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}

We compute the third-order Bondi shear ${\sigma }_3^{+}$ in the null surface formulation (NSF) of general relativity with definite graviton helicities. The quantum operator $\text{daout}3,\pm$ 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 $⟨\text{daout}3,+\text{daout}3,-⟩$ for the process $h^{+}(K_1)+h^{-}(K_2)\to h^{+}(K_3^{\mathrm{\prime }})+h^{-}(K_4^{\mathrm{\prime }})$ 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 $1\mathrm{/}(tu)$ and the $s^3$ dependence are exact consequences of the null-cone geometry; the coefficient $16\pi G$ 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 ${\mathcal{M}}^{(24)}=⟨\text{daout}2\cdot \text{daout}4⟩$ is carried out in Part~III~\cite{PRD2026partIII}. Unitarity at order ${\epsilon }^2$ is verified; the operators $\text{daout}n,\pm$ are identified as the terms of the Baker-Campbell-Hausdorff expansion of $S^{†}\text{ain}\pm S$, establishing unitarity as a structural consequence of the recursive NSF equations~\cite{SmatrixNSF}.