Geometric origin of the non-supersymmetric Yang-Mills loop integrand
In plain words
For gluons without supersymmetry, a 2024 construction using curves drawn on surfaces gives a natural loop integrand (the function integrated over loop momenta) in the limit of many colors. Whether it is the volume-like form of an actual geometric shape is not known.
Precise statement
Arkani-Hamed, Cao, Dong, Figueiredo and He (Phys. Rev. Lett. 134, 171601, 2025) defined a gauge-invariant, cut-consistent planar (leading-color) Yang-Mills loop integrand using surface kinematics, building on curve-integral formulas for $\operatorname{Tr}\phi^{3}$ theory, with explicit results at one loop for all multiplicities and a simplest two-loop case. Determine whether this integrand, for all multiplicities n and loop orders L in the planar limit, is the canonical form of a positive geometry, and whether a 4D helicity-basis version exists.
What would settle it
Construction of a positive geometry with this canonical form, or a proof that the integrand has non-logarithmic singularities incompatible with one.
Status in the literature
Unverified note
The planar integrand was defined in 2024 (published 2025); no positive geometry with this canonical form is known as of 2026.