ξ/ξ2nd ratio as a tool to refine effective 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 ξ 2nd . The ratio ξ/ξ 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 ξ and ξ 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 ξ/ξ 2nd in the confining phase of the (D = 3 + 1) 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.

Finite-density transition line for QCD with 695 MeV dynamical fermions

“…Conversely, if the ratio ξ=ξ 2nd is greater than one, then this is evidence that there is a spectrum of excited states, not just the lowest lying excitation, that make a non-negligible contribution to the sum. For SU(2) pure gauge theory, lattice Monte Carlo simulations well away from the deconfinement transition show a much more substantial deviation, on the order 40%-50% [10].…”

“…Caselle and Nada [10] have introduced an interesting observable which tests for the presence, in an effective Polyakov line action, of a spectrum of excitations contributing to two-point correlators. We briefly review their idea in this section, as implemented at μ ¼ 0.…”

“…In that case ξ defined in (A3) equals ξ 1 , and we have ξ=ξ 2nd ¼ 1. In the Ising model 3 We adhere in this appendix to the notation of [10], but GðτÞ should not be confused with the usual Polyakov line correlator defined in Eq. (19).…”

“…[10]. The fit to a single correlation length (A3) is expected to be very accurate at large τ, and the influence of higher excitations would only be evident at smaller τ.…”

“…IV. A result for a recently introduced observable ξ=ξ 2nd [10], which is sensitive to excitations above the lowest lying excitation, is presented in Appendix.…”