quantum field theory - Gauge symmetries and elementary particles

The Weinberg-Witten theorem (disclaimer: I don't know this wikipedia entry) is usually mentioned as the reason why gravitons may not be composite particles. I do understand the proof of the theorem, but not the previous conclusion.

The theorem states that in an interacting and Poincare invariant quantum field theory, there are not massless, spin-2 particles unless there exists a gauge symmetry — which makes the energy-momentum tensor non-covariant (actually covariant up to a gauge transformation) under Lorentz transformations in the Fock space. So the immediate conclusion of the theorem is that the existence of a massless, spin-2 particle (like a graviton) requires linearized diffeomorphisms.

My question is: why do linearized diffeomorphisms imply that gravitons are elementary particles? Or, more in general, why the particle corresponding to a gauge field must be elementary (I know that a gauge symmetry must be exact, but why this implies that the corresponding particle must be elementary?).

Answer

I have followed this reference

The Weinberg-Witten theorem states that a theory containing a Poincaré covariant conserved tensor $T_{\mu\nu}$ forbids massless particles of spin $j > 1$ for which $P_\nu = \int T_{0\nu}dx$ is the conserved energy-momentum four-vector.

Consider a composite graviton made of $2$ particles of spin $1$.

Each of the spin-$1$ particles will be possibly have a non-vanishing charge current, in this case the Poincaré covariant conserved tensor $T_{\mu\nu}$ (this is authorized for a spin-$1$ particle)

But this means that the composite graviton, being the "sum" of these 2 spin-1 particles, will have also a non-vanishing Poincaré covariant conserved tensor $T_{\mu\nu}$

But this is forbidden by the Weinberg-Witten theorem, because the spin of the graviton is 2.

So the graviton cannot be a composite particle.

In the full General Relativity, the covariant stress-energy tensor $T_{\mu\nu}$ is not conserved, and the conserved stress-energy quantity $(T_{\mu\nu} + \tau_{\mu\nu})$, is not a full covariant tensor.

If we linearized the Einstein equation, so as to have a conserved stress-energy tensor, we have:

$$(G_{\mu\nu})_{linearized} = \chi [(T_{\mu\nu} + \tau_{\mu\nu})] $$

The gauge symmetries, for the linear graviton as :

$$h_{\mu\nu} \rightarrow h_{\mu\nu} + \partial_\mu \phi_\nu + \partial_\nu \phi_\mu$$

and could be interpreted as "linear diffeomorphisms".

But in fact, the $\tau_{\mu\nu}$ term is not invariant, by the gauge symmetry, so the full conserved stress-energy quantity $(T_{\mu\nu} + \tau_{\mu\nu})$ is not gauge-invariant, and so we escape from the Weinberg-Witten theorem.