Normalization of boson mass term in Lagrangian field theory

Question

Why for charged boson, it's mass term is $$m^2 W^+_{\mu}W^{-\mu}~?$$ While for neutral boson, it's mass term is $$\color{red}{\frac{1}{2}}m^2 Z_{\mu}Z^{\mu}.$$ Is there a mathematical reason that charged boson mass must in this way, also have a $\color{red}{\frac{1}{2}}$ difference? For example, the $W$ and $Z$ boson mass term in Glashow-Weinberg-Salam Theory $$\mathcal{L}=m_W^2W^+_{\mu}W^{-\mu} + \color{red}{\frac{1}{2}}m_Z^2 Z_{\mu}Z^{\mu}.$$ And I find that we cannot spit this into $W^{+2}+W^{-2}$, I think this violate charge conservation in single term.

Answer 1

It's conventions. Terms in the Lagrangian density are conventionally divided by their symmetry factor. Hence a factor $1/2$ for the mass term of a real field, and a factor $1$ for a complex field, cf. e.g. this Phys.SE post.