Molecular $P_{\psi }$ pentaquarks from light-meson exchange saturation

Theoretical predictions for the spectrum of heavy meson-baryon bound states are a fundamental tool for disentangling the nature of the different pentaquark states that have been observed in experimental facilities. Here we explore this spectrum in a phenomenological model that describes the heavy meson-baryon interaction in terms of a contact-range interaction, where the coupling strength is saturated by the exchange of light scalar and vector mesons, i.e. $\sigma$, $\rho$ and $\omega$ exchanges. Saturation determines the couplings modulo an unknown proportionality constant that can be calibrated from a molecular candidate. If we use the $P_{\psi }^N(4312)$ as input, we predict a series of molecular pentaquarks including the $P_{\psi }^N(4440)$ and $P_{\psi }^N(4457)$, the recent $P_{\psi s}^{\mathrm{\Lambda }}(4338)$ and the $P_{\psi s}^{\mathrm{\Lambda }}(4459)$.