Universal scaling dimensions for highly irrelevant operators in the Local Potential Approximation - Vlad-Mihai Mandric; Morris, Tim R; Stulga, Dalius (arXiv.org, 2023)
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 \(\mathcal{O}_n(φ)\) around a non-trivial fixed point. We find that \(d_n = n(d-d_φ)\) to leading order in \(n\) as \(n → ∞\), where \(d_φ=\frac{1}{2}(d-2+η)\) is the scaling dimension of the field, \(φ\), 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 \(N≥0\) and additionally for \(N=-2,-4,… \,.\) These results are universal, independent of the choice of cutoff function which we keep general throughout, subject only to some weak constraints.