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

TL;DR

This paper analytically derives the universal scaling dimensions of highly irrelevant operators in scalar field theories within the Local Potential Approximation, revealing a linear growth pattern with respect to operator index n.

Contribution

It provides a universal analytical expression for the scaling dimensions of high-dimension operators in scalar field theories using Sturm-Liouville and WKB methods.

Findings

Scaling dimension $d_n$ grows linearly with n as $d_n = n(d - d_)$.

Results are universal, independent of cutoff function choice.

Scaling dimensions for $O(N)$ theories are double those of single-field cases.

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 $\mathcal{O}_n(\varphi)$ around a non-trivial fixed point. We find that $d_n = n(d-d_\varphi)$ to leading order in $n$ as $n \to \infty$, where $d_\varphi=\frac{1}{2}(d-2+\eta)$ is the scaling dimension of the field, $\varphi$, 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 $N\geq0$ and additionally for $N=-2,-4,\ldots \,.$ These results are universal, independent of the choice of cutoff function which we keep general throughout, subject only to some weak constraints.