On differential operators for scalar-scaffolded gluons

TL;DR

This paper introduces differential operators that transform scalar-scaffolded gluon amplitudes into simpler forms, revealing new structural properties and reducing redundancy in Yang-Mills amplitude expansions.

Contribution

It develops novel differential operators that convert scalar-scaffolded gluons into scalar-only or mixed amplitudes, connecting them to $ ext{phi}^3$ diagrams and generalizing the uniqueness theorem.

Findings

Operators convert n-gluon amplitudes into single $ ext{phi}^3$ diagrams.

Number of mixed amplitudes matches Catalan number $ ext{C}_{r-2}$.

Reduced redundancy in Yang-Mills amplitude expansion.

Abstract

Recently, based on the curve-integral formulation for stringy Tr$\phi^3$ amplitudes, a combinatorial formulation for Yang-Mills amplitudes has been proposed which describes gluons using pairs of scalars and produces the $n$-gluon amplitude from simple kinematical shift of stringy Tr$\phi^3$ amplitudes with $2n$ scalars. It has revealed a variety of new properties and structures even for tree-level gluon amplitudes such as hidden zeros and splits, and in this note we provide another example: we study differential operators acting on Yang-Mills amplitudes with respect to $2n$-scalar kinematic variables, which convert such scalar-scaffolded gluons into scalars. In particular, we find $(n{-}1)$-fold differential operators (using $2n$-scalar variables) that turn the $n$-gluon amplitude into a single planar $\phi^3$ diagram; we then generalize such operators to those that convert $n$ gluons to mixed amplitudes with $r$ scalars and $n{-}r$ gluons (the latter can be viewed as insertions on $\phi^3$ diagrams). We also show that the number of linearly independent mixed amplitudes with $r$ scalars and $n-r$ gluons is given by the number of $\phi^3$ diagrams, the Catalan number $\mathcal{C}_{r-2}$, which can be viewed as a generalization of the ``uniqueness" theorem of gluon amplitudes (with $r=0$). Finally, our construction leads to a planar version of the universal expansion of Yang-Mills amplitudes into a sum of gauge-invariant prefactors built from nested commutators, each accompanied by an mixed amplitude in the natural basis. This formulation significantly reduces the redundancy present in the original expansion.