Leading singularities and chambers of Correlahedron

[2505.09808] Leading singularities and chambers of Correlahedron Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search High Energy Physics - Theory arXiv:2505.09808 (hep-th) [Submitted on 14 May 2025 ( v1 ), last revised 11 Apr 2026 (this version, v2)] Title: Leading singularities and chambers of Correlahedron Authors: Song He , Yu-tin Huang , Chia-Kai Kuo View a PDF of the paper titled Leading singularities and chambers of Correlahedron, by Song He and 2 other authors View PDF HTML (experimental) Abstract: In this paper, we explore the chamber dissection of the loop-geometry of Correlahedron, which encodes the loop integrand of four-point stress-energy correlators in planar $\mathcal{N}=4$ super Yang-Mills. We demonstrate that at four loops, continuing the pattern of lower loops, the integrand of the four-point correlation function can be written as a sum over products of chamber-forms and local loop integrands. The chambers and their associated forms are identical to those of three loops, indicating that the dissection may be complete to all loop orders. Furthermore, this suggests that the leading singularities at all loops are simply linear combinations of these chamber forms. This is especially intriguing at four loops since it contains elliptic functions. Interestingly, each elliptic function appears in a subset of chambers. Our geometric approach motivates us to ``diagonalize" the representation, where the local integrals only possess a single leading singularity or elliptic cut. In such a representation, all integrands must evaluate to pure functions, including a single pure elliptic integrand. Inspired by this picture, we also present a simplified form of the three-loop correlator in terms of two independent pure functions (weight-$6$ single-valued multiple polylogarithms), which are directly computed from local integrands with unit leading singularities, multiplied by the leading singularities from chamber forms. Comments: 49 pages, 4 figures, JHEP published version Subjects: High Energy Physics - Theory (hep-th) Report number: MPP-2025-80 Cite as: arXiv:2505.09808 [hep-th] (or arXiv:2505.09808v2 [hep-th] for this version) https://doi.org/10.48550/arXiv.2505.09808 Focus to learn more arXiv-issued DOI via DataCite Journal reference: JHEP 03 (2026) 071 Related DOI : https://doi.org/10.1007/JHEP03%282026%29071 Focus to learn more DOI(s) linking to related resources Submission history From: Chia-Kai Kuo [ view email ] [v1] Wed, 14 May 2025 21:21:38 UTC (6,050 KB) [v2] Sat, 11 Apr 2026 20:45:58 UTC (6,044 KB) Full-text links: Access Paper: View a PDF of the paper titled Leading singularities and chambers of Correlahedron, by Song He and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: hep-th < prev | next > new | recent | 2025-05 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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