Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

EFT Asymptotics: the Growth of Operator Degeneracy

Tom Melia, Sridip Pal

TL;DR

The paper develops analytic, asymptotic formulae for the growth of EFT operator degeneracy as a function of scaling dimension by mapping operator counting to Hilbert-series generating functions and applying Meinardus' theorem. It unifies 2D modular methods with higher-dimensional, spinful, and gauge-extended EFT content, introducing a practical trick to extract leading behavior and validating it against plane partitions and SMEFT data. The main contributions include bosonic and fermionic Meinardus theorems in arbitrary dimensions, explicit leading exponents for operator growth, and a detailed treatment of Hilbert-series projections (spin, IBP, and internal singlets) with phase-transition-like saddle subtleties. The SMEFT application demonstrates remarkable agreement with exact counts at low mass dimensions and provides a concrete framework for estimating the growth of independent EFT measurements at high orders, informing phenomenology and high-precision EFT program design.

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.

EFT Asymptotics: the Growth of Operator Degeneracy

TL;DR

The paper develops analytic, asymptotic formulae for the growth of EFT operator degeneracy as a function of scaling dimension by mapping operator counting to Hilbert-series generating functions and applying Meinardus' theorem. It unifies 2D modular methods with higher-dimensional, spinful, and gauge-extended EFT content, introducing a practical trick to extract leading behavior and validating it against plane partitions and SMEFT data. The main contributions include bosonic and fermionic Meinardus theorems in arbitrary dimensions, explicit leading exponents for operator growth, and a detailed treatment of Hilbert-series projections (spin, IBP, and internal singlets) with phase-transition-like saddle subtleties. The SMEFT application demonstrates remarkable agreement with exact counts at low mass dimensions and provides a concrete framework for estimating the growth of independent EFT measurements at high orders, informing phenomenology and high-precision EFT program design.

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 -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.

Paper Structure

This paper contains 31 sections, 199 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Rigour in two dimensions
  3. Analysis via modular properties of the Dedekind eta function
  4. Analysis via integer partitions and the Hardy-Ramanujan formula
  5. A new trick
  6. Reproducing $d=2$ asymptotics
  7. Plane partitions
  8. Relation of plane partitions to $d=3$ Hilbert series
  9. Asymptotic growth of operator degeneracy from Meinardus' theorem
  10. Bosonic Meinardus theorems
  11. Fermionic Meinardus theorems
  12. Leading behaviour in arbitrary dimension
  13. Hilbert Series Projections: Spin, internal symmetry, and IBP
  14. A worked example in $d=4$
  15. General lessons
  16. ...and 16 more sections

Figures (7)

  • Figure 1: The growth of operators for a single real scalar in $d=4$. Upper panel: The thick dashed line corresponds to exact data, obtained by a direct expansion of the PE. The thin grey curve that is indistinguishable from the data in the upper panel is the asymptotic result. Lower panel: Relative error between data and the asymptotic result.
  • Figure 2: The growth of operators for a fermion in $d=4$. Upper panel: The thick dashed line corresponds to exact data, obtained by a direct expansion of the PE. The thin grey curve that is indistinguishable from the data in the upper panel is the asymptotic result. Lower panel: Relative error between data and the asymptotic result.
  • Figure 3: Calculation using $SU(2)\times SU(2)$. The figure depicts saddles on $(\omega_1,\omega_2)$ plane. The center of the square is at $(0,0)$. The corners are at $(1/2,1/2), (1/2,-1/2), (-1/2,1/2), (-1/2,1/2)$. For the bosonic case, all the saddles contribute. When we add fermions in the mix, the green colored saddles at $(0,\pm1/2), (\pm 1/2, 0)$ don't contribute anymore, leading to an overall factor of $1/2$. We have drawn circular regions arounds the saddle to denote how much the fluctuation around the saddle contributes. For example, each of the corner ones contributes one quarter of the center one.
  • Figure 4: The effect of a linear shift in $\Delta$ to capture sub-leading corrections to the asymptotic formula for the growth of operators in the SMEFT, with one generation of fermions, $N_g=1$. The thick dashed line is the exact results for the number of operators in the SMEFT as a function of $\Delta$. The thin dashed curve indicates the asymptotic result given in \ref{['eq:thesmeft']}. The solid grey curve indicates \ref{['eq:thesmeft']} evaluated with the shifted $\widetilde{\Delta}$ given by \ref{['eq:smeftshift']}.
  • Figure 5: The growth of operators in the SMEFT, comparing exact data with the asymptotic formulae. The thick dashed curve is data for one generation of fermions, $N_g=1$; the thick dotted curve is data for three generations of fermions $N_g=3$. The solid grey curves indicate the asymptotic result \ref{['eq:thesmeft']} evaluated with the shifted $\widetilde{\Delta}$ given by \ref{['eq:smeftshift']}.
  • ...and 2 more figures