Classifying causal nonlinear electrodynamics via φ-parity and irrelevant deformations

A bstract We investigate the classification of self-dual nonlinear electrodynamic (NED) theories based on their analyticity properties, which are directly linked to invariance under a discrete φ -parity transformation. This classification is expressed through the structure of the irrelevant TT̄ T T ¯ -like deformations that generate the theories from a Maxwell seed. Using both closed-form and perturbative methods within the Courant-Hilbert (CH) and Russo-Townsend auxiliary field formalisms, we demonstrate a precise correspondence: φ -parity-invariant, analytic theories are generated by irrelevant deformations built from integer powers of the energy-momentum tensor scalars, Oλ O λ ~ Σ C m ( T μν T μν ) 1− m ( T μ μ T ν ν ) m . Conversely, φ -parity-violating, non-analytic theories require deformations involving both integer and half-integer powers, Oλ O λ ~ Σ C m ( T μν T μν ) 1− m /2 ( T μ μ T ν ν ) m /2 . We prove this result in generality via a perturbative CH framework, showing that φ -parity invariance imposes specific constraints on the expansion coefficients of the CH function ℓ ( τ ) which, in turn, force all half-integer powers in the deformation to vanish. The classification is explicitly verified for known closed-form theories: the analytic generalized Born-Infeld model and the non-analytic examples of the q = 3/4-deformed and “no τ -maximum” theories. Furthermore, we show how the φ -parity transformation is consistently generalized in the presence of a marginal root- TT̄ T T ¯ coupling γ , and we derive the corresponding marginal and irrelevant flow equations for the studied theories.