Scalar-Scaffolded Gluons and the Combinatorial Origins of Yang-Mills Theory

TL;DR

The paper develops a universal, combinatorial framework to compute Yang–Mills amplitudes by recasting gluons as scalars in a scalar scaffolding, linked to Tr φ^3 theory via a simple kinematic shift. Central to the construction are the u-variables and their binary geometry on surfaces, which organize tree and loop amplitudes, factorization, and gauge invariance in a coordinate-free way. Tree-level amplitudes reproduce bosonic-string-like results under scaffolding, while loop-level amplitudes require controlled inclusion of self-intersecting curves and closed Δ-curves, yielding well-defined YM integrands and matching leading singularities. A key outcome is a consistent, triangulation-independent description of gluon amplitudes that unifies scalars, pions, and gluons in a single kinematic language, with potential for all-loop Yang–Mills recursion in the planar limit. The work suggests new computational strategies and deeper connections between combinatorial geometry in kinematic space and real-world gauge dynamics, with broader implications for gravity and beyond.

Abstract

We present a new formulation for Yang-Mills scattering amplitudes in any number of dimensions and at any loop order, based on the same combinatorial and binary-geometric ideas in kinematic space recently used to give an all-order description of Tr $φ^3$ theory. We propose that in a precise sense the amplitudes for a suitably "stringy" form of these two theories are identical, up to a simple shift of kinematic variables. This connection is made possible by describing the amplitudes for $n$ gluons via a "scalar scaffolding", arising from the scattering of $2n$ colored scalars coming in $n$ distinct pairs of flavors fusing to produce the gluons. Fundamental properties of the "$u$-variables", describing the "binary geometry" for surfaces appearing in the topological expansion, magically guarantee that the kinematically shifted Tr $φ^3$ amplitudes satisfy the physical properties needed to be interpreted as scaffolded gluons. These include multilinearity, gauge invariance, and factorization on tree- and loop- level gluon cuts. Our "stringy" scaffolded gluon amplitudes coincide with amplitudes in the bosonic string for extra-dimensional gluon polarizations at tree-level, but differ (and are simpler) at loop-level. We provide many checks on our proposal, including matching non-trivial leading singularities through two loops. The simple counting problem underlying the $u$ variables autonomously "knows" about everything needed to convert colored scalar to gluon amplitudes, exposing a striking "discovery" of Yang-Mills amplitudes from elementary combinatorial ideas in kinematic space.

Figures (38)

• Figure 1: Binary behaviour of $u$-variables and factorizations.
• Figure 2: From colored scalars to scalar-scaffolded gluons.
• Figure 3: Triangulation of the disk with 5 marked points on the boundary and corresponding dual fat graph.
• Figure 4: Box triangulation of the punctured disk with 4 marked points on the boundary and corresponding dual fat graph.
• Figure 5: Determining momentum of curve $X_{2,5}$.