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 ξ2nd. The ratio ξ/ξ2nd 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 ξexp and ξ2nd 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.
Highlights
IntroductionIn the past few years there has been a growing interest on the effective Polyakov loop (EPL) models for the description of QCD at finite temperature and finite chemical potential [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]
In EPL models the original theory regularized on the lattice is mapped to a three-dimensional, center-symmetric, effective Polyakov loop spin model obtained by integration over the gauge and matter degrees of freedom
The simplest possible EPL model for the SU(2) lattice gauge theory discussed in the previous section is the d = 3 Ising model, which corresponds to the case in which in Eq (1) we truncate the action to the nearest-neighbour term, choose only the fundamental representation and approximate the Polyakov loop with its sign
Summary
In the past few years there has been a growing interest on the effective Polyakov loop (EPL) models for the description of QCD at finite temperature and finite chemical potential [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. The second issue is that the original lattice gauge theory (LGT) is characterized by a rich spectrum of excitations, with an “Hagedorn” type of dependence on the energy, which, again, is typical of gauge theories and not to reproducible with a spin model These two features are deeply related, since it is precisely the accumulation of an infinite number of excitations that leads to the 1/R correction in the interquark potential. The ξ/ξ2nd ratio represents a simple and easy way to keep track of the spectrum of a statistical model: one can use this computable quantity to understand if the spectrum, both in the original theory and in the effective model, is dominated by a single mass or contains several masses in competition among them This information could help in the selection of the terms to be included in the effective action and, possibly, even in the fine-tuning of the couplings obtained with existing approaches.
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