Abstract
The energy-energy correlation (EEC) measures the angular distribution of the energy that flows through two calorimeters separated by some relative angle in the final state created by a source. We study this observable in the limit of small and large angles when it describes the correlation between particles belonging, respectively, to the same jet and to two almost back-to-back jets. We present a new approach to resumming large logarithmically enhanced corrections in both limits that exploits the relation between the energy correlations and four-point correlation functions of conserved currents. At large angle, we derive the EEC from the behaviour of the correlation function in the limit when four operators are light-like separated in a sequential manner. At small angle, in a conformal theory, we obtain the EEC from resummation of the conformal partial wave expansion of the correlation function at short-distance separation between the calorimeters. In both cases, we obtain a concise representation of the EEC in terms of the conformal data of twist-two operators and verify it by comparing with the results of explicit calculation at next-to-next-to-leading order in maximally supersymmetric Yang-Mills theory. As a byproduct of our analysis, we predict the maximal weight part of the analogous QCD expression in the back-to-back limit.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C.L. Basham, L.S. Brown, S.D. Ellis and S.T. Love, Energy correlations in electron-positron annihilation: testing QCD, Phys. Rev. Lett.41 (1978) 1585 [INSPIRE].
C.L. Basham, L.S. Brown, S.D. Ellis and S.T. Love, Energy correlations in electron-positron annihilation in quantum chromodynamics: asymptotically free perturbation theory, Phys. Rev.D 19 (1979) 2018 [INSPIRE].
SLD collaboration, Measurement of α s (\( {M}_Z^2 \)) from hadronic event observables at the Z 0 resonance, Phys. Rev.D 51 (1995) 962 [hep-ex/9501003] [INSPIRE].
L.J. Dixon et al., Analytical computation of energy-energy correlation at next-to-leading order in QCD, Phys. Rev. Lett.120 (2018) 102001 [arXiv:1801.03219] [INSPIRE].
M.-X. Luo, V. Shtabovenko, T.-Z. Yang and H.X. Zhu, Analytic next-to-leading order calculation of energy-energy correlation in gluon-initiated Higgs decays, JHEP06 (2019) 037 [arXiv:1903.07277] [INSPIRE].
V. Del Duca et al., Three-jet production in electron-positron collisions at next-to-next-to-leading order accuracy, Phys. Rev. Lett.117 (2016) 152004 [arXiv:1603.08927] [INSPIRE].
K. Konishi, A. Ukawa and G. Veneziano, A simple algorithm for QCD jets, Phys. Lett.B 78 (1978) 243.
K. Konishi, A. Ukawa and G. Veneziano, On the transverse spread of QCD jets, Phys. Lett.B 80 (1979) 259.
K. Konishi, A. Ukawa and G. Veneziano, Jet calculus: a simple algorithm for resolving QCD jets, Nucl. Phys.B 157 (1979) 45 [INSPIRE].
D.G. Richards, W.J. Stirling and S.D. Ellis, Energy-energy correlations to second order in quantum chromodynamics, Nucl. Phys.B 229 (1983) 317 [INSPIRE].
J.C. Collins and D.E. Soper, Back-to-back jets in QCD, Nucl. Phys.B 193 (1981) 381 [Erratum ibid.B 213 (1983) 545] [INSPIRE].
J. Kodaira and L. Trentadue, Summing Soft Emission in QCD, Phys. Lett.B 112 (1982) 66.
S.D. Ellis, D.G. Richards and W.J. Stirling, Fixed order perturbation theory and leading logarithms, Phys. Lett.B 136 (1984) 99.
D. de Florian and M. Grazzini, The back-to-back region in e +e −energy-energy correlation, Nucl. Phys.B 704 (2005) 387 [hep-ph/0407241] [INSPIRE].
I. Moult and H.X. Zhu, Simplicity from recoil: the three-loop soft function and factorization for the energy-energy correlation, JHEP08 (2018) 160 [arXiv:1801.02627] [INSPIRE].
A.V. Belitsky et al., Event shapes in \( \mathcal{N} \) = 4 super-Yang-Mills theory, Nucl. Phys.B 884 (2014) 206 [arXiv:1309.1424] [INSPIRE].
A.V. Belitsky et al., From correlation functions to event shapes, Nucl. Phys.B 884 (2014) 305 [arXiv:1309.0769] [INSPIRE].
N.A. Sveshnikov and F.V. Tkachov, Jets and quantum field theory, Phys. Lett.B 382 (1996) 403 [hep-ph/9512370] [INSPIRE].
G.P. Korchemsky, G. Oderda and G.F. Sterman, Power corrections and nonlocal operators, AIP Conf. Proc.407 (1997) 988 [hep-ph/9708346] [INSPIRE].
G.P. Korchemsky and G.F. Sterman, Power corrections to event shapes and factorization, Nucl. Phys.B 555 (1999) 335 [hep-ph/9902341] [INSPIRE].
A.V. Belitsky, G.P. Korchemsky and G.F. Sterman, Energy flow in QCD and event shape functions, Phys. Lett.B 515 (2001) 297 [hep-ph/0106308] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
G. Mack, D-independent representation of conformal field theories in D dimensions via transformation to auxiliary dual resonance models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
A.V. Belitsky et al., Energy-energy correlations in N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett.112 (2014) 071601 [arXiv:1311.6800] [INSPIRE].
J.M. Henn, E. Sokatchev, K. Yan and A. Zhiboedov, Energy-energy correlation in N = 4 super Yang-Mills theory at next-to-next-to-leading order, Phys. Rev.D 100 (2019) 036010 [arXiv:1903.05314] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech.01 (2007) P01021 [hep-th/0610251] [INSPIRE].
L. Freyhult, A. Rej and M. Staudacher, A generalized scaling function for AdS/CFT, J. Stat. Mech.07 (2008) P07015 [arXiv:0712.2743] [INSPIRE].
L. Freyhult and S. Zieme, The virtual scaling function of AdS/CFT, Phys. Rev.D 79 (2009) 105009 [arXiv:0901.2749] [INSPIRE].
D. Fioravanti, P. Grinza and M. Rossi, Beyond cusp anomalous dimension from integrability, Phys. Lett.B 675 (2009) 137 [arXiv:0901.3161] [INSPIRE].
L.J. Dixon, I. Moult and H.X. Zhu, The collinear limit of the energy-energy correlator, to appear.
A.V. Belitsky, S. Hohenegger, G.P. Korchemsky and E. Sokatchev, N = 4 superconformal Ward identities for correlation functions, Nucl. Phys.B 904 (2016) 176 [arXiv:1409.2502] [INSPIRE].
G.P. Korchemsky and E. Sokatchev, Four-point correlation function of stress-energy tensors in \( \mathcal{N} \) = 4 superconformal theories, JHEP12 (2015) 133 [arXiv:1504.07904] [INSPIRE].
O. Erdoğan and G. Sterman, Ultraviolet divergences and factorization for coordinate-space amplitudes, Phys. Rev.D 91 (2015) 065033 [arXiv:1411.4588] [INSPIRE].
O. Erdoğan and G. Sterman, Path description of coordinate-space amplitudes, Phys. Rev.D 95 (2017) 116015 [arXiv:1705.04539] [INSPIRE].
L.F. Alday et al., From correlation functions to Wilson loops, JHEP09 (2011) 123 [arXiv:1007.3243] [INSPIRE].
L.F. Alday and A. Bissi, Higher-spin correlators, JHEP10 (2013) 202 [arXiv:1305.4604] [INSPIRE].
A.H. Mueller, On the asymptotic behavior of the Sudakov form-factor, Phys. Rev.D 20 (1979) 2037 [INSPIRE].
A. Sen, Asymptotic behavior of the Sudakov form-factor in QCD, Phys. Rev.D 24 (1981) 3281 [INSPIRE].
G.P. Korchemsky, Sudakov form-factor in QCD, Phys. Lett.B 220 (1989) 629 [INSPIRE].
J.C. Collins, Sudakov form-factors, Adv. Ser. Direct. High Energy Phys.5 (1989) 573 [hep-ph/0312336] [INSPIRE].
G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov evolution kernels of parton distributions, Mod. Phys. Lett.A 4 (1989) 1257 [INSPIRE].
B. Eden, Three-loop universal structure constants in N = 4 SUSY Yang-Mills theory, arXiv:1207.3112 [INSPIRE].
G.P. Korchemsky, On level crossing in conformal field theories, JHEP03 (2016) 212 [arXiv:1512.05362] [INSPIRE].
A.V. Kotikov, L.N. Lipatov, A.I. Onishchenko and V.N. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett.B 595 (2004) 521 [Erratum ibid.B 632 (2006) 754] [hep-th/0404092] [INSPIRE].
G. Arutyunov, B. Eden, A.C. Petkou and E. Sokatchev, Exceptional nonrenormalization properties and OPE analysis of chiral four point functions in N = 4 SYM(4), Nucl. Phys.B 620 (2002) 380 [hep-th/0103230] [INSPIRE].
F.A. Dolan and H. Osborn, Superconformal symmetry, correlation functions and the operator product expansion, Nucl. Phys.B 629 (2002) 3 [hep-th/0112251] [INSPIRE].
B. Eden et al., Five-loop Konishi in N = 4 SYM, Nucl. Phys.B 862 (2012) 123 [arXiv:1202.5733] [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
R. Koekoek, P.A. Lesky and R.F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer, Germany (2010).
O.T. Engelund and R. Roiban, Correlation functions of local composite operators from generalized unitarity, JHEP03 (2013) 172 [arXiv:1209.0227] [INSPIRE].
M. Koloǧlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, The light-ray ope and conformal colliders, to appear.
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, Shocks, superconvergence and a stringy equivalence principle, arXiv:1904.05905 [INSPIRE].
A.V. Belitsky, S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Superconformal operators in N = 4 super-Yang-Mills theory, Phys. Rev.D 70 (2004) 045021 [hep-th/0311104] [INSPIRE].
A.V. Kotikov and V.N. Velizhanin, Analytic continuation of the Mellin moments of deep inelastic structure functions, hep-ph/0501274 [INSPIRE].
A.V. Kotikov et al., Dressing and wrapping, J. Stat. Mech.10 (2007) P10003 [arXiv:0704.3586] [INSPIRE].
Z. Bajnok, R.A. Janik and T. Lukowski, Four loop twist two, BFKL, wrapping and strings, Nucl. Phys.B 816 (2009) 376 [arXiv:0811.4448] [INSPIRE].
B. Eden and F. Paul, Half-BPS half-BPS twist two at four loops in N = 4 SYM, arXiv:1608.04222 [INSPIRE].
B. Basso and G.P. Korchemsky, Anomalous dimensions of high-spin operators beyond the leading order, Nucl. Phys.B 775 (2007) 1 [hep-th/0612247] [INSPIRE].
S. Caron-Huot, private communication.
Yu.L. Dokshitzer, G. Marchesini and G.P. Salam, Revisiting parton evolution and the large-x limit, Phys. Lett.B 634 (2006) 504 [hep-ph/0511302] [INSPIRE].
G. Marchesini, Relating small Feynman and Bjoken x, in the proceedings of the 41stRencontres de Moriond, 2006 QCD and High Energy Hadronic Interactions, March 18–25, La Thuile, Italy (2006), hep-ph/0605262 [INSPIRE].
K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, New approach to evaluation of multiloop Feynman integrals: the Gegenbauer polynomial x space technique, Nucl. Phys.B 174 (1980) 345 [INSPIRE].
S.G. Gorishnii, A.L. Kataev and S.A. Larin, The O(\( {\alpha}_s^3 \))-corrections to σ tot (e +e −→ hadrons) and Γ(τ −→ ν τ + hadrons) in QCD, Phys. Lett.B 259 (1991) 144 [INSPIRE].
L.R. Surguladze and M.A. Samuel, Total hadronic cross-section in e +e −annihilation at the four loop level of perturbative QCD, Phys. Rev. Lett.66 (1991) 560 [Erratum ibid.66 (1991) 2416] [INSPIRE].
P.A. Baikov, K.G. Chetyrkin, J.H. Kuhn and J. Rittinger, Adler function, sum rules and Crewther relation of order O(\( {\alpha}_s^4 \)): the singlet case, Phys. Lett.B 714 (2012) 62 [arXiv:1206.1288] [INSPIRE].
F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, On Higgs decays to hadrons and the R-ratio at N 4LO, JHEP08 (2017) 113 [arXiv:1707.01044] [INSPIRE].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, QCD and resonance physics. Theoretical foundations, Nucl. Phys.B 147 (1979) 385 [INSPIRE].
A.H. Mueller, On the structure of infrared renormalons in physical processes at high-energies, Nucl. Phys.B 250 (1985) 327 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1905.01444
G.P. Korchemsky Unité Mixte de Recherche 3681 du CNRS.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Korchemsky, G. Energy correlations in the end-point region. J. High Energ. Phys. 2020, 8 (2020). https://doi.org/10.1007/JHEP01(2020)008
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2020)008