# What does this Lagrangian represents?

I came across this expression while doing exercises and I was wondering if it was a 'real' expression.

$$\mathcal{L}=\frac{1}{2}\partial_\mu \phi \partial^\mu \phi -\frac{m^2}{2}\phi ^2 +\overline{\psi}(i\gamma^\mu\partial_\mu -M)\psi + i\lambda \phi \overline{\psi}\gamma_5\psi$$

I can recognize the first term is a kinetic term, the second as a potential term and the middle one as the Dirac Lagrangian. The last one is an interaction term, so it involves the interaction between electrons and a spin 0 particle, which one?

Does this Lagrangian density represent a specific theory?

The Lagrangian could be a simplified version of a part of the standard model. $\phi$ may be a simplified version of Higgs boson (however Higgs potential $V(\psi)$ does not appear). The last term $$i\lambda \phi \overline{\psi}\gamma_5\psi$$ is a (pseudoscalar) Yukawa interaction. See http://en.wikipedia.org/wiki/Yukawa_coupling
That is a simplified version of QED that uses a scalar field instead of an electromagnetic field. It is usually used first to introduce the concept without having to worry about gauge invariance and other other such things, but still has spinors unlike the $\varphi^4$ theory.
• It's not what people usually call "scalar electrodynamics" because of the $\gamma_5$ ($\phi$ becomes a pseudoscalar this way). – Vibert Jan 18 '14 at 0:03