Relevant Gluonic Momentum for Confinement and Gauge-Invariant Formalism with Dirac-mode Expansion

We investigate the relevant gluon-momentum region for confinement in lattice QCD on $16^4$ at $\beta$=5.7, 5.8 and 6.0, based on the Fourier expansion. We find that the string tension $\sigma$, i.e., the confining force, is almost unchanged even after removing the high-momentum gluon component above 1.5GeV in the Landau gauge. In fact, the confinement property originates from the low-momentum gluon component below 1.5GeV, which is the upper limit to contribute to $\sigma$. In the relevant region, smaller gluon momentum component is more important for confinement. Next, we develop a manifestly gauge-covariant expansion of the QCD operator such as the Wilson loop, using the eigen-mode of the QCD Dirac operator $\gamma^\mu D^\mu$. With this method, we perform a direct analysis of the correlation between confinement and chiral symmetry breaking in lattice QCD on $6^4$ at $\beta$=5.6. As a remarkable fact, the confinement force is almost unchanged even after removing the low-lying Dirac modes, which are responsible to chiral symmetry breaking. This indicates that one-to-one correspondence does not hold for between confinement and chiral symmetry breaking in QCD.