Differential equations for one-loop generalized Feynman integrals

Barucchi, G.; Ponzano, G.

doi:10.1063/1.1666327

Cited by 29 publications

(41 citation statements)

References 9 publications

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“…It is very illuminating but seems to be accompanied with many technical difficulties, as Professor Regge himself points out in the report. This important property of the Feynman integral has also been conjectured and proved in simple cases by Sato [16] independently and in a little different context See also Barucchi-Ponzano [1], Kawai-Stapp [11], [12] and references cited there. Note that , [12]) discusses the ^-matrix itself, not the individual Feynman integral, as Sato [16] originally proposed.…”

Section: Holonomic Systems Of Linear Differential Equations and Feynmmentioning

confidence: 58%

Holonomic Systems of Linear Differential Equations and Feynman Integrals

Kashiwara¹,

Kawai²

1976

Publ. Res. Inst. Math. Sci.

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Supported by Miller Institute for Basic Research in Science. *** It is called "maximally overdetermined system" in Sato-Kawai-Kashiwara [17], Kashiwawa-Kawai [7] and Kawai-Stapp [11]. f A denning function of a hyperfunction is by definition a holomorphic function whose boundary value attains the hyperfunction in question. See Theorem 4 below for the precise definition. * Though the well-definedness of ®D is discussed only for external diagram D in Sato-Miwa-Oshima-Jimbo [18], their argument applies to the general case. ** Note that \f\ l is a linear combination of (jf+z"0) A and (f-ity*' as long as /I is not an integer.

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Section: Holonomic Systems Of Linear Differential Equations and Feynmmentioning

confidence: 58%

Holonomic Systems of Linear Differential Equations and Feynman Integrals

Kashiwara¹,

Kawai²

1976

Publ. Res. Inst. Math. Sci.

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“…At the same time, IBP-decomposition algorithms can be applied to special integrands, built by acting on the master integrand with differential operators (w.r.t. kinematic invariants), or by multiplying their numerators by polynomials which modify their dimensions, or by considering arbitrary denominator powers, respectively turning the decomposition formulas into differential equations [3][4][5][6][7][8][9][10], dimensional recurrence relations [11,12], and finite difference equations [13,14] obeyed by MIs. Solving them amounts to the actual determination of the MIs themselves, as an alternative to the use of direct integration techniques.…”

Section: Jhep05(2019)153mentioning

confidence: 99%

Decomposition of Feynman integrals on the maximal cut by intersection numbers

Frellesvig

Gasparotto

Laporta

et al. 2019

J. High Energ. Phys.

128

147

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We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss 2 F 1 hypergeometric function, and the Appell F 1 function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decompositionby-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to n-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.

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“…In this work, we explore the latter idea, and we elaborate on a new method for establishing relations among Feynman integrals in arbitrary space-time dimensions, and for projecting them onto a basis. An archetype of such a basis reduction is the Gauss's contiguous relation, e.g., 2 F 1 (a, b, c+1; z) = c 2 F 1 (a, b; c; z) c − a + a 2 F 1 (a+1, b; c+1; z) a − c .…”

Section: Introductionmentioning

confidence: 99%

Feynman integrals and intersection theory

Mastrolia

Mizera

2019

J. High Energ. Phys.

183

297

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We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-called intersection numbers and describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a two-loop non-planar triangle diagram with a massive loop. * pierpaolo.mastrolia@pd.infn.it † smizera@pitp.ca arXiv:1810.03818v2 [hep-th]

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Differential equations for one-loop generalized Feynman integrals

Cited by 29 publications

References 9 publications

Holonomic Systems of Linear Differential Equations and Feynman Integrals

Holonomic Systems of Linear Differential Equations and Feynman Integrals

Decomposition of Feynman integrals on the maximal cut by intersection numbers

Feynman integrals and intersection theory

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