Ultraviolet Cancellations in Half-Maximal Supergravity as a Consequence of the Double-Copy Structure

TL;DR

This work analyzes ultraviolet cancellations in half-maximal supergravity through BCJ color–kinematics duality and the gravity–gauge double-copy framework. By relating gravity divergences to forbidden color structures in pure Yang–Mills amplitudes, it proves four- and five-point one-loop finiteness for $D<8$ and two-loop finiteness for $D<6$, while providing explicit divergences in select dimensions. A central finding is that cancellations arise when gauge-theory divergences lack certain color tensors, mirroring gravity finiteness, which substantiates a deep gauge–gravity correspondence. The explicit one- and two-loop results, including $D=8$ one-loop and $D=6$ two-loop divergences, constrain possible counterterms and guide future higher-loop investigations in theories with 16 or more supercharges.

Abstract

We show that the double-copy structure of gravity forbids divergences in pure half-maximal (16 supercharge) supergravity at four and five points at one loop in D<8 and at two loops in D<6. We link the cancellations that render these supergravity amplitudes finite to corresponding ones that eliminate forbidden color factors from the divergences of pure nonsupersymmetric Yang-Mills theory. The vanishing of the two-loop four-point divergence in D=5 half-maximal supergravity is an example where a valid counterterm satisfying the known symmetries exists, yet is not present. We also give explicit forms of divergences in half-maximal supergravity at one loop in D=8 and at two loops in D=6.

Figures (6)

• Figure 1: Diagram (a) specifies the four-point color factor $c^{(1)}_{1234}$ used in eq. (\ref{['OneLoopFourPtGauge']}), and diagram (b) specifies the color factor $c^{(1)}_{12345}$ in eq. (\ref{['OneLoopFivePtGauge']}). Diagram (a) and its permutations appear in the four-point amplitude of maximal super-Yang-Mills theory. At five points both (b) and (c) and their permutations appear.
• Figure 2: The two-loop planar and nonplanar double-box diagrams.
• Figure 3: The four-point diagrams generated by the $F^3$ counterterm in pure Yang-Mills at one loop $D=6$ or at two loops in $D=5$. The large dot indicate an insertion of a counterterm vertex, while a vertex without a dot represents an ordinary Yang-Mills vertex. In (a) a three-point counterterm vertex appears while in (b) a four-point counterterm vertex appears.
• Figure 4: The counterterm diagrams describing the one-loop divergences of either pure Yang-Mills theory or half-maximal supergravity in $D=8$. The large dots indicate an insertion of a counterterm vertex generated by either an $F^4$ operator in Yang-Mills theory or an $R^4$ operator in supergravity.
• Figure 5: Diagrams contributing to the five-point two-loop amplitude of maximal super-Yang-Mills theory. From ref. BCJ5Point.