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\title{The $D_s^+$ decay into $\pi ^+ K_S^0 K_S^0 $ reaction and the $I=1$ partner of the $f_0$(1710) state}
\author{L. R. Dai\from{ins:x}\thanks{dailianrong@zjhu.edu.cn}\ETC,
E. Oset \from{ins:y} \atque
L. S. Geng\from{ins:z}}
\instlist{\inst{ins:x} School of science, Huzhou University, Huzhou, 313000, Zhejiang, China
\inst{ins:y} Departamento de F\'{\i}sica Te\'orica and IFIC, Centro Mixto Universidad de Valencia-CSIC Institutos de Investigaci\'on de Paterna, Aptdo.22085, 46071 Valencia, Spain
\inst{ins:z} School of Physics, Beihang University, Beijing, 102206, China}
%% When only one author is present, please do not use the command \from{} near the author name.
\begin{document}
\maketitle
\begin{abstract} Two identified decay modes of the $D_s^+ \to \pi^+ K^{*+} K^{*-} , \pi^+ K^{*0} \bar{K}^{*0}$ reactions producing a pion and two vector mesons are discussed in this talk. The posterior vector-vector interaction generates two resonances that we associate to the $f_0(1710)$ and the $a_0(1710)$ recently claimed, and they decay to the observed $K^+ K^-$ or $K_S^0 K_S^0$ pair, leading to the reactions $D_s^+ \to \pi^+ K^+ K^- , \pi^+ K_S^0 K_S^0$. The results depend on two parameters related to external and internal emission. We determine a narrow region of the parameters consistent with the large $N_c$ limit within uncertainties which gives rise to decay widths in agreement with experiment. With this scenario we make predictions for the branching ratio of the $a_0(1710)$ contribution to the $D_s^+ \to \pi^0 K^+ K_S^0$ reaction, finding values within the range of $(1.3 \pm 0.4)\times 10^{-3}$. We are happy to see that we have a fair prediction with BESIII experiments, confirming the new $a_0(1710)$ $[a_0(1817)]$ resonance. This is an important state and will shed light into the structure of scalar mesons in light quark sector and other relevant issues currently under debate in hadron physics.
\end{abstract}
\section{Motivation}
An isospin $I=0$, $f_0(1710)$ resonance has been known for quite some time~\cite{ref:pdg}.
It was found from the recent BESIII experiments, the branching fraction \cite{ref:prd1}
\begin{eqnarray}
{\mathrm{Br}}[D_s^+ \to \pi^+ ``f_0(1710)";~``f_0(1710)" \to K^+ K^-]=(1.0 \pm 0.2\pm 0.3)\times 10^{-3} \,,\nonumber
\end{eqnarray}
and in another work it was found that \cite{ref:prd2}
\begin{eqnarray}\label{eq:nBr}
{\mathrm{Br}}[D_s^+ \to \pi^+ ``f_0(1710)";~ ``f_0(1710)" \to K_S^0 K_S^0]=(3.1 \pm 0.3\pm 0.1)\times 10^{-3} \,,\nonumber
\end{eqnarray}
where $``f_0(1710)"$ was supposed to be the $f_0(1710)$ resonance. Thus one finds
\begin{eqnarray}\label{eq:exR1}
R_1=\frac{\Gamma(D_s^+ \to \pi^+ ``f_0(1710)" \to \pi^+ K^0 \bar{K}^0)}
{\Gamma(D_s^+ \to \pi^+ ``f_0(1710)" \to \pi^+ K^+ K^-)} =6.20 \pm 0.67\,.
\end{eqnarray}
But, it is easy to proof that if $``f_0(1710)"$ was the $f_0(1710)$ resonance, this latter ratio should be $1$. Therefore, hidden below or around the $f_0(1710)$, there should be an $I=1$ resonance responsible for this surprising large ratio.
We think a mixture of the two resonances and their interference would be
responsible for a different $K^{+} K^{-}$ or $K^{0}\bar{K}^{0}$ production, due to
\begin{eqnarray}
|K \bar{K}, I=0,I_3=0\rangle &=& -\frac{1}{\sqrt{2}} \big(K^{0}\bar{K}^{0} \, + \,K^{+} K^{-} \big) \,,\nonumber \\
|K \bar{K}, I=1,I_3=0\rangle &=& \frac{1}{\sqrt{2}} \big(K^{0}\bar{K}^{0}- K^{+} K^{-} \big)\,.
\end{eqnarray}
%This is quite similar
As we know in the chiral unitary approach, $a_0(980)$ is dynamically generated as the interaction of the coupled channels $\pi\eta$ and $K\bar{K}$. Then an extension of these ideas to the interaction of vector mesons was done, and interestingly two resonances of $f_0(1710)$ and $a_0(1710)$ were predicted qualifying roughly as $K^*\bar{K}^*$ molecules~\cite{ref:a980,ref:geng}.
\section{Formalism}
Fig.~\ref{fig.ex} shows the Cabibbo-favored decay mode of $D_s^+$ at the quark level and the hadronization with the vacuum quantum numbers $(\bar{q}q=\bar{u}u+\bar{d}d+\bar{s}s)$. Fig.~\ref{fig.in} shows the internal emission and hadronization, which is suppressed by a color factor $1/N_c$. The external and internal emissions will produce the
$f_0(1710)$ and $a_0(1710)$ resonances~\cite{ref:dai}.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.75]{fig1.eps}
\caption{External emission of $D_s^+$ decay with $\pi^+$ production at the quark level (a) and hadronization of the $s\bar{s}$ component (b) and the $u\bar{d}$ component (c) with the vacuum quantum numbers. }
\label{fig.ex}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.82]{fig2.eps}
\caption{Internal emission of $D_s^+$ decay and hadronization of the $s\bar{d}$ pair (a) and the $u\bar{s}$ pair (b). }
\label{fig.in}
\end{figure}
From Figs.~\ref{fig.ex} and \ref{fig.in} and because the $G_{\omega\phi}$ and $G_{\rho\phi}$ loop functions are remarkably similar to $G_{K^* \bar{K}^*}$, finally the amplitudes ${\tilde{t}_{f_0}}$ and ${\tilde{t}_{a_0}}$
can be written
\begin{eqnarray}
{\tilde{t}_{f_0}} &=& A \lbrace -\sqrt{2}\, G_{K^* \bar{K}^*}(M_{\rm inv})\, g_{f_0,K^* \bar{K}^*}
+ G_{\phi\phi}(M_{\rm inv})\sqrt{2}\, g_{f_0,\phi\phi} \nonumber \\
& -& \sqrt{2}\, \gamma' \, G_{K^* \bar{K}^*}(M_{\rm inv})\, g_{f_0,K^* \bar{K}^*} \rbrace\,, \\
{\tilde{t}_{a_0}} &=& -A \sqrt{2}\, \delta' \,G_{K^* \bar{K}^*}(M_{\rm inv})\, g_{a_0,K^* \bar{K}^*} \,.\nonumber
\end{eqnarray}
with the two effective parameters
\begin{eqnarray}
\gamma' = \gamma- \alpha \, \frac{g_{f_0,\omega\phi}}{g_{f_0,K^* \bar{K}^*}}\,, \qquad \delta' =\delta- \beta \frac{g_{f_0,\rho\phi}} {g_{a_0,K^* \bar{K}^*}} \,. \nonumber
\end{eqnarray}
and the global factor $A$ will disappear when we evaluate the ratios of production.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.67]{fig6.eps}
\caption{Amplitude for $R \to K\bar{K}$ for a resonance build up from the $V_i,V_i'$ channels. Diagrams with $\bar{K}K$ instead of $K \bar{K}$ in the final state appear with $\rho,\omega,\phi$
vector mesons but not for the $V_i,V_i' \equiv K^* \bar{K}^*$. }
\label{fig.sb1}
\end{figure}
We also need amplitudes of the two resonances decay into $K\bar{K}$, shown in
Fig.~\ref{fig.sb1}, with $K^* \bar{K}^* \to K\bar{K}$ transitions driven by $\pi$ exchange, and
$\phi (\rho,\omega,\phi) \to K\bar{K}$ transitions driven by $K$ exchange,
the weights are given by
\begin{eqnarray}
W_{f_0} = \sum_i g_{f_0,i} \, \widetilde{W}_i \, G_i(M_{\rm inv})\,, \quad
W_{a_0} = \sum_i g_{a_0,i} \, \widetilde{W}_i \, G_i(M_{\rm inv}) \,.
\end{eqnarray}
where $g_{f_0,i}$ and $g_{a_0,i}$ are the couplings of the $f_0(1710)$ and $a_0(1710)$ resonances to
the different coupled channels that build up the resonance, the $\widetilde{W}_i$ coefficients can be
evaluated from Lagrangian for $V \to PP$, and the sum over $i$ goes over the channels of $I=0$ and $I=1$ respectively.
Finally we obtain the $t_i$ amplitudes in the following
\begin{eqnarray}
t_{K^+ K^-} &=& - {\tilde{t}_{f_0}} \frac{1}{M^2_{\rm inv}-M^2_{f_0}+i M_{f_0} \Gamma_{f_0}} W_{f_0} \frac{1}{\sqrt{2}} g_{K\bar{K}}
-{\tilde{t}_{a_0}} \frac{1}{M^2_{\rm inv}-M^2_{a_0}+i M_{a_0} \Gamma_{a_0}} W_{a_0} \frac{1}{\sqrt{2}}g_{K\bar{K}}\,, \nonumber \\
%
t_{K^0 \bar{K}^0} &=&- {\tilde{t}_{f_0}} \frac{1}{M^2_{\rm inv}-M^2_{f_0}+i M_{f_0} \Gamma_{f_0}} W_{f_0} \frac{1}{\sqrt{2}} g_{K\bar{K}}
+ {\tilde{t}_{a_0}} \frac{1}{M^2_{\rm inv}-M^2_{a_0}+i M_{a_0} \Gamma_{a_0}} W_{a_0}\, \frac{1}{\sqrt{2}} g_{K\bar{K}} \,, \nonumber \\
%
t_{K^+ \bar{K}^0} &=& {\tilde{t}_{a_0}} \frac{1}{M^2_{\rm inv}-M^2_{a_0}+i M_{a_0} \Gamma_{a_0}} W_{a_0} g_{K\bar{K}} \,, \nonumber \\
t_{K^+ K_S^0} &=& -\frac{1}{\sqrt{2}}t_{K^+ \bar{K}^0} \,.\nonumber
\end{eqnarray}
The differential decay width
\begin{eqnarray} \label{eq:tii}
\frac{d\Gamma_i}{dM_{\rm inv}(K\bar{K})}=\frac{1}{(2\pi)^3}\,\frac{1}{4 M^2_{D_s}} \, p_{\pi} \, \tilde{p}_k \, |t_i|^2\,.
\end{eqnarray}
The ratios are defined
\begin{eqnarray}
R_1 =\frac{\Gamma(D_s^+ \to \pi^+ K^0 \bar{K}^0)}{\Gamma(D_s^+ \to \pi^+ K^+ K^-)} \,,\quad
R_2 =\frac{\Gamma(D_s^+ \to \pi^0 K^+ K_S^0)}{\Gamma(D_s^+ \to \pi^+ K^+ K^-)} \,. \end{eqnarray}
%\begin{eqnarray}
% T(s)=\frac{1}{[V_{0}+\beta(s-s_{0})]^{-1} -G(s)}
%\end{eqnarray}
% Fig.\ref{fig.mm1}
%\nonumber
\section{Results}
For the two effective parameters, a narrow region of the parameters
$\gamma' \in [-1,0.1]$, $\delta' \in [-1.3,1.3]$, are obtained and shown in Fig.~\ref{fig.sb}, which are consistent with the large $N_c$ limit within uncertainties.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{fig7.eps}
\caption{The range of two effective parameters}
\label{fig.sb}
\end{figure}
Using the above parameters, we obtain the ratio of $R_1=6.20 \pm 0.67 $, which is in good agreement with BESIII experimental data \cite{ref:prd1,ref:prd2}.
Next, the big challenge of the approach is to make prediction of $R_2$. In \cite{ref:dai},
$R^{\rm theo}_2 \simeq 1.31 \pm 0.12$ was obtained and from this ratio we have evaluated
\begin{eqnarray}
{\mathrm{Br}}[D_s^+ \to \pi^0 a_0(1710)^+; a_0(1710)^+ \to K^+ K_S^0] \simeq (1.3 \pm 0.4)\times 10^{-3} \,,
\end{eqnarray}
which was a prediction before this ratio was measured.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.6]{fig8.eps}
\caption{Mass distributions $d\Gamma/dM_{\rm inv}$ for the cases of Eq.~(\ref{eq:tii}).}
\label{fig.sb2}
\end{figure}
We make a further analysis by taking $\gamma'=-0.5$, $\delta'=-0.75$ (middle of the allowed region) in Fig.~\ref{fig.sb2}, finding that in the $K^0 \bar{K}^0$ mass distribution there has been a constructive interference from the two resonances of $I=0$ and $I=1$,
while in the $K^+ K^-$ mass distribution the interference has been destructive. This is exactly the reason suggested in the experimental analysis to justify the existence of the $a_0(1710)$ resonance \cite{ref:prd1,ref:prd2}, because it should give the same $K^+ K^-$ or $K^0 \bar{K}^0$ mass distributions should there
be only the $f_0(1710)$ state. Hence, we give a boost to the molecular interpretation on the nature of these two $f_0(1710)$ and $a_0(1710)$ resonances.
\section{Summary}
Based on the prediction of $f_0(1710)$ and $a_0(1710)$ as a molecular states of $K^* \bar{K}^*$ and other vector-vector coupled channels, we investigate the two $D_s^+ \to \pi^+ K^+ K^-$ and $D_s^+ \to \pi^+ K_S^0 K_S^0$ reactions.
Two effective parameters related to external and internal emission are obtained with a narrow region, which is consistent with the large $N_c$ limit within uncertainties. Using the allowed parameters, we can reasonably explain the surprising large ratio of $R_1$, in good agreement with recent BESIII experiments. We further made a prediction of
$ {\mathrm{Br}}[D_s^+ \to \pi^0 a_0(1710)^+; a_0(1710)^+ \to K^+ K_S^0] \simeq (1.3 \pm 0.4)\times 10^{-3}$.
Now we are happy to see a fair prediction with the coming data of the branching fraction
$ {\mathrm{Br}}[D_s^+ \to \pi^0 a_0(1710)^+; a_0(1710)^+ \to K^+ K_S^0] \simeq (3.44 \pm 0.52 \pm 0.32)\times 10^{-3} $ \cite{ref:prl}, confirming the existence of new $a_0(1817)$ resonance. Our predicted state of $a_0(1710)$ [new $a_0(1817)$], as an important state, will shed light into the structure of scalar mesons in the light quark sector and other relevant issues currently under debate in hadron physics.
% eqs.~(\ref{e.x})
\acknowledgments
The author acknowledge supports partly from the National Natural Science Foundation of China under Grants Nos. 12175066 and 11975009.
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\end{thebibliography}
\end{document}
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{\small obtained in the limit of \blue{very small binding} and \blue{zero range interaction} in r-space.