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Title: Harmonic analysis and mean field theory

Abstract

We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weight-shifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and “seed” operators in 4d, and MFT for currents and stress-tensors in 3d.

Authors:
 [1]; ORCiD logo [2];  [3]
  1. Inst. of Physics (EPFL), Lausanne (Switzerland)
  2. California Inst. of Technology (CalTech), Pasadena, CA (United States); Inst. for Advanced Study, Princeton, NJ (United States)
  3. California Inst. of Technology (CalTech), Pasadena, CA (United States)
Publication Date:
Research Org.:
California Inst. of Technology (CalTech), Pasadena, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); Swiss National Science Foundation (SNF)
OSTI Identifier:
1594253
Report Number(s):
arXiv:1809.05111
Journal ID: ISSN 1029-8479
Grant/Contract Number:  
SC0019085; SC0009988
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: 2019; 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; Conformal and W Symmetry; Conformal Field Theory

Citation Formats

Karateev, Denis, Kravchuk, Petr, and Simmons-Duffin, David. Harmonic analysis and mean field theory. United States: N. p., 2019. Web. doi:10.1007/JHEP10(2019)217.
Karateev, Denis, Kravchuk, Petr, & Simmons-Duffin, David. Harmonic analysis and mean field theory. United States. doi:10.1007/JHEP10(2019)217.
Karateev, Denis, Kravchuk, Petr, and Simmons-Duffin, David. Mon . "Harmonic analysis and mean field theory". United States. doi:10.1007/JHEP10(2019)217. https://www.osti.gov/servlets/purl/1594253.
@article{osti_1594253,
title = {Harmonic analysis and mean field theory},
author = {Karateev, Denis and Kravchuk, Petr and Simmons-Duffin, David},
abstractNote = {We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weight-shifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and “seed” operators in 4d, and MFT for currents and stress-tensors in 3d.},
doi = {10.1007/JHEP10(2019)217},
journal = {Journal of High Energy Physics (Online)},
number = 10,
volume = 2019,
place = {United States},
year = {2019},
month = {10}
}

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Works referenced in this record:

Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies
journal, March 2016

  • Bzowski, Adam; McFadden, Paul; Skenderis, Kostas
  • Journal of High Energy Physics, Vol. 2016, Issue 3
  • DOI: 10.1007/JHEP03(2016)066

1/n Expansion: Calculation of the exponents ? and? in the order 1/n2 for arbitrary number of dimensions
journal, June 1981

  • Vasil'ev, A. N.; Pis'mak, Yu. M.; Khonkonen, Yu. R.
  • Theoretical and Mathematical Physics, Vol. 47, Issue 3
  • DOI: 10.1007/BF01019296

Comments on operators with large spin
journal, November 2007


Conformal partial wave expansions for <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mrow><mml:mi mathvariant="script">N</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math> chiral four-point functions
journal, March 2006


The analytic bootstrap and AdS superhorizon locality
journal, December 2013

  • Fitzpatrick, A. Liam; Kaplan, Jared; Poland, David
  • Journal of High Energy Physics, Vol. 2013, Issue 12
  • DOI: 10.1007/JHEP12(2013)004

Casimir recursion relations for general conformal blocks
journal, February 2018


Spinning conformal blocks
journal, November 2011

  • Costa, Miguel S.; Penedones, João; Poland, David
  • Journal of High Energy Physics, Vol. 2011, Issue 11
  • DOI: 10.1007/JHEP11(2011)154

Gauge theory correlators from non-critical string theory
journal, May 1998


An algebraic approach to the analytic bootstrap
journal, April 2017

  • Alday, Luis F.; Zhiboedov, Alexander
  • Journal of High Energy Physics, Vol. 2017, Issue 4
  • DOI: 10.1007/JHEP04(2017)157

The spectrum in the Sachdev-Ye-Kitaev model
journal, April 2016

  • Polchinski, Joseph; Rosenhaus, Vladimir
  • Journal of High Energy Physics, Vol. 2016, Issue 4
  • DOI: 10.1007/JHEP04(2016)001

Integrability of conformal fishnet theory
journal, January 2018

  • Gromov, Nikolay; Kazakov, Vladimir; Korchemsky, Gregory
  • Journal of High Energy Physics, Vol. 2018, Issue 1
  • DOI: 10.1007/JHEP01(2018)095

Fermion conformal bootstrap in 4d
journal, June 2019

  • Karateev, Denis; Kravchuk, Petr; Serone, Marco
  • Journal of High Energy Physics, Vol. 2019, Issue 6
  • DOI: 10.1007/JHEP06(2019)088

Spinning conformal correlators
journal, November 2011

  • Costa, Miguel S.; Penedones, João; Poland, David
  • Journal of High Energy Physics, Vol. 2011, Issue 11
  • DOI: 10.1007/JHEP11(2011)071

Analyticity in spin in conformal theories
journal, September 2017


Bootstrapping 3D fermions with global symmetries
journal, January 2018

  • Iliesiu, Luca; Kos, Filip; Poland, David
  • Journal of High Energy Physics, Vol. 2018, Issue 1
  • DOI: 10.1007/JHEP01(2018)036

Anti de Sitter space and holography
journal, January 1998


More on supersymmetric and 2d analogs of the SYK model
journal, August 2017

  • Murugan, Jeff; Stanford, Douglas; Witten, Edward
  • Journal of High Energy Physics, Vol. 2017, Issue 8
  • DOI: 10.1007/JHEP08(2017)146

Factorization of Mellin amplitudes
journal, October 2015

  • Gonçalves, Vasco; Penedones, João; Trevisani, Emilio
  • Journal of High Energy Physics, Vol. 2015, Issue 10
  • DOI: 10.1007/JHEP10(2015)040

Scalar-fermion analytic bootstrap in 4D
journal, June 2019

  • Elkhidir, Emtinan; Karateev, Denis
  • Journal of High Energy Physics, Vol. 2019, Issue 6
  • DOI: 10.1007/JHEP06(2019)026

Seed conformal blocks in 4D CFT
journal, February 2016

  • Echeverri, Alejandro Castedo; Elkhidir, Emtinan; Karateev, Denis
  • Journal of High Energy Physics, Vol. 2016, Issue 2
  • DOI: 10.1007/JHEP02(2016)183

Light-ray operators in conformal field theory
journal, November 2018

  • Kravchuk, Petr; Simmons-Duffin, David
  • Journal of High Energy Physics, Vol. 2018, Issue 11
  • DOI: 10.1007/JHEP11(2018)102

Counting conformal correlators
journal, February 2018

  • Kravchuk, Petr; Simmons-Duffin, David
  • Journal of High Energy Physics, Vol. 2018, Issue 2
  • DOI: 10.1007/JHEP02(2018)096

The 3d stress-tensor bootstrap
journal, February 2018

  • Dymarsky, Anatoly; Kos, Filip; Kravchuk, Petr
  • Journal of High Energy Physics, Vol. 2018, Issue 2
  • DOI: 10.1007/JHEP02(2018)164

Recent developments in conformal invariant quantum field theory
journal, August 1978


A generalization of Sachdev-Ye-Kitaev
journal, February 2017

  • Gross, David J.; Rosenhaus, Vladimir
  • Journal of High Energy Physics, Vol. 2017, Issue 2
  • DOI: 10.1007/JHEP02(2017)093

A spacetime derivation of the Lorentzian OPE inversion formula
journal, July 2018

  • Simmons-Duffin, David; Stanford, Douglas; Witten, Edward
  • Journal of High Energy Physics, Vol. 2018, Issue 7
  • DOI: 10.1007/JHEP07(2018)085

Conformal partial waves and the operator product expansion
journal, February 2004


Holography from conformal field theory
journal, October 2009


General three-point functions in 4D CFT
journal, January 2015

  • Elkhidir, Emtinan; Karateev, Denis; Serone, Marco
  • Journal of High Energy Physics, Vol. 2015, Issue 1
  • DOI: 10.1007/JHEP01(2015)133

Implications of conformal invariance in momentum space
journal, March 2014

  • Bzowski, Adam; McFadden, Paul; Skenderis, Kostas
  • Journal of High Energy Physics, Vol. 2014, Issue 3
  • DOI: 10.1007/JHEP03(2014)111

Bootstrapping 3D fermions
journal, March 2016

  • Iliesiu, Luca; Kos, Filip; Poland, David
  • Journal of High Energy Physics, Vol. 2016, Issue 3
  • DOI: 10.1007/JHEP03(2016)120

Deconstructing conformal blocks in 4D CFT
journal, August 2015

  • Echeverri, Alejandro Castedo; Elkhidir, Emtinan; Karateev, Denis
  • Journal of High Energy Physics, Vol. 2015, Issue 8
  • DOI: 10.1007/JHEP08(2015)101

Renormalised CFT 3-point functions of scalars, currents and stress tensors
journal, November 2018

  • Bzowski, Adam; McFadden, Paul; Skenderis, Kostas
  • Journal of High Energy Physics, Vol. 2018, Issue 11
  • DOI: 10.1007/JHEP11(2018)159

Large Spin Perturbation Theory for Conformal Field Theories
journal, September 2017


Exact correlation functions in conformal fishnet theory
journal, August 2019

  • Gromov, Nikolay; Kazakov, Vladimir; Korchemsky, Gregory
  • Journal of High Energy Physics, Vol. 2019, Issue 8
  • DOI: 10.1007/JHEP08(2019)123

Weight shifting operators and conformal blocks
journal, February 2018

  • Karateev, Denis; Kravchuk, Petr; Simmons-Duffin, David
  • Journal of High Energy Physics, Vol. 2018, Issue 2
  • DOI: 10.1007/JHEP02(2018)081

Renormalised 3-point functions of stress tensors and conserved currents in CFT
journal, November 2018

  • Bzowski, Adam; McFadden, Paul; Skenderis, Kostas
  • Journal of High Energy Physics, Vol. 2018, Issue 11
  • DOI: 10.1007/JHEP11(2018)153

Interwining Operators for Semisimple Groups
journal, May 1971

  • Knapp, A. W.; Stein, E. M.
  • The Annals of Mathematics, Vol. 93, Issue 3
  • DOI: 10.2307/1970887

Convexity and liberation at large spin
journal, November 2013

  • Komargodski, Zohar; Zhiboedov, Alexander
  • Journal of High Energy Physics, Vol. 2013, Issue 11
  • DOI: 10.1007/JHEP11(2013)140

Conformal correlators of mixed-symmetry tensors
journal, February 2015


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


General bootstrap equations in 4D CFTs
journal, January 2018

  • Cuomo, Gabriel Francisco; Karateev, Denis; Kravchuk, Petr
  • Journal of High Energy Physics, Vol. 2018, Issue 1
  • DOI: 10.1007/JHEP01(2018)130

All unitary ray representations of the conformal group SU(2,2) with positive energy
journal, February 1977

  • Mack, G.
  • Communications in Mathematical Physics, Vol. 55, Issue 1
  • DOI: 10.1007/BF01613145

Unitarity and the holographic S-Matrix
journal, October 2012

  • Fitzpatrick, A. Liam; Kaplan, Jared
  • Journal of High Energy Physics, Vol. 2012, Issue 10
  • DOI: 10.1007/JHEP10(2012)032

The lightcone bootstrap and the spectrum of the 3d Ising CFT
journal, March 2017


Projectors, shadows, and conformal blocks
journal, April 2014


Conformal four point functions and the operator product expansion
journal, April 2001


Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory
journal, June 1977

  • Mack, G.
  • Communications in Mathematical Physics, Vol. 53, Issue 2
  • DOI: 10.1007/BF01609130