Abstract
The four-loop Sudakov form factor in maximal super Yang-Mills theory is analysed in detail. It is shown explicitly how to construct a basis of integrals that have a uniformly transcendental expansion in the dimensional regularisation parameter, further elucidating the number-theoretic properties of Feynman integrals. The physical form factor is expressed in this basis for arbitrary colour factor. In the nonplanar sector the required integrals are integrated numerically using a mix of sector-decomposition and Mellin-Barnes representation methods. Both the cusp as well as the collinear anomalous dimension are computed. The results show explicitly the violation of quadratic Casimir scaling at the four-loop order. A thorough analysis concerning the reliability of reported numerical uncertainties is carried out.
Highlights
Review and setupThe colour structure of the form factor at l loops in N = 4 SYM theory with matter in the adjoint representation is (CA)l up to l = 3
The maximally supersymmetric, N = 4, Yang-Mills (SYM) theory based on the SU(Nc) gauge group, in ’t Hooft’s planar limit [3]
This sector contains among others information on the nonplanar correction to the cusp anomalous dimension
Summary
The colour structure of the form factor at l loops in N = 4 SYM theory with matter in the adjoint representation is (CA)l up to l = 3. Starting from four loops, the quartic Casimir invariant arises in addition, and in SU(Nc) gauge theory one has, besides the planar (i.e. Ncl leadingcolour) contribution a nonplanar (i.e. Ncl−2 subleading-colour) correction. The form factor exhibits a Laurent expansion in the dimensional regularisation parameter In this expansion, each term is expected to be a rational-coefficient polynomial of Riemann Zeta values ζn, or their multi-index generalizations, ζn1,n2,..., known as multiple zeta values (MZVs) We will provide strong evidence that the nonplanar form factor and in particular the CAD are of uniform transcendentality at four loops. The analytic value of this quantity was presented [90]
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