The self-dual condition, which ensures invariance under electromagnetic duality, manifests as a partial differential equation in nonlinear electromagnetism theories. The general solution to this equation is expressed in terms of an auxiliary field, τ, and Courant-Hilbert functions, (τ), which depend on τ. Recent studies have shown that duality-invariant nonlinear electromagnetic theories fulfill the principle of causality under the conditions τ 1 and ² τ² 0. In this paper, we investigate theories with two coupling constants that also comply with the principle of causality. We demonstrate that these theories possess a new universal representation of the root-TT operator. Additionally, we derive marginal and irrelevant flow equations for the logarithmic causal self-dual electrodynamics and identify a symmetry referred to as α-symmetry, which is present in all these models.
Babaei-Aghbolagh et al. (Mon,) studied this question.