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Gauge-Invariant Formalism with a Dirac-mode Expansion for Confinement and Chiral Symmetry Breaking

Using the eigen-mode of the QCD Dirac operator $\Slash D=γ^μD^μ$, we develop a manifestly gauge-covariant expansion and projection of the QCD operators such as the Wilson loop and the Polyakov loop. With this method, we perform a direct analysis of the correlation between confinement and chiral symmetry breaking in lattice QCD Monte Carlo calculation on $6^4$ at $β$=5.6. Even after removing the low-lying Dirac modes, which are responsible to chiral symmetry breaking, we find that the Wilson loop obeys the area law, and the slope parameter corresponding to the string tension or the confinement force is almost unchanged. We find also that the Polyakov loop remains to be almost zero even without the low-lying Dirac modes, which indicates the $Z_3$-unbroken confinement phase. These results indicate that one-to-one correspondence does not hold for between confinement and chiral symmetry breaking in QCD.