Three-body molecules $\bar{D}{\bar{D}}^{\ast }{\Sigma }_c$- understanding the nature of $T_{cc}$, $P_c(4312)$, $P_c(4440)$ and $P_c(4457)$

Abstract

The nature of the three pentaquark states, $P_c(4312)$, $P_c(4440)$ and $P_c(4457)$, discovered by the LHCb Collaboration in 2019, is still under debate, although the ${\bar{D}}^{(\ast )}{\Sigma }_c$ molecular interpretation seems to be the most popular. In this work, by adding a $\bar{D}$ meson into the ${\bar{D}}^{\ast }{\Sigma }_c$ pair, we investigate the mass and decay width of the three-body molecules $\bar{D}{\bar{D}}^{\ast }{\Sigma }_c$ and explore the correlation between the existence of the $\bar{D}{\bar{D}}^{\ast }{\Sigma }_c$ molecules with the existence of ${\bar{D}}^{(\ast )}{\Sigma }_c$ and ${\bar{D}}^{\ast }\bar{D}$ two-body molecules. The latter can be identified with the doubly charmed tetraquark state $T_{cc}$ recently discovered by the LHCb Collaboration. Based on the molecular nature of $P_c(4312)$, $P_c(4440)$, $P_c(4457)$, and $T_{cc}$, our results indicate that there exist two three-body bound states of $\bar{D}{\bar{D}}^{\ast }{\Sigma }_c$ with $I(J^P)=1(1/2^{+})$ and $I(J^P)=1(3/2^{+})$, and binding energies $37.24$ MeV and $29.63$ MeV below the $\bar{D}{\bar{D}}^{\ast }{\Sigma }_c$ mass threshold. In addition, we find that the mass splitting of these two three-body molecules are correlated to the mass splitting of $P_c(4440)$ and $P_c(4457)$, which offers a non-trivial way to reveal the molecular nature of these states. The partial widths of two $\bar{D}{\bar{D}}^{\ast }{\Sigma }_c$ molecules decaying into $J/\psi p\bar{D}$ and $J/\psi p{\bar{D}}^{\ast }$ are found to be several MeV. We recommend the experimental searches for the $\bar{D}{\bar{D}}^{\ast }{\Sigma }_c$ molecules in the $J/\psi p\bar{D}$ and $J/\psi p{\bar{D}}^{\ast }$ invariant mass distributions.

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