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Taming Calabi-Yau Feynman integrals: The four-loop equal-mass banana integral

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arxiv 2211.04292 v2 pith:C2PLREWJ submitted 2022-11-08 hep-th hep-phmath-phmath.MP

Sebastian Pögel , Xing Wang , Stefan Weinzierl This is my paper · ORCID
classification hep-thhep-phmath-phmath.MP
keywords feynmanintegralscalabi-yauintegralbananafour-loopdifferentialequal-mass
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abstract

Certain Feynman integrals are associated to Calabi-Yau geometries. We demonstrate how these integrals can be computed with the method of differential equations. The four-loop equal-mass banana integral is the simplest Feynman integral whose geometry is a non-trivial Calabi-Yau manifold. We show that its differential equation can be cast into an $\varepsilon$-factorised form. This allows us to obtain the solution to any desired order in the dimensional regularisation parameter $\varepsilon$. The method generalises to other Calabi-Yau Feynman integrals. Our calculation also shows that the four-loop banana integral is only minimally more complicated than the corresponding Feynman integrals at two or three loops.

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Cited by 3 Pith papers

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    hep-ph 2026-07 accept novelty 6.5 of 10

    Analytic master-integral results through O(ε²) are obtained for a three-loop family containing elliptic and K3 geometries by building and solving a mixed-sector ε-factorized differential equation.

  2. First look at the evaluation of two-loop Feynman integrals for radiative return processes

    hep-ph 2026-07 accept novelty 6.0 of 10

    Planar two-loop four-point master integrals for massive radiative-return QED, including elliptic and nested-root sectors, are reduced to polynomial-in-ε differential equations that evaluate stably in the physical region.

  3. Special Fano geometry from Feynman integrals

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    Special Fano varieties, which include Calabi-Yau spaces as the Q=1 case, arise from the Symanzik polynomials of several families of Feynman integrals.

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