Functional reduction of Feynman integrals

Tarasov, O. V.

doi:10.1007/jhep02(2019)173

Cited by 8 publications

(15 citation statements)

References 52 publications

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“…Another quite interesting and still algorithmically open problem is the transformation of multiple Horn-type hypergeometric functions with reducible monodromy to hypergeometric functions with irreducible monodromy. In the application to Feynman diagrams, such transformations correspond to functional relations, studied recently by Oleg Tarasov [233,234,235], and by Andrei Davydychev [101].…”

Section: Discussionmentioning

confidence: 99%

Hypergeometric Functions and Feynman Diagrams

Kalmykov¹,

Kniehl²,

Moch³

et al. 2020

Preprint

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The relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniques for the construction of the -expansion. As an example, we present a detailed discussion of the construction of the -expansion of the Appell function 3 around rational values of parameters via an iterative solution of differential equations. Another interesting example is the Puiseux-type solution involving a differential operator generated by a hypergeometric function of three variables. The holonomic properties of the hypergeometric functions are briefly discussed.

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Section: Discussionmentioning

confidence: 99%

Hypergeometric Functions and Feynman Diagrams

Kalmykov¹,

Kniehl²,

Moch³

et al. 2020

Preprint

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“…A systematic method for solving functional equations for Feynman integrals was presented in ref. [20]. In a sense, this is a generalization of the method that is used to solve the usual Sincov's functional equation [21], [22], [23] f (x, y) = f (x, z) − f (y, z).…”

Section: Methods Of Functional Reductionmentioning

confidence: 99%

“…For instance, these methods can be used in combination with approach proposed in refs. [18], [19], [20]. In ref.…”

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confidence: 99%

Functional reduction of one-loop Feynman integrals with arbitrary masses

Tarasov

2022

Preprint

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A method of functional reduction for the dimensionally regularized one-loop Feynman integrals with massive propagators is described in detail.The method is based on a repeated application of the functional relations proposed by the author. Explicit formulae are given for reducing one-loop scalar integrals to a simpler ones, the arguments of which are the ratios of polynomials in the masses and kinematic invariants. We show that a general scalar n-point integral, depending on n(n + 1)/2 generic masses and kinematic variables, can be expressed as a linear combination of integrals depending only on n variables. The latter integrals are given explicitly in terms of hypergeometric functions of (n − 1) dimensionless variables. Analytic expressions for the 2-, 3-and 4-point integrals, that depend on the minimal number of variables, were also obtained by solving the dimensional recurrence relations. The resulting expressions for these integrals are given in terms of Gauss' hypergeometric function 2 F 1 , the Appell function F 1 and the hypergeometric Lauricella -Saran function F S . A modification of the functional reduction procedure for some special values of kinematical variables is considered.13.2.1 Series representation 13.2.2 Integral representations 13.3 Differential relations for the Lauricella-Saran function F S

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“…For instance, these methods can be used in combination with the approach proposed in refs. [18][19][20]. In ref.…”

Section: Jhep06(2022)155 1 Introductionmentioning

confidence: 99%

Functional reduction of one-loop Feynman integrals with arbitrary masses

Tarasov

2022

J. High Energ. Phys.

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A method of functional reduction for the dimensionally regularized one-loop Feynman integrals with massive propagators is described in detail.The method is based on a repeated application of the functional relations proposed by the author. Explicit formulae are given for reducing one-loop scalar integrals to a simpler ones, the arguments of which are the ratios of polynomials in the masses and kinematic invariants. We show that a general scalar n-point integral, depending on n(n + 1)/2 generic masses and kinematic variables, can be expressed as a linear combination of integrals depending only on n variables. The latter integrals are given explicitly in terms of hypergeometric functions of (n − 1) dimensionless variables. Analytic expressions for the 2-, 3- and 4-point integrals, that depend on the minimal number of variables, were also obtained by solving the dimensional recurrence relations. The resulting expressions for these integrals are given in terms of Gauss’ hypergeometric function 2F1, the Appell function F1 and the hypergeometric Lauricella — Saran function FS. A modification of the functional reduction procedure for some special values of kinematic variables is considered.

show abstract

Functional reduction of Feynman integrals

Cited by 8 publications

References 52 publications

Hypergeometric Functions and Feynman Diagrams

Hypergeometric Functions and Feynman Diagrams

Functional reduction of one-loop Feynman integrals with arbitrary masses

Functional reduction of one-loop Feynman integrals with arbitrary masses

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