Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA

TL;DR

CANONICA provides the first publicly available, Mathematica-based implementation of a multi-scale canonicalization algorithm for differential equations of Feynman integrals. By constructing finite, rational ansatz spaces for diagonal and off-diagonal blocks and proving canonical forms are unique up to constants, it enables robust, linear-order solutions to transform complex multi-loop systems into $\epsilon$-form. The package supports recursive sector decompositions, detailed control over ansatz size, and handles multi-scale topologies previously inaccessible to canonical-form automation. This advances automated multi-loop calculations with potential impact on NNLO QCD predictions and other precision collider processes.

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.

Figures (1)

• Figure 1: Hierarchy of the main functions in CANONICA. Each block lists the public functions called by the function in the blocks title.

Theorems & Definitions (2)

• Claim 1
• Claim 2