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The kinetic term in the standard Lagrangian for a vector field, whether massive or not, is $$-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}=-\frac{1}{2}(\partial_\alpha V_\beta \partial^\alpha V^\beta - \partial_\alpha V_\beta \partial^\beta V^\alpha).$$ It is well-known that this is the only "allowed" possibility; i.e. that the ratio between the two terms must be exactly $-1$, or else the theory is invalid. If I understand correctly, the issue is that otherwise there will be a mode with negative kinetic energy, making the vacuum unstable.

Inverting this kinetic term, in the massive case, gives us the propagator $$\frac{{\eta}_{\alpha\beta}+\frac{p_\alpha p_\beta}{m^2}}{p^2+m^2-i\epsilon},$$ again with a fixed ratio between two terms.

What's bothering me is that this fixed ratio of $-1$ will presumably not be preserved once we include interactions, unless there is some symmetry enforcing that - which I do not want to assume. There will be various loop diagrams contributing to the two-point function; and they will be proportional to various combinations of $p_\alpha p_\beta$ and ${\eta}_{\alpha\beta}$, without any fixed ratio. Does this mean that the theory is doomed to become unstable?

Of course, the massive vector theory is not renormalizable (unless the vector couples to a conserved current, which again, I am not assuming). Nevertheless, I would expect that it should be usable as an EFT, for energy scales not too far above the mass. This issue with the kinetic terms looks like it could ruin that.

To be a bit more specific, suppose we include a very heavy scalar, which we want to integrate out since it's beyond the desired energy scale. Then doing an EFT-matching between the theories with and without the scalar will give us a low-energy theory with Wilson coefficients for both $\partial_\alpha V_\beta \partial^\alpha V^\beta$ and $\partial_\alpha V_\beta \partial^\beta V^\alpha$, and the ratio between them will not in general equal $-1$. (I have tried this and confirmed it, but my work is too messy to be helpful. But even without detailed calculations; there seems to be nothing that would enforce the ratio.)

So is there any way to make such a theory work?

EDIT: It has now been explained to me that massive vector fields cannot in fact be used in an EFT, because the coefficients of the high-mass-dimensional operators (as generated by RG flow from the loop divergences involving whatever interaction you start with) will be proportional to negative powers of the spin-1 particle mass, and hence not negligible at any energy scale comparable with the mass. So infinitely many parameters are required to make predictions, even at low energy. [The exception being when the mass is generated by spontaneous symmetry breaking, so that the $p_a p_b$ term can be eliminated from the propagator because of the conserved current. And in that case we have nice Ward identities protecting things, which I assume will fix the ratio of the kinetic terms as well. Though I wouldn't mind seeing an explicit calculation of this in some theory at one loop, if there's one written down somewhere. Maybe in QED with a Higgs?] I would appreciate resources that discuss this topic, especially on whether there is any other way to make a massive vector theory usable.

Nevertheless, I believe the massive vector theory should at least be usable at energies well below the mass, with the vector as a virtual particle mediating interactions between the lighter fields. In that case we could also integrate out the vector - if and only if the theory is stable and consistent, which is what I'm trying to find out.

And even for somewhat higher energies, it makes sense to suppose that we picked a fixed energy scale, say $\mu=m$, and that it turned out, miraculously, that all but one or two of the infinitely many couplings vanish (say in MS-bar) at this energy. This is absurdly unnatural, but not impossible. Then making predictions for vector scattering in this theory should be straightforward - again assuming the theory makes sense and has a stable vacuum. So does it, or does the unprotected ratio between the kinetic terms ruin that?

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You are correct that for generic interactions, it is not guaranteed that this "-1 ratio" will be preserved. The point is that the kind of interactions that you add matters, because certain interactions will not preserve the degrees of freedom of the massive spin 1 particle. If the interactions don't preserve the particle's degrees of freedom, the particle will no longer belong to a unitary representation of the Poincare group. This is related to the "-1 ratio", because if the theory is no longer unitary, there is nothing which preserves this ratio under renormalization.

There is a way to figure out which interactions preserve the particle's degrees of freedom, and this is by requiring that the interactions are invariant under some gauge symmetry which guarantees the particle has the correct degrees of freedom. For massless spin 1 particles, this is done through the usual $U(1)$ gauge symmetry $\delta A_{\mu}=\partial_{\mu}\epsilon$. For massive spin 1 particles, the right way to formulate the theory with a gauge symmetry is by introducing an auxiliary scalar field $\phi$ and demanding the theory be invariant under the transformations $\delta A_{\mu}=\partial_{\mu}\epsilon$ and $\delta\phi = m\epsilon$, where $m$ is the mass of the spin 1 particle. Look at the Stueckelberg action for an example of a free theory which is invariant under this symmetry.

Actually, a treatment of how to introduce interactions in a way which respects the gauge symmetry that preserves the massive particle's degrees of freedom for any spin was made recently https://arxiv.org/abs/2309.03901.

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  • $\begingroup$ I awarded the bounty to this answer, because it is the first that I found helpful. The (hot off the press!) article you linked seems to give a very useful and enlightening approach to solving the issue. $\endgroup$ Sep 20, 2023 at 9:30
  • $\begingroup$ Nevertheless, I don't think I will mark this answer as "accepted", because I still need some exposition on what exactly "preserving the degrees of freedom" means and why this problem exists. The vector fields seem to be nice, well-defined operators on the Fock space; why shouldn't an arbitrary product of them make sense as an interaction? $\endgroup$ Sep 20, 2023 at 9:39

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