CRITICAL (Phi^{4}_{3,ε})

Abstract

The Euclidean $(ϕ^{4})_{3,ε$ model in $R^3$ corresponds to a perturbation by a ${\varphi }^4$ 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 $\epsilon =0$ ${\varphi }^4$ 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.

49 pages, plain tex, macros included

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