# Feynman Diagrams for Yukawa Theory

I am trying to draw the Feynman diagram for the following scattering amplitude (f a fermion) $$i\mathcal{M}(f\overline{f}\phi\phi\phi)$$

Given the following interaction term in the Lagrangian:

$$\mathcal{L}_I = -g\phi\psi\overline{\psi}$$

Now, I am trying to construct a feynman diagram from the rules of the theory, but I am having trouble recognising which diagrams are not allowed.

Which symmetries should I be respecting at each vertex?

Should I think of this as fermion/anti-fermion annihilation which produces two scalars, one of which then decays, producing another scalar?

• Could you perhaps try to indicate more clearly what the initial and final particles are? Do you mean $i\mathcal M (f\bar f \to \phi\phi\phi)$? – Danu Oct 6 '15 at 16:44
• I guess so, but actually I wouldn't mind knowing if any other initial/final states are even allowed by the theory. For example, I assume that $f\phi\longrightarrow\overline{f}\phi\phi$ isn't allowed, but not exactly certain on the reasons why. – user94752 Oct 6 '15 at 17:30
• when in doubt go back to wick contractions and commutation relations and derive the Feynman rules by your self! – Ali Moh Oct 6 '15 at 18:32

Assuming that $g\phi\bar{\psi}\psi$ is the only interaction term you will need at least 3 vertices to produce 3 external $\phi$-legs. Therefore, the amplitude at leading order in perturbation theory is going to be of the form $$\mathcal{M}\propto \bar{v}(p_1)\Delta_F(p_1-p_3)\Delta_F(p_1-p_3-p_4)u(p_2)+\ldots$$ where $\Delta_F$ is the fermion propagator, and the $\ldots$ refers to the permutations of the three bosons momenta $p_3$, $p_4$ and $p_5$. It comes from the Feynmann diagrams where 3 boson legs attach to the internal fermion line in t-channel (corresponding i.e. to 2 fermionic propagators).