Rationalizability of square roots
Marco Besier, Dino Festi
TL;DR
Problem: rationalizing square roots appearing in Feynman integral computations to express results in terms of multiple polylogarithms. The authors define rationalizability for $\sqrt{p/q}$ and connect it to the unirationality of two geometric objects: the projective hypersurface $\overline{V}$ and the associated double cover $\overline{S}$, providing a unified framework that reduces the problem to squarefree polynomials. They establish precise criteria in the one-variable and two-variable cases (deg $f$ bounds; del Pezzo reductions) and describe a dimension-reduction approach for higher variables; they also discuss how to decide non-rationalizability for alphabets of several square roots. The results yield practical, geometry-based tests for rationalizability in high energy physics computations and guide when a single variable substitution can rationalize all square roots, with computational tools for singularity analysis.
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.