EFT Asymptotics: the Growth of Operator Degeneracy

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.

arXiv comment35+9 pages; v2: resubmission for review process in SciPost, minor changes

References in corpus (49)

• Dimension-Six Terms in the Standard Model Lagrangian
• Bounding scalar operator dimensions in 4D CFT
• Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology
• An Index for 4 dimensional Super Conformal Theories
• The Hagedorn Transition, Deconfinement and N=4 SYM Theory
• Counting Gauge Invariants: the Plethystic Program
• Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics
• OPE Convergence in Conformal Field Theory
• One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts
• The S-matrix Bootstrap I: QFT in AdS
• Cardy Formulae for SUSY Theories in d=4 and d=6
• Operator bases, $S$-matrices, and their partition functions
• Low-derivative operators of the Standard Model effective field theory via Hilbert series methods
• A first order deconfinement transition in large N Yang-Mills theory on a small 3-sphere
• Hilbert Series for Constructing Lagrangians: expanding the phenomenologist's toolbox
• Hilbert series and operator bases with derivatives in effective field theories
• The S-matrix Bootstrap II: Two Dimensional Amplitudes
• SQCD: A Geometric Apercu
• W-symmetry, topological vertex and affine Yangian
• The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices
• The Analytic Functional Bootstrap II: Natural Bases for the Crossing Equation
• Character Formulae and Partition Functions in Higher Dimensional Conformal Field Theory
• A 4d N=1 Cardy Formula
• A tauberian theorem for the conformal bootstrap
• Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation
• On the CFT Operator Spectrum at Large Global Charge
• Partition Sums and Entropy Bounds in Weakly Coupled CFT
• Landau diagrams in AdS and S-matrices from conformal correlators
• Modular Invariance, Tauberian Theorems, and Microcanonical Entropy
• Effective Field Theory of Gravity to All Orders
• Hilbert Series for Flavor Invariants of the Standard Model
• The S-matrix Bootstrap IV: Multiple Amplitudes
• Modular forms and a generalized Cardy formula in higher dimensions
• Charged structure constants from modularity
• Tauberian-Cardy formula with spin
• Beurling-Selberg Extremization and Modular Bootstrap at High Energies
• Hilbert Series and Operator Basis for NRQED and NRQCD/HQET
• Adding flavour to the S-matrix bootstrap
• Bootstrapping Massive Quantum Field Theories
• Modular invariance on $S^1\times S^3$ and circle fibrations
• Level Density of a Bose Gas and Extreme Value Statistics
• The O(N) S-matrix Monolith
• Anomalies, Black strings and the charged Cardy formula
• Bounds on the density of states and the spectral gap in CFT${}_2$
• ..., 83106786, 114382724, 1509048322, 2343463290, 27410087742, ... Efficient Hilbert Series for Effective Theories
• Distributions in CFT I. Cross-Ratio Space
• Twisted sectors from plane partitions
• Asymptotic counting of BPS operators in superconformal field theories
• On the density of states in a free CFT and finite volume corrections

Cited by in corpus (13)

• Universal Asymptotics for High Energy CFT Data
• Spontaneously Broken Boosts in CFTs
• Symmetry-resolved Entanglement Entropy, Spectra & Boundary Conformal Field Theory
• Flavor Invariants and Renormalization-group Equations in the Leptonic Sector with Massive Majorana Neutrinos
• A universal formula for the density of states with continuous symmetry
• Renormalization and non-renormalization of scalar EFTs at higher orders
• Flat space gravity at finite cutoff
• Noninvertible Symmetry-Resolved Affleck-Ludwig-Cardy Formula and Entanglement Entropy from the Boundary Tube Algebra
• M2-branes and plane partitions
• Discrete symmetries and Efficient Counting of Operators
• Asymptotic degeneracies of M2-brane SCFTs
• Supersymmetric zeta functions and determinants
• A Large-$N$ Expansion for Minimum Bias