Linear Sigma model in the Gaussian wave functional approximation II: Analyticity of the S-matrix and the effective potential/action

TL;DR

This paper explores the analytic structure of the S-matrix and effective potential in the linear Sigma model using Gaussian approximation, revealing complex solutions and their physical implications across different Riemann sheets.

Contribution

It establishes a connection between Gaussian equations of motion, the effective potential, and dispersion theory, analyzing the analytic properties and physical solutions of the model.

Findings

Only one physical state per quantum number is found, corresponding to CDD poles.

Solutions include complex mass squared values on multiple Riemann sheets.

Numerical analysis covers both strong and weak coupling regimes.

Abstract

We show an explicit connection between the solution to the equations of motion in the Gaussian functional approximation and the minimum of the (Gaussian) effective potential/action of the linear $\Sigma$ model, as well as with the N/D method in dispersion theory. The resulting equations contain analytic functions with branch cuts in the complex mass squared plane. Therefore the minimum of the effective action may lie in the complex mass squared plane. Many solutions to these equations can be found on the second, third, etc. Riemann sheets of the equation, though their physical interpretation is not clear. Our results and the established properties of the S-matrix in general, and of the N/D solutions in particular, guide us to the correct choice of the Riemann sheet. We count the number of states and find only one in each spin-parity and isospin channel with quantum numbers corresponding to the fields in the Lagrangian, i.e. to Castillejo-Dalitz-Dyson (CDD) poles. We examine the numerical solutions in both the strong and weak coupling regimes and calculate the Kallen-Lehmann spectral densities and then use them for physical interpretation.