Bhabha scattering: Simplifying steps

The completeness relations for the four-spinors u and v are

$\sum_{s=1,2}{u^{(s)}_p \bar{u}^{(s)}_p} = p\!\!\!/ + m \,$

$\sum_{s=1,2}{v^{(s)}_p \bar{v}^{(s)}_p} = p\!\!\!/ - m \,$

where

$p\!\!\!/ = \gamma^\mu p_\mu \,$ (see Feynman slash notation)

$\bar{u} = u^{\dagger} \gamma^0 \,$

Main article: Trace identities

To simplify the trace of the Dirac gamma matrices, one must use trace identities. Three used in this article are:

The Trace of any product of an odd number of $\gamma_\mu \,$ 's is zero
$\operatorname{tr} (\gamma^\mu\gamma^\nu) = 4\eta^{\mu\nu}$
$\operatorname{Tr}\left( \gamma_\rho \gamma_\mu \gamma_\sigma \gamma_\nu \right) = 4 \left( \eta_{\rho\mu}\eta_{\sigma\nu}-\eta_{\rho\sigma}\eta_{\mu\nu}+\eta_{\rho\nu}\eta_{\mu\sigma} \right) \,$

Using these two one finds that, for example,

$\operatorname{Tr}\left( (p\!\!\!/' + m) \gamma_\mu (p\!\!\!/ + m) \gamma_\nu \right) \,$	$= \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu p\!\!\!/ \gamma_\nu \right) + \operatorname{Tr}\left(m \gamma_\mu p\!\!\!/ \gamma_\nu \right) \,$
	$+ \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu m \gamma_\nu \right) + \operatorname{Tr}\left(m^2 \gamma_\mu \gamma_\nu \right) \,$
	(the two middle terms are zero because of (1))
	$= \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu p\!\!\!/ \gamma_\nu \right) + m^2 \operatorname{Tr}\left(\gamma_\mu \gamma_\nu \right) \,$
	(use identity (2) for the term on the right)
	$= {p'}^{\rho} p^\sigma \operatorname{Tr}\left( \gamma_\rho \gamma_\mu \gamma_\sigma \gamma_\nu \right) + m^2 \cdot 4\eta_{\mu\nu} \,$
	(now use identity (3) for the term on the left)
	$= {p'}^{\rho} p^\sigma 4 \left( \eta_{\rho\mu}\eta_{\sigma\nu}-\eta_{\rho\sigma}\eta_{\mu\nu}+\eta_{\rho\nu}\eta_{\mu\sigma} \right) + 4 m^2 \eta_{\mu\nu} \,$
	$=4 \left( {p'}_\mu p_\nu - \mathbf{p' \cdot p}\eta_{\mu\nu} + p'_\nu p_\mu \right) + 4 m^2 \eta_{\mu\nu} \,$

Bhabha scattering