I'm trying to build an interaction Lagrangian (${\cal L}_{int}$) with 2 fields, a complex $\varphi$ and another $\chi$ that can be real or complex. The key is to do it granting $U(1)$ symmetry and in such a way that the process $\varphi + \varphi \rightarrow \varphi + \varphi$ is purely s-channel and $\varphi + \bar\varphi \rightarrow \varphi + \bar\varphi$ has no s-channel ($\bar\varphi$ is the antiparticle created by $\varphi$).

In the first case, vertex s-channel leads to a charged particle, so $\chi$ has to be complex due to charge conservation. For this reason, t, u-channels aren't allowed. For the second case, we need to fix $\chi$ as complex again due to s-diagram would lead to a non-charged particle, this is, scalar field.

So, ok, we know how $\chi$ should be but if I want $U(1)$ symmetry I would need ${\cal L}_{int}$ constructed by products of the form $g\chi^\dagger\chi\varphi^\dagger\varphi$, but at $\mathcal{O}(g^2)$ I get a loop of $\chi$ particles, not the propagator of just one.

Therefore, this Lagrangian doesn't work. How do I fix it?


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