Four-loop screened perturbation theory

TL;DR

This paper applies four-loop screened perturbation theory to massless phi-fourth theory, improving convergence in thermodynamic calculations by reorganizing the perturbative expansion with a thermal mass term.

Contribution

It extends screened perturbation theory to four loops, providing detailed calculations of the free energy and evaluating complex sum-integrals for better accuracy.

Findings

Improved convergence over weak-coupling expansion

Successful calculation of four-loop free energy

Evaluation of complex three-loop sum-integral

Abstract

We study the thermodynamics of massless phi-fourth theory using screened perturbation theory. In this method, the perturbative expansion is reorganized by adding and subtracting a thermal mass term in the Lagrangian. We calculate the free energy through four loops expanding in a double power expansion in m/T and g^2, where m is the thermal mass and g is the coupling constant. The expansion is truncated at order g^7 and the loop expansion is shown to have better convergence properties than the weak-coupling expansion. The free energy at order g^6 involves the four-loop triangle sum-integral evaluated by Gynther, Laine, Schroeder, Torrero, and Vuorinen using methods developed by Arnold and Zhai. The evaluation of the free energy at order g^7 requires the evaluation of a nontrivial three-loop sum-integral, which we calculate by the same methods.