Simplified self-dual electrodynamics

TL;DR

The paper develops a unified auxiliary-field framework for self-dual nonlinear electrodynamics by introducing a scalar field $\phi$ with potential $W(\phi)$, yielding a Lagrangian $\mathcal{L}(S,P;\phi)=(\cosh\phi)S+(\sinh\phi)\sqrt{S^2+P^2}-W(\phi)$ and a corresponding Hamiltonian derived via a Legendre transform. Causality imposes simple, strong constraints on the CH-function and the $W$-potential, guaranteeing a unique solution for the auxiliary field and enabling a full analytic treatment; in particular, the condition $W(\phi)$ even in $\phi$ selects the analytic (i.e., $S$- and $P^2$-analytic) subclass, with ties to a generalized Roek-Tseytlin (BI) construction. The authors establish a general, constructive map between Lagrangian and Hamiltonian pictures, present explicit causal examples (including a constant-$A_-/A_+$ case leading to MMB/BI), and derive energy conditions that follow from causality. The framework generalizes known theories (e.g., Born–Infeld, ModMax) while offering a versatile route to new analytic, self-dual NLED models, with potential extensions to supersymmetry, higher-dimensional chiral forms, and $T\bar{T}$-like deformations.

Abstract

We present a new formulation of self-dual nonlinear electrodynamics in which interactions are determined by an auxiliary-field potential, with causality ensuring a unique solution to the auxiliary-field equation. The long-standing problem of an explicit Lagrangian for the generic `analytic' theory is simply solved by restriction to potentials that are even functions of the auxiliary field. In this case the Lagrangian can be linearised in quadratic field-strength scalars by the introduction of an additional pseudoscalar auxiliary field; this generalises, to all analytic self-dual theories, a well-known construction of the Born-Infeld theory.