Feynman diagrams for Yukawa Interaction I'm solving some QFT problems and one of them deals with the Yukawa coupling. It asks to consider three processes, namely, $\psi\psi\to\psi\psi$, $\psi\bar{\psi}\to\psi\bar{\psi}$ and $\phi\phi\to\phi\phi$, and then find the Feynman diagrams to second order and derive the matrix elements.
The issue is that if I understood the coupling, there is one single vertex type and it requires a fermion particle, a fermion anti-particle and a scalar particle to meet at the vertex.
For the first and last processes I simply can't imagine a diagram with the given external legs, given number of internal points (namely one and two) and with the internal pointa subject to the constraint that a $\psi$ line a $\bar{\psi}$ line and a $\phi$ line must meet there. Not even the simple tree level diagrams for the $s,t,u$ channels seems to work.
How can I find the Feynman diagrams for the given processes in Yukawa theory? What is the correct reasoning and the correct diagrams?
 A: I'm assuming you're using the lagrangian
$$\mathcal{L}=\overline{\psi}(i\gamma^{\mu}\partial_{\mu}-m_{\psi})\psi+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m_{\phi}-g\overline{\psi}\psi\phi,$$
or something similar.
The diagrams don't require you to have a particle-antiparticle-scalar triple to make a vertex. It just requires that all interaction vertices must have two $\psi$ propagators and one $\phi$ propagator. So long as the charge flow into each vertex is valid, you are fine. For instance, think of the first process,
$$\psi\psi\to\psi\psi$$
as a $t$ channel process. It's just the
$$\psi\overline{\psi}\to\psi\overline{\psi}$$
diagram, but rotated on its side (that is, related by a crossing symmetry).
The $\phi\phi\to\phi\phi$ process is actually a bit more complicated, and there isn't a diagram to second order that contributes to it. The lowest order diagram would be a box diagram (see below, with curvy lines representing $\phi$ propagators and straight lines representig $\psi$ propagators).

I hope this helped!
