On the U(1) duality anomaly and the S-matrix of N=4 supergravity

TL;DR

The paper demonstrates that a U(1) subgroup of the SU(1,1) duality in ${\cal N}=4$ supergravity is anomalous at one loop, linking the anomaly to the conformal supergravity sector via the superconformal formulation. It derives both local and nonlocal anomalous terms in the one-loop effective action and shows that this anomaly manifests in the S-matrix as one-loop amplitudes that violate the tree-level U(1) symmetry, with soft-scalar limits revealing a universal single-multiplet soft scalar function. By employing color/kinematics duality and the double-copy construction, the authors compute explicit anomalous amplitudes for theories with and without vector multiplets, and show the anomaly’s dependence on the number of vector multiplets ${n_{ m v}}$ through a simple scaling $k_{\rm PSG+VM}=6(2+n_{ m v})$. They further construct all-multiplicity anomalous amplitudes via inverse soft limits and relate the resulting local amplitude structures to the anomaly-induced effective action, demonstrating a concrete correspondence between amplitude data and the underlying duality anomaly. The work also outlines implications for higher-loop amplitudes and for related truncations of ${\cal N}=8$ supergravity, highlighting a potential link between duality anomalies and ultraviolet properties in theories with reduced supersymmetry.

Abstract

N=4 Poincare supergravity has a global SU(1,1) duality symmetry that acts manifestly only on shell as it involves duality rotations of vector fields. A U(1) subgroup of this symmetry is known to be anomalous at the quantum level in the presence of a non-trivial gravitational background. We first derive this anomaly from a novel perspective, by relating it to a similar anomaly in conformal supergravity where SU(1,1) acts off shell, using the fact that N=4 Poincare supergravity has a superconformal formulation. We explicitly construct the corresponding local and nonlocal anomalous terms in the one-loop effective action. We then study how this anomaly is reflected in the supergravity S-matrix. Calculating one-loop N=4 supergravity scattering amplitudes (with and without additional matter multiplets) using color/kinematics duality and the double-copy construction we find that a particular U(1) symmetry which was present in the tree-level amplitudes is broken at the quantum level. This breaking manifests itself in the appearance of new one-loop N=4 supergravity amplitudes that have non-vanishing soft-scalar limits (these amplitudes are absent in N>4 supergravities). We discuss the relation between these symmetry-violating amplitudes and the corresponding U(1) anomalous term in the one-loop supergravity effective action.

Figures (2)

• Figure 1: Cuts of the three- and four-loop four-graviton amplitude in ${{\cal N}=4}$ Poincaré supergravity which reveal the contribution of anomalous amplitudes.
• Figure 2: The two integral topologies that appear in the five-point one-loop amplitudes.