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Quantum Field Theory. Note « 11 »
Feynman diagrams in momentum space
  1. Free lines
    • scalar: 1/√(2ω)
    • fermion: u in the final state, u in the initial state
    • antifermion: v in the final state, v in the initial state
    • photon: √(4π)/√(2ω)εμ
    • vector: 1/√(2ω)εμ
  2. Propogators
    • scalar: i/k2-μ2+i0
    • spinor: i/pγ-m+i0 = i[ pγ+m/p2-m2+i0 ]
    • electromagnetic: 4π -igμν/k2+i0
    • massive vector: -i(gμν-kμkν/m2)/k2-μ2+i0
Exercises
  • Consider the interaction Lagrangian L=-gψγ5ψφ between fermions ψ with mass m and neutral pseudoscalar particles φ with mass μ. The lowest order matrix element of the elastic scattering of fermions in the nonrelativistic limit is
    Mfi(q) = (g2/m2)φ1+φ2+ ((s1q)(s2q)/q2+μ2) φ2φ1 .
    As we showed the nonrelativistic potential V(r) is simply a Fourie transform of the scattering amplitude
    V(r) = -∫ d3q/(2π)3 Mfi(q) e-iqr.
    Exercise: perform this Fourie transform and find the so called "one boson exchange potential" (OBEP).
    Hint: use the assumption
    ∫ d3q/(2π)3 qjqk/q2+μ2 e-iqr = A(r)δjk + B(r)rjrk/r2
    and find A(r) and B(r).
    Answer:
    A = μ2/4π exp(-μr)/r (1/μr+1/(μr)2)
    B = - μ2/4π exp(-μr)/r (1 + 3/μr + 3/(μr)2)
    • Copyleft © 2000-2002 D.V.Fedorov (fedorov@ifa.au.dk)