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 $V\left(\overrightarrow{k}\right)$ and of the vacuum polarization $P\left(\overrightarrow{k}{,k}_4\right).\mathrm{}$ 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 $k^{2}V\left(\overrightarrow{k}{)=x}_0{g}^2\right(\overrightarrow{k}/{\Lambda }_{\mathrm{Coul}}),$ and of ${\alpha }_s\equiv g^2(\overrightarrow{k}/{\Lambda }_{\mathrm{Coul}})/4\mathrm{}\pi ,$ where $x_0=12N/\left(11N-{2N}_f\right),$ and ${\Lambda }_{\mathrm{Coul}}$ is a finite QCD mass scale. We also show how to calculate the coefficients in the expansion of the invariant $\beta$ function $\beta \mathrm{}\left(g\right)\mathrm{}\equiv |\overrightarrow{k}|(\partial g/\partial |\overrightarrow{k}|)=-{(b}_0{g}^3{+b}_1{g}^5{+b}_2{g}^7+\dots ),$ where all coefficients are scheme independent.