The $\sigma$ and $\rho$ coupling constants for the charmed and beauty mesons

Abstract

We investigate the $\sigma$ and $\rho$ coupling constants for the $DD$ and $D^{\ast }{D}^{\ast }$ interactions, based on correlated $2\pi$ exchange in the $DD$ and $D^{\ast }{D}^{\ast }$ interactions. Starting from the $D\bar{D}\to \pi \pi$ and $D^{\ast }{\bar{D}}^{\ast }\to \pi \pi$ amplitudes derived in the pseudophysical region ($4m_{\pi }^2\le t\le 52m_{\pi }^2$) with the $S$- and $P$-wave $2\pi$ correlations considered, we obtain the spectral functions for the $DD\to DD$ and $D^{\ast }{D}^{\ast }\to D^{\ast }{D}^{\ast }$ amplitude with correlated $S$- and $P$-wave $2\pi$ exchanges. Using the pole approximation, we estimate the $DD\sigma$, $DD\rho$, $D^{\ast }{D}^{\ast }\sigma$, and $D^{\ast }{D}^{\ast }\rho$ coupling constants. We extended phenomenologically the present results to the region in $t\le 0$ and compare them with those from lattice QCD. The results are also compared with those of other models. We also present the results of the $BB\sigma$, $BB\rho$, $B^{\ast }{B}^{\ast }\sigma$, and $B^{\ast }{B}^{\ast }\rho$ coupling constants. We observe that it is unlikely that the $\sigma$ and $\rho$ coupling constants for the $B$ and $B^{\ast }$ mesons are the same as those for the $D$ and $D^{\ast }$ mesons. On the contrary, they are quite larger than those for the charmed mesons.

Comments: 19 pages. 10 figures. Discussion part was revised