TripleK: A Mathematica package for evaluating triple-K integrals and conformal correlation functions

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 summary

Program Title: TripleK

CPC Library link to program files: http://dx.doi.org/10.17632/5sz4bt28vr.1

Developer’s repository link: https://triplek.hepforge.org/

Licensing provisions: GNU General Public License v3.0

Programming 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 [email protected] or [email protected]. [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/1304.7760 arXiv:1304.7760, https://doi.org/10.1007/JHEP03(2014)111 doi:10.1007/JHEP03(2014)111 [3] A. Bzowski, P. McFadden, K. Skenderis, Evaluation of conformal integrals, JHEP 02 (2016) 068. http://arxiv.org/abs/1511.02357 arXiv:1511.02357, https://doi.org/10.1007/JHEP02(2016)068 doi:10.1007/JHEP02(2016)068