Renormalization-group Calculation of Color-Coulomb Potential

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.