Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond

TL;DR

The paper confirms an iterative structure for planar N=4 SYM amplitudes by explicitly computing the three-loop four-point amplitude and showing it can be expressed in terms of one- and two-loop amplitudes. It uses unitarity to construct the integrand and Mellin-Barnes techniques to evaluate the remaining three-loop integrals, with the result consistent with IR exponentiation and Sterman–Tejeda-Yeomans predictions. Building on this, the authors propose an all-orders exponentiation for planar MH V n-point amplitudes, derive the form of finite remainders, and connect the soft anomalous dimensions to leading-twist QCD data via leading transcendentality. The work also provides independent support for the KLOV results and highlights links to integrability and AdS/CFT, suggesting a tractable structure for the full planar perturbative series.

Abstract

We compute the leading-color (planar) three-loop four-point amplitude of N=4 supersymmetric Yang-Mills theory in 4 - 2 epsilon dimensions, as a Laurent expansion about epsilon = 0 including the finite terms. The amplitude was constructed previously via the unitarity method, in terms of two Feynman loop integrals, one of which has been evaluated already. Here we use the Mellin-Barnes integration technique to evaluate the Laurent expansion of the second integral. Strikingly, the amplitude is expressible, through the finite terms, in terms of the corresponding one- and two-loop amplitudes, which provides strong evidence for a previous conjecture that higher-loop planar N = 4 amplitudes have an iterative structure. The infrared singularities of the amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on resummation. Based on the four-point result and the exponentiation of infrared singularities, we give an exponentiated ansatz for the maximally helicity-violating n-point amplitudes to all loop orders. The 1/epsilon^2 pole in the four-point amplitude determines the soft, or cusp, anomalous dimension at three loops in N = 4 supersymmetric Yang-Mills theory. The result confirms a prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and Vogt. Following similar logic, we are able to predict a term in the three-loop quark and gluon form factors in QCD.

Figures (7)

• Figure 1: The result for the leading-color two-loop amplitude in terms of scalar integral functions, given in eq. (\ref{['TwoloopPlanarResult']}).
• Figure 2: The rung insertion rule for generating higher-loop integrands from lower loop ones, given in ref. BRY.
• Figure 3: Mondrian diagrams for the three-loop four-point MSYM planar amplitude given in eq. (\ref{['ThreeLoopPlanarResult']}). The second and third diagrams have identical values, as do the fifth and sixth. The factors of $(l_1 + l_2)^2$ denote numerator factors appearing in the integrals, where $l_1$ and $l_2$ are the momenta carried by the lines marked by arrows.
• Figure 4: The two integrals appearing in the three-loop amplitude. The "ladder" integral (a) has no factors in the numerator. The "tennis court" integral (b) contains a factor of $(p+r)^2$ in the numerator.
• Figure 5: Infrared structure of leading-color scattering amplitudes for particles in the adjoint representation. The straight lines represent hard external states, while the curly lines carry soft or collinear virtual momenta. At leading color, soft exchanges are confined to wedges between the hard lines.