Renormalization-group calculation of the 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(k→) and of the vacuum polarization P(k→,k₄). 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 k→²V(k→)=x₀g ²(k→/Λ_Coul), and of α_s≡ g²(k→/Λ_Coul)/4π, where x₀= 12N/(11N -2N_f), and Λ_Coul is a finite QCD mass scale. We also show how to calculate the coefficients in the expansion of the invariant β function β(g)≡\k→\(∂g/∂\k→\)=- (b₀g³ + b₁g⁵ + b₂g ⁷+ ...), where all coefficients are scheme independent.