Perturbative Approach at Finite Temperature and the $\phi^4$ model

TL;DR

This paper questions the validity of the $^4$ model at high temperatures, suggesting it may be an approximation of a more fundamental theory, and demonstrates this with a comparison in the Nambu-Goto string model.

Contribution

It introduces a conjecture that the $^4$ model's high-temperature behavior may be inaccurate, supported by a comparison with the Nambu-Goto string model's exact results.

Findings

Perturbative expansion fails near the critical temperature in the Nambu-Goto model.

High-temperature features are not captured accurately by the two-loop calculation.

The $^4$ model might require reconsideration for high-temperature regimes.

Abstract

We suggest that the $\phi^4$ model is only a polynomial approximation to a more fundamental theory. As a consequence the high temperature regime might not be correctly described by this model. If this turns out to be true then several results concerning e.g., critical temperatures, symmetry restoration at high temperature and high temperature expansions should be reconsidered. We illustrate our conjecture by using the Nambu-Goto string model. We compare a two-loop calculation of the free energy or quark-antiquark static potential at finite temperature with a previous exact calculation in the large-d limit and show how the perturbative expansion fails to reproduce important features in the neighborhood of the critical temperature. It becomes clear why this happens in the Nambu-Goto model and we suggest that perhaps something similar occurs with the $\phi^4$ model.