Rationalization Questions in Particle Physics
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Abstract
Theoretical predictions in high energy particle physics require the computation of Feynman integrals. Certain steps in these computations generate square roots in the kinematic variables. One way to express Feynman integrals in terms of multiple polylogarithms is to rationalize all occurring square roots by a suitable variable change. Although such a variable change does not always exist, there are many examples from recent high energy physics that admit a rationalization.
In this thesis, we study the question of how to rationalize a given set of square roots in detail. On the one hand, not all square roots are rationalizable. For these cases, we establish criteria that allow us to prove non-rationalizability in a rigorous manner. On the other hand, many square roots admit a rationalization. For these cases, we give a rationalization algorithm that is applicable whenever the hypersurface associated to the square root has a point of multiplicity d−1, where d is the degree of the hypersurface. Furthermore, we present the F-decomposition theorem, which expands the scope of the algorithm to square roots whose rationalization would otherwise be out of reach. Lastly, we present the RationalizeRoots software package, which implements our rationalization methods for Mathematica and Maple. We clarify all of our techniques through several examples from modern high energy physics.