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

• Published in: The journal of high energy physics 2022-09, Vol.2022 (9), p.156-119, Article 156
• Main Authors: Bönisch, Kilian, Duhr, Claude, Fischbach, Fabian, Klemm, Albrecht, Nega, Christoph
• Format: Article
• Language: English
• Subjects: Boundary conditions; Classical and Quantum Gravitation; Differential and Algebraic Geometry; Differential equations; Elementary Particles; High energy physics; Integrals; Mathematical analysis; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Regularization; Relativity Theory; Representations; Scattering Amplitudes; String Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP09(2022)156

A bstract 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.