On differential operators for scalar-scaffolded gluons
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911325192454144 |
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| author | Dong, Jin Su, Yong-Xiang Yang, Dongyu |
| author_facet | Dong, Jin Su, Yong-Xiang Yang, Dongyu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_15882 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On differential operators for scalar-scaffolded gluons Dong, Jin Su, Yong-Xiang Yang, Dongyu High Energy Physics - Theory 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. |
| title | On differential operators for scalar-scaffolded gluons |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2512.15882 |