The $ξ/ξ_{2nd}$ ratio as a test for Effective Polyakov Loop Actions

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 (3+1) dimensional LGTs and are affected by a milder sign problem. However it is not clear to which extent they really capture the rich spectrum of the original theories, a feature which is instead of great importance if one aims to address the sign problem. 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 becomes larger and larger as the complexity of the spectrum increases. Since both $\xi_{exp}$ and $\xi_{2nd}$ are easy to measure on the lattice, this is an economic and effective way to keep track of the spectrum of the theory. In this respect we show using both numerical simulation and effective string calculations that this ratio increases dramatically as the temperature decreases. This non-trivial behaviour should be reproduced by the Polyakov loop effective action.