The static QCD potential in coordinate space with quark masses through two loops

TL;DR

This work addresses the static QCD potential for infinitely heavy quarks in a color singlet in the perturbative regime with finite quark masses. It reconstructs an analytic form of the momentum-space coupling alpha_V(Q,m) from two-loop results and uses a Fourier transform to obtain the coordinate-space potential V(r,m) and the force F(r,m), defining couplings alpha_V(r,m) and alpha_F(r,m). The paper provides explicit two-loop expressions with mass dependence, discusses decoupling behavior, renormalization-scale choices, and mass-scheme effects, and demonstrates notable mass effects at intermediate distances. The results enable quantitative comparisons with lattice data in the overlap region and have practical relevance for bottom-quark mass determinations and perturbative–nonperturbative cross-checks.

Abstract

The potential between infinitely heavy quarks in a color singlet state is of fundamental importance in QCD. While the confining long distance part is inherently non-perturbative, the short-distance (Coulomb-like) regime is accessible through perturbative means. In this paper we present new results of the short distance potential in coordinate space with quark masses through two loops. The results are given in explicit form based on reconstructed solutions in momentum space in the Euclidean regime. Thus, a comparison with lattice results in the overlap region between the perturbative and non-perturbative regime is now possible with massive quarks. We also discuss the definition of the strong coupling based on the force between the static sources.

Figures (13)

• Figure 1: The Wilson-loop $\Gamma$ with large temporal extent ($T \longrightarrow \infty$) from which the position space potential is defined. Through two loops in four dimensions, gluons connecting the spatial source lines can be neglected.
• Figure 2: The momentum space Feynman rules used in the calculation of Ref. mel. The $i \varepsilon$-prescription is analogous to the conventional quark propagator. For anti-sources $v \longrightarrow - v$ must be used.
• Figure 3: The two-loop massive fermionic corrections to the heavy quark potential in the Feynman-gauge (from mel). Double lines denote the heavy quarks, single lines the "light" quarks with mass $m$. The first two rows contain diagrams with a typical non-Abelian topology. The middle line includes the infrared divergent "Abelian" Feynman diagrams. They contribute to the potential only in the non-Abelian theory due to color factors $\propto C_F C_A$. In addition, although each diagram is infrared divergent, their sum is infrared finite. The infrared finite Feynman diagrams with an Abelian topology plus the diagrams consisting of one-loop insertions with non-Abelian terms are shown in the last two rows.
• Figure 4: The numerical results for the gauge-invariant $N_{F,V}^{(1)}$ in QED (open circles) and QCD (triangles) with the best $\chi^2$ fits of Eqs. (\ref{['eq:psi1qed']}) and (\ref{['eq:psi1qcd']}) superimposed respectively (from Ref. bmr). The dashed line shows the one-loop $N_{F,V}^{(0)}$ function of Eq. (\ref{['eq:nf0']}). For comparison we also show the gauge dependent two-loop result obtained in MOM schemes (dash-dot) yhjt. At large $\frac{Q}{m}$ the theory becomes effectively massless, and both schemes agree as expected. The figure also illustrates the decoupling of heavy quarks at small $\frac{Q}{m}$.
• Figure 5: The comparison between the exact results from Ref. mel (open symbols) and the reconstructed solution in Eq. \ref{['eq:aVmsb']} (solid lines) for the bottom (triangles) and charm (circles) quarks. The absolute value of the fermionic contributions (proportional to $T_F$) times $\frac{1}{16 \pi^4}$ is shown. The scale $\mu$ was chosen to coincide with the quark masses in the MS-scheme of Ref. mel. It can clearly be seen that Eq. \ref{['eq:aVmsb']} is in good agreement with the full analytical result over all perturbative values of the momentum transfer $Q$ within the statistical Monte Carlo and fitting errors of a few percent in each case.