Numerical tests of the Kugo-Ojima color confinement criterion

The Kugo-Ojima color confinement criterion, which is based on the BRST symmetry of QCD is numerically tested by the lattice Landau gauge simulation. We first discuss the BRST symmetry on the lattice and the Gribov copy problem. The horizon function defined by Zwanziger based on the U-linear definition of the gauge field and that based on log U are compared. The parameter u which is expected to be $-1\delta^a_b$ in the continuum theory was found to be $-0.7\delta^a_b$ in the strong coupling region. There are about 20% difference between the two horizon functions, but in the weak coupling region, the expectation value of the horizon function is suggested to be zero or negative.