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Manuscript Title: APart 2: A Generalized Mathematica Apart Function | ||

Authors: Feng Feng | ||

Program title: APart 2.0 | ||

Catalogue identifier: AEMK_v2_0Distribution format: tar.gz | ||

Journal reference: Comput. Phys. Commun. 198(2016)260 | ||

Programming language: Mathematica. | ||

Computer: Any computer with Mathematica installed. | ||

Operating system: Any capable of running Mathematica. | ||

Keywords: APart, Partial Fraction. | ||

PACS: 12.38.Bx. | ||

Classification: 11.1. | ||

Does the new version supersede the previous version?: Yes | ||

Nature of problem:As discussed in [1], the general procedure to compute a cross section for a physical process in perturbative quantum field theory involves generating the corresponding amplitude via Feynman diagram and performing the loop integrals in dimensional regularization [7]. The essential part in the computation is to reduce these loop integrals to a small number of standard integrals, which are called master integrals (MI), via the systematic methods of integration by parts (IBP) identities [8, 9] and Lorentz invariance (LI) identities [10]. The basic reduction algorithm is introduced by Laporta [11], which defines an ordering for Feynman integrals, generates IBP identities and solves the corresponding linear equations. Alternative methods to exploit IBP and LI identities for reductions can be found in [12-17]. There are many public computer programs for implementations of different reduction algorithms: AIR [18], FIRE [19] and Reduze [20]. To facilitate the input for Fire[19], Reduze[20], etc. we need to decompose the linear independent propagators to independent ones, this precodure can be done by the APart package [1] which generalizes the Mathematica function APart from one dimension to any N dimensions. | ||

Solution method:We have proven that all linear independent propagators can be decomposed into the summation of linear independent ones in [1], APart is such an Mathematica package that implements such a reduction method and generalizes the Mathematica Apart function from 1 to any N dimensions. | ||

Reasons for new version:The Mathematica pattern matching in the last version may become very slow when the number of variables becomes large, this calls for a revised version with a more efficient reduction. The feature with all positive or negative sign of some variables is favored in combined usage of FIRE [2] and FIESTA [3]. | ||

Summary of revisions:We introduce an abstract and compact representation for the linear composition of the independent variables, this results in a more efficient and fast reduction during the APart partial fraction, we also introduce an extra feature to make the sign of some variables always positive or negative during the reduction. | ||

Running time:A few seconds or less. | ||

References: | ||

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[19] | A. V. Smirnov, Algorithm FIRE - Feynman Integral REduction, JHEP 0810, 107 (2008) [arXiv:0807.3243 [hep-ph]] | |

[20] | C. Studerus, Reduze - Feynman Integral Reduction in C++, Comput. Phys. Commun. 181, 1293 (2010) [arXiv:0912.2546 [physics.comp-ph]]. |

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