Generalisations of the Russo-Townsend formulation

As a generalisation of the recent construction by Russo and Townsend, we propose a new approach to generate $\mathsf{U}(1)$ duality-invariant models for nonlinear electrodynamics. It is based on the use of two building blocks: (i) a fixed (but otherwise arbitrary) model for self-dual nonlinear electrodynamics with Lagrangian $L(F_{\mu\nu};g)$ depending on a duality-invariant parameter $g$; and (ii) an arbitrary potential $W(\psi)$, with $\psi$ an auxiliary scalar field. It turns out that the model $\mathfrak{L}(F_{\mu\nu};\psi) = L(F_{\mu\nu};\psi) + W(\psi)$ leads to a self-dual theory for nonlinear electrodynamics upon elimination of $\psi$. As an illustration, we work out two examples in which the seed Lagrangian $L(F_{\mu\nu};g)$ corresponds to the Born-Infeld model and two particular potentials $W(\psi)$ are chosen such that integrating out $\psi$ gives: (i) the ModMaxBorn theory; and (ii) the ModMax theory. We also briefly discuss supersymmetric generalisations of the proposed formulation.