Counterterms in type I Supergravities

TL;DR

The paper analyzes the ultraviolet behavior of D=10, N=1 supergravity and its dimensional reductions by computing one-loop four-graviton amplitudes. It shows that beyond D=8, divergences arise in the form of $R^4$ invariants in D=8 and $partial^2R^4$ invariants in D=10, with explicit tensor structures and factorization into left and right moving factors that mirror string theory. The authors derive the complete counterterm structures for various supermultiplets, connect them to $t_8$-based invariants, and discuss their interpretation in the context of string theory and the gravity–Yang–Mills double copy. The results illuminate how supersymmetry and dimensional regularization constrain one-loop UV behavior in extended supergravities and reveal a deep connection between field theory and string-inspired invariants.

Abstract

We compute the one-loop divergences of D=10, N=1 supergravity and of its reduction to D=8. We study the tensor structure of the counterterms appearing in D=8 and D=10 and compare these to expressions previously found in the low energy expansion of string theory. The infinities have the primitive Yang-Mills tree amplitude as a common factor.

Figures (4)

• Figure 1: Helicity configuration for the $M(1^-,2^-,3^+,4^+)$$s$-channel cut
• Figure 2: Boxes which contribute to the $M^{1-loop}(1^-,2^-,3^+,4^+)$ calculation.
• Figure 3: Helicity configurations for the $M(1^-,2^-,3^+,4^+)$$t$-channel cut
• Figure 4: Boxes and triangles contributing to $M(1^-,2^-,3^+,4^+)$$t$-channel cut