Analytical relation between quark confinement and chiral symmetry breaking in odd-number lattice QCD

Abstract

To clarify the relation between confinement and chiral symmetry breaking in QCD, we consider a temporally odd-number lattice, with the temporal lattice size $N_t$ being odd. We here use an ordinary square lattice with the normal (nontwisted) periodic boundary condition for link-variables in the temporal direction. By considering $\mathrm{Tr}({\hat{U}}_4{\widehat{/D}}^{N_t-1})$, we analytically derive a gauge-invariant relation between the Polyakov loop $⟨L_P⟩$ and the Dirac eigenvalues ${\lambda }_n$ in QCD, i.e., $⟨L_P⟩\propto {\sum }_n{\lambda }_n^{N_t-1}⟨n|{\hat{U}}_4|n⟩$, which is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes $|n⟩$. Owing to the factor ${\lambda }_n^{N_t-1}$ in the Dirac spectral sum, this relation generally indicates fairly small contribution of low-lying Dirac modes to the Polyakov loop, while the low-lying Dirac modes are essential for chiral symmetry breaking. Also in lattice QCD calculations in both confined and deconfined phases, we numerically confirm the analytical relation, non-zero finiteness of $⟨n|{\hat{U}}_4|n⟩$ for each Dirac mode, and negligibly small contribution from low-lying Dirac modes to the Polyakov loop, i.e., the Polyakov loop is almost unchanged even by removing low-lying Dirac-mode contribution from the QCD vacuum generated by lattice QCD simulations. We thus conclude that low-lying Dirac modes are not essential modes for confinement, which indicates no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD.

Comments: 8 pages, 5 figures, Proc. of 2nd Int. Conf. on New Frontiers in Physics (ICNFP 2013). arXiv admin note: substantial text overlap with arXiv:1311.3838