Universal scaling dimensions for highly irrelevant operators in the local potential approximation

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 Schrödinger-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 O_n(φ ) around a nontrivial fixed point. We find that d_n=n (d -d_φ) to leading order in n as n →∞ , where d_φ=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.