Logarithms of alpha in QED bound states from the renormalization group

TL;DR

The velocity renormalization group is used to determine lnalpha contributions to QED bound state energies and shows for the first time that these logarithms can be computed from the renormalized group.

Abstract

The velocity renormalization group is used to determine ln(alpha) contributions to QED bound state energies. The leading order anomalous dimension for the potential gives the alpha^5 ln(alpha) Bethe logarithm in the Lamb shift. The next-to-leading order anomalous dimension determines the alpha^6 ln(alpha), alpha^7 ln^2(alpha), and alpha^8 ln^3 (alpha) corrections to the energy. These are used to obtain the alpha^8 ln^3(alpha) Lamb shift and alpha^7 ln^2(alpha) hyperfine splitting for Hydrogen, muonium and positronium, as well as the alpha^2 ln(alpha) and alpha^3 ln^2(alpha) corrections to the ortho- and para-positronium lifetimes.

Figures (3)

• Figure 1: Graphs contributing to the soft (a) and ultrasoft (b,c,...) anomalous dimensions at leading order. The potential is denoted by $\otimes$, and one sums over all possible ultrasoft exchanges (including wavefunction renormalization).
• Figure 2: Examples of graphs contributing to the NLO anomalous dimensions. The $\otimes$ denotes insertions of terms in the potential such as $U_c$, $U_2$, etc.
• Figure 3: Fig. (a) is the $\alpha^8 \ln^3\! \alpha$ contribution to the Lamb shift computed by Karshenboim. Fig. (b) is the additional contribution of Yerokhin. The dashed lines are Coulomb exchanges.