Analytical Properties and Unitarization

  • Rafael Delgado LópezEmail author
Part of the Springer Theses book series (Springer Theses)


This chapter includes a detailed study of the mathematical properties of the S-matrix, encoded in dispersion relations. Several unitarization procedures (IAM, N/D, old K-matrix and improved K-matrix) are studied in detail. We also consider a weak coupling of 2-\(\gamma \) and \(t\bar{t}\) states with a strongly interacting EWSBS. This requires a modification of the considered unitarization procedures, which is also providen.


  1. 1.
    B.R. Martin, D. Morgan, G. Shaw, Pion-Pion Interactions in Particle Physics (Academic Press, London, 1976)Google Scholar
  2. 2.
    J.R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover Books on Engineering (Dover Publ, Newburyport, MA, 2012)Google Scholar
  3. 3.
    R.J. Eden, P.V. Landshoff, D.I. Olive, J.C. Polkinghorne, The Analytic S-matrix (Cambridge University Press, Cambridge, 1966)zbMATHGoogle Scholar
  4. 4.
    A. Dobado, A. Gómez-Nicola, A.L. Maroto, J.R. Peláez, Effective Lagrangians for the Standard Model, Texts and Monographs in Physics (Springer, Berlin, 1997)CrossRefzbMATHGoogle Scholar
  5. 5.
    M. Chaichian, C. Montonen, A. Tureanu, Tree unitarity and partial wave expansion in noncommutative quantum field theory. Phys. Lett. B 566, 263–270 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    V. Gribov, Strong Interactions of Hadrons at High Energies: Gribov Lectures on Theoretical Physics, Cambridge Monographs on Particle Physics, Nuclear Physics, and Cosmology (Cambridge University Press, Cambridge, 2008)CrossRefGoogle Scholar
  7. 7.
    M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory, 1995th edn. (Westview, Boulder, CO, 1995)Google Scholar
  8. 8.
    R.E. Cutkosky, P.V. Landshoff, D.I. Olive, J.C. Polkinghorne, A non-analytic S matrix. Nucl. Phys. B 12, 281–300 (1969)ADSCrossRefGoogle Scholar
  9. 9.
    B. Grinstein, D. O’Connell, M.B. Wise, Causality as an emergent macroscopic phenomenon: the Lee–Wick O(N) model. Phys. Rev. D 79, 105019 (2009)ADSCrossRefGoogle Scholar
  10. 10.
    T. Figy, R. Zwicky, The other Higgses, at resonance, in the Lee–Wick extension of the standard model. JHEP 10, 145 (2011)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    T.G. Rizzo, Searching for Lee–Wick gauge bosons at the LHC. JHEP 06, 070 (2007)ADSCrossRefGoogle Scholar
  12. 12.
    R.L. Delgado, A. Dobado, F.J. Llanes-Estrada, Light ‘Higgs’, yet strong interactions. J. Phys. G41, 025002 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    D. Espriu, F. Mescia, Unitarity and causality constraints in composite Higgs models. Phys. Rev. D 90, 015035 (2014)ADSCrossRefGoogle Scholar
  14. 14.
    R.L. Delgado, A. Dobado, F.J. Llanes-Estrada, Unitarity, analyticity, dispersion relations, and resonances in strongly interacting WLWL, ZLZL, and hh scattering. Phys. Rev. D 91, 075017 (2015)ADSCrossRefGoogle Scholar
  15. 15.
    R.L. Delgado, A. Dobado, F.J. Llanes-Estrada, Possible new resonance from \(W_{L}W_{L}-hh\) interchannel coupling. Phys. Rev. Lett. 114, 221803 (2015)Google Scholar
  16. 16.
    S.G. Krantz, Handbook of Complex Variables (Birkhäuser, Basel, 1999)CrossRefzbMATHGoogle Scholar
  17. 17.
    J.E. Marsden, M. Buchner, M. Hoffman, C. Risk, Basic Complex Analysis (Freeman, San Francisco, CA, 1973)Google Scholar
  18. 18.
    R.V. Churchill, J.W. Brown, Complex Variables and Applications, 4th edn. (McGraw-Hill, New York, NY, 1984)zbMATHGoogle Scholar
  19. 19.
    T.N. Truong, Chiral perturbation theory and final state theorem. Phys. Rev. Lett. 61, 2526 (1988)ADSCrossRefGoogle Scholar
  20. 20.
    A. Dobado, M.J. Herrero, T.N. Truong, Unitarized chiral perturbation theory for elastic pion-pion scattering. Phys. Lett. B 235, 134 (1990)ADSCrossRefGoogle Scholar
  21. 21.
    A. Dobado, M.J. Herrero, T.N. Truong, Study of the strongly interacting higgs sector. Phys. Lett. B 235, 129 (1990)ADSCrossRefGoogle Scholar
  22. 22.
    A. Dobado, J.R. Pelaez, A global fit of pi–pi and pi-K elastic scattering in ChPT with dispersion relations. Phys. Rev. D 47, 4883–4888 (1993)ADSCrossRefGoogle Scholar
  23. 23.
    A. Dobado, J.R. Pelaez, The inverse amplitude method in chiral perturbation theory. Phys. Rev. D 56, 3057–3073 (1997)ADSCrossRefGoogle Scholar
  24. 24.
    J.A. Oller, E. Oset, J.R. Pelaez, Nonperturbative approach to effective chiral Lagrangians and meson interactions. Phys. Rev. Lett. 80, 3452–3455 (1998)ADSCrossRefGoogle Scholar
  25. 25.
    A.G. Nicola, J.R. Pelaez, Meson meson scattering within one loop chiral perturbation theory and its unitarization. Phys. Rev. D 65, 054009 (2002)ADSCrossRefGoogle Scholar
  26. 26.
    A. Dobado, M.J. Herrero, Phenomenological Lagrangian approach to the symmetry breaking sector of the standard model. Phys. Lett. B 228, 495 (1989)ADSCrossRefGoogle Scholar
  27. 27.
    A. Dobado, M.J. Herrero, Testing the hypothesis of strongly interacting longitudinal weak bosons in electron—positron collisions at Tev energies. Phys. Lett. B 233, 505 (1989)ADSCrossRefGoogle Scholar
  28. 28.
    J.F. Donoghue, C. Ramirez, Symmetry breaking schemes and WW scattering. Phys. Lett. B 234, 361 (1990)ADSCrossRefGoogle Scholar
  29. 29.
    T. Corbett, O.J.P. Éboli, M.C. Gonzalez-Garcia, Inverse amplitude method for the perturbative electroweak symmetry breaking sector: the singlet Higgs portal as a study case. Phys. Rev. D 93, 015005 (2016)ADSCrossRefGoogle Scholar
  30. 30.
    A. Gomez Nicola, J.R. Pelaez, G. Rios, The inverse amplitude method and Adler zeros. Phys. Rev. D 77, 056006 (2008)ADSCrossRefGoogle Scholar
  31. 31.
    J.D. Bjorken, Construction of coupled scattering and production amplitudes satisfying analyticity and unitarity. Phys. Rev. Lett. 4, 473–474 (1960)ADSCrossRefGoogle Scholar
  32. 32.
    W. Heitler, The inuence of radiation damping on the scattering of light and mesons by free particles. I. Math. Proc. Camb. Philos. Soc. 37, 291–300 (1941)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    J.S. Schwinger, Quantum electrodynamics. I a covariant formulation. Phys. Rev. 74, 1439 (1948)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    S.N. Gupta, Quantum Electrodynamics (Gordon and Breach, New York, NY, 1977)Google Scholar
  35. 35.
    S.N. Gupta, J.M. Johnson, W.W. Repko, W, Z and Higgs scattering at SSC energies. Phys. Rev. D 48, 2083–2096 (1993)ADSCrossRefGoogle Scholar
  36. 36.
    W. Kilian, T. Ohl, J. Reuter, M. Sekulla, High-energy vector boson scattering after the Higgs discovery. Phys. Rev. D 91, 096007 (2015)ADSCrossRefGoogle Scholar
  37. 37.
    R.L. Delgado, A. Dobado, F.J. Llanes-Estrada, Coupling WW, ZZ unitarized amplitudes to in the TeV region. Eur. Phys. J. C 77, 205 (2017)ADSCrossRefGoogle Scholar
  38. 38.
    A. Castillo, R. L. Delgado, A. Dobado, F. J. Llanes-Estrada, Top-antitop production from \(W_{L}^{ + } W_{L}^{ - }\) and \(Z_{L}Z_{L}\) scattering under a strongly-interacting symmetrybreaking sector, (2016), arXiv:1607.01158 [hep-ph]

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Facultad de Ciencias Físicas, Theoretical Physics I DepartmentComplutense University of MadridMadridSpain

Personalised recommendations