Broken symmetry phase solution of the phi^4 model at two-loop level of the Phi-derivable approximation

TL;DR

This paper numerically solves the broken symmetry phase of the phi^4 model at two-loop level using Phi-derivable approximation, deriving explicitly finite equations and improving convergence with evolving counterterms.

Contribution

It introduces a high-accuracy numerical solution for the phi^4 model's broken symmetry phase at two-loop level, utilizing explicitly finite equations and a novel iterative counterterm evolution method.

Findings

Explicitly finite equations are derived for the model.

Evolving counterterms accelerate convergence.

Comparison with renormalized equations confirms consistency.

Abstract

The set of coupled equations for the self-consistent propagator and the field expectation value is solved numerically with high accuracy in Euclidean space at zero temperature and in the broken symmetry phase of the phi^4 model. Explicitly finite equations are derived with the adaptation of the renormalization method of van Hees and Knoll [H. van Hees, J. Knoll, Phys. Rev. D65, 025010 (2001)] to the case of non-vanishing field expectation value. The set of renormalization conditions used in this method leads to the same set of counterterms obtained recently in A. Patkos, Zs. Szep, Nucl. Phys. A811, 329-352 (2008). This makes possible the direct comparison of the accurate solution of explicitly finite equations with the solution of renormalized equations containing counterterms. The numerically efficient way of solving iteratively these latter equations is obtained by deriving at each order of the iteration new counterterms which evolve during the iteration process towards the counterterms determined based on the asymptotic behavior of the converged propagator. As shown at different values of the coupling, the use of these evolving counterterms accelerates the convergence of the solution of the equations.