The Sunrise Integral Around Two and Four Space-time Dimensions in Terms of Elliptic Polylogarithms

Adams, Luise; Bogner, Christian; Weinzierl, Stefan

doi:10.5506/aphyspolb.46.2131

Cited by 4 publications

(4 citation statements)

References 24 publications

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“…These irreducible, 'elliptic', sectors 1 are, in turn, responsible for the complications in the sectors lying above in the hierarchy 2 . A lot of efforts have been made to bring an order in these irreducible cases [2][3][4][5][6][7][8][9]. Nevertheless, it looks fair to say that our understanding of the elliptic cases is still far from being complete.…”

Section: Introductionmentioning

confidence: 99%

“…1 In the present paper we use the term 'elliptic' when referring to the sectors in which the differential systems can not be reduced to -form, unrelated to whether the leading order solutions can be written in terms of elliptic integrals or not. 2 As usual, by sector we mean the integrals with a fixed set of denominators. The latter is defined, of course, up to shift of loop momenta.…”

Section: Introductionmentioning

confidence: 99%

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Symmetric ϵ- and (ϵ + 1/2)-forms and quadratic constraints in “elliptic” sectors.

Lee

2018

J. High Energ. Phys.

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Within the differential equation method for multiloop calculations, we examine the systems irreducible to -form. We argue that for many cases of such systems it is possible to obtain nontrivial quadratic constraints on the coefficients of -expansion of their homogeneous solutions. These constraints are the direct consequence of the existence of symmetric ( + 1/2)-form of the homogeneous differential system, i.e., the form where the matrix in the right-hand side is symmetric and its -dependence is localized in the overall factor ( + 1/2). The existence of such a form can be constructively checked by available methods and seems to be common to many irreducible systems, which we demonstrate on several examples. The obtained constraints provide a nontrivial insight on the structure of general solution in the case of the systems irreducible to -form. For the systems reducible to -form we also observe the existence of symmetric form and derive the corresponding quadratic constraints.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symmetric ϵ- and (ϵ + 1/2)-forms and quadratic constraints in “elliptic” sectors.

Lee

2018

J. High Energ. Phys.

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“…|ω1s. 19 Near four-dimension (ε Ñ 0) higher order terms in ε are suppressed [122], so we are mainly interested in the symbol up to Opε 2 q.…”

Section: Jhep03(2023)155mentioning

confidence: 99%

Loop-by-loop differential equations for dual (elliptic) Feynman integrals

Giroux

Pokraka

2023

J. High Energ. Phys.

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We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an ε-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of q-series. Then, we discuss interesting properties satisfied by our dual basis as well as its simple relation to the known ε-form basis of Feynman integrands. The underlying K3-geometry of the three-loop four-mass sunrise integral is also discussed. Finally, we speculate on how to construct a “good” loop-by-loop basis at three-loop.

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“…4 we compare results for the nondegenerate two-loop sunrise diagram with results from Refs. [8,9,10,11] for D 0 = 2 to obtain again an integral identity. Our conclusions are given in Sec.…”

Section: Introductionmentioning

confidence: 99%

Coordinate space calculation of two- and three-loop sunrise-type diagrams, elliptic functions and truncated Bessel integral identities

Groote

Körner

2019

Nuclear Physics B

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We integrate three-loop sunrise-type vacuum diagrams in D 0 = 4 dimensions with four different masses using configuration space techniques. The finite parts of our results are in numerical agreement with corresponding three-loop calculations in momentum space. Using some of the closed form results of the momentum space calculation we arrive at new integral identities involving truncated integrals of products of Bessel functions. For the non-degenerate finite two-loop sunrise-type vacuum diagram in D 0 = 2 dimensions we make use of the known closed form p-space result to express the moment of a product of three Bessel functions in terms of a sum of Clausen polylogarithms. Using results for the nondegenerate two-loop sunrise diagram from the literature in D 0 = 2 dimensions we obtain a Bessel function integral identity in terms of elliptic functions.

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The Sunrise Integral Around Two and Four Space-time Dimensions in Terms of Elliptic Polylogarithms

Cited by 4 publications

References 24 publications

Symmetric ϵ- and (ϵ + 1/2)-forms and quadratic constraints in “elliptic” sectors.

Symmetric ϵ- and (ϵ + 1/2)-forms and quadratic constraints in “elliptic” sectors.

Loop-by-loop differential equations for dual (elliptic) Feynman integrals

Coordinate space calculation of two- and three-loop sunrise-type diagrams, elliptic functions and truncated Bessel integral identities

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