2013/12/21 by Hideo Suganuma, Takahiro M. Doi, Takumu Iritani
Physics and Astronomy · #hep-lat #hep-ph #hep-th
paper · pdf · doi:10.1051/epjconf/20147100129
published as EPJ Web Conf. 71 (2014) 00129 · 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
arxiv created 2013/12/21 · arxiv updated 2014/05/06
To clarify the relation between confinement and chiral symmetry breaking in QCD, we consider a temporally odd-number lattice, with the temporal lattice size Nt 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 \rm Tr (U4\not DNt-1), we analytically derive a gauge-invariant relation between the Polyakov loop ⟨ LP ⟩ and the Dirac eigenvalues λn in QCD, i.e., ⟨ LP ⟩ ∝ ∑n λnNt -1 ⟨ n| U4|n ⟩, which is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes |n⟩. Owing to the factor λnNt -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| U4|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.