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The five-loop sunset endpoint constants: a Mahler measure and a second Domb series

Year: 2026
Type: preprint
Abstract: . Maennel and Vanhove reduced the endpoint values of the equal-mass sunset integral in two dimensions to finitely many real constants. With six propagators (five loops) two constants remain. The first is −2 m(1 + x1 + ⋯ + x5), a Mahler measure, by a result that holds for any number of propagators. We prove that the second is C2(∞) = (1/8) Σk≥0 binom(2k + 2, k + 1) Dk/(64k(k + 1)2), where Dk are the Domb numbers; without the binomial coefficient this is the four-loop constant. Both are cases of one formula, C2(∞) = (8/π2) ∫0∞ (I0n−4 − 1) I0K03 dt/t for n = 4, 5, 6 propagators, which is proved by a single contour rotation. Combined with evaluations of Zhou and with the regularized quadratic relations of Fresán, Sabbah and Yu, for which we also give an elementary proof, this yields ∫0∞ tK0(t)6 dt = 3π2L(f, 2) m6 − (45/32) L(f, 3) S3, where f is the newform of weight 4 and level 6, m6 is the Mahler measure and S3 is the series above. For five and for six Bessel functions alike, the two endpoint constants are the coordinates of the vacuum moments in the basis of the mixed moments. All the work was done by AI. Generative-AI disclosure and author responsibility. The evaluation of the second constant and its proof were first obtained with Claude Fable 5.2, which was also used for computations, for audit and for drafting this note. OpenAI ChatGPT-6 Pro was used for independent proofs, source checks and certified numerical verification in the research round behind the note. The author directed and audited the work throughout and takes full responsibility for the mathematics and the final text. This record holds the note (9 pages) and its verification package Matuzas_Five_Loop_Sunset_Endpoint_Constants_verification.zip: the supplement proof_details.pdf (with its LaTeX source), which gives the constants of the rotation lemma, the asymptotics and tail bound of the series S3, the second proof of Theorem 2 in full with a bound for every remainder, and the four matrices of the first proof entry by entry; exact SymPy scripts in rational arithmetic; mpmath scripts that recompute the moments, the regularized moments and the endpoint constants independently from their definitions; ball-arithmetic values (python-flint) of the moments, the regularized moments, the endpoint constants and L(f, 1), …, L(f, 6); the output log of every script; and a SHA-256 manifest. README.md in the package is the entry point and maps every statement of the note to the script that checks it. The package was checked with Python 3.13, mpmath 1.3.0 and SymPy 1.14.0. Companion note: J. Matuzas, The four-loop sunset endpoint constants: a Mahler measure and a Domb series, doi:10.5281/zenodo.24122507. 2020 Mathematics Subject Classification: Primary 33C10; Secondary 11R06, 11F67, 81T18, 81Q30. Preprint. External peer review is not claimed. Website: paper and verification guide. ResearchGate: public preprint.
DOI: 10.5281/zenodo.24139788
Institution: Vilnius University
Language: English
Open Access status: green
OpenAlex ID: W7225817153
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