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Rationalizing roots: an algorithmic approach

Marco Besier, Duco van Straten, Stefan Weinzierl

TL;DR

The authors present an algorithmic framework rooted in elementary algebraic geometry to rationalize square-root kernels that appear in Feynman integral evaluations, enabling expression in terms of multiple polylogarithms. The method associates each root with an irreducible algebraic hypersurface and uses a high-multiplicity point to parametrize the hypersurface via a line family, yielding explicit rational changes of variables. They define and exploit the notion of a perfect root (irreducible, degree $d$, with a $d-1$ multiplicity point) and show the approach can be iterated to handle several roots simultaneously, with clear demonstrations on physics-relevant examples. Notably, the approach does not apply to non-rationalizable cases, such as those linked to elliptic curves or K3 surfaces, which underlines the method’s scope and limitations for simplifying Feynman integrals.

Abstract

In the computation of Feynman integrals which evaluate to multiple polylogarithms one encounters quite often square roots. To express the Feynman integral in terms of multiple polylogarithms, one seeks a transformation of variables, which rationalizes the square roots. In this paper, we give an algorithm for rationalizing roots. The algorithm is applicable whenever the algebraic hypersurface associated with the root has a point of multiplicity $(d-1)$, where $d$ is the degree of the algebraic hypersurface. We show that one can use the algorithm iteratively to rationalize multiple roots simultaneously. Several examples from high energy physics are discussed.

Rationalizing roots: an algorithmic approach

TL;DR

The authors present an algorithmic framework rooted in elementary algebraic geometry to rationalize square-root kernels that appear in Feynman integral evaluations, enabling expression in terms of multiple polylogarithms. The method associates each root with an irreducible algebraic hypersurface and uses a high-multiplicity point to parametrize the hypersurface via a line family, yielding explicit rational changes of variables. They define and exploit the notion of a perfect root (irreducible, degree , with a multiplicity point) and show the approach can be iterated to handle several roots simultaneously, with clear demonstrations on physics-relevant examples. Notably, the approach does not apply to non-rationalizable cases, such as those linked to elliptic curves or K3 surfaces, which underlines the method’s scope and limitations for simplifying Feynman integrals.

Abstract

In the computation of Feynman integrals which evaluate to multiple polylogarithms one encounters quite often square roots. To express the Feynman integral in terms of multiple polylogarithms, one seeks a transformation of variables, which rationalizes the square roots. In this paper, we give an algorithm for rationalizing roots. The algorithm is applicable whenever the algebraic hypersurface associated with the root has a point of multiplicity , where is the degree of the algebraic hypersurface. We show that one can use the algorithm iteratively to rationalize multiple roots simultaneously. Several examples from high energy physics are discussed.

Paper Structure

This paper contains 13 sections, 3 theorems, 146 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Warm-up exercise
  4. Mathematical preliminaries
  5. The algorithm
  6. Applications in physics
  7. Summary
  8. Limitations of the algorithm
  9. Roots without rational parametrizations
  10. Roots associated with elliptic curves
  11. Roots associated with K3 surfaces
  12. Non-rationalizability of alphabets and Kummer coverings
  13. A theorem on a particular type of square roots

Key Result

Corollary 4.6

Whenever we encounter a hypersurface $V:f(x_1,\ldots,x_n)=0$ of degree $d$ with a point $p=(p_1,\ldots,p_n)$ of multiplicity $r$ and move $p=(p_1,\ldots,p_n)$ to the origin by considering then $g(x_1,\ldots,x_n)$ can always be written as where $g_k(x_1,\ldots,x_n)$ with $k=r,\ldots,d$ are the homogeneous components of $g(x_1,\ldots,x_n)$.

Figures (3)

  • Figure 1: A one-loop Feynman diagram contributing to the gauge boson self-energy with a massive fermion loop.
  • Figure 2: Parametrizing the circle by a 1-parameter family of lines.
  • Figure 3: The nodal cubic $V: y^2-x^3-x^2=0$.

Theorems & Definitions (19)

  • Remark 3.1
  • Definition 4.1
  • Definition 4.2
  • Example 4.3
  • Definition 4.4
  • Remark 4.5
  • Corollary 4.6
  • Definition 4.7
  • Example 5.1
  • Remark 5.2
  • ...and 9 more