Here the ellipses denote all terms in the low-energy amplitude that are not the graviton pole. This includes contributions that are regular in $t$ coming from general relativity, as well as higher-dimension operator corrections in the effective field theory. Notably, all of these terms exhibit different $t$ dependence than the graviton pole, so there are always values of $t$ for which the $s^2$ coefficient of the low-energy amplitude is nonzero, for instance if $t$ is below the mass gap.

In Ref. [17], Regge bounds were derived for gravitational scattering amplitudes as a consequence of analyticity, unitarity, and the partial wave expansion in various contexts. The most conservative of these bounds, merely assuming dominance of single-graviton exchange at large impact parameter, showed in generality that, for tree-level amplitudes, $A(s,t)<s^{2-\frac{D-7}{2(D-4)}}$ for $t<0$, enabling twice-subtracted dispersion relations in $D>7$. By including eikonalization, even better scaling is possible. More generally, Refs. [18, 19] outlined causality arguments that bound $A(s,t)<s^2$.

Throughout this work, we will assume that $G$ is a single group factor. For a product group, $G={\otimes }_i{G}_i$, completeness of each $G_i$ factor implies completeness of $G$ under certain conditions. In particular, consider the case in which a starting spectrum $Q_i$ is sufficient to prove completeness for $G_i$. First, we require an initial spectrum $Q$ for $G$ composed of all states with charges in $Q_i$ under $G_i$ and neutral under all others, for each $i$. Second, we assume that each $G_i$ has no outer automorphisms, which would induce a relative alignment between charges under different group factors. If these conditions are met, completeness of the product group $G$ is guaranteed.

This can also be derived using the null constraints introduced in Ref. [34].

For earlier work bootstrapping Ansätze for theories with color and gravity, see Ref. [38].

Our results on charge completeness can be understood in this language. By scattering the charges $\overrightarrow{q}$ and ${\overrightarrow{q}}^{\prime }$, we have used that there is a state of charge $\overrightarrow{q}+{\overrightarrow{q}}^{\prime }$ or $\overrightarrow{q}-{\overrightarrow{q}}^{\prime }$, corresponding to the $s$ or $u$ channel. This implies that there exists some choice of $v$ for which Eq. (4) is satisfied where the set of exchanged representations $\mathcal{R}$ contains neither $\overrightarrow{q}+{\overrightarrow{q}}^{\prime }$ nor $\overrightarrow{q}-{\overrightarrow{q}}^{\prime }$.