Collinear limit of the energy-energy correlator

Dixon, Lance J.; Moult, Ian; Zhu, Hua Xing

doi:10.1103/PhysRevD.100.014009

Title: Collinear limit of the energy-energy correlator

Abstract

The energy-energy-correlator (EEC) observable in e⁺e^– annihilation measures the energy deposited in two detectors as a function of the angle between the detectors. The collinear limit, where the angle between the two detectors approaches zero, is of particular interest for describing the substructure of jets produced at hadron colliders as well as in e⁺e^– annihilation. We derive a factorization formula for the leading power asymptotic behavior in the collinear limit of a generic quantum field theory, which allows for the resummation of logarithmically enhanced terms to all orders by renormalization group evolution. The relevant anomalous dimensions are expressed in terms of the timelike data of the theory, in particular the moments of the timelike splitting functions, which are known to high perturbative orders. We relate the small angle and back-to-back limits to each other via the total cross section and an integral over intermediate angles. This relation, for the EEC in e⁺e^– and in Higgs decay to gluons, provides us with the initial conditions for quark and gluon jet functions at order α²_s. In QCD and in N = 1 super-Yang-Mills theory, we then perform the resummation to next-to-next-to-leading logarithm, improving previous calculations by two perturbative orders. We highlight themore »« less

Authors:: Dixon, Lance J.; Moult, Ian; Zhu, Hua Xing

Publication Date:: 2019-07-09

Research Org.:: SLAC National Accelerator Lab., Menlo Park, CA (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1532582

Alternate Identifier(s):: OSTI ID: 1546714

Grant/Contract Number:: AC02-76SF00515; AC02-05CH11231; 107201*172210191

Resource Type:: Published Article

Journal Name:: Physical Review D

Additional Journal Information:: Journal Name: Physical Review D Journal Volume: 100 Journal Issue: 1; Journal ID: ISSN 2470-0010

Publisher:: American Physical Society

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

Citation Formats


                    Dixon, Lance J., Moult, Ian, and Zhu, Hua Xing. Collinear limit of the energy-energy correlator.  United States: N. p., 2019. 
Web.  https://doi.org/10.1103/PhysRevD.100.014009.

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                    Dixon, Lance J., Moult, Ian, & Zhu, Hua Xing. Collinear limit of the energy-energy correlator.  United States.  https://doi.org/10.1103/PhysRevD.100.014009

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                    Dixon, Lance J., Moult, Ian, and Zhu, Hua Xing. Tue .  
"Collinear limit of the energy-energy correlator".  United States.  https://doi.org/10.1103/PhysRevD.100.014009.

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@article{osti_1532582,

  title        = {Collinear limit of the energy-energy correlator},

  author       = {Dixon, Lance J. and Moult, Ian and Zhu, Hua Xing},

  abstractNote = {The energy-energy-correlator (EEC) observable in e+e– annihilation measures the energy deposited in two detectors as a function of the angle between the detectors. The collinear limit, where the angle between the two detectors approaches zero, is of particular interest for describing the substructure of jets produced at hadron colliders as well as in e+e– annihilation. We derive a factorization formula for the leading power asymptotic behavior in the collinear limit of a generic quantum field theory, which allows for the resummation of logarithmically enhanced terms to all orders by renormalization group evolution. The relevant anomalous dimensions are expressed in terms of the timelike data of the theory, in particular the moments of the timelike splitting functions, which are known to high perturbative orders. We relate the small angle and back-to-back limits to each other via the total cross section and an integral over intermediate angles. This relation, for the EEC in e+e– and in Higgs decay to gluons, provides us with the initial conditions for quark and gluon jet functions at order α2s. In QCD and in N = 1 super-Yang-Mills theory, we then perform the resummation to next-to-next-to-leading logarithm, improving previous calculations by two perturbative orders. We highlight the important role played by the nonvanishing β function in these theories, which while subdominant for Higgs decays to gluons, dominates the behavior of the EEC in the collinear limit for e+e– annihilation, and in N = 1 super-Yang-Mills theory. In conformally invariant N = 4 super-Yang-Mills theory, reciprocity between timelike and spacelike evolution can be used to express our factorization formula as a power law with exponent equal to the spacelike twist-two spin-three anomalous dimensions, thus providing a connection between timelike and spacelike approaches.},

  doi          = {10.1103/PhysRevD.100.014009},

  journal      = {Physical Review D},

  number       = 1,

  volume       = 100,

  place        = {United States},

  year         = {2019},

  month        = {7}

}

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We present the analytic formula for the Energy-Energy Correlation (EEC) in electron-positron annihilation computed in perturbative QCD to next-to-next-to-next-to-leading order (N³LO) in the back-to-back limit. In particular, we consider the EEC arising from the annihilation of an electron-positron pair into a virtual photon as well as a Higgs boson and their subsequent inclusive decay into hadrons. Our computation is based on a factorization theorem of the EEC formulated within Soft-Collinear Effective Theory (SCET) for the back-to-back limit. We obtain the last missing ingredient for our computation — the jet function — from a recent calculation of the transverse-momentum dependent fragmentationmore »« less
https://doi.org/10.1007/jhep08(2021)022

Full Text Available
Subleading power resummation of rapidity logarithms: the energy-energy correlator in $$ \mathcal{N}$$ = 4 SYM

Journal Article Moult, Ian ; Vita, Gherardo ; Yan, Kai - Journal of High Energy Physics (Online)

We derivemore »« less
Cited by 2
https://doi.org/10.1007/jhep07(2020)005

Full Text Available
Origin of the Six-Gluon Amplitude in Planar $N = 4$ Supersymmetric Yang-Mills Theory

Journal Article Basso, Benjamin ; Dixon, Lance J. ; Papathanasiou, Georgios - Physical Review Letters

We smore »« less
Cited by 1
https://doi.org/10.1103/PhysRevLett.124.161603
The four-loop remainder function and multi-Regge behavior at NNLLA in planar $$ \mathcal{N} $$ = 4 super-Yang-Mills theory

Journal Article - Journal of High Energy Physics (Online)

We present the four-loop remainder function for six-gluon scattering with maximal helicity violation in planar NN = 4 super-Yang-Mills theory, as an analytic function of three dual-conformal cross ratios. The function is constructed entirely from its analytic properties, without ever inspecting any multi-loop integrand. We employ the same approach used at three loops, writing an ansatz in terms of hexagon functions, and fixing coefficients in the ansatz using the multi-Regge limit and the operator product expansion in the near-collinear limit. We express the result in terms of multiple polylogarithms, and in terms of the coproduct for the associated Hopf algebra.more »« less
https://doi.org/10.1007/JHEP06(2014)116

Full Text Available
Analytical Computation of Energy-Energy Correlation at Next-to-Leading Order in QCD

Journal Article Dixon, Lance J. ; Luo, Ming-xing ; Shtabovenko, Vladyslav ; ... - Physical Review Letters

Here, the energy-energy correlation (EEC) between two detectors in e⁺e^– annihilation was computed analytically at leading order in QCD almost 40 years ago, and numerically at next-to-leading order (NLO) starting in the 1980s. We present the first analytical result for the EEC at NLO, which is remarkably simple, and facilitates analytical study of the perturbative structure of the EEC. We provide the expansion of the EEC in the collinear and back-to-back regions through next-to-leading power, information which should aid resummation in these regions.
Cited by 1
https://doi.org/10.1103/PhysRevLett.120.102001

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Title: Collinear limit of the energy-energy correlator

Abstract

Citation Formats

Analytical Computation of Energy-Energy Correlation at Next-to-Leading Order in QCD journal, March 2018

On the Drell–Levy–Yan relation to O(α2) journal, October 2000

On the quark distribution in an on-shell heavy quark and its all-order relations with the perturbative fragmentation function journal, February 2005

Analytic next-to-leading order calculation of energy-energy correlation in gluon-initiated Higgs decays journal, June 2019

Chiral-odd parton distributions and Drell-Yan processes journal, May 1992

From correlation functions to event shapes journal, July 2014

Erratum to: “DGLAP and BFKL equations in the N=4 supersymmetric gauge theory” journal, May 2004

Four-point correlation function of stress-energy tensors in N = 4 $$ \mathcal{N}=4 $$ superconformal theories journal, December 2015

Three-Jet Production in Electron-Positron Collisions at Next-to-Next-to-Leading Order Accuracy journal, October 2016

Five-loop anomalous dimension of twist-two operators journal, May 2010

Quantum spectral curve for arbitrary state/operator in AdS5/CFT4 journal, September 2015

Relating e + e − annihilation to high energy scattering at weak and strong coupling journal, November 2008

Three-loop universal anomalous dimension of the Wilson operators in N=4 SUSY Yang–Mills model journal, August 2004

An effective field theory for collinear and soft gluons: Heavy to light decays journal, May 2001

Power counting to better jet observables journal, December 2014

Anomalous dimensions of Wilson operators in N=4 SYM theory journal, March 2003

Four loop twist two, BFKL, wrapping and strings journal, August 2009

QCD corrections to semi-inclusive hadron production in electron–positron annihilation at two loops journal, September 2006

A simple algorithm for QCD jets journal, September 1978

Anomalous dimensions of high-spin operators beyond the leading order journal, July 2007

Quantum spectral curve at work: from small spin to strong coupling in N $$ \mathcal{N} $$ = 4 SYM journal, July 2014

Precise determination of $$\alpha _{S}(M_Z)$$ α S ( M Z ) from a global fit of energy–energy correlation to NNLO+NNLL predictions journal, June 2018

Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron-positron annihilation journal, March 1997

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

Energy-Energy Correlations in N = 4 Supersymmetric Yang-Mills Theory journal, February 2014

Non-planar anomalous dimension of twist-2 operators: Higher moments at four loops journal, August 2014

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

N = 4 superconformal Ward identities for correlation functions journal, March 2016

Next-to-next-to-leading order evolution of non-singlet fragmentation functions journal, June 2006

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

Revisiting parton evolution and the large-x limit journal, March 2006

Twist 3 of the sector of SYM and reciprocity respecting evolution journal, August 2007

Implications of supersymmetry for QCD conformal operators journal, March 1999

Analytic boosted boson discrimination journal, May 2016

On the next-to-next-to-leading order evolution of flavour-singlet fragmentation functions journal, January 2012

Resolving QCD jets beyond leading order: Quark decay probabilities journal, April 1981

On third-order timelike splitting functions and top-mediated Higgs decay into hadrons journal, January 2008

Anomalous dimension of non-singlet Wilson operators at in deep inelastic scattering journal, February 1994

Positivity restriction on anomalous dimensions journal, June 1974

Large spin systematics in CFT journal, November 2015

On Higgs decays to hadrons and the R-ratio at N4LO journal, August 2017

Invariant operators in collinear effective theory journal, September 2001

Back-to-back jets in QCD journal, December 1981

Summing Sudakov logarithms in B → X s γ in effective field theory journal, December 2000

Specific features of heavy quark production: Local parton-hadron duality approach to heavy particle spectra journal, January 1996

Complete O ( α s 4 ) QCD Corrections to Hadronic Z Decays journal, May 2012

Energy Correlations in Electron-Positron Annihilation: Testing Quantum Chromodynamics journal, December 1978

Jet production in the CoLoRFulNNLO method: Event shapes in electron-positron collisions journal, October 2016

Energy correlation functions for jet substructure journal, June 2013

The back-to-back region in energy–energy correlation journal, January 2005

New angles on energy correlation functions journal, December 2016

Theory of Deep-Inelastic Lepton-Nucleon Scattering and Lepton-Pair Annihilation Processes. I journal, November 1969

Infrared factorization of tree-level QCD amplitudes at the next-to-next-to-leading order and beyond journal, March 2000

DGLAP and BFKL equations in the N=4 supersymmetric gauge theory journal, June 2003

Twist-2 at five loops: wrapping corrections without wrapping computations journal, June 2014

Simplicity from recoil: the three-loop soft function and factorization for the energy-energy correlation journal, August 2018

Six-loop anomalous dimension of twist-two operators in planar N = 4 $$ \mathcal{N}=4 $$ SYM theory journal, July 2015

Double unresolved approximations to multiparton scattering amplitudes journal, August 1998

On the phase structure of vector-like gauge theories with massless fermions journal, March 1982

Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS 5 /CF T 4 journal, June 2016

Jet calculus: A simple algorithm for resolving QCD jets journal, September 1979

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

Energy correlations in electron-positron annihilation in quantum chromodynamics: Asymptotically free perturbation theory journal, April 1979

Energy–energy correlation in electron–positron annihilation at NNLL + NNLO accuracy journal, November 2017

Multiplicity and hadron distributions in QCD jets journal, November 1983

Erratum to: “Three-loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang–Mills model” [Phys. Lett. B 595 (2004) 521] journal, January 2006

Quantum Spectral Curve for Planar N = 4 Super-Yang-Mills Theory journal, January 2014

Second order corrections to the energy-energy correlation function in quantum chromodynamics journal, December 1982

Twist-2 at seven loops in planar N $$ \mathcal{N} $$ = 4 SYM theory: full result and analytic properties journal, November 2016

Four-loop non-singlet splitting functions in the planar limit and beyond journal, October 2017

On the transverse spread of QCD jets journal, January 1979

Dressing and wrapping journal, October 2007

Renormalization of the Wilson loops beyond the leading order journal, January 1987

On the highest transcendentality in SUSY journal, May 2007

Soft-collinear factorization in effective field theory journal, February 2002

Multiplicity and hadron distributions in QCD jets journal, March 1983

Four-loop non-singlet splitting functions in the planar limit and beyond text, January 2017

Analytical Computation of Energy-Energy Correlation at Next-to-Leading Order in QCD
journal, March 2018

On the Drell–Levy–Yan relation to O(α2)
journal, October 2000

On the quark distribution in an on-shell heavy quark and its all-order relations with the perturbative fragmentation function
journal, February 2005

Analytic next-to-leading order calculation of energy-energy correlation in gluon-initiated Higgs decays
journal, June 2019

Chiral-odd parton distributions and Drell-Yan processes
journal, May 1992

From correlation functions to event shapes
journal, July 2014

Erratum to: “DGLAP and BFKL equations in the N=4 supersymmetric gauge theory”
journal, May 2004

Four-point correlation function of stress-energy tensors in N = 4 $$ \mathcal{N}=4 $$ superconformal theories
journal, December 2015

Three-Jet Production in Electron-Positron Collisions at Next-to-Next-to-Leading Order Accuracy
journal, October 2016

Five-loop anomalous dimension of twist-two operators
journal, May 2010

Quantum spectral curve for arbitrary state/operator in AdS5/CFT4
journal, September 2015

Relating e ⁺ e ⁻ annihilation to high energy scattering at weak and strong coupling
journal, November 2008

Three-loop universal anomalous dimension of the Wilson operators in N=4 SUSY Yang–Mills model
journal, August 2004

An effective field theory for collinear and soft gluons: Heavy to light decays
journal, May 2001

Power counting to better jet observables
journal, December 2014

Anomalous dimensions of Wilson operators in N=4 SYM theory
journal, March 2003

Four loop twist two, BFKL, wrapping and strings
journal, August 2009

QCD corrections to semi-inclusive hadron production in electron–positron annihilation at two loops
journal, September 2006

A simple algorithm for QCD jets
journal, September 1978

Anomalous dimensions of high-spin operators beyond the leading order
journal, July 2007

Quantum spectral curve at work: from small spin to strong coupling in N $$ \mathcal{N} $$ = 4 SYM
journal, July 2014

Precise determination of $$\alpha _{S}(M_Z)$$ α S ( M Z ) from a global fit of energy–energy correlation to NNLO+NNLL predictions
journal, June 2018

Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron-positron annihilation
journal, March 1997

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

Energy-Energy Correlations in $N = 4$ Supersymmetric Yang-Mills Theory
journal, February 2014

Non-planar anomalous dimension of twist-2 operators: Higher moments at four loops
journal, August 2014

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

$N = 4$ superconformal Ward identities for correlation functions
journal, March 2016

Next-to-next-to-leading order evolution of non-singlet fragmentation functions
journal, June 2006

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

Revisiting parton evolution and the large-x limit
journal, March 2006

Twist 3 of the sector of SYM and reciprocity respecting evolution
journal, August 2007

Implications of supersymmetry for QCD conformal operators
journal, March 1999

Analytic boosted boson discrimination
journal, May 2016

On the next-to-next-to-leading order evolution of flavour-singlet fragmentation functions
journal, January 2012

Resolving QCD jets beyond leading order: Quark decay probabilities
journal, April 1981

On third-order timelike splitting functions and top-mediated Higgs decay into hadrons
journal, January 2008

Anomalous dimension of non-singlet Wilson operators at in deep inelastic scattering
journal, February 1994

Positivity restriction on anomalous dimensions
journal, June 1974

Large spin systematics in CFT
journal, November 2015

On Higgs decays to hadrons and the R-ratio at N4LO
journal, August 2017

Invariant operators in collinear effective theory
journal, September 2001

Back-to-back jets in QCD
journal, December 1981

Summing Sudakov logarithms in $\vec{B} X_{s} γ$ in effective field theory
journal, December 2000

Specific features of heavy quark production: Local parton-hadron duality approach to heavy particle spectra
journal, January 1996

Complete $O (α_{s}^{4})$ QCD Corrections to Hadronic $Z$ Decays
journal, May 2012

Energy Correlations in Electron-Positron Annihilation: Testing Quantum Chromodynamics
journal, December 1978

Jet production in the CoLoRFulNNLO method: Event shapes in electron-positron collisions
journal, October 2016

Energy correlation functions for jet substructure
journal, June 2013

The back-to-back region in energy–energy correlation
journal, January 2005

New angles on energy correlation functions
journal, December 2016

Theory of Deep-Inelastic Lepton-Nucleon Scattering and Lepton-Pair Annihilation Processes. I
journal, November 1969

Infrared factorization of tree-level QCD amplitudes at the next-to-next-to-leading order and beyond
journal, March 2000

DGLAP and BFKL equations in the N=4 supersymmetric gauge theory
journal, June 2003

Twist-2 at five loops: wrapping corrections without wrapping computations
journal, June 2014

Simplicity from recoil: the three-loop soft function and factorization for the energy-energy correlation
journal, August 2018

Six-loop anomalous dimension of twist-two operators in planar N = 4 $$ \mathcal{N}=4 $$ SYM theory
journal, July 2015

Double unresolved approximations to multiparton scattering amplitudes
journal, August 1998

On the phase structure of vector-like gauge theories with massless fermions
journal, March 1982

Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS 5 /CF T 4
journal, June 2016

Jet calculus: A simple algorithm for resolving QCD jets
journal, September 1979

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

Energy correlations in electron-positron annihilation in quantum chromodynamics: Asymptotically free perturbation theory
journal, April 1979

Energy–energy correlation in electron–positron annihilation at NNLL + NNLO accuracy
journal, November 2017

Multiplicity and hadron distributions in QCD jets
journal, November 1983

Erratum to: “Three-loop universal anomalous dimension of the Wilson operators in $N = 4$ SUSY Yang–Mills model” [Phys. Lett. B 595 (2004) 521]
journal, January 2006

Quantum Spectral Curve for Planar $N = 4$ Super-Yang-Mills Theory
journal, January 2014

Second order corrections to the energy-energy correlation function in quantum chromodynamics
journal, December 1982

Twist-2 at seven loops in planar N $$ \mathcal{N} $$ = 4 SYM theory: full result and analytic properties
journal, November 2016

Four-loop non-singlet splitting functions in the planar limit and beyond
journal, October 2017

On the transverse spread of QCD jets
journal, January 1979

Dressing and wrapping
journal, October 2007

Renormalization of the Wilson loops beyond the leading order
journal, January 1987

On the highest transcendentality in SUSY
journal, May 2007

Soft-collinear factorization in effective field theory
journal, February 2002

Multiplicity and hadron distributions in QCD jets
journal, March 1983

Four-loop non-singlet splitting functions in the planar limit and beyond
text, January 2017

Landau-Khalatnikov-Fradkin transformation and the mystery of even $ζ$ -values in Euclidean massless correlators
journal, November 2019