Exact Stress Tensor Flows in Causal Electrodynamics: Auxiliary Field Formalism

Causal self-dual nonlinear electrodynamics admits a simple auxiliary-field description: the Lagrangian is linear in the two Lorentz invariants of the field strength, and the dynamics is encoded in a potential $W(φ)$ of a single auxiliary variable. We study non-perturbative $T\bar T$-like flows of these theories in this formalism. The irrelevant (stress-tensor) deformation acts on the potential by a rescaling, while the marginal ModMax-type deformation translates its vacuum. We derive the resulting flow equations exactly, show that the irrelevant exact $T\bar T$-like flow takes the autonomous form $\partial_λ\mathcal L=X\,F(r)$, where the closed form kernel $F$ depends only on the ratio $r=-T^μ_μ/\sqrt{4X-Y}$ of stress-tensor invariants. We also give an explicit reconstruction of the auxiliary potential from a given kernel. Causality imposes the differential inequality $D(r)>1/(1-r)$, while a globally single-valued kernel requires the ratio $r$ to be monotonic along the auxiliary branch. We examine the relation between the evenness of the exact $T\bar T$-like flow , the $φ$-parity of the potential, and the analyticity of the Lagrangian in the electromagnetic invariants, including the effect of a marginal ModMax parameter. We construct the linear auxiliary and Hamiltonian formulations and show that the Hamiltonian flow is governed by the same exact $T\bar T$-like flow with the opposite sign. We also express Born's Legendre self-duality, the two characteristic cones, and the energy conditions in terms of the kernel. In particular, $r=(\varepsilon-p_\perp)/(\varepsilon+p_\perp)$ has a direct physical interpretation. Exact $T\bar T$-like flow s are obtained for generalized Born-Infeld, logarithmic, q-deformed, and no-$τ$-maximum electrodynamics.