Lagrangian formalism for boson mass

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 $\frac{1}{2}m^2 Z_{\mu}Z^{\mu}$. (Is their a mathematical reason that charged boson mass must in this way, also have a $\frac{1}{2}$ difference?) For example, the $W$ and $Z$ boson mass term in Glashow-Weinberg-Salam Theory. $$\mathcal{L}=m_W^2W^+_{\mu}W^{-\mu} + \frac{1}{2}m_Z^2 Z_{\mu}Z^{\mu}$$ 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.