Matter dependence of the four-loop cusp anomalous dimension

TL;DR

This work analytically determines the matter-dependent (n_f, n_s) quartic Casimir contributions to the four-loop, light-like cusp anomalous dimension in QCD by constructing a uniform-transcendental-weight basis of pure master integrals and solving differential equations for non-planar integrals. An infrared-region analysis guides the selection of integrals with controlled divergences, enabling a computation up to weight 8 and an exact analytic expression for the linear-in-$n_f$ terms. The authors provide explicit formulas for $K_4|_{\,\mathrm{quartic}}^{n_f}$ and $K_4|_{\,\mathrm{quartic}}^{n_s}$, combine them with known results to obtain the full $n_f$ dependence, and obtain a precise $\,\mathcal{N}=4$ SYM prediction that agrees with prior numerical results. The methodology, leveraging UT integrals and canonical differential equations, reduces computational complexity and offers a framework applicable to broader multi-loop, non-planar problems in gauge theories.

Abstract

We compute analytically the matter-dependent contributions to the quartic Casimir term of the four-loop light-like cusp anomalous dimension in QCD, with $n_f$ fermion and $n_s$ scalar flavours. The result is extracted from the double pole of a scalar form factor. We adopt a new strategy for the choice of master integrals with simple analytic and infrared properties, which significantly simplifies our calculation. To this end we first identify a set of integrals whose integrands have a dlog form, and are hence expected to have uniform transcendental weight. We then perform a systematic analysis of the soft and collinear regions of loop integration and build linear combinations of integrals with a simpler infrared pole structure. In this way, only integrals with ten or fewer propagators are needed for obtaining the cusp anomalous dimension. These integrals are then computed via the method of differential equations through the addition of an auxiliary scale. Combining our result with that of a parallel paper, we obtain the complete $n_{f}$ dependence of the four-loop cusp anomalous dimension in QCD. Finally, using known numerical results for the gluonic contributions, we obtain an improved numerical prediction for the cusp anomalous dimension in $\mathcal{N}=4$ super Yang-Mills theory.

Figures (3)

• Figure 1: Feynman diagrams contributing to the $n_f {d_R^{abcd} d_F^{abcd}}/{N_R}$ term of the four-loop cusp anomalous dimension. These diagrams also define the top sector topologies of the associated integral families listed in Table \ref{['tab:integrals']}.
• Figure 2: Integral used to illustrate our infrared analysis.
• Figure 3: Typical form factor integral, which we compute for $p_1^2 \neq 0$. We then extract its value at $p_1^2=0$.