Universal scaling dimensions for highly irrelevant operators in the Local Potential Approximation

Abstract

We study $d$-dimensional scalar field theory in the Local Potential Approximation of the functional renormalization group. Sturm-Liouville methods allow the eigenoperator equation to be cast as a Schrodinger-type equation. Combining solutions in the large field limit with the Wentzel-Kramers-Brillouin approximation, we solve analytically for the scaling dimension $d_n$ of high dimension potential-type operators ${\mathcal{O}}_n(\phi )$ around a non-trivial fixed point. We find that $d_n=n(d-d_{\phi })$ to leading order in $n$ as $n\to \infty$, where $d_{\phi }=\frac{1}{2}(d-2+\eta )$ is the scaling dimension of the field, $\phi$, and determine the power-law growth of the subleading correction. For $O(N)$ invariant scalar field theory, the scaling dimension is just double this, for all fixed $N\ge 0$ and additionally for $N=-2,-4,\dots .$ These results are universal, independent of the choice of cutoff function which we keep general throughout, subject only to some weak constraints.

arXiv comment23 pages, no figures. Clarifications added. Version published in PRD

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