Even-point Multi-loop Unitarity and its Applications: Exponentiation, Anomalies and Evanescence

Abstract

We identify novel structure in newly computed multi-loop amplitudes and quantum actions for even-point effective field theories, including both the nonlinear sigma model (NLSM) and double-copy gauge theories such as Born-Infeld and its supersymmetric generalizations. We exploit special properties of all even-point theories towards efficient unitarity based amplitude construction. In doing so, we find evidence that the leading IR divergence of NLSM amplitudes exponentiates when the symmetry group is $\mathbb{CP}^1\cong SU(2)/U(1)$. We then systematically compute the two-loop anomalous behavior of Born-Infeld, and find that the counterterms needed to restore $U(1)$ invariant behavior at loop-level can be constructed via a symmetric-structure double-copy. We also demonstrate that the divergent part of the one-minus $(-+++)$ two-loop anomaly vanishes upon introducing an evanescent operator. In addition to these pure photon counterterms, we verify through explicit calculation that the anomalous matrix elements that violate $U(1)$ duality invariance can be alternatively cancelled by summing over internal $\mathcal{N}=4$ DBIVA superfields. Finally we find that $\mathcal{N}=4$ Dirac-Born-Infeld-Volkov-Akulov (DBIVA) amplitudes permit double-copy construction through two-loop order by reproducing our unitarity based result with a double copy between color-dual $\mathcal{N}=4$ super-Yang-Mills and our two-loop NLSM amplitudes. This result supports the possibility of color-dual representations for NLSM beyond one-loop. We conclude with an overview of how $D$-dimensional four-photon counterterms can be constructed in generality with the symmetric-structure double-copy, and outline a convenient way of counting evanescent operators using Hilbert series as generating functions.