Proof of the Dual Conformal Anomaly of One-Loop Amplitudes in N=4 SYM

TL;DR

This work proves the one-loop dual conformal anomaly for generic ${\cal N}=4$ SYM superamplitudes using two complementary approaches based on unitarity and analytic properties. The first proof shows that multi-particle discontinuities of the anomaly vanish in four dimensions, enforcing the conjectured anomaly form involving one-mass triangles. The second proof links the anomaly to infrared divergences via two-particle cuts, uplifting the cut to full loop integrals and leveraging the no-triangle/no-bubble property to recover the same anomaly expression, thereby tying the anomaly directly to IR structure. Together, the results establish the universality of the one-loop dual conformal anomaly for all helicity configurations and external particle numbers in maximally supersymmetric Yang-Mills theory, and they illuminate the precise role of infrared physics in the anomaly. The findings reinforce the deep connection between dual conformal symmetry, Wilson-loop duality, and infrared behavior in ${\cal N}=4$ SYM.

Abstract

We provide two derivations of the one-loop dual conformal anomaly of generic n-point superamplitudes in maximally supersymmetric Yang-Mills theory. Our proofs are based on simple applications of unitarity, and the known analytic properties of the amplitudes.

Figures (4)

• Figure 1: A generic box function. $K_1, K_2, K_3$ and $K_4$ denote the external momenta, and $x_1, x_2, x_3$ and $x_4$ the corresponding region momenta, with $K_i = x_i - x_{i+1}$, $i=1, \ldots , 4$.
• Figure 2: A cut diagram reproducing the discontinuity of the anomaly for a generic superamplitude in a kinematic channel $x_{i\,j+1}^2$. When $x_{i\,j+1}^2$ is a multi-particle invariant, the phase space integral corresponding to this cut diagram is finite, and vanishes in four dimensions due to the factor of $D-4$ on the right hand side of \ref{['anomalycut']}.
• Figure 3: The cut diagram reproducing the discontinuity of the anomaly for a generic superamplitude in the two-particle channel $s_{i i+1}=x^2_{ii+2}$. In this case the superamplitude on the left hand side has four particles and, hence, must be a MHV superamplitude. This diagram has an infrared divergence arising from the region of integration where $l_1\sim -p_i$ and $l_2\sim -p_{i+1}$. In this region we also have that $y\sim x_{i+1}$ where $y$ is the region momentum between the two cut legs.
• Figure 4: Graphic representation of the factorised structure of the two-particle channel discontinuities of the amplitude and of the anomaly in \ref{['ampcuttri']} and \ref{['anomcuttri']}.