# Scattering of spinless fields

I'm learning Feynman diagrams and I'm currently working on an exercise with Feynman diagrams where I have the following Lagrangian interaction:

$$L_{int}=-\lambda\phi^3\Phi^*\Phi$$

That is, a scalar real field coupled with a complex scalar field. From what I've studied, $$\Phi$$ is a charged field (since it's complex) and it's propagators must be specified with lines to account for particles and antiparticles, while $$\phi$$ are just normal lines. Obviously we have vertices of 5 legs for the Feynman rules.

I'm trying to compute the scattering process for two incoming particles of $$\phi$$ and $$\Phi$$, and I'm trying to find the most simple non-trivial final state.

According to me, since we have a 5 vertex field, two incoming particles $$\phi$$ and $$\Phi$$ should lead to three outcoming particles $$\phi$$, $$\phi$$, and $$\Phi$$. However, I'm unsure if it's possible due to the conservation of 4-momenta, and this just seems to be the trivial case.

Therefore, considering one loop in the diagram, we would have a process where two incoming $$\phi$$ and $$\Phi$$ particles come, a loop of $$\phi$$, $$\phi$$, and $$\Phi$$ occurs, and then come two $$\phi$$ and $$\Phi$$ particles.

To illustrate my idea, I included this picture for the trivial and first non-trivial case (single-lines are $$\phi$$ fields, double-lines are $$\Phi$$ fields):

It seems reasonable, since during a scattering process I'm expecting the incoming particles to interact and then go away, but I'm not sure if this is correct as I'm still struggling to understand Feynman diagrams. Moreover, I'm unsure about the direction of the lines (for example, why can't I have an antiparticle inside of the loop?)

• Have you tried writing the complex field as two real fields and see what diagrams you can make? But, more generally, how do you define simplicity of the final state here? – Oбжорoв Nov 22 '18 at 19:12