# Toy field theories

## Main Question or Discussion Point

Hi all,

Can you help me come up with toy field theories to practice calculating cross-sections? I'm not considering theories with derivative couplings like scalar electrodynamics.

The interaction vertices I'm aware of are $\bar{\psi} \gamma^\mu A_\mu \psi$ (QED), $\bar{\psi}\psi\varphi$ (Yukawa), $\bar{\psi}\gamma_5 \psi \varphi$ (pseudoscalar theory), $\bar{\psi}\gamma^\mu B_\mu \gamma_5 \psi$ (axial vector), $\bar{\psi}\psi\varphi^2$ (don't know if this has a name). What are some other creative field theory interactions?

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Haelfix
Try a Yukawa interaction with a Majorana field and say a scalar

Btw there are some typos in your interaction terms and some overlap, but anyway.

Really, I don't see any typos in those interaction lagrangians? The placement of gamma-5 to the right of the gamma's is consistent with $\gamma_5 = i\gamma^0\gamma^1\gamma^2\gamma^3$. Are you referring to the fact that I wrote B instead of A for the vector field? I guess there should be an i in front of the pseudoscalar to ensure Hermiticity. What do you mean when you say some of them overlap?

I'm not familiar with Majorana fields, how would one write down such an interaction?

Thanks.

There's always $$\phi^4$$ theory, and the theory of a complex scalar field with interaction $$(\phi^{\dag}\phi)^2$$. Another popular one is $$\phi^3$$ theory in six spacetime dimensions -- Srednicki uses that in his QFT book since its one-loop renormalization is more interesting than that of $$\phi^4$$. Yes, it's sick (no ground state), but it can still be used as a toy field theory.

Also from Srednicki (chapter 11): a theory of two real scalar fields $$A$$ and $$B$$ with an interaction $$gAB^2$$; a theory of a real scalar field and a complex scalar field $$\chi$$ with an interaction $$g\phi\chi^{\dag}\chi$$; a theory of three real scalar fields $$A$$, $$B$$ and $$C$$ with interaction $$gABC$$.

Srednicki's chapter 49 walks you through Feynman rules for Majorana fields, considering specifically Yukawa theory. The chapter 49 problem considers a SUSY-inspired theory of two different complex scalar fields $$E_L$$ and $$E_R$$, a Dirac field $$\Psi$$ and a Majorana field $$X$$ with interaction
$$\sqrt 2e E_L^{\dag}\bar XP_L\Psi + \sqrt 2e E_R^{\dag}\bar XP_R\Psi + h.c.$$
where $$P_{L, R}$$ are the projections $$(1 \pm \gamma_5) / 2$$.