AMBRE - Automatic Mellin-Barnes Representation

and calculation of Feynman integrals


Ievgen Dubovyk (DESY, e.a.dubovyk,gmail.com), J. Gluza (U. of Silesia, janusz.gluza,us.edu.pl), K. Kajda (U. of Silesia, now at IT company, kajda.krzysztof,gmail.com), T. Riemann (DESY, tord.riemann,desy.de)

Publications on the AMBRE project

AMBRE v3.1, AMBREv2.1, AMBREv1.3 [Updates for versions 1,2,3]

Authors: I. Dubovyk, J. Gluza, K. Kajda, T. Riemann

"Numerical integration of massive two-loop Mellin-Barnes integrals in Minkowskian regions “,
Ievgen Dubovyk, Janusz Gluza, Tord Riemann, Johann Usovitsch,
http://arxiv.org/abs/arXiv:1607.07538

AMBRE v3.0 [Global approach, optimized for non-planar two-loop cases, see Features in v3.0 description for details.]

Authors: I. Dubovyk, J. Gluza, K. Kajda, T. Riemann

Publication: section „Construction of Mellin-Barnes integrals in „Non-planar Feynman integrals, Mellin-Barnes , representations, multiple sums “,
J. Blümlein, I. Dubovyk, J. Gluza, M. Ochman, C.G. Raab, T. Riemann, C. Schneider, PoS(LL2014)052
http://arxiv.org/abs/arXiv:1407.7832

AMBREv2.0 [Loop-by-loop approach, optimized for planar cases, covers also features of v1.2]

Authors: J. Gluza, K. Kajda, T. Riemann

Publication: „ Numerical Evaluation of Tensor Feynman Integrals in Euclidean Kinematics“,
J. Gluza, K. Kajda, T. Riemann, V. Yundin, Eur. Phys. J. C71 (2011) 1516, DOI: http://dx.doi.org/10.1140/epjc/s10052-010-1516-y
http://arxiv.org/abs/arXiv:1010.1667

AMBRE v1.0

Authors: J. Gluza, K. Kajda, T. Riemann

Publication:„AMBRE: A Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals“
J. Gluza, K. Kajda, T. Riemann, Comput. Phys. Commun. 177 (2007) 879, DOI: http://dx.doi.org/10.1016/j.cpc.2007.07.001
http://arxiv.org/abs/arXiv:0704.2423


Further publications on the AMBRE project

„ Some Remarks on Non-planar Feynman Diagrams“, K. Bielas, I. Dubovyk, J. Gluza, T. Riemann,
Contrib. to Int. Conf. „Matter to the deepest“, 1-6 Sep. 2013, Ustron, Poland
Acta Phys.Polon. B44 (2013) 2249, DOI: http://dx.doi.org/10.5506/APhysPolB.44.2249
http://arxiv.org/abs/arXiv:1312.5603

„News on Ambre [v1.2] and CSectors“, J. Gluza, K. Kajda, T. Riemann, V. Yundin
Contrib. to Proc. of „Loops and Legs in Quantum Field Theory“', 25-30 April 2010, Wörlitz, Germany
Nucl. Phys. Proc. Suppl. 205-206 (2010) 147, DOI: http://dx.doi.org/10.1016/j.nuclphysbps.2010.08.034
http://arxiv.org/abs/arXiv:1006.4728

„New results for loop integrals: AMBRE [v1.0], CSectors, hexagon“ , J. Gluza, K. Kajda, T. Riemann, V. Yundin
Contrib. to Proc. of „Workshop on advanced computing and analysis techniques in physics research“, ACAT 2008), 3-7 Nov. 2008 Erice, Italy
PoS ACAT08 (2008) 124, http://pos.sissa.it/archive/conferences/070/124/ACAT08_124.pdf
http://arxiv.org/abs/arXiv:0902.4830


HEPFORGE

HEPFORGE is a development environment for high energy physics software projects.

See: https://www.hepforge.org/.

AMBRE is part of a collection of tools devoted to the evaluation of Mellin-Barnes integrals collected at HEPFORGE. Most of the AMBRE applications make use of the Mathematica package MB by M. Czakon. There you can also find links to the numerical libraries CUBA of T. Hahn and CERNlib (in fact, only libmathlib.a and libkernlib.a are required) which are used by the MB package.

For MB, see: https://mbtools.hepforge.org/.

For CUBA, see: http://www.feynarts.de/cuba/.

For CERNlib, see: http://cernlib.web.cern.ch/cernlib/.


Related and auxiliary Software

MBnumerics

Project: I. Dubovyk, T. Riemann, J. Usovitsch (jusovitsch,googlemail.com)
Software: Johann Usovitsch
Publications: https://doi.org/10.18452/19484 , https://doi.org/10.1016/j.cpc.2006.07.002, https://doi.org/10.1016/j.physletb.2016.09.012
To be cited by users in publications, for details see README_copyright in the downloaded tarball.
Features: MBnumerics is a software for evaluation of MB integrals in the Minkowski kinematics
Download: packages/mbnumerics.tgz

MBsums

Project: Michal Ochman, Tord Riemann (ochman.michal,gmail.com)
Software: Michal Ochman
(i) When using MBsums, please quote [1].
(ii) Please do not share/distribute/publish derivatives of MBsums without written permission of the project authors.
Main publication:
[1] M. Ochman, T. Riemann, "MBsums, a Mathematica package for the 2 representation of Mellin-Barnes integrals by multiple sums", Acta Phys. Polon. 46 (2015) 2117, arXiv:1511.01323.
[2] An extended version of the description of MBsums, DESY 15-209, was submitted to the arXiv: hep-ph/1511.01323.
Features: MBsums is a Mathematica package representing multiple MB-integrals by multiple sums.
Download: http://jgluza.us.edu.pl/ambre/MBsums/MBsums.v1.0.3.m (CC BY-ND, the CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html ).
Examples: 1lbox.nb [1lbox.pdf], 2lbox-m0.nb [2lbox-m0.pdf], 2lbox.nb [2lbox.pdf], dim3.nb [dim3.pdf]

Csectors

Project: J. Gluza, K. Kajda, T. Riemann, V. Yundin (yuvaleriy,gmail.com)
Software: J. Gluza, K. Kajda, V. Yundin
Main publication: J. Gluza, K. Kajda, T. Riemann, V. Yundin, PoS ACAT08 (2008) 124. To be cited by users in publications.
Features: CSectors is a Mathematica interface to sector_decomposition, a Ginac package for the numerical calculation of multiloop tensor integrals.
Download:
 For sector_decomposition, see: http://wwwthep.physik.uni-mainz.de/~stefanw/sector_decomposition/
For Csectors, see: http://jgluza.us.edu.pl/csectors/ (CC BY-ND)
For Ginac see: http://www.ginac.de/

PlanarityTest

Project: K. Bielas (krzysiek.bielas,gmail.com), I. Dubovyk, J. Gluza, T. Riemann
Software: K. Bielas, I. Dubovyk
Main publication: K. Bielas, I. Dubovyk, J. Gluza, T. Riemann, Acta Phys.Polon. B44 (2013) (2013) 2249. To be cited by users in publications.
Features: PlanarityTest is a Mathematica package for testing the planarity of Feynman diagrams.
Download: http://jgluza.us.edu.pl/ambre/planarity/ (CC BY-ND).

KinematicsGen

Project and Software: K. Kajda, J. Gluza
Features: Kinematics generator for 4-, 5-, and 6-point functions with arbitrary external legs.
Download: http://jgluza.us.edu.pl/ambre/packages/KinematicsGen.m (CC BY-ND).
Examples: http://jgluza.us.edu.pl/ambre/examples/MomentaGenExamples.nb

HPL4

Package: T. Riemann (tordriemann,gmail.com)
Main publication: M. Czakon, J. Gluza, T. Riemann, Phys.Rev. D71 (2005) 073009, hep-ph/0412164. To be cited by users in publications.
Features: HPL4 is a numerical package for harmonic polylogarithms of weights 1-4.
Download: http://jgluza.us.edu.pl/ambre/packages/HPLnum.m (CC BY-ND).


The AMBRE project and software: Packages and Samples

Project authors:

I. Dubovyk, J. Gluza, K. Kajda, T. Riemann

Copyright statement:

The authors teams of publications, projects, and software packages related to AMBRE are not necessarily identical. The individual software packages have their own copyright statements = conditions of use, formulated by the authors of that software. They should be reproduced in the distributed package.

Globally, the authors expect credit by:

(i) Appropriately mentioning that software of the AMBRE project has been used.

(ii) Appropriately citing, for that use, the references which have been named by the authors.

Users may, of course, contact the authors for explanations or for special regulations.

For a recent report on the ACAT2013 round table discussion on „Open-source software, knowledge sharing and scientific collaboration“ by F. Carminati, D. Perret-Gallix, T. Riemann see: http://arxiv.org/abs/arXiv:1407.0540. Copyright issues are discussed there.

Out of the family of Creative Commons licences the following two are close to our conditions of use:

The version CC BY-ND - if the authors want to exclude non-authorized derivations.

The version CC BY-SA - if the authors allow for derivations, by retaining the original attribution and the original conditions of use. For Creative Commons licences see e.g.:

https://creativecommons.org/licenses/?lang=pl, https://creativecommons.org/licenses/?lang=en,

https://creativecommons.org/licenses/?lang=pl, https://creativecommons.org/licenses/?lang=de,

https://creativecommons.org/licenses/?lang=en, https://creativecommons.org/licenses/?lang=ru.


AMBRE software is distributed in different ways.


If the software is available for anonymous download by everybody, please „right click“ and „save link as“. In other cases, please contact one of the authors of this webpage by email.

The Mathematica package AMBRE v3.1.1

Author of the software: I. Dubovyk

This version is an extension of v3.1. It works also for non-planar diagrams with numerators in a form of inverse propagators.

Download: Package available on request from the authors of this webpage (I. Dubovyk, J. Gluza or T. Riemann). The distribution of the software or of its derivatives by users is not allowed without explicit permission by the authors (CC BY-ND).
Main publication:
Ievgen Dubovyk, Janusz Gluza, Tord Riemann, Johann Usovitsch, http://arxiv.org/abs/arXiv:1607.07538 To be cited by users in publications.

Features:
Extension of version 3.0 - it works also for non-planar diagrams with numerators in a form of inverse propagators. We would like to thank Johannes Henn for a suggestion in this direction. For planar cases, inverse propagators work with versions < 3, see for instance example6.nb from ver1.0 below.

Example: Inverse_propagator_examples.nb [Inverse_propagator_examples.pdf]

The Mathematica package AMBRE v3.1

Author of the software: I. Dubovyk

This version is an extension of v3.0 to 3-loop non-planar cases.

Download: Package available on request from the authors of this webpage (I. Dubovyk, J. Gluza or T. Riemann). The distribution of the software or of its derivatives by users is not allowed without explicit permission by the authors (CC BY-ND).
Main publication:
Ievgen Dubovyk, Janusz Gluza, Tord Riemann, Johann Usovitsch, http://arxiv.org/abs/arXiv:1607.07538 To be cited by users in publications.

AMBRE v3.1 uses parts of earlier versions of AMBRE software.

Features:
The algorithm implemented in v3.0 is extended to 3-loop non-planar cases. The 2-loop part is identical to v3.0, the 3-loop part can treat only non-planar integrals excluding non-planar cases with a planar subloop. Its usage might be limited by dimensionality of obtained representations (will be improved in future releases).

Example: AMBREv3.1_examples.nb [AMBREv3.1_examples.pdf]

The Mathematica package AMBRE v3.0

Author of the software: I. Dubovyk

The version has been developed for the treatment of basic, non-planar cases.

Download: Package available on request from the authors of this webpage (I. Dubovyk, J. Gluza or T. Riemann). The distribution of the software or of its derivatives by users is not allowed in this preliminary stage of the project without explicit permission by the authors (CC BY-ND).
Main publication:
J. Blümlein, I. Dubovyk, J. Gluza, M. Ochman, C.G. Raab, T. Riemann, C. Schneider, PoS(LL2014)052. To be cited by users in publications.

AMBRE v3.0 uses parts of earlier versions of AMBRE software.

Features:
The first AMBRE version for generation in an automatic way Mellin-Barnes representations for non-planar tensor Feynman integrals, including two loop cases. The automatic processing of three loop cases by the hybrid approach will be made available soon. AMBRE v3.0 is not yet merged with AMBRE v2.0 (using the loop-by-loop approach, and thus often limited to planar cases). That's why the function call is specific: MBreprNP. The package was tested intensively on a variety of massive two-loop tensor vertex integrals with tensors up to rank six and with some two-loop box integrals.

Example files:


The Mathematica package AMBRE v2.1.1

Author of the software: K. Kajda, I. Dubovyk

Download: http://jgluza.us.edu.pl/ambre/packages/AMBREv2.1.1.m (CC BY-ND).
Main publication:

J. Gluza, K. Kajda, T. Riemann, V. Yundin, Eur. Phys. J. C71 (2011) 1516. To be cited by users in publications.

Features:
For planar [and few non-planar] tensor Feynman diagrams, complete, automatic approach. Improvements over v2.0: (i) extension to general d-dimensions; (ii) automatic simplification of the F polynomial; (iii) function BarnesLemma[] works now correctly in Mathematica 10.
Improved simplifications in Mathematica connected with the extension to general d-dimensions.

Examples: see v2.1

The Mathematica package AMBRE v2.1

Author of the software: K. Kajda, I. Dubovyk

Download: http://jgluza.us.edu.pl/ambre/packages/AMBREv2.1.m (CC BY-ND).
Main publication:

J. Gluza, K. Kajda, T. Riemann, V. Yundin, Eur. Phys. J. C71 (2011) 1516. To be cited by users in publications.

Features:
For planar [and few non-planar] tensor Feynman diagrams, complete, automatic approach. Improvements over v2.0: (i) extension to general d-dimensions; (ii) automatic simplification of the F polynomial; (iii) function BarnesLemma[] works now correctly in Mathematica 10.

Example: AMBREv2.1_examples.nb [AMBREv2.1_examples.pdf]

The Mathematica package AMBRE v2.0

Author of the software: K. Kajda

Download: http://jgluza.us.edu.pl/ambre/packages/AMBRE.m (CC BY-ND).
Main publication:

J. Gluza, K. Kajda, T. Riemann, V. Yundin, Eur. Phys. J. C71 (2011) 1516. To be cited by users in publications.

Features:
For planar [and few non-planar] tensor Feynman diagrams, complete, automatic approach.

This version allows to generate in an automatic way Mellin-Barnes representations for multiloop planar tensor integrals. It might work also for non-planar topologies. If you want to control manually the loop-by-loop dimensionality of the constructed MB representations, use one of previous versions. Due to this functionality, they are not replaced by v2.0.
To make analytic continuations and numerical tests, the program needs the auxiliary files MBnum.m or MBresolve.m (see HEPFORGE/mbtools, by Alex and Volodya Smirnov), and in order to reduce the dimensionality of integrals as much as possible, the package barnesroutines.m (see HEPFORGE/mbtools, by David Kosower) can be useful.

Examples:

Use of MBnum: MB_SE5l0m.m, out_SE5l0m
Use of MBresolve: MB_SE5l0m_MBresolve.m, out_SE5l0m_Mbresolve,
MB_B1_massless.m, out_B1_massless,
MB_B1_massive.m, out_B1_massive,
Use of barnesroutines: MB_B1_massive_BL.m, out_B1_massive_BL;
Rank 2 cases: MB_B1_massive_rank2.m, out_B1_massive_rank2,
MB_B1_massless_rank2.m, out_B1_massless_rank2.

Pentabox of rank 3 (for a figure see example10.nb below): MB_PBox.m, out_PBox

Four loop self-energy (for a figure see example9.nb below): MB_SE4loop.m, out_SE4loop

tar file with all above examples: ex.tgz


The Mathematica package AMBRE v1.3.1

Author of the software : K. Kajda, I. Dubovyk

Download: http://jgluza.us.edu.pl/ambre/packages/AMBREv1.3.1.m (CC BY-ND).

Main reference: J. Gluza, K. Kajda, T. Riemann, V. Yundin, Eur. Phys. J. C71 (2011) 1516. To be cited by users in publications.

Features:
For planar cases, complete, manual approach. Improvements over v1.3: (i) extension to general d-dimensions; (ii) automatic simplification of the F polynomial.
Improved simplifications in Mathematica connected with the extension to general d-dimensions.

Examples: see ver1.3

The Mathematica package AMBRE v1.3

Author of the software : K. Kajda, I. Dubovyk

Download: http://jgluza.us.edu.pl/ambre/packages/AMBREv1.3.m (CC BY-ND).

Main reference: J. Gluza, K. Kajda, T. Riemann, V. Yundin, Eur. Phys. J. C71 (2011) 1516. To be cited by users in publications.

Features:
For planar cases, complete, manual approach. Improvements over v1.3: (i) extension to general d-dimensions; (ii) automatic simplification of the F polynomial

Example: AMBREv1.3_examples.nb [AMBREv1.3_examples.pdf]

The Mathematica package AMBRE v1.2

Author of the software : K. Kajda

Download (The function BarnesLemma[] works now correctly in Mathematica10, modified by I. Dubovyk):
http://jgluza.us.edu.pl/ambre/packages/AMBREv1.2.1.m (CC BY-ND).

Download: http://jgluza.us.edu.pl/ambre/packages/AMBREv1.2.m (CC BY-ND).

Main reference: J. Gluza, K. Kajda, T. Riemann, V. Yundin, Eur. Phys. J. C71 (2011) 1516. To be cited by users in publications.

Features:
For planar cases, complete, manual approach.

This version allows to generate MB representations for tensor integrals containing not only scalar products of internal and external momenta, but also internal momenta with indices only. Additionally new options were added, among others it allows to generate representations without doing the Feynman parameter, or x-, integrations (here we would like to thank Pierpaolo Mastrolia for this suggestion). Detailed description of new features is available in the following

Examples:

description of new features: mathematica file

example of QED vertex with the following numerators: (k1.k1)^2, numerator with general external momenta, numerator without external momenta - example file


The Mathematica package AMBRE v1.1
Author of the software: K. Kajda

Download: http://jgluza.us.edu.pl/ambre/packages/AMBREv1.1.m (CC BY-ND).

Main reference: J. Gluza, K. Kajda, T. Riemann, Comput. Phys. Commun. 177 (2007) 879. To be cited by users in publications.

Features:

This version allows to obtain MB-representations for direct products of Feynman integrals like e.g. tadpole*box, SE*vertex, etc.
We thank Stefan Bekavac for indicating the interest in this extension.


The Mathematica package AMBRE v1.0
Authors of the software: J. Gluza, K. Kajda

Download: http://jgluza.us.edu.pl/ambre/packages/AMBREv1.0.m (CC BY-ND).

Main reference: J. Gluza, K. Kajda, T. Riemann, Comput. Phys. Commun. 177 (2007) 879. To be cited by users in publications.

Features:
Basic version, examples below use this package.

Tarball with examples given below: http://jgluza.us.edu.pl/ambre/examples.tar.gz

Some of the examples shown below, including those with numerators, are calculated by sector decomposition. See: http://jgluza.us.edu.pl/csectors/.

Remark. Examples (numerics) works with:
  • MB.m, which can be downloaded using this link: MB tools
There you can also find links to CUBA and CERNlib (only libmathlib.a and libkernlib.a are actually required) which are used by MB.m