Operator bases, $S$-matrices, and their partition functions

Abstract

Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where $S$-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the $S$-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (on-shell) and integration by parts (momentum conservation). We introduce a partition function, termed the Hilbert series, to enumerate the operator basis--correspondingly, the $S$-matrix--and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries. In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to $n=5$ scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of $n$-point amplitudes. We discuss in detail the operator basis for non-linearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the $S$-matrix in the form of soft limits. The most naïve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations.

75 pages plus appendices

References in corpus (57)

• The Large N Limit of Superconformal Field Theories and Supergravity
• Causality, Analyticity and an IR Obstruction to UV Completion
• Bounding scalar operator dimensions in 4D CFT
• Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology
• Holography from Conformal Field Theory
• Conformal collider physics: Energy and charge correlations
• Writing CFT correlation functions as AdS scattering amplitudes
• The Analytic Bootstrap and AdS Superhorizon Locality
• Spinning Conformal Correlators
• Analyticity in Spin in Conformal Theories
• Convexity and Liberation at Large Spin
• EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions
• The Mesonic Chiral Lagrangian of Order $p^6$
• A Natural Language for AdS/CFT Correlators
• Effective Field Theories from Soft Limits
• Mellin amplitudes for $Ad{S}_5\times S^5$
• Effective Conformal Theory and the Flat-Space Limit of AdS
• Leading proton production in deep inelastic scattering at HERA
• Gauge Fields and Space-Time
• Loops in AdS from Conformal Field Theory
• Extension of the Chiral Perturbation Theory Meson Lagrangian to Order $P^6$
• Positivity Bounds for Scalar Theories
• Scaled momentum distributions of charged particles in dijet photoproduction at HERA
• A Periodic Table of Effective Field Theories
• Bounding the Space of Holographic CFTs with Chaos
• Low-derivative operators of the Standard Model effective field theory via Hilbert series methods
• A Cheap Alternative to the Lattice?
• Hilbert Series for Constructing Lagrangians: expanding the phenomenologist's toolbox
• Hilbert series and operator bases with derivatives in effective field theories
• Non-renormalization Theorems without Supersymmetry
• What Maxwell Theory in D<>4 teaches us about scale and conformal invariance
• Einstein gravity 3-point functions from conformal field theory
• Character Formulae and Partition Functions in Higher Dimensional Conformal Field Theory
• Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation
• Symmetries, Sum Rules and Constraints on Effective Field Theories
• 2D CFT Partition Functions at Late Times
• Conformal Collider Physics from the Lightcone Bootstrap
• More Holography from Conformal Field Theory
• A Numerical Approach to Virasoro Blocks and the Information Paradox
• Field Transformations and the Classical Equation of Motion in Chiral Perturbation Theory
• Non-Abelian Binding Energies from the Lightcone Bootstrap
• A Conformal Truncation Framework for Infinite-Volume Dynamics
• On the field theory expansion of superstring five point amplitudes
• On the four-point function of the stress-energy tensors in a CFT
• Hilbert Series and Operator Basis for NRQED and NRQCD/HQET
• Unitarity and positivity constraints for CFT at large central charge
• 2, 84, 30, 993, 560, 15456, 11962, 261485, ...: Higher dimension operators in the SM EFT
• Graviton and gluon scattering from first principles
• The Spectrum of Yang Mills on a Sphere
• On Information Loss in AdS${}_3$/CFT${}_2$
• Minimal generating sets of non-modular invariant rings of finite groups
• On HQET and NRQCD Operators of Dimension 8 and Above
• Conformal Fields in Higher Dimensions
• Lessons from crossing symmetry at large N
• CFT4 as SO(4,2)-invariant TFT2
• Counting Conformal Correlators
• Holomorphic primary fields in free CFT4 and Calabi-Yau orbifolds

Cited by in corpus (111)

• Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching
• Updated Global SMEFT Fit to Higgs, Diboson and Electroweak Data
• The SMEFTsim package, theory and tools
• Energy flow polynomials: A complete linear basis for jet substructure
• The Standard Model effective field theory at work
• On the impact of dimension-eight SMEFT operators on Higgs measurements
• Lepton-Number-Charged Scalars and Neutrino Beamstrahlung
• Effective Field Theory Amplitudes the On-Shell Way: Scalar and Vector Couplings to Gluons
• BasisGen: automatic generation of operator bases
• The electroweak effective field theory from on-shell amplitudes
• An explicit construction of the dimension-9 operator basis in the standard model effective field theory
• The critical O(N) CFT: Methods and conformal data
• Tidal effects in quantum field theory
• Effective Field Theory of Gravity to All Orders
• Renormalization group evolution of Higgs effective field theory
• Bounds on Amplitudes in Effective Theories with Massive Spinning Particles
• Standard Model Effective Field Theory from On-shell Amplitudes
• Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds
• Enumerating the operators of an effective field theory
• Constructing effective field theories via their harmonics
• Enumerating higher-dimensional operators with on-shell amplitudes
• Constructing massive on-shell contact terms
• DEFT: A program for operators in EFT
• From Amplitudes to Contact Cosmological Correlators
• Tidal effects for spinning particles
• 2, 12, 117, 1959, 45171, 1170086, ...: A Hilbert series for the QCD chiral Lagrangian
• Renormalization Group Evolution of Dimension-seven Operators in Standard Model Effective Field Theory and Relevant Phenomenology
• Composing Effective Prediction at Five Points
• Hilbert Series, the Higgs Mechanism, and HEFT
• NLO corrections to $h\to b\bar{b}$ decay in SMEFT
• Soft Bootstrap and Effective Field Theories
• FCNC $B$ and $K$ Meson Decays with Light Bosonic Dark Matter
• The order $p^8$ mesonic chiral Lagrangian
• Amplitude bases in generic EFTs
• Spontaneously Broken Boosts in CFTs
• Symmetry-resolved Entanglement Entropy, Spectra & Boundary Conformal Field Theory
• Exploding operators for Majorana neutrino masses and beyond
• ..., 83106786, 114382724, 1509048322, 2343463290, 27410087742, ... Efficient Hilbert Series for Effective Theories
• Systematic construction of basis invariants in the 2HDM
• Characters and Group Invariant Polynomials of (Super)fields: Road to "Lagrangian"
• Flavor Invariants and Renormalization-group Equations in the Leptonic Sector with Massive Majorana Neutrinos
• Ansätze for Scattering Amplitudes from $p$-adic Numbers and Algebraic Geometry
• Beyond Jarlskog: 699 invariants for CP violation in SMEFT
• The Black Hole Weak Gravity Conjecture with Multiple Charges
• Low-Energy Effective Field Theory below the Electroweak Scale: Dimension-8 Operators
• Consistent Searches for SMEFT Effects in Non-Resonant Dijet Events
• Effective field theory approach to lepton number violating decays $K^{\pm }\to {\pi }^{\mp }{l}^{\pm }{l}^{\pm }$: short-distance contribution
• Hilbert Series and Plethystics: Paving the path towards 2HDM- and MLRSM-EFT
• Constraints on a Gravitational Higgs Mechanism
• EFT Asymptotics: the Growth of Operator Degeneracy
• Effective Operator Bases for Beyond Standard Model Scenarios: An EFT compendium for discoveries
• Operator Counting and Soft Blocks in Chiral Perturbation Theory
• CP violation and flavor invariants in the seesaw effective field theory
• Cutting Multiparticle Correlators Down to Size
• Equations of Motion for the Standard Model Effective Field Theory: Theory and Applications
• Renormalization and non-renormalization of scalar EFTs at higher orders
• Monolepton production in SMEFT to $\mathcal{O}(1/{\Lambda }^4)$ and beyond
• An Effective Cosmological Collider
• Non-perturbative renormalization scheme for the CP-odd three-gluon operator
• Effective Theories with Dark Matter Applications
• Unitarity Bounds on Effective Field Theories at the LHC
• Jet Bundle Geometry of Scalar Field Theories
• Nonrelativistic effective field theories with enhanced symmetries and soft behavior
• Primary Observables for Indirect Searches at Colliders
• Outer Automorphism Anomalies
• Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory
• 23, 381, 6242, 103268, 1743183, ...: Hilbert series for CP-violating operators in SMEFT
• Free field primaries in general dimensions: Counting and construction with rings and modules
• The Geometric $\nu$SMEFT: Operators and Connections
• Unitarity and Universality in non relativistic Conformal Field theory
• Primary Observables for Top Quark Collider Signals
• Hilbert series and higher-order Lagrangians for the $O(N)$ model
• Reparameterization Invariant Operator Basis for NRQED and HQET
• Conformal Structure of the Heavy Particle EFT Operator Basis
• Constructing Operator Basis in Supersymmetry: A Hilbert Series Approach
• The tricritical Ising CFT and conformal bootstrap
• M2-branes and plane partitions
• XEFT, the challenging path up the hill: dim = 6 and dim = 8
• Discrete symmetries and Efficient Counting of Operators
• An Index for Quantum Integrability
• On the systematic construction of basis invariants
• Counting Operators in $N=1$ Supersymmetric Gauge Theories
• LEX-EFT: The Light Exotics Effective Field Theory
• Anomalous Dimensions in Hypercubic Theories
• Catalogues of Cosmologically Self-Consistent Hadronic QCD Axion Models
• Simple models with both baryon and lepton number violation by two units
• From Spinning Primaries to Permutation Orbifolds
• Froggatt-Nielsen Meets the SMEFT
• Custodial Symmetry (Violation) in SMEFT
• Lorentz and permutation invariants of particles I
• Asymptotic degeneracies of M2-brane SCFTs
• Primary Observables for Electroweak Gauge Boson Collider Signals
• Perturbative 4D conformal field theories and representation theory of diagram algebras
• Non-renormalizable theories and finite formulation of QFT
• General Signals for Charged Lepton Flavor Violating Decays
• Assessing (H)EFT theory errors by pitting EoM against Field Redefinitions
• One Formula To Match Them All: The Bispinor Universal One-Loop Effective Action
• Basis for Non-Factorizable Superamplitudes in N = 1 Supersymmetry
• Extending low energy effective field theory with a complete set of dimension-7 operators
• Two-loop dimension Six Effective Action: Integrating Out Heavy Scalar
• Universal fine grained asymptotics of free and weakly coupled Quantum Field Theory
• $D$-Dimensional Modular Assembly of Higher-Derivative Four-Point Contact Amplitudes Involving Fermions
• Supersymmetric zeta functions and determinants
• Notes on index of quantum integrability
• Operator Dimension Parity Fractionalization
• Holography, Supergravity, and the Weak Gravity Conjecture
• Searches for new interactions within the SMEFT framework at present and future colliders
• A Large-$N$ Expansion for Minimum Bias
• Flavour Invariants of the N Higgs Doublet Model
• BV Formalism and Partition Functions
• Non-minimal non-Abelian quantum vector fields in curved spacetime