Remarks on the Gribov problem in the direct maximal center gauge - art. no. 034511

We review the equivalence of maximal center gauge fixing to the problem of finding the best fit. to a given lattice gauge field, by a thin vortex conf iguration. This fit is necessarily worst at the location of P plaquettes. W e then compare the fits achieved in Gribov copies generated by (i) over-rel axation, (ii) over-relaxation after Landau gauge preconditioning, and (iii) simulated annealing. Simulated annealing yields the best fit if all links on the lattice are included but the situation changes if we consider only t he lattice volume exterior to P plaquettes. In this exterior region, the fi t is best for Gribov copies generated by over-relaxation. and worst for Gri bov copies generated after Landau gauge preconditioning. The two fitting cr iteria (including or not including the P plaquettes) yield string tensions differing by -34% to +20%, respectively, relative to the full string tensio n. Our usual procedure ("quenched minimization") seems to be a compromise b etween these criteria, and yields string tensions at an intermediate value close to the full string tension.