Consider the Lagrangian $$\mathcal{L}=\bar\psi_1\left(i\partial\!\!\!/-m_1\right)\psi_1 + \bar\psi_2\left(i\partial\!\!\!/-m_2\right)\psi_2 - g\bar\psi_1\gamma_\mu\psi_1\bar\psi_2\gamma^\mu\psi_2.$$ The Feynman rule for the vertex (see my previous PSE question) is $$-ig\left(\gamma_\mu\right)_{ab}\left(\gamma^\mu\right)_{cd},$$ so there's no matrix multiplication.

I'm now trying to calculate the following scattering amplitude: $$\psi_1(p_1)+\psi_2(p_2)\rightarrow\psi_1(p'_1)+\psi_2(p'_2).$$ At tree-level, it is $$\left(-ig\right)\left[\bar u(p'_1)\gamma_\mu u(p_1)\right]\left[\bar u(p'_2)\gamma^\mu u(p_2)\right];$$ but I have troubles for order $g^2$, since there are 4 gamma matrices and I'm not sure how to insert them between spinors and propagators.


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