Scattering amplitudes in quantum field theory
This open access book bridges a gap between introductory Quantum Field Theory (QFT) courses and state-of-the-art research in scattering amplitudes. It covers the path from basic definitions of QFT to amplitudes, which are relevant for processes in the Standard Model of particle physics. The book beg...
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | eBook Book |
| Language: | English |
| Published: |
Cham
Springer
2024
Springer Nature Springer International Publishing AG |
| Edition: | 1 |
| Series: | Lecture Notes in Physics |
| Subjects: | |
| ISBN: | 3031469860, 9783031469862, 9783031469879, 3031469879 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- Intro -- Preface -- Acknowledgements -- Contents -- Acronyms -- 1 Introduction and Foundations -- 1.1 Poincaré 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=4-2ε 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 ε -- 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
- 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
- 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