β Scribed by Simon Badger; Johannes Henn; Jan Christoph Plefka; Simone Zoia
No coin nor oath required. For personal study only.
Preface
Acknowledgements
Contents
Acronyms
1 Introduction and Foundations
1.1 PoincarΓ© Group and Its Representations
1.2 Weyl and Dirac Spinors
1.3 Non-Abelian Gauge Theories
1.4 Feynman Rules for Non-Abelian Gauge Theories
1.5 Scalar QCD
1.6 Perturbative Quantum Gravity
1.7 Feynman Rules for Perturbative Quantum Gravity
1.8 Spinor-Helicity Formalism for Massless Particles
1.9 Polarisations of Massless Particles of Spin 12, 1 and 2
1.10 Colour Decompositions for Gluon Amplitudes
1.10.1 Trace Basis
1.10.2 Structure Constant Basis
1.11 Colour-Ordered Amplitudes
1.11.1 Vanishing Tree Amplitudes
1.11.2 The Three-Gluon Tree-Amplitudes
1.11.3 Helicity Weight
1.11.4 Vanishing Graviton Tree-Amplitudes
References
2 On-Shell Techniques for Tree-Level Amplitudes
2.1 Factorisation Properties of Tree-Level Amplitudes
2.1.1 Collinear Limits
2.1.2 Soft Theorems
2.1.3 Spinor-Helicity Formulation of Soft Theorems
2.1.4 Subleading Soft Theorems
2.2 BCFW Recursion for Gluon Amplitudes
2.2.1 Large z Falloff
2.3 BCFW Recursion for Gravity and Other Theories
2.4 MHV Amplitudes from the BCFW Recursion Relation
2.4.1 Proof of the Parke-Taylor Formula
2.4.2 The Four-Graviton MHV Amplitude
2.5 BCFW Recursion with Massive Particles
2.5.1 Four-Point Amplitudes with Gluons and MassiveScalars
2.6 Symmetries of Scattering Amplitudes
2.7 Double-Copy Relations for Gluon and Graviton Amplitudes
2.7.1 Lower-Point Examples
2.7.2 Colour-Kinematics Duality: A Four-Point Example
2.7.3 The Double-Copy Relation
References
3 Loop Integrands and Amplitudes
3.1 Introduction to Loop Amplitudes
3.2 Unitarity and Cut Construction
3.3 Generalised Unitarity
3.4 Reduction Methods
3.4.1 Tensor Reduction
3.4.2 Transverse Spaces and Transverse Integration
3.5 General Integral and Integrand Bases for One-Loop Amplitudes
3.5.1 The One-Loop Integral Basis
3.5.2 A One-Loop Integrand Basis in Four Dimensions
3.5.2.1 The Box Integrand in Four Dimensions
3.5.2.2 The Triangle Integrand in Four Dimensions
3.5.2.3 The Bubble Integrand in Four Dimensions
3.5.3 D-Dimensional Integrands and Rational Terms
3.5.3.1 The Pentagon Integrand
3.5.3.2 Extending the Box, Triangle and Bubble Integrand Basis to D=4-2Ξ΅ Dimensions
3.5.4 Final Expressions for One-Loop Amplitudes in D-Dimensions
3.5.5 The Direct Extraction Method
3.6 Project: Rational Terms of the Four-Gluon Amplitude
3.7 Outlook: Rational Representations of the External Kinematics
3.8 Outlook: Multi-Loop Amplitude Methods
References
4 Loop Integration Techniques and Special Functions
4.1 Introduction to Loop Integrals
4.2 Conventions and Basic Methods
4.2.1 Conventions for Minkowski-Space Integrals
4.2.2 Divergences and Dimensional Regularisation
4.2.3 Statement of the General Problem
4.2.4 Feynman Parametrisation
4.2.5 Summary
4.3 Mellin-Barnes Techniques
4.3.1 Mellin-Barnes Representation of the One-Loop Box Integral
4.3.2 Resolution of Singularities in Ξ΅
4.4 Special Functions, Differential Equations, and Transcendental Weight
4.4.1 A First Look at Special Functionsin Feynman Integrals
4.4.2 Special Functions from Differential Equations: The Dilogarithm
4.4.3 Comments on Properties of the Defining Differential Equations
4.4.4 Functional Identities and Symbol Method
4.4.5 What Differential Equations Do Feynman Integrals Satisfy?
4.5 Differential Equations for Feynman Integrals
4.5.1 Organisation of the Calculation in Terms of Integral Families
4.5.2 Obtaining the Differential Equations
4.5.3 Dimensional Analysis and Integrability Check
4.5.4 Canonical Differential Equations
4.5.5 Solving the Differential Equations
4.6 Feynman Integrals of Uniform Transcendental Weight
4.6.1 Connection to Differential Equationsand (Unitarity) Cuts
4.6.2 Integrals with Constant Leading Singularities and Uniform Weight Conjecture
References
5 Solutions to the Exercises
Exercise 1.1: Manipulating Spinor Indices
Exercise 1.2: Massless Dirac Equation and Weyl Spinors
Exercise 1.3: SU(Nc) Identities
Exercise 1.4: Casimir Operators
Exercise 1.5: Spinor Identities
Exercise 1.6: Lorentz Generators in the Spinor-Helicity Formalism
Exercise 1.7: Gluon Polarisations
Exercise 1.8: Colour-Ordered Feynman Rules
Exercise 1.9: Independent Gluon Partial Amplitudes
Exercise 1.10: The MHV3 Amplitude
Exercise 1.11: Four-Point Quark-Gluon Scattering
Exercise 2.1: The Vanishing Splitting Function Splittree+(x,a+,b+)
Exercise 2.2: Soft Functions in the Spinor-Helicity Formalism
Exercise 2.3: A qggg Amplitude from Collinear and Soft Limits
Exercise 2.4: The Six-Gluon Split-Helicity NMHV Amplitude
Exercise 2.5: Soft Limit of the Six-Gluon Split-Helicity Amplitude
Exercise 2.6: Mixed-Helicity Four-Point Scalar-Gluon Amplitude
Exercise 2.7: Conformal Algebra
Exercise 2.8: Inversion and Special Conformal Transformations
Exercise 2.9: Kinematical Jacobi Identity
Exercise 2.10: Five-Point KLT Relation
Exercise 3.1: The Four-Gluon Amplitude in N=4 Super-Symmetric Yang-Mills Theory
Exercise 3.2: Quadruple Cuts of Five-Gluon MHV Scattering Amplitudes
Exercise 3.3: Tensor Decomposition of the Bubble Integral
Exercise 3.4: Spurious Loop-Momentum Space for the Box Integral
Exercise 3.5: Reducibility of the Pentagon in Four Dimensions
Exercise 3.6: Parametrising the Bubble Integrand
Exercise 3.7: Dimension-Shifting Relation at One Loop
Exercise 3.8: Projecting Out the Triangle Coefficients
Exercise 3.9: Rank-One Triangle Reduction with Direct Extraction
Exercise 3.10: Momentum-Twistor Parametrisations
Exercise 4.1: The Massless Bubble Integral
Exercise 4.2: Feynman Parametrisation
Exercise 4.3: Taylor Series of the Log-Gamma Function
Exercise 4.4: Finite Two-Dimensional Bubble Integral
Exercise 4.5: Laurent Expansion of the Gamma Function
Exercise 4.6: Massless One-Loop Box with Mellin-Barnes Parametrisation
Exercise 4.7: Discontinuities
Exercise 4.8: The Symbol of a Transcendental Function
Exercise 4.9: Symbol Basis and Weight-Two Identities
Exercise 4.10: Simplifying Functions Using the Symbol
Exercise 4.11: The Massless Two-Loop Kite Integral
Exercise 4.13: ``d log'' Form of the Massive Bubble Integrand with D=2
Exercise 4.14: An Integrand with Double Poles: The Two-Loop Kite in D=4
Exercise 4.16: The Box Integrals with the Differential Equations Method
References
A Conventions and Useful Formulae
Reference
π SIMILAR VOLUMES
<p><p>At the fundamental level, the interactions of elementary particles are described by quantum gauge field theory. The quantitative implications of these interactions are captured by scattering amplitudes, traditionally computed using Feynman diagrams. In the past decade tremendous progress has b
Providing a comprehensive, pedagogical introduction to scattering amplitudes in gauge theory and gravity, this book is ideal for graduate students and researchers. It offers a smooth transition from basic knowledge of quantum field theory to the frontier of modern research. Building on basic quantum
<p>Axiomatic and constructive approaches to quantum field theory first aim to establish it on precise, non-perturbative bases: general axioms and rigorous definition of specific theories respectively. From the viewpoint of particle physics, the goal is then to develop a relativistic scattering theor
<br> <p>Axiomatic and constructive approaches to quantum field theory first aim to establish it on precise, non-perturbative bases: general axioms and rigorous definition of specific theories respectively. From the viewpoint of particle physics, the goal is then to develop a relativistic scatteri
Leonard Parker is a Distinguished Professor of physics and director of the Center for Gravitation and Cosmology at the University of Wisconsin-Milwaukee. He is basically the founder of the study of quantum field theory in curved space-time. His has work formed the basis of research by hundreds of ph