Constraining the Higgs self-couplings at e+e colliders

  • Fabio Maltoni
  • Davide PaganiEmail author
  • Xiaoran Zhao
Open Access
Regular Article - Theoretical Physics


We study the sensitivity to the shape of the Higgs potential of single, double, and triple Higgs production at future e+e colliders. Physics beyond the Standard Model is parameterised through the inclusion of higher-dimensional operators (ΦΦ−v2/2)n/Λ(2n−4) with n = 3, 4, which allows a consistent treatment of independent deviations of the cubic and quartic self-couplings beyond the tree level. We calculate the effects induced by a modified potential up to one loop in single and double Higgs production and at the tree level in triple Higgs production, for both Z boson associated and W boson fusion production mechanisms. We consider two different scenarios. First, the dimension six operator provides the dominant contribution (as expected, for instance, in a linear effective-field-theory (EFT)); we find in this case that the corresponding Wilson coefficient can be determined at \( \mathcal{O}\left(10\%\right) \) accuracy by just combining accurate measurements of single Higgs cross sections at \( \sqrt{\widehat{s}}=240-250 \) GeV and double Higgs production in W boson fusion at higher energies. Second, both operators of dimension six and eight can give effects of similar order, i.e., independent quartic self-coupling deviations are present. Constraints on Wilson coefficients can be best tested by combining measurements from single, double and triple Higgs production. Given that the sensitivity of single Higgs production to the dimension eight operator is presently unknown, we consider double and triple Higgs production and show that combining their information colliders at higher energies will provide first coarse constraints on the corresponding Wilson coefficient.


NLO Computations Phenomenological Models 


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.


  1. [1]
    CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
  2. [2]
    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 (2012) 1 [arXiv:1207.7214] [INSPIRE].
  3. [3]
    ATLAS and CMS collaborations, Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at \( \sqrt{s}=7 \) and 8 TeV, JHEP 08 (2016) 045 [arXiv:1606.02266] [INSPIRE].
  4. [4]
    U. Baur, T. Plehn and D.L. Rainwater, Probing the Higgs selfcoupling at hadron colliders using rare decays, Phys. Rev. D 69 (2004) 053004 [hep-ph/0310056] [INSPIRE].
  5. [5]
    M.J. Dolan, C. Englert and M. Spannowsky, Higgs self-coupling measurements at the LHC, JHEP 10 (2012) 112 [arXiv:1206.5001] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    A. Papaefstathiou, L.L. Yang and J. Zurita, Higgs boson pair production at the LHC in the \( b\overline{b}{W}^{+}{W}^{-} \) channel, Phys. Rev. D 87 (2013) 011301 [arXiv:1209.1489] [INSPIRE].
  7. [7]
    J. Baglio, A. Djouadi, R. Gröber, M.M. Mühlleitner, J. Quevillon and M. Spira, The measurement of the Higgs self-coupling at the LHC: theoretical status, JHEP 04 (2013) 151 [arXiv:1212.5581] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    W. Yao, Studies of measuring Higgs self-coupling with \( HH\to b\overline{b}\gamma \gamma \) at the future hadron colliders, in Proceedings, 2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on the Mississippi (CSS2013): Minneapolis, MN, U.S.A., July 29 - August 6, 2013, 2013, arXiv:1308.6302 [INSPIRE].
  9. [9]
    V. Barger, L.L. Everett, C.B. Jackson and G. Shaughnessy, Higgs-Pair Production and Measurement of the Triscalar Coupling at LHC(8,14), Phys. Lett. B 728 (2014) 433 [arXiv:1311.2931] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    D.E. Ferreira de Lima, A. Papaefstathiou and M. Spannowsky, Standard model Higgs boson pair production in the \( \left(b\overline{b}\right)\left(b\overline{b}\right) \) final state, JHEP 08 (2014) 030 [arXiv:1404.7139] [INSPIRE].
  11. [11]
    C. Englert, F. Krauss, M. Spannowsky and J. Thompson, Di-Higgs phenomenology in \( t\overline{t}hh \) : The forgotten channel, Phys. Lett. B 743 (2015) 93 [arXiv:1409.8074] [INSPIRE].
  12. [12]
    D. Wardrope, E. Jansen, N. Konstantinidis, B. Cooper, R. Falla and N. Norjoharuddeen, Non-resonant Higgs-pair production in the \( b\overline{b}\ b\overline{b} \) final state at the LHC, Eur. Phys. J. C 75 (2015) 219 [arXiv:1410.2794] [INSPIRE].
  13. [13]
    T. Liu and H. Zhang, Measuring Di-Higgs Physics via the \( t\overline{t}hh\to t\overline{t}b\overline{b}b\overline{b} \) Channel, arXiv:1410.1855 [INSPIRE].
  14. [14]
    A. Azatov, R. Contino, G. Panico and M. Son, Effective field theory analysis of double Higgs boson production via gluon fusion, Phys. Rev. D 92 (2015) 035001 [arXiv:1502.00539] [INSPIRE].ADSGoogle Scholar
  15. [15]
    Q. Li, Z. Li, Q.-S. Yan and X. Zhao, Probe Higgs boson pair production via the 32j+ Open image in new window mode, Phys. Rev. D 92 (2015) 014015 [arXiv:1503.07611] [INSPIRE].
  16. [16]
    C.-T. Lu, J. Chang, K. Cheung and J.S. Lee, An exploratory study of Higgs-boson pair production, JHEP 08 (2015) 133 [arXiv:1505.00957] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    Q.-H. Cao, B. Yan, D.-M. Zhang and H. Zhang, Resolving the Degeneracy in Single Higgs Production with Higgs Pair Production, Phys. Lett. B 752 (2016) 285 [arXiv:1508.06512] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    Q.-H. Cao, Y. Liu and B. Yan, Measuring trilinear Higgs coupling in WHH and ZHH productions at the high-luminosity LHC, Phys. Rev. D 95 (2017) 073006 [arXiv:1511.03311] [INSPIRE].ADSGoogle Scholar
  19. [19]
    J.K. Behr, D. Bortoletto, J.A. Frost, N.P. Hartland, C. Issever and J. Rojo, Boosting Higgs pair production in the \( b\overline{b}b\overline{b} \) final state with multivariate techniques, Eur. Phys. J. C 76 (2016) 386 [arXiv:1512.08928] [INSPIRE].
  20. [20]
    Q.-H. Cao, G. Li, B. Yan, D.-M. Zhang and H. Zhang, Double Higgs production at the 14 TeV LHC and a 100 TeV pp collider, Phys. Rev. D 96 (2017) 095031 [arXiv:1611.09336] [INSPIRE].ADSGoogle Scholar
  21. [21]
    A. Adhikary, S. Banerjee, R.K. Barman, B. Bhattacherjee and S. Niyogi, Revisiting the non-resonant Higgs pair production at the HL-LHC, arXiv:1712.05346 [INSPIRE].
  22. [22]
    CMS collaboration, Search for Higgs boson pair production in the final state containing two photons and two bottom quarks in proton-proton collisions at \( \sqrt{s}=13 \) TeV, CMS-PAS-HIG-17-008 [INSPIRE].
  23. [23]
    ATLAS collaboration, Prospects for measuring Higgs pair production in the channel \( H\left(\to \gamma \gamma \right)H\left(\to b\overline{b}\right) \) using the ATLAS detector at the HL-LHC, ATL-PHYS-PUB-2014-019 (2014).
  24. [24]
    T. Plehn and M. Rauch, The quartic Higgs coupling at hadron colliders, Phys. Rev. D 72 (2005) 053008 [hep-ph/0507321] [INSPIRE].
  25. [25]
    T. Binoth, S. Karg, N. Kauer and R. Ruckl, Multi-Higgs boson production in the Standard Model and beyond, Phys. Rev. D 74 (2006) 113008 [hep-ph/0608057] [INSPIRE].
  26. [26]
    F. Maltoni, E. Vryonidou and M. Zaro, Top-quark mass effects in double and triple Higgs production in gluon-gluon fusion at NLO, JHEP 11 (2014) 079 [arXiv:1408.6542] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    J. Baglio, A. Djouadi and J. Quevillon, Prospects for Higgs physics at energies up to 100 TeV, Rept. Prog. Phys. 79 (2016) 116201 [arXiv:1511.07853] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    C.-Y. Chen, Q.-S. Yan, X. Zhao, Y.-M. Zhong and Z. Zhao, Probing triple-Higgs productions via 4b2γ decay channel at a 100 TeV hadron collider, Phys. Rev. D 93 (2016) 013007 [arXiv:1510.04013] [INSPIRE].ADSGoogle Scholar
  29. [29]
    W. Kilian, S. Sun, Q.-S. Yan, X. Zhao and Z. Zhao, New Physics in multi-Higgs boson final states, JHEP 06 (2017) 145 [arXiv:1702.03554] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    B. Fuks, J.H. Kim and S.J. Lee, Scrutinizing the Higgs quartic coupling at a future 100 TeV proton-proton collider with taus and b-jets, Phys. Lett. B 771 (2017) 354 [arXiv:1704.04298] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    M. McCullough, An Indirect Model-Dependent Probe of the Higgs Self-Coupling, Phys. Rev. D 90 (2014) 015001 [Erratum ibid. D 92 (2015) 039903] [arXiv:1312.3322] [INSPIRE].
  32. [32]
    J.R. Ellis, M.K. Gaillard and D.V. Nanopoulos, A Phenomenological Profile of the Higgs Boson, Nucl. Phys. B 106 (1976) 292 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    B.W. Lee, C. Quigg and H.B. Thacker, Weak Interactions at Very High-Energies: The Role of the Higgs Boson Mass, Phys. Rev. D 16 (1977) 1519 [INSPIRE].ADSGoogle Scholar
  34. [34]
    B.L. Ioffe and V.A. Khoze, What Can Be Expected from Experiments on Colliding e+ e- Beams with e Approximately Equal to 100-GeV?, Sov. J. Part. Nucl. 9 (1978) 50 [Fiz. Elem. Chast. Atom. Yadra 9 (1978) 118].Google Scholar
  35. [35]
    M. Gorbahn and U. Haisch, Indirect probes of the trilinear Higgs coupling: ggh and hγγ, JHEP 10 (2016) 094 [arXiv:1607.03773] [INSPIRE].
  36. [36]
    G. Degrassi, P.P. Giardino, F. Maltoni and D. Pagani, Probing the Higgs self coupling via single Higgs production at the LHC, JHEP 12 (2016) 080 [arXiv:1607.04251] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    W. Bizon, M. Gorbahn, U. Haisch and G. Zanderighi, Constraints on the trilinear Higgs coupling from vector boson fusion and associated Higgs production at the LHC, JHEP 07 (2017) 083 [arXiv:1610.05771] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    G. Degrassi, M. Fedele and P.P. Giardino, Constraints on the trilinear Higgs self coupling from precision observables, JHEP 04 (2017) 155 [arXiv:1702.01737] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    G.D. Kribs, A. Maier, H. Rzehak, M. Spannowsky and P. Waite, Electroweak oblique parameters as a probe of the trilinear Higgs boson self-interaction, Phys. Rev. D 95 (2017) 093004 [arXiv:1702.07678] [INSPIRE].ADSGoogle Scholar
  40. [40]
    F. Maltoni, D. Pagani, A. Shivaji and X. Zhao, Trilinear Higgs coupling determination via single-Higgs differential measurements at the LHC, Eur. Phys. J. C 77 (2017) 887 [arXiv:1709.08649] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    S. Di Vita, C. Grojean, G. Panico, M. Riembau and T. Vantalon, A global view on the Higgs self-coupling, JHEP 09 (2017) 069 [arXiv:1704.01953] [INSPIRE].CrossRefGoogle Scholar
  42. [42]
    T. Barklow, K. Fujii, S. Jung, M.E. Peskin and J. Tian, Model-Independent Determination of the Triple Higgs Coupling at e + e Colliders, Phys. Rev. D 97 (2018) 053004 [arXiv:1708.09079] [INSPIRE].ADSGoogle Scholar
  43. [43]
    S. Di Vita et al., A global view on the Higgs self-coupling at lepton colliders, JHEP 02 (2018) 178 [arXiv:1711.03978] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    D.R.T. Jones and S.T. Petcov, Heavy Higgs Bosons at LEP, Phys. Lett. B 84 (1979) 440 [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    CEPC-SPPC Group collaboration, CEPC-SPPC Preliminary Conceptual Design Report. 1. Physics and Detector, IHEP-CEPC-DR-2015-01 IHEP-TH-2015-01 IHEP-EP-2015-01 [INSPIRE].
  46. [46]
    TLEP Design Study Working Group collaboration, M. Bicer et al., First Look at the Physics Case of TLEP, JHEP 01 (2014) 164 [arXiv:1308.6176] [INSPIRE].
  47. [47]
    H. Baer et al., The International Linear Collider Technical Design Report - Volume 2: Physics, arXiv:1306.6352 [INSPIRE].
  48. [48]
    CLICdp and CLIC collaborations, M.J. Boland et al., Updated baseline for a staged Compact Linear Collider, arXiv:1608.07537 [INSPIRE].
  49. [49]
    H. Abramowicz et al., Higgs physics at the CLIC electron-positron linear collider, Eur. Phys. J. C 77 (2017) 475 [arXiv:1608.07538] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    F. Boudjema and E. Chopin, Double Higgs production at the linear colliders and the probing of the Higgs selfcoupling, Z. Phys. C 73 (1996) 85 [hep-ph/9507396] [INSPIRE].
  51. [51]
    A. Denner, Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200, Fortsch. Phys. 41 (1993) 307 [arXiv:0709.1075] [INSPIRE].ADSGoogle Scholar
  52. [52]
    A.A. Sokolov and I.M. Ternov, On polarization and spin effects in the theory of synchrotron radiation, Sov. Phys. Dokl. 8 (1964) 1203 [INSPIRE].ADSGoogle Scholar
  53. [53]
    Particle Data Group collaboration, C. Patrignani et al., Review of Particle Physics, Chin. Phys. C 40 (2016) 100001 [INSPIRE].
  54. [54]
    A. Denner, S. Dittmaier, M. Roth and M.M. Weber, Electroweak radiative corrections to e + e nu anti-nu H, Nucl. Phys. B 660 (2003) 289 [hep-ph/0302198] [INSPIRE].
  55. [55]
    A. Denner, S. Dittmaier, M. Roth and M.M. Weber, Electroweak radiative corrections to single Higgs boson production in e + e annihilation, Phys. Lett. B 560 (2003) 196 [hep-ph/0301189] [INSPIRE].
  56. [56]
    L. Di Luzio, R. Gröber and M. Spannowsky, Maxi-sizing the trilinear Higgs self-coupling: how large could it be?, Eur. Phys. J. C 77 (2017) 788 [arXiv:1704.02311] [INSPIRE].ADSGoogle Scholar
  57. [57]
    J. Tian, Study of Higgs self-coupling at the ILC based on the full detector simulation at \( \sqrt{s}=500 \) GeV and \( \sqrt{s}=1 \) TeV, in Helmholtz Alliance Linear Collider Forum: Proceedings of the Workshops Hamburg, Munich, Hamburg 2010-2012, Germany, pp. 224-247, DESY, (2013).Google Scholar
  58. [58]
    G. Bélanger et al., Full 0(alpha) electroweak corrections to double Higgs strahlung at the linear collider, Phys. Lett. B 576 (2003) 152 [hep-ph/0309010] [INSPIRE].
  59. [59]
    T. Barklow et al., Improved Formalism for Precision Higgs Coupling Fits, Phys. Rev. D 97 (2018) 053003 [arXiv:1708.08912] [INSPIRE].ADSGoogle Scholar
  60. [60]
    H. Abramowicz et al., The International Linear Collider Technical Design Report — Volume 4: Detectors, arXiv:1306.6329 [INSPIRE].
  61. [61]
    A. Sirlin and R. Zucchini, Dependence of the Quartic Coupling H m on M H and the Possible Onset of New Physics in the Higgs Sector of the Standard Model, Nucl. Phys. B 266 (1986) 389 [INSPIRE].
  62. [62]
    C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter, UFO — The Universal FeynRules Output, Comput. Phys. Commun. 183 (2012) 1201 [arXiv:1108.2040] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    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].ADSCrossRefGoogle Scholar
  64. [64]
    T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260] [INSPIRE].
  65. [65]
    T. Hahn and M. Pérez-Victoria, Automatized one loop calculations in four-dimensions and D-dimensions, Comput. Phys. Commun. 118 (1999) 153 [hep-ph/9807565] [INSPIRE].
  66. [66]
    R.K. Ellis and G. Zanderighi, Scalar one-loop integrals for QCD, JHEP 02 (2008) 002 [arXiv:0712.1851] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    S. Carrazza, R.K. Ellis and G. Zanderighi, QCDLoop: a comprehensive framework for one-loop scalar integrals, Comput. Phys. Commun. 209 (2016) 134 [arXiv:1605.03181] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Centre for Cosmology, Particle Physics and Phenomenology (CP3)Université catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Technische Universität MünchenGarchingGermany

Personalised recommendations