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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 $L(F_{μν};g)$ depending on a duality-invariant parameter $g$; and (ii) an arbitrary potential $W(ψ)$, with $ψ$ an auxiliary scalar field. It turns out that the model $\mathfrak{L}(F_{μν};ψ) = L(F_{μν};ψ) + W(ψ)$ leads to a self-dual theory for nonlinear electrodynamics upon elimination of $ψ$. As an illustration, we work out two examples in which the seed Lagrangian $L(F_{μν};g)$ corresponds to the Born-Infeld model and two particular potentials $W(ψ)$ are chosen such that integrating out $ψ$ gives: (i) the ModMaxBorn theory; and (ii) the ModMax theory. We also briefly discuss supersymmetric generalisations of the proposed formulation.