Quantum graviton scattering with definite helicities in the null surface formulation, Part II: Third-order scattering and the exchange channels \author{C.~N.~Kozameh \and G.~O.~Depaola}

Abstract

We compute the third-order Bondi shear $σ^+_3$ in the null surface formulation (NSF) of general relativity with definite graviton helicities. The quantum operator $\daout{3,\pm}$ is derived explicitly in terms of the four helicity channels (I)--(IV) of the scattering equation, and compared with the helicity-summed result of Ref.~\cite{PRL2026}. Applied to two-graviton scattering, the contribution $\langle\daout{3,+}\,\daout{3,-}\rangle$ for the process $h^+(K_1)+h^-(K_2)\to h^+(K'_3)+h^-(K'_4)$ generates simultaneously the $t$- and $u$-channel poles of the tree-level graviton amplitude. An explicit Wick-contraction calculation (Appendix~\ref{app:Wick}) shows that the NSF kernels yield \begin{equation*} \mathcal{M}^{(33)}\big|_{(+,-\to+,-)} = 16πG\,\frac{s^3}{tu}, \end{equation*} from first principles, with the angular integration over $S^2$ manifestly finite and no propagator introduced. The pole structure $1/(tu)$ and the $s^3$ dependence are exact consequences of the null-cone geometry; the coefficient $16πG$ follows from the Ashtekar normalization of the asymptotic modes, in analogy with Ref.~\cite{PRD2026partI}. Both $t$- and $u$-channel poles arise simultaneously from a single Wick contraction; in covariant perturbation theory they arise from two separate Feynman diagrams. The completion of the tree-level amplitude via $\mathcal{M}^{(24)}=\langle\daout{2}\cdot\daout{4}\rangle$ is carried out in Part~III~\cite{PRD2026partIII}. Unitarity at order $\varepsilon^2$ is verified; the operators $\daout{n,\pm}$ are identified as the terms of the Baker-Campbell-Hausdorff expansion of $S^\dagger\ain{\pm}S$, establishing unitarity as a structural consequence of the recursive NSF equations~\cite{SmatrixNSF}.