2026/05/07 by Carlos Kozameh, Gerardo Depaola
#hep-th
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 (\mathscrI), and the metric is reconstructed from the cut function. Matching between \mathscrI+ and \mathscrI- yields Ztotal = Zcut + Zcone, where Zcut contains free shear data and Zcone 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 Z2 and Ω2, we derive the second-order Bondi shear and outgoing operators δa2,±out. Written via on-shell phase-space data at \mathscrI, 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 → 1) tail amplitude and (2 → 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.