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}=<br /> -2\gamma^{\sigma}\gamma^{\rho}\gamma^{\nu}
\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}=<br /> 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&#039;_{\alpha}k_{\beta}p_{\gamma}p&#039;_{\delta}<br /> ]=-2Tr[ \gamma^{\delta}\gamma^{\beta}\gamma^{\mu}\gamma^{\alpha} \gamma^{\gamma}\gamma_{\mu}k&#039;_{\alpha}k_{\beta}p_{\gamma}p&#039;_{\delta}]<br /> =
=-8Tr[ \gamma^{\delta}\gamma^{\beta}k_{\beta}p&#039;_{\delta}(k&#039; \cdot p)<br /> ] =-32(k&#039;\cdot p) (k\cdot p&#039;)