Two-loop polygon Wilson loops in 𝒩 = 4 SYM

Abstract

We compute for the first time the two-loop corrections to arbitrary n-gon lightlike Wilson loops in N = 4 supersymmetric Yang-Mills theory, using efficient numerical methods. The calculation is motivated by the remarkable agreement between the finite part of planar six-point MHV amplitudes and hexagon Wilson loops which has been observed at two loops. At n = 6 we confirm that the ABDK/BDS ansatz must be corrected by adding a remainder function, which depends only on conformally invariant ratios of kinematic variables. We numerically compute remainder functions for n = 7, 8 and verify dual conformal invariance. Furthermore, we study simple and multiple collinear limits of the Wilson loop remainder functions and demonstrate that they have precisely the form required by the collinear factorisation of the corresponding two-loop n-point amplitudes. The number of distinct diagram topologies contributing to the n-gon Wilson loops does not increase with n, and there is a fixed number of "master integrals", which we have computed. Thus we have essentially computed general polygon Wilson loops, and if the correspondence with amplitudes continues to hold, all planar n-point two-loop MHV amplitudes in the N = 4 theory. 1

Six-point gluon scattering amplitudes from $$ {\mathbb{Z}_4} $$ -symmetric integrable model

“…computations [10,[33][34][35] would be of interest. One can expect that the physical picture of the amplitude to all order will emerge through further investigations both at weak and strong coupling.…”

Null-polygonal minimal surfaces in AdS4 from perturbed W minimal models

“…Denoting λ a,s by a and ρ adj = λ 2,s by 2, the ratios of the modular S-matrix elements appearing in t ( Collecting these results, we find 32) where κ 6 G is given by (4.8) with G =M 1 = 1. In addition, substituting the expansion (5.24) into the T-system (2.1) with T 1,1 = T 3,1 , we also find that t (0,2)…”

“…In these cases, the input from the perturbed W minimal models completely determines the leading-order expansion. For n = 6, we also compare the rescaled remainder function with the two-loop one which is read off from [32][33][34][35], to observe that they are close to each other. This paper is organized as follows: in section 2, we review the remainder function corresponding to the minimal surfaces in AdS 4 , and the associated TBA system.…”

“…On the weak-coupling side, the remainder function at two loops for n = 6 in the AdS 4 case is read off from the results in the AdS 5 case [32][33][34][35]. In particular, one can find the UV expansion of the remainder function from a very concise expression in [35] and the expansion of the T-functions in the previous section: R 2-loop 6 ≈ 1.08917 − 0.0487985l 8/3 .…”

“…In particular, one can find the UV expansion of the remainder function from a very concise expression in [35] and the expansion of the T-functions in the previous section: R 2-loop 6 ≈ 1.08917 − 0.0487985l 8/3 . The value in the UV limit l → 0 has been given in [32,34]. The rescaled remainder function is defined similarly to (6.9).…”