The unequal-mass three-loop banana integral

2025/07/31 by Sebastian Pögel, Toni Teschke, Pögel, Sebastian +5 · 3 citations
Agricultural and Biological Sciences · Mathematics · Engineering · #Banana Cultivation and Research #Mathematics and Applications #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2507.23594

Abstract

We compute the three-loop banana integral with four unequal masses in dimensional regularisation. This integral is associated to a family of K3 surfaces, thus representing an example for Feynman integrals with geometries beyond elliptic curves. We evaluate the integral by deriving an ε-factorised differential equation, for which we rely on the algorithm presented in a recent publication. Equipping the space of differential forms in Baikov representation by a set of filtrations inspired by Hodge theory, we first obtain a differential equation with entries as Laurent polynomials in ε. Via a sequence of basis rotations we then remove any non-ε-factorising terms. This procedure is algorithmic and at no point relies on prior knowledge of the underlying geometry.

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Cited by

• From d log to d E: Canonical Elliptic Integrands and Modular Symbol Letters with Pure eMPLs
• Three-loop banana integrals with three equal masses
• D-ideal of generic mass banana integrals in dimensional regularization
• Three-loop banana integrals with four unequal masses

Related

• Three-loop banana integrals with three equal masses
• Three-loop banana integrals with four unequal masses
• Feynman integrals, geometries and differential equations
• Feynman integrals, elliptic integrals and two-parameter K3 surfaces
• A Feynman integral via higher normal functions