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.

 

[denian]

I find this toy theory intriguing and potentially useful in understanding certain physical processes. The inclusion of a quartic self-interaction term in the Lagrangian allows for the description of contact processes, which are important in understanding particle interactions. The Feynman rule derived for this interaction provides a clear understanding of the mathematical representation of these processes.

The observation that the process \psi + \bar{\psi} \to \psi \bar{\psi} has a Feynman rule of zero is interesting and warrants further investigation. It is possible that this result is a consequence of the normal ordering of the operators, which may lead to a cancellation of contributions. However, it is important to consider the physical implications of this result and whether it is consistent with experimental observations.

In conclusion, the psi^4 theory presents an interesting and potentially useful framework for understanding certain physical processes. Further research and analysis of this theory may lead to a better understanding of particle interactions and their underlying mechanisms.