Non perturbative renormalisation group and momentum dependence of $n$-point functions (I)

TL;DR

This paper introduces an approximation scheme for solving non-perturbative renormalization group equations to obtain full momentum dependence of n-point functions, applied to the O(N) model and tested on Bose gas transition temperature shifts.

Contribution

It develops an iterative approximation method exploiting derivative expansion and mode decoupling to accurately compute momentum-dependent n-point functions.

Findings

Accurate self-energy in leading order across regimes

Agreement with lattice results for Bose gas transition temperature shift

Theoretical uncertainty around 25% in results

Abstract

We present an approximation scheme to solve the Non Perturbative Renormalization Group equations and obtain the full momentum dependence of the $n$-point functions. It is based on an iterative procedure where, in a first step, an initial ansatz for the $n$-point functions is constructed by solving approximate flow equations derived from well motivated approximations. These approximations exploit the derivative expansion and the decoupling of high momentum modes. The method is applied to the O($N$) model. In leading order, the self energy is already accurate both in the perturbative and the scaling regimes. A stringent test is provided by the calculation of the shift $\Delta T_c$ in the transition temperature of the weakly repulsive Bose gas, a quantity which is particularly sensitive to all momentum scales. The leading order result is in agreement with lattice calculations, albeit with a theoretical uncertainty of about 25%.