What am I missing here? (QFT: ##e^- - p## scattering)

[Wrichik Basu]

I am currently reading Particle Physics by Palash Pal. In one place, the author shows the Feynman diagram for the electron-proton scattering:

​

Then, he writes the Feynman amplitude for the process: $$i \mathcal{M} \ = \ \left[ \bar{u}(\vec{k'}) i e \gamma^\mu u(\vec{k}) \right] \frac{-ig_{\mu\nu}}{q^2} \left[\bar{u}(\vec{p'}) i e \Gamma^\nu u(\vec{p}) \right]$$ Then he writes,

where,

and

After this, he proceeds to find the expression for the $\Gamma$ in $h_{\mu\nu}$.

My question is, how does he write the terms for $l^{\mu\nu}$ and $h_{\mu\nu}$ just by looking at $\mathcal{M}$? I am sure I am missing something, and the answer is very trivial, but since I have none to ask, I hope you will help me out.

Notation used: According to the author, bold font means a 3-vector, while normal font means a 4-vector. So, k is a 4-vector, while k is the corresponding 3-vector.

 

[DarMM]

Has the book gone through the identities for traces and contractions with the gamma matrices and polarization spinors?

 

[Wrichik Basu]

Has the book gone through the identities for traces and contractions with the gamma matrices and polarization spinors?

Yes, it has, but not in great details. I might be forgetting something. I need to look those up.

 

[DarMM]

Yes, it has, but not in great details. I might be forgetting something. I need to look those up.

They're essentially responsible for the result. I'd look them up again and remember what indices are being contracted.

 

[Wrichik Basu]

They're essentially responsible for the result. I'd look them up again and remember what indices are being contracted.

Seems that I have to take up a different book.