One-Loop n-Point Helicity Amplitudes in (Self-Dual) Gravity

TL;DR

Bern, Dixon, Perelstein, and Rozowsky derive an explicit ansatz for all one-loop, n-point all-plus gravity amplitudes using a half-soft function framework inspired by self-dual gravity. The amplitudes are finite, rational, and respect soft and collinear limits, with explicit validation against D-dimensional unitarity for n ≤ 6. The work draws deep connections to analogous all-plus Yang-Mills amplitudes via KLT relations, suggesting a universal structure linking gravity and gauge theories in their self-dual sectors. It also discusses the limitations of current proofs for all n and explores implications for string theory and potential anomalies, pointing to avenues for future research. Overall, it provides a concrete, symmetry-driven construction that illuminates the loop-level structure of gravity amplitudes and their gauge-theory correspondences.

Abstract

We present an ansatz for all one-loop amplitudes in pure Einstein gravity for which the n external gravitons have the same outgoing helicity. These loop amplitudes, which are rational functions of the momenta, also arise in the quantization of self-dual gravity in four-dimensional Minkowski space. Our ansatz agrees with explicit computations via D-dimensional unitarity cuts for n less than or equal to 6. It also has the expected analytic behavior, for all n, as a graviton becomes soft, and as two momenta become collinear. The gravity results are closely related to analogous amplitudes in (self-dual) Yang-Mills theory.

Figures (2)

• Figure 1: The configurations of external legs that are summed over in eq. (\ref{['AllPlusSimple']}).
• Figure 2: As two momenta become collinear the gravity $S$-matrix develops a phase singularity which can be detected by rotating the two momenta about the axis formed by their sum.