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 λ_n in QCD, i.e., L_P∝∑_nλ_n^N_t−1〈n|U-^₄|n〉, in ordinary periodic square lattice QCD with odd-number temporal size N_t. Here, |n〉 denotes the Dirac eigenstate, and U-^₄ the temporal link-variable operator. This formula is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes |n〉. Because of the factor λ_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 σ, 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