Why the proca lagrangian has an inverted sign
I've recently finished reading <PERSON> "Modern Particle Physics". There is a question which is not answered in his book, and to which I couldn't find on the internet in the "introductory" parts of QFT.
My problem is with the form of the Proca Lagrangian. Indeed, for a massless field (Say in QED) the lagrangian is given in the book as :
$$L = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$$
I am placing myself in the vacuum here. As we can see, there is a minus sign in front of the "kinetic" term. This does not affect dynamics in this case, but it looked already unusual to me since usually the kinetic term is taken with a positive sign. 298
If we generalise to a massive boson, we get : $$L = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} +\frac{1}{2}m^2A^\mu A_\mu$$
Again, the overall sign is the opposite of the usual convention, otherwise nothing really strange here.
However, my problem arises when we consider the lagrangian of spin-half particle interacting with QED (but imagining the photon has a mass, I know it's non-physical but it's an easier example for me :D). Then, the total Lagrangian becomes, if I'm not mistaken :
$$ L =i\overline{\psi}\gamma^\mu D_\mu\psi-m\psi\overline{\psi}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^2A^\mu A_\mu$$
And here, the seemingly arbitrary sign isn't anymore... What is the justification for this inverted kinetic and mass sign ? I guess that for massless bosons, the sign is still arbitrary since the $F_{\mu\nu}$ term decouples from the rest of the lagrangian, but I don't think it is with the mass term (haven't done the calculations though, so shame on my lazyness if that is the answer). I know my lagrangian is not U(1) gauge invariant, so my guess is maybe that strange sign comes from the spontaneous symmetry breaking from the Higgs mechanism ?
Edit : Some precisions about what I mean by "Inverted sign" : If we take say a scalar field with a mass, it's lagrangian will be composed of a kinetic term $\frac{1}{2}\partial_\mu\phi\partial^\mu\phi$ and a mass term $\frac{1}{2}m^2\phi^2$. The lagrangian then is $L = T-V = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2$
If I interpret $F_{\mu\nu}F^{\mu\nu}$ as a kinetic term, then we see that in the lagrangian I wrote up, the signs of the terms are inverted. I would like to know if this is a purely conventional choice (which I know it is, in the case of the isolated lagrangian, but not sure in the combined one), and if it's not, why it is written like that. 298