A renormalization group improved computation of correlation functions in theories with non-trivial phase diagram

We present a simple and consistent way to compute correlation functions in interacting theories with non-trivial phase diagram. As an example we show how to consistently compute the four-point function in three dimensional $\mathbb{Z}_2$-scalar theories. The idea is to perform the path integral by weighting the momentum modes that contribute to it according to their renormalization group (RG) relevance, i.e. we weight each mode according to the value of the running couplings at that scale. In this way, we are able encode in a loop computation the information regarding the RG trajectory along which we are integrating. We show that depending on the initial condition, or initial point in the phase diagram, we obtain different behaviors of the four-point function at the end point of the flow.