What the paper is about: The authors compute how the three gauge forces of the Standard Model change with energy at four-loop order. These changes are given by “beta functions,” which tell how each coupling runs with energy, and by “anomalous dimensions,” which describe how the normalization of the gauge fields evolves. The calculation is done in the unbroken Standard Model, that is, before electroweak symmetry breaking and without masses.

What the researchers did: They extended previous three-loop, direct calculations to obtain the complete four-loop gauge-coupling beta functions and the four-loop anomalous dimensions of the gauge fields. The work is performed in the background field gauge, a setup that keeps a simple relation between the renormalization of the background gauge fields and the renormalization of the gauge couplings. The authors keep the full dependence on the gauge-fixing parameters and work in an MS-like renormalization scheme (MS = Minimal Subtraction). They also provide ancillary files with renormalization constants in a general Rξ gauge.

How the calculation works at a high level: The result comes from evaluating Feynman diagrams for gauge-boson two-point functions up to four loops and extracting the ultraviolet renormalization constants. To do that they generate diagrams and algebraically manipulate them with automated tools (DIANA and qgraf for diagram generation, FORM for algebra, COLOR for color algebra, and FORCER for the four-loop integrals). The computation requires knowledge of lower-order quantities as input: four-loop renormalization constants for the gauge couplings, three-loop results for the Yukawa couplings, and two-loop input for the Higgs quartic coupling. The authors use dimensional regularization and renormalization-group equations (RGEs) to convert the renormalization constants into the beta functions and anomalous dimensions.