# Feynman rule for a vertex with spinorial structure

Consider the interaction term $$\mathcal{L}_{\rm{int}}=-g\bar\psi_1\gamma_\mu\psi_1 \bar\psi_2\gamma^\mu\psi_2,$$ where $$\psi_i$$ are fermions. I would like to calculate the Feynman rule for the vertex.

Is it equal to $$-\left(i\right)g\gamma_\mu\gamma^\mu=-4ig,$$ or am I missing something about spinorial structure of the vertices?

• @AccidentalFourierTransform, where do you get $2\times 2$ from? – Stig Oct 24 '18 at 14:29
• @AccidentalFourierTransform shouldn't $\bar\psi_i$ and $\psi_i$ be distinguishable? – Stig Oct 24 '18 at 14:45
• @AccidentalFourierTransform shouldn't we distinguish $\bar\psi_1$, $\psi_1$, $\bar\psi_2$ and $\psi_2$? Should we have 4 distinguishable particles? – Stig Oct 24 '18 at 15:22
• More like $-i(\gamma_\mu)_{ab}(\gamma^\mu)_{cd}$ (no matrix multiplication). – AccidentalFourierTransform Oct 24 '18 at 15:24
• @AccidentalFourierTransform by "no matrix multiplication" do you mean that $\gamma$s will be separately sandwiched between spinors? (And this separation prevents us to contract tensor indices) Sorry, this is the first time I see this kind of stuff – Vincenzo Ventriglia Oct 24 '18 at 15:42