Interference term in Bhabha scattering

[karangovil]

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[$$\displaystyle{\not}p'\gamma^{\nu}\displaystyle{\not}k'\gamma^{\mu}\displaystyle{\not}k\gamma_{\nu}\displaystyle{\not}p\gamma_{\mu}$$]
(here first two momenta are p' and k')

 

[Final]

Hi...
You can use some contraction identity (Peskin p. 805):
$$\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma_{\mu}= -2\gamma^{\sigma}\gamma^{\rho}\gamma^{\nu}$$
$$\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}= 4 g^{\nu \rho}$$.
Your term is:
$$Tr[ \gamma^{\delta}\gamma^{\nu}\gamma^{\alpha}\gamma^{\mu}\gamma^{\beta} \gamma_{\nu}\gamma^{\gamma}\gamma_{\mu}k'_{\alpha}k_{\beta}p_{\gamma}p'_{\delta} ]=-2Tr[ \gamma^{\delta}\gamma^{\beta}\gamma^{\mu}\gamma^{\alpha} \gamma^{\gamma}\gamma_{\mu}k'_{\alpha}k_{\beta}p_{\gamma}p'_{\delta}] =$$
$$=-8Tr[ \gamma^{\delta}\gamma^{\beta}k_{\beta}p'_{\delta}(k' \cdot p) ] =-32(k'\cdot p) (k\cdot p')$$