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.

16 pages, 4 figures

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Cited by in corpus (6)

• Relating Quark Confinement and Chiral Symmetry Breaking in QCD
• Different faces of confinement
• Local correlation among the chiral condensate, monopoles, and color magnetic fields in Abelian projected QCD
• Some Relations for Quark Confinement and Chiral Symmetry Breaking in QCD
• Lattice QCD analysis for relation between quark confinement and chiral symmetry breaking
• Dirac spectrum representation of Polyakov loop fluctuations in lattice QCD