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  • Open Access

Novel implementation of the matrix element method at next-to-leading order for the measurement of the Higgs self-coupling λ3H

Matthias Tartarin* and Jan Stark†

  • *Contact author: matthias.tartarin@l2it.in2p3.fr
  • †Contact author: jan.stark@l2it.in2p3.fr
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Phys. Rev. D 113, 113010 – Published 15 June, 2026

DOI: https://doi.org/10.1103/r6p3-d5wg

Abstract

The determination of the Higgs boson trilinear self-coupling λ3H is a key goal of the LHC physics program. Its precise measurement will provide unique insight into the scalar potential and the mechanism of electroweak symmetry breaking. Higgs boson pair production in the gg→HH process, and particularly in the HH→bb¯γγ final state, offers direct sensitivity to λ3H. We present the first implementation of the matrix element method (MEM) at next-to-leading order (NLO) for this process, which is publicly available. The MEM is a statistically optimal approach that maximizes information extraction from collision events. Extending it to NLO represents a major methodological challenge, which we address with a new formalism integrated into the momemta framework. Results with simulated pseudoexperiments demonstrate, in a proof-of-principle study, the strong discriminating power of the method and its ability to extract the coupling modifier κλ=λ3H/λ3HSM with high precision.

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References (47)

  1. ATLAS Collaboration, Observation of a new particle in the search for the standard model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716, 1 (2012).
  2. CMS Collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716, 30 (2012).
  3. ATLAS Collaboration, The ATLAS experiment at the CERN large hadron collider, J. Instrum. 3, S08003 (2008).
  4. CMS Collaboration, The CMS experiment at the CERN LHC, J. Instrum. 3, S08004 (2008).
  5. ATLAS Collaboration, Projected sensitivity of searches for Higgs boson pair production in final states with light leptons, taus, and photons with the ATLAS detector at the HL-LHC, Technical Report No. ATL-PHYS-PUB-2025-004, CERN, Geneva, 2025, https://cds.cern.ch/record/2925675.
  6. ATLAS and CMS Collaborations, Highlights of the HL-LHC physics projections by ATLAS and CMS, arXiv:2504.00672.
  7. M. Reichert, A. Eichhorn, H. Gies, J. M. Pawlowski, T. Plehn, and M. Scherer, Probing baryogenesis through the Higgs boson self-coupling, Phys. Rev. D 97, 075008 (2018).
  8. ATLAS Collaboration, Combination of searches for Higgs boson pair production in pp collisions at s=13  TeV with the ATLAS detector, Phys. Rev. Lett. 133, 101801 (2024).
  9. CMS Collaboration, Combination of searches for nonresonant Higgs boson pair production in proton-proton collisions at s=13  TeV, arXiv:2510.07527. CMS Collaboration[Rep. Prog. Phys. (to be published)].
  10. ATLAS Collaboration, Study of Higgs boson pair production in the HH→bb¯γγ final state with 308  fb−1 of data collected at s=13  TeV and 13.6 TeV by the ATLAS experiment, Phys. Lett. B 876, 140280 (2026).
  11. K. Kondo, Dynamical likelihood method for reconstruction of events with missing momentum. I. Method and toy models, J. Phys. Soc. Jpn. 57, 4126 (1988).
  12. K. Kondo, Dynamical likelihood method for reconstruction of events with missing momentum. II. Mass spectra for 2→2 processes, J. Phys. Soc. Jpn. 60, 836 (1991).
  13. DØ Collaboration, Measurement of the top quark mass using dilepton events, Phys. Rev. Lett. 80, 2063 (1998).
  14. F. Canelli, Helicity of the W boson in single-lepton tt¯ events, Ph.D. thesis, Rochester University, 2003, 10.1142/S0217751X03016689.
  15. DØCollaboration, A precision measurement of the mass of the top quark, Nature (London) 429, 638 (2004).
  16. CMS Collaboration, Measurement of the Higgs boson mass and σ(pp→H) using the H→ZZ→4ℓ and H→γγ channels, Phys. Lett. B 759, 321 (2016).
  17. ATLAS Collaboration, Evidence for single top-quark production in the s-channel in proton–proton collisions at s=8  TeV with the ATLAS detector using the matrix element method, Phys. Lett. B 756, 228 (2016).
  18. J. Alwall, A. Freitas, and O. Mattelaer, Matrix element method and QCD radiation, Phys. Rev. D 83, 074010 (2011).
  19. J. M. Campbell, W. T. Giele, and C. Williams, The matrix element method at next-to-leading order, J. High Energy Phys. 11 (2012) 043.
  20. T. Martini and P. Uwer, The matrix element method at next-to-leading order QCD for hadronic collisions: Single top-quark production at the LHC as an example application, J. High Energy Phys. 05 (2018) 141.
  21. T. Martini, T. Nuraliyev, and P. Uwer, Determination of the top-quark mass from top-quark pair events with the matrix element method at next-to-leading order: Potential and prospects, arXiv:2301.03280.
  22. M. Tartarin, Measurement of the Higgs self-coupling using the matrix element method at next-to-leading order (NLO) in the gg→HH→bb¯γγ channel for the ATLAS experiment at the LHC, Ph.D. thesis, Université de Toulouse, 2025, 10.17181/8hbdr-mgg27.
  23. F. Eble and J. Stark, Prospective study for the development of an analysis to measure the tri-linear Higgs coupling using the matrix element method with the ATLAS experiment at the HL-LHC, Master thesis, KTH Royal Institute of Technology, 2020, https://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-291100.
  24. S. Wertz et al., momemta: A modular matrix element method framework, Eur. Phys. J. C 77, 708 (2017).
  25. P. Artoisenet, V. Lemaitre, F. Maltoni, and O. Mattelaer, Automation of the matrix element reweighting method, J. High Energy Phys. 12 (2010) 068.
  26. S. Alioli, P. Nason, C. Oleari, and E. Re, A general framework for implementing NLO calculations in shower Monte Carlo programs: The POWHEG BOX, J. High Energy Phys. 06 (2010) 043.
  27. G. Heinrich, S. P. Jones, M. Kerner, G. Luisoni, and E. Vryonidou, NLO predictions for Higgs boson pair production with full top quark mass dependence matched to parton showers, J. High Energy Phys. 08 (2017) 088.
  28. G. Heinrich, S. P. Jones, M. Kerner, G. Luisoni, and L. Scyboz, Probing the trilinear Higgs boson coupling in di-Higgs production at NLO QCD including parton shower effects, J. High Energy Phys. 06 (2019) 066.
  29. T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, and P. Z. Skands, An introduction to pythia 8.2, Comput. Phys. Commun. 191, 159 (2015).
  30. M. Tartarin and J. Stark, MEM@NLO ggHH auxiliary material and code (GitLab repository), https://gitlab.com/mtartarin/phd_share.
  31. J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H.-S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, J. High Energy Phys. 07 (2014) 079.
  32. T. Hahn, cuba—A library for multidimensional numerical integration, Comput. Phys. Commun. 168, 78 (2005).
  33. M. Ubiali, Modern machine learning and particle physics phenomenology at the LHC, Summary Talk Given at EuCAIFCon 2025 (European AI for Fundamental Physics Conference), Cagliari, Italy, 2025, https://agenda.infn.it/event/43565/contributions/268339/.
  34. F. Bishara and M. Montull, (Machine) learning amplitudes for faster event generation, Phys. Rev. D 107, L071901 (2023).
  35. S. Badger and J. Bullock, Using neural networks for efficient evaluation of high multiplicity scattering amplitudes, J. High Energy Phys. 06 (2020) 114.
  36. A. Buckley, A. Kvellestad, Å. Raklev, P. Scott, J. V. Sparre, J. Van den Abeele, and I. A. Vazquez-Holm, Xsec: The cross-section evaluation code, Eur. Phys. J. C 80, 1106 (2020).
  37. D. L. B. Sombillo, Y. Ikeda, T. Sato, and A. Hosaka, Model independent analysis of coupled-channel scattering: A deep learning approach, Phys. Rev. D 104, 036001 (2021).
  38. J. Aylett-Bullock, S. Badger, and R. Moodie, Optimising simulations for diphoton production at hadron colliders using amplitude neural networks, J. High Energy Phys. 08 (2021) 066.
  39. K. Danziger, T. Janßen, S. Schumann, and F. Siegert, Accelerating Monte Carlo event generation—rejection sampling using neural network event-weight estimates, SciPost Phys. 12, 164 (2022).
  40. A. Alnuqaydan, S. Gleyzer, and H. Prosper, SYMBA: Symbolic computation of squared amplitudes in high energy physics with machine learning, Mach. Learn. Sci. Tech. 4, 015007 (2023).
  41. D. Maître and H. Truong, One-loop matrix element emulation with factorisation awareness, J. High Energy Phys. 05 (2023) 159.
  42. T. Janßen, D. Maître, S. Schumann, F. Siegert, and H. Truong, Unweighting multijet event generation using factorisation-aware neural networks, SciPost Phys. 15, 107 (2023).
  43. T. Heimel, N. Huetsch, R. Winterhalder, T. Plehn, and A. Butter, Precision-machine learning for the matrix element method, SciPost Phys. 17, 129 (2024).
  44. A. Butter, T. Heimel, T. Martini, S. Peitzsch, and T. Plehn, Two invertible networks for the matrix element method, SciPost Phys. 15, 094 (2023).
  45. P. Chang, S. Gleyzer, M. Neubauer, and D. Zhong, Sustainable matrix element method through deep learning, Zenodo, 10.5281/zenodo.4008241 (2017).
  46. F. Bury and C. Delaere, Matrix element regression with deep neural networks—breaking the CPU barrier, J. High Energy Phys. 04 (2021) 020.
  47. M. Feickert, M. Katare, M. Neubauer, and A. Roy, Deep learning for the matrix element method, Proc. Sci. ICHEP2022 (2022) 246 [arXiv:2211.11910].

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Phys. Rev. D 113, 113010– Published 15 June, 2026
  • Received 2 February 2026
  • Accepted 12 May 2026
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