Gaussian functional approximation to ’t Hooft’s extension of the linearΣmodel

Abstract

We apply a self-consistent relativistic mean-field variational ``Gaussian functional'' (or optimized one-loop perturbation theory, or $\mathrm{Hartree}+\mathrm{RPA}$) approximation to the extended ${N}_{f}=2$ linear $\ensuremath{\sigma}$ model with spontaneously and explicitly broken chiral $\mathrm{S}{\mathrm{U}}_{\mathrm{R}}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{S}{\mathrm{U}}_{\mathrm{L}}(2)\ifmmode\times\else\texttimes\fi{}{\mathrm{U}}_{\mathrm{A}}(1)\ensuremath{\equiv}\mathrm{O}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ 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. Dmitra\ifmmode \check{s}\else \v{s}\fi{}inovi\ifmmode \acute{c}\else \'{c}\fi{} 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 $\mathrm{S}{\mathrm{U}}_{\mathrm{R}}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{S}{\mathrm{U}}_{\mathrm{L}}(2)\ifmmode\times\else\texttimes\fi{}{\mathrm{U}}_{\mathrm{A}}(1)$ chiral symmetry breaking first we consider how the ${\mathrm{U}}_{\mathrm{A}}(1)$ 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 $\ensuremath{\eta}$ meson channel Bethe-Salpeter equation may coexist.