Analytic Continuation of the Kite Family

Bogner, Christian; Schweitzer, Armin; Weinzierl, Stefan

doi:10.1007/978-3-030-04480-0_4

Cited by 5 publications

(4 citation statements)

References 71 publications

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“…This analysis extends naturally to the higher loop sunset integrals [55]. The elliptic polylogarithm representation generalises to other two-loop integrals like the kite integral [75][76][77] or the all equal masses three-loop sunset [61]. This representation leads to fast numerical evaluation [76].…”

Section: Resultsmentioning

confidence: 99%

Feynman Integrals, Toric Geometry and Mirror Symmetry

Vanhove

2019

Texts &Amp; Monographs in Symbolic Computation

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This expository text is about using toric geometry and mirror symmetry for evaluating Feynman integrals. We show that the maximal cut of a Feynman integral is a GKZ hypergeometric series. We explain how this allows to determine the minimal differential operator acting on the Feynman integrals. We illustrate the method on sunset integrals in two dimensions at various loop orders. The graph polynomials of the multi-loop sunset Feynman graphs lead to reflexive polytopes containing the origin and the associated variety are ambient spaces for Calabi-Yau hypersurfaces. Therefore the sunset family is a natural home for mirror symmetry techniques. We review the evaluation of the two-loop sunset integral as an elliptic dilogarithm and as a trilogarithm. The equivalence between these two expressions is a consequence of 1) the local mirror symmetry for the non-compact Calabi-Yau three-fold obtained as the anti-canonical hypersurface of the del Pezzo surface of degree 6 defined by the sunset graph polynomial and 2) that the sunset Feynman integral is expressed in terms of the local Gromov-Witten prepotential of this del Pezzo surface. * IPHT-t18/096 arXiv:1807.11466v2 [hep-th]

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Section: Resultsmentioning

confidence: 99%

Feynman Integrals, Toric Geometry and Mirror Symmetry

Vanhove

2019

Texts &Amp; Monographs in Symbolic Computation

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“…The solution for these Feynman integrals in terms of iterated integrals of modular forms follows now directly from the differential equation ( 85). The q-expansion of the iterated integrals provides an efficient method for the numerical evaluation [25,93]. Let us close this paragraph with the observation that the integration kernels ω 0 = dx x , ω 0 = dx x − 1 (90) may also be expressed as modular forms:…”

Section: Feynman Integrals Evaluating To Iterated Integrals Of Modula...mentioning

confidence: 99%

On a Class of Feynman Integrals Evaluating to Iterated Integrals of Modular Forms

Adams

Weinzierl

2019

Texts &Amp; Monographs in Symbolic Computation

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In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic curves and modular forms. Feynman integrals, which evaluate to iterated integrals of modular forms go beyond the class of multiple polylogarithms. Nevertheless, we may bring for all examples considered the associated system of differential equations by a non-algebraic transformation to an ε-form, which makes a solution in terms of iterated integrals immediate.

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“…For instance, the kite diagram shown in Figure 7 contributes to the self-energy of the electron in QED when all three internal masses are equal (m 1 = m 2 = m 3 ). This diagram was first recognized to involve an elliptic integral nearly sixty years ago [108], and has now been evaluated in terms of iterated integrals over products of elliptic integrals and polylogarithms [124], as iterated integrals over modular forms [123,126,127,132,217], and as elliptic multiple polylogarithms [143]. Moreover, the full two-loop contribution to the self-energy of the electron has been evaluated in terms of iterated integrals of modular forms, which can be used to obtain q-expansions for efficient numerical evaluation [136].…”

Section: Further Integrals At Two and Three Loopsmentioning

confidence: 99%

Functions Beyond Multiple Polylogarithms for Precision Collider Physics

Bourjaily¹,

Broedel²,

Chaubey³

et al. 2022

Preprint

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Feynman diagrams constitute one of the essential ingredients for making precision predictions for collider experiments. Yet, while the simplest Feynman diagrams can be evaluated in terms of multiple polylogarithms-whose properties as special functions are well understood-more complex diagrams often involve integrals over complicated algebraic manifolds. Such diagrams already contribute at NNLO to the self-energy of the electron, t t production, γγ production, and Higgs decay, and appear at two loops in the planar limit of maximally supersymmetric Yang-Mills theory. This makes the study of these more complicated types of integrals of phenomenological as well as conceptual importance.In this white paper contribution to the Snowmass community planning exercise, we provide an overview of the state of research on Feynman diagrams that involve special functions beyond multiple polylogarithms, and highlight a number of research directions that constitute essential avenues for future investigation.

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Analytic Continuation of the Kite Family

Cited by 5 publications

References 71 publications

Feynman Integrals, Toric Geometry and Mirror Symmetry

Feynman Integrals, Toric Geometry and Mirror Symmetry

On a Class of Feynman Integrals Evaluating to Iterated Integrals of Modular Forms

Functions Beyond Multiple Polylogarithms for Precision Collider Physics

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