Scope and convergence of the hopping parameter expansion in finite-temperature quantum chromodynamics with heavy quarks around the critical point

Abstract

Abstract Hopping parameter expansion is a useful tool for investigating heavy dynamical quarks in lattice quantum chromodynamics (QCD), though its range of applicability has sometimes been questioned. We study the convergence and the valid range of the hopping parameter expansion in determining the critical point (critical quark mass) of QCD with heavy quarks at finite temperature and density. On lattices with sufficiently large spatial extent, the terms in the hopping parameter expansion are classified into Wilson loop terms and Polyakov-type loop terms. We first study the case of the worst convergence in which all the gauge link variables are unit matrices and thus the Wilson loops and the Polyakov-type loops get their maximum values. We perform explicit calculation up to more than the 100th order of the hopping parameter expansion. We show that the hopping parameter expansion is convergent up to the chiral limit of free Wilson quarks. We then perform a Monte Carlo simulation to measure correlation among Polyakov-type loop terms up to the 20th order of the hopping parameter expansion. In previous studies, strong correlation between the leading-order Polyakov loop term and the next-to-leading-order bent Polyakov loop terms was reported and used to construct an effective theory to incorporate the next-to-leading-order effect by a shift of the leading-order coupling parameter. We establish that the strong correlation among Polyakov-type loop terms also holds at higher orders of the hopping parameter expansion, and extend the effective theory to incorporate higher-order effects up to high orders. Using the effective theory, we study the truncation error of the hopping parameter expansion. We find that the previous next-to-leading-order result for the critical point for Nt = 4 is reliable. For Nt ≥ 6, we need to incorporate higher-order effects in the effective theory.