Applications of an isotopy theorem

Fotiadi, Dimitri; Froissart, M.; Lascoux, Jean; Pham, Frédéric

doi:10.1016/0040-9383(65)90063-7

Cited by 59 publications

(79 citation statements)

References 2 publications

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“…The homology groups relevant to one-loop integrals 10 have been studied in refs. [31,32,45], where it was shown that they are one-dimensional and generated by all the cycles that wind around 11 P j , with j = ∞. The rest of this subsection consists of a short review of this result.…”

Section: Homology Groups Associated To One-loop Integralsmentioning

confidence: 99%

“…The values of these discontinuities are computed by cut integrals [1]. This idea can be described rigorously through the Picard-Lefschetz theorem and the multivariate residue theorem of section 3 [31][32][33]. Although these results have long been known in the mathematical physics literature, we feel that many of these ideas have slipped into oblivion.…”

Section: Jhep06(2017)114mentioning

confidence: 99%

“…In order to extend these concepts to the singularities of the second type, the usual momentum representation of loop integrals is not the most convenient, because the pinch happens at infinite loop momentum. We therefore use a representation of one-loop integrals as integrals over a compact quadric in the complex projective space CP D+1 [31,46,47], which we review in this subsection. Throughout this section we work in Euclidean kinematics, and we strictly follow the conventions of ref.…”

Section: Compactification Of One-loop Integralsmentioning

confidence: 99%

“…In the compactified picture, singularities of the first and second types can be treated on the same footing. Furthermore, for one-loop integrals, the so-called Decomposition Theorem [31,32,45] implies that cut integrals for singularities of the two types are not independent, leading to various linear relations among cut integrals.…”

Section: Compactification and Singularities Of The Second Typementioning

confidence: 99%

“…We focus on cut integrals at one loop, though we expect that many of the concepts we introduce in this paper are generic and will carry through to higher loops. In fact, many of these concepts have been introduced into the mathematical physics literature in the 60s [1,2,[31][32][33] (albeit without the machinery of dimensional regularization), but they have since slipped into oblivion. The cornerstone of our approach to cut integrals is the multivariate residue calculus of Leray [34].…”

Section: Jhep06(2017)114mentioning

confidence: 99%

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Cuts from residues: the one-loop case

Abreu

Britto

Duhr

et al. 2017

J. High Energ. Phys.

140

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Using the multivariate residue calculus of Leray, we give a precise definition of the notion of a cut Feynman integral in dimensional regularization, as a residue evaluated on the variety where some of the propagators are put on shell. These are naturally associated to Landau singularities of the first type. Focusing on the one-loop case, we give an explicit parametrization to compute such cut integrals, with which we study some of their properties and list explicit results for maximal and next-to-maximal cuts. By analyzing homology groups, we show that cut integrals associated to Landau singularities of the second type are specific combinations of the usual cut integrals, and we obtain linear relations among different cuts of the same integral. We also show that all one-loop Feynman integrals and their cuts belong to the same class of functions, which can be written as parametric integrals.

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Section: Homology Groups Associated To One-loop Integralsmentioning

confidence: 99%

Section: Jhep06(2017)114mentioning

confidence: 99%