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

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We derive an analytical gauge-invariant formula between the Polyakov loop Lₚ and the Dirac eigenvalues λₙ in QCD, i.e., Lₚ ∝ ∑ₙ λₙⁿᵗ ⁻¹ n| U₄|n, in ordinary periodic square lattice QCD with odd-number temporal size Nₜ. Here, |n denotes the Dirac eigenstate, and U₄ 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 λₙⁿᵗ ⁻¹ in the Dirac spectral sum, this formula indicates 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.