Rationalizability of square roots

TL;DR

This paper provides a rigorous framework and practical criteria for determining when square roots of polynomial ratios can be rationalized, aiding the computation of Feynman integrals in particle physics.

Contribution

It introduces a formal definition of rationalizability for polynomial ratio square roots and offers geometric criteria and strategies to assess rationalizability in physics-related problems.

Findings

Criteria for rationalizability in one and two variables

Geometric reformulation of the rationalizability problem

Application to real-world Feynman integral examples

Abstract

Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a solution in terms of these functions is to rationalize all occurring square roots by a suitable variable change. In this paper, we give a rigorous definition of rationalizability for square roots of ratios of polynomials. We show that the problem of deciding whether a single square root is rationalizable can be reformulated in geometrical terms. Using this approach, we give easy criteria to decide rationalizability in most cases of square roots in one and two variables. We also give partial results and strategies to prove or disprove rationalizability of sets of square roots. We apply the results to many examples from actual computations in high energy particle physics.