These lecture notes bridge a gap between introductory quantum field theory (QFT) courses and state-of-the-art research in scattering amplitudes. They cover the path from basic definitions of QFT to amplitudes relevant for processes in the Standard ModelÂ of particle physics. The book begins with a concise yet self-contained introduction into QFT, including perturbative quantum gravity. It then presents modern methods for calculating scattering amplitudes, focusing on tree-level amplitudes, loop-level integrands and loop integration techniques. These methods help reveal intriguing relations between gauge and gravity amplitudes, and are of increasing importance for obtaining high-precision predictions for collider experiments, such as those at CERN's Large Hadron Collider, as well as for foundational mathematical physics studies in QFT, including recent applications to gravitational wave physics.

These course-tested lecture notes include numerous exercises with detailed solutions. Requiring only minimal knowledge of QFT, they are well-suited for MSc and PhD students as a preparation for research projects in theoretical particle physics. They can be used as a one-semester graduate level course, or as a self-study guide for researchers interested in fundamental aspects of QFT.

##### Download our lecture notes from arxiv.org/abs/2306.05976 or pre-order the softcover book from Springer

## Table of contents

##### 1. Introduction and foundations

- PoincarÃ© group and its representations
- Weyl and Dirac spinors
- Non-Abelian gauge theories
- Feynman rules for non-Abelian gauge theories
- Scalar QCD
- Perturbative quantum gravity
- Feynman rules for perturbative quantum gravity
- Spinor-helicity formalism for massless particles
- Polarisations of massless particles of spin 1/2, 1 and 2
- Colour decompositions for gluon amplitudes
- Colour-ordered amplitudes

##### 2. On-shell techniques for tree-level amplitudes

- Factorisation properties of tree-level amplitudes
- BCFW recursion for gluon amplitudes
- BCFW recursion for gravity and other theories
- MHV amplitudes from the BCFW recursion relation
- BCFW recursion with massive particles
- Symmetries of scattering amplitudes
- Double-copy relations for gluon and graviton amplitudes

##### 3. Loop integrands and amplitudes

- Introduction to loop amplitudes
- Unitarity and cut construction
- Generalised unitarity
- Reduction methods
- General integral and integrand bases for one-loop amplitudes
- Project: Rational terms of the four-gluon amplitude
- Outlook: Rational representations of the external kinematics
- Outlook: Multi-loop amplitudes methods

##### 4. Loop integration techniques and special functions

- Introduction to loop integrals
- Conventions and basic methods
- Mellin-Barnes techniques
- Special functions, differential equations, and transcendental weight
- Differential equations for Feynman integrals
- Feynman integrals of uniform transcendental weight