Past exam of the theoretical physics major option course of the University of Oxford / 2018 / 3 / b / i

Counting the mass dimensions of each interaction operator givesIndeed $\Phi (\partial a{)}^2$ and $\Phi F^2$ have dimension five, whereas $\Phi b^2(\partial b{)}^2$ has dimension seven. These are interactions of an effective field theory with dimensionful couplings.

The direct one-vertex decay channels are $\Phi \to aa$, $\Phi \to bbbb$ and $\Phi \to \gamma \gamma$. Each leading scattering amplitude is linear in its corresponding coupling, and a decay width is quadratic in the amplitude. Therefore, holding the parent mass fixed,There are no tree-level interference terms between these different final states.