Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops

TL;DR

This work constructs the five-loop planar four-point amplitude of maximally supersymmetric Yang-Mills theory using a basis of pseudo-conformal integrals identified via dual diagrams and conformal weights. It develops and applies a maximal-cut, integrand-level unitarity program to determine the integral coefficients, with extensive four- and D-dimensional consistency checks. The results support a remarkably simple coefficient pattern (often ±1) and reinforce the conjectured all-loop iterative structure and connections to AdS/CFT via cusp anomalous dimensions and strong-ccoupling string computations. The approach also provides a framework for extending to non-planar contributions and to gravity via double-copy relations, offering a path toward deeper tests of holography and UV behavior in supergravity.

Abstract

We present an ansatz for the planar five-loop four-point amplitude in maximally supersymmetric Yang-Mills theory in terms of loop integrals. This ansatz exploits the recently observed correspondence between integrals with simple conformal properties and those found in the four-point amplitudes of the theory through four loops. We explain how to identify all such integrals systematically. We make use of generalized unitarity in both four and D dimensions to determine the coefficients of each of these integrals in the amplitude. Maximal cuts, in which we cut all propagators of a given integral, are an especially effective means for determining these coefficients. The set of integrals and coefficients determined here will be useful for computing the five-loop cusp anomalous dimension of the theory which is of interest for non-trivial checks of the AdS/CFT duality conjecture. It will also be useful for checking a conjecture that the amplitudes have an iterative structure allowing for their all-loop resummation, whose link to a recent string-side computation by Alday and Maldacena opens a new venue for quantitative AdS/CFT comparisons.

Figures (20)

• Figure 1: The two-loop planar double-box integral (a) and its dual (b) overlaying a faded version of (a). In (b) the dashed lines represent a numerator factor of $(x_{24}^2)^2 x_{13}^2 = s^2 t$. This inserted numerator factor is needed for conformal invariance of the integral.
• Figure 2: The rung-rule for generating higher-loop integrands from lower-loop ones.
• Figure 3: The rung rule maintains conformal weight. If the dual diagram prior to applying the rung rule has the proper conformal weight so will the resulting diagram.
• Figure 4: The four-loop "window" diagram. The lighter colored line running through the diagrams is a five-particle cut which separates the diagram into a product of tree diagrams.
• Figure 5: The dual diagram corresponding to the window diagram. The lines with the arrows indicate the cut which separates the dual diagram into a product of tree dual diagrams.