# 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 α^{2}_{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 »

- Authors:

- Publication Date:

- 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. doi:10.1103/PhysRevD.100.014009.
```

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

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

```
@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}

}

DOI: 10.1103/PhysRevD.100.014009

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Works referenced in this record:

##
Analytical Computation of Energy-Energy Correlation at Next-to-Leading Order in QCD

journal, March 2018

- Dixon, Lance J.; Luo, Ming-xing; Shtabovenko, Vladyslav
- Physical Review Letters, Vol. 120, Issue 10

##
On the Drell–Levy–Yan relation to O(α2)

journal, October 2000

- Blümlein, J.; Ravindran, V.; van Neerven, W. L.
- Nuclear Physics B, Vol. 586, Issue 1-2

##
<mml:math altimg="si1.gif" 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:mi mathvariant="script">N</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math> SUSY Yang–Mills: Three loops made simple(r)

journal, March 2007

- Dokshitzer, Yu. L.; Marchesini, G.
- Physics Letters B, Vol. 646, Issue 4

##
On the quark distribution in an on-shell heavy quark and its all-order relations with the perturbative fragmentation function

journal, February 2005

- Gardi, Einan
- Journal of High Energy Physics, Vol. 2005, Issue 02

##
Analytic next-to-leading order calculation of energy-energy correlation in gluon-initiated Higgs decays

journal, June 2019

- Luo, Ming-xing; Shtabovenko, Vladyslav; Yang, Tong-Zhi
- Journal of High Energy Physics, Vol. 2019, Issue 6

##
Chiral-odd parton distributions and Drell-Yan processes

journal, May 1992

- Jaffe, R. L.; Ji, Xiangdong
- Nuclear Physics B, Vol. 375, Issue 3

##
From correlation functions to event shapes

journal, July 2014

- Belitsky, A. V.; Hohenegger, S.; Korchemsky, G. P.
- Nuclear Physics B, Vol. 884

##
Erratum to: “DGLAP and BFKL equations in the N=4 supersymmetric gauge theory”

journal, May 2004

- Kotikov, A. V.; Lipatov, L. N.
- Nuclear Physics B, Vol. 685, Issue 1-3

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

journal, December 2015

- Korchemsky, G. P.; Sokatchev, E.
- Journal of High Energy Physics, Vol. 2015, Issue 12

##
Three-Jet Production in Electron-Positron Collisions at Next-to-Next-to-Leading Order Accuracy

journal, October 2016

- Del Duca, Vittorio; Duhr, Claude; Kardos, Adam
- Physical Review Letters, Vol. 117, Issue 15

##
Five-loop anomalous dimension of twist-two operators

journal, May 2010

- Łukowski, T.; Rej, A.; Velizhanin, V. N.
- Nuclear Physics B, Vol. 831, Issue 1-2

##
Quantum spectral curve for arbitrary state/operator in AdS5/CFT4

journal, September 2015

- Gromov, Nikolay; Kazakov, Vladimir; Leurent, Sébastien
- Journal of High Energy Physics, Vol. 2015, Issue 9

##
Relating *e* ^{+} *e* ^{−} annihilation to high energy scattering at weak and strong coupling

journal, November 2008

- Hatta, Yoshitaka
- Journal of High Energy Physics, Vol. 2008, Issue 11

##
Three-loop universal anomalous dimension of the Wilson operators in N=4 SUSY Yang–Mills model

journal, August 2004

- Kotikov, A. V.; Lipatov, L. N.; Onishchenko, A. I.
- Physics Letters B, Vol. 595, Issue 1-4

##
An effective field theory for collinear and soft gluons: Heavy to light decays

journal, May 2001

- Bauer, Christian W.; Fleming, Sean; Pirjol, Dan
- Physical Review D, Vol. 63, Issue 11

##
Power counting to better jet observables

journal, December 2014

- Larkoski, Andrew J.; Moult, Ian; Neill, Duff
- Journal of High Energy Physics, Vol. 2014, Issue 12

##
Anomalous dimensions of Wilson operators in N=4 SYM theory

journal, March 2003

- Kotikov, A. V.; Lipatov, L. N.; Velizhanin, V. N.
- Physics Letters B, Vol. 557, Issue 1-2

##
Four loop twist two, BFKL, wrapping and strings

journal, August 2009

- Bajnok, Zoltan; Janik, Romuald A.; Łukowski, Tomasz
- Nuclear Physics B, Vol. 816, Issue 3

##
QCD corrections to semi-inclusive hadron production in electron–positron annihilation at two loops

journal, September 2006

- Mitov, A.; Moch, S.
- Nuclear Physics B, Vol. 751, Issue 1-2

##
A simple algorithm for QCD jets

journal, September 1978

- Konishi, K.; Ukawa, A.; Veneziano, G.
- Physics Letters B, Vol. 78, Issue 2-3

##
Anomalous dimensions of high-spin operators beyond the leading order

journal, July 2007

- Basso, B.; Korchemsky, G. P.
- Nuclear Physics B, Vol. 775, Issue 1-2

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

journal, July 2014

- Gromov, Nikolay; Levkovich-Maslyuk, Fedor; Sizov, Grigory
- Journal of High Energy Physics, Vol. 2014, Issue 7

##
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

- Kardos, Adam; Kluth, Stefan; Somogyi, Gábor
- The European Physical Journal C, Vol. 78, Issue 6

##
Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron-positron annihilation

journal, March 1997

- Rijken, P. J.; van Neerven, W. L.
- Nuclear Physics B, Vol. 487, Issue 1-2

##
Event shapes in <mml:math altimg="si1.gif" 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" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd"><mml:mi mathvariant="script">N</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math> super-Yang–Mills theory

journal, July 2014

- Belitsky, A. V.; Hohenegger, S.; Korchemsky, G. P.
- Nuclear Physics B, Vol. 884

##
Energy flow polynomials: a complete linear basis for jet substructure

journal, April 2018

- Komiske, Patrick T.; Metodiev, Eric M.; Thaler, Jesse
- Journal of High Energy Physics, Vol. 2018, Issue 4

##
Energy-Energy Correlations in $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory

journal, February 2014

- Belitsky, A. V.; Hohenegger, S.; Korchemsky, G. P.
- Physical Review Letters, Vol. 112, Issue 7

##
Non-planar anomalous dimension of twist-2 operators: Higher moments at four loops

journal, August 2014

- Velizhanin, V. N.
- Nuclear Physics B, Vol. 885

##
Light-ray operators in conformal field theory

journal, November 2018

- Kravchuk, Petr; Simmons-Duffin, David
- Journal of High Energy Physics, Vol. 2018, Issue 11

##
$\mathcal{N}=4$ superconformal Ward identities for correlation functions

journal, March 2016

- Belitsky, A. V.; Hohenegger, S.; Korchemsky, G. P.
- Nuclear Physics B, Vol. 904

##
Next-to-next-to-leading order evolution of non-singlet fragmentation functions

journal, June 2006

- Mitov, A.; Moch, S.; Vogt, A.
- Physics Letters B, Vol. 638, Issue 1

##
Conformal collider physics: energy and charge correlations

journal, May 2008

- Hofman, Diego M.; Maldacena, Juan
- Journal of High Energy Physics, Vol. 2008, Issue 05

##
Revisiting parton evolution and the large-x limit

journal, March 2006

- Dokshitzer, Yu. L.; Marchesini, G.; Salam, G. P.
- Physics Letters B, Vol. 634, Issue 5-6

##
Twist 3 of the sector of SYM and reciprocity respecting evolution

journal, August 2007

- Beccaria, M.; Dokshitzer, Yu. L.; Marchesini, G.
- Physics Letters B, Vol. 652, Issue 4

##
Implications of supersymmetry for QCD conformal operators

journal, March 1999

- Belitsky, A. V.; Müller, D.; Schäfer, A.
- Physics Letters B, Vol. 450, Issue 1-3

##
Analytic boosted boson discrimination

journal, May 2016

- Larkoski, Andrew J.; Moult, Ian; Neill, Duff
- Journal of High Energy Physics, Vol. 2016, Issue 5

##
On the next-to-next-to-leading order evolution of flavour-singlet fragmentation functions

journal, January 2012

- Almasy, A. A.; Moch, S.; Vogt, A.
- Nuclear Physics B, Vol. 854, Issue 1

##
Resolving QCD jets beyond leading order: Quark decay probabilities

journal, April 1981

- Kalinowski, J.; Konishi, K.; Scharbach, P. N.
- Nuclear Physics B, Vol. 181, Issue 2

##
On third-order timelike splitting functions and top-mediated Higgs decay into hadrons

journal, January 2008

- Moch, S.; Vogt, A.
- Physics Letters B, Vol. 659, Issue 1-2

##
Anomalous dimension of non-singlet Wilson operators at in deep inelastic scattering

journal, February 1994

- Gracey, J. A.
- Physics Letters B, Vol. 322, Issue 1-2

##
Positivity restriction on anomalous dimensions

journal, June 1974

- Ferrara, S.; Gatto, R.; Grillo, A.
- Physical Review D, Vol. 9, Issue 12

##
Large spin systematics in CFT

journal, November 2015

- Alday, Luis F.; Bissi, Agnese; Lukowski, Tomasz
- Journal of High Energy Physics, Vol. 2015, Issue 11

##
On Higgs decays to hadrons and the R-ratio at N4LO

journal, August 2017

- Herzog, F.; Ruijl, B.; Ueda, T.
- Journal of High Energy Physics, Vol. 2017, Issue 8

##
Invariant operators in collinear effective theory

journal, September 2001

- Bauer, Christian W.; Stewart, Iain W.
- Physics Letters B, Vol. 516, Issue 1-2

##
Back-to-back jets in QCD

journal, December 1981

- Collins, John C.; Soper, Davison E.
- Nuclear Physics B, Vol. 193, Issue 2

##
Summing Sudakov logarithms in $\overrightarrow{B}{X}_{s}\gamma $ in effective field theory

journal, December 2000

- Bauer, Christian W.; Fleming, Sean; Luke, Michael
- Physical Review D, Vol. 63, Issue 1

##
Specific features of heavy quark production: Local parton-hadron duality approach to heavy particle spectra

journal, January 1996

- Dokshitzer, Yu. L.; Khoze, V. A.; Troyan, S. I.
- Physical Review D, Vol. 53, Issue 1

##
Complete $O({\alpha}_{s}^{4})$ QCD Corrections to Hadronic $Z$ Decays

journal, May 2012

- Baikov, P. A.; Chetyrkin, K. G.; Kühn, J. H.
- Physical Review Letters, Vol. 108, Issue 22

##
Energy Correlations in Electron-Positron Annihilation: Testing Quantum Chromodynamics

journal, December 1978

- Basham, C. Louis; Brown, Lowell S.; Ellis, Stephen D.
- Physical Review Letters, Vol. 41, Issue 23

##
Jet production in the CoLoRFulNNLO method: Event shapes in electron-positron collisions

journal, October 2016

- Del Duca, Vittorio; Duhr, Claude; Kardos, Adam
- Physical Review D, Vol. 94, Issue 7

##
Energy correlation functions for jet substructure

journal, June 2013

- Larkoski, Andrew J.; Salam, Gavin P.; Thaler, Jesse
- Journal of High Energy Physics, Vol. 2013, Issue 6

##
The back-to-back region in energy–energy correlation

journal, January 2005

- de Florian, Daniel; Grazzini, Massimiliano
- Nuclear Physics B, Vol. 704, Issue 1-2

##
New angles on energy correlation functions

journal, December 2016

- Moult, Ian; Necib, Lina; Thaler, Jesse
- Journal of High Energy Physics, Vol. 2016, Issue 12

##
Theory of Deep-Inelastic Lepton-Nucleon Scattering and Lepton-Pair Annihilation Processes. I

journal, November 1969

- Drell, Sidney D.; Levy, Donald J.; Yan, Tung-Mow
- Physical Review, Vol. 187, Issue 5

##
Infrared factorization of tree-level QCD amplitudes at the next-to-next-to-leading order and beyond

journal, March 2000

- Catani, Stefano; Grazzini, Massimiliano
- Nuclear Physics B, Vol. 570, Issue 1-2

##
DGLAP and BFKL equations in the N=4 supersymmetric gauge theory

journal, June 2003

- Kotikov, A. V.; Lipatov, L. N.
- Nuclear Physics B, Vol. 661, Issue 1-2

##
Twist-2 at five loops: wrapping corrections without wrapping computations

journal, June 2014

- Velizhanin, V. N.
- Journal of High Energy Physics, Vol. 2014, Issue 6

##
Simplicity from recoil: the three-loop soft function and factorization for the energy-energy correlation

journal, August 2018

- Moult, Ian; Zhu, Hua Xing
- Journal of High Energy Physics, Vol. 2018, Issue 8

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

journal, July 2015

- Marboe, Christian; Velizhanin, Vitaly; Volin, Dmytro
- Journal of High Energy Physics, Vol. 2015, Issue 7

##
Double unresolved approximations to multiparton scattering amplitudes

journal, August 1998

- Campbell, J. M.; Glover, E. W. N.
- Nuclear Physics B, Vol. 527, Issue 1-2

##
On the phase structure of vector-like gauge theories with massless fermions

journal, March 1982

- Banks, T.; Zaks, A.
- Nuclear Physics B, Vol. 196, Issue 2

##
Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS 5 /CF T 4

journal, June 2016

- Gromov, Nikolay; Levkovich-Maslyuk, Fedor; Sizov, Grigory
- Journal of High Energy Physics, Vol. 2016, Issue 6

##
Jet calculus: A simple algorithm for resolving QCD jets

journal, September 1979

- Konishi, K.; Ukawa, A.; Veneziano, G.
- Nuclear Physics B, Vol. 157, Issue 1

##
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

##
Energy correlations in electron-positron annihilation in quantum chromodynamics: Asymptotically free perturbation theory

journal, April 1979

- Basham, C. Louis; Brown, Lowell S.; Ellis, Stephen D.
- Physical Review D, Vol. 19, Issue 7

##
Energy–energy correlation in electron–positron annihilation at NNLL + NNLO accuracy

journal, November 2017

- Tulipánt, Zoltán; Kardos, Adam; Somogyi, Gábor
- The European Physical Journal C, Vol. 77, Issue 11

##
Multiplicity and hadron distributions in QCD jets

journal, November 1983

- Mueller, A. H.
- Nuclear Physics B, Vol. 228, Issue 2

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

journal, January 2006

- Kotikov, A. V.; Lipatov, L. N.; Onishchenko, A. I.
- Physics Letters B, Vol. 632, Issue 5-6

##
Quantum Spectral Curve for Planar $\mathcal{N}=4$ Super-Yang-Mills Theory

journal, January 2014

- Gromov, Nikolay; Kazakov, Vladimir; Leurent, Sébastien
- Physical Review Letters, Vol. 112, Issue 1

##
Second order corrections to the energy-energy correlation function in quantum chromodynamics

journal, December 1982

- Richards, D. G.; Stirling, W. J.; Ellis, S. D.
- Physics Letters B, Vol. 119, Issue 1-3

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

journal, November 2016

- Marboe, Christian; Velizhanin, Vitaly
- Journal of High Energy Physics, Vol. 2016, Issue 11

##
Four-loop non-singlet splitting functions in the planar limit and beyond

journal, October 2017

- Moch, S.; Ruijl, B.; Ueda, T.
- Journal of High Energy Physics, Vol. 2017, Issue 10

##
On the transverse spread of QCD jets

journal, January 1979

- Konishi, K.; Ukawa, A.; Veneziano, G.
- Physics Letters B, Vol. 80, Issue 3

##
Dressing and wrapping

journal, October 2007

- Kotikov, A. V.; Lipatov, L. N.; Rej, A.
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2007, Issue 10

##
Renormalization of the Wilson loops beyond the leading order

journal, January 1987

- Korchemsky, G. P.; Radyushkin, A. V.
- Nuclear Physics B, Vol. 283

##
On the highest transcendentality in SUSY

journal, May 2007

- Kotikov, A. V.; Lipatov, L. N.
- Nuclear Physics B, Vol. 769, Issue 3

##
Soft-collinear factorization in effective field theory

journal, February 2002

- Bauer, Christian W.; Pirjol, Dan; Stewart, Iain W.
- Physical Review D, Vol. 65, Issue 5