CRITICAL (Phi^{4}_{3,\epsilon})

TL;DR

This paper investigates the critical behavior of the $(^{4})_{3, ext{epsilon}}$ model in three dimensions, establishing the existence of a non-Gaussian fixed point for small positive epsilon through Renormalization Group analysis.

Contribution

It proves the existence of a non-Gaussian fixed point for the Euclidean $(^{4})_{3, ext{epsilon}}$ model in three dimensions for small epsilon, advancing understanding of critical phenomena in quantum field theory.

Findings

Existence of a non-Gaussian fixed point for small epsilon

Convergence of RG iterations to the fixed point on its stable manifold

Construction of the critical manifold for the model

Abstract

The Euclidean $(\phi^{4})_{3,\epsilon$ model in $R^3$ corresponds to a perturbation by a $\phi^4$ interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter $\epsilon$ in the range $0\le \epsilon \le 1$. For $\epsilon =1$ one recovers the covariance of a massless scalar field in $R^3$. For $\epsilon =0$ $\phi^{4}$ is a marginal interaction. For $0\le \epsilon < 1$ the covariance continues to be Osterwalder-Schrader and pointwise positive. After introducing cutoffs we prove that for $\epsilon > 0$, sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. These iterations converge to the fixed point on its stable (critical) manifold which is constructed.