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

Abstract

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.

Meromorphic modular forms and the three-loop equal-mass banana integral

“…The fact that L ban,(3) x is a symmetric square has a geometric origin: the l-loop equal-mass banana integral is associated to a one-parameter family of Calabi-Yau (l−1)-folds [64,65,[68][69][70], and the maximal cuts of the l-loop equal-mass banana integral are annihilated by the Picard-Fuchs operator for this family, which has degree l. It is expected that the degree-three Picard-Fuchs operator of a one-parameter family of Calabi-Yau two-folds (also known as K3 surfaces) is always the symmetric square of a Picard-Fuchs operator describing a one-parameter family of elliptic curves, cf., e.g., ref. [80].…”

“…Finally, and probably most importantly, holomorphic modular forms are not sufficient to cover even the simplest cases of Feynman integrals depending on one variable. Indeed, it is known that, while in general higher-loop analogues of the sunrise integral -the so-called l-loop banana integrals -are associated to families of Calabi-Yau (l − 1)-folds [64][65][66][67][68][69][70], the three-loop equal-mass banana integral in D = 2 dimensions can be expressed in terms of the same class of functions as the two-loop equal-JHEP02(2022)184 mass sunrise integral [64,65,71]. However, if higher terms in the -expansion in dimensional regularisation are considered, new classes of iterated integrals are required, which cannot be expressed in terms of modular forms alone.…”

Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges

“…The factorization implies a motivic relationship between the periods on the Calabi-Yau and the periods on particular elliptic curves. This phenomenon also occurs in the period integrals on Calabi-Yau manifolds that are related to Feynman integrals [130,131]. Given the relation between the complex structure moduli space and the stringy Kähler moduli space via mirror symmetry, it is natural to ask for possible connections between the modular properties that appear on both sides of the duality.…”

Localizing non-linear ${\cal N}=(2,2)$ sigma model on $S^2$

“…In fact in the ratio of gamma functions only terms with odd zeta-values survive, and the transcendental weight is related to the loop order at which corrections to the sigma model appear. This pattern of seemingly complicated transcendental loop corrections summing up into a simple expression indicates once again, why the study of the transcendental structure of loop integrals [34][35][36] is important. Let us comment on the non-analytical dependence in (170) for a general Kähler case.…”