The same conclusion holds even for mesons which leads to a weaker bound. The bound for mesons can be obtained from (2) by replacing $N$ by $\sqrt{N}$.

The leading $1/N$ correction comes from the phase-shift associated with the mixed amplitude $G_J{G}_J\to h_{\mu \nu }{h}_{\mu \nu }$ which led to the interference bound of Ref. [6].

It is also possible that width of glueball states, for large but finite $N$, increases significantly as one moves up the spectrum. So, these states may not be approximated by well-separated poles. However, this effect does not change the results of this article (unless stated explicitly) as long as glueballs states are well-separated from the graviton pole.

The only nonvanishing kinetic mixing term that can be written has the structure $h_{\mu \nu }{\square }^k{G}_{J=2}^{\mu \nu }$.

We thank J. Maldacena for pointing this out to us.

The mixing between gravitons and spin-2 mesons/glueballs is simply a spin-2 analogue of the mixing of photons with $\rho$ mesons that has been familiar in QCD for several decades.

For external gravitons, as explained in Ref. [9], even an infinite tower of massive spin-2 exchanges cannot solve the problem. This is true for any external massless particle with spin $\ge \frac{3}{2}$.

It is also possible that interactions (4), however fine tuned, cannot actually solve the problem by itself.

In a gravitational theory, free particles should be defined carefully. To be precise, by free particle we mean that the spin-$\frac{3}{2}$ particle interacts (not necessarily minimally) only with gravitons. Of course, because of the mixing (4) interactions like $⟨X_{-}{X}_{+}{G}_{J=2}⟩$ are also nonzero. However, all other interactions $⟨X_{3/2}{X}_{3/2}{G}_{J\ne 2}⟩$ are still absent at the leading order in $N$.

The argument is essentially the same as that in Ref. [9] for the decay of a massive spin-2 particle into two gravitons.

Strictly speaking, the theorem of Ref. [25] is applicable only for external scalars. However, it is expected that the same argument holds even for spinning external states. This is supported by the fact that high energy string amplitudes are independent of the exact quantum numbers of external states.

In Ref. [5], a parametric bound on $N$ was derived under some additional assumptions. One of the assumptions was that the scale of new physics, such as ${\mathrm{\Lambda }}_{\mathrm{HS}}$, is parametrically higher than ${\mathrm{\Lambda }}_{\mathrm{QCD}}$. From the bound (2) it is clear that any parametric separation between ${\mathrm{\Lambda }}_{\mathrm{HS}}$ and ${\mathrm{\Lambda }}_{\mathrm{QCD}}$ necessarily requires $N\lesssim \frac{M_{\mathrm{pl}}}{{\mathrm{\Lambda }}_{\mathrm{QCD}}}$ which agrees with the bound of Ref. [5].

The connection between glueballs and closed strings has a long and interesting history. For example see Refs. [49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63].

There is a small loophole that we must close. In certain situation [for example the situation (iii)] causality, in principle, can be restored by one or more spin-2 states. However, since these states are now in the gravity sector, they will appear in the 4-graviton scattering as well. Spin-2 exchanges lead to additional causality violations [9] in the 4-graviton scattering unless they are accompanied by other higher-spin $(J\ge 3)$ states in the gravity sector.

Let us emphasize that this conclusion is valid only when the UV completion is weakly coupled. In fact, there are M-theory compactifications on manifolds with singularities that give rise to confining large $N$ gauge theories which have strongly coupled UV completion.

Note that the lower cut-off $N_{\min }$ is defined after Eq. (2).

To be clear, if large $N$ confining theories somehow do not contain higher spin glueballs in their spectra, none of our arguments would apply.

It is possible that a combination of scenario (I) and (II) restores causality in such a way that the bound (8) can be relaxed. For example, before we hit the scale on the right hand side of (8), there may be a transition from scenario (II) to scenario (I). In this case, the bound (8) on the string scale is no longer reliable.