Elsevier Science Home
Computer Physics Communications Program Library
Full text online from Science Direct
Programs in Physics & Physical Chemistry
CPC Home

[Licence| Download | New Version Template] aemk_v2_0.tar.gz(864 Kbytes)
Manuscript Title: APart 2: A Generalized Mathematica Apart Function
Authors: Feng Feng
Program title: APart 2.0
Catalogue identifier: AEMK_v2_0
Distribution 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.

[1] F. Feng, $Apart: A Generalized Mathematica Apart Function, Comput. Phys. Commun. 183 (2012) 2158 [arXiv:1204.2314 [hep-ph]].
[2] A. V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2014) 182 [arXiv:1408.2372 [hep-ph]].
[3] A. V. Smirnov, FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions, Comput. Phys. Commun. 185 (2014) 2090 [arXiv:1312.3186 [hep-ph]].
[4] T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260].
[5] R. Mertig, M. Bohm and A. Denner, FeynCalc: Computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345.
[6] F. Feng and R. Mertig, FormLink/FeynCalcFormLink : Embedding FORM in Mathematica and FeynCalc, arXiv:1212.3522.
[7] G.'t Hooft and M. J. G. Veltman, Regularization And Renormalization Of Gauge Fields, Nucl. Phys. B 44 (1972) 189.
[8] F. V. Tkachov, A Theorem On Analytical Calculability Of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65.
[9] K. G. Chetyrkin and F. V. Tkachov, Integration By Parts: The Algorithm To Calculate Beta Functions In 4 Loops, Nucl. Phys. B 192 (1981) 159.
[10] T. Gehrmann and E. Remiddi, Differential equations for two-loop four-point functions, Nucl. Phys. B 580 (2000) 485 [arXiv:hep-ph/9912329].
[11] S. Laporta, High-precision calculation of multi-loop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [arXiv:hep-ph/0102033].
[12] A. V. Smirnov and V. A. Smirnov, Applying Gröbner bases to solve reduction problems for Feynman integrals, JHEP 0601 (2006) 001 [arXiv:hep-lat/0509187].
[13] A. V. Smirnov, An Algorithm to construct Gröbner bases for solving integration by parts relations, JHEP 0604 (2006) 026 [arXiv:hep-ph/0602078].
[14] J. Gluza, K. Kajda and D. A. Kosower, Towards a Basis for Planar Two-Loop Integrals, Phys. Rev. D 83 (2011) 045012 [arXiv:1009.0472 [hep-th]].
[15] R. M. Schabinger, A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations, arXiv:1111.4220 [hep-ph].
[16] R. N. Lee, Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals, JHEP 0807 (2008) 031 [arXiv:0804.3008 [hep-ph]]
[17] A. G. Grozin, Integration by parts: An Introduction, Int. J. Mod. Phys. A 26 (2011) 2807 [arXiv:1104.3993 [hep-ph]].
[18] C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations, JHEP 0407 (2004) 046 [arXiv:hep-ph/0404258]
[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]].