Explicit parametrizations for the $n$-body phase space in terms of spinor-helicity variables are available in [11, 33, 34, 35, 36].

Public Mathematica packages that allow one to perform this task are available in [28, 38].

We use the convention ${\epsilon }^{12}={\epsilon }^{\dot{1}\dot{2}}=1$, where spinor indices are raised and lowered as ${\lambda }^{\alpha }={\epsilon }^{\alpha \beta }{\lambda }_{\beta }$ and ${\tilde{\lambda }}_{\dot{\alpha }}={\epsilon }_{\dot{\alpha }\dot{\beta }}{\tilde{\lambda }}^{\dot{\beta }}$. Angle and square brackets are then defined as $⟨ij⟩={\lambda }_i^{\alpha }{\lambda }_{j,\alpha }$ and $[ij]={\tilde{\lambda }}_{i,\dot{\alpha }}{\tilde{\lambda }}_j^{\dot{\alpha }}$, respectively.

The convention for the Levi-Civita tensor is ${\epsilon }^{0123}=1/\sqrt{-g}$.

In this case, the phase space integral can be achieved, e.g., by parametrizing the final-state spinors as [21, 34] $$\left(\begin{array}{c}{\lambda }_3\\ {\lambda }_4\end{array}\right)=\frac{1}{\sqrt{1+z\overline{z}}}\left(\begin{array}{cc}1& \overline{z}\\ -z& 1\end{array}\right)\left(\begin{array}{c}{\lambda }_1\\ {\lambda }_2\end{array}\right)$$and writing the phase space measure as $$\int \mathrm{d}{\mathrm{\Phi }}_{34}=-\frac{1}{2}\frac{1}{8\pi }\oint \frac{\mathrm{d}z}{2\pi i}\int \frac{\mathrm{d}\overline{z}}{(1+z\overline{z}{)}^2},$$where we included the $1/2$ symmetry factor due to indistinguishable particles. This yields the factors $$-{\text{Res}}_{z=0}\int \mathrm{d}\overline{z}\frac{1+(z\overline{z}{)}^{2S}+(1+z\overline{z}{)}^{2S}}{(1+z\overline{z}{)}^{2S+2}}=\frac{2S+3}{2S+1}$$that appear in Eq. (24).

In deriving these bounds for the gravitational EFT ($S=2$), we explicitly neglected the universal long-range contribution from single-graviton exchange $$|{\mathcal{A}}_{1^{+2},2^{+2}\to 3^{+2},4^{+2}}⟩=\frac{1}{M_{\mathrm{P}}^2}\frac{⟨12{⟩}^4[34{]}^4}{stu},$$responsible for the well-known forward and backward singularities that induce infrared divergences in the partial wave projections of the scattering amplitude. Such divergences are a well-understood feature of long-range interactions and can be systematically handled by factoring out a universal infrared-divergent phase [55] common to all partial waves. This universal phase, which arises from the resummation of iterated single-graviton exchange (the gravitational Born series), does not affect physical observables and can be stripped from the $S$ matrix, as done for example in [56, 57], yielding infrared-finite amplitudes suitable for unitarity and positivity analyses, and leaving the bounds presented here robust. The resolution scale $\mathcal{E}$ introduced by this procedure [57] admits a physical interpretation as the energy resolution of a macroscopic detector for soft gravitons. Alternative approaches to deal with the above singularity include the smearing method of Ref. [58] and the dimensional reduction technique of Ref. [59].

$C_2(G)$ is defined by $f^{ACD}{f}^{BCD}=C_2(G){\delta }^{AB}$. In particular, the structure constants of the $SU(N)$ group, which has dimension $d(SU(N))=N^2-1$, are normalized such that $C_2(SU(N))=N$.