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Peer-reviewedCC BYJournal of High Energy Physics

Kinematics, cluster algebras and Feynman integrals

  • Song Lin He1,2,3,4,5
  • Zhenjie Li1,5
  • Qinglin Yang1,5
5 affiliations
  1. 1Chinese Academy of Sciences
  2. 2ICTP-AP International Centre for Theoretical Physics Asia-Pacific
  3. 3Peng Huanwu Center for Fundamental Theory
  4. 4UCAS
  5. 5University of Chinese Academy of Sciences
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Abstract

A bstract We identify cluster algebras for planar kinematics of conformal Feynman integrals in four dimensions, as sub-algebras of cluster algebras for Grassmannian G (4, n ) corresponding to n -point massless kinematics. We classify such algebras for cases through eight points, and provide evidence that singularities of the corresponding Feynman integrals are given by cluster variables and their algebraic generalizations. By sending a point to infinity, our results have implications for symbology of non-conformal Feynman integrals. We also find that dimensional reduction is achieved by folding and the resulting cluster algebras encode singularities of Feynman integrals in three dimensions. As a highly-nontrivial application, we study a eight-point three-loop wheel integral whose kinematics correspond to two-mass-easy box in the non-conformal limit: in addition to nine rational letters that are cluster variables, we also find three algebraic letters containing a new square root, in accordance with results from differential equations for two-loop case. Based on this alphabet, we bootstrap its symbol, which turns out to be strongly constrained by cluster adjacency conditions.

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Cite

BibTeXRIS
BibTeX
@article{he2026kinematics,
  title = {{Kinematics, cluster algebras and Feynman integrals}},
  author = {Song Lin He and Zhenjie Li and Qinglin Yang},
  journal = {Journal of High Energy Physics},
  year = {2026},
  month = jun,
  doi = {10.1007/jhep06(2026)067},
  url = {https://doi.org/10.1007/jhep06(2026)067},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Kinematics, cluster algebras and Feynman integrals
AU  - Song Lin He
AU  - Zhenjie Li
AU  - Qinglin Yang
PY  - 2026
DA  - 2026/06/05
JO  - Journal of High Energy Physics
DO  - 10.1007/jhep06(2026)067
UR  - https://doi.org/10.1007/jhep06(2026)067
AB  - A bstract We identify cluster algebras for planar kinematics of conformal Feynman integrals in four dimensions, as sub-algebras of cluster algebras for Grassmannian G (4, n ) corresponding to n -point massless kinematics. We classify such algebras for cases through eight points, and provide evidence that singularities of the corresponding Feynman integrals are given by cluster variables and their algebraic generalizations. By sending a point to infinity, our results have implications for symbology of non-conformal Feynman integrals. We also find that dimensional reduction is achieved by folding and the resulting cluster algebras encode singularities of Feynman integrals in three dimensions. As a highly-nontrivial application, we study a eight-point three-loop wheel integral whose kinematics correspond to two-mass-easy box in the non-conformal limit: in addition to nine rational letters that are cluster variables, we also find three algebraic letters containing a new square root, in accordance with results from differential equations for two-loop case. Based on this alphabet, we bootstrap its symbol, which turns out to be strongly constrained by cluster adjacency conditions.
SN  - 1029-8479
SN  - 1126-6708
SN  - 1127-2236
ER  -