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

TL;DR

This study investigates the convergence and applicability of the hopping parameter expansion in finite temperature lattice QCD with heavy quarks, demonstrating its reliability near the chiral limit and extending effective theories to higher orders.

Contribution

The paper provides a detailed analysis of the convergence range of the hopping parameter expansion and extends the effective theory to include higher-order Polyakov loop effects in heavy quark QCD.

Findings

Hopping parameter expansion converges up to the chiral limit of free Wilson quarks.

Strong correlation among Polyakov-type loop terms persists at higher orders.

Previous next-to-leading order results for the critical point are reliable for Nt=4, but higher orders are needed for Nt≥6.

Abstract

Hopping parameter expansion is a useful tool to investigate heavy dynamical quarks in lattice QCD, while the range of its applicability has been sometimes questioned. We study the convergence and the valid range of the hopping parameter expansion in the determination of 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 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 holds also 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 of the critical point for $N_t=4$ are well reliable. For $N_t \ge 6$, we need to incorporate higher-order effects in the effective theory.