Four-loop cusp anomalous dimension in QED

Abstract

The 4-loop $C_F^3 T_F n_l$ and 5-loop $C_F^4 T_F n_l$ terms in the HQET field anomalous dimension $γ_h$ are calculated analytically (the 4-loop one agrees with the recent numerical result [arXiv:1801.08292]). The 4-loop $C_F^3 T_F n_l$ and 5-loop $C_F^4 T_F n_l$ terms in the cusp anomalous dimension $Γ(φ)$ are calculated analytically, exactly in $φ$ (the $φ\to\infty$ asymptotics of the 4-loop one agrees with the recent numerical result [arXiv:1707.08315]). Combining these results with the recent 4-loop $d_{FF} n_l$ contributions to $γ_h$ and to the small-$φ$ expansion of $Γ(φ)$ up to $φ^4$ [arXiv:1708.01221] (recently extended to $φ^6$ [arXiv:1807.05145]) we now have the complete analytical 4-loop result for the Bloch--Nordsieck field anomalous dimension in QED, and the small-$φ$ expansion of the 4-loop QED cusp anomalous dimension up to $φ^6$.