The D^{2k} R^4 Invariants of N=8 Supergravity

TL;DR

This work constructs the complete linearized ${\cal N}=8$ SUSY completions of higher-derivative invariants ${D^{2k}R^4}$ in four-dimensional supergravity by deriving the 4-point MHV superamplitudes and extracting a detailed component expansion. The main achievement is the explicit 51-term expansion of the ${R^4}$ invariant, built from SU(8) invariants and guaranteed to be linearized SUSY; higher invariants ${D^{2k}R^4}$ are obtained by applying symmetric polynomials in Mandelstam variables, translating to spacetime derivatives on the same operator basis. The method clarifies how linearized SUSY, spinor-helicity, and SU(8) structure constrain the possible terms, and it provides a practical framework for assessing UV properties and potential counterterms in ${\cal N}=8$ supergravity, including connections to Bel-Robinson tensor forms and classical superspace results. The results offer a concrete, extensible toolkit for exploring higher-derivative invariants and their implications for ultraviolet behavior in maximal supergravity.

Abstract

The existence of a linearized SUSY invariant for N=8 supergravity whose gravitational components are usually called R^4 was established long ago by on-shell superspace arguments. Superspace and string theory methods have also established analogous higher dimensional D^{2k} R^4 invariants. However, very little is known about the SUSY completions of these operators which involve other fields of the theory. In this paper we find the detailed component expansion of the linearized R^4 invariant starting from the corresponding superamplitude which generates all component matrix elements of the operator. It is then quite straightforward to extend results to the entire set of D^{2k} R^4 operators.