Logarithmic Singularities and Maximally Supersymmetric Amplitudes

TL;DR

The paper investigates whether nonplanar ${ m N}=4$ sYM amplitudes share the planar property of having only logarithmic singularities and no poles at infinity, proposing that these analytic constraints encode dual conformal information. It constructs a complete basis of three-loop four-point integrands with only $dlog$ forms and uses unitarity to fix the full amplitude, providing substantial higher-loop evidence and explicit $dlog$ representations. It then connects these planar-like singularity constraints to the nonplanar sector, arguing they are more restrictive than dual conformal invariance and testing at up to seven loops, while extending to ${ m N}=8$ supergravity via BCJ double copy. The results suggest a shared analytic structure between planar and nonplanar sectors, hinting at a generalized symmetry or formulation that governs four-point amplitudes across these theories and offering insights into ultraviolet behavior and integrability.

Abstract

The dual formulation of planar N = 4 super-Yang-Mills scattering amplitudes makes manifest that the integrand has only logarithmic singularities and no poles at infinity. Recently, Arkani-Hamed, Bourjaily, Cachazo and Trnka conjectured the same singularity properties hold to all loop orders in the nonplanar sector as well. Here we conjecture that to all loop orders these constraints give us the key analytic information contained in dual conformal symmetry. We also conjecture that to all loop orders, while N = 8 supergravity has poles at infinity, at least at four points it has only logarithmic singularities at finite locations. We provide nontrivial evidence for these conjectures. For the three-loop four-point N = 4 super-Yang-Mills amplitude, we explicitly construct a complete basis of diagram integrands that has only logarithmic singularities and no poles at infinity. We then express the complete amplitude in terms of the basis diagrams, with the coefficients determined by unitarity. We also give examples at three loops showing how to make the logarithmic singularity properties manifest via dlog forms. We give additional evidence at four and five loops supporting the nonplanar logarithmic singularity conjecture. Furthermore, we present a variety of examples illustrating that these constraints are more restrictive than dual conformal symmetry. Our investigations show that the singularity structures of planar and nonplanar amplitudes in N = 4 super-Yang-Mills are strikingly similar. While it is not clear how to extend either dual conformal symmetry or a dual formulation to the nonplanar sector, these results suggest that related concepts might exist and await discovery. Finally, we describe the singularity structure of N = 8 supergravity at three loops and beyond.

Figures (22)

• Figure 1: The (a) bubble, (b) triangle and (c) box one-loop diagrams.
• Figure 2: The left diagram is a generic $L$-loop contribution to the four-point ${\cal N} = 4$ sYM amplitude. The thick (red) highlighting indicates propagators replaced by on-shell conditions. After this replacement, the highlighted propagators leave behind the simplified diagram on the right multiplied by an inverse Jacobian, Eq. (\ref{['eqn:CutBoxJacobian']}). The four momenta $K_1, \ldots , K_4$ can correspond either to external legs or propagators of the higher-loop diagram.
• Figure 3: Two-loop four-point parent diagrams for ${\cal N} = 4$ sYM theory.
• Figure 4: The distinct parent diagrams for three-loop four-point amplitudes. The remaining parent diagrams are obtained by relabeling external legs.
• Figure 5: The rung rule gives the relative coefficient between an $L$-loop diagram and an $(L-1)$-loop diagram. The dotted shaded (red) line represents the propagator at $L$ loops that is removed to obtain the $(L-1)$-loop diagram. As indicated on the second row, if one of the lines is twisted around, as can occur in nonplanar diagrams, there is an additional sign from the color antisymmetry.