Massive Fermionic Corrections to the Heavy Quark Potential Through Two Loops

TL;DR

This work computes the complete massive fermionic two-loop corrections to the heavy quark potential, defining a physical coupling $\alpha_V({q^2},m^2)$ that remains analytic across quark flavor thresholds. Using a mixed analytic, computer-algebraic, and numerical approach (including Monte Carlo VEGAS) within HQET in the Feynman gauge, the authors derive and renormalize the relevant amplitudes, carefully canceling non-local divergences and validating the massless limit. They find that at charm and bottom thresholds the massive corrections increase $\alpha_V$ by roughly $33\%$ relative to the massless case, with non-Abelian contributions dominating the effect. The results provide a threshold-aware framework for evolving the strong coupling and offer insights for lattice QCD analyses through a continuous flavor function $n_F(-q^2,m^2)$ and related observables.

Abstract

A physically defined effective charge can incorporate quark masses analytically at the flavor thresholds. Therefore, no matching conditions are required for the evolution of the strong coupling constant through these thresholds. In this paper, we calculate the massive fermionic corrections to the heavy quark potential through two loops. The calculation uses a mixed approach of analytical, computer-algebraic and numerical tools including Monte Carlo integration of finite terms. Strong consistency checks are performed by ensuring the proper cancellation of all non-local divergences by the appropriate counterterms and by comparing with the massless limit. The size of the effect for the (gauge invariant) fermionic part of $α_V (q^2,m^2) $ relative to the massless case at the charm and bottom flavor thresholds is found to be of order 33%.

Figures (9)

• Figure 1: The Feynman rules for heavy quark effective theory used in this work for the source propagator and the source gluon vertex. For anti sources one has to make the replacement $v \longrightarrow - v$. The $i$-$\varepsilon$ prescription is the same as for the usual fermion propagator.
• Figure 2: The non-Abelian Feynman diagrams contributing to the massive fermionic corrections to the heavy quark potential at the two loop level. The first two rows contain diagrams with a typical non-Abelian topology. Double lines denote the heavy quarks, single lines the "light" quarks. Color and Lorentz indices are suppressed in the first graph. The notation for the remaining digrams is analogous. The last line includes the infra-red divergent "Abelian" Feynman diagrams. While the topology of these three diagrams is the same as in QED, they contribute to the potential only in the non-Abelian theory due to color factors $C_F C_A$. In addition, although each diagram is infra-red divergent, their sum is infra-red finite.
• Figure 3: The infra-red finite Feynman diagrams with an Abelian topology (upper line) contributing to the massive fermionic corrections to the heavy quark potential at the two loop level plus diagrams consisting of one loop insertions with non-Abelian terms (lower line).
• Figure 4: The two loop counterterms corresponding to the diagrams in Figs. \ref{['fig:hqpfig1']} and \ref{['fig:hqpfig1a']}. Adding these contributions to the original graphs removes all non-local functions from the occurring pole terms. The only exception are $\frac{m^2}{ \epsilon}$ terms in the two point functions which only cancel in the sum of all two point diagrams as explained in the text. The fact that the tadpole diagram has no counterterm is already indicative of this cancellation.
• Figure 5: The sum of the $\lambda^2$-dependent amplitudes and counterterms ${\cal M}^{k_0}_{cl}+{\cal M}^{k_0}_{vc_3}+{\cal M}_{olvc}+ {\cal M}^{k_0}_{cl_{ct}}+{\cal M}^{k_0}_{{vc_3}_{ct}}$. Circles correspond to a choice of $q^2 = -10 GeV^2$ and $m = m_c$, triangles to $q^2 = -100 GeV^2$ and $m = m_c$ while the lower curve (squares) has $q^2 = -100 GeV^2$ and $m = m_b$. The overall normalization neglects color factors and the coupling strength. All data are obtained by using $10^6$ evaluations per iteration with VEGAS and 100 iterations. The statistical error is indicated and smaller than the symbols where invisible. The sum for each of the displayed sets of parameters is clearly independent of the IR-gluon mass regulator $\lambda$ as expected.