Skyrmions – A Theory of Nuclei surveys 60 years of research into the brilliant and imaginative idea of Tony Skyrme that atomic nuclei can be modelled as Skyrmions, topologically stable states in an effective quantum field theory of pions. Skyrme theory emerges as a low-energy approximation to the more fundamental theory of quarks and gluons – quantum chromodynamics (QCD). Skyrmions give spatial structure to the protons and neutrons inside nuclei, and capture the interactions of these basic particles, allowing them to partially merge. Skyrme theory also gives a topological explanation for the conservation of baryon number, a fundamental principle of physics. The book summarises the particle and field theory background, then presents Skyrme field theory together with the mathematics needed to understand it. Many beautiful and surprisingly symmetric Skyrmions are described and illustrated in colour. Quantized Skyrmion motion models the momentum, energy and spin of nuclei, and also their isospin, the quantum number distinguishing protons and neutrons. Skyrmion vibrations also need to be quantized, and the book reviews how the complicated energy spectra of several nuclei, including Carbon-12 and Oxygen-16, are accurately modelled by rotational/vibrational states of Skyrmions. A later chapter explores variants of Skyrme theory, incorporating mesons heavier than pions, and extending the basic theory to include particles like kaons that contain strange quarks. The final chapter introduces the Sakai–Sugimoto model, which relates Skyrmions to gauge theory instantons in a higher-dimensional framework inspired by string theory.
Author(s): Nicholas S. Manton
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 321
City: London
Contents
Foreword
Preface
1. Introduction
2. Fields and Particles
2.1 The Classical Notions of Particles and Fields
2.1.1 Quantum theory
2.2 Quantum Field Theory and the Standard Model
2.3 Pions and Nucleons
2.4 Nuclei and Isospin
2.5 Effective Field Theory
2.6 Skyrme’s Effective Field Theory
3. Lagrangians and Symmetries
3.1 Finite-Dimensional Systems
3.2 Symmetries and Conservation Laws
3.3 Lagrangian Field Theory
3.4 Noether’s Theorem in Field Theory
4. Skyrme Theory
4.1 SU(2)
4.2 The Skyrme Lagrangian and Field Equation
4.3 Skyrme Field Topology
4.3.1 Topological degree of a map
4.4 Skyrme Field Energy
4.4.1 Elastic strain formulation
4.5 Hedgehog Skyrmions
4.6 Visualising Skyrmions
4.7 Asymptotic Interactions of Hedgehogs
4.8 Adding a Pion Mass Term
5. Quantization of Skyrmions
5.1 Quantization of Skyrme Fields
5.2 Topology and Quantization
5.3 Rigid-Body Quantization
5.4 Quantized Hedgehog Skyrmion – Proton and Neutron
5.5 Classical Interpretation of Quantized Skyrmion States
5.6 The Need for Fermionic Quantization
6. Skyrmions with Higher B – Massless Pions
6.1 Skyrmions with Baryon Numbers B ≤ 8
6.2 The Rational Map Ansatz
6.3 Symmetric Rational Maps
6.3.1 Platonic symmetries
6.4 Skyrmions from Rational Maps
6.5 Skyrmions up to B = 22
6.6 Rigorous Investigation of Skyrmions
6.7 The Skyrmion Crystal
7. Rigid-Body Skyrmion Quantization
7.1 Collective Coordinate Quantization
7.2 Rational Maps and Finkelstein–Rubinstein Signs
7.3 Parity of States
7.4 The Quantized Toroidal B = 2 Skyrmion
7.5 The Quantized B = 3 Skyrmion
7.6 The B = 4 Skyrmion and the α-Particle
7.7 B = 5 and B = 7
7.8 Quantization of the B = 6 Skyrmion
8. Skyrmions with Higher B – Massive Pions
8.1 Massive Pions
8.2 The Double Rational Map Ansatz
8.3 Skyrmions from B = 8 to B = 32
8.3.1 B = 8
8.3.2 B = 12
8.3.3 B = 16
8.3.4 B = 24 and B = 32
8.4 Geometrical Construction of Rational Maps
8.4.1 B = 24 to B = 31 solutions by corner cutting
8.5 Rational Maps with Oh and Td Symmetry
8.6 Skyrmions up to Baryon Number 256
8.7 Summary
9. Quantized Skyrmions with Even B ≤ 12
9.1 Masses, Charge Radii and Calibration
9.2 B = 4
9.3 B = 6
9.4 B = 8
9.5 B = 10
9.6 B = 12
9.6.1 Quantizing the D3h-symmetric Skyrmion
9.6.2 Comparison with experimental data
9.6.3 The chain Skyrmion and the Hoyle band
9.6.4 Matter radii
10. Skyrmion Deformations and Vibrations
10.1 The Need to Consider Vibrations
10.2 The α-Particle and its Vibrational Excitations
10.3 Deformations of the B = 7 Skyrmion
10.4 B = 12 Skyrmion Deformations and Carbon-12
10.4.1 A multiphonon model for Carbon-12
11. Modelling Oxygen-16
11.1 Introduction
11.2 The Vibrational E-Manifold
11.2.1 The Hamiltonian and quantum states
11.3 E-Manifold States
11.3.1 E-phonons
11.3.2 Rovibrational states
11.4 Beyond E-Vibrations
11.5 The Complete Oxygen-16 Energy Spectrum
12. Modelling Calcium-40
12.1 Tetrahedral Structure of Calcium-40
12.2 Rovibrational Bands
12.3 Interpreting the Calcium-40 Spectrum
12.4 Summary
13. Electromagnetic Transition Strengths
13.1 B(E2) Transition Strengths in Skyrme Theory
13.1.1 Beryllium-8
13.1.2 Carbon-12 and the Hoyle state
13.1.3 Oxygen-16
13.1.4 Neon-20, Magnesium-24 and Sulphur-32
13.2 Beryllium-12
13.3 Further Transitions
13.4 Summary
14. Variants of Skyrme Theory
14.1 Adding a Sextic Term
14.1.1 The lightly-bound Skyrme model
14.2 The BPS Skyrme Model
14.3 Including Heavier Mesons
14.3.1 Skyrme model with ρ mesons
14.3.2 Skyrme model with ω mesons
14.4 The SU(3) Extension of Skyrme Theory
15. The Sakai–Sugimoto Model
15.1 Skyrmions from Instantons
15.2 An Expansion for 4-Dimensional Gauge Potentials
15.3 Baryon Resonances in the Sakai–Sugimoto Model
Bibliography
Index
About the Author
