Abstract
By quantifying the distance between two collider events, one can triangulate a metric space and reframe collider data analysis as computational geometry. One popular geometric approach is to first represent events as an energy flow on an idealized celestial sphere and then define the metric in terms of optimal transport in two dimensions. In this paper, we advocate for representing events in terms of a spectral function that encodes pairwise particle angles and products of particle energies, which enables a metric distance defined in terms of one-dimensional optimal transport. This approach has the advantage of automatically incorporating obvious isometries of the data, like rotations about the colliding beam axis. It also facilitates first-principles calculations, since there are simple closed-form expressions for optimal transport in one dimension. Up to isometries and event sets of measure zero, the spectral representation is unique, so the metric on the space of spectral functions is a metric on the space of events. At lowest order in perturbation theory in electron-positron collisions, our metric is simply the summed squared invariant masses of the two event hemispheres. Going to higher orders, we present predictions for the distribution of metric distances between jets in fixed-order and resummed perturbation theory as well as in parton-shower generators. Finally, we speculate on whether the spectral approach could furnish a useful metric on the space of quantum field theories.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.J. Larkoski, I. Moult and B. Nachman, Jet Substructure at the Large Hadron Collider: A Review of Recent Advances in Theory and Machine Learning, Phys. Rept. 841 (2020) 1 [arXiv:1709.04464] [INSPIRE].
D. Guest, K. Cranmer and D. Whiteson, Deep Learning and its Application to LHC Physics, Ann. Rev. Nucl. Part. Sci. 68 (2018) 161 [arXiv:1806.11484] [INSPIRE].
K. Albertsson et al., Machine Learning in High Energy Physics Community White Paper, J. Phys. Conf. Ser. 1085 (2018) 022008 [arXiv:1807.02876] [INSPIRE].
A. Radovic et al., Machine learning at the energy and intensity frontiers of particle physics, Nature 560 (2018) 41 [INSPIRE].
G. Carleo et al., Machine learning and the physical sciences, Rev. Mod. Phys. 91 (2019) 045002 [arXiv:1903.10563] [INSPIRE].
D. Bourilkov, Machine and Deep Learning Applications in Particle Physics, Int. J. Mod. Phys. A 34 (2020) 1930019 [arXiv:1912.08245] [INSPIRE].
M.D. Schwartz, Modern Machine Learning and Particle Physics, arXiv:2103.12226 [DOI:10.1162/99608f92.beeb1183] [INSPIRE].
G. Karagiorgi et al., Machine Learning in the Search for New Fundamental Physics, arXiv:2112.03769 [INSPIRE].
T. Plehn, A. Butter, B. Dillon and C. Krause, Modern Machine Learning for LHC Physicists, arXiv:2211.01421 [INSPIRE].
P.T. Komiske, E.M. Metodiev and J. Thaler, Metric Space of Collider Events, Phys. Rev. Lett. 123 (2019) 041801 [arXiv:1902.02346] [INSPIRE].
G. Monge, Mémoire sur la théorie des déblais et des remblais, Mem. Math. Phys. Acad. Royale Sci. (1781) 666.
L.V. Kantorovich, The mathematical method of production planning and organization, Manage. Sci. 6 (1939) 363.
L.N. Wasserstein, Markov processes on countable product space describing large systems of automata, Probl. Pered. Inform 5 (1969) 64.
R.L. Dobrushin, Prescribing a system of random variables by conditional distributions, Theory Probab. Appl. 15 (1970) 458.
P.T. Komiske, E.M. Metodiev and J. Thaler, The Hidden Geometry of Particle Collisions, JHEP 07 (2020) 006 [arXiv:2004.04159] [INSPIRE].
C. Cesarotti and J. Thaler, A Robust Measure of Event Isotropy at Colliders, JHEP 08 (2020) 084 [arXiv:2004.06125] [INSPIRE].
D. Ba et al., SHAPER: can you hear the shape of a jet?, JHEP 06 (2023) 195 [arXiv:2302.12266] [INSPIRE].
P.T. Komiske et al., Exploring the Space of Jets with CMS Open Data, Phys. Rev. D 101 (2020) 034009 [arXiv:1908.08542] [INSPIRE].
P.T. Komiske, S. Kryhin and J. Thaler, Disentangling quarks and gluons in CMS open data, Phys. Rev. D 106 (2022) 094021 [arXiv:2205.04459] [INSPIRE].
T. Cai, J. Cheng, N. Craig and K. Craig, Linearized optimal transport for collider events, Phys. Rev. D 102 (2020) 116019 [arXiv:2008.08604] [INSPIRE].
T. Cai, J. Cheng, K. Craig and N. Craig, Which metric on the space of collider events?, Phys. Rev. D 105 (2022) 076003 [arXiv:2111.03670] [INSPIRE].
A. Mullin et al., Does SUSY have friends? A new approach for LHC event analysis, JHEP 02 (2021) 160 [arXiv:1912.10625] [INSPIRE].
M. Crispim Romão et al., Use of a generalized energy Mover’s distance in the search for rare phenomena at colliders, Eur. Phys. J. C 81 (2021) 192 [arXiv:2004.09360] [INSPIRE].
A. Davis, T. Menzo, A. Youssef and J. Zupan, Earth mover’s distance as a measure of CP violation, JHEP 06 (2023) 098 [arXiv:2301.13211] [INSPIRE].
S. Alipour-Fard, P.T. Komiske, E.M. Metodiev and J. Thaler, Pileup and Infrared Radiation Annihilation (PIRANHA): A Paradigm for Continuous Jet Grooming, arXiv:2305.00989 [INSPIRE].
S. Tsan et al., Particle Graph Autoencoders and Differentiable, Learned Energy Mover’s Distance, in the proceedings of the 35th Conference on Neural Information Processing Systems, Online Canada, December 6–14 (2021) [arXiv:2111.12849] [INSPIRE].
O. Kitouni, N. Nolte and M. Williams, Finding NEEMo: Geometric Fitting using Neural Estimation of the Energy Mover’s Distance, arXiv:2209.15624 [INSPIRE].
A.J. Larkoski and T. Melia, Covariantizing phase space, Phys. Rev. D 102 (2020) 094014 [arXiv:2008.06508] [INSPIRE].
De investigando ordine systematis aequationum differentialium vulgarium cujuscunque, in C. G. J. Jacobi’s Gesammelte Werke, Karl Weierstrass ed., (1890) [Cambridge University Press (2013), p. 191–216] [DOI:10.1017/cbo9781139567992.003].
De aequationum differentialium systemate non normali ad formam normalem revocando, in C. G. J. Jacobi’s Gesammelte Werke, Karl Weierstrass ed., (1890) [Cambridge University Press (2013), pp. 483–513] [DOI:10.1017/cbo9781139567992.008].
H.W. Kuhn, The hungarian method for the assignment problem, Nav. Res. Logist. Q. 2 (1955) 83.
J. Rabin, G. Peyré, J. Delon and M. Bernot, Wasserstein barycenter and its application to texture mixing, in International Conference on Scale Space and Variational Methods in Computer Vision, Springer (2011), pp. 435–446 [DOI:https://doi.org/10.1007/978-3-642-24785-9_37].
N. Bonneel, J. Rabin, G. Peyré and H. Pfister, Sliced and radon wasserstein barycenters of measures, J. Math. Imaging Vision 51 (2015) 22.
O. Pele and B. Taskar, The tangent earth mover’s distance, in Geometric Science of Information, F. Nielsen and F. Barbaresco eds., Springer Berlin Heidelberg (2013), pp. 397–404 [DOI:https://doi.org/10.1007/978-3-642-40020-9_43].
S. Catani and M.H. Seymour, A General algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291 [hep-ph/9605323] [INSPIRE].
M. Dasgupta and G.P. Salam, Resummation of nonglobal QCD observables, Phys. Lett. B 512 (2001) 323 [hep-ph/0104277] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
D. Kutasov, Geometry on the Space of Conformal Field Theories and Contact Terms, Phys. Lett. B 220 (1989) 153 [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].
F.V. Tkachov, Measuring multi-jet structure of hadronic energy flow or What is a jet?, Int. J. Mod. Phys. A 12 (1997) 5411 [hep-ph/9601308] [INSPIRE].
M. Jankowiak and A.J. Larkoski, Jet Substructure Without Trees, JHEP 06 (2011) 057 [arXiv:1104.1646] [INSPIRE].
H. Chen, I. Moult, X.Y. 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].
A. Chakraborty, S.H. Lim and M.M. Nojiri, Interpretable deep learning for two-prong jet classification with jet spectra, JHEP 07 (2019) 135 [arXiv:1904.02092] [INSPIRE].
M. Boutin and G. Kemper, On reconstructing n-point configurations from the distribution of distances or areas, math/0304192 [DOI:10.48550/arXiv.math/0304192].
A. Banfi, G.P. Salam and G. Zanderighi, Principles of general final-state resummation and automated implementation, JHEP 03 (2005) 073 [hep-ph/0407286] [INSPIRE].
A.J. Larkoski, G.P. Salam and J. Thaler, Energy Correlation Functions for Jet Substructure, JHEP 06 (2013) 108 [arXiv:1305.0007] [INSPIRE].
R. Kelley, M.D. Schwartz, R.M. Schabinger and H.X. Zhu, The two-loop hemisphere soft function, Phys. Rev. D 84 (2011) 045022 [arXiv:1105.3676] [INSPIRE].
A. Hornig et al., Non-global Structure of the \( \mathcal{O}\left({\alpha}_s^2\right) \) Dijet Soft Function, JHEP 08 (2011) 054 [Erratum ibid. 10 (2017) 101] [arXiv:1105.4628] [INSPIRE].
J. Alwall et al., The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079 [arXiv:1405.0301] [INSPIRE].
C. Bierlich et al., A comprehensive guide to the physics and usage of PYTHIA 8.3, arXiv:2203.11601 [DOI:10.21468/SciPostPhysCodeb.8] [INSPIRE].
S. Catani et al., New clustering algorithm for multi-jet cross-sections in e+e− annihilation, Phys. Lett. B 269 (1991) 432 [INSPIRE].
M. Cacciari, G.P. Salam and G. Soyez, FastJet User Manual, Eur. Phys. J. C 72 (2012) 1896 [arXiv:1111.6097] [INSPIRE].
L. Clavelli, Jet Invariant Mass in Quantum Chromodynamics, Phys. Lett. B 85 (1979) 111 [INSPIRE].
S. Catani, L. Trentadue, G. Turnock and B.R. Webber, Resummation of large logarithms in e+e− event shape distributions, Nucl. Phys. B 407 (1993) 3 [INSPIRE].
E. Farhi, A QCD Test for Jets, Phys. Rev. Lett. 39 (1977) 1587 [INSPIRE].
C.F. Berger, T. Kucs and G.F. Sterman, Event shape/energy flow correlations, Phys. Rev. D 68 (2003) 014012 [hep-ph/0303051] [INSPIRE].
S.D. Ellis et al., Jet Shapes and Jet Algorithms in SCET, JHEP 11 (2010) 101 [arXiv:1001.0014] [INSPIRE].
Y.L. Dokshitzer and B.R. Webber, Calculation of power corrections to hadronic event shapes, Phys. Lett. B 352 (1995) 451 [hep-ph/9504219] [INSPIRE].
C. Lee and G.F. Sterman, Universality of nonperturbative effects in event shapes, eConf C0601121 (2006) A001 [hep-ph/0603066] [INSPIRE].
G. Salam, The E∞ Scheme, unpublished.
D. Bertolini, T. Chan and J. Thaler, Jet Observables Without Jet Algorithms, JHEP 04 (2014) 013 [arXiv:1310.7584] [INSPIRE].
A.J. Larkoski, D. Neill and J. Thaler, Jet Shapes with the Broadening Axis, JHEP 04 (2014) 017 [arXiv:1401.2158] [INSPIRE].
S. Brandt and H.D. Dahmen, Axes and Scalar Measures of Two-Jet and Three-Jet Events, Z. Phys. C 1 (1979) 61 [INSPIRE].
I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, N-Jettiness: An Inclusive Event Shape to Veto Jets, Phys. Rev. Lett. 105 (2010) 092002 [arXiv:1004.2489] [INSPIRE].
J. Thaler and K. Van Tilburg, Identifying Boosted Objects with N-subjettiness, JHEP 03 (2011) 015 [arXiv:1011.2268] [INSPIRE].
J.-H. Kim, Rest Frame Subjet Algorithm With SISCone Jet For Fully Hadronic Decaying Higgs Search, Phys. Rev. D 83 (2011) 011502 [arXiv:1011.1493] [INSPIRE].
J. Thaler and K. Van Tilburg, Maximizing Boosted Top Identification by Minimizing N-subjettiness, JHEP 02 (2012) 093 [arXiv:1108.2701] [INSPIRE].
L. de Oliveira et al., Jet-images — deep learning edition, JHEP 07 (2016) 069 [arXiv:1511.05190] [INSPIRE].
M.R. Douglas, Spaces of Quantum Field Theories, J. Phys. Conf. Ser. 462 (2013) 012011 [arXiv:1005.2779] [INSPIRE].
J. Erdmenger, K.T. Grosvenor and R. Jefferson, Information geometry in quantum field theory: lessons from simple examples, SciPost Phys. 8 (2020) 073 [arXiv:2001.02683] [INSPIRE].
J. Polchinski, Renormalization and Effective Lagrangians, Nucl. Phys. B 231 (1984) 269 [INSPIRE].
J. Cotler and S. Rezchikov, Renormalization group flow as optimal transport, Phys. Rev. D 108 (2023) 025003 [arXiv:2202.11737] [INSPIRE].
H. Whitney, Differentiable manifolds, Annals Math. (1936) 645.
V.V. Prasolov, Elements of homology theory, American Mathematical Society (2007) [DOI:https://doi.org/10.1090/gsm/081/03].
M. Adachi, Embeddings and immersions, American Mathematical Society (2012) [DOI:https://doi.org/10.1090/mmono/124].
J. Nash, C1 isometric imbeddings, Annals Math. 60 (1954) 383.
N.H. Kuiper, On c1-isometric imbeddings. I, Indag. Math. (Proceedings) 58 (1955) 545.
N.H. Kuiper, On c1-isometric imbeddings. II, Indag. Math. (Proceedings) 58 (1955) 683.
J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, math/0412127.
J. Lott, Some geometric calculations on Wasserstein space, math/0612562.
C. Villani, Optimal Transport: Old and New, Grundlehren der mathematischen Wissenschaften 338, Springer Berlin Heidelberg (2008) [DOI:https://doi.org/10.1007/978-3-540-71050-9].
D. Neill and W.J. Waalewijn, Entropy of a Jet, Phys. Rev. Lett. 123 (2019) 142001 [arXiv:1811.01021] [INSPIRE].
C. Cheung, T. He and A. Sivaramakrishnan, On Entropy Growth in Perturbative Scattering, arXiv:2304.13052 [INSPIRE].
J. Holguin, I. Moult, A. Pathak and M. Procura, New paradigm for precision top physics: Weighing the top with energy correlators, Phys. Rev. D 107 (2023) 114002 [arXiv:2201.08393] [INSPIRE].
J. Batson, C.G. Haaf, Y. Kahn and D.A. Roberts, Topological Obstructions to Autoencoding, JHEP 04 (2021) 280 [arXiv:2102.08380] [INSPIRE].
J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, math/0412127 [DOI:10.48550/arXiv.math/0412127].
B. Henning, X. Lu, T. Melia and H. Murayama, Hilbert series and operator bases with derivatives in effective field theories, Commun. Math. Phys. 347 (2016) 363 [arXiv:1507.07240] [INSPIRE].
P.T. Komiske, E.M. Metodiev and J. Thaler, Energy flow polynomials: A complete linear basis for jet substructure, JHEP 04 (2018) 013 [arXiv:1712.07124] [INSPIRE].
P.T. Komiske, E.M. Metodiev and J. Thaler, Cutting Multiparticle Correlators Down to Size, Phys. Rev. D 101 (2020) 036019 [arXiv:1911.04491] [INSPIRE].
P. Cal, J. Thaler and W.J. Waalewijn, Power counting energy flow polynomials, JHEP 09 (2022) 021 [arXiv:2205.06818] [INSPIRE].
Acknowledgments
We thank Rikab Gambhir for comments on the manuscript and Nathaniel Craig for suggesting comparing to the 2-Wasserstein distance. This work was supported in part by the UC Southern California Hub, with funding from the UC National Laboratories division of the University of California Office of the President. J.T. was supported by the U.S. Department of Energy (DOE) Office of High Energy Physics under Grant Contract No. DE-SC0012567, by the DOE QuantISED program through the theory consortium “Intersections of QIS and Theoretical Particle Physics” at Fermilab (FNAL 20-17), and by the National Science Foundation under Cooperative Agreement No. PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, http://iaifi.org/).
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: 2305.03751
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
Larkoski, A.J., Thaler, J. A spectral metric for collider geometry. J. High Energ. Phys. 2023, 107 (2023). https://doi.org/10.1007/JHEP08(2023)107
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2023)107