I want to show that the scattering amplitude of Compton scattering at the lowest order:
$$i(-ie)^2u(p)\left[\frac{\gamma_\mu (\gamma^ap_a + \gamma^aq_a + m)\gamma_\nu }{(p+q)^2 - m^2}+ \frac{\gamma_\mu (\gamma^ap_a - \gamma^aq'_a + m)\gamma_\nu}{(p-q')^2-m^2}\right] \bar{u}(p') \epsilon^\mu_{in}\epsilon^\nu_{out}$$
remains unchanged upon the polarisation shift:
$$\epsilon^\mu_{in} \rightarrow \epsilon^\mu_{in}+\alpha \, k^\mu$$
where k is the incoming photon's momentum. I tried playing with the expressions but I can see at all how this new term is to vanish.
homework-and-exercises
policy, i.e., it's off-topic. $\endgroup$