Psi^4 Theory: A Fermion with Quartic Self-Interaction

[jdstokes]

I've been analysing the following toy theory which I've called psi^4 theory for want of a better name.

$\mathcal{L} = :i\bar{\psi}\gamma^\mu\partial_\mu \psi - m\bar{\psi}\psi + \lambda (\bar{\psi}\psi)^2:$.

Ie a fermion with quartic self-interaction. This interaction can describe contact processes such as $\psi + \bar{\psi} \to \psi + \bar{\psi}$ whose Feynman rule I derived to be $4i\lambda$.

Interestingly, the process $\psi + \bar{\psi} \to \psi \bar{\psi}$ has a Feynman rule of zero and consequently zero scattering amplitude. This comes about because if one expands the interaction Lagrangian in positive and negative frequency parts, there are four operator contributions which cancel after normal ordering. Does anyone know why this might be expected physically (ignoring the obvious unphysicality of the Lagrangian).

 

[lbrits]

Phi^4 is a scalar theory. I think this is more like a four-Fermi interaction. I was unaware that the amplitude were zero. I would think that it is equal to $$4i\lambda$$, after renormalization, but I haven't actually done the calculation. It might have to do with antisymmetry after exchange of external legs, but I'm not sure how.