Renormalization-group Calculation of Color-Coulomb Potential

Abstract

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 ${\alpha }_s\equiv \frac{g^2(\vec{k}/{\Lambda }_{coul})}{4\pi }$, where $x_0=\frac{12N}{11N-2N_f}$, 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 $\beta (g)\equiv |\vec{k}|\frac{\partial g}{\partial |\vec{k}|}=-(b_0{g}^3+b_1{g}^5+b_2{g}^7+...)$, where all coefficients are scheme-independent.

Comments: 24 pages, 1 figure, TeX file. Minor modifications, incorporating referee's suggestions