Novel Mathematical Techniques for Scattering Amplitudes

Abstract

In this thesis, we discuss mathematical methods to compute families of Feynman integrals. Central to our approach are two mathematical structures: the geometries underlying integral families, and their associated twisted (co-)homology groups. We use the latter to derive general relations among integrals and understand the so-called canonical form of a basis of master integrals better. Additionally, we present the analytic computation of several integral families: specific Lauricella integrals associated with hyperelliptic curves, the unequal-mass kite integral family, and two-dimensional (fishnet) integrals. <br/>

We begin by reviewing the physical and mathematical background. Starting with a brief summary of the role of Feynman integrals in perturbative quantum field theory, we introduce key tools such as their parametric representations, their cuts, and the method of differential equations, which relies on finding a so-called canonical basis of an integral family. Then we introduce the mathematical background: We discuss the geometries appearing in the integral families studied in this thesis – including hyperelliptic curves and Calabi–Yau varieties – and review the twisted (co-)homology groups, with an emphasis on the computation of intersection and period matrices. <br/>

We conclude the first part by discussing how a matrix of cuts of Feynman integrals can be interpreted as a period matrix of twisted (co-)homology groups. This interpretation provides a transition into one of the central results of this thesis: We explore what can be learned from the so-called twisted Riemann bilinear relations for the intersection and period matrices of twisted (co-)homology groups associated to a Feynman integral family. These relations not only explain known and provide new relations, but also enable us to gain a deeper understanding of the canonical basis. Then we use these insights to construct canonical differential equations for Lauricella integral families associated to genus-one and genus-two hyperelliptic curves which serve as models for hyperelliptic maximal cuts. Next we present the analytic computation of the unequal-mass kite integral family, which is related to two distinct elliptic curves. A key step is the parametrisation of the five kinematic variables on two elliptic tori, enabling the solution in terms of iterated integrals on these tori. Then we study single-valued Feynman integrals in two spacetime dimensions. We show how these integrals naturally are bilinear combinations – effectively, double copies – of twisted periods. This construction is equivalent to the one of the Kähler potential of a Calabi–Yau geometry, which we exploit to compute the conformal fishnet integrals directly from the data of their associated Calabi-Yau varieties. <br/>

The techniques developed here provide a conceptual bridge between the abstract mathematics of (twisted) periods and the analytic computation of Feynman integrals, e.g., for scattering amplitudes.