Charge assignments for a cubic complex-scalar interaction (${\varphi }^2\Phi :2q_{\varphi }+q_{\Phi }=0$)

The interaction $-\frac{1}{2}(h{\varphi }^2\Phi +h^{\ast }{\varphi }^{†2}{\Phi }^{†})$ is invariant under a global phase symmetry precisely when the displayed charge relation holds. With $q_{\varphi }=1$, a heavy particle annihilated by $\Phi$ has charge $-2$, so its allowed decay channel is $\Phi \to \bar{\varphi }\bar{\varphi }$. The reverse heavy antiparticle decays into $\varphi \varphi$. Crossing only one heavy external state changes the exchange channel of mixed-species quantum scattering.