Scattering Amplitudes in Quantum Field Theory.

Scattering Amplitudes in Quantum Field Theory.

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Bibliographic Details
Superior document:Lecture Notes in Physics Series ; v.1021
:
Badger, Simon.
TeilnehmendeR:
Henn, Johannes M.
Plefka, Jan C.
Zoia, Simone.
Place / Publishing House:Cham : : Springer International Publishing AG,, 2024.
©2024.
Year of Publication:2024
Edition:First edition.
Language:English
Series:Lecture notes in physics
Physical Description:1 online resource (312 pages)
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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.
  • 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.

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