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<p>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 <i>ε</i>-factorised differential equation, for which we rely on the algorithm presented in a recent publication [<CitationRef CitationID="CR1">1</CitationRef>]. 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 <i>ε</i>. Via a sequence of basis rotations we then remove any non-<i>ε</i>-factorising terms. This procedure is algorithmic and at no point relies on prior knowledge of the underlying geometry.</p>

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The unequal-mass three-loop banana integral

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月5日

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  • Sebastian Pögel,
  • Toni Teschke,
  • Xing Wang,
  • Stefan Weinzierl

摘要

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 [1]. 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.