Analytical Relation between Quark Confinement and Chiral Symmetry Breaking in QCD

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We study the relation between quark confinement and spontaneous chiral-symmetry breaking directly in QCD. In lattice QCD formalism, we derive an analytical gauge-invariant relation between the Polyakov loop Lₚ and the Dirac eigenvalues λₙ, i.e., Lₚ ∝ ∑ₙ λₙⁿᵗ ⁻¹ n| U₄|n, on a temporally odd-number lattice, where the temporal lattice size Nₜ is odd. Here, |n denotes the Dirac eigenmode, i.e., D|n =iλₙ|n, and U₄ the temporal link-variable operator. We here use an ordinary square lattice with the normal periodic boundary condition for link-variables Uμ(s) in the temporal direction. Because of the factor λₙⁿᵗ ⁻¹ in the analytical relation, the contribution of low-lying Dirac modes to the Polyakov loop is negligibly small in both confined and deconfined phases, while the low-lying Dirac modes are essential for chiral symmetry breaking. Also, in lattice QCD simulations, we numerically confirm the analytical relation, non-zero finiteness of n| U₄|n for each Dirac mode, and negligibly small contribution of low-lying Dirac modes to the Polyakov loop. Thus, we conclude that low-lying Dirac modes are not essential for confinement, which indicates no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD.