I Why is there a -1/2 in the Lagrangian density for ω mesons?

[izzi wekwek]

My question is why there is -1/2 for ω meson?

 

[my2cts]

Where did you find these lagrangians?
I would remove the (-)1/2( ...) bit.
Then it can be correct if σ is complex and ω is real valued.

 

[vanhees71]

No, you shouldn't remove the $-1/2$ ;-), because it's correct. You obviously work in the east-coast metric, where $(\eta_{\mu \nu})=\mathrm{diag}(1,-1,-1,-1)$. The physical degrees of freedom of $\omega^{\mu}$ are three space like vectors, and thus to get the right sign for the kinetic and the mass term there must be that additional - sign.