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

• Published in: arXiv.org
• Main Authors: Vlad-Mihai Mandric, Morris, Tim R, Stulga, Dalius
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 14.11.2023
• Subjects: Approximation; Field theory; Fixed points (mathematics); Mathematical analysis; Operators (mathematics); Physics - High Energy Physics - Theory; Scalars; Scaling; Schrodinger equation
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2306.14643

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.