Abstract
The integration of differential equations of Feynman integrals can be greatly facilitated by using a canonical basis. This paper presents the Mathematica package CANONICA, which implements a recently developed algorithm to automatize the transformation to a canonical basis. This represents the first publicly available implementation suitable for differential equations depending on multiple scales. In addition to the presentation of the package, this paper extends the description of some aspects of the algorithm, including a proof of the uniqueness of canonical forms up to constant transformations. Program summaryProgram Title: CANONICAProgram Files doi:http://dx.doi.org/10.17632/fmwnmmhn77.1Licensing provisions: GNU General Public License version 3Programming language: Wolfram Mathematica, version 10 or higherNature of problem: Computation of a rational basis transformation of master integrals leading to a canonical form of the corresponding differential equation.Solution method: The transformation law is expanded in the dimensional regulator. The resulting differential equations for the expansion coefficients of the transformation are solved with a rational ansatz.
Highlights
The calculation of higher order corrections to the cross-sections measured at the LHC is crucial in order to improve the understanding of both the background reactions as well as the signal processes
While the calculation of Feynman integrals can be attempted with numerous approaches, the method of differential equations [1,2,3] has been successful in the recent years [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]
The directory of each example contains a .m file with the corresponding differential equation and a .nb notebook file illustrating the application of CANONICA to this example
Summary
The calculation of higher order corrections to the cross-sections measured at the LHC is crucial in order to improve the understanding of both the background reactions as well as the signal processes. The systematic application [57, 58] of integration by parts relations [59, 60] to reduce all scalar integrals to a finite number of master integrals has been automated in a variety of publicly available tools [61,62,63,64,65,66,67,68,69] This leaves the process of choosing a canonical basis as the step to be automated. The occurrence of non-linear polynomial equations in the parameters of the ansatz is addressed with a procedure to extract all relevant information by solving only linear equations, while maintaining all of the algorithms generality The latter relies on the uniqueness of canonical forms up to constant transformations. The uniqueness of canonical forms up to constant transformations is proven Building on this result, the treatment of non-linear polynomial equations in the parameters of the ansatz is discussed. The global variables and protected symbols of the package are listed in Appendix C
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