RENORMALIZATION IN THE COULOMB GAUGE AND ORDER-PARAMETER FOR CONFINEMENT IN QCD

Renormalization of the Coulomb gauge is studied in the phase space for malism, where one integrates over both the vector potential A, and its canonical momentum Pi, as well as the usual Faddeev-Popov auxiliary f ields. A proof of renormalizability is not attempted. Instead, algebra ic identities are derived from BRST invariance which renormalization m ust satisfy if the Coulomb gauge is renormalizable. In particular, a W ard identity is derived which holds at a fixed time t, and which is an analog of Gauss's law in the BRST formalism. and which we call the Ga uss-BRST identify. The familiar Zinn-Justin equation results when this identity is integrated over all t. It is shown that in the Coulomb ga uge, g(2)D(0,0) is a renormalization-group invariant, as is its instan taneous part V(R), which we call the color-Coulomb potential. (Here D- 0,D-0 is the time-time component of the gluon propagator.) The contrib ution of V(R) to the Wilson loop exponentiates. It is proposed that th e string tension defined by K-Coul = lim(R --> infinity) CV(R)/R may s erve as an order parameter for confinement, where C = (2N)(-1) (N-2 - 1) for SU(N) gauge theory. A remarkable consequence of the above-menti oned Ward identity is that the Fourier transform V(k) of V(R) is of th e product form V(k) = [k(2)D(C,C)(k)]L-2(k), where D-C,D-C* (k) is th e ghost propagator, and L(k) is a correlation function of longitudinal gluons. This exact equation combines with a previous analysis of the Gribov problem according to which k(2)D(C,C)(k) diverges at k = 0, to provide a scenario for confinement. (C) 1998 Elsevier Science B.V.