Abstract
We consider the higher-order resummation of Sudakov double logarithms in the presence of multiple coupled gauge in. The associated evolution equations depend on the coupled β functions of two (or more) coupling constants αa and αb, as well as anomalous dimensions that have joint perturbative series in αa and αb. We discuss possible strategies for solving the system of evolution equations that arises. As an example, we obtain the complete three-loop (NNLL) QCD⊗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 N3LL and in some cases even NNLL. Therefore, one should be cautious when using approximate analytic evolution kernels for high-precision analyses.
Highlights
As part of our analysis we revisit the case of a single gauge interaction, 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 N3LL and in some cases even next-to-next-to-leading logarithmic (NNLL)
It implicitly depends on the β functions controlling the renormalization group evolution (RGE) of both gauge coupling constants αa(μ) and αb(μ), which in general are a coupled system of differential equations, as well as anomalous dimensions (Γcusp, γ) whose perturbative expansions are themselves joint series in αa,b
Summary
Before discussing the evaluation of the Sudakov evolution kernel, it will be important to analyze one of its key ingredients, namely the solution of the β-function RGE of the coupling constant.
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