EFT Asymptotics: the Growth of Operator Degeneracy

TL;DR

This paper derives asymptotic formulas for the growth of operator degeneracy in effective field theories across dimensions, using Hilbert series and generalised theorems, with strong numerical validation.

Contribution

It introduces a generalised theorem and applies it to Hilbert series to analytically estimate operator growth in EFTs, including complex theories like the Standard Model EFT.

Findings

Asymptotic formulas match low-order numerical results

Identification of phase transition-like behavior in Hilbert series

Potential for tighter bounds and rigorous error estimates

Abstract

We establish formulae for the asymptotic growth (with respect to the scaling dimension) of the number of operators in effective field theory, or equivalently the number of $S$-matrix elements, in arbitrary spacetime dimensions and with generic field content. This we achieve by generalising a theorem due to Meinardus and applying it to Hilbert series -- partition functions for the degeneracy of (subsets of) operators. Although our formulae are asymptotic, numerical experiments reveal remarkable agreement with exact results at very low orders in the EFT expansion, including for complicated phenomenological theories such as the standard model EFT. Our methods also reveal phase transition-like behaviour in Hilbert series. We discuss prospects for tightening the bounds and providing rigorous errors to the growth of operator degeneracy, and of extending the analytic study and utility of Hilbert series to EFT.