Piotr Zdybel
Active scalar field theories break detailed balance in two physically distinct ways: through a field-dependent noise-to-mobility ratio $Θ(φ)=D(φ)/M(φ)$, and through the gradient activity of Active Model~B+. We classify the resulting perturbations of the conserved Wilson-Fisher fixed point in three dimensions. Working from the Martin-Siggia-Rose-Janssen-De~Dominicis (MSRJD) action, a functional renormalization-group (FRG) calculation shows that the transport sector has a block-triangular stability matrix whose leading odd mode has $y_{Θ_1}=-Δ_φ$, while quotienting the gradient sector by its detailed-balance-preserving tangent leaves a chemical and a current-like mode. Two smooth regulators give $y_{Θ_1}\simeq-0.52$, $y_J\simeq-0.56$ and $y_{\rm ch}\simeq-0.89$. The translation Ward identity represents the current operator by a stress-tensor divergence modulo chemical gradients; conservation and causality then make the active stability matrix triangular to all loop orders, and an explicit two-loop calculation in $d=4-ε$ finds no additional contact counterterm. The two slowest modes are separated only by the Ising anomalous dimension, $y_J-y_{Θ_1}=-η$. This relation is a dimension count that holds provided neither operator acquires a further contact anomaly, a condition we establish within the present truncation for $\mathcal O_Θ$ and verify through two loops for $\mathcal O_J$; under it the $d=3$ reference Ising value gives $y_{Θ_1}\simeq-0.5181$ and $y_J\simeq-0.5544$. We derive the parity-resolved finite-size scaling of this near degeneracy, identify block observables that separate the two modes, and show that when the current amplitude dominates the crossover scale lies far beyond any accessible system size.