Analytical relation between quark confinement and chiral symmetry breaking in odd-number lattice QCD

¹ Department of Physics & Division of Physics and Astronomy, Graduate School of Science, Kyoto University, Kitashirakawaoiwake, Sakyo, Kyoto 606-8502, Japan
² High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan

^a e-mail: suganuma@scphys.kyoto-u.ac.jp

Published online: 29 April 2014

Abstract

To clarify the relation between confinement and chiral symmetry breaking in QCD, we consider a temporally odd-number lattice, with the temporal lattice size N_t being odd. We here use an ordinary square lattice with the normal (nontwisted) periodic boundary condition for link-variables in the temporal direction. By considering Tr() we analytically derive a gauge-invariant relation between the Polyakov loop 〈L_P〉 and the Dirac eigenvalues λ_n in QCD, i.e., 〈L_P〉 ∝ Σ_n λ_n^N_t-1 〈n |Û₄|n〉, which is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes |n〉. Owing to the factor λ_n^N_t−1 in the Dirac spectral sum, this relation generally indicates fairly small contribution of low-lying Dirac modes to the Polyakov loop, while the low-lying Dirac modes are essential for chiral symmetry breaking. Also in lattice QCD calculations in both confined and deconfined phases, we numerically confirm the analytical relation, 〈n|Û₄|n〉 non-zero finiteness of for each Dirac mode, and negligibly small contribution from low-lying Dirac modes to the Polyakov loop, i.e., the Polyakov loop is almost unchanged even by removing low-lying Dirac-mode contribution from the QCD vacuum generated by lattice QCD simulations. We thus conclude that low-lying Dirac modes are not essential modes for confinement, which indicates no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD.