Gaussian functional approximation to ’t Hooft’s extension of the linear Σ model

We apply a self-consistent relativistic mean-field variational “Gaussian functional” (or optimized one-loop perturbation theory, or Hartree+RPA) approximation to the extended N_f=2 linear σ model with spontaneously and explicitly broken chiral SU_R(2)×SU_L(2)×U_A(1)≡O(4)×O(2) symmetry. We set up the self-consistency, or gap equations that dress up the bare fields with “cactus tree” loop diagrams, and the Bethe-Salpeter equations that provide further dressing with one-loop irreducible diagrams. In a previous publication [ V. Dmitrašinović and I. Nakamura J. Math. Phys. (N.Y.) 44 2839 (2003)] we have already shown the ability of this approximation to create composite (i.e., bound and/or resonance) states. With explicit SU_R(2)×SU_L(2)×U_A(1) chiral symmetry breaking first we consider how the U_A(1) symmetry induced scalar-pseudoscalar meson mass relation that is known to hold in fermionic chiral models is modified by the bosonic gap equations. Then we solve the gap and Bethe-Salpeter equations numerically and discuss the solutions’ properties and the particle content of the theory. We show that in the strong-coupling regime two, sometimes even three solutions to the η meson channel Bethe-Salpeter equation may coexist