GROUP THEORY IN PARTICLE, NUCLEAR, AND HADRON PHYSICS www.com GROUP THEORY IN PARTICLE, NUCLEAR, AND HADRON PHYSICS SYED AFSAR ABBAS c& CRC PressTaylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business A C H A P M A N & HALL BOOK www.com CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U. Government works Printed on acid-free paper Version Date: 20160222 International Standard Book Number-13: 978-1-4987-0466-3 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained.
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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Names: Abbas, Syed Afsar, 1946- author. Title: Group theory in particle, nuclear, and hadron physics / Syed Afsar Abbas. Description: Boca Raton, FL : CRC Press, Taylor & Francis Group, [2016] | ©2016 | Includes bibliographical references and index.
Identifiers: LCCN 2016006097| ISBN 9781498704663 (hardback ; alk. paper) | ISBN 1498704662 (hardback ; alk. paper) Subjects: LCSH: Group theory. | Particles (Nuclear physics) | Nuclear physics.
Classification: LCC QC20.701/5122--dc23 LC record available at http://lccn.gov/2016006097 Visit the Taylor & Francis Web site at http://www.com and the CRC Press Web site at http://www.com To Ratna www.com Contents List of Figures xi List of Tables xiii Preface xv Author xvii 1 Basic Symmetry Concepts 1 1.2 Elementary Concepts of Symmetry .3 Kepler’s Construction: A Mistake? .4 Symmetries and Conservation Laws .5 Appendix A: Mathematics as the Language of Nature .6 Solutions of Problems .1 Definitions and Examples .2 Subgroups and Cyclic Groups .3 Cosets and Normal Subgroups .5 Homomorphism and Isomorphism .6 Torsion Group and Betti Number .8 Solutions of Problems .1 Lie Groups and Lie Algebras .2 Some Specific Lie Groups .1 SO(2): Special Orthogonal Group in Two Dimensions 73 3.2 SO(3): Special Orthogonal Group in Three Dimensions 76 3.3 SU(2): Special Unitary Group in Two Dimensions .4 SU(3): The Special Unitary Group in Three Dimesions 80 3.com viii Contents 3.4 Adjoint Representations of SU(2) .5 Cartan Subalgebra and Roots/Weight Space .6 Fractional Charges in SU(n) .7 Appendix C: Linear Vector Space .8 Solutions of Problems .1 Indistinguishable Particles in Quantum Mechanics .2 Transpositions and Permutations .3 Disjoint Cycles and Signatures .6 Classes and Partitions .7 Solutions of Problems .1 Symmetries and Young Diagrams .3 Young Diagram and Unitary Symmetry .4 Product of Irreducible Representations .5 Multiplets of the SU (n − 1) Subgroup of SU (n) .6 The Reduction SU (m + n) → SU (m) ⊗ SU (n) .7 Reduction of SU (mn) → SU (m) ⊗ SU (n) .8 Coefficients of Fractional Parentage and Isoscalar Factors .9 Appendix D: Representation Theory .10 Appendix E: Symmetry in Quantum Mechanics .11 Solutions of Problems .1 Why the SU (3)c Group? .2 QCD Langrangian and Asymptotic Freedom .3 Colour Singlet States and Confinement .5 Large Nc Colour QCD .6 Solutions of Problems .1 Current and Constituent Quarks .2 The Eightfold Way Model .3 SU (3)F Flavour Model .4 Quark Model Calculations .5 SU (6)SF Model .com Contents ix 7.6 Solutions of Problems .2 Confinement in a Spherically Static Bag .3 MIT Bag Model .4 Finite Mass Quarks in a Bag .2 Axial Vector Coupling Constant .3 Spin Structure of the Nucleon .5 Scalar and Vector Confining Potentials .6 Solutions of Problems .1 Harmonic Oscillator Model .6 Spin of Nucleon in the Quark Model .1 Spin of a Deformed Nucleon .2 Spin of Nucleon with Configuration Mixed Wave Func- tion .7 Colour Confinement in QCD and Deformed Baryons .8 Solutions of Problems. 359 10 Glashow-Salam-Weinberg Model 367 10.1 Its Chiral and Non-Chiral Structure .2 Spontaneous Symmetry Breaking .5 Second Generation Fermions .6 Right-Handed Neutrino .7 Gauge Bosons and Neutral Currents .9 Appendix F: Lorentz and Poincare Groups .10Solutions of Problems. 407 11 Symmetry in Nuclei 411 11.1 Isospsin Symmetry in Nuclei .2 SU (4) Symmetry and Saturation in Nuclei .3 Distinguishable Protons and Neutrons in Nuclei .4 Gamow-Teller Strengths in Nuclei .5 SU (2)A Nusospin Symmetry in Nuclei .6 Quantum Groups in Nuclei. 431 12 Quarks in Nuclei 437 12.1 The EMC Effect and the Nucleus .2 Hidden Colour in Multiquarks in Nuclei .3 Quarks in A=3 Nuclei .4 ∆ Excitations in the Nucleus .5 M1 Strength in Nuclei .6 Gamow-Teller (GT) Strength in Nuclei .7 Solutions of Problems.
458 13 Quark Gluon Plasma(QGP) 463 13.1 Basics for QGP .2 Finite Lie Group Transformations .3 Group Characters of the Lie Group .4 Measure Function of SU(n) .5 Symmetries and Partition Functions .6 Colour Singlet and Coloured QGP States .7 Solutions of Problems. 488 14 Topology for Hadrons 493 14.3 Linear and Non-Linear Sigma Models .5 SU(3) Adjoint Representation Skyrme Model .6 Appendix G: Introduction to Topology .7 Solutions of Problems. 513 Bibliography 515 Index 523 www.com List of Figures 1.1 Reflection symmetry of a neck-tie and symmetry of an equi- lateral triangle under rotation .2 Rotation and reflection symmetry of a dumbbell and an irreg- ular figure .3 Symmetry of cyclic groups C4 (a), C3 (b), C2 (c), C1 (d) .4 Rotation and reflection symmetry of the dihedral group DN 6 1.5 The five Platonic solids .6 Two identical particles with a two–body potential .1 Symmetries of an equilateral triangle and a square .2 Lattice diagram of the subgroups of the groups Z4 , V and D3 30 2.3 Lattice diagram showing the subgroup structure of D4 .1 Rotation of a two-dimensional vector – SO(2) group .2 Root-vector representation of the adjoint representation of SU (2) .3 Weight vectors of the j-representation of SU (2) .4 Weights of the baryon octet states in SU (3) .5 Roots of the baryon octet states in SU (3) .6 Passive transformation on a two-dimensional vector .1 Regular tetrahedron symmetry as isomorphic to the symmet- ric group S4 .1 Schematic plot of the ratio R which proves the presence of three colours .1 Infinite momentum frame in deep inelastic scattering .2 Spin 1/2 baryons plus pseudoscalar and vector meson octets 243 7.3 Spin 3/2 baryon decuptet .1 Jacobi coordinates for three-dimensional space .2 Baryon spectrum under the lowest-order perturbation theory 320 9.3 Locations of the three quarks at the positions 1, 2 and 3 .com xii List of Figures 11.1 One and two triton separation energies as a function of triton number .1 Experimentally determined central hole in the charge density distribution of A=3 and A=4 nuclei and the standard theo- retical expectation .1 Schematic picture of high energy heavy ion collisons to create QGP .2 Weight diagram of the triplet, antitriplet, and octet represen- tations used to calculate partition functions .3 Def f for singlet, octet, and 27-plet representation for two flavours .1 Topological mapping and winding numbers .2 Open interval on a real number line .3 An open set on a real number line .4 An open ball .5 Path-connected points .7 Square and circle topology .8 Euler number for a cube and a tetrahedron .9 Tetrahedron on a sphere .10 Betti number of networks .11 Euler number of S 2 and a torus .com List of Tables 1.1 Invariance and conservation laws connection .1 Composition table of the cube roots of unity .2 Composition table of the cyclic group C3 .3 Composition table for the group D3 .4 Reflections and rotations as blocks in the D3 group .5 Composition table of D4 .6 Composition table of addition modulo 4 .7 Viergruppe V or 4-group .8 Factor group of D3 .9 C2 and Z2 composition table .11 Order three group .12 Composition table of square roots of unity .13 Composition table of fourth roots of unity .14 Z6 the addition modulo 6 composition table .15 Left cosets of {0,3} of Z6 .17 Factor group of V .18 Factor group of D4 .19 Blocked composition table of D4 .1 Fractional charges of h-electrons in the groups SU (2)h2 to SU (4)h4 as in FQHE .1 Composition table for the symmetric group S3 .2 Number of elements in classes of S4 and S5 .1 Hidden colour components in the multiquark systems .1 Parallel structure of 12 baryons and 0− mesons leading to the eightfold way model .2 The complete set of SU(3) commutation relations .3 Mixed symmetric wave functions for the spin-1/2 octet baryons .com xiv List of Tables 7.4 Magnetic moments of spin-1/2 baryon octet in the quark model .1 Eigenvalues in a central potential .2 Experiment [E142] compared to predictions of MIT bag model (for massless quarks w0 = 2.04 and for m 6= 0, ER=2.01); the mR → ∞ case corresponds to non-relativistic quark model; best configuration mixed cases (1) and (2); and the deformed nucleon model case .1 Baryon space wave functions in the Harmonic Oscillator Model .2 Symmetric SU (6)F S wave functions for N and ∆, octet and decuptet only for spins, J = 12 and 32 , respectively .3 Semi-leptonic decays of baryons with deformation (only single parameter for all these fits) .4 Results for u↑↓ and d↑↓ terms arising from different parts of the configuration mixed quark model wave functions .5 Polarized structure function values of different quantities – the experimental values (E); the configuration mixed quark model values (1 and 2, Equation 9.151); and the deformed nucleon results (D) .1 Inter-triton cluster bond energies of neutron-rich nuclei .2 Experimental and theoretical data for Superdeformed Bands in 194 Hg(2) compared with our results .3 Supersymmetry in nucleus – fitting of identical Superde- formed Bands in the neighbouring nuclei, 152 Dy and 151 T b∗ 436 12.1 Colour singlet components in 6-, 9- and 12-quark systems - the rest are hidden colour .com Preface This book is primarily designed for those who are embarking on a research career in particle, nuclear or hadron physics. The single-minded goal is to present group theory in the manner it is practically applied in these disci- plines in contemporary research.
Group theory is presented here in a simple, consistent and unified fashion, so that the field does not appear too threaten- ing to the novice. However, sufficient advanced material has also been supplied in this work, so as to provide a source of valuable information and present tan- talizing challenges to the experienced researcher as well. Though the book is targeted at theorists, the direct and lucid approach adopted throughout may very well render it accessible to experimentalists too. A unique feature of this book is the large number of solved problems, in addition to the few unsolved ones.