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\documentclass{cimento}
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\usepackage[dvipdfmx]{graphicx}
%%%%\usepackage{graphicx} % got figures? uncomment this
\title{Molecular states of $D^* D^* \bar K^*$ and $B^* B^* K^*$ nature}
\author{ N.~Ikeno\from{tottori},
M.~Bayar\from{turkey},
L.~Roca\from{murcia},
\atque
E.~Oset\from{valencia}
}
\instlist{
%\inst{nara} Department of Physics, Nara Women's University, Nara 630-8506, Japan
\inst{tottori} Department of Agricultural, Life and Environmental Sciences, Tottori University, Tottori 680-8551, Japan
\inst{turkey} Department of Physics, Kocaeli University, 41380, Izmit, Turkey
\inst{murcia} Departamento de F\'isica, Universidad de Murcia, E-30100 Murcia, Spain
\inst{valencia} Departamento de F\'{i}sica Te\'{o}rica and IFIC, Centro Mixto Universidad de Valencia - CSIC, Institutos de Investigaci\'{o}n de Paterna, Aptdo. 22085, 46071 Valencia, Spain
}
%% When only one author is present, please do not use the command \from{} near the author name.
\begin{document}
\maketitle
\begin{abstract}
We report the theoretical study of the three-body system composed of $D^* D^* \bar K^*$ and $B^* B^* K^*$.
We study the interaction of two $ D^*$ (or two $\bar B^*$) and one $\bar{K}^{*}$ by using the fixed center approximation to the Faddeev equations to search for bound states of the three-body system.
Since the $D^* D^*$ interaction is attractive and the $D^* \bar{K}^{*}$ interaction is also attractive, we can expect to obtain the bound state of the three-body system $ D^* D^* \bar{K}^{*}$ which is manifestly exotic state with $ccs$ open quarks.
Using the same analogy of the $D^* D^* \bar K^*$ system, we also study the $\bar B^* \bar B^* \bar K^*$ system containing the $bbs$ open quarks since both interactions of $\bar B^* \bar B^*$ and $\bar B^* \bar K^*$ are attractive.
We obtain the bound states of isospin $I=1/2$, negative parity, and total spin $J=0$, 1 and 2.
\end{abstract}
\section{Introduction}
Many exotic mesons, which cannot be explained as the ordinary mesons of $q \bar q$, have been observed in the experiments.
The recent experimental findings of the $X_0(2900)$ in the $D \bar K$ invariant mass and the $T_{cc}(3875)$ in the $D D \pi$ spectrum revealed clear exotic mesonic structures, since one has $cs$ quarks in the first case and $cc$ quarks in the second one.
% The theoretical interpretations of the exotic mesons range from the picture of the tetraquraks to a molecular structure or even a kinematic triangle singularity.
Based on the theoretical interpretations of the molecular picture, the $X_0(2900)$ and the $T_{cc}(3875)$ are identified as the $D^* \bar K^*$ and $D^* D$ bound states, respectively.
In this article, we report the theoretical studies~\cite{Ikeno:2022jbb,Bayar:2023itf} of the three-body systems $D^* D^* \bar K^*$ and $\bar B^* \bar B^* \bar K^*$, which are manifestly exotic bound states with $ccs$ and $bbs$ open quarks.
The reason to choose the systems is that the $D^* D^*$ and $\bar B^* \bar B^*$ interactions with $I(J^P) = 0(1^+)$ were found to bind in Refs.~\cite{Dai:2021vgf,Dai:2022ulk}, and the $D^* \bar K^*$ and $\bar B^* \bar K^*$ interactions were also found to be attractive in Refs.~\cite{Molina:2020hde,Oset:2022xji}. Especially, the $D^* \bar K^*$ bound state with $J^P = 0^+$ in Ref.~\cite{Molina:2020hde} is identified as the $X_0(2900)$.
Therefore, these exotic three-body systems are expected to exist and we calculate the binding energy and width of the possible bound states.
% The $D^* D^* \bar K^*$ interaction is attractive in this paper, and the bound state with the spin $J=0$ is identified with the $X_0$(2900) state.
% And, the D*D* interaction is also attractive and gives a bound state in this paper. There, they use the same cut-off parameter (to regularize the loops) of D*D interaction. This parameter is fixed to describe the Tcc by the $D^*D$ interaction in this paper.
% Therefore, we can expect to obtain the bound states of the three-body system $D^* D^* \bar K^*$. Then we studied the bound states using these interactions.
The three body systems of molecular nature have been also studied recently.
%(see Table~1 of Ref.~\cite{MartinezTorres:2020hus}).
One of the methods to solve the three-body system is the fixed center approximation (FCA) to the Faddeev equations.
%The FCA has been successfully applied to many three-body systems.
In the study of $D \bar D K$ system, the FCA has been compared to the variational method and similar results have been found in Refs.~\cite{Wu:2020job,Wei:2022jgc}.
Thus, we use the FCA to study the $D^* D^* \bar K^*$ and $\bar B^* \bar B^* \bar K^*$ systems.
\section{Formalism}
%We use the Fixed Center Approximation (FCA) to the Faddeev equation to study the three-body systems of $D^* D^* \bar K^*$ and $B^* B^* K^*$.
First, we briefly explain the FCA formalism of the $D^* D^* \bar K^*$ system.
In this picture, we assumed that there is a cluster of two bound particles $D^*D^*$, and the third one ($\bar K^*$) collides with the components of this cluster without modifying its wave function.
The $D^* D^*$ system was found to be bound with about 4-6~MeV in $I(J^P)= 0(1^+)$ in Ref.~\cite{Dai:2021vgf}.
%(Certainly, if the third particle is lighter than the constituents of the cluster, the approximation is better.)
In Fig.~\ref{fig:faddeev}, we show the corresponding diagrams.
The total three-body scattering amplitude $T$ is written by the sum of the partition functions $T_1$ and $T_2$.
$T_1$ is the sum of all diagrams in the upper part of Fig.~\ref{fig:faddeev} where the $\bar K^*$ collides first with the particle 1 of the cluster, while $T_2$ is the sum of all diagrams in the lower part of Fig.~\ref{fig:faddeev} where the $\bar K^*$ collides first with the particle 2 of the cluster.
We can write as
\begin{eqnarray}
T &=& T_1 + T_2, \nonumber \\
T_1&=&t_1+t_1G_0T_2, \label{eq:FCA1}\\
T_2&=&t_2+t_2G_0T_1 \nonumber ,
\end{eqnarray}
where $G_0$ is the $\bar K^*$ propagator folded with the cluster wave function and $t_i$ is the amplitude for two-body scattering $D^*(i) \bar K^*$ ($i=1,2$).
For the evaluation of the two-body $t_i$ amplitudes, we consider the combination of the isospin and the spin decomposition of the $D^* \bar K^*$.
We use the $D^* \bar K^*$ amplitude in Ref.~\cite{Molina:2020hde} for the different isospin $I=0,1$ and spin $J=0,1,2$.
In the $D^* D^* \bar K^*$ system, we can have three total spins $J=0, 1, 2$, and we obtain the final contribution of $t_i$ for different total spin $J$ in Ref.~\cite{Ikeno:2022jbb}.
In addition, we consider the normalization of the amplitudes when mixing two-body amplitudes with three-body amplitudes in the same expression.
We replace $t_i$ into $\tilde{t_i} = \frac{m_C}{m_{D^*}} t_i$ with the cluster mass $m_C$ and the $D^*$ mass $m_{D^*}$, thus Eq.~(\ref{eq:FCA1}) leads to
% We should note that, since $ t_1 = t_2 $, then $ T_1 = T_2 $ and, thus, Eq. (\ref{eq:tildeT1T2} ) leads to
%
\begin{equation}
\tilde{T_1} =\tilde{t_1} + \tilde{t_1} \tilde{G_0} \tilde{T_1};~~~
\tilde{T_1}=\frac{1}{\tilde{t_1}^{-1}-\tilde{G_0}} ;~~~~
~~ \tilde{T}= \tilde{T_1}+ \tilde{T_2}=2 \tilde{T_1},
\label{eq:tildeTtotal}
\end{equation}
where we used $ t_1 = t_2 $, then $ T_1 = T_2 $.
We plot $| \tilde{T} |^{2} $ for the three-body invariant mass energy $\sqrt{s}$ and we look for the peaks to deduce the mass and width of the bound states.
In the $\bar B^* \bar B^* \bar K^*$ system, it was found that the $\bar B^* \bar B^*$ in $I(J^P) = 0(1^+)$ was bound with a binding energy of about 40 MeV in Ref.~\cite{Dai:2022ulk}. In addition, the $ \bar B^* \bar K^*$ was also found to be strongly attractive in Ref.~\cite{Oset:2022xji}.
Thus, we perform a similar calculation to the $D^*D^*\bar K^*$.
\begin{figure}[ht]
\begin{center}
% \includegraphics[width=0.6\columnwidth]{fig1_faddeev.PNG}
\includegraphics[width=9cm]{fig1_faddeev.PNG}
% \includegraphics[width=10cm]{FCADsDsKbars.eps} %{fig5with2deg.pdf}
\caption{ Diagrams involved in the Fixed center approximation (FCA) for the collision of the $\bar K^*$with the cluster of $D^* D^*$.}
\label{fig:faddeev}
\end{center}
\end{figure}
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=\columnwidth]{J012.eps}
\caption{ The three-body amplitude $|\tilde{T}|^2$ for the $ D^* D^* \bar{K}^{*}$ system as a function of the three-body invariant mass energy $\sqrt{s}$ for the different total spin $J$. The dotted vertical line indicates the $ D^* D^* \bar{K}^{*}$ threshold ($2 m_{D^*} + m_{\bar K^*}$).
}
\label{fig:T}
\end{center}
\end{figure}
\section{Numerical results and discussions}
In Fig.~\ref{fig:T}, we show the calculated three-body amplitude $|\tilde{T} |^2$ for the $D^* D^* \bar{K}^{*}$ system as a function of the three-body invariant mass energy $\sqrt{s}$.
For the total spin $J=0$, we find a clear peak around 4845 MeV, about 61 MeV below the $ D^* D^* \bar{K}^{*}$ threshold. The width is about 80 MeV.
The $D^* D^*$ state is bound by about 4–6 MeV, while the $D^* \bar K^*$ state, corresponding to the $X_0 (2900)$, is bound by about 30 MeV.
This means that the interaction of $\bar K^*$ with two $D^*$ would lead to a binding about twice as big as that of $D^* \bar K^*$.
We also calculated the wave function for the $\bar K^*$ in the $D^* D^* \bar K^*$ system at rest in Ref.~\cite{Ikeno:2022jbb}.
We found that the mean square radius is about 1~fm, which is larger than the mean square radius of the proton, 0.84~fm, and smaller than that of the deuteron, 2.1~fm.
%
For the total spin $J=1,2$, we can see two peaks indicating two states.
We can easily trace the origin of the peaks from the $D^* \bar K^*$ amplitude $t_i$ %for the isospin and spin
as discussed in Ref.~\cite{Ikeno:2022jbb}.
This is because the calculation of the three-body total spin $J=1$ is tied to the $J=0,1,2$ of $D^* \bar K^*$. %, the two peak structures showed up.
On the other hand, the calculation of the total spin $J=0$ appears as one peak because it has only $J=1$ of $D^* \bar K^*$.
Thus, in total, we find five states for the total spin $J=0,1,2$ and summarize their binding energy and width in Table~\ref{tab:BE} (upper).
In Table~\ref{tab:BE} (lower), we summarize the binding energy and width for the three-body systems $\bar B^* \bar B^* \bar K^*$ obtained.
In the $\bar B^* \bar B^* \bar K^*$ system, one bound state is obtained for each $J$, which is different from the case of the $D^* D^* \bar K^*$ system.
This is the effect of an overlap of the different states due to the $\bar B^* \bar K^*$ large width.
We also find that the binding energy and width for the $\bar B^* \bar B^* \bar K^*$ are relatively larger than those of the $D^* D^* \bar K^*$.
\begin{table}[h]
\caption{ The calculated binding ($B$), width ($\Gamma$) of the three-body systems $D^* D^* \bar K^*$ and $\bar B^* \bar B^* \bar K^*$ states for the different possible total spins $J$.
The binding energy $B$ is obtained with respect to the threshold energy, $2 m_{D^*} + m_{\bar K^*}$ and $2 m_{\bar B^*} + m_{\bar K^*}$ for $D^* D^* \bar K^*$ and $\bar B^* \bar B^* \bar K^*$ respectively.
Numbers are taken from Refs.~\cite{Ikeno:2022jbb} and \cite{Bayar:2023itf}.
}\label{tab:BE}
\begin{tabular}{ccccc}
\hline
& $J$&~$B$ ~[MeV] &$ \Gamma$~[MeV] \\ \hline
$ D^* D^* \bar K^*$ & $0$~& 61 & 80 \\
& $1$ (State~I)~~& 56 & 94 \\
& $1$ (State~II)& 152 & 100 \\
& $2$ (State~I)~~& 66 & 85 \\
& $2$ (State~II)& 151 & 100 \\ \hline
$\bar B^* \bar B^* \bar K^*$ & $0$~&~109--150 & 72--104 \\
& $1$~&~118--158&106--153 \\
& $2$~&~130--174&103--149 \\
\hline
\end{tabular}
\end{table}
% \begin{table}[t]
% \centering
% \begin{tabular}[t]{c|c|c}
% \hline\hline
% % & Presc. I& Presc. II \\
% &~$E_B$ &$ \Gamma$ \\ \hline
% $J=0$~&~109--150 & 72--104 \\
% $J=1$~&~118--158&106--153 \\
% $J=2$~&~130--174&103--149 \\
% \hline\hline
% \end{tabular}\\\vspace{3mm}
% \end{table}
\section{Conclusion}
We have reported the theoretical study of a search for possible bound states of the three-body systems $D^* D^* \bar K^*$ and $\bar B^* \bar B^* \bar K^*$ based on Refs.~\cite{Ikeno:2022jbb} and \cite{Bayar:2023itf}.
The $D^* D^*$ and $\bar B^* \bar B^*$ interactions with $I(J^P) = 0(1^+)$ were found to bind, and the $D^* \bar K^*$ and $\bar B^* \bar K^*$ interactions were also found to be attractive. For this, we applied the fixed center approximation (FCA) to Faddeev equations where the $\bar K^*$ interact with each of the particles in the $D^* D^*$ and $\bar B^* \bar B^*$ cluster.
From the numerical results, we found that the bound states obtained have relatively large binding energy for different total spin $J=0,1,2$.
% since the flavor
% is conserved in the strong interactions,
% we can expect
% new states made of many mesons, relatively stable, which
% cannot decay to mesons with smaller meson number.
Thus, we hope that these exotic mesons, with open strange and double-charm(bottom) flavors, can be experimentally found in the near future.
\acknowledgments
This work was partly supported by JSPS KAKENHI Grant Number JP P19K14709.
\begin{thebibliography}{0}
\bibitem{Ikeno:2022jbb}
\BY{Ikeno N., Bayar M. \atque Oset E.}
%``Molecular states of D*D*K\textasciimacron{}* nature,''
\IN{Phys. Rev. D}{107}{2023}{034006};
doi:10.1103/PhysRevD.107.034006.
\bibitem{Bayar:2023itf}
\BY{Bayar M., Ikeno N. \atque Roca L.}
%``Predictions of superexotic heavy mesons from K*B(*)B* interactions,''
\IN{Phys. Rev. D}{107}{2023}{054042};
doi:10.1103/PhysRevD.107.054042
%\cite{Dai:2021vgf}
\bibitem{Dai:2021vgf}
\BY{Dai L.~R., Molina R. \atque Oset E.}
%``Prediction of new Tcc states of D*D* and Ds*D* molecular nature,''
\IN{Phys. Rev. D}{105} {2022}{016029};
doi:10.1103/PhysRevD.105.016029
%[arXiv:2110.15270 [hep-ph]].
%\cite{Dai:2022ulk}
\bibitem{Dai:2022ulk}
\BY{Dai L.~R., Oset E., Feijoo A., Molina R., Roca L., Torres A.~M. \atque Khemchandani K.~P.}
%``Masses and widths of the exotic molecular B(s)(*)B(s)(*) states,''
\IN{Phys. Rev. D}{105} {2022}{074017};
[erratum: Phys. Rev. D \textbf{106} (2022), 099904];
doi:10.1103/PhysRevD.105.074017
%[arXiv:2201.04840 [hep-ph]].
%\cite{Molina:2020hde}
\bibitem{Molina:2020hde}
\BY{Molina R. \atque Oset E.}
%``Molecular picture for the $X_0(2866)$ as a $D^* \bar{K}^*$ $J^P=0^+$ state and related $1^+,2^+$ states,''
\IN{Phys. Lett. B}{811} {2020} {135870};
doi:10.1016/j.physletb.2020.135870
%[arXiv:2008.11171 [hep-ph]].
%\cite{Oset:2022xji}
\bibitem{Oset:2022xji}
\BY{Oset E. \atque Roca L.}
%``Exotic molecular meson states of $B^{(*)} K^{(*)}$ nature,''
\IN{Eur. Phys. J. C}{82} {2022} {882};
[erratum: Eur. Phys. J. C \textbf{82} (2022), 1014];
doi:10.1140/epjc/s10052-022-10850-8
%[arXiv:2207.08538 [hep-ph]].
%\cite{Wu:2020job}
\bibitem{Wu:2020job}
\BY{Wu T.~W., ~Liu M.~Z. \atque Geng L.~S.}
%``Excited $K$ meson, $K_c(4180)$ , with hidden charm as a $D\bar D K$ bound state,''
\IN{Phys. Rev. D}{103} {2021} {L031501};
doi:10.1103/PhysRevD.103.L031501
%\cite{Wei:2022jgc}
\bibitem{Wei:2022jgc}
\BY{Wei X., Shen Q.~H. \atque Xie J.~J.}
%``Faddeev fixed-center approximation to the $D\bar{D}K$ system and the hidden charm $K_{c\bar{c}}(4180)$ state,''
\IN{Eur. Phys. J. C}{82} {2022}{718};
doi:10.1140/epjc/s10052-022-10675-5
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% \BY{Hirenzaki S. \atque Ikeno N.}
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% %111 citations counted in INSPIRE as of 02 Oct 2023
%\bibitem{ref:apo} \BY{Einstein A. \atque Fermi E.}
% \IN{Phys. Rev. A}{13}{1999}{12};
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%\bibitem{ref:pul} \BY{Newton I.}
% preprint INFN 8181.
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% \TITLE{Complete Works}, in \TITLE{Workers Playtime}, edited by \NAME{Tizio A. \atque Caio B.} (Unexeditor, Bologna) 1997, pp.~1-10.
\end{thebibliography}
\end{document}
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