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

Abstract

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(Û 4ˆ D Nt−1 ), 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 ∝ n λ Nt−1 n n|Û 4 |n , which is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes |n . Owing to the factor λ Nt−1 n 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, non-zero finiteness of n|Û 4 |n 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.