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We derive an analytical gauge-invariant formula between the Polyakov loop$L_P$ and the Dirac eigenvalues $\lambda_n$ in QCD, i.e., $L_P \propto \sum_n\lambda_n^{N_t -1} \langle n|\hat U_4|n \rangle$, in ordinary periodic squarelattice QCD with odd-number temporal size $N_t$. Here, $|n\rangle$ denotes theDirac eigenstate, and $\hat U_4$ temporal link-variable operator. This formulais a Dirac spectral representation of the Polyakov loop in terms of Diraceigenmodes $|n\rangle$. Because of the factor $\lambda_n^{N_t -1}$ in the Diracspectral sum, this formula indicates negligibly small contribution of low-lyingDirac modes to the Polyakov loop in both confinement and deconfinement phases,while these modes are essential for chiral symmetry breaking. Next, we find asimilar formula between the Wilson loop and Dirac modes on arbitrary squarelattices, without restriction of odd-number size. This formula suggests a smallcontribution of low-lying Dirac modes to the string tension $\sigma$, or theconfining force. These findings support no crucial role of low-lying Diracmodes for confinement, i.e., no direct one-to-one correspondence betweenconfinement and chiral symmetry breaking in QCD, which seems to be naturalbecause heavy quarks are also confined even without light quarks or the chiralsymmetry.

Analytical Formulae of the Polyakov and the Wilson Loops with Dirac Eigenmodes in Lattice QCD