Spinless $e^{-}\gamma\rightarrow e^{-}$ Cross section

Question

I was trying to figure out the cross section $\frac{d\sigma}{d\Omega}$ for spinless $e^{-}\gamma\rightarrow e^{-}$ scattering. First I wrote the terms associated with each component.

Vertex: $$ie(P_A+P_B)^{\mu}$$ External Boson: $1$

Photon: $\epsilon_{\mu}$

Multiplying these will give the invariant amplitude. $$i\mathcal{M} =ie(P_A+P_B)^{\mu}\epsilon_{\mu}$$ Now consider the momenta in high energy approximation $$P_A =(p,P)$$ $$P_B=(p,P')$$ Such that $|P|=|P'|=p$ Then $$P_A+P_B=(2p,P+P')$$ Now squaring $\mathcal{M}$ $$\mathcal{M}^2 = e²(6p^2+2p^2\cos\theta)\epsilon^2$$ The differential cross section will become: $$\frac{d\sigma}{d\Omega}=\frac{p^2e²}{32\pi^2s}(3+\cos\theta)\epsilon^2$$

Now I have two questions:

1) What have I done wrong? I couldn't find the answer anywhere online , is there something obvious that I am missing? I know I am wrong because $\epsilon^2$ is a $3\times 3$ matrix. A cross section can't be a matrix (As far as I know).

2) What will $s$ be? In the book Martin and Halzen the definition $s$ was simply $$s=(P_A+P_B)^2$$ But $s$ in Martin and Halzen was defined in the case of two vertex diagram. What will be the definition of $s$ in single vertex diagram?

Answer 1

Your expression for $\mathcal{M}^2$ is wrong. Inside $\mathcal{M}$ polarisation vectors are contracted with the momenta so for example $$\left|(P + P')^\mu \epsilon_\mu\right|^2 =(P + P')^\mu \epsilon_\mu \, (P + P')^\nu \epsilon_\nu =(P + P') \cdot \epsilon \, \,(P + P') \cdot \epsilon$$ It seems that you incorrectly contracted the $(P + P')$ factors with themselves and were left with $\epsilon$ vectors you didn't know what to do with.