Existence and nonexistence of a ground state for the massless Nelson model under binding condition

We consider a model describing N nonrelativistic particles coupled to a massless quantum scalar field, called Nelson model, under a binding condition on the external potential. We prove that this model does not admit ground state in the Fock representation of the canonical commutation relations, but it does in another not unitarily equivalent coherent representation. We remark that the binding condition is satisfied for small values of the coupling constant.