# Invariance of the scattering amplitude under photon polarization shift

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.

• Hi Piotr. Please note that exercise-solving questions are typically considered off-topic here, unless they ask about a concept rather than a calculation. In its current state (v1), your post is simply stating that you don't know how to solve this particular problem, and as such, it falls under the homework-and-exercises policy, i.e., it's off-topic. – AccidentalFourierTransform May 25 '17 at 17:19
• I read the meta and I'm not sure what the problem is. Is it that I didn't supply the homework tag? Or that I'm asking about a calculation? If it's the latter, then how is this different from asking about the concept of the longitudinally-polarized photon not influencing the QED amplitudes? Would that be better? But then, is it not better to put that sort of question in some context? Sooner or later, to make the explanation concrete, one has to transition into mathematics as that's where the solution is - I'm looking to understand how the QED formalism deals with longitudinal photons. – Piotr May 25 '17 at 20:16

The result is guaranteed to be unchanged from the Ward Identity, $k_\mu\mathcal{M}^\mu=0$, which is a result of gauge invariance.