Scaling Behaviors in Active Model B+ via the Functional Renormalization Group

We study the scaling behaviors of the active model B+ using the functional renormalization group (FRG) approach, based on the nonequilibrium effective action formulated via the Martin-Siggia-Rose path-integral formalism. We derive the $\beta$ functions for all couplings of the system in generic $d$ dimensions, revealing regulator independence in various contributions to the renormalization group (RG) flow at specific values for $d$. After identifying specific regions of the parameter space that define submodels closed under RG transformations, we determine all fixed points of potential physical relevance. We confirm the existence of a bicritical fixed point, which was conjectured within the perturbative momentum-shell RG method for being responsible for the transition from bulk phase separation to microphase separation in active systems. We argue that, within the FRG approach, global flows significantly differ from those obtained in its perturbative counterpart.