Quantum graviton scattering with definite helicities in the null surface formulation

Abstract

We develop a helicity-resolved description of quantum graviton scattering in the Null Surface Formulation (NSF) of gravity. Dynamical data are Bondi shear modes at null infinity ($\mathcal{I}$), and the metric is reconstructed from the cut function. Matching between ${\mathcal{I}}^{+}$ and ${\mathcal{I}}^{-}$ yields $Z_{\text{total}}=Z_{\text{cut}}+Z_{\text{cone}}$, where $Z_{\text{cut}}$ contains free shear data and $Z_{\text{cone}}$ is the nonlinear cone source solution--analogous to the scalar retarded-advanced relation with on-shell free-data differences. Using second-order NSF equations for $Z_2$ and $\hat{I}{\text{c◯}}_2$, we derive the second-order Bondi shear and outgoing operators $\hat{I}´a_{2,\pm }^{\text{out}}$. Written via on-shell phase-space data at $\mathcal{I}$, their kernels contain spin-weighted angular Green functions and 3D momentum constraints, with energy fixed by positive frequency. The quadratic cone source decomposes into normal-ordered sectors, allowing matrix elements to select distinct operator components. We obtain the $(2\to 1)$ tail amplitude and $(2\to 2)$ four-graviton matrix elements. The tail amplitude is generated by the two-annihilation sector, selecting helicities via the NSF kernels' spin-weight structure. For four-point scattering, products of two outgoing operators reconstruct spatial momentum conservation; 4D conservation is recovered by imposing the positive-energy on-shell Poincaré sector. This reproduces standard tree-level amplitudes, with Mandelstam poles arising from angular Green functions and helicity projections. This yields an intrinsically on-shell formulation of graviton scattering without off-shell bulk propagators, with celestial sphere spectral-angular distributions as natural observables.