Recursion Relations for One-Loop Gravity Amplitudes

TL;DR

This work investigates the use of on-shell recursion, inspired by BCFW methods, to compute finite one-loop gravity amplitudes. It demonstrates that all-plus amplitudes for five- and six-point gravitons, as well as the four-graviton $-+++$ amplitude, can be derived from recursion using appropriate shifts and a gravity analogue of the one-loop three-point vertex, with explicit forms and consistency checks provided. A key result is the introduction of a one-loop graviton $+++$ vertex $W_3^{(1)}$ and its role in reconstructing the $-+++$ amplitude, including the interplay with gravity soft factors. However, the paper also highlights limitations: the $-++++$ gravity amplitude cannot be fully constructed with the current recursive framework due to nonstandard factorisations and double-pole structures, signaling a need for a deeper understanding of complex-momentum factorisation in gravity and possibly new recursion techniques. Overall, the results illustrate both the power of on-shell recursion for gravity in several finite cases and the challenges that remain for more intricate helicity configurations.

Abstract

We study the application of recursion relations to the calculation of finite one-loop gravity amplitudes. It is shown explicitly that the known four, five, and six graviton one-loop amplitudes for which the external legs have identical outgoing helicities, and the four graviton amplitude with helicities (-,+,+,+) can be derived from simple recursion relations. The latter amplitude is derived by introducing a one-loop three-point vertex of gravitons of positive helicity, which is the counterpart in gravity of the one-loop three-plus vertex in Yang-Mills. We show that new issues arise for the five point amplitude with helicities (-,+,+,+,+), where the application of known methods does not appear to work, and we discuss possible resolutions.

Figures (12)

• Figure 1: The diagram in the recursive expression for $M_5^{(1)}(1^+,2^+,3^+,4^+,5^+)$ associated with the pole $\langle\hat{1}\hat{2}\rangle=0$. The amplitude labelled by a T is a tree-level amplitude and the one labelled by L is a one-loop amplitude.
• Figure 2: The diagram in the recursive expression for $M_5^{(1)}(1^+,2^+,3^+,4^+,5^+)$ associated with the pole $\langle\hat{1}5\rangle=0$.
• Figure 3: The diagram in the recursive expression for $M_5^{(1)}(1^+,2^+,3^+,4^+,5^+,6^+)$ associated with the pole $\langle\hat{2}\hat{3}\rangle=0$.
• Figure 4: The diagram in the recursive expression for $M_5^{(1)}(1^+,2^+,3^+,4^+,5^+,6^+)$ associated with the pole $\langle\hat{1}6\rangle=0$.
• Figure 5: The diagram in the recursive expression for $A_4^{(1)}(1^-,2^+,3^+,4^+)$.