Renormalization in the Coulomb gauge and order parameter for confinement in QCD

Renormalization of the Coulomb gauge is studied in the phase space formalism, where one integrates over both the vector potential A, and its canonical momentum II, as well as the usual Faddeev-Popov auxiliary fields. A proof of renormalizability is not attempted. Instead, algebraic identities are derived from BRST invariance which renormalization must satisfy if the Coulomb gauge is renormalizable. In particular, a Ward 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 Gauss-BRST identity. The familiar Zinn-Justin equation results when this identity is integrated over all t. It is shown that in the Coulomb gauge, g²D_0,0 is a renormalization-group invariant, as is its instantaneous part V(R), which we call the color-Coulomb potential. (Here D_0,0 is the time-time component of the gluon propagator.) The contribution of V(R) to the Wilson loop exponentiates. It is proposed that the string tension defined by K_Coul = lim_R→∞ CV(R)/R may serve as an order parameter for confinement, where C = (2N)^-1(N² - 1) for SU(N) gauge theory. A remarkable consequence of the above-mentioned Ward identity is that the Fourier transform V(k) of V(R) is of the product form V(k) = [k²D^c,c* (k)]²L(k), where D^c,c* (k) is the 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 k²D^c,c* (k) diverges at k = 0, to provide a scenario for confinement.