Fisher's Zeros and Perturbative Series in Gluodynamics

TL;DR

This paper investigates Fisher's zeros in the complex beta plane for SU(2) and SU(3) gluodynamics, introducing new methods to locate these zeros and analyzing their impact on perturbative series behavior.

Contribution

It presents novel techniques to estimate Fisher's zeros in regions where Monte Carlo reweighting fails, improving understanding of their locations and effects in gluodynamics.

Findings

Identified Fisher's zeros for SU(2) at beta = 2.18 ± i0.18 and 2.18 ± i0.22

Confirmed zeros for SU(3) at beta = 5.54 ± i0.10 and found additional at 5.54 ± i0.16

Provided new estimates of zero locations on a 4^4 lattice

Abstract

We study the zeros of the partition function in the complex beta plane (Fisher's zeros) in SU(2) and SU(3) gluodynamics. We discuss their effects on the asymptotic behavior of the perturbative series for the average plaquette. We present new methods to infer the existence of these zeros in region of the complex beta plane where MC reweighting is not reliable. These methods are based on the assumption that the plaquette distribution can be approximated by a phi^4 type distribution. We give new estimates of the locations for a 4^4 lattice. For SU(2), we found zeros at beta =2.18(1) \pm i0.18(2) (which differs from previous estimates), and at beta =2.18(1) \pm i0.22(2). For SU(3), we confirm beta =5.54(2)\pm i0.10(2) and found additional zeros at beta =5.54(2)\pm i0.16(2). Some of the technical material can be found in recent preprints, in the following we emphasize the motivations (why it is important to know the locations of the zeros) and the challenges (why it is difficult to locate the zeros when the volume increases)