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

To clarify the relation between confinement and chiral symmetry breaking in QCD, we consider a temporally odd-number lattice, with the temporal lattice size Nt 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(${\hat U_4}{\hat \not D^{{N_t} - 1}}$) we analytically derive a gauge-invariant relation between the Polyakov loop 〈LP〉 and the Dirac eigenvalues λn in QCD, i.e., 〈LP〉 ∝ Σn λnNt-1 〈n |Û4|n〉, which is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes |n〉. Owing to the factor λnNt−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