This work presents some essential techniques that constitute the modern strategy for computing scattering amplitudes. It begins with an introductory chapter to fill the gap between a standard QFT course and the latest developments in the field. The author then tackles the main bottleneck: the computation of the loop Feynman integrals. The most efficient technique for their computation is the method of the differential equations. This is discussed in detail, with a particular focus on the mathematical aspects involved in the derivation of the differential equations and their solution. Ample space is devoted to the special functions arising from the differential equations, to their analytic properties, and to the mathematical techniques which allow us to handle them systematically. The thesis also addresses the application of these techniques to a cutting-edge problem of importance for the physics programme of the Large Hadron Collider: five-particle amplitudes at two-loop order. It presents the first analytic results for complete two-loop five-particle amplitudes, in supersymmetric theories and QCD. The techniques discussed here open the door to precision phenomenology for processes of phenomenological interest, such as three-photon, three-jet, and di-photon + jet production.
Author(s): Simone Zoia
Series: Springer Theses
Publisher: Springer
Year: 2022
Language: English
Pages: 220
City: Cham
Supervisor’s Foreword
Abstract
Publications Related to This Thesis
Acknowledgments
Contents
1 Introduction
Bibliography
2 Scattering Amplitudes
2.1 Scattering Amplitudes and the Phenomenon
2.2 Scattering Amplitudes and the Noumenon
2.3 Loop Scattering Amplitudes
2.3.1 Dimensional Regularisation
2.3.2 Ultraviolet Divergences and Renormalisation
2.3.3 Infrared Divergences
2.4 Analytic Structure of Scattering Amplitudes
2.4.1 Poles and Locality
2.4.2 Discontinuities and Unitarity
Bibliography
3 The Art of Integrating by Differentiating
3.1 Feynman Integrals and Differential Equations
3.1.1 Integral Families and Integration-by-Parts Identities
3.1.2 Differential Equations
3.2 Differential Equations in the Canonical Form
3.2.1 A Note on the Choice of the Letters
3.3 Special Functions
3.3.1 Chen's Iterated Integrals
3.3.2 Classical Polylogarithms
3.3.3 Goncharov Polylogarithms
3.3.4 The Transcendental Weight
3.3.5 On the Naturalness of the Canonical Form
3.3.6 The Symbol
3.4 Solving the Differential Equations
3.4.1 Kinematic Region and Analytic Continuation
3.4.2 Boundary Constants
3.4.3 Solution in Terms of Goncharov Polylogarithms
3.4.4 Solution in Terms of Chen's Iterated Integrals
3.4.5 Solution in Terms of a Basis of Functions
3.5 Asymptotic Solution of the Differential Equations
3.5.1 General Procedure
3.5.2 Soft Limit of the One-Loop Three-Mass Triangle Integrals
3.6 How to Find a Canonical Basis
3.6.1 Leading Singularities
3.6.2 dlog Integrands
Bibliography
4 Two-Loop Five-Particle Scattering Amplitudes
4.1 Kinematics
4.2 Feynman Integrals
4.2.1 Pure Integrals from D-Dimensional Leading Singularities
4.2.2 Pentagon Functions
4.2.3 Boundary Values
4.2.4 Non-trivial Analytic Behaviour at the Boundary
4.3 Maximally Supersymmetric Amplitudes
4.3.1 Notation
4.3.2 Expected Structure of the Two-Loop Amplitudes
4.3.3 Integrating the Integrands
4.3.4 Divergence Structure and Hard Functions
4.3.5 Further Validation of the Results
4.4 Multi-Regge Limit of the Maximally Supersymmetric Amplitudes
4.4.1 Multi-Regge Kinematics
4.4.2 Multi-Regge Limit of the Pentagon Functions
4.4.3 Basis of Transcendental Functions in the Multi-Regge Limit
4.4.4 Multi-Regge Limit of the mathcalN=4 Super Yang-Mills Amplitude
4.4.5 Multi-Regge Limit of the mathcalN=8 Supergravity Amplitude
4.4.6 Discussion
4.5 The All-Plus Amplitude in Pure Yang-Mills Theory
4.5.1 Notation
4.5.2 Divergence Structure and Hard Function
4.5.3 Expected Structure of the Hard Function
4.5.4 How to Express the Integrand in Terms of Inverse Propagators
4.5.5 Computation of the Hard Function
4.5.6 Result
4.5.7 Discussion
Bibliography
5 Conclusions and Outlook
Bibliography
