FIRE4, LiteRed and accompanying tools to solve integration by parts relations

doi:10.1016/J.CPC.2013.06.016

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FIRE4, LiteRed and accompanying tools to solve integration by parts relations

Journal Article•DOI•

FIRE4, LiteRed and accompanying tools to solve integration by parts relations

Alexander V. Smirnov¹, Vladimir A. Smirnov¹, Vladimir A. Smirnov²•Institutions (2)

Moscow State University¹, Humboldt University of Berlin²

01 Dec 2013-Computer Physics Communications (North-Holland)-Vol. 184, Iss: 12, pp 2820-2827

TL;DR: The new version of the Mathematica code FIRE can be applied together with the recently developed code LiteRed by Lee in order to provide an integration by parts reduction to master integrals for quite complicated families of Feynman integrals.

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About: This article is published in Computer Physics Communications.The article was published on 2013-12-01 and is currently open access. It has received 212 citations till now. The article focuses on the topics: Order of integration (calculus) & Dimensional regularization.

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Journal Article•DOI•

New Developments in FeynCalc 9.0

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Vladyslav Shtabovenko¹, Rolf Mertig, Frederik Orellana²•Institutions (2)

Technische Universität München¹, Technical University of Denmark²

01 Oct 2016-Computer Physics Communications

TL;DR: The main features of version 9.0 are: improved tensor reduction and partial fractioning of loop integrals, new functions for using FeynCalc together with tools for reduction of scalar loop Integrals using integration-by-parts (IBP) identities, better interface to FeynArts and support for S U ( N ) generators with explicit fundamental indices.

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795 citations

Cites methods from "FIRE4, LiteRed and accompanying too..."

...ose tool for IBP reduction (the built-in TARCER [10] is suitable only for 2-loop self-energy type integrals), this omission can be compensated by using one of the publicly available IBP-packages (FIRE[50], LiteRED[55], Reduze[56], AIR[57]). However, one should keep in mind that such tools usually expect their input to contain only loop integrals with linearly independent propagators that form a basis....
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...used at some stages of NNLO [48, 49] calculations. Indeed, FeynCalc can be well employed for small or medium size multi-loop processes if one connects it to suitable tools for IBP-reduction (e.g. FIRE[50]) and numeric evaluation of multi-loop integrals (e.g. FIESTA [51] or SecDec[52]). Last but not least, FeynCalc can be also useful for educational purposes. The possibility of easily getting hands-on ...
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Journal Article•DOI•

FIRE5: A C++ implementation of Feynman Integral REduction

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Alexander V. Smirnov¹•Institutions (1)

Moscow State University¹

01 Apr 2015-Computer Physics Communications

TL;DR: The C++ version of FIRE is presented — a powerful program performing Feynman integral reduction to master integrals that can be treated as a task to solve a huge system of sparse linear equations with polynomial coefficients.

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490 citations

Journal Article•DOI•

Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals

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Erik Panzer¹•Institutions (1)

Humboldt University of Berlin¹

01 Mar 2015-Computer Physics Communications

TL;DR: These algorithms for symbolic integration of hyperlogarithms multiplied by rational functions are implemented in Maple and their application to the computation of Feynman integrals is discussed.

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430 citations

Journal Article•DOI•

Kira—A Feynman integral reduction program

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P. Maierhöfer¹, Johann Usovitsch², Johann Usovitsch³, Peter Uwer²•Institutions (3)

University of Freiburg¹, Humboldt University of Berlin², Trinity College, Dublin³

01 Sep 2018-Computer Physics Communications

TL;DR: A new implementation of the Laporta algorithm to reduce scalar multi-loop integrals appearing in quantum field theoretic calculations to a set of master integrals is presented by using an additional algorithm based on modular arithmetic to remove linearly depen- dent equations from the system of equations arising from integration-by-parts and Lorentz identities.

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343 citations

Journal Article•DOI•

FIRE6: Feynman Integral REduction with Modular Arithmetic

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Alexander V. Smirnov¹, Alexander V. Smirnov², F.S. Chukharev²•Institutions (2)

Karlsruhe Institute of Technology¹, Moscow State University²

01 Feb 2020-Computer Physics Communications

TL;DR: The goal of this paper is to present the current version of FIRE, a program performing reduction of Feynman integrals to master integrals, and to perform reduction with modular arithmetic.

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334 citations

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Journal Article•DOI•

Integration by parts: The algorithm to calculate β-functions in 4 loops

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K.G. Chetyrkin¹, F.V. Tkachov¹•Institutions (1)

Russian Academy of Sciences¹

23 Nov 1981-Nuclear Physics

TL;DR: In this paper, it was proved that the counterterm for an arbitrary 4-loop Feynman diagram in an arbitrary model is calculable within the minimal subtraction scheme in terms of rational numbers and the Riemann ζ-function in a finite number of steps via a systematic "algebraic" procedure involving neither integration of elementary, special, or any other functions, nor expansions in and summation of infinite series of any kind.

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1,928 citations

"FIRE4, LiteRed and accompanying too..." refers methods in this paper

...A classical approach is to apply the so-called integration by parts (IBP) relations [1] (see Chapter 6 of [2] for a recent review) and reduce all integrals to a smaller set, the master integrals(1)....
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...The integration by parts relations [1]...
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Journal Article•DOI•

High-precision calculation of multiloop feynman integrals by difference equations

[...]

S. Laporta¹•Institutions (1)

University of Bologna¹

30 Dec 2000-International Journal of Modern Physics A

TL;DR: Algorithms for the construction of the systems using integration-by-parts identities and methods of solutions by means of expansions in factorial series and Laplace transformation and procedures for generating and solving systems of differential equations in masses and momenta for master integrals are shown.

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Abstract: We describe a new method of calculation of generic multiloop master integrals based on the numerical solution of systems of difference equations in one variable. We show algorithms for the construction of the systems using integration-by-parts identities and methods of solutions by means of expansions in factorial series and Laplace transformation. We also describe new algorithms for the identification of master integrals and the reduction of generic Feynman integrals to master integrals, and procedures for generating and solving systems of differential equations in masses and momenta for master integrals. We apply our method to the calculation of the master integrals of massive vacuum and self-energy diagrams up to three loops and of massive vertex and box diagrams up to two loops. Implementation in a computer program of our approach is described. Important features of the implementation are: the ability to deal with hundreds of master integrals and the ability to obtain very high precision results expanded at will in the number of dimensions.

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1,023 citations

"FIRE4, LiteRed and accompanying too..." refers methods in this paper

...A Laporta algorithm [10] in a given sector is solving IBP’s with a Gauss elimination after choosing an ordering....
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Journal Article•DOI•

High-precision calculation of multi-loop Feynman integrals by difference equations

[...]

S. Laporta

03 Feb 2001-arXiv: High Energy Physics - Phenomenology

TL;DR: In this paper, a new method of calculation of generic multi-loop master integrals based on the numerical solution of systems of difference equations in one variable is described. But this method is not suitable for the calculation of complex systems.

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Abstract: We describe a new method of calculation of generic multi-loop master integrals based on the numerical solution of systems of difference equations in one variable. We show algorithms for the construction of the systems using integration-by-parts identities and methods of solutions by means of expansions in factorial series and Laplace's transformation. We also describe new algorithms for the identification of master integrals and the reduction of generic Feynman integrals to master integrals, and procedures for generating and solving systems of differential equations in masses and momenta for master integrals. We apply our method to the calculation of the master integrals of massive vacuum and self-energy diagrams up to three loops and of massive vertex and box diagrams up to two loops. Implementation in a computer program of our approach is described. Important features of the implementation are: the ability to deal with hundreds of master integrals and the ability to obtain very high precision results expanded at will in the number of dimensions.

...read moreread less

689 citations

Journal Article•DOI•

Algorithm FIRE—Feynman Integral REduction

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Alexander V. Smirnov¹•Institutions (1)

Moscow State University¹

01 Oct 2008-Journal of High Energy Physics

TL;DR: The recently developed algorithm FIRE performs the reduction of Feynman integrals to master integrals based on a number of strategies, such as applying the Laporta algorithm, the s-bases algorithm, region-Bases and integrating explicitly over loop momenta when possible.

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Abstract: The recently developed algorithm FIRE performs the reduction of Feynman integrals to master integrals. It is based on a number of strategies, such as applying the Laporta algorithm, the s-bases algorithm, region-bases and integrating explicitly over loop momenta when possible. Currently it is being used in complicated three-loop calculations.

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555 citations

"FIRE4, LiteRed and accompanying too..." refers background in this paper

...0 was the first public version of FIRE [4]....
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...A few years ago one of the present authors developed a program named FIRE [4] performing reduction of Feynman integrals to master integrals....
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...As it has been explained in [4], there are basically two ways to perform reduction:...
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Journal Article•DOI•

The analytical value of the electron (g − 2) at order α3 in QED

[...]

S. Laporta¹, Ettore Remiddi¹•Institutions (1)

University of Bologna¹

27 Jun 1996-Physics Letters B

TL;DR: In this article, the authors evaluated the contribution of the three-loop non-planar triplescross diagrams contributing to the electron (g−2) in QED; their value, omitting the already known infrared divergent part, is a e (3−cross)= α π 3 83 72 π 2 e(3)− 215 24 e(5)+ 100 3.

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448 citations