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

Abstract

We derive an analytical gauge-invariant formula between the Polyakov loop $L_P$ and the Dirac eigenvalues ${\lambda }_n$ in QCD, i.e., $L_P\propto {\sum }_n{\lambda }_n^{N_t-1}⟨n|{\hat{U}}_4|n⟩$, in ordinary periodic square lattice QCD with odd-number temporal size $N_t$. Here, $|n⟩$ denotes the Dirac eigenstate, and ${\hat{U}}_4$ 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 ${\lambda }_n^{N_t-1}$ 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 $\sigma$, 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.

Comments: 16 pages, 4 figures