Relation between confinement and chiral symmetry breaking in temporally odd-number lattice QCD

Abstract

In the lattice QCD formalism, we investigate the relation between confinement and chiral symmetry breaking. A gauge-invariant analytical relation connecting the Polyakov loop and the Dirac modes is derived on a temporally odd-number lattice, where the temporal lattice size is odd, with the normal (nontwisted) periodic boundary condition for link-variables. This analytical relation indicates that low-lying Dirac modes have little contribution to the Polyakov loop, and it is numerically confirmed at the quenched level in both confinement and deconfinement phases. This fact indicates no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD. Using the relation, we also investigate the contribution from each Dirac mode to the Polyakov loop. In the confinement phase, we find a new "positive/negative symmetry" of the Dirac-mode matrix element of the link-variable operator, and this symmetry leads to the zero value of the Polyakov loop. In the deconfinement phase, there is no such symmetry and the Polyakov loop is nonzero. Also, we develop a new method for spin-diagonalizing the Dirac operator on the temporally odd-number lattice modifying the Kogut-Susskind formalism.

Polyakov loop fluctuations in the Dirac eigenmode expansion

“…However, a very large (4 × N c × V ) 2 -dimension of a Dirac operator implies also a large cost of numerical calculations. This can be, however, remarkably reduced by use of the modified Kogut-Susskind(KS) formalism [26].…”

“…(18) and (19) one finds that, on the temporally odd-number lattice, there is a direct relation between the Polyakov loop and the Dirac modes [25,26],…”

“…To derive the analytic relation between the Polyakov loop and the Dirac modes, we consider the temporally odd-number lattice with the normal non-twisted periodic boundary condition for link-variables, in both temporal and spatial directions [25,26].…”

“…On the other hand, there exist several observations that unbroken chiral symmetry does not dictate deconfinement: Given a tower of hadron spectra with eliminating the low-lying Dirac modes [23], the hadrons keep their identity even in chirally restored phase where parity doublers are all degenerate. In addition, it has been shown in the SU(3) lattice simulations that the low-lying modes have little contribution to the Polyakov loop and to the confining force, indicating that the two features are rather independent [24][25][26].…”

“…In the context of the interplay between confinement and chiral symmetry breaking [15,18,20,21,[23][24][25][26][27][28][29][30], it is important to make a reliable separation of one from another, whereas those phenomena are supposed to be correlated. The apparent coincidence in change of properties of the Polyakov loop fluctuation ratios and the chiral condensate and its susceptibility near the chiral crossover might indicate that there is a tied relation between confinement and chiral symmetry breaking in QCD.…”