The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory

TL;DR

This work computes the planar four-loop four-gluon amplitude in N=4 SYM using D-dimensional unitarity, expressing the result as eight integral functions and verifying the infrared pole structure up to 1/ε^2. It directly tests the Eden–Staudacher all-orders cusp anomalous dimension, finding a discrepancy that can be reconciled by a sign modification of the ζ3^2 term, consistent with a dressing-factor adjustment discussed by BES. By leveraging strong-coupling inputs and KLV-type interpolations, the authors obtain highly accurate estimates of the cusp anomalous dimension across couplings and demonstrate a smooth weak-to-strong coupling transition, lending support to AdS/CFT while highlighting the need for refined integrability-based frameworks. The analysis also clarifies the role of conformal invariance and the no-triangle hypothesis in constraining higher-loop amplitudes and sets the stage for analytic checks of the four-loop iteration at finite terms. Overall, the results illuminate the interplay between perturbative gauge theory, integrability, and string-theory duals in a highly symmetric quantum field theory.

Abstract

We present an expression for the leading-color (planar) four-loop four-point amplitude of N=4 supersymmetric Yang-Mills theory in 4-2 e dimensions, in terms of eight separate integrals. The expression is based on consistency of unitarity cuts and infrared divergences. We expand the integrals around e=0, and obtain analytic expressions for the poles from 1/e^8 through 1/e^4. We give numerical results for the coefficients of the 1/e^3 and 1/e^2 poles. These results all match the known exponentiated structure of the infrared divergences, at four separate kinematic points. The value of the 1/e^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 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 non-trivial 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 one percent 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.

Figures (15)

• Figure 1: Integrals required for $gg \to gg$ scattering in planar MSYM at one loop (1), two loops (2) and three loops ((3)a and (3)b). The box (1), planar double box (2) and three-loop ladder (3)a integrals are scalar integrals, with no loop-momentum dependent factors in the numerator. The tennis-court integral (3)b contains a factor of $(l_1+l_2)^2$, where $l_1$ and $l_2$ are marked with arrows in the figure.
• Figure 2: "Rung-rule" contributions to the leading-color four-loop amplitude, in terms of integral functions given in eqs. (\ref{['MBIntegralA']})--(\ref{['MBIntegralF']}). An overall factor of $st$ has been suppressed in each figure, compared with the definitions in eqs. (\ref{['MBIntegralA']})--(\ref{['MBIntegralF']}).
• Figure 3: Generalized cuts that provide information about the planar four-loop amplitude. (i) A two-particle cut separating a tree amplitude from a three-loop amplitude. (ii) A two-particle cut separating a one-loop amplitude from a two-loop amplitude. (iii) A "3--3" cut separating the amplitude into a product of three tree amplitudes. (iv) An "upper-2--3--lower-2" cut separating the amplitude into a product of four tree amplitudes. (v) A "lower-2--3--lower-2" cut. (vi) A "3--lower-3" cut.
• Figure 4: Non-rung-rule contributions to the leading-color four-loop amplitude, in terms of integral functions defined in eqs. (\ref{['MBIntegralD2']}) and (\ref{['MBIntegralF2']}). Integral (d${}_2$) follows the labeling of integral (d) in fig. \ref{['rrFigure']}, and integral (f${}_2$) follows the labeling of integral (f). An overall factor of $st$ has been suppressed in each figure, compared with eqs. (\ref{['MBIntegralD2']}) and (\ref{['MBIntegralF2']}).
• Figure 5: The only one-particle-irreducible purely-cubic four-loop four-point graph with no triangle or bubble subgraphs, besides the rung-rule graphs in fig. \ref{['rrFigure']}.