QCD cusp anomalous dimension: current status

Abstract

Calculation results for the HQET field anomalous dimension and the QCD cusp anomalous dimension, as well as their properties, are reviewed. The HQET field anomalous dimension $γ_h$ is known up to 4 loops. The cusp anomalous dimension $Γ(φ)$ is known up to 3 loops, and its small-angle and large-angle asymptotics -- up to 4 loops. Some (but not all) color structures at 4 loops are known with the full $φ$ dependence. Some simple contributions are known at higher loops. For the $φ\to\infty$ asymptotics of $Γ(φ)$ (the light-like cusp anomalous dimension) and the $φ^2$ term of the small-$φ$ expansion (the Bremsstrahlung function), the $\mathcal{N}=4$ SYM results are equal to the highest-weight parts of the QCD results. There is an interesting conjecture about the structure of $Γ(φ)$ which holds up to 3 loops; at 4 loops it holds for some color structures and breaks down for other ones. In cases when it holds it related highly non-trivial functions of $φ$, and it cannot be accidental; however, the reasons of this conjecture and its failures are not understood. The cusp anomalous dimension at Euclidean angle $ϕ\toπ$ is related to the static quark-antiquark potential due to conformal symmetry; in QCD this relation is broken by an anomalous term proportional to the $β$ function. Some new results are also presented. Using the recent 4-loop result for $γ_h$, here we obtain analytical expressions for some terms in the 4-loop on-shell renormalization constant of the massive quark field $Z_Q^{\text{os}}$ which were previously known only numerically. We also present 2 new contribution to $γ_h$, $Γ(φ)$ at 5 loops and to the quark-antiquark potential at 4 loops.