Analytical Formulae of the Polyakov and the Wilson Loops with Dirac Eigenmodes in Lattice QCD

TL;DR

This paper derives analytical formulas linking Polyakov and Wilson loops to Dirac eigenmodes in lattice QCD, revealing that low-lying Dirac modes have negligible impact on confinement indicators despite their importance for chiral symmetry breaking.

Contribution

It introduces new gauge-invariant spectral formulas connecting Dirac eigenmodes with Polyakov and Wilson loops, clarifying their roles in confinement and chiral symmetry breaking.

Findings

Low-lying Dirac modes contribute negligibly to Polyakov and Wilson loops.

The formulas are valid for arbitrary lattice sizes, not just odd-numbered temporal extents.

Results suggest no direct link between confinement and chiral symmetry breaking.

Abstract

We derive an analytical gauge-invariant formula between the Polyakov loop $L_P$ and the Dirac eigenvalues $\lambda_n$ in QCD, i.e., $L_P \propto \sum_n \lambda_n^{N_t -1} \langle n|\hat U_4|n \rangle$, in ordinary periodic square lattice QCD with odd-number temporal size $N_t$. Here, $|n\rangle$ denotes the Dirac eigenstate, and $\hat U_4$ temporal link-variable operator. This formula is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes $|n\rangle$. Because of the factor $\lambda_n^{N_t -1}$ in the Dirac spectral sum, this formula indicates negligibly small contribution of low-lying Dirac modes to the Polyakov loop in both confinement and deconfinement phases, while these modes are essential for chiral symmetry breaking. Next, we find a similar formula between the Wilson loop and Dirac modes on arbitrary square lattices, without restriction of odd-number size. This formula suggests a small contribution of low-lying Dirac modes to the string tension $\sigma$, or the confining force. These findings support no crucial role of low-lying Dirac modes for confinement, i.e., no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD, which seems to be natural because heavy quarks are also confined even without light quarks or the chiral symmetry.