On differential operators for scalar-scaffolded gluons

A bstract 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 2 n 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 2 n -scalar kinematic variables, which convert such scalar-scaffolded gluons into scalars. In particular, we find ( n − 1)-fold differential operators (using 2 n -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 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 a mixed amplitude in the natural basis. This formulation significantly reduces the redundancy present in the original expansion.