Four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory

Abstract

We present an expression for the leading-color (planar) four-loop four-point amplitude of $\mathcal{N}=4$ supersymmetric Yang-Mills theory in $4\ensuremath{-}2ϵ$ dimensions, in terms of eight separate integrals. The expression is based on consistency of unitarity cuts and infrared divergences. We expand the integrals around $ϵ=0$, and obtain analytic expressions for the poles from $1/{ϵ}^{8}$ through $1/{ϵ}^{4}$. We give numerical results for the coefficients of the $1/{ϵ}^{3}$ and $1/{ϵ}^{2}$ poles. These results all match the known exponentiated structure of the infrared divergences, at four separate kinematic points. The value of the $1/{ϵ}^{2}$ coefficient allows us to test a conjecture of Eden and Staudacher for the four-loop cusp (soft) anomalous dimension. We find that the conjecture is incorrect, although our numerical results suggest that a simple modification of the expression, flipping the sign of the term containing ${\ensuremath{\zeta}}_{3}^{2}$, may yield the correct answer. Our numerical value can be used, in a scheme proposed by Kotikov, Lipatov, and Velizhanin, to estimate the two constants in the strong-coupling expansion of the cusp anomalous dimension that are known from string theory. The estimate works to 2.6% and 5% accuracy, providing nontrivial evidence in support of the AdS/CFT correspondence. We also use the known constants in the strong-coupling expansion as additional input to provide approximations to the cusp anomalous dimension which should be accurate to under 1% for all values of the coupling. When the evaluations of the integrals are completed through the finite terms, it will be possible to test the iterative, exponentiated structure of the finite terms in the four-loop four-point amplitude, which was uncovered earlier at two and three loops.