Heavy-hadron molecular spectrum from light-meson exchange saturation

If known, the spectrum of heavy-hadron molecules will be a key tool to disentangle the nature of the exotic states that are being discovered in experiments. Here we argue that the general features of the molecular spectrum can be deduced from the idea that the short-range interaction between the heavy hadrons is effectively described by scalar and vector meson exchange, i.e. the $\sigma$, $\omega$ and $\rho$ mesons. By means of a contact-range theory where the couplings are saturated by the aforementioned light mesons we are indeed able to postdict the $X(3872)$ (as a $D^* \bar{D}$ molecule) from the three $P_c(4312)$, $P_c(4440)$ and $P_c(4457)$ pentaquarks (as $\bar{D}\Sigma_c$ and $\bar{D}^* \Sigma_c$ molecules). We predict a $J^{PC} = 1^{--}$ $D \bar{D}_1$ molecule at $4240-4260\,{\rm MeV}$ which might support the hypothesis that the $Y(4260)$ is at least partly molecular. The extension of these ideas to the light baryons requires minor modifications, after which we recover approximate SU(4)-Wigner symmetry in the two-nucleon system and approximately reproduce the masses of the deuteron and the virtual state.