Genus drop involving nonhyperelliptic curves in Feynman integrals

Source abstract

For both theoretical and phenomenological studies, it is important to analyze the function types of Feynman integrals. The phenomenon of genus drop between different representations of hyperelliptic Feynman integrals was discussed in [1]. In this paper, we reformulate the extra-involution mechanism of [1] as a special case of an unramified double covering between algebraic curves, and show that this covering mechanism also explains genus drops accompanied by a curve-type change from nonhyperelliptic to hyperelliptic for a class of three-loop Feynman diagrams. We also demonstrate that within a specific framework, the origin of the discrete spacetime symmetry that leads to the genus drop in hyperelliptic cases is manifest. This work also points out that there exist nonhyperelliptic Feynman integrals that exhibit no apparent genus drop.