Global phase symmetry (${\phi }_j\mapsto e^{-i{q}_j\alpha }{\phi }_j$)

An internal circle group action with spacetime-independent parameter $\alpha$. For canonical complex scalar fields, Noether's theorem gives $j^{\mu }=i{\sum }_j{q}_j{\phi }_j^{†}\overleftrightarrow{{\partial }^{\mu }}{\phi }_j$. The Noether charge of the particle annihilated by ${\phi }_j$ is $q_j$, and its antiparticle has charge $-q_j$.

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.