Four-point Amplitudes in N=8 Supergravity and Wilson Loops

TL;DR

The paper investigates four-point MHV amplitudes in N=8 supergravity and their relation to Wilson loops. It confirms that infrared divergences exponentiate at two loops but the finite part does not, and it reveals a uniform transcendental weight in the epsilon expansion. It analyzes gravity Wilson loops, finding a Christoffel-based definition fails to match the amplitude, but a metric-based Wilson loop in a conformal gauge reproduces the full one-loop result. The results illuminate connections with N=4 SYM results, AdS/CFT intuition, and suggest gauge choices that enable Wilson-loop approaches to gravity amplitudes, with implications for higher-loop extensions.

Abstract

Prompted by recent progress in the study of N=4 super Yang-Mills amplitudes, and evidence that similar approaches might be relevant to N=8 supergravity, we investigate possible iterative structures and applications of Wilson loop techniques in maximal supergravity. We first consider the two-loop, four-point MHV scattering amplitude in N=8 supergravity, confirming that the infrared divergent parts exponentiate, and we give the explicit expression which represents the failure for this to occur for the finite part. We observe that each term in the expansion of the one- and two-loop amplitudes in the dimensional regularisation parameter epsilon has a uniform degree of transcendentality. We then turn to consider Wilson loops in supergravity, showing that a natural definition of the loop, involving the Christoffel connection, fails to reproduce the one-loop amplitude. An alternative expression, which involves the metric explicitly, is shown to have a close relationship with the physical amplitude. We find that in a gauge in which the cusp diagrams vanish, the remaining diagrams for this Wilson loop correctly generate the full one-loop, four-point N=8 supergravity amplitude.

Figures (4)

• Figure 1: A one-loop correction to the Wilson loop bounded by momenta $p_1,\cdots , p_4$, where a graviton is exchanged between two lightlike momenta meeting at a cusp. Diagrams in this class generate infrared-divergent contributions to the four-point amplitude which, after summing over the appropriate permutations give rise to \ref{['MIR']}.
• Figure 2: Diagrams in this class, where a graviton stretches between two non-adjacent edges of the loop, are finite, and give in the four-point case a contribution equal to the finite part of the zero-mass box function $F^{(1)} (s,t)$ multiplied by $u$.
• Figure 3: A one-loop correction for a cusped contour. We show in the text that, when evaluated in the conformal gauge, the result of this diagram vanishes.
• Figure 4: A one-loop diagram where a gluon connects two non-adjacent segments. In the Feynman gauge employed in bht, the result of this diagram is equal to the finite part of a two-mass easy box function $F^{\rm 2me} (p, q, P, Q)$, where $p$ and $q$ are the massless legs of the two-mass easy box, and correspond to the segments which are connected by the gluon. In the conformal gauge, this diagram is equal to the full box function. The diagram depends on the other gluon momenta only through the combinations $P$ and $Q$. In this example, $P=p_3 + p_4$, $Q= p_6 + p_7 + p_1$.