$\xi/\xi_{2nd}$ ratio as a tool to refine Effective Polyakov Loop models

TL;DR

This paper introduces a ratio of correlation lengths as a simple, effective tool to evaluate how well Polyakov loop models capture the spectrum of lattice gauge theories, demonstrated through SU(2) gauge theory analysis.

Contribution

It proposes using the ratio of exponential to second moment correlation lengths to assess the spectrum complexity in effective Polyakov loop models, providing a practical lattice measurement approach.

Findings

The ratio $\xi/\xi_{2nd}$ is close to 1 near the deconfinement transition.

The ratio increases significantly at lower temperatures, indicating spectrum complexity.

The behavior aligns with an effective string description, offering a test for Polyakov loop models.

Abstract

Effective Polyakov line actions are a powerful tool to study the finite temperature behaviour of lattice gauge theories. They are much simpler to simulate than the original lattice model and are affected by a milder sign problem, but it is not clear to which extent they really capture the rich spectrum of the original theories. We propose here a simple way to address this issue based on the so called second moment correlation length $\xi_{2nd}$. The ratio $\xi/\xi_{2nd}$ between the exponential correlation length and the second moment one is equal to 1 if only a single mass is present in the spectrum, and it becomes larger and larger as the complexity of the spectrum increases. Since both $\xi$ and $\xi_{2nd}$ are easy to measure on the lattice, this is a cheap and efficient way to keep track of the spectrum of the theory. As an example of the information one can obtain with this tool we study the behaviour of $\xi/\xi_{2nd}$ in the confining phase of the ($D=3+1$) $\mathrm{SU}(2)$ gauge theory and show that it is compatible with 1 near the deconfinement transition, but it increases dramatically as the temperature decreases. We also show that this increase can be well understood in the framework of an effective string description of the Polyakov loop correlator. This non-trivial behaviour should be reproduced by the Polyakov loop effective action; thus, it represents a stringent and challenging test of existing proposals and it may be used to fine-tune the couplings and to identify the range of validity of the approximations involved in their construction.