The Sudakov form factor at four loops in maximal super Yang-Mills theory

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.

Non-planar data of $$ \mathcal{N} $$ = 4 SYM

“…This implies that terms with µνρλ can be generated in twistor space calculations. In this work we were able to write down an ansatz for these terms and found the contribution given in (35). This terms must integrate to zero and they do not appear in the final integrand, but in principle it can be difficult to isolate the epsilon terms from numerical calculations.…”

“…These results are important for understanding non-planar integrability or possible formulations of a non-planar quantum spectral curve [32], see [33] for progress in this direction. One should also stress that the non-planar cusp anomalous dimension was computed numerically by studying Sudakov form-factors with a suitable rewriting in terms of uniformly transcendental integrals [34][35][36].…”

The principle of maximal transcendentality and the four-loop collinear anomalous dimension

“…The first order at which G N =4 0 can have a subleading-color term is four loops. Recently this term has been computed numerically [28,29], Could one try to get an analytic value for this quantity using the methods in this paper? One issue is that the principle of maximal transcendentality has not really been tested yet for cases where there is a subleading-color contribution to N = 4 SYM, but one could try nevertheless.…”

“…In planar N = 4 SYM, it is known analytically through three loops [23,26], and it was computed numerically at four loops a decade ago [27]. The nonplanar contribution to the four-loop collinear anomalous dimension was computed numerically very recently [28,29]. The collinear anomalous dimension also enters the Regge trajectory for forward scattering [10,30,31].…”

On-shell methods for form factors in $$\mathcal{N}=4$$ SYM and their applications

“…The four-loop integrand reduction for the N = 4 Sudakov form factor was achieved in [137]. The computation of integrals was realized by expanding the integrand in a set of uniform transcendentality basis [138,139]. We will not discuss the details of these computation since they go beyond the on-shell methods we focus in this review.…”