Abstract
I present a Mathematica package designed for manipulations and evaluations of triple-K integrals and conformal correlation functions in momentum space. Additionally, the program provides tools for evaluation of a large class of 2- and 3-point massless multi-loop Feynman integrals with generalized propagators. The package is accompanied by five Mathematica notebooks containing detailed calculations of numerous conformal 3-point functions in momentum space. Program summaryProgram Title: TripleKCPC Library link to program files:http://dx.doi.org/10.17632/5sz4bt28vr.1Developer’s repository link:https://triplek.hepforge.org/Licensing provisions: GNU General Public License v3.0Programming language: Wolfram Language [1] (Mathematica 10.0 or higher)Supplementary material: The package includes five Mathematica notebooks containing bulk of the results regarding the structure of conformal 3-point functions.Nature of problem: Triple-K integrals were introduced in [2] as a convenient tool for the analysis of conformal 3-point functions in momentum space. All 3-point functions of scalar operators, conserved currents and stress tensor can be expressed in terms of triple-K integrals. Furthermore, a large class of 2- and 3-point massless multi-loop Feynman integrals with generalized propagators is expressible in terms of triple-K integrals as well. Since the expressions are usually long and unwieldy, an automated tool is essential for efficient manipulations.Solution method: In [3] an effective reduction algorithm was provided for expressing a large class of triple-K integrals in terms of master integrals. The presented package implements this reduction scheme. As far as the multi-loop Feynman integrals are concerned, the conversion to multiple-K integrals proceeds by means of Schwinger parameterization.Additional comments including restrictions and unusual features: Despite extensive testing, this package is a one man job, therefore bugs are unavoidable. Please, report all issues at adam.bzowski@physics.uu.se or abzowski@gmail.com. [1] Wolfram Research Inc., Mathematica, Version 11.2, 12.0, Champaign, IL, 2020 [2] A. Bzowski, P. McFadden, K. Skenderis, Implications of conformal invariance in momentum space, JHEP 03 (2014) 111. http://arxiv.org/abs/1304.7760arXiv:1304.7760, https://doi.org/10.1007/JHEP03(2014)111doi:10.1007/JHEP03(2014)111 [3] A. Bzowski, P. McFadden, K. Skenderis, Evaluation of conformal integrals, JHEP 02 (2016) 068. http://arxiv.org/abs/1511.02357arXiv:1511.02357, https://doi.org/10.1007/JHEP02(2016)068doi:10.1007/JHEP02(2016)068
Highlights
Problem of analytical or numerical evaluation of Feynman diagrams has been at the heart of high energy research and has been tackled by numerous authors
A large class of 2- and 3-point massless multi-loop Feynman integrals with generalized propagators is expressible in terms of triple-K integrals as well
Was. ✩✩ This paper and its associated computer program are available via the Computer Physics Communication homepage on ScienceDirect
Summary
Problem of analytical or numerical evaluation of Feynman diagrams has been at the heart of high energy research and has been tackled by numerous authors. A number of packages designed for analytic evaluation and manipulations of amplitudes and Feynman. The main focus of the standard set-up is to consider Feynman diagrams in (close to) 4 spacetime dimensions and composed from a number of bosonic or fermionic massive propagators, with the usual 1/(p2 + m2) factor. As shown recently, [10], all conformal (scalar) correlation functions in momentum space can be expressed as momentum integrals with massless, generalized propagators 1/p2ν , with ν not necessarily equal to one. While standard methods employed by the aforementioned programs can be used to some extent in the analysis of such problems, new methods aimed at conformal correlators can be developed. In subsequent papers [12,13,14] a detailed analysis of 3-point functions involving scalar operators as well as conserved currents and stress tensors was presented. The package includes a number of notebooks containing results constituting bulk of the material published in [11,12,13,14]
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