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

TL;DR

The paper establishes a precise link between analyticity in self-dual nonlinear electrodynamics and a discrete $\varphi$-parity symmetry, showing that analytic theories are generated from Maxwell by irrelevant $T\bar{C}$-like deformations built from integer powers of the EMT invariants $X=T_{\mu\nu}T^{\mu\nu}$ and $Y={T_{\mu}}^{\mu}{T_{\nu}}^{\nu}$, while non-analytic, φ-parity-violating theories require deformations with both integer and half-integer powers. This classification is proven generally using a perturbative Courant–Hilbert framework and corroborated by closed-form examples (analytic generalized Born–Infeld; non-analytic $q=3/4$-deformed and no $\tau$-maximum theories). The auxiliary-field (Russo–Townsend) formulation clarifies the role of $\varphi$-parity through a master potential $\Omega(y)$ and its inversion symmetry, and the analysis extends to a marginal root-$T\bar{T}$ coupling $\gamma$, yielding explicit marginal and irrelevant flow equations. A perturbative CH construction demonstrates how parity-invariant theories impose coefficient constraints that remove half-integer powers, thus selecting analytic, integer-power deformations and offering a structured path to discover new analytic NEDs. The results deepen the understanding of duality, causality, and deformation dynamics in higher-dimensional TT-like deformations and suggest avenues for supersymmetric embeddings and RG analyses.

Abstract

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, $\mathcal{O}_λ\sim \sum C_m (T_{μν}T^{μν})^{1-m}({T_μ}^μ{T_ν}^ν)^{m}$. Conversely, $\varphi$-parity-violating, non-analytic theories require deformations involving both integer and half-integer powers, $\mathcal{O}_λ\sim \sum C_m (T_{μν}T^{μν})^{1-m/2}({T_μ}^μ{T_ν}^ν)^{m/2}$. We prove this result in generality via a perturbative CH framework, showing that $\varphi$-parity invariance imposes specific constraints on the expansion coefficients of the CH function $\ell(τ)$ 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 $\varphi$-parity transformation is consistently generalized in the presence of a marginal root-$T\bar{T}$ coupling $γ$, and we derive the corresponding marginal and irrelevant flow equations for the studied theories.