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

We investigate the $\sigma$ and $\rho$ coupling constants for the $DD$ and $D^*D^*$ interactions, based on correlated $2\pi$ exchange in the $DD$ and $D^*D^*$ interactions. Starting from the $D\bar{D}\to \pi\pi$ and $D^*\bar{D}^*\to \pi\pi$ 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 $D^*D^*\to D^*D^*$ amplitude with correlated $S$- and $P$-wave $2\pi$ exchanges. Using the pole approximation, we estimate the $DD\sigma$, $DD\rho$, $D^*D^*\sigma$, and $D^*D^*\rho$ 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 $BB\sigma$, $BB\rho$, $B^*B^*\sigma$, and $B^*B^*\rho$ coupling constants. We observe that it is unlikely that the $\sigma$ and $\rho$ coupling constants for the $B$ and $B^*$ mesons are the same as those for the $D$ and $D^*$ mesons. On the contrary, they are quite larger than those for the charmed mesons.