Causal self-dual nonlinear electrodynamics from the Born-Infeld theory

Recently we have proposed a new auxiliary-field formulation for self-dual nonlinear electrodynamics (NLED) which makes use of two building blocks: (i) a seed self-dual theory $L(F_{\mu\nu};g)$, where $F_{\mu \nu}$ is the electromagnetic field strength and $g$ a duality-invariant coupling constant; and (ii) a scalar potential $W(\psi)$. Our formulation is based on the Lagrangian $\mathfrak{L}(F_{\mu\nu};\psi) = L(F_{\mu\nu};\psi) + W(\psi)$, where $\psi$ is an auxiliary scalar field. Integrating out $\psi$, using its equation of motion, one obtains a $\mathsf{U}(1)$ duality-invariant NLED. Different self-dual NLEDs are derived by choosing different potentials $W(\psi)$. In the case that the seed Lagrangian defines the Born-Infeld theory, in this paper we demonstrate that the resulting models for self-dual NLED are causal and provide a general solution of the self-duality equation. We also elaborate on the procedure to relate our formulation to that developed by Russo and Townsend.