Functional RG of higher derivative flows beyond the LPA approximation scheme

In this work we discuss the conceptual framework of a method developed in the last decades: the Functional Renormalization Group (FRG). The FRG encapsulates the typical approach of the Wilsonian RG to the problem of Renormalization, together with the Functional methods of QFT. We consider here the version given by Wetterich and Morris, where the fundamental object, the Effective Average Action, is a functional whose dynamics is governed by an exact non linear integro-differential flow equation, called indeed the Wetterich-Morris equation. Unfortunately, it is not possible to solve this equation exactly, so different approximation schemes are usually introduced and a truncation of the functional in some (infinite) basis of operators is typically employed.This thesis deals with a novel approximating framework which consists in trasforming the exact RG equation which is first order in the RG time into another exact equation of higher order in RG time, here at second order. When one applies a truncation in the so called derivative expansion the higher order equation contains therefore more information than the original one. The lowest order (LPA and LPA’) truncations are studied considering the Wilson-Fisher problem of the critical behaviour of a scalar field with Z2 symmetry in d=3 dimensions. The numerical analysis has been conducted with an extensive use of Chebyshev Pseudospectral method: a very powerful tool that allows to obtain a global convergence on the entire domain for the fixed point solutions.The so called LPA’ truncation is known to be inconsistent for the computation of the field anomalous dimension and its definition related to the flow of the Renormalization function Zk is also discussed. We consider this fact as a hint for the need of increasing the order of the truncation. Therefore, in the second part of this work the second order in Derivative Expansion for the higher order flow is considered, leaving the regulator undetermined.