Higher-Order Sudakov Resummation in Coupled Gauge Theories

We consider the higher-order resummation of Sudakov double logarithms in the presence of multiple coupled gauge interactions. The associated evolution equations depend on the coupled $\beta$ functions of two (or more) coupling constants $\alpha_a$ and $\alpha_b$, as well as anomalous dimensions that have joint perturbative series in $\alpha_a$ and $\alpha_b$. We discuss possible strategies for solving the system of evolution equations that arises. As an example, we obtain the complete three-loop (NNLL) QCD$\otimes$QED Sudakov evolution factor. Our results also readily apply to the joint higher-order resummation of electroweak and QCD Sudakov logarithms. As part of our analysis we also revisit the case of a single gauge interaction (pure QCD), and study the numerical differences and reliability of various methods for evaluating the Sudakov evolution factor at higher orders. We find that the approximations involved in deriving commonly used analytic expressions for the evolution kernel can induce noticeable numerical differences of several percent or more at low scales, exceeding the perturbative precision at N$^3$LL and in some cases even NNLL. Therefore, one should be cautious when using approximate analytic evolution kernels for high-precision analyses.