The static potential in QCD - a full Two-Loop Calculation

TL;DR

This work delivers a complete analytic two-loop calculation of the QCD static potential and uses renormalization-group improvement to obtain the three-loop momentum-space potential in the V-scheme, including the coefficient $\beta_2^{\mathrm{V}}$, with a subsequent Fourier transform to position space. It clarifies the exponentiation structure in the non-Abelian case, establishing the universality of $\alpha_{\mathrm{V}}$ across heavy color sources and providing explicit expressions for the coefficients $a_1$ and $a_2$, as well as the relation between β-functions in different schemes. The analysis shows that $\alpha_{\mathrm{V}}$ runs faster than $\alpha_{\overline{\mathrm{MS}}}$ and that perturbation theory remains reliable only up to $r\Lambda_{\mathrm{QCD}} \lesssim 0.07$, with strong scheme dependence beyond this range, offering insight into the perturbative–nonperturbative transition and implications for quarkonium spectroscopy. Overall, the results illuminate the perturbative structure of the QCD potential at NNLO and establish a foundation for precise comparisons with lattice results and phenomenology.

Abstract

A full analytic calculation of the two-loop diagrams contributing to the static potential in QCD is presented in detail. Using a renormalization group improvement, the ``three-loop'' potential in momentum space is thus derived and the third coefficient of the $β$-function for the V-scheme is given. The Fourier transformation to position space is then performed, and the result is briefly discussed.

Figures (8)

• Figure 1: One-loop ladder diagrams. The double-lines represent the static sources.
• Figure 2: Two-loop ladder diagrams in QCD.
• Figure 3: Two-gluon exchange diagrams in QCD that remain in Feynman gauge.
• Figure 4: Comparison of the results for $\alpha_{\rm V}({\bf q}^2)$ at different loop-orders, where $\alpha_{\rm V}({\bf q}^2)$ is determined from $\alpha_{\overline{\rm MS}}({\bf q}^2)$. Input parameters are $\alpha_{\overline{\rm MS}}(M_Z^2)=0.118$, $n_f=5$.
• Figure 5: Comparison of the running of $\alpha_{\rm V}({\bf q}^2)$ (solid line) and $\alpha_{\overline{\rm MS}}({\bf q}^2)$ (dashed line) at three loops. The dotted curve displays $\alpha_{\rm V}({\bf q}^2)$ with the initial value $\alpha_{\rm V}(M_Z^2)=\alpha_{\overline{\rm MS}} (M_Z^2)$