Mesons and Baryons

The following sections are included:

• Preface

• Contents

The following sections are included:

• Constituent Quarks, Effective Gluons and Hadrons

• Naive Quark Model
  • Spin-flavour SU(6) symmetry for mesons
  • Low-lying baryons
  • Spin-avour SU(6) symmetry for baryons

• Estimation of Masses of the Constituent Quarks in the Quark Model
  • Magnetic moments of baryons
  • Radiative meson decays V → P + γ
  • Empirical mass formulae

• Light Quarks and Highly Excited Hadrons
  • Hadron systematisation
  • Diquarks

• Scalar and Tensor Glueballs
  • Low-lying σ-meson

• High Energies: The Manifestation of the Two- and Three-Quark Structure of Low-Lying Mesons and Baryons
  • Ratios of total cross sections in nucleon-nucleon and pion-nucleon collisions
  • Diffraction cone slopes in elastic nucleon-nucleon and pion-nucleon diffraction cross sections
  • Multiplicities of secondary hadrons in e⁺e⁻ and hadron-hadron collisions
  • Multiplicities of secondary hadrons in πA and pA collisions
  • Momentum fraction carried by quarks at moderately high energies

• Constituent Quarks, QCD-Quarks, QCD-Gluons and the Parton Structure of Hadrons
  • Moderately high energies and constituent quarks
  • Hadron collisions at superhigh energies

• Appendix 1.A: Metrics and SU(N) Groups
  • Metrics
  • SU(N) groups

• References

The following sections are included:

• Classification of Mesons in the (n,M²) Plane
  • Kaon states

• Trajectories on (J,M²) Plane
  • Kaon trajectories on (J,M²) plane

• Assignment of Mesons to Nonets

• Baryon Classiffication on (n,M²) and (J,M²) Planes

• Assignment of Baryons to Multiplets

• Sectors of the 2⁺⁺ and 0⁺⁺ Mesons — Observation of Glueballs
  • Tensor mesons
    • Systematisation of tensor mesons on the (n,M²) trajectories
    • Mixing of the quarkonium and gluonium states
    • Quark combinatorial relations for decay constants
    • Sum rules for decay couplings
    • The broad state f₂(2000): the tensor glueball
    • Ratios between coupling constants of f₂(2000) → π⁰π⁰, ηη, ηη′ as indication of the glueball nature of this state
    • Mixing of the glueball with neighbouring -resonances
    • The content of f₂-mesons, observed in the reactions , ηη, ηη′
    • The analysis of the quarkonium-gluonium contents of the f₂(1920), f₂(2020), f₂(2240), f₂(2300)
    • The content of the -mesons
    • The broad f₂(2000) as a glueball state
  • Scalar states
    • Overlapping of f₀-resonances in the mass region 1200-1700 MeV: accumulation of widths of the states by the glueball
    • The (n,M²) plot for scalar-isoscalar states and the glueball
    • Hadronic decays and estimation of the quark-gluonium content of the f₀ resonances
    • The f₀(980) and a₀(980): are they the quark-antiquark states?
    • The light σ-meson: Is there a pole of the 00⁺⁺-wave amplitude?
    • The σ as the white component of the confinement singularity

• References

The following sections are included:

• Scattering in Quantum Mechanics
  • Schrödinger equation and the wave function of two scattering particles
  • Scattering process
  • Free motion: plane waves and spherical waves
  • Scattering process: cross section, partial wave expansion and phase shifts
  • K-matrix representation, scattering length approximation and the Breit-Wigner resonances
  • Scattering with absorption

• Analytical Properties of the Amplitudes
  • Propagator function in quantum mechanics: the coordinate representation
  • Propagator function in quantum mechanics:the momentum representation
  • Equation for the scattering amplitude f(k,p)
  • Propagators in the description of the two-particle scattering amplitude
  • Relativistic propagator for a free particle
  • Mandelstam plane
  • Dalitz plot

• Dispersion Relation N/D-Method and Bethe-Salpeter Equation
  • N/D-method for the one-channel scattering amplitude of spinless particles
  • N/D-amplitude and K-matrix
  • Dispersion relation representation and light-cone variables
  • Bethe-Salpeter equations in the momentum representation
  • Spectral integral equation with separable kernel in the dispersion relation technique
  • Composite system wave function, its normalization condition and additive model for form factors
    • Separable interaction in the N/D method and the prospects of its application to the calculation of radiative decays

• The Matrix of Propagators
  • The mixing of two unstable states
  • The case of many overlapping resonances: construction of propagator matrices
  • A complete overlap of resonances: the effect of accumulation of widths by a resonance

• K-Matrix Approach
  • One-channel amplitude
  • Multichannel amplitude
  • The problem of short and large distances
  • Overlapping resonances: broad locking states and their role in the formation of the confinement barrier
    • Accumulation of widths in the K-matrix approach

• Elastic and Quasi-Elastic Meson-Meson Reactions
  • Pion exchange reactions
  • Regge pole propagators

• Appendix 3.A: The f₀(980) in Two-Particle and Production Processes

• Appendix 3.B: K-Matrix Analyses of the (IJ^PC = 00⁺⁺)-Wave Partial Amplitude for Reactions ππ → ππ, , ηη, ηη′, ππππ
  • Scattering amplitude
  • The fitting procedure
  • Description of data and the results for the 00⁺⁺-wave
  • Bare f₀-states and resonances
  • f₀-resonances: masses, decay couplings and partial widths

• Appendix 3.C: The K-Matrix Analyses of the ()-Wave Partial Amplitude for Reaction πK → πK

• Appendix 3.D: The Low-Mass σ-Meson
  • Dispersion relation solution for the ππ-scattering amplitude below 900 MeV

• Appendix 3.E: Cross Sections and Amplitude Discontinuities
  • Exclusive and inclusive cross sections
  • Amplitude discontinuities and unitary condition

• References

The following sections are included:

• Two-Baryon States and Their Scattering Amplitudes
  • Spin-1/2 wave functions
  • Baryon-antibaryon scattering
    • Baryons with isospin I = 0
    • Nucleon-antinucleon scattering amplitude
  • Baryon–baryon scattering
    • The pΛ → pΛ cattering amplitude
    • Amplitude for ΛΛ → ΛΛ scattering
    • Nucleon-nucleon scattering amplitude
  • Unitarity conditions and K-matrix representations of the baryon-antibaryon and baryon-baryon scattering amplitudes
    • scattering
    • ΛΛ scattering
    • The K-matrix representation for nucleon-antinucleon scattering amplitude
    • The K-matrix representation for the nucleon-nucleon scattering amplitude
  • Nucleon-nucleon scattering amplitude in the dispersion relation technique with separable vertices
    • The S = 0, L = 0 partial wave amplitudes
    • Generalisation for S = 0 and arbitrary L = J
    • Two-channel amplitude with S = 1 and J = L ± 1
  • Comments on the spectral integral equation

• Inelastic Processes in Collisions Production of Mesons
  • Reaction two pseudoscalar mesons
  • Reaction

• Inelastic Processes in NN Collisions: the Production of Δ-Resonances
  • Spin- wave functions
    • Wave functions for Δ
    • Wave function for
  • Processes NN → NΔ NNπ. Triangle singularity
    • Pole singularity in the NN → NΔ NNπ reaction
    • The decay width of Δ
    • Triangle-diagram amplitude with pion-nucleon rescattering: the logarithmic singularity
  • The NN → ΔΔ → NNππ process. Box singularity
    • The ΔΔ-production from (NN)_D-state with J^P = 2⁺

• The process with j > 3/2

• NN Scattering Amplitude at Moderately High Energies — the Reggeon Exchanges
  • Reggeon-quark vertices in the two-component spinor technique
  • Four-component spinors and reggeon vertices
    • Scalar vertex
    • Vector reggeon vertex
    • Pseudoscalar reggeon vertex
    • Pseudovector reggeon vertex

• Production of Heavy Particles in the High Energy Hadron-Hadron Collisions: Effects of New Thresholds
  • Impact parameter representation of the scattering amplitude
    • Example: Elastic scattering amplitude for two spinless particles
    • K-matrix representation of the impact parameter amplitude

• Appendix 4.A. Angular Momentum Operators
  • Projection operators and denominators of the boson propagators
  • Useful relations for and

• Appendix 4.B. Vertices for Fermion-Antifermion States
  • Operators for ¹L_J states
  • Operators for ³L_J states with J = L
  • Operators for ³L_J states with L<J and L>J

• Appendix 4.C. Spectral Integral Approach with Separable Vertices: Nucleon-Nucleon Scattering Amplitude NN → NN, Deuteron Form Factors and Photodisintegration and the Reaction NN → NΔ
  • The pp → pp and pn → pn scattering amplitudes
    • Deuteron form factors
    • Deuteron disintegration
    • Reaction NN → NΔ at energies T_N ≤ 1.5 GeV

• Appendix D. NΔ One-Loop Diagrams

• Appendix 4.E. Analysis of the Reactions , ηη, ηη′ Search for f_J-Mesons

• Appendix 4.F. New Thresholds and the Data for ρ = Im A\Re A of the UA4 Collaboration at

• Appendix 4.G. Rescattering Effects in Three-Particle States: Triangle Diagram Singularities and the Schmid Theorem
  • Visual rules for the determination of positions of the triangle-diagram singularities
  • Calculation of the triangle diagram in terms of the dispersion relation N/D-method
  • The Breit-Wigner pole and triangle diagrams: interference effects

• Appendix 4.H. Excited Nucleon States N(1440) and N(1710) — Position of Singularities in the Complex-M Plane

• References

The following sections are included:

• Production and Decay of Baryon States
  • The classification of the baryon states
  • The photon and baryon wave functions
    • The photon projection operator
    • Baryon projection operators
  • Pion-nucleon and photon-nucleon vertices
    • πN vertices
    • πN scattering
  • Photon-nucleon vertices
    • Operators for a photon-nucleon system
    • Operators for γ^*N states with
    • Operators for 1/2⁺,3/2⁻,5/2⁺,… states
    • Vertices for γN states

• Single Meson Photoproduction
  • Photoproduction amplitudes for 1/2⁻,3/2⁺,5/2⁻,… states
  • Photoproduction amplitudes for 1/2⁺,3/2⁻,5/2⁺,… states
  • Relations between the amplitudes in the spin-orbit and helicity representation

• The Decay of Baryons into a Pseudoscalar Particle and a 3/2 State
  • Operators for ‘+’ states
  • Operators for 1/2⁺,3/2⁻,5/2⁺,… states
  • Operators for the decays J⁺ → 0⁻ + 3/2⁺, J⁺ → 0⁺ + 3/2⁻, J⁻ → 0⁺ + 3/2⁺, J⁻ → 0⁻ + 3/2⁻

• Double Pion Photoproduction Amplitudes
  • Amplitudes for baryons states decaying into a 1/2 state and a pion
  • Photoproduction amplitudes for baryon states decaying into a 3/2 state and a pseudoscalar meson

• πN and γN Partial Widths of Baryon Resonances
  • πN partial widths of baryon resonances
  • The γN widths and helicity amplitudes
    • The ‘+’ states
    • The ‘−’ states
  • Three-body partial widths of the baryon resonances
  • Miniconclusion

• Photoproduction of Baryons Decaying into Nπ and Nη
  • The experimental situation — an overview
    • Parametrisations of amplitudes
  • Fits to the data
    • Fit to the pη channel

• Hyperon Photoproduction γp → ΛK⁺ and γp → ΣK⁺

• Analyses of γp → π⁰π⁰p and γp → π⁰ηp Reactions

• Summary

• Appendix 5.A. Legendre Polynomials and Convolutions of Angular Momentum Operators
  • Some properties of Legendre polynomials
  • Convolutions of angular momentum operators

• Appendix 5.B: Cross Sections and Partial Widths for the Breit-Wigner Resonance Amplitudes
  • The Breit-Wigner resonance and rescattering of particles in the resonance state
  • Blatt-Weisskopf form factors

• Appendix 5.C. Multipoles

• References

The following sections are included:

• Three-Particle Production at Intermediate Energies
  • Isobar model
  • The (J^P = 0⁻)-state → P₁P₂P₃ transition
  • The transition for an arbitrary spin state
• Dispersion integral equation for a three-body system
  • Two-particle interactions in the 0⁻-state → P₁P₂P₃ decay
  • Dispersion relation equations for a three-body system with resonance interaction in the two-particle states of the outgoing hadrons
• Description of the three-meson production in the K-matrix approach
  • Resonances in the scattering amplitude P₁P₂ → P₁P₂: spectral integral representation and the K-matrix approach
  • Three-meson production in the K-matrix approach
  • Results of the K-matrix fit of annihilation reactions at rest , into πππ, ππη, πηη,

• Meson-Nucleon Collisions at High Energies: Peripheral Two-Meson Production in Terms of Reggeon Exchanges
  • Reggeon exchange technique and the K-matrix analysis of meson spectra in the waves J^PC = 0⁺⁺, 1⁻⁻, 2⁺⁺, 3⁻⁻, 4⁺⁺ in high energy reactions πN → two mesons + N
    • The t-channel exchanges of pion trajectories in the reaction π⁻p → ππn
    • Amplitudes with a₁-trajectory exchanges
    • reaction with -exchange by ρ-meson trajectories
  • Results of the K-matrix fit of two-meson systems produced in the peripheral productions
    • The basic formulae
    • Fit to the data

• Appendix 6.A. Three-meson production , ππη, πηη

• Appendix 6.B. Reggeon Exchanges in the Two-Meson Production Reactions — Calculation Routine and Some Useful Relations

• Reggeised pion exchanges
  • Calculation routine for the reggeised pion exchange
  • Calculations related to the expansion of the differential cross section πp → ππ + N over spherical functions for the reggeised π₂-exchange

• References

The following sections are included:

• A System of Two Vector Particles
  • General structure of spin-orbital operators for the system of two vector mesons
  • Transitions γ^*γ^* → hadrons
    • Structure of the spin-orbital operators for a γ^*(q₁)γ^*(q₂) system when ,
    • The production of the nucleon-antinucleon pair
    • The production of two pseudoscalar mesons γ^*(q₁)γ^* (q₂) → P₁(k₁)P₂(k₂)
  • Quark structure of meson production processes

• Nilpotent Operators — Production of Scalar States
  • Gauge invariance and orthogonality of the operators
  • Transition amplitude γγ^* → S when one of the photons is real
    • Ambiguities in the representation of the spin operator
    • Analytical properties of the amplitude for the emission of a real photon

• Reaction e⁺e⁻ → γ^* → γππ
  • Analytical structure of amplitudes in the reactions e⁺e⁻ → γ^* → φ → γ(ππ)_s, φ → γf₀ and φ → γ(ππ)_s
    • The amplitude for the transition γ^* → γ(ππ)_S and poles corresponding to subprocesses φ → f⁰ and φ → γ(ππ)_S
    • Example of idealistic description of φ(1020) → γππ
    • Description of the reaction φ(1020) → γ(ππ)_S
  • Decay >φ(1020) → γππ: Non-relativistic quark model calculation of the form factor and the K-matrix consideration of the transition
    • The V → γ^S process in the framework of the non-relativistic quark model
    • Quantum mechanical consideration of the reaction φ → γf₀ with the simplifying assumption of φ and f₀ being stable particles
    • K-matrix calculation of the decay amplitude φ(1020) → γf₀(980)
    • Comparison of data
    • The additive quark model, does it work?
  • Form factors in the additive quark model and confinement
    • Triangle diagrams in non-relativistic approximation
    • Requirement for quark confinement

• Spectral Integral Technique in the Additive Quark Model: Transition Amplitudes and Partial Widths of the Decays
  • Radiative transitions P → μV and S → φV
    • The decay of the pseudoscalar meson P → μ V
    • Decay of the scalar meson S → μ V
    • Normalisation conditions for wave functions of states
  • Transitions T(2⁺⁺) → μV and A(1⁺⁺) → μV
    • Transition
    • Transition A → μV

• Determination of the Quark-Antiquark Component of the Photon Wave Function for u; d; s-Quarks
  • Transition form factors π⁰, η,
  • e⁺e⁻-annihilation
    • Partial decay widths ω, ρ⁰, φ → e⁺e⁻
    • The ratio R(s) = σ(e⁺e⁻ → hadrons)/σ(e⁺e⁻ → μ⁺μ⁻ at energies below the open charm production
  • Photon wave function
  • Transitions S → γγ and T → γγ

• Nucleon Form Factors
  • Quark-nucleon vertex
  • Nucleon form factor — relativistic description
  • Nucleon form factors — non-relativistic calculation

• Appendix 7.A: Pion Charge Form Factor and Pion Wave Function

• Appendix 7.B: Two-Photon Decay of Scalar and Tensor Mesons
  • Decay of scalar mesons
  • Tensor-meson decay amplitudes for the process

• Appendix 7.C: Comments about Efficiency of QCD Sum Rules

• References

The following sections are included:

• Basic Standings in the Consideration of Light Meson Levels in the Framework of the Spectral Integral Equation

• Spectral Integral Equation

• Light Quark Mesons
  • Short-range interactions and confinement
  • Masses and mean radii squared of mesons with L ≤ 4
    • Mesons of the (L = 0) group
    • Mesons of the (L = 1) group
    • Mesons of the (L = 2) group
    • Mesons of the (L = 3) group
    • Mesons of the (L = 4) group
  • Trajectories on the (n,M²) planes

• Radiative decays
  • Wave functions of the quark-antiquark states

• Appendix 8.A: Bottomonium States Found from Spectral Integral Equation and Radiative Transitions
  • Masses of the states
  • Radiative decays
  • The component of the photon wave function and the and -meson→ γγ transitions
    • Determination of the photon vertex
    • Photon-photon decays of -states

• Appendix 8.B: Charmonium States
  • Radiative transitions
  • The component of the photon wave function and two-photon radiative decays

• Appendix 8.C: The Fierz Transformation and the Structure of the t-Channel Exchanges

• Appendix 8.D: Spectral Integral Equation for Composite Systems Built by Spinless Constituents
  • Spectral integral equation for a vertex function with L = 0
    • The spectral integral equation for states with angular momentum L

• Appendix 8.E: Wave Functions in the Sector of the Light Quarks

• Appendix 8.F: How Quarks Escape from the Confinement Trap?

• References

The following sections are included:

• Quark Structure of Mesons and Baryons

• Systematics of the -Mesons and Baryons

• Additive Quark Model, Radiative Decays and Spectral Integral Equation

• Resonances and Their Characteristics

• Exotic States — Glueballs

• White Remnants of the Confinement Singularities

• Quark Escape from Confinement Trap

• References

The following sections are included:

• Index