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Manuscript Title: $Apart: A Generalized Mathematica Apart Function
Authors: Feng Feng
Program title: $Apart
Catalogue identifier: AEMK_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 183(2012)2158
Programming language: Mathematica.
Computer: Any computer where Mathematica is running.
Operating system: Any capable of running Mathematica.
Keywords: Next-to-Leading Order(NLO), Integrate By Part(IBP), Apart.
PACS: 12.38.Bx.
Classification: 11.1.

External routines: FeynCalc, FeynArts, Fire (all included in the distribution file).

Nature of problem:
The traditional method of computing cross sections for a physical process in perturbative quantum field theory involves generating the amplitudes via Feynman diagrams and performing the dimensionally regularized loop integrals [1]. Simplifications of the expressions are performed at the analytical level, there an essential part is the reduction of these loop integrals to a small number of standard integrals. This step can be performed at the amplitude level for tensor integrals or, after contraction of Lorentz indices, at the level of interferences for scalar integrals. Considering the case of scalar integrals, integration by parts (IBP) identities [2, 3] and Lorentz invariance (LI) identities [4] may be used for a systematic reduction to a set of independent integrals, called master integrals (MI). The standard reduction algorithm by Laporta [5] defines an ordering for Feynman integrals, generates identities and solves the resulting system of linear equations. Alternative methods to exploit IBP and LI identities for reductions have been proposed [6-9], see also [10, 11] and references therein. Public implementations of different reduction algorithms are available with the computer programs AIR [12], FIRE [13] and Reduze [14]. The usage of Fire[13], Reduze[14], etc. requires that the propagators must be decomposed to independent ones, as for 1 dimension, there is a Mathematica function Apart to do this, while for N dimensions there is no such package yet, we want to generalize the Mathematica function Apart to $Apart in N dimensions.

Solution method:
We first prove that any linear dependent elements in V*x can be decomposed into the summation of linear independent ones, the procedure of the proof gives us a method to perform the decomposition, $Apart is such an Mathematica package that implements this method and generalizes the Mathematica Apart function from 1 to N dimensions.

Running time:
Depends on the complexity of the system.

References:
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[2] F. V. Tkachov, A Theorem On Analytical Calculability Of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65.
[3] 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.
[4] T. Gehrmann and E. Remiddi, Differential equations for two-loop four-point functions, Nucl. Phys. B 580 (2000) 485 [arXiv:hep-ph/9912329].
[5] 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].
[6] 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].
[7] A. V. Smirnov, An Algorithm to construct Gröbner bases for solving integration by parts relations, JHEP 0604 (2006) 026 [arXiv:hep-ph/0602078].
[8] 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]].
[9] R. M. Schabinger, A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations, arXiv:1111.4220 [hep-ph].
[10] 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]].
[11] A. G. Grozin, Integration by parts: An Introduction, Int. J. Mod. Phys. A 26 (2011) 2807 [arXiv:1104.3993 [hep-ph]].
[12] C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations, JHEP 0407 (2004) 046 [arXiv:hep-ph/0404258].
[13] A. V. Smirnov, Algorithm FIRE - Feynman Integral REduction, JHEP 0810, 107 (2008) [arXiv:0807.3243 [hep-ph]].
[14] C. Studerus, Reduze - Feynman Integral Reduction in C++, Comput. Phys. Commun. 181, 1293 (2010) [arXiv:0912.2546 [physics.comp-ph]].