The n-point MHV one-loop Amplitude in N=4 Supergravity

TL;DR

This paper derives an explicit all-$n$ expression for the rational part $R_n$ of the $n$-point MHV one-loop amplitude in $ ext{N}=4$ supergravity by exploiting soft and collinear factorisations. The full amplitude is written as $M^{N=4}_n = M^{N=8}_n - 4 M^{N=6, ext{matter}}_n + 2 M^{N=4, ext{matter}}_n$ with the rational piece decomposed as $R_n = (-1)^n rac{raket{m_1 m_2}^4}{2} (R_n^0 + extstyleigsum_{r=3}^{n-2} R_n^r)$, where $R_n^0$ collects box-integral–driven terms and $R_n^r$ encodes higher-subset structures via $C_r$ and $hat{S}^{n-2-r}$. The construction leverages gravity-specific soft/collinear factorisation, including explicit splitting functions and half-soft functions, and provides an alternative description in terms of $hat{S}$ and $hat{C}$ to ensure correct limits. The authors verify soft and collinear constraints (with numerical checks up to $n=10$) and argue that this all-$n$ rational piece satisfies the necessary limits and symmetries, contributing a crucial explicit example to the otherwise sparse all-$n$ loop results in gravity. Overall, this work advances understanding of the perturbative structure of quantum gravity and informs gauge–gravity duality considerations by supplying a concrete, testable all-$n$ amplitude.

Abstract

We present an explicit formula for the n-point MHV one-loop amplitude in a N=4 supergravity theory. This formula is derived from the soft and collinear factorisations of the amplitude.