Analytical relation between quark confinement and chiral symmetry breaking in odd-number lattice QCD
1 Department of Physics & Division of Physics and Astronomy, Graduate School of Science, Kyoto University, Kitashirakawaoiwake, Sakyo, Kyoto 606-8502, Japan
2 High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan
a e-mail: email@example.com
Published online: 29 April 2014
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 Tr() we analytically derive a gauge-invariant relation between the Polyakov loop 〈LP〉 and the Dirac eigenvalues λn in QCD, i.e., 〈LP〉 ∝ Σn λnNt-1 〈n |Û4|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, 〈n|Û4|n〉 non-zero finiteness of 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.
© Owned by the authors, published by EDP Sciences, 2014
This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.