triplek is hosted by Hepforge, IPPP Durham


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


Triple-K integrals were introduced in [1304.7760] as a convenient tool for the analysis of conformal 3-point functions in momentum space. They provide new solutions to old problems:

  • Triple-K integrals provide natural way of expressing solutions to conformal Ward identities in momentum space. This includes any 2- and 3-point functions of conformal operators of arbitrary spin such as conserved currents or stress tensor.
  • All 2- and 3-point massless multi-loop Feynman diagrams with generalized propagators are expressible in terms of triple-K integrals.
  • A large class of triple-K integrals can be analytically expressed in terms of almost elementary functions. In particular 3-point functions of operators of integral conformal dimensions in odd and even-dimensional spacetimes can be evaluated explicitly!

The package

The package provides tools for manipulations and evaluations of triple-K integrals and conformal correlation functions. The package consists of 2 Wolfram Language (Mathematica) files as well as 5 Mathematica notebooks. Mathematica version 10.0 or higher is required. The files are as follows:

  • TripleK.wl is the heart of the package. This file contains procedures for manipulations and evaluations of triple-K integrals. Import it by Mathematica's Get command: << TripleK`
  • BasicExamples.nb contains examples regarding the use of TripleK.wl. It contains 5 sections presenting the package. It is a good point to start.
  • Konformal.wl and the remaining Mathematica notebooks serve as a repository of a number of results on the structure of conformal 3-point functions.

Click HERE to download the package. The package is licensed under the GNU General Public License v3.0.


The package is accompanied by a number of expamples. The basic features of the package are described in my article [2005.10841]. A short video tutorial with the applications in cosmology can be found below.


If you use the package, please cite:

Adam Bzowski, TripleK: A Mathematica package for evaluating triple-K integrals and conformal correlation functions, Comput. Phys. Commun. 258 (2021) 107538, arXiv: [2005.10841]

For further reference on triple-K integrals and conformal correlation functions consult [1304.7760]. The reduction algorithm used by the package for the evaluation of triple-K integrals is based on [1511.02357].

If you have any questions/comments/suggestions, please don't hesitate to contact me at abzowski[at]