2014/01/17 by Hideo Suganuma, Takahiro M. Doi, Takumi Iritani
Physics and Astronomy · #hep-lat #hep-ph #hep-th
published as PoS (Hadron 2013) 121 · 5 pages, Proc. of Hadron 2013. arXiv admin note: substantial text overlap with arXiv:1311.3838, arXiv:1401.2414
arxiv created 2014/01/17 · arxiv updated 2014/01/20
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., \not 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.