QCD cusp anomalous dimension: current status

TL;DR

This review summarizes the current state of calculations for the QCD cusp anomalous dimension and related quantities, highlighting known results up to 4 loops, conjectures, and new analytical findings at higher loops.

Contribution

It provides a comprehensive review of the perturbative calculations of the cusp anomalous dimension in QCD, including new analytical results at 4 and 5 loops and discusses conjectures on its structure.

Findings

Known up to 4 loops for HQET and cusp anomalous dimensions.

Conjecture about the structure of mma() holds at 3 loops, partially at 4 loops.

New analytical expressions for 4-loop on-shell renormalization constants and contributions at 5 loops.

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 $\gamma_h$ is known up to 4 loops. The cusp anomalous dimension $\Gamma(\varphi)$ 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 $\varphi$ dependence. Some simple contributions are known at higher loops. For the $\varphi\to\infty$ asymptotics of $\Gamma(\varphi)$ (the light-like cusp anomalous dimension) and the $\varphi^2$ term of the small-$\varphi$ 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 $\Gamma(\varphi)$ 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 $\varphi$, 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 $\gamma_h$, $\Gamma(\varphi)$ at 5 loops and to the quark-antiquark potential at 4 loops.