I have a few questions related to the following interaction Lagrangian (no use of crossing symmetry in the following) involving the uncharged scalar $\chi$ and the charged scalar $\phi$:
$$\tag{1}\mathcal{L}_{\rm int} ~=~ \sqrt\lambda· g· \chi\phi^\dagger \phi - \frac{\lambda}{4}(\phi^\dagger\phi)^2.$$
Can we use (1) to say anything about the following reactions.
a) $\phi\phi\rightarrow\phi\phi$
b) $\phi\phi^\dagger\rightarrow \chi\chi$
c) $\chi\phi\rightarrow\chi\phi$
My thoughts: To say anything about a) shouldn't we have a term in $\mathcal{L}_{\rm int}$ not involving $\phi^\dagger~$?
If we use the "usual" rules of letting a dashed line represent the $\chi$-propagator and (properly directed) arrowed lines the charged $\phi$ then it seems to me that the first term in (1) involves three lines; one dashed, one particle $(\phi)$ and one anti-particle. The second term involves four lines with incoming antiparticle ($\phi^\dagger$)-line pointing away from the vertex and outgoing antiparticle pointing towards the vertex etc..
Using these rules it's easy to write down the rule say for c) but what is the vertex factor then since we don't have a term proportional to $\chi\phi^2$ (say) in $\mathcal{L}_{\rm int}$?
Please feel free to comment and explain in detail if you can or suggest examples of similar Lagrangians and reactions/interactions.