Evaluating $$\alpha \longrightarrow \beta + \overline{\beta}$$ Feynman Diagram

[c++guru]

If I have a scalar field $$\alpha$$ and a Dirac particle $$\beta$$ and its anti particle $$\overline{\beta}$$ such that the three couple to give a vertex factor of $$-ik$$ when evaluating the Feynman diagram (where k is an arbitrary constant).
How do I evaluate the first order diagram of $$\alpha \longrightarrow \beta + \overline{\beta}$$

 

[Orodruin]

With that vertex factor, I assume you have something like $-k \bar\beta \beta \alpha$ in the Lagrangian density. It should be a simple matter of writing down the vertex factor and adding spinors for the outgoing fermions.

 

[c++guru]

in the Lagrangian density. It should be a simple matter of writing down the vertex factor and adding spinors for the outgoing fermions.

Could you please give me an explicit expression? I'm not sure mine is correct.

 

[Orodruin]

Since this is a homework-like question, it will be in more accordance with forum guidelines if you first show your attempt including your reasoning. I (or someone else) can then help you to iron out any misunderstandings or misconceptions.

 

[c++guru]

Since this is a homework-like question, it will be in more accordance with forum guidelines if you first show your attempt including your reasoning. I (or someone else) can then help you to iron out any misunderstandings or misconceptions.

Of Course

I believe the amplitude simply evaluates to

$$ k\overline{U}^{(s)}V^{(s)}$$

where U is the spinor of the Beta and V is the spinor of the anti Beta, just from simply accounting for the spinors and the vertex factor.