Matter dependence of the four-loop QCD cusp anomalous dimension: from small angles to all angles

TL;DR

This work computes the fermionic four-loop contributions to the QCD cusp anomalous dimension $\Gamma_\text{cusp}(\phi,\alpha_s)$ via a small-angle expansion, simultaneously yielding analytic results for the HQET field anomalous dimension. It critically tests the Grozin–Kna conjecture that full angle dependence at this order can be inferred from lower-loop data, finding exact agreement for most fermionic color structures but explicit violations for two particular quartic-Casimir structures. The authors also extend the small-angle results to higher order, extract new analytic results for the light-like limit, and analyze the anti-parallel lines limit to illuminate the conformal anomaly contributions to the static quark–antiquark potential. Collectively, the results advance precision tests of perturbative QCD at four loops and provide new analytic benchmarks for future multi-loop studies.

Abstract

We compute the fermionic contributions to the cusp anomalous dimension in QCD at four loops as an expansion for small cusp angle. As a byproduct we also obtain the respective terms of the four-loop HQET wave function anomalous dimension. Our new results at small angles provide stringent tests of a recent conjecture for the exact angle dependence of the matter terms in the four-loop cusp anomalous dimension. We find that the conjecture does not hold for two of the seven fermionic color structures, but passes all tests for the remaining terms. This provides strong support for the validity of the corresponding conjectured expressions with full angle dependence. Taking the limit of large Minkowskian angle, we extract novel analytic results for certain terms of the light-like cusp anomalous dimension. They agree with the known numerical results. Finally, we study the anti-parallel lines limit of the cusp anomalous dimension. In a conformal theory, the latter is proportional to the static quark-antiquark potential. We use the new four-loop results to determine parts of the conformal anomaly term.

Figures (2)

• Figure 1: Wilson line with two straight line segments forming a cusp with Euclidean cusp angle $\phi$.
• Figure 2: Integral topologies of the master integrals of all $n_f$ dependent color structures of the cusp anomalous dimension in the small angle expansion.