The Running Coulomb Potential and Lamb Shift in QCD

TL;DR

This work delivers the three-loop anomalous dimension for the Coulomb potential within the vNRQCD framework, revealing a fundamental difference from the static potential caused by the coupling of energy and momentum scales in non-relativistic bound states. By employing the velocity renormalization group, the authors obtain NNLL running for the Coulomb coefficient and compute the NNLL energy for heavy quark-antiquark bound states, including all terms up to m α_s^4 (α_s ln α_s)^k. The results clarify how ultrasoft and soft dynamics interplay at high orders and provide quantitative predictions for bottomonium-like systems, including a QCD analog of the Lamb shift. These findings enhance control over perturbative predictions near threshold and bear on processes such as near-threshold e+e− → tt̄ cross sections and heavy-quark sum rules.

Abstract

The QCD beta-function and the anomalous dimensions for the Coulomb potential and the static potential first differ at three loop order. We evaluate the three loop ultrasoft anomalous dimension for the Coulomb potential and give the complete three loop running. Using this result, we calculate the leading logarithmic Lamb shift for a heavy quark-antiquark bound state, which includes all contributions to the binding energies of the form mα_s^4(α_s lnα_s)^k, k\ge 0.

Figures (10)

• Figure 1: Graphs contributing to the three-loop IR divergence of the QCD static potential.
• Figure 2: Loop with an insertion of the $1/(m|{\bf k})$ and $1/{\bf k^2}$ potentials.
• Figure 3: Compton scattering graphs that contribute to the soft vertex.
• Figure 4: Examples of vertices involving soft gluons.
• Figure 5: Order $\alpha_s^3/v$ diagrams with potential iterations. The $\times$ denotes an insertion of the ${\bf p^4}/8m^3$ relativistic correction to the kinetic term.