<PERSON> diagram for interactions in $\phi^3$ complex scalar field theory
I recently encountered the Lagrangian density
$$\mathcal L = \partial_\mu \phi^\dagger \partial^\mu \phi - m^2 \phi^\dagger \phi - \frac{\lambda}{3!} \phi^3 - \frac{\lambda}{3!} \phi^{\dagger 3}.$$
I thought I understood complex scalar field theory, but after trying to figure out this one I've realized that I obviously don't. In standard complex $\phi^4$ theory, the interaction term is $\frac{\lambda_4}{4!}(\phi^\dagger \phi)^2$. As such, I interpret a vertex, e.g.
to mean that two $\phi$'s are coming into the vertex, while two $\phi$'s are leaving the vertex. So I interpret the complex conjugation of the $\phi$ to indicate negating the momentum. Thus $(\phi^\dagger \phi)^2 = \phi^\dagger \phi^\dagger \phi \phi$ would have two particles with incoming momentum and two particles with outgoing momentum.
However, in the Lagrangian I posted above, there are two interaction terms $\frac{\lambda}{3!}\phi^3$ and $\frac{\lambda}{3!}\phi^{\dagger 3}$. 281
What does this mean? In my interpretation above, it would indicate a vertex with either three particles incoming or three particles outgoing. But I'm struggling to understand if that is the case.
How would one write down the <PERSON> vertex rules for this interaction? When I try to write out $\langle f \vert T \exp\left[-i \int \frac{\lambda}{3!}\phi^{\dagger 3}(x) d^4x \right] \vert i \rangle$ for example, I find that this gives a vertex term of $-i\lambda$ if $\vert i \rangle = \vert \phi \phi \phi\rangle$ and $\vert f \rangle = \vert 0 \rangle$, and otherwise $0$. But I'm assuming this is incorrect as it means that we start with three $\phi$'s and end with zero.
Any help would be appreciated, thanks!
Edit
It seems that with this Lagrangian, we must keep track of two particles $\phi$ and $\phi^\dagger$. In particular, unlike in real $\phi^3$ theory, it seems to me that we don't have vertices of the form $\phi \to 2 \phi$, but instead have $\phi \to 2 \phi^\dagger$ and $\phi^\dagger \to 2 \phi$. So for example a vertex may look like
If I were to look for the vertex correction, real $\phi^3$ theory, I would just add a loop by connecting the two incoming legs. But is this true for complex $\phi^3$ theory? When I add such a line, the vertex becomes $2\phi^\dagger \to \phi^\dagger$ instead of $2\phi \to \phi^\dagger$,
Is this correct? Am I correct in saying that vertex corrections for complex $\phi^3$ theory require more complicated diagrams? 669