Descent equations for superamplitudes

TL;DR

This paper addresses the breaking of dual superconformal symmetry at loop level in planar $\mathcal{N}=4$ SYM by analyzing the dual super Wilson loop. It introduces descent equations that express the $\bar{Q}_{\text{ext}}$ anomaly as a single integral over lower-loop data, enabling a recursive, all-loop construction where transcendentality is manifest as $2\ell$ for $\ell$-loop amplitudes. The authors verify the framework in the Abelian 1-loop case, reproducing the 1-loop MHV symbol, and extend the descent to non-Abelian amplitudes, relating higher-loop variations to insertions on an $(n{+}1)$-point superloop. They further interpret the anomaly in terms of corrections to the original superconformal generator $\bar{s}$ and connect the descent to all-loop BCFW-type structures, providing a coherent picture of how loop corrections arise while preserving the full dual supersymmetry of the S-matrix.

Abstract

At loop level in planar N=4 super Yang-Mills, the dual superconformal symmetry of tree amplitudes is lost. This is true even if one uses a supersymmetry preserving regulator, and even for finite quantities that remain dual conformally invariant. We examine this breaking from the dual point of view of the super Wilson Loop, tracing it to the difference between supersymmetries of the self-dual and of the full theories. We show that the anomaly is controlled by a descent equation that determines the derivative of an L-loop amplitude in terms of a single non-trivial integral of an (L-1)-loop amplitude. We propose that this equation can be used recursively to construct multi-loop amplitudes in a way that makes their transcendentality manifest.

Figures (4)

• Figure 1: The framed Wilson Loop. To 1-loop order, we only need consider two copies of the Wilson Loop, obtained by translating the original polygon along a nowhere null normal vector field $v$. In our conventions, the vertices of the original Wilson Loop are labelled by $\{x_i\}$, whereas the vertices of the translated loop are labelled by $\{x_j\}$. Since $v$ is nowhere null, $x_{i,j}^2\neq0$.
• Figure 2: In the Abelian theory at order ${\rm g}^2$, the $\bar{Q}^{(1)}$ variation receives contributions only from a single fermion propagator stretched between the two copies of the framed loop. In Caron-Huot:2011ky, Caron-Huot showed that exactly this diagram resides at order $\chi\bar{\chi}$ in the non-chiral extension of the supersymmetric Wilson Loop. Here we have discovered the same object purely within the chiral superloop dual to the superamplitude.
• Figure 3: The $(n\!+\!1)$-point superloop is integrated over a contour that fixes $\mathcal{Z}\to\mathcal{Z}_i$ and also causes the line X to move in the plane $(i\!-\!1,i,i\!+\!1)$ between the lines $(i\!-\!1,i)$ and $(i,i\!+\!1)$. This corresponds to a point $x$ that is integrated along the edge of the space-time Wilson Loop between $x_i$ and $x_{i+1}$.
• Figure 4: From the point of view of the amplitudes, corrections to the original superconformal generator $\bar{s}=\bar{Q}$ arise when a loop momentum becomes collinear with some external momentum $p_i$. At one loop this contribution may be represented by a dispersion integral of the cut diagram on the left. From the point of view of the superloop, the same contribution arises as a particular BCFW decomposition of the $(n+1)$-point tree amplitude inside the descent equations.