Positive singularities and volumes in scattering amplitudes

Mazzucchelli, Elia (2026): Positive singularities and volumes in scattering amplitudes. Dissertation, LMU München: Faculty of Physics

Abstract

Recent advances have shown that scattering amplitudes in certain quantum field theories admit a geometric formulation in terms of positive geometries. In this framework, tree-level amplitudes and loop-level integrands are associated with regions in kinematic space whose boundary structure encodes physical principles such as locality, unitarity, and factorization. This thesis provides a self-contained introduction to positive geometries, with particular emphasis on the Amplituhedron. This geometric object encodes amplitudes in planar maximally supersymmetric Yang-Mills theory through a distinguished differential form, known as the canonical form. We study applications of positive geometries to scattering amplitudes, with emphasis on different incarnations of positivity, volumes, and singularities, developing three related directions. First, we investigate positivity properties of canonical forms through dual volume representations. For polytopes, canonical functions compute the volume of the dual polytope. We extend this picture to nonlinear positive geometries, uncovering non-negative transcendental measures that reveal a relation to a strong positivity property called complete monotonicity. We discuss first steps toward analogous constructions for Amplituhedra, supporting the idea that amplitudes compute volumes of geometric objects. Second, we study the implications of positive geometries for loop-level amplitudes and related quantities. These are typically expressed in terms of complicated transcendental functions, but their singularities already carry essential physical information. By combining the geometric constraints of the Amplituhedron with Landau analysis, we classify leading singularities of the Wilson loop with Lagrangian insertion, a finite quantity closely related to the logarithm of the amplitude, and constrain the set of possible singularities. Third, we investigate conjectural structures relating Landau singularities, positivity, and cluster algebras. Amplitudes in planar supersymmetric Yang--Mills theory are expected to be well behaved in a distinguished positive region of kinematic space, and their singularities are conjectured to factorize into special building blocks known as cluster variables. We formulate Landau analysis in momentum-twistor and Grassmannian language, and use the geometric framework to uncover recursive structures in these singularities across loop orders. We prove these conjectured properties for several families across all loop orders and propose a strategy for establishing the general conjectures. Overall, this thesis develops the idea that positive geometry organizes many notions of positivity, as well as the structure of singularities resulting after loop integration.

• Item Type: Theses (Dissertation, LMU Munich)
• Keywords: scattering amplitudes, positive geometry, Amplituhedron, Landau singularities, canonical forms, cluster algebras
• Subjects: 500 Natural sciences and mathematics; 500 Natural sciences and mathematics > 530 Physics
• Faculties: Faculty of Physics
• Language: English
• Date of oral examination: 16. July 2026
• 1. Referee: Henn, Johannes
• Signature of the printed copy: 0001/UMC 32141