Picard–Vessiot groups of Lauricella's hypergeometric systems EC and Calabi–Yau varieties arising integral representations

Gotō, Yoshiaki; Koike, Kazuhiko

doi:10.1112/jlms.12311

Cited by 3 publications

(1 citation statement)

References 26 publications

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“…For more (mathematically rigorous) studies of the Lauricella system see also refs. [240,[246][247][248].…”

Section: Discussionmentioning

confidence: 99%

Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives

Bönisch¹,

Duhr²,

Fischbach³

et al. 2021

Preprint

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We provide a comprehensive summary of concepts from Calabi-Yau motives relevant to the computation of multi-loop Feynman integrals. From this we derive several consequences for multi-loop integrals in general, and we illustrate them on the example of multi-loop banana integrals. For example, we show how Griffiths transversality, known from the theory of variation of mixed Hodge structures, leads quite generically to a set of quadratic relations among maximal cut integrals associated to Calabi-Yau motives. These quadratic relations then naturally lead to a compact expression for l-loop banana integrals in D = 2 dimensions in terms of an integral over a period of a Calabi-Yau (l − 1)-fold. This new integral representation generalizes in a natural way the known representations for l ≤ 3 involving logarithms with square root arguments and iterated integrals of Eisenstein series. In a second part, we show how the results obtained by some of the authors in earlier work can be extended to dimensional regularization. We present a method to obtain the differential equations for banana integrals with an arbitrary number of loops in dimensional regularization without the need to solve integration-by-parts relations. We also present a compact formula for the leading asymptotics of banana integrals with an arbitrary number of loops in the large momentum limit. This generalizes the novel Γ-class introduced by some of the authors to dimensional regularization and provides a convenient boundary condition to solve the differential equations for the banana integrals. As an application, we present for the first time numerical results for equal-mass banana integrals with up to four loops and up to second order in the dimensional regulator.

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“…For more (mathematically rigorous) studies of the Lauricella system see also refs. [240,[246][247][248].…”

Section: Discussionmentioning

confidence: 99%

Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives

Bönisch¹,

Duhr²,

Fischbach³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

Feynman integrals in dimensional regularization and extensions of Calabi-Yau motives

BÃ¶nisch

Duhr²,

Fischbach³

et al. 2022

J. High Energ. Phys.

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We provide a comprehensive summary of concepts from Calabi-Yau motives relevant to the computation of multi-loop Feynman integrals. From this we derive several consequences for multi-loop integrals in general, and we illustrate them on the example of multi-loop banana integrals. For example, we show how Griffiths transversality, known from the theory of variation of mixed Hodge structures, leads quite generically to a set of quadratic relations among maximal cut integrals associated to Calabi-Yau motives. These quadratic relations then naturally lead to a compact expression for l-loop banana integrals in D = 2 dimensions in terms of an integral over a period of a Calabi-Yau (l − 1)-fold. This new integral representation generalizes in a natural way the known representations for l ≤ 3 involving logarithms with square root arguments and iterated integrals of Eisenstein series. In a second part, we show how the results obtained by some of the authors in earlier work can be extended to dimensional regularization. We present a method to obtain the differential equations for banana integrals with an arbitrary number of loops in dimensional regularization without the need to solve integration-by-parts relations. We also present a compact formula for the leading asymptotics of banana integrals with an arbitrary number of loops in the large momentum limit. This generalizes the novel $$ \hat{\Gamma} $$ Γ ̂ -class introduced by some of the authors to dimensional regularization and provides a convenient boundary condition to solve the differential equations for the banana integrals. As an application, we present for the first time numerical results for equal-mass banana integrals with up to four loops and up to second order in the dimensional regulator.

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Lauricella's $F_C$ with finite irreducible monodromy group

Gotō

2022

J. Math. Soc. Japan

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Picard–Vessiot groups of Lauricella's hypergeometric systems EC and Calabi–Yau varieties arising integral representations

Cited by 3 publications

References 26 publications

Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives

Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives

Feynman integrals in dimensional regularization and extensions of Calabi-Yau motives

Lauricella's $F_C$ with finite irreducible monodromy group

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