Operator bases, S-matrices, and their partition functions

Henning, Brian; Lu, Xiaochuan; Melia, Tom; Murayama, Hitoshi

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.  https://doi.org/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.  https://doi.org/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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Systematic construction of basis invariants in the 2HDM
journal, May 2019

Trautner, Andreas
Journal of High Energy Physics, Vol. 2019, Issue 5
DOI: 10.1007/jhep05(2019)208

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

Ellis, John; Murphy, Christopher W.; Sanz, Verónica
Journal of High Energy Physics, Vol. 2018, Issue 6
DOI: 10.1007/jhep06(2018)146

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

de Mello Koch, Robert; Ramgoolam, Sanjaye
Journal of High Energy Physics, Vol. 2018, Issue 8
DOI: 10.1007/jhep08(2018)088

NLO corrections to h → bb¯$$ \overline{b} $$ decay in SMEFT
journal, August 2019

Cullen, Jonathan M.; Pecjak, Benjamin D.; Scott, Darren J.
Journal of High Energy Physics, Vol. 2019, Issue 8
DOI: 10.1007/jhep08(2019)173

Reparameterization invariant operator basis for NRQED and HQET
journal, November 2019

Kobach, Andrew; Pal, Sridip
Journal of High Energy Physics, Vol. 2019, Issue 11
DOI: 10.1007/jhep11(2019)012

Soft bootstrap and effective field theories
journal, November 2019

Low, Ian; Yin, Zhewei
Journal of High Energy Physics, Vol. 2019, Issue 11
DOI: 10.1007/jhep11(2019)078

The SMEFTsim package, theory and tools
journal, December 2017

Brivio, Ilaria; Jiang, Yun; Trott, Michael
Journal of High Energy Physics, Vol. 2017, Issue 12
DOI: 10.1007/jhep12(2017)070

Constraints on a gravitational Higgs mechanism
journal, October 2019

Bonifacio, James; Hinterbichler, Kurt; Rosen, Rachel A.
Physical Review D, Vol. 100, Issue 8
DOI: 10.1103/physrevd.100.084017

Renormalization group evolution of Higgs effective field theory
journal, February 2018

Alonso, Rodrigo; Kanshin, Kirill; Saa, Sara
Physical Review D, Vol. 97, Issue 3
DOI: 10.1103/physrevd.97.035010

Lepton-number-charged scalars and neutrino beamstrahlung
journal, April 2018

Berryman, Jeffrey M.; de Gouvêa, André; Kelly, Kevin J.
Physical Review D, Vol. 97, Issue 7
DOI: 10.1103/physrevd.97.075030

Bounds on amplitudes in effective theories with massive spinning particles
journal, August 2018

Bonifacio, James; Hinterbichler, Kurt
Physical Review D, Vol. 98, Issue 4
DOI: 10.1103/physrevd.98.045003

BasisGen: automatic generation of operator bases
journal, March 2019

Criado, Juan Carlos
The European Physical Journal C, Vol. 79, Issue 3
DOI: 10.1140/epjc/s10052-019-6769-5

DEFT: A program for operators in EFT
text, January 2019

Gripaios, Ben; Sutherland, Dave
Apollo - University of Cambridge Repository
DOI: 10.17863/cam.36615

Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching
text, January 2017

Jenkins, Elizabeth E.; Manohar, Aneesh V.; Stoffer, Peter
arXiv
DOI: 10.48550/arxiv.1709.04486

The SMEFTsim package, theory and tools
text, January 2017

Brivio, Ilaria; Jiang, Yun; Trott, Michael
arXiv
DOI: 10.48550/arxiv.1709.06492

Renormalization group evolution of Higgs effective field theory
text, January 2017

Alonso, R.; Kanshin, K.; Saa, S.
arXiv
DOI: 10.48550/arxiv.1710.06848

Consistent Searches for SMEFT Effects in Non-Resonant Dijet Events
text, January 2017

Alte, Stefan; König, Matthias; Shepherd, William
arXiv
DOI: 10.48550/arxiv.1711.07484

Energy flow polynomials: A complete linear basis for jet substructure
text, January 2017

Komiske, Patrick T.; Metodiev, Eric M.; Thaler, Jesse
arXiv
DOI: 10.48550/arxiv.1712.07124

Free field primaries in general dimensions: Counting and construction with rings and modules
text, January 2018

Koch, Robert de Mello; Ramgoolam, Sanjaye
arXiv
DOI: 10.48550/arxiv.1806.01085

DEFT: A program for operators in EFT
text, January 2018

Gripaios, Ben; Sutherland, Dave
arXiv
DOI: 10.48550/arxiv.1807.07546

The order $p^8$ mesonic chiral Lagrangian
text, January 2018

Bijnens, Johan; Hermansson-Truedsson, Nils; Wang, Si
arXiv
DOI: 10.48550/arxiv.1810.06834

BasisGen: automatic generation of operator bases
text, January 2019

Criado, Juan Carlos
arXiv
DOI: 10.48550/arxiv.1901.03501

Renormalization Group Evolution of Dimension-seven Operators in Standard Model Effective Field Theory and Relevant Phenomenology
text, January 2019

Liao, Yi; Ma, Xiao-Dong
arXiv
DOI: 10.48550/arxiv.1901.10302

Soft Bootstrap and Effective Field Theories
text, January 2019

Low, Ian; Yin, Zhewei
arXiv
DOI: 10.48550/arxiv.1904.12859

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

Abstract

Citation Formats

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

The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large n Gauge Theories conference, January 2012

Gauge Fields and Space-Time conference, November 2011

Tensor Representations of Conformal Algebra and Conformally Covariant Operator Product Expansion text, January 1971

Holography from Conformal Field Theory text, January 2009

More Holography from Conformal Field Theory text, January 2010

Effective Conformal Theory and the Flat-Space Limit of AdS text, January 2010

A Natural Language for AdS/CFT Correlators text, January 2011

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

The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large n Gauge Theories
conference, January 2012

Gauge Fields and Space-Time
conference, November 2011

Tensor Representations of Conformal Algebra and Conformally Covariant Operator Product Expansion
text, January 1971

Holography from Conformal Field Theory
text, January 2009

More Holography from Conformal Field Theory
text, January 2010

Effective Conformal Theory and the Flat-Space Limit of AdS
text, January 2010

A Natural Language for AdS/CFT Correlators
text, January 2011

Symmetries, Sum Rules and Constraints on Effective Field Theories
text, January 2014

Lessons from crossing symmetry at large N
preprint, January 2014

Non-renormalization Theorems without Supersymmetry
text, January 2015