The Four-Loop Cusp Anomalous Dimension from the $\mathcal{N} = 4$ Sudakov Form Factor

We present an analytic derivation of the full four-loop cusp anomalous dimension of $\mathcal{N}=4$ supersymmetric Yang-Mills theory from the Sudakov form factor. To extract the cusp anomalous dimension, we calculate the $\epsilon^{-2}$ pole of the form factor using parametric integrations of finite integrals. We provide uniformly transcendental results for the master integrals through to weight six and confirm a very recent independent analytic result for the full four-loop cusp anomalous dimension of the $\mathcal{N}=4$ model.


Introduction
The cusp anomalous dimension of N = 4 super Yang-Mills theory has long been recognized as an important probe of the infrared structure of massless gauge theory scattering amplitudes [1]. In fact, the four-loop correction to the cusp anomalous dimension represents the first non-trivial check of the so-called Casimir scaling conjecture [2][3][4], a proposal which would imply that the leading infrared poles of massless gauge theory scattering amplitudes have a universal, semi-classical origin. From numerical studies in N = 4 super Yang-Mills theory [5,6] and QCD [7,8] it is now clear that the Casimir scaling conjecture must be generalized [9,10] to accommodate color structures built out of quartic Casimir operators. While an analytic expression for the leading-color four-loop cusp anomalous dimension in N = 4 super Yang-Mills was proposed long ago in [11] (and subsequently confirmed in [12,13]), the analytic sub-leading-color corrections were first discussed in the literature only recently in [14].
A possible method to obtain the cusp anomalous dimension is via the calculation of the ǫ −2 pole of the N = 4 Sudakov form factor. The latter was derived to three loops in [15], while its four-loop integrand [16], reduction [17], transformation to a uniformly transcendental basis, and numerical integration [5,6] became available over the years. The ǫ −1 pole of the Sudakov form factor is related to the collinear anomalous dimension, whose planar-color part is known both numerically [18] and analytically [19], while the non-planar-color part was computed numerically in [6]. Ongoing efforts to compute the finite (i.e. O(ǫ 0 )) part of massless form factors both in N = 4 super Yang-Mills theory and QCD can be found in [20][21][22][23][24][25][26][27].
In this paper, we confirm the result of [14] by providing an alternative, fully-independent analytic derivation of the full four-loop cusp anomalous dimension of the N = 4 model. From the ǫ −2 pole of the N = 4 Sudakov form factor, we use the infrared evolution equation satisfied by the form factor [28] to read off the cusp anomalous dimension. In order to calculate the Laurent expansions of the 55 ten-, eleven-, and twelve-line four-loop form factor master integrals which contribute to the form factor, the method of parametric integrations for a basis of finite integrals [29,30] is used. In Section 2, we give our conventions for the Sudakov form factor of N = 4 super Yang-Mills theory and, in Section 3, we recall the compact formula for the integrand derived in reference [6]. In Section 4, we present analytic results for all 55 master integrals through to O ǫ −2 . As expected from the analysis of [6], our results are built out of Riemann zeta values of uniform weight 8 + k at O ǫ k . We present the main results of this paper, the Laurent expansion of the Sudakov form factor and the associated cusp anomalous dimension, in Section 5. Finally, we conclude in Section 6.

Notation and Conventions
Up to an unimportant overall normalization factor, the Sudakov form factor of N = 4 super Yang-Mills theory may be defined as in reference [15] in terms of scalar fields, In Eq. (2.1), field superscripts denote adjoint SU (N c ) color indices and field subscripts denote SU (4) R indices. It is convenient to expand the Sudakov form factor in a modified bare 't Hooft coupling, where N c is the number of colors, g N =4 is the bare coupling of the model, γ E is Euler's constant, and ǫ = (4 − d)/2 is the parameter of dimensional regularization. At each order in perturbation theory, the perturbative expansion coefficients of the Sudakov form factor depend on a virtuality parameter, q 2 = (p 1 + p 2 ) 2 , which may be set to −1 without loss of generality. In terms of the modified bare 't Hooft coupling, we then have for the Sudakov form factor.

Master Integrals to Weight Six
In this section, we provide results for the (conjectured) uniform-transcendentality master integrals which appear on the right-hand side of Eq. (3.1), up to and including terms of transcendental weight six. From the definitions of Section 2, it is evident that we employ the MS normalization convention for our master integrals. In total, 32 integrals contribute to the planar-color part and 23 integrals contribute to the non-planar-color part.
Note that non-trivial non-planar master integrals also contribute to the leading-color part.
p,10 = 1 The above expressions were derived from results for finite master integrals calculated in reference [34] for the determination of the cusp anomalous dimensions of massless QCD. A key feature of the finite integral analysis is that the most complicated, non-linearly reducible master integrals, e.g. I (19) p, 16 , I p,17 , I , may be computed through to weight six by judiciously choosing finite integrals in the relevant integral topologies which first contribute to the form factor at transcendental weight seven (i.e. at the level of the ǫ −1 pole). The finite integrals were defined allowing for shifted dimensions and additional powers of the propagators (dots) [29,30,35,36] using the integral finder in Reduze 2 [37] and, in linearlyreducible cases, integrated in the Feynman parametric representation using HyperInt [38]. In order to express the above 55 integrals from reference [6] in terms of finite integrals, linear relations between integrals were computed using finite field arithmetic [21,39]. Moreover, syzygies were employed to avoid numerators in the reductions of integrals with many dots [24,40,41]. Note that analytic results for a subset of the integrals discussed here were presented in previous works [20,22,[24][25][26].

The N = 4 Four-Loop Sudakov Form Factor to Weight Six
Combining the formulae contained in Sections 3 and 4, we find It is worth pointing out that, via the principle of maximal transcendentality [42,43], the ǫ −8 − ǫ −3 poles may be inferred in a straightforward manner from the renormalization group predictions of reference [28] for the four-loop quark form factor of massless QCD. Alternatively, they can be predicted by the requirement that the logarithm of the form factor has at most a double pole in ǫ.

The N = 4 Four-Loop Cusp Anomalous Dimension
The renormalization group analysis of reference [28], together with the known higher-orderin-ǫ results for the one-, two-, and three-loop form factors [15], we see that the ǫ −2 pole of the N = 4 Sudakov form factor must be − is the four-loop cusp anomalous dimension of the N = 4 model. Its relation to γ (4) cusp defined in [5,6] is γ

Conclusions
We calculated the full four-loop cusp anomalous dimension of N = 4 supersymmetric Yang-Mills theory analytically. Our result was derived from the four-loop Sudakov form factor using parametric integrations of finite master integrals calculated in [34] for the determination of the cusp anomalous dimensions in QCD. In our approach, the most complicated integral topologies decouple from the calculation of the cusp because their finite master integrals may be judiciously selected to first contribute to the ǫ −1 pole of the Sudakov form factor. Our calculation confirms the result of the very recent independent calculation of the N = 4 cusp anomalous dimension in [14] based on the Wilson loop picture. Our findings are in agreement with the earlier semi-numerical analysis of [5,6] at the level of the master integrals, for which we provide uniformly transcendental analytic results through to weight six. The analytic results for the master integrals strongly suggest that the full four-loop form factor in N = 4 super Yang-Mills is uniformly transcendental.