Classifying Causal Nonlinear Electrodynamics via $\varphi$-Parity and Irrelevant Deformations

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 $\varphi$-parity transformation. This classification is expressed through the structure of the irrelevant $T\bar{T}$-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: $\varphi$-parity-invariant, analytic theories are generated by irrelevant deformations built from integer powers of the energy-momentum tensor scalars, $\mathcal{O}_\lambda \sim \sum C_m (T_{\mu\nu}T^{\mu\nu})^{1-m}({T_{\mu}}^{\mu}{T_{\nu}}^{\nu})^{m}$. Conversely, $\varphi$-parity-violating, non-analytic theories require deformations involving both integer and half-integer powers, $\mathcal{O}_\lambda \sim \sum C_m (T_{\mu\nu}T^{\mu\nu})^{1-m/2}({T_{\mu}}^{\mu}{T_{\nu}}^{\nu})^{m/2}$. We prove this result in generality via a perturbative CH framework, showing that $\varphi$-parity invariance imposes specific constraints on the expansion coefficients of the CH function $\ell(\tau)$ 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 $\tau$-maximum" theories. Furthermore, we show how the $\varphi$-parity transformation is consistently generalized in the presence of a marginal root-$T\bar{T}$ coupling $\gamma$, and we derive the corresponding marginal and irrelevant flow equations for the studied theories.