Hi. I'm reading about non-abelian theories and have thus far an understanding that a gauge invariant Lagrangian is something to strive for. I previously thought that the Yang-Mills gauge boson free field term ##-1/4 F^2 ## was gauge invariant, but now after realizing that the field strength transform homogeneously under gauge transformation U(x) s.t $$ F \to UFU^\dagger$$ it seems like this term also transform like $$F^2 \to UF^2U^\dagger.$$ Am I right there? If so isn't this a problem for the physics? Why, why not?