Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams

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This paper studies the global structure of algebraic curves defined by generalized unitarity cut of four-dimensional three-loop diagrams with eleven propagators.

• The global structure is a topological invariant that is characterized by the geometric genus of the algebraic curve.
• We use the Riemann-Hurwitz formula to compute the geometric genus of algebraic curves with the help of techniques involving convex hull polytopes and numerical algebraic geometry.
• Some interesting properties of genus for arbitrary loop orders are also explored where computing the genus serves as an initial step for integral or integrand reduction of three-loop amplitudes via an algebraic geometric approach.