 Charge selection rule for complex scalar correlators

In a phase-invariant vacuum, a correlator with $r$ insertions of $\varphi$ and $s$ of ${\varphi }^{\ast }$ gains the phase $e^{i\alpha (r-s)}$. Its invariance for all real $\alpha$ forces it to vanish unless $r=s$. Balanced local interactions and the absence of same-charge free contractions give the same rule order by order in Wick's theorem. A symmetry-breaking vacuum is an essential exception to the vacuum-invariance premise.