2026/05/31 by C. N. Kozameh, G. O. Depaola
Physics and Astronomy · #hep-th
arxiv created 2026/07/30 · arxiv updated 2026/07/31
We extend the perturbative null-surface formulation (NSF) scattering map to fourth order and derive an all-order recursion for the quantum cut. After the antipodal matching, both cone sources are evaluated on the same retarded solution determined by the free incoming radiative data. The perturbative NSF equations therefore determine every coefficient Zn from that data, without introducing new independent asymptotic information. The partial cut Z[N]=∑j=1N\epj Zj defines the cumulative operator Uω,[N]=exp[-\iiωZ[N]]. An exact factor recursion for this operator gives a generating formula for δan,λout in terms of the order-n cone source and lower-order operators. The scalar flux term Σ, which begins quadratically, is included on the cut side of the matching equation; its free quadratic part cancels between future and past infinity and it never introduces a new order-n radiative operator. For smooth smeared radiative data, every finite-order cut is well defined and self-adjoint, so Uω,[N] and its recursive factors are unitary and bounded. The frequency powers in the perturbative coefficients are thus part of the expansion of a bounded unitary operator, rather than separate ultraviolet enhancements. At fourth order we formally determine δa4,λout and identify the mixed one-loop sector \mathcal M24=\mathcal M(24)+\mathcal M(42). A general radial power-counting proposition proves ultraviolet finiteness at arbitrary perturbative order for the flat-cone two-vertex sectors. In particular, \mathcal M24 and the previously obtained \mathcal M33 both scale as ∫^∞ \dd K/K4 in the uniform radial ultraviolet region.