Curvature-Squared Multiplets, Evanescent Effects and the U(1) Anomaly in N = 4 Supergravity

TL;DR

This work analyzes the one-loop four-point amplitude of $\mathcal{N}=4$ supergravity in $D$ dimensions via a BCJ color-kinematics double-copy construction, uncovering evanescent curvature-squared contributions and their link to the $U(1)$ duality anomaly. By building two gauge-invariant tensor bases and associated projectors, the authors map YM and MSYM structures to gravity, obtaining an explicit amplitude expression that separates an infrared divergent piece, a finite $R^2$-type term, and curvature-squared- or $F^3$-driven components. A key finding is that evanescent curvature-squared matrix elements have finite coefficients in this theory and are intertwined with the anomaly, implying nontrivial implications for higher-loop ultraviolet properties. The results also establish the existence of $\mathcal{N}=4$ curvature-squared multiplets in $D\le 10$ through the double copy, while arguing that $\mathcal{N}\ge 5$ supersymmetry does not permit such completions. The accompanying tensor bases and projectors provide broadly applicable tools for analyzing four-gluon amplitudes in gauge theories and their gravity counterparts.

Abstract

We evaluate one-loop amplitudes of N = 4 supergravity in D dimensions using the double-copy procedure that expresses gravity integrands in terms of corresponding ones in Yang--Mills theory.We organize the calculation in terms of a set of gauge-invariant tensors, allowing us to identify evanescent contributions. Among the latter, we find the matrix elements of supersymmetric completions of curvature-squared operators. In addition, we find that such evanescent terms and the U(1)-anomalous contributions to one-loop N = 4 amplitudes are tightly intertwined. The appearance of evanescent operators in N = 4 supergravity and their relation to anomalies raises the question of their effect on the known four-loop divergence in this theory. We provide bases of gauge-invariant tensors and corresponding projectors useful for Yang--Mills theories as a by-product of our analysis.

Figures (4)

• Figure 1: Box diagrams of the one-loop four-point amplitude of ${\mathcal{N}=4}$ supergravity.
• Figure 2: Representative diagrams for (a) three- and (b) four-point $F^3$ insertions.
• Figure 3: Representative diagram for the insertion of the evanescent $R^2$ counterterm, affecting the two-loop divergence in pure-graviton amplitudes GBPureGravity.
• Figure 4: Representative diagrams for insertions of the supersymmetric $R^2$ operator at three loops that could affect the four-loop divergence.