{B}(\equiv g_{B}^{2}/4\pi)$ and mass $m_{B}$, set by experiment.
For that, we write the amplitude for these decays as the coherent sum
of the vector, scalar and $B$-boson exchange contributions,
$\mathcal{A}=\mathcal{A}_{\rm{VMD}}+\mathcal{A}_{\rm{L\sigma M}}+\mathcal{A}_{B\,\rm{ boson}}$.
%such that the partial decay widths depend on $\alpha_{B}$ and $m_{B}$.
We start with the $\eta\to\pi^{0}\gamma\gamma$ decay using the $\rm{BR}$ measurements by KLOE-2 (preliminary) and the PDG value.
In Fig.~\ref{Fig:ExclusionPlot}, we show the limits in the $\alpha_{B}$--$m_{B}$ plane,
which are found by requiring our predictions to not exceed the corresponding $\rm{BR}$ at $2\sigma$.
The grey area is excluded by KLOE-2, which yield a more stringent limit than the resulting one from the PDG (solid red line).
%This is as expected given that the BR from KLOE is found to be in good agreement with
%our SM prediction from Ref.~\cite{Escribano:2018cwg}, $\rm{BR}=(1.35\pm0.08)\times10^{-4}$,
%and, thus, the KLOE constraints on the $B$ boson turn out to be stronger.
The dashed lines in the figure are found setting the SM (or, equivalently, QCD) contributions to zero.
Clearly, these contributions are not negligible as the limits on $\alpha_{B}$ could be up to an order of magnitude weaker when their effects are turned off (labelled QCD off in the plots).
\begin{figure}[h]
\centering\includegraphics[scale=0.38]{ExclusionPlotEta.pdf}
\vspace{-0.35cm}
\caption{Limits on the leptophobic $B$-boson coupling $\alpha_{B}$ for different $m_{B}$ masses from the $\eta\to\pi^{0}\gamma\gamma$ BR measurements by KLOE-2~\cite{Cao:2022rxo} (black line) and the PDG~\cite{Workman:2022ynf} (red line). Figure taken from Ref.~\cite{Escribano:2022njt} APS copyright.}
\label{Fig:ExclusionPlot}
\end{figure}
\section{Outlook}\label{Sec:Outlook}
In this work, we have presented predictions for the decays $\eta^{(\prime)}\to\pi^{0}\gamma\gamma$ and $\eta^{\prime}\to\eta\gamma\gamma$ using the VMD and L$\sigma$M frameworks to account for the vector and scalar meson exchange contributions, respectively.
%Our results for the branching ratios are summarized in Table~\ref{table:BRpredictions}, while our predictions for the invariant mass distribution are shown in Fig.~\ref{Fig:SMdistributions}.
On the one hand, our predictions for the decays of the $\eta^{\prime}$ meson are compatible with the experimental measurements from BESIII.
On the other hand, our prediction for $\eta\to\pi^0\gamma\gamma$ is in agreement with the new (preliminary) measurement from KLOE-2, but in tension with the previous measurement from the A2 and CrystalBall collaborations.
There is a growing interest in resolving this discrepancy.
On the theory side, Ref.~\cite{Lu:2020qeo} suggests that contributions from the $a_{2}(1320)$ tensor resonance could be relevant at low diphoton invariant mass, while on the experimental side future analysis with improved statistics, {\it{e.g.}} from A2 or the Jefferson Lab Eta Factory (JEF) experiment, could help clarify the experimental situation.
We have also studied the sensitivity of these decays to a leptophobic $B$ boson.
Adding the explicit $B$-boson exchange contribution to the SM amplitude
has allowed us to place stringent limits on the $B$-boson parameters $\alpha_B$ and $m_B$
by comparing with current experimental data.
In particular, from the analysis of the decay $\eta\to\pi^{0}\gamma\gamma$,
we have strengthened by one order of magnitude the current constraints.
%, reaching $\alpha_B\sim 10^{-6}$.
Our results are relevant for studies of these decays at existing (A2, BESIII, KLOE-2) and forthcoming $\eta/\eta^{\prime}$-factories, such as the JEF and REDTOP experiments.
\acknowledgments
S.~G-S would like to thank the HADRON23 Organizing Committee for the opportunity of presenting this work.
S.~G-S is a Serra H\'{u}nter Fellow at the University of Barcelona.
The work of R.~E. and E.~R. has been supported by the Spanish Ministry of Science and Innovation (project no. PID2020-112965GB-I00), and by the European Union's Horizon 2020 Research and Innovation Programme (grant no. 824093 (H2020-INFRAIA-2018-1)). IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.
They also acknowledge the support from the Departament de Recerca i Universitats from Generalitat de Catalunya to the Grup de Recerca 00649 (Codi: 2021 SGR 00649).
\begin{thebibliography}{0}
\bibitem{Trento23}
ECT$^{*}$ workshop, Precision test of fundamental physics with light mesons, https://indico.ectstar.eu/event/168/
\bibitem{KLOEhadron23}
G.~Mandaglio (KLOE-2 Collaboration), Latest hadron physics results at KLOE-2, Talk at HADRON2023 workshop:\\ https://agenda.infn.it/event/33110/contributions/197413/attachments/106296/149678/hadron23$_$Mandaglio.pdf
%\cite{Escribano:2018cwg}
\bibitem{Escribano:2018cwg}
R.~Escribano, S.~Gonz\`alez-Sol\'\i{}s, R.~Jora and E.~Royo,
%``Theoretical analysis of the doubly radiative decays $\eta^{(\prime)}\to\pi^0\gamma\gamma$ and $\eta^\prime\to\eta\gamma\gamma$,''
Phys. Rev. D \textbf{102} (2020) no.3, 034026
%doi:10.1103/PhysRevD.102.034026
[arXiv:1812.08454 [hep-ph]].
%21 citations counted in INSPIRE as of 13 Oct 2023
%\cite{BESIII:2016oet}
\bibitem{BESIII:2016oet}
M.~Ablikim \textit{et al.} [BESIII],
%``Observation of the doubly radiative decay $\eta^{\prime}\to \gamma\gamma\pi^0$,''
Phys. Rev. D \textbf{96} (2017) no.1, 012005
%doi:10.1103/PhysRevD.96.012005
[arXiv:1612.05721 [hep-ex]].
%23 citations counted in INSPIRE as of 13 Oct 2023
%\cite{BESIII:2019ofm}
\bibitem{BESIII:2019ofm}
M.~Ablikim \textit{et al.} [BESIII],
%``Search for the decay $\eta'\to\gamma\gamma\eta$,''
Phys.\,Rev.\,D\,\textbf{100}\,(2019)\,no.5, 052015
%doi:10.1103/PhysRevD.100.052015
[arXiv:1906.10346 [hep-ex]].
%14 citations counted in INSPIRE as of 13 Oct 2023
%\cite{Workman:2022ynf}
\bibitem{Workman:2022ynf}
R.~L.~Workman \textit{et al.} [Particle Data Group],
%``Review of Particle Physics,''
PTEP \textbf{2022}, 083C01 (2022)
%doi:10.1093/ptep/ptac097
%3 citations counted in INSPIRE as of 12 Jul 2022
%\cite{A2atMAMI:2014zdf}
\bibitem{A2atMAMI:2014zdf}
B.~M.~K.~Nefkens \textit{et al.} [A2 at MAMI],
%``New measurement of the rare decay $\eta \to \pi^0\gamma\gamma$ with the Crystal Ball/TAPS detectors at the Mainz Microtron,''
Phys. Rev. C \textbf{90} (2014) no.2, 025206
%doi:10.1103/PhysRevC.90.025206
[arXiv:1405.4904 [hep-ex]].
%38 citations counted in INSPIRE as of 13 Oct 2023
%\cite{Tulin:2014tya}
\bibitem{Tulin:2014tya}
S.~Tulin,
%``New weakly-coupled forces hidden in low-energy QCD,''
Phys. Rev. D \textbf{89} (2014) no.11, 114008
%doi:10.1103/PhysRevD.89.114008
[arXiv:1404.4370 [hep-ph]].
%99 citations counted in INSPIRE as of 13 Oct 2023
%\cite{Escribano:2022njt}
\bibitem{Escribano:2022njt}
R.~Escribano, S.~Gonz\`alez-Sol\'\i{}s and E.~Royo,
%``Sensitivity of the \ensuremath{\eta}(')\textrightarrow{}\ensuremath{\pi}0\ensuremath{\gamma}\ensuremath{\gamma} and \ensuremath{\eta}'\textrightarrow{}\ensuremath{\eta}\ensuremath{\gamma}\ensuremath{\gamma} decays to a sub-GeV leptophobic U(1)B boson,''
Phys. Rev. D \textbf{106} (2022) no.11, 114007
%doi:10.1103/PhysRevD.106.114007
[arXiv:2207.14263 [hep-ph]].
%1 citations counted in INSPIRE as of 13 Oct 2023
%\cite{Ametller:1991dp}
%\bibitem{Ametller:1991dp}
%L.~Ametller, J.~Bijnens, A.~Bramon and F.~Cornet,
%``Chiral perturbation theory for eta ---\ensuremath{>} pi0 gamma gamma,''
%Phys. Lett. B \textbf{276} (1992), 185-190
%doi:10.1016/0370-2693(92)90561-H
%97 citations counted in INSPIRE as of 13 Oct 2023
%\cite{Bramon:1997va}
\bibitem{Bramon:1997va}
A.~Bramon, R.~Escribano and M.~D.~Scadron,
%``The eta - eta-prime mixing angle revisited,''
Eur. Phys. J. C \textbf{7} (1999), 271-278
%doi:10.1007/s100529801009
[arXiv:hep-ph/9711229 [hep-ph]].
%194 citations counted in INSPIRE as of 06 Dec 2021
%\cite{Bramon:1992kr}
\bibitem{Bramon:1992kr}
A.~Bramon, A.~Grau and G.~Pancheri,
%``Intermediate vector meson contributions to V0 ---\ensuremath{>} P0 P0 gamma decays,''
Phys. Lett. B \textbf{283} (1992), 416-420
%doi:10.1016/0370-2693(92)90041-2
%140 citations counted in INSPIRE as of 13 Oct 2023
%\cite{Prakhov:2008zz}
\bibitem{Prakhov:2008zz}
S.~Prakhov, \textit{et al.} [CrystalBall]
%``Measurement of the invariant-mass spectrum for the two photons from the eta --\ensuremath{>} pi0 gamma gamma decay,''
Phys. Rev. C \textbf{78} (2008), 015206.
%doi:10.1103/PhysRevC.78.015206
%49 citations counted in INSPIRE as of 13 Oct 2023
%\cite{Cao:2022rxo}
\bibitem{Cao:2022rxo}
B.~Cao [KLOE-2],
%``Update on hadron physics at KLOE/KLOE-2,''
PoS \textbf{EPS-HEP2021} (2022), 409.
%doi:10.22323/1.398.0409
%0 citations counted in INSPIRE as of 18 Apr 2022
%\cite{Lu:2020qeo}
\bibitem{Lu:2020qeo}
J.~Lu and B.~Moussallam,
%``The $\pi \eta $ interaction and $a_0$ resonances in photon\textendash{}photon scattering,''
Eur. Phys. J. C \textbf{80} (2020) no.5, 436
%doi:10.1140/epjc/s10052-020-7969-8
[arXiv:2002.04441 [hep-ph]].
%23 citations counted in INSPIRE as of 23 Nov 2023
\end{thebibliography}
\end{document}
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