The cut equation
- 1School of Natural Sciences, Institute for Advanced Study
- 2Werner-Heisenberg-Institut
Abstract
A bstract Scattering amplitudes for colored theories have recently been formulated in a new way, in terms of curves on surfaces. In this note we describe a canonical set of functions we call surface functions , associated to all orders in the topological expansion, that are naturally suggested by this point of view. Surface functions are generating functions for all inequivalent triangulations of the surface. They generalize matrix model correlators, and in the planar limit, coincide with field theoretic loop integrands. We show that surface functions satisfy a universal recursion relation, the cut equation , that can be solved without introducing spurious poles, to all orders in the genus expansion. The formalism naturally extends to include triangulations with closed curves, corresponding to theories with uncolored particles. This new recursion is quite different from the topological recursion relations satisfied by matrix models. Applied to field theory, the new recursion efficiently computes all-order planar integrands for general colored theories, together with uncolored theories at tree-level. As an example we give the all-order recursion for the planar NLSM integrand. We attach a Mathematica notebook for the efficient computation of these planar integrands, with illustrative examples through four loops.
BibTeX
@article{arkanihamed2026equation,
title = {{The cut equation}},
author = {Nima Arkani-hamed A and Hadleigh Frost and Giulio Salvatori},
journal = {Journal of High Energy Physics},
year = {2026},
month = jun,
doi = {10.1007/jhep06(2026)102},
url = {https://doi.org/10.1007/jhep06(2026)102},
note = {CC BY},
}
RIS
TY - JOUR
TI - The cut equation
AU - Nima Arkani-hamed A
AU - Hadleigh Frost
AU - Giulio Salvatori
PY - 2026
DA - 2026/06/10
JO - Journal of High Energy Physics
DO - 10.1007/jhep06(2026)102
UR - https://doi.org/10.1007/jhep06(2026)102
AB - A bstract Scattering amplitudes for colored theories have recently been formulated in a new way, in terms of curves on surfaces. In this note we describe a canonical set of functions we call surface functions , associated to all orders in the topological expansion, that are naturally suggested by this point of view. Surface functions are generating functions for all inequivalent triangulations of the surface. They generalize matrix model correlators, and in the planar limit, coincide with field theoretic loop integrands. We show that surface functions satisfy a universal recursion relation, the cut equation , that can be solved without introducing spurious poles, to all orders in the genus expansion. The formalism naturally extends to include triangulations with closed curves, corresponding to theories with uncolored particles. This new recursion is quite different from the topological recursion relations satisfied by matrix models. Applied to field theory, the new recursion efficiently computes all-order planar integrands for general colored theories, together with uncolored theories at tree-level. As an example we give the all-order recursion for the planar NLSM integrand. We attach a Mathematica notebook for the efficient computation of these planar integrands, with illustrative examples through four loops.
SN - 1029-8479
SN - 1126-6708
SN - 1127-2236
ER -