Operator bases, S-matrices, and their partition functions

doi:10.1007/JHEP10(2017)199

Title: 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. Here 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-pointmore »« less

Authors:

Henning, Brian ^[1];

^[2]; Melia, Tom ^[3]; Murayama, Hitoshi ^[3]

Yale Univ., New Haven, CT (United States). Dept. of Physics
Univ. of California, Davis, CA (United States). Dept. of Physics
Univ. of California, Berkeley, CA (United States). Dept. of Physics; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of Tokyo (Japan). Kavli Inst. for the Physics and Mathematics of the Universe (WPI), Todai Inst. for Advanced Study

Publication Date:: 2017-10-27

Research Org.:: Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC); Japan Society for the Promotion of Science (JSPS)

OSTI Identifier:: 1421808

Grant/Contract Number:: AC02-05CH11231; SC0009999; 291377; PHY-1316783; PHY-1638509; 26400241; 17K05409; 15H05887; 15K21733

Resource Type:: Accepted Manuscript

Journal Name:: Journal of High Energy Physics (Online)

Additional Journal Information:: Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 10; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Effective Field Theories; Differential and Algebraic Geometry; Scattering Amplitudes

Citation Formats


                    Henning, Brian, Lu, Xiaochuan, Melia, Tom, and Murayama, Hitoshi. Operator bases, S-matrices, and their partition functions.  United States: N. p., 2017. 
        Web.  doi:10.1007/JHEP10(2017)199.

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                    Henning, Brian, Lu, Xiaochuan, Melia, Tom, & Murayama, Hitoshi. Operator bases, S-matrices, and their partition functions.  United States.  doi:10.1007/JHEP10(2017)199.

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                    Henning, Brian, Lu, Xiaochuan, Melia, Tom, and Murayama, Hitoshi. Fri .  
        "Operator bases, S-matrices, and their partition functions".  United States.  doi:10.1007/JHEP10(2017)199.  https://www.osti.gov/servlets/purl/1421808.

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@article{osti_1421808,

  title        = {Operator bases, S-matrices, and their partition functions},

  author       = {Henning, Brian and Lu, Xiaochuan and Melia, Tom and Murayama, Hitoshi},

  abstractNote = {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. Here 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. Finally, although primarily discussed in the language of EFT, some of our results — conceptual and quantitative — may be of broader use in studying conformal field theories as well as the AdS/CFT correspondence.},

  doi          = {10.1007/JHEP10(2017)199},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 10,

  volume       = 2017,

  place        = {United States},

  year         = {2017},

  month        = {10}

}

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DOI: 10.1007/JHEP10(2017)199

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The soft bootstrap is an on-shell method to constrain the landscape of effective field theories (EFTs) of massless particles via the consistency of the low-energy S-matrix. Given assumptions on the on-shell data (particle spectra, linear symmetries, and low-energy theorems), the soft bootstrap is an efficient algorithm for determining the possible consistency of an EFT with those properties. The implementation of the soft bootstrap uses the recently discovered method of soft subtracted recursion. We derive a precise criterion for the validity of these recursion relations and show that they fail exactly when the assumed symmetries can be trivially realized by independentmore »« less
Cited by 18
DOI: 10.1007/jhep01(2019)195

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Does there exist a sensible quantum theory of an ''algebra-valued'' scalar field

Journal Article Anco, S C ; Wald, R M - Phys. Rev. D; (United States)

Consider a scalar field phi in Minkowski spacetime, but let phi be valued in an associative, commutative algebra openA rather than openR. One may view the resulting theory as describing a collection of coupled real scalar fields. At the classical level, theories of this type are completely well behaved and have a global symmetry group which is a nontrivial enlargement of the Poincare group. (They are analogs of the new class of gauge theories for massless spin-2 fields found recently by one of us, whose gauge group is a nontrivial enlargement of the usual diffeomorphism group.) We investigate the quantizationmore »« less
DOI: 10.1103/PhysRevD.39.2297

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Title: Operator bases, S-matrices, and their partition functions

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Structure of Phenomenological Lagrangians. II journal, January 1969

Minimal generating sets of non-modular invariant rings of finite groups journal, January 2013

Causality, analyticity and an IR obstruction to UV completion journal, October 2006

The analytic bootstrap and AdS superhorizon locality journal, December 2013

Unitarity and positivity constraints for CFT at large central charge journal, July 2017

Field transformations and the classical equation of motion in chiral perturbation theory journal, December 1995

Gauge Fields and Space-Time journal, October 2002

Chiral perturbation theory: Expansions in the mass of the strange quark journal, January 1985

Nonrenormalization Theorems without Supersymmetry journal, August 2015

A periodic table of effective field theories journal, February 2017

Spinning conformal correlators journal, November 2011

Analyticity in spin in conformal theories journal, September 2017

Bounding the space of holographic CFTs with chaos journal, October 2016

Graviton and Gluon Scattering from First Principles journal, February 2017

All free Conformal Representations in all Dimensions journal, May 1989

Extended affine root systems. I. Coxeter transformations journal, January 1985

The spectrum of Yang Mills on a sphere journal, January 2006

Hilbert series and operator bases with derivatives in effective field theories journal, December 2015

Group Theoretical Discussion of Relativistic Wave Equations journal, May 1948

Renormalization group evolution of the Standard Model dimension six operators III: gauge coupling dependence and phenomenology journal, April 2014

Departures from chiral symmetry journal, March 1975

What Maxwell theory in teaches us about scale and conformal invariance journal, July 2011

Change of variables and equivalence theorems in quantum field theories journal, November 1961

Chiral perturbation theory to one loop journal, November 1984

More holography from conformal field theory journal, September 2010

Conformal collider physics: energy and charge correlations journal, May 2008

Effective Field Theories from Soft Limits of Scattering Amplitudes journal, June 2015

Algebraic invariants of graphs; a study based on computer exploration journal, September 2000

Invariants of finite groups and their applications to combinatorics journal, May 1979

Extension of the chiral perturbation theory meson Lagrangian to order p 6 journal, January 1996

Lie Algebra Cohomology and the Generalized Borel-Weil Theorem journal, September 1961

Quantum Field Theories with Composite Particles and Asymptotic Conditions journal, October 1958

Character formulas and partition functions in higher dimensional conformal field theory journal, June 2006

Tensor representations of conformal algebra and conformally covariant operator product expansion journal, March 1973

On the four-point function of the stress-energy tensors in a CFT journal, October 2015

A numerical approach to Virasoro blocks and the information paradox journal, September 2017

Holography from conformal field theory journal, October 2009

2, 84, 30, 993, 560, 15456, 11962, 261485, . . .: higher dimension operators in the SM EFT journal, August 2017

Structure of Phenomenological Lagrangians. I journal, January 1969

Counting and construction of holomorphic primary fields in free CFT4 from rings of functions on Calabi-Yau orbifolds journal, August 2017

2D CFT partition functions at late times journal, August 2017

Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay journal, June 1974

Writing CFT correlation functions as AdS scattering amplitudes journal, March 2011

A natural language for AdS/CFT correlators journal, November 2011

Non-abelian binding energies from the lightcone bootstrap journal, February 2016

On HQET and NRQCD operators of dimension 8 and above journal, July 2017

Invariants ofS 4 and the shape of sets of vectors journal, January 1996

Conformal collider physics from the lightcone bootstrap journal, February 2016

On information loss in AdS3/CFT2 journal, May 2016

Covariant expansion of the conformal four -point function journal, December 1972

Convexity and liberation at large spin journal, November 2013

A conformal truncation framework for infinite-volume dynamics journal, July 2016

The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories journal, January 2004

Hilbert functions of graded algebras journal, April 1978

The mesonic chiral lagrangean of order p 6 journal, February 1999

Bounding scalar operator dimensions in 4 D CFT journal, December 2008

Restrictions imposed by superconformal invariance on quantum field theories journal, January 1998

Symmetries, sum rules and constraints on effective field theories journal, September 2014

Hilbert series and operator basis for NRQED and NRQCD/HQET journal, September 2017

On the general theory of collisions for particles with spin journal, August 1959

Effective conformal theory and the flat-space limit of AdS journal, July 2011

Loops in AdS from conformal field theory journal, July 2017

Low-derivative operators of the Standard Model effective field theory via Hilbert series methods journal, February 2016

Operator content and modular properties of higher-dimensional conformal field theories journal, December 1991

Phenomenological Lagrangians journal, April 1979

On the field theory expansion of superstring five point amplitudes journal, November 2013

Lessons from crossing symmetry at large N journal, June 2015

Mellin Amplitudes for Supergravity on AdS 5 × S 5 journal, March 2017

Algebraic structure of lepton and quark flavor invariants and CP violation journal, October 2009

On the General Theory of Collisions for Particles with Spin journal, April 2000

Consistent searches for SMEFT effects in non-resonant dijet events journal, January 2018

The order p8 mesonic chiral Lagrangian journal, January 2019

DEFT: a program for operators in EFT journal, January 2019

Effective field theory amplitudes the on-shell way: scalar and vector couplings to gluons journal, February 2019

Low-energy effective field theory below the electroweak scale: operators and matching journal, March 2018

Renormalization group evolution of dimension-seven operators in standard model effective field theory and relevant phenomenology journal, March 2019

Energy flow polynomials: a complete linear basis for jet substructure journal, April 2018

Structure of Phenomenological Lagrangians. II
journal, January 1969

Minimal generating sets of non-modular invariant rings of finite groups
journal, January 2013

Causality, analyticity and an IR obstruction to UV completion
journal, October 2006

The analytic bootstrap and AdS superhorizon locality
journal, December 2013

Unitarity and positivity constraints for CFT at large central charge
journal, July 2017

Field transformations and the classical equation of motion in chiral perturbation theory
journal, December 1995

Gauge Fields and Space-Time
journal, October 2002

Chiral perturbation theory: Expansions in the mass of the strange quark
journal, January 1985

Nonrenormalization Theorems without Supersymmetry
journal, August 2015

A periodic table of effective field theories
journal, February 2017

Spinning conformal correlators
journal, November 2011

Analyticity in spin in conformal theories
journal, September 2017

Bounding the space of holographic CFTs with chaos
journal, October 2016

Graviton and Gluon Scattering from First Principles
journal, February 2017

All free Conformal Representations in all Dimensions
journal, May 1989

Extended affine root systems. I. Coxeter transformations
journal, January 1985

The spectrum of Yang Mills on a sphere
journal, January 2006

Hilbert series and operator bases with derivatives in effective field theories
journal, December 2015

Group Theoretical Discussion of Relativistic Wave Equations
journal, May 1948

Renormalization group evolution of the Standard Model dimension six operators III: gauge coupling dependence and phenomenology
journal, April 2014

Departures from chiral symmetry
journal, March 1975

What Maxwell theory in teaches us about scale and conformal invariance
journal, July 2011

Change of variables and equivalence theorems in quantum field theories
journal, November 1961

Chiral perturbation theory to one loop
journal, November 1984

More holography from conformal field theory
journal, September 2010

Conformal collider physics: energy and charge correlations
journal, May 2008

Effective Field Theories from Soft Limits of Scattering Amplitudes
journal, June 2015

Algebraic invariants of graphs; a study based on computer exploration
journal, September 2000

Invariants of finite groups and their applications to combinatorics
journal, May 1979

Extension of the chiral perturbation theory meson Lagrangian to order $p^{6}$
journal, January 1996

Lie Algebra Cohomology and the Generalized Borel-Weil Theorem
journal, September 1961

Quantum Field Theories with Composite Particles and Asymptotic Conditions
journal, October 1958

Character formulas and partition functions in higher dimensional conformal field theory
journal, June 2006

Tensor representations of conformal algebra and conformally covariant operator product expansion
journal, March 1973

On the four-point function of the stress-energy tensors in a CFT
journal, October 2015

A numerical approach to Virasoro blocks and the information paradox
journal, September 2017

Holography from conformal field theory
journal, October 2009

2, 84, 30, 993, 560, 15456, 11962, 261485, . . .: higher dimension operators in the SM EFT
journal, August 2017

Structure of Phenomenological Lagrangians. I
journal, January 1969

Counting and construction of holomorphic primary fields in free CFT4 from rings of functions on Calabi-Yau orbifolds
journal, August 2017

2D CFT partition functions at late times
journal, August 2017

Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay
journal, June 1974

Writing CFT correlation functions as AdS scattering amplitudes
journal, March 2011

A natural language for AdS/CFT correlators
journal, November 2011

Non-abelian binding energies from the lightcone bootstrap
journal, February 2016

On HQET and NRQCD operators of dimension 8 and above
journal, July 2017

Invariants ofS 4 and the shape of sets of vectors
journal, January 1996

Conformal collider physics from the lightcone bootstrap
journal, February 2016

On information loss in AdS3/CFT2
journal, May 2016

Covariant expansion of the conformal four -point function
journal, December 1972

Convexity and liberation at large spin
journal, November 2013

A conformal truncation framework for infinite-volume dynamics
journal, July 2016

The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories
journal, January 2004

Hilbert functions of graded algebras
journal, April 1978

The mesonic chiral lagrangean of order p ⁶
journal, February 1999

Bounding scalar operator dimensions in 4 D CFT
journal, December 2008

Restrictions imposed by superconformal invariance on quantum field theories
journal, January 1998

Symmetries, sum rules and constraints on effective field theories
journal, September 2014

Hilbert series and operator basis for NRQED and NRQCD/HQET
journal, September 2017

On the general theory of collisions for particles with spin
journal, August 1959

Effective conformal theory and the flat-space limit of AdS
journal, July 2011

Loops in AdS from conformal field theory
journal, July 2017

Low-derivative operators of the Standard Model effective field theory via Hilbert series methods
journal, February 2016

Operator content and modular properties of higher-dimensional conformal field theories
journal, December 1991

Phenomenological Lagrangians
journal, April 1979

On the field theory expansion of superstring five point amplitudes
journal, November 2013

Lessons from crossing symmetry at large N
journal, June 2015

Mellin Amplitudes for Supergravity on ${AdS}_{5} \times S^{5}$
journal, March 2017

Algebraic structure of lepton and quark flavor invariants and CP violation
journal, October 2009

On the General Theory of Collisions for Particles with Spin
journal, April 2000

Consistent searches for SMEFT effects in non-resonant dijet events
journal, January 2018

The order p8 mesonic chiral Lagrangian
journal, January 2019

DEFT: a program for operators in EFT
journal, January 2019

Effective field theory amplitudes the on-shell way: scalar and vector couplings to gluons
journal, February 2019

Low-energy effective field theory below the electroweak scale: operators and matching
journal, March 2018

Renormalization group evolution of dimension-seven operators in standard model effective field theory and relevant phenomenology
journal, March 2019

Energy flow polynomials: a complete linear basis for jet substructure
journal, April 2018

Systematic construction of basis invariants in the 2HDM
journal, May 2019

Updated global SMEFT fit to Higgs, diboson and electroweak data
journal, June 2018

Free field primaries in general dimensions: counting and construction with rings and modules
journal, August 2018