Abstract
The energy correlator measures the energy deposited in multiple detectors as a function of the angles among them. In this paper, an analytic formula is given for the three-point energy correlator with full angle dependence at leading order in electron-positron annihilation. This is the first analytic computation of trijet event shape observables in QCD, which provides valuable data for phenomenological studies. The result is computed with direct integration, where appropriate parameterizations of both phase space and kinematic space are adopted to simplify the calculation. With full shape dependence, our result provides the expansions in various kinematic regions such as equilateral, triple collinear and squeezed limits, which benefit studies on both factorization and large logarithm resummation.
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 in Quantum Chromodynamics: Asymptotically Free Perturbation Theory, Phys. Rev. D 19 (1979) 2018 [INSPIRE].
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].
H. Chen, M.-X. Luo, I. Moult, T.-Z. Yang, X. Zhang and H.X. Zhu, Three point energy correlators in the collinear limit: symmetries, dualities and analytic results, JHEP 08 (2020) 028 [arXiv:1912.11050] [INSPIRE].
H. Chen, I. Moult, X. Zhang and H.X. Zhu, Rethinking jets with energy correlators: Tracks, resummation, and analytic continuation, Phys. Rev. D 102 (2020) 054012 [arXiv:2004.11381] [INSPIRE].
K. Yan and X. Zhang, Three-Point Energy Correlator in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 129 (2022) 021602 [arXiv:2203.04349] [INSPIRE].
L.J. Dixon, M.-X. Luo, V. Shtabovenko, T.-Z. Yang and H.X. Zhu, 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, JHEP 06 (2019) 037 [arXiv:1903.07277] [INSPIRE].
J. Gao, V. Shtabovenko and T.-Z. Yang, Energy-energy correlation in hadronic Higgs decays: analytic results and phenomenology at NLO, JHEP 02 (2021) 210 [arXiv:2012.14188] [INSPIRE].
A.V. Belitsky, S. Hohenegger, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, 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].
C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An Effective field theory for collinear and soft gluons: Heavy to light decays, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE].
C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in \( \overrightarrow{B} \) → Xsγ in effective field theory, Phys. Rev. D 63 (2000) 014006 [hep-ph/0005275] [INSPIRE].
C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].
C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].
M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power, Nucl. Phys. B 643 (2002) 431 [hep-ph/0206152] [INSPIRE].
L.J. Dixon, I. Moult and H.X. Zhu, Collinear limit of the energy-energy correlator, Phys. Rev. D 100 (2019) 014009 [arXiv:1905.01310] [INSPIRE].
G.P. Korchemsky, Energy correlations in the end-point region, JHEP 01 (2020) 008 [arXiv:1905.01444] [INSPIRE].
J.C. Collins and D.E. Soper, Back-To-Back Jets in QCD, Nucl. Phys. B 193 (1981) 381 [Erratum ibid. 213 (1983) 545] [INSPIRE].
S.D. Ellis, D.G. Richards and W.J. Stirling, Fixed Order Perturbation Theory and Leading Logarithms, Phys. Lett. B 136 (1984) 99 [INSPIRE].
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].
Z. Tulipánt, A. Kardos and G. Somogyi, Energy-energy correlation in electron-positron annihilation at NNLL + NNLO accuracy, Eur. Phys. J. C 77 (2017) 749 [arXiv:1708.04093] [INSPIRE].
I. Moult, G. Vita and K. Yan, Subleading power resummation of rapidity logarithms: the energy-energy correlator in \( \mathcal{N} \) = 4 SYM, JHEP 07 (2020) 005 [arXiv:1912.02188] [INSPIRE].
I. Moult and H.X. Zhu, Simplicity from Recoil: The Three-Loop Soft Function and Factorization for the Energy-Energy Correlation, JHEP 08 (2018) 160 [arXiv:1801.02627] [INSPIRE].
M.A. Ebert, B. Mistlberger and G. Vita, The Energy-Energy Correlation in the back-to-back limit at N3LO and N3LL’, JHEP 08 (2021) 022 [arXiv:2012.07859] [INSPIRE].
I. Moult, H.X. Zhu and Y.J. Zhu, The Four Loop QCD Rapidity Anomalous Dimension, arXiv:2205.02249 [INSPIRE].
C. Duhr, B. Mistlberger and G. Vita, The Four-Loop Rapidity Anomalous Dimension and Event Shapes to Fourth Logarithmic Order, arXiv:2205.02242 [INSPIRE].
A. Gao, H.T. Li, I. Moult and H.X. Zhu, Precision QCD Event Shapes at Hadron Colliders: The Transverse Energy-Energy Correlator in the Back-to-Back Limit, Phys. Rev. Lett. 123 (2019) 062001 [arXiv:1901.04497] [INSPIRE].
W. Chen, Y. Li, Z. Xu, X. Zhang and H.X. Zhu, Projected three-point energy correlator at nnll, to be published soon.
P.T. Komiske, I. Moult, J. Thaler and H.X. Zhu, Analyzing N-point Energy Correlators Inside Jets with CMS Open Data, arXiv:2201.07800 [INSPIRE].
Y. Li, I. Moult, S.S. van Velzen, W.J. Waalewijn and H.X. Zhu, Extending Precision Perturbative QCD with Track Functions, Phys. Rev. Lett. 128 (2022) 182001 [arXiv:2108.01674] [INSPIRE].
M. Jaarsma, Y. Li, I. Moult, W. Waalewijn and H.X. Zhu, Renormalization group flows for track function moments, JHEP 06 (2022) 139 [arXiv:2201.05166] [INSPIRE].
H.-M. Chang, M. Procura, J. Thaler and W.J. Waalewijn, Calculating Track-Based Observables for the LHC, Phys. Rev. Lett. 111 (2013) 102002 [arXiv:1303.6637] [INSPIRE].
J. Holguin, I. Moult, A. Pathak and M. Procura, A New Paradigm for Precision Top Physics: Weighing the Top with Energy Correlators, arXiv:2201.08393 [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [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].
M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, The light-ray OPE and conformal colliders, JHEP 01 (2021) 128 [arXiv:1905.01311] [INSPIRE].
C.-H. Chang, M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, Transverse spin in the light-ray OPE, JHEP 05 (2022) 059 [arXiv:2010.04726] [INSPIRE].
H. Chen, I. Moult and H.X. Zhu, Spinning Gluons from the QCD Light-Ray OPE, arXiv:2104.00009 [INSPIRE].
H. Chen, I. Moult and H.X. Zhu, Quantum Interference in Jet Substructure from Spinning Gluons, Phys. Rev. Lett. 126 (2021) 112003 [arXiv:2011.02492] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
H. Chen, I. Moult, J. Sandor and H.X. Zhu, Celestial Blocks and Transverse Spin in the Three-Point Energy Correlator, arXiv:2202.04085 [INSPIRE].
C.-H. Chang and D. Simmons-Duffin, Three-point energy correlators and the celestial block expansion, arXiv:2202.04090 [INSPIRE].
H. Chen, I. Moult, J. Thaler and H.X. Zhu, Non-Gaussianities in collider energy flux, JHEP 07 (2022) 146 [arXiv:2205.02857] [INSPIRE].
K. Lee, B. Meçaj and I. Moult, Conformal Colliders Meet the LHC, arXiv:2205.03414 [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
T. Gehrmann and E. Remiddi, Two loop master integrals for γ* → 3 jets: The Planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
P. Nogueira, Automatic feynman graph generation, J. Comput. Phys. 105 (1993) 279.
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
T. van Ritbergen, A.N. Schellekens and J.A.M. Vermaseren, Group theory factors for Feynman diagrams, Int. J. Mod. Phys. A 14 (1999) 41 [hep-ph/9802376] [INSPIRE].
T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260] [INSPIRE].
R. Mertig, M. Böhm and A. Denner, FEYN CALC: Computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [INSPIRE].
V. Shtabovenko, R. Mertig and F. Orellana, New Developments in FeynCalc 9.0, Comput. Phys. Commun. 207 (2016) 432 [arXiv:1601.01167] [INSPIRE].
A. Pak, The Toolbox of modern multi-loop calculations: novel analytic and semi-analytic techniques, J. Phys. Conf. Ser. 368 (2012) 012049 [arXiv:1111.0868] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann and G. Heinrich, Four particle phase space integrals in massless QCD, Nucl. Phys. B 682 (2004) 265 [hep-ph/0311276] [INSPIRE].
E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148 [arXiv:1403.3385] [INSPIRE].
A.B. Goncharov, Multiple polylogarithms and mixed tate motives, math/0103059.
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
J.M. Borwein, D.M. Bradley, D.J. Broadhurst and P. Lisonek, Special values of multiple polylogarithms, Trans. Am. Math. Soc. 353 (2001) 907 [math/9910045] [INSPIRE].
C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].
C. Duhr and F. Dulat, PolyLogTools — polylogs for the masses, JHEP 08 (2019) 135 [arXiv:1904.07279] [INSPIRE].
H. Frellesvig, D. Tommasini and C. Wever, On the reduction of generalized polylogarithms to Lin and Li2,2 and on the evaluation thereof, JHEP 03 (2016) 189 [arXiv:1601.02649] [INSPIRE].
M. Heller and A. von Manteuffel, MultivariateApart: Generalized partial fractions, Comput. Phys. Commun. 271 (2022) 108174 [arXiv:2101.08283] [INSPIRE].
Z. Wojtkowiak, Functional equations of iterated integrals with regular singularities, Nagoya Math. J. 142 (1996) 145.
H.R.P. Ferguson, D.H. Bailey and S. Arno, Analysis of pslq, an integer relation finding algorithm, Math. Comput. 68 (1999) 351.
D.H. Bailey and D.J. Broadhurst, Parallel integer relation detection: Techniques and applications, Math. Comput. 70 (2001) 1719 [math/9905048] [INSPIRE].
C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, J. Symb. Comput. 33 (2002) 1 [cs/0004015].
Y. Wang, L.L. Yang and B. Zhou, FastGPL: a C++ library for fast evaluation of generalized polylogarithms, arXiv:2112.04122 [INSPIRE].
L.D. Landau, On analytic properties of vertex parts in quantum field theory, Nucl. Phys. 13 (1959) 181 [INSPIRE].
F.C.S. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 [INSPIRE].
J.L. Bourjaily, H. Hannesdottir, A.J. McLeod, M.D. Schwartz and C. Vergu, Sequential Discontinuities of Feynman Integrals and the Monodromy Group, JHEP 01 (2021) 205 [arXiv:2007.13747] [INSPIRE].
H.S. Hannesdottir, A.J. McLeod, M.D. Schwartz and C. Vergu, Implications of the Landau equations for iterated integrals, Phys. Rev. D 105 (2022) L061701 [arXiv:2109.09744] [INSPIRE].
A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].
A. Dersy, M.D. Schwartz and X. Zhang, Simplifying Polylogarithms with Machine Learning, arXiv:2206.04115 [INSPIRE].
S. Catani and M.H. Seymour, The Dipole formalism for the calculation of QCD jet cross-sections at next-to-leading order, Phys. Lett. B 378 (1996) 287 [hep-ph/9602277] [INSPIRE].
S. Catani and M.H. Seymour, A General algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291 [Erratum ibid. 510 (1998) 503] [hep-ph/9605323] [INSPIRE].
Z. Nagy, Next-to-leading order calculation of three jet observables in hadron hadron collision, Phys. Rev. D 68 (2003) 094002 [hep-ph/0307268] [INSPIRE].
T. Sjöstrand, S. Mrenna and P.Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05 (2006) 026 [hep-ph/0603175] [INSPIRE].
T. Sjöstrand et al., An introduction to PYTHIA 8.2, Comput. Phys. Commun. 191 (2015) 159 [arXiv:1410.3012] [INSPIRE].
C. Bierlich et al., A comprehensive guide to the physics and usage of PYTHIA 8.3, arXiv:2203.11601 [INSPIRE].
A. Bhattacharya, M.D. Schwartz and X. Zhang, Sudakov Shoulder Resummation for Thrust and Heavy Jet Mass, arXiv:2205.05702 [INSPIRE].
V.V. Sudakov, Vertex parts at very high-energies in quantum electrodynamics, Sov. Phys. JETP 3 (1956) 65 [INSPIRE].
A. Banfi, Y.L. Dokshitzer, G. Marchesini and G. Zanderighi, Near-to-planar three jet events in and beyond QCD perturbation theory, Phys. Lett. B 508 (2001) 269 [hep-ph/0010267] [INSPIRE].
A. Banfi, Y.L. Dokshitzer, G. Marchesini and G. Zanderighi, QCD analysis of D parameter in near to planar three jet events, JHEP 05 (2001) 040 [hep-ph/0104162] [INSPIRE].
A.J. Larkoski and A. Procita, New Insights on an Old Problem: Resummation of the D-parameter, JHEP 02 (2019) 104 [arXiv:1810.06563] [INSPIRE].
S. Catani and B.R. Webber, Infrared safe but infinite: Soft gluon divergences inside the physical region, JHEP 10 (1997) 005 [hep-ph/9710333] [INSPIRE].
T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].
T.D. Lee and M. Nauenberg, Degenerate Systems and Mass Singularities, Phys. Rev. 133 (1964) B1549 [INSPIRE].
I. Balitsky and A. Tarasov, Power corrections to TMD factorization for Z-boson production, JHEP 05 (2018) 150 [arXiv:1712.09389] [INSPIRE].
M.A. Ebert, I. Moult, I.W. Stewart, F.J. Tackmann, G. Vita and H.X. Zhu, Subleading power rapidity divergences and power corrections for qT , JHEP 04 (2019) 123 [arXiv:1812.08189] [INSPIRE].
L. Cieri, C. Oleari and M. Rocco, Higher-order power corrections in a transverse-momentum cut for colour-singlet production at NLO, Eur. Phys. J. C 79 (2019) 852 [arXiv:1906.09044] [INSPIRE].
M.A. Ebert, J.K.L. Michel, I.W. Stewart and F.J. Tackmann, Drell-Yan qT resummation of fiducial power corrections at N3LL, JHEP 04 (2021) 102 [arXiv:2006.11382] [INSPIRE].
V. Moos and A. Vladimirov, Calculation of transverse momentum dependent distributions beyond the leading power, JHEP 12 (2020) 145 [arXiv:2008.01744] [INSPIRE].
C. Oleari and M. Rocco, Power corrections in a transverse-momentum cut for vector-boson production at NNLO: the qg-initiated real-virtual contribution, Eur. Phys. J. C 81 (2021) 183 [arXiv:2012.10538] [INSPIRE].
M.A. Ebert, A. Gao and I.W. Stewart, Factorization for azimuthal asymmetries in SIDIS at next-to-leading power, JHEP 06 (2022) 007 [arXiv:2112.07680] [INSPIRE].
A. Gehrmann-De Ridder and E.W.N. Glover, A Complete O(ααs) calculation of the photon +1 jet rate in e+e− annihilation, Nucl. Phys. B 517 (1998) 269 [hep-ph/9707224] [INSPIRE].
M. Ritzmann and W.J. Waalewijn, Fragmentation in Jets at NNLO, Phys. Rev. D 90 (2014) 054029 [arXiv:1407.3272] [INSPIRE].
J.M. Campbell and E.W.N. Glover, Double unresolved approximations to multiparton scattering amplitudes, Nucl. Phys. B 527 (1998) 264 [hep-ph/9710255] [INSPIRE].
S. Catani and M. Grazzini, Collinear factorization and splitting functions for next-to-next-to-leading order QCD calculations, Phys. Lett. B 446 (1999) 143 [hep-ph/9810389] [INSPIRE].
T. Gehrmann, A. von Manteuffel and T.-Z. Yang, Renormalization of twist-two operators in QCD and its application to singlet splitting functions, in 16th DESY Workshop on Elementary Particle Physics: Loops and Legs in Quantum Field Theory 2022, Ettal Germany, April 25–30 2022 [arXiv:2207.10108] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2208.01051
Supplementary Information
ESM 1
(ZIP 1401 kb)
Rights and permissions
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.
About this article
Cite this article
Yang, TZ., Zhang, X. Analytic Computation of three-point energy correlator in QCD. J. High Energ. Phys. 2022, 6 (2022). https://doi.org/10.1007/JHEP09(2022)006
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2022)006