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So, I am giving the following interaction term in a real-complex scalar Yukawa theory, $$\mathcal{L}_{int} = -g\phi^\dagger\phi\chi^2$$ with $\chi$ the real scalar and $\phi$ the complex scalar. I have seen everywhere the usual interaction term for the real scalar part on the interaction term is not squared, so if I am computing the interaction, $\phi\chi\rightarrow\phi\chi$ how does it change and what new does the potential in this form give? $\textbf{Or where should I start in finding the proper vertex interaction for this potential?}$

The name of the theory (as I have read online) is called "Scalar Yukawa Theory" with the following Lagrangian, \begin{equation} \mathcal{L} = \partial_\nu\phi^\dagger\partial^\nu\phi - m^2\phi^\dagger\phi + \frac{1}{2}(\partial_\nu\chi\partial^\nu\chi - \mu^2\chi) - \lambda\phi^\dagger\phi\chi^2. \end{equation}

We (the class I am in) has not even been told what does this theory describe and why modify the potential is such a way.

Any hints are helpful, thanks.

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    $\begingroup$ 1) Your interaction Lagrangian should probably read $-g \bar{\psi} \psi \phi^2$ in order to be Lorentz invariant. 2) Your reaction $\psi \phi \to \psi \phi$ answers already your question: the Lagrangian describes an interaction vertex with two fermions and two real scalars. $\endgroup$
    – Hyperon
    Commented Nov 21, 2022 at 8:03
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    $\begingroup$ @Hyperon $\psi$ in this case is not a fermion, I changed the symbols to clarify this, and so the interaction term gives what each loop of a Feynman diagram should look like correct? $\endgroup$
    – MathZilla
    Commented Nov 21, 2022 at 13:03
  • $\begingroup$ In a "Yukawa theory" you should have a fermion field and a scalar field. In the new version of your question you seem to have a real scalar $\phi$ and a complex scalar $\chi$? If these are the only fields in your theory it has nothing to do with a Yukawa theory. Please specify the full particle content of your theory and write down the complete Lagrangian. In its present form it is unclear what your actually asking. $\endgroup$
    – Hyperon
    Commented Nov 21, 2022 at 13:57
  • $\begingroup$ @Hyperon Completed. $\endgroup$
    – MathZilla
    Commented Nov 21, 2022 at 14:16
  • $\begingroup$ Such 4-point interaction terms are never called Yukawa terms. They specify a simple vertex for the scattering reaction you wrote down. Write the amplitude down. $\endgroup$ Commented Nov 21, 2022 at 14:54

2 Answers 2

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  1. A "Yukawa interaction" refers to a model with a fermion field and a scalar field and not to a theory with a real scalar field $\chi$ and a complex scalar field $\phi$ (as in your case).

  2. Your Lagrangian is incomplete in the sense that your interaction term $-\lambda \phi^\dagger \phi \chi^2$ will generate divergent contributions of the form $\chi^4$ and $(\phi^\dagger \phi)^2$ at one loop. For a consistent renormalizable theory you have to add the interaction terms $g_1 \chi^4$ and $ g_2 (\phi^\dagger \phi)^2$ with two additional coupling constants $g_1$ and $g_2$ in your Lagrangian.

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Field theory, as you probably noticed by now, is difficult. Many, if not most, real-life models are actually quite complicated and even more difficult to perform actual computations with. The Standard Model, for example, is one of the most beautiful theories ever obtained, but it is considerably tricky to do calculations with if you're not experienced.

Hence, it is extremely common to use models that don't describe anything in real life as a way of practicing. For example, the point of working out $\phi^4$ theory is not because it ends up being a model for the Higgs boson, but rather just because it is a simple enough theory that you actually can do the usual calculations sort of fast. In this way, it plays the role of a toy model. It is a model that you can use for practicing before going on to tackle actual real-life problems.

Think of it as a boxer practicing with a punch bag before going to an actual fight. The punch bag is not a real fighter and won't punch you back, but it works out very well as a practice device for you to get the right muscles used to the movements they should perform. Similarly, we use simple quantum field theories that might not represent anything real with the goal of getting the right concepts in order and learning how to do the calculations. After we learn how to do the calculations, we can go on to understand accurate, real-life models.

To be completely fair, it might be that the Lagrangian you wrote represents something real. Perhaps it has some application in Condensed Matter, or in a proposal for an extension of the Standard Model, but that's not the point. The reason your professor proposed this Lagrangian is most likely because it is a sort of simple Lagrangian representing an interaction between two different fields, which allows you to do calculations and learn important stuff. For example, this Lagrangian is a good example of how two new terms will need to be included if you want to do one-loop calculations, as was mentioned in another answer.

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