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 ${\gamma }_h$ is known up to 4 loops. The cusp anomalous dimension $\Gamma (\phi )$ 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 $\phi \to \infty$ asymptotics of $\Gamma (\phi )$ (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 $\Gamma (\phi )$ 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 $\varphi \to \pi$ 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 ${\gamma }_h$, $\Gamma (\phi )$ at 5 loops and to the quark-antiquark potential at 4 loops.

54 pages; v2: 1 reference added, more detailed comment on [92]; v3: more information included, more references, 58 pages; v4: small text changes, 1 reference added

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Cited by in corpus (1)

• Generalized eikonal identities for charged currents