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I am having difficulties applying the concept of Ward-identities to the amplitude Compton scattering at tree-level.

To my knowledge, Ward-identity implies that the scattering amplitude of any (abelian) process with an external photon vanishes upon contraction with the photon momentum and all fermion lines amputated, set on-shell and contracted with bi-spinors $u,\bar{u}, v, \bar{v}$. So diagrammatically Ward identity implies:

enter image description here

In particular the photons do not have to be on-shell (but I don't know if that also includes $q$). However, if I want to show explicitly that the amplitude of Compton scattering at tree-level is transverse, \begin{equation} k_\mu i \mathcal{M}^{\mu \nu} = k_\mu \left[ \bar{u} (p^\prime) (-ie\gamma^\mu ) \frac{i(\not{p} + \not{k} + m)}{(p + k)^2 - m^2} (-ie \gamma^\nu) u(p) + \bar{u} (p^\prime) (-ie\gamma^\nu ) \frac{i(\not{p} - \not{k}^\prime + m)}{(p + k^\prime)^2 - m^2} (-ie \gamma^\mu) u(p) \right] \end{equation} I always need to assume that $k^2 = k^{\prime 2} = 0$.

So I would like to know,

  1. does the momentum of the photon that is contracted ($q$ in the picture) needs to be on-shell?

  2. Is the amplitude of Compton scattering transverse without assuming that the photon(s) are(is) on-shell and how can I show it?

EDIT: I have shown that the amplitude is transverse without assuming that $k_1, k_2$ are on-shell by now.

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The Ward identity says that if you replace the polarisation vector of a photon by its momentum, then the corresponding amplitude vanishes. So the photon needs to be external.

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  • $\begingroup$ But then, you kind of assumed the photon to be on-shell in the first place, don't you? If you look back on the derivation of the Ward-identity, the momentum enters through a derivative $\partial_\mu$ which becomes $p_\mu$ in momentum space. So I don't really see any reason why the momentum should be regarded on-shell. $\endgroup$
    – tomtom1-4
    Jun 21 '21 at 11:27
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    $\begingroup$ You are talking yourself about an external photon in your question: "To my knowledge, Ward-identity implies that the scattering amplitude of any (abelian) process with an external photon vanishes upon contraction with the photon momentum..." $\endgroup$ Jun 21 '21 at 11:50
  • $\begingroup$ Touche! Still don't think it's necessary though. $\endgroup$
    – tomtom1-4
    Jun 21 '21 at 12:42

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