Renormalized $g-log (g)$ double expansion for the invariant $\phi^4$-trajectory in three dimensions

TL;DR

This paper investigates the invariant unstable manifold related to the $eta$-function of the $ ext{phi}^4$ theory in three dimensions, introducing a renormalized double expansion involving the coupling and its logarithm.

Contribution

It presents a novel renormalized double expansion framework for analyzing the $ ext{phi}^4$-trajectory near the fixed point in three dimensions.

Findings

Established a parametrization of the unstable manifold using a running coupling.

Derived a linear $eta$-function for the coupling.

Developed a renormalized double expansion involving the coupling and its logarithm.

Abstract

We study the invariant unstable manifold of the trivial renormalization group fixed point tangent to the $\phi^{4}$-vertex in three dimensions. We parametrize it by a running $\phi^{4}$-coupling with linear step $\beta$-function. It is shown to have a renormalized double expansion in the running coupling and its logarithm.