The toolbox of modern multi-loop calculations: novel analytic and semi-analytic techniques
Alexey Pak
TL;DR
The paper tackles the automation of challenging multi-loop calculations by introducing three algorithms: (i) a canonical, alpha-representation–based method to identify Feynman integral topologies via a canonical permutation of parameters; (ii) a practical tensor reduction approach that fixes tensor index counts and precomputes reduction coefficients, enabling reuse across integrals and efficient Gauss-like solving; and (iii) a Gröbner-basis–driven partial fractioning framework that handles linearly dependent denominators and yields a rewrite system translatable to FORM. These methods are demonstrated to be practical in real calculations, with capabilities extending to multi-loop (up to four or five loops) problems and a pipeline that generates code for symbolic engines. The contributions promise improved automation and performance for high-loop computations relevant to LHC-era phenomenology. The work integrates canonicalization, fixed-index tensor algebra, and algebraic geometry to streamline reduction to master integrals and simpler topologies.
Abstract
We describe three algorithms for computer-aided symbolic multi-loop calculations that facilitated some recent novel results. First, we discuss an algorithm to derive the canonical form of an arbitrary Feynman integral in order to facilitate their identification. Second, we present a practical solution to the problem of multi-loop analytical tensor reduction. Finally, we discuss the partial fractioning of polynomials with external linear relations between the variables. All algorithms have been tested and used in real calculations.