We consider a quantum mechanical system, which is modeled by a Hamiltonian acting on a
finite dimensional space with degenerate eigenvalues interacting with a field of
relativistic bosons. Provided a mild infrared assumption holds, we prove existence
of the ground state eigenvalues and ground state eigenvectors using operator theoretic
renormalization. We show that the eigenvectors and eigenvalues are analytic functions
of the coupling constant in a cone with apex at the origin.