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We identify novel structure in newly computed multi-loop amplitudes andquantum actions for even-point effective field theories, including both thenonlinear sigma model (NLSM) and double-copy gauge theories such as Born-Infeldand its supersymmetric generalizations. We exploit special properties of alleven-point theories towards efficient unitarity based amplitude construction.In doing so, we find evidence that the leading IR divergence of NLSM amplitudesexponentiates when the symmetry group is $\mathbb{CP}^1\cong SU(2)/U(1)$. Wethen systematically compute the two-loop anomalous behavior of Born-Infeld, andfind that the counterterms needed to restore $U(1)$ invariant behavior atloop-level can be constructed via a symmetric-structure double-copy. We alsodemonstrate that the divergent part of the one-minus $(-+++)$ two-loop anomalyvanishes upon introducing an evanescent operator. In addition to these purephoton counterterms, we verify through explicit calculation that the anomalousmatrix elements that violate $U(1)$ duality invariance can be alternativelycancelled by summing over internal $\mathcal{N}=4$ DBIVA superfields. Finallywe find that $\mathcal{N}=4$ Dirac-Born-Infeld-Volkov-Akulov (DBIVA) amplitudespermit double-copy construction through two-loop order by reproducing ourunitarity based result with a double copy between color-dual $\mathcal{N}=4$super-Yang-Mills and our two-loop NLSM amplitudes. This result supports thepossibility of color-dual representations for NLSM beyond one-loop. We concludewith an overview of how $D$-dimensional four-photon counterterms can beconstructed in generality with the symmetric-structure double-copy, and outlinea convenient way of counting evanescent operators using Hilbert series asgenerating functions.

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