Analytical Relation between Quark Confinement and Chiral Symmetry Breaking in QCD

Abstract

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 $⟨L_P⟩$ and the Dirac eigenvalues ${\lambda }_n$, i.e., $⟨L_P⟩\propto {\sum }_n{\lambda }_n^{N_t-1}⟨n|{\hat{U}}_4|n⟩$, on a temporally odd-number lattice, where the temporal lattice size $N_t$ is odd. Here, $|n⟩$ denotes the Dirac eigenmode, i.e., $/D|n⟩=i{\lambda }_n|n⟩$, and ${\hat{U}}_4$ the temporal link-variable operator. We here use an ordinary square lattice with the normal periodic boundary condition for link-variables $U_{\mu }(s)$ in the temporal direction. Because of the factor ${\lambda }_n^{N_t-1}$ 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 $⟨n|{\hat{U}}_4|n⟩$ 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.

Comments: 5 pages, Proc. of Hadron 2013. arXiv admin note: substantial text overlap with arXiv:1311.3838, arXiv:1401.2414