The possible $B\pi$ molecular state and its radiative decay

Recently, several exotic bosons have been confirmed as multi-quark states, but there are violent disputes about their inner structures, namely if they are molecular states or tetraquarks, or even mixtures of the two structures. It would be interesting to experimentally search for non-strange four-quark states with open charm or bottom which are lighter than $\Lambda_c$ or $\Lambda_b$. Reasonable arguments indicate that they are good candidates of pure molecular states $D\pi$ or $B\pi$ because pions are the lightest boson. Both $B\pi$ and $D\pi$ bound states do not decay via strong interaction. The $B\pi$ molecule may decay into $B^*$ by radiating a photon, whereas $D\pi$ molecule can only decay via weak interaction. In this paper we explore the mass spectra of $B\pi$ molecular statesby solving the corresponding B-S equation. Then the rate of radiative decay $|\frac{3}{2},\frac{1}{2}\rangle\to B^*\gamma$ is calculated and our numerical results indicate that the processes can be measured by the future experiment. We also briefly discuss the $D\pi$ case, due to the constraint of the final state phase space, it can only decay via weak interaction.