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

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

Interestingly, the process [itex]\psi + \bar{\psi} \to \psi \bar{\psi}[/itex] 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).