On differential operators for scalar-scaffolded gluons

TL;DR

<3-5 sentence high-level summary>

Abstract

Recently, based on the curve-integral formulation for stringy Tr$φ^3$ 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$φ^3$ 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 $φ^3$ 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 $φ^3$ 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 $φ^3$ 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.

Figures (20)

• Figure 1: Scaffolding from 8-scalar amplitude $\mathcal{A}_8^{\text{scalar}}(1,2,3,4,5,6,7,8)$ to 4-gluon amplitude $\mathcal{A}_4^{\rm YM}(1,3,5,7)$.
• Figure 2: $\mathcal{A}_n^{\rm YM}$ factorizes on $X_{a,b}$, forming a pair of sub-amplitudes $\mathcal{A}_L$ and $\mathcal{A}_R$.
• Figure 3: 4-point $\phi ^3$ diagrams and corresponding differential operators. The propagators are represented by black diagonals, while differential operators are indicated by blue arrows. For example, the operator $[1,6]$ is depicted as an arrow starting from vertex-1 and pointing between vertices 5 and 7.
• Figure 4: 5-point $\phi ^3$ diagram and corresponding differential operator.
• Figure 5: 6-point $\phi ^3$ diagrams and corresponding differential operators for ray-like, zigzag and cyclic triangulations.

Theorems & Definitions (14)

• Example 3.1
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• Example 3.10