One-loop W L W L and Z L Z L scattering from the electroweak Chiral Lagrangian with a light Higgs-like scalar


By including the recently discovered Higgs-like scalar φ in the Electroweak Chiral Lagrangian, and using the Equivalence Theorem, we carry out the complete one-loop computation of the elastic scattering amplitude for the longitudinal components of the gauge bosons V = W, Z at high energy. We also compute φφφφ and the inelastic process VVφφ, and identify the counterterms needed to cancel the divergences, namely the well known a 4 and a 5 chiral parameters plus three additional ones only superficially treated in the literature because of their dimension 8. Finally we compute all the partial waves and discuss the limitations of the one-loop computation due to only approximate unitarity.


  1. [1]

    J.M. Cornwall, D.N. Levin and G. Tiktopoulos, Derivation of Gauge Invariance from High-Energy Unitarity Bounds on the s Matrix, Phys. Rev. D 10 (1974) 1145 [Erratum ibid. D 11 (1975) 972] [INSPIRE].

    ADS  Google Scholar 

  2. [2]

    C. Vayonakis, Born Helicity Amplitudes and Cross-Sections in Nonabelian Gauge Theories, Lett. Nuovo Cim. 17 (1976) 383 [INSPIRE].

    Article  Google Scholar 

  3. [3]

    B.W. Lee, C. Quigg and H. Thacker, Weak Interactions at Very High-Energies: The Role of the Higgs Boson Mass, Phys. Rev. D 16 (1977) 1519 [INSPIRE].

    ADS  Google Scholar 

  4. [4]

    M.S. Chanowitz and M.K. Gaillard, The TeV physics of strongly interacting Ws and Zs, Nucl. Phys. 261 (1985) 379.

    ADS  Article  Google Scholar 

  5. [5]

    M.S. Chanowitz, M. Golden and H. Georgi, Low-Energy Theorems for Strongly Interacting Ws and Zs, Phys. Rev. D 36 (1987) 1490 [INSPIRE].

    ADS  Google Scholar 

  6. [6]

    A. Dobado and J. Pelaez, On The Equivalence theorem in the chiral perturbation theory description of the symmetry breaking sector of the standard model, Nucl. Phys. B 425 (1994) 110 [Erratum ibid. B 434 (1995) 475] [hep-ph/9401202] [INSPIRE].

    ADS  Article  Google Scholar 

  7. [7]

    A. Dobado and J.R. Pelaez, The Equivalence theorem for chiral lagrangians, Phys. Lett. B 329 (1994) 469 [Addendum ibid. B 335 (1994) 554] [hep-ph/9404239] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  8. [8]

    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].

    ADS  Google Scholar 

  9. [9]

    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].

    ADS  Google Scholar 

  10. [10]

    ATLAS collaboration, Observation and study of the Higgs boson candidate in the two photon decay channel with the ATLAS detector at the LHC, ATLAS-CONF-2012-168 (2012).

  11. [11]

    CMS collaboration, 1460419, CMS-HIG-12-015 (1460419).

  12. [12]

    CMS collaboration, Search for Resonances in the Dijet Mass Spectrum from 7 TeV pp Collisions at CMS, Phys. Lett. B 704 (2011) 123 [arXiv:1107.4771] [INSPIRE].

    ADS  Google Scholar 

  13. [13]

    ATLAS collaboration, Search for heavy vector-like quarks coupling to light quarks in proton-proton collisions at \( \sqrt{s} \) = 7 TeV with the ATLAS detector, Phys. Lett. B 712 (2012) 22 [arXiv:1112.5755] [INSPIRE].

    ADS  Google Scholar 

  14. [14]

    ATLAS collaboration, Search for long-lived, multi-charged particles in pp collisions at \( \sqrt{s} \) = 7 TeV using the ATLAS detector, Phys. Lett. B 722 (2013) 305 [arXiv:1301.5272] [INSPIRE].

    ADS  Google Scholar 

  15. [15]

    D. Espriu, F. Mescia and B. Yencho, Radiative corrections to WL WL scattering in composite Higgs models, Phys. Rev. D 88 (2013) 055002 [arXiv:1307.2400] [INSPIRE].

    ADS  Google Scholar 

  16. [16]

    A. Azatov, R. Contino and J. Galloway, Model-Independent Bounds on a Light Higgs, JHEP 04 (2012) 127 [Erratum ibid. 1304 (2013) 140] [arXiv:1202.3415] [INSPIRE].

    ADS  Article  Google Scholar 

  17. [17]

    I. Brivio, et al., Disentangling a dynamical Higgs, arXiv:1311.1823 [INSPIRE].

  18. [18]

    R. Alonso, M. Gavela, L. Merlo, S. Rigolin and J. Yepes, The Effective Chiral Lagrangian for a Light DynamicalHiggs Particle”, Phys. Lett. B 722 (2013) 330 [Erratum ibid. B 726 (2013) 926] [arXiv:1212.3305] [INSPIRE].

    ADS  Article  Google Scholar 

  19. [19]

    A. Pich, I. Rosell and J.J. Sanz-Cillero, Strongly Coupled Models with a Higgs-like Boson, EPJ Web Conf. 60 (2013) 19009 [arXiv:1307.1958] [INSPIRE].

    Article  Google Scholar 

  20. [20]

    E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence, JHEP 10 (2013) 087 [arXiv:1308.2627] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  21. [21]

    C. Degrande, N. Greiner, W. Kilian, O. Mattelaer, H. Mebane et al., Effective Field Theory: A Modern Approach to Anomalous Couplings, Annals Phys. 335 (2013) 21 [arXiv:1205.4231] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  22. [22]

    G. Buchalla, O. Catà and C. Krause, Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO, arXiv:1307.5017 [INSPIRE].

  23. [23]

    G. Buchalla and O. Catà, Effective Theory of a Dynamically Broken Electroweak Standard Model at NLO, JHEP 07 (2012) 101 [arXiv:1203.6510] [INSPIRE].

    ADS  Article  Google Scholar 

  24. [24]

    T. Appelquist and C.W. Bernard, Strongly Interacting Higgs Bosons, Phys. Rev. D 22 (1980) 200 [INSPIRE].

    ADS  Google Scholar 

  25. [25]

    A.C. Longhitano, Heavy Higgs Bosons in the Weinberg-Salam Model, Phys. Rev. D 22 (1980) 1166 [INSPIRE].

    ADS  Google Scholar 

  26. [26]

    A.C. Longhitano, Low-Energy Impact of a Heavy Higgs Boson Sector, Nucl. Phys. B 188 (1981) 118 [INSPIRE].

    ADS  Article  Google Scholar 

  27. [27]

    A. Dobado, D. Espriu and M.J. Herrero, Chiral Lagrangians as a tool to probe the symmetry breaking sector of the SM at LEP, Phys. Lett. B 255 (1991) 405 [INSPIRE].

    ADS  Article  Google Scholar 

  28. [28]

    B. Holdom and J. Terning, Large corrections to electroweak parameters in technicolor theories, Phys. Lett. B 247 (1990) 88 [INSPIRE].

    ADS  Article  Google Scholar 

  29. [29]

    A. Dobado, D. Espriu and M.J. Herrero, Chiral Lagrangians as a tool to probe the symmetry breaking sector of the SM at LEP, Phys. Lett. B 255 (1991) 405 [INSPIRE].

    ADS  Article  Google Scholar 

  30. [30]

    M. Golden and L. Randall, Radiative Corrections to Electroweak Parameters in Technicolor Theories, Nucl. Phys. B 361 (1991) 3 [INSPIRE].

    ADS  Article  Google Scholar 

  31. [31]

    S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].

    ADS  Article  Google Scholar 

  32. [32]

    J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142 [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  33. [33]

    J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].

    ADS  Article  Google Scholar 

  34. [34]

    R.L. Delgado, A. Dobado and F.J. Llanes-Estrada, LightHiggs, yet strong interactions, J. Phys. G 41 (2014) 025002 [arXiv:1308.1629] [INSPIRE].

    ADS  Article  Google Scholar 

  35. [35]

    R. Contino, C. Grojean, M. Moretti, F. Piccinini and R. Rattazzi, Strong Double Higgs Production at the LHC, JHEP 05 (2010) 089 [arXiv:1002.1011] [INSPIRE].

    ADS  Article  Google Scholar 

  36. [36]

    R. Contino, The Higgs as a Composite Nambu-Goldstone Boson, arXiv:1005.4269 [INSPIRE].

  37. [37]

    R. Grober and M. Muhlleitner, Composite Higgs Boson Pair Production at the LHC, JHEP 06 (2011) 020 [arXiv:1012.1562] [INSPIRE].

    ADS  Article  Google Scholar 

  38. [38]

    G. Bélanger, B. Dumont, U. Ellwanger, J. Gunion and S. Kraml, Global fit to Higgs signal strengths and couplings and implications for extended Higgs sectors, Phys. Rev. D 88 (2013) 075008 [arXiv:1306.2941] [INSPIRE].

    ADS  Google Scholar 

  39. [39]

    T. Corbett, O. Eboli, J. Gonzalez-Fraile and M. Gonzalez-Garcia, Constraining anomalous Higgs interactions, Phys. Rev. D 86 (2012) 075013 [arXiv:1207.1344] [INSPIRE].

    ADS  Google Scholar 

  40. [40]

    T. Corbett, O. Éboli, J. Gonzalez-Fraile and M. Gonzalez-Garcia, Robust determination of the scalar boson couplings, arXiv:1306.0006 [INSPIRE].

  41. [41]

    J. Ellis and T. You, Updated Global Analysis of Higgs Couplings, JHEP 06 (2013) 103 [arXiv:1303.3879] [INSPIRE].

    ADS  Article  Google Scholar 

  42. [42]

    P.P. Giardino, K. Kannike, I. Masina, M. Raidal and A. Strumia, The universal Higgs fit, arXiv:1303.3570 [INSPIRE].

  43. [43]

    A. Falkowski, F. Riva and A. Urbano, Higgs at last, JHEP 11 (2013) 111 [arXiv:1303.1812] [INSPIRE].

    ADS  Article  Google Scholar 

  44. [44]

    E. Halyo, Technidilaton or Higgs?, Mod. Phys. Lett. A 8 (1993) 275 [INSPIRE].

    ADS  Article  Google Scholar 

  45. [45]

    W.D. Goldberger, B. Grinstein and W. Skiba, Distinguishing the Higgs boson from the dilaton at the Large Hadron Collider, Phys. Rev. Lett. 100 (2008) 111802 [arXiv:0708.1463] [INSPIRE].

    ADS  Article  Google Scholar 

  46. [46]

    K. Agashe, R. Contino and A. Pomarol, The Minimal composite Higgs model, Nucl. Phys. B 719 (2005) 165 [hep-ph/0412089] [INSPIRE].

    ADS  Article  Google Scholar 

  47. [47]

    R. Contino, L. Da Rold and A. Pomarol, Light custodians in natural composite Higgs models, Phys. Rev. D 75 (2007) 055014 [hep-ph/0612048] [INSPIRE].

    ADS  Google Scholar 

  48. [48]

    D. Barducci, A. Belyaev, M. Brown, S. De Curtis, S. Moretti et al., The 4-Dimensional Composite Higgs Model (4DCHM) and the 125 GeV Higgs-like signals at the LHC, JHEP 09 (2013) 047 [arXiv:1302.2371] [INSPIRE].

    ADS  Article  Google Scholar 

  49. [49]

    D.B. Kaplan and H. Georgi, SU(2) x U(1) Breaking by Vacuum Misalignment, Phys. Lett. B 136 (1984) 183 [INSPIRE].

    ADS  Article  Google Scholar 

  50. [50]

    S. Dimopoulos and J. Preskill, Massless Composites With Massive Constituents, Nucl. Phys. B 199 (1982) 206 [INSPIRE].

    ADS  Article  Google Scholar 

  51. [51]

    T. Banks, Constraints on SU(2) × U (1) breaking by vacuum misalignment, Nucl. Phys. B 243 (1984) 125 [INSPIRE].

    ADS  Google Scholar 

  52. [52]

    D.B. Kaplan, H. Georgi and S. Dimopoulos, Composite Higgs Scalars, Phys. Lett. B 136 (1984) 187 [INSPIRE].

    ADS  Article  Google Scholar 

  53. [53]

    H. Georgi, D.B. Kaplan and P. Galison, Calculation of the Composite Higgs Mass, Phys. Lett. B 143 (1984) 152 [INSPIRE].

    ADS  Article  Google Scholar 

  54. [54]

    H. Georgi and D.B. Kaplan, Composite Higgs and Custodial SU(2), Phys. Lett. B 145 (1984) 216 [INSPIRE].

    ADS  Article  Google Scholar 

  55. [55]

    M.J. Dugan, H. Georgi and D.B. Kaplan, Anatomy of a Composite Higgs Model, Nucl. Phys. B 254 (1985) 299 [INSPIRE].

    ADS  Article  Google Scholar 

  56. [56]

    G. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, The Strongly-Interacting Light Higgs, JHEP 06 (2007) 045 [hep-ph/0703164] [INSPIRE].

    ADS  Article  Google Scholar 

  57. [57]

    D. Espriu and B. Yencho, Longitudinal WW scattering in light of theHiggs bosondiscovery, Phys. Rev. D 87 (2013) 055017 [arXiv:1212.4158] [INSPIRE].

    ADS  Google Scholar 

  58. [58]

    S.L. Glashow, Partial Symmetries of Weak Interactions, Nucl. Phys. 22 (1961) 579.

    Article  Google Scholar 

  59. [59]

    S. Weinberg, A Model of Leptons, Phys. Rev. Lett. 19 (1967) 1264 [INSPIRE].

    ADS  Article  Google Scholar 

  60. [60]

    A. Salam, Elementary Particle Physics, in Proc. 8th Nobel Symp., N. Svartholm ed., pg. 367, Stockholm, Almqvist and Wiksells, 1968.

  61. [61]

    A. Dobado and M.J. Herrero, Phenomenological Lagrangian Approach to the Symmetry Breaking Sector of the Standard Model, Phys. Lett. B 228 (1989) 495 [INSPIRE].

    ADS  Article  Google Scholar 

  62. [62]

    A. Dobado and M.J. Herrero, Testing the Hypothesis of Strongly Interacting Longitudinal Weak Bosons in Electron - Positron Collisions at TeV Energies, Phys. Lett. B 233 (1989) 505 [INSPIRE].

    ADS  Article  Google Scholar 

  63. [63]

    J.F. Donoghue and C. Ramirez, Symmetry Breaking Schemes and W W Scattering, Phys. Lett. B 234 (1990) 361 [INSPIRE].

    ADS  Article  Google Scholar 

  64. [64]

    R.S. Gupta, H. Rzehak and J.D. Wells, How well do we need to measure the Higgs boson mass and self-coupling?, Phys. Rev. D 88 (2013) 055024 [arXiv:1305.6397] [INSPIRE].

    ADS  Google Scholar 

  65. [65]

    A. Alloul, N.D. Christensen, C. Degrande, C. Duhr and B. Fuks, FeynRules 2.0 - A complete toolbox for tree-level phenomenology, arXiv:1310.1921 [INSPIRE].

  66. [66]

    T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  67. [67]

    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].

    ADS  Article  Google Scholar 

  68. [68]

    J. Kuipers, T. Ueda, J. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].

    ADS  Article  Google Scholar 

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Delgado, R.L., Dobado, A. & Llanes-Estrada, F.J. One-loop W L W L and Z L Z L scattering from the electroweak Chiral Lagrangian with a light Higgs-like scalar. J. High Energ. Phys. 2014, 121 (2014).

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  • Beyond Standard Model
  • Chiral Lagrangians
  • Spontaneous Symmetry Breaking
  • Scattering Amplitudes