Analytical formulae of the Polyakov and Wilson loops with Dirac eigenmodes in lattice QCD

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} \big \langle n \big |\hat U_4 \big |n \big \rangle $ , in ordinary periodic square lattice QCD with odd-number temporal size $N_t$ . Here, $|n\rangle $ denotes the Dirac eigenstate, and $\hat U_4$ the 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 a 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.