Fisher's Zeros and Perturbative Series in Gluodynamics

Abstract

We study the zeros of the partition function in the complex β 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 β plane where MC reweighting is not reliable. These methods are based on the assumption that the plaquette distribution can be approximated by a φ 4 type distribution. We give new estimates of the locations for a 4 4 lattice. For SU(2), we found zeros at β = 2.18(1) ± i0.18(2) (which differs from previous estimates), and at β = 2.18(1) ± i0.22(2). For SU(3), we confirm β = 5.54(2)±i0.10(2) and found additional zeros at β = 5.54(2)±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).