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Homework Help: Interference term in Bhabha scattering

  1. May 31, 2008 #1
    Hi guys...I am trying the problem 5.2 from Peskin to calculate cross section for Bhabha scattering. In the interference (cross) term, I'm getting a term involving trace of 8 gamma matrices and I am having some trouble in evaluating it. So can anyone help???

    The term is Tr[[tex]\displaystyle{\not}p'\gamma^{\nu}\displaystyle{\not}k'\gamma^{\mu}\displaystyle{\not}k\gamma_{\nu}\displaystyle{\not}p\gamma_{\mu}[/tex]]
    (here first two momenta are p' and k')
     
  2. jcsd
  3. Aug 15, 2008 #2
    Hi...
    You can use some contraction identity (Peskin p. 805):
    [tex]\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma_{\mu}=
    -2\gamma^{\sigma}\gamma^{\rho}\gamma^{\nu} [/tex]
    [tex]\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}=
    4 g^{\nu \rho} [/tex].
    Your term is:
    [tex]Tr[ \gamma^{\delta}\gamma^{\nu}\gamma^{\alpha}\gamma^{\mu}\gamma^{\beta} \gamma_{\nu}\gamma^{\gamma}\gamma_{\mu}k'_{\alpha}k_{\beta}p_{\gamma}p'_{\delta}
    ]=-2Tr[ \gamma^{\delta}\gamma^{\beta}\gamma^{\mu}\gamma^{\alpha} \gamma^{\gamma}\gamma_{\mu}k'_{\alpha}k_{\beta}p_{\gamma}p'_{\delta}]
    =[/tex]
    [tex]=-8Tr[ \gamma^{\delta}\gamma^{\beta}k_{\beta}p'_{\delta}(k' \cdot p)
    ] =-32(k'\cdot p) (k\cdot p') [/tex]
     
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