We report here on the application of the perturbative renormalization-group to the Coulomb gauge in QCD. We use it to determine the high-momentum asymptotic form of the instantaneous color-Coulomb potential $V(\vec{ k})$ and of the vacuum polarization $P(\vec{ k}, k_4)$. These quantities are renormalization-group invariants, in the sense that they are independent of the renormalization scheme. A scheme-independent definition of the running coupling constant is provided by $\vec{ k}^2 V(\vec{ k}) = x_0 g^2(\vec{ k}/\Lambda_{ coul})$, and of $\alpha_s \equiv { { g^2(\vec{ k} / \Lambda_{ coul})} \over { 4\pi}}$, where $x_0 = { { 12N} \over { 11N - 2N_f}}$, and $\Lambda_{ coul}$ is a finite QCD mass scale. We also show how to calculate the coefficients in the expansion of the invariant $\beta$-function $\beta(g) \equiv |\vec{ k}| { { \partial g} \over{ \partial |\vec{ k}|}} = -(b_0 g^3 + b_1 g^5 +b_2 g^7 + ...)$, where all coefficients are scheme-independent.
Categories: hep-th hep-lat