Singularities of Multiple Scattering Processes


It will be useful to start this general study of Landau singularities by reviewing the simplest among them—the most important ones for many practical purposes:
  1. 1.

    The one-particle exchange graph (Fig. 1) is currently interpreted as representing the “exchange” of a “virtual” particle; it contributes a pole for an unphysical value t = m 2 of the momentum transfer squared: t ≡ (p xp 3)2 (m is the mass of the exchanged particle).†

  2. 2.

    The N-particle threshold (or normal threshold) graph (Fig. 2) contributes a square root (N even) or logarithmic (N odd) branch point, at the value s = (m 1 + m 2 + m 3 + … + m N)2 of the total energy squared: s = (p 1 + p 2)2. Thus this singularity occurs for a physical value of the total energy, corresponding to the “opening of an N-particle channel” (m 1,m 2, …, m N, are the masses of these N-particles).



Multiple Scattering Mass Shell Landau Equation Internal Line Effective Contact 


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References and Comments

  1. 1.
    “Singularités des processus de diffusion multiple,” Ann. Inst. Henri Poincaré, Vol. VI, No. 2, p. 89–204, 1967. The presentation is made more accessible, and improved in some respects: For instance, in Section 2, the reasoning leading to Proposition 4 is made much simpler, thanks to the systematic study presented in Ref. 2.Google Scholar
  2. 2.
    F. Pham. “Some Notions of Local Differential Topology,” (this volume p. 65).Google Scholar
  3. 3.
    R.J. Eden, P.V. Landshoff, D.I. Olive, and J.C. Polkinghorne, “The Analytic S-Matrix,” Cambridge University Press, (1966).Google Scholar
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    S. Coleman and R.E. Norton. Nuovo Cimento 38: 438 (1965).CrossRefGoogle Scholar
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    R.P. Feynman, Rev. Mod. Phys. 20: 367 (1948).MathSciNetADSCrossRefGoogle Scholar
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    M.L. Goldberger and K.M. Watson, Phys. Rev. 127: 2284 (1962).MathSciNetADSMATHCrossRefGoogle Scholar
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    K. Hepp, J. Math. Phys. 6: 1762 (1965).MathSciNetADSCrossRefGoogle Scholar
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    P.V. Landshoff and D.I. Olive, J. Math. Phys. 7: 1464 (1966).ADSCrossRefGoogle Scholar
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    P.V. Landshoff, D.I. Olive, and J.C. Polkinghorne, “The Hierarchical Principle in Perturbation Theory,” Cambridge preprint, (1965).Google Scholar

Copyright information

© Plenum Press 1968

Authors and Affiliations

  • F. Pham
    • 1
    • 2
  1. 1.C.E.N.SaclayFrance
  2. 2.C.E.R.NGenevaSwitzerland

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