# Mixed two-point vertex in QFT

I am considering a theory with two fields, say $\phi$ and $\psi$. The Lagrangian contains quadratic terms, i.e., propagators for both fields and a quartic interaction term for one of the fields. However, I have also the terms $\psi^*\phi + \phi^*\psi$. I have never encountered such terms before and I don't really know how to handle them in perturbation theory. If you would perform perturbation theory in these terms, I guess up to second order they are just corrections to the propagators for both fields, is that right?

I was wondering if somebody knows a reference in which such a theory is treated.

• Why do you believe such a two-point vertex should behave different from any other vertex? – ACuriousMind Oct 15 '14 at 10:35
• Are both fields scalars? I guess so from the context but we usually reserve $\psi$ for fermions. Could you please clarify? – Heterotic Oct 16 '14 at 8:38
• @Heterotic yes they are both scalar fields. Sorry for the confusion. – Funzies Oct 16 '14 at 15:17
• It is just mass mixing, $\Phi^\dagger X \Phi$, where $\Phi=(\psi,\phi)^T$ and $X_{ij}=X_{ji}=1$ for $i\neq j$ and zero otherwise. Make a 45 degree rotation to get a diagonal quadratic term with standard propagators. All the mixing will be moved to the interactions via the quadratic term. – TwoBs Oct 16 '14 at 20:08
• @TwoBs Yes, this works. I should have thought of this myself. – Funzies Oct 24 '14 at 9:30