QCD cusp anomalous dimension: current status

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 ${\gamma }_h$ is known up to 4 loops. The cusp anomalous dimension $\mathrm{\Gamma }(\phi )$ 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 $\phi$ dependence. Some simple contributions are known at higher loops. For the $\phi \to \mathrm{\infty }$ asymptotics of $\mathrm{\Gamma }(\phi )$ (the light-like cusp anomalous dimension) and the ${\phi }^2$ term of the small-$\phi$ 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 $\mathrm{\Gamma }(\phi )$ 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 $\phi$, and it cannot be accidental; however, the reasons of this conjecture and its failures are not understood. The cusp anomalous dimension at Euclidean angle $\phi \to \pi$ 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 $\beta$ function. Some new results are also presented. Using the recent 4-loop result for ${\gamma }_h$, 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 ${\gamma }_h$, $\mathrm{\Gamma }(\phi )$ at 5 loops and to the quark-antiquark potential at 4 loops.