Analytical relation between confinement and chiral symmetry breaking in terms of Polyakov loop and Dirac eigenmodes in odd-number lattice QCD

In lattice QCD formalism, we derive an analytical gauge-invariant relation between the Polyakov loop $\langle L_P \rangle$ and the Dirac eigenvalues $\lambda_n$ in QCD, i.e., $\langle L_P \rangle \propto \sum_n \lambda_n^{N_t -1} \langle n|\hat U_4|n \rangle$, by considering ${\rm Tr} (\hat{U}_4\hat{\not{D}}^{N_t-1})$ on a temporally odd-number lattice, where the temporal lattice size $N_t$ is odd. This formula is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes $|n\rangle$. We here use an ordinary square lattice with the normal (nontwisted) periodic boundary condition for link-variables $U_\mu(s)$ in the temporal direction. From this relation, one can estimate each contribution of the Dirac eigenmode to the Polyakov loop. Because of the factor $\lambda_n^{N_t -1}$ in the Dirac spectral sum, this analytical relation generally indicates quite small contribution of low-lying Dirac modes to the Polyakov loop in both confined and deconfined phases, while the low-lying Dirac modes are essential for chiral symmetry breaking. Also in lattice QCD calculations in confined and deconfined phases, we numerically confirm the analytical relation, non-zero finiteness of $\langle n|\hat U_4|n \rangle$ for each Dirac mode, and negligibly small contribution of 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. Thus, we 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.