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Sigma model and applications

Sigma model and applications. 1. The linear sigma model (& NJL model) 2. Chiral perturbation 3. Applications. Summary. • The law of the nature is simple . • But spontaneous symmetry breaking must occur, which brings us varieties . • For hadrons it also generates mass .

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Sigma model and applications

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  1. Sigma model and applications 1. The linear sigma model (& NJL model) 2. Chiral perturbation 3. Applications

  2. Summary • The law of the nature is simple. • But spontaneous symmetry breaking must occur, which brings us varieties. • For hadrons it also generates mass. • SSB induces collective (Nambu-Goldstone) mode (Pion) which governs the low energy dynamics of the broken world. => Chiral dynamics • The chiral dynamics can be extended to resonance physics • This might predict new form of hadronic matter.

  3. 1. The linear sigma model Chirality Left Right c c Velocity Spin

  4. Lagrangian The simplest that contain all the essences. Potential: Nucleon (or quark) Scalar meson (sigma) Pseudoscalar meson (pion) Chiral mesons • The Lagrangian has chiral symmetry U(1)L x U(1)R • Before SSB: the fermion is massless • After SSB: the fermion obtains a finite mass the pion becomes massless Structure of The vacuum

  5. are eigenstates of (chirality) Left and Right

  6. U(1)LxU(1)RChiral symmetry

  7. U(1)LxU(1)RChiral symmetry Kinetic term is invariant under U(1)L x U(1)R

  8. U(1)LxU(1)RChiral symmetry Kinetic term is invariant under U(1)L x U(1)R The chiral mesons transform The total Lagrangian is invariant under U(1)L x U(1)R

  9. The ground state: For f2 < 0 The minimum energy configuration

  10. The ground state: For f2 < 0 The minimum energy configuration

  11. The ground state: For f2 < 0 The minimum energy configuration Minimum energy (density) is given by

  12. Invariance of the vacuum Minimum energy (density) is given by This vacuum is invariant under the chiral transformation This corresponds to Translation causes nothing Uniform density

  13. The ground state: For f2 > 0 Minimum energy (density) is given by an any point on the circle This vacuum is not invariant under the chiral transformation This corresponds to Translation changes the location of the cluster Localize Clusterize

  14. Symmetry nature is determined by the parameter f A microscopic model is needed to determine f => Nambu (NJL) model Liniear sigma model <=> Ginzburg-Landau model NJL model <=> BCS model Attractive interaction causes the instability of the ground state Cooperative phenomena of infinitely many-body systems

  15. Particle properties Fluctuations around the vacuum For f2 < 0 Masses: Degenerate between P=+,- particles

  16. For f2 < 0 The Goldberger-Treiman relation Masses: Fermion acquires a mass, and the pion becomes massless Nambu-Goldstone theorem

  17. Two modes in the broken phase Massive mode Massless mode They correspond to

  18. 2. Chiral perturbation At low energy, massless modes dominates => implies the use of an anglevariable The constraint

  19. Nonlinear model Also introduce

  20. Pion interactions All pion terms contan derivatives ~ momentum Small momentum can be a small expansion parameter => Chiral perturbation theory • NN vertex •  interaction (needs isospin) in

  21. b u c t s d e m t nm nt ne 3. Application Force Matter gluon Quarks photon, g weak boson, W, Z Leptons graviton

  22. Octet mesons and baryons Their interactions are dictated by chiral symmetry and may reproduce resonances => New type of hadrons ~ hadron mokecules

  23. Observation of exotic hadron resonances Θ+, N*(1670), Λ(1405), …, X(3872), Z+(4430), etc Pentaquarks Hadronic molecule Tetraquarks Key question: What multiquark configurations are possible? Meson-baryon molecule Diquark Triquark Colorless correlation Colored correlation

  24. (1405) … … Spin, parity; 1/2– l = 1 l = 0 • Quark model ~ uds, one of them is in p-state • But this state is the lightest among the family of 1/2– • Also small LS splitting with (1520) It could be KN (hadron-hadron) molecule, a new form of matter

  25. Solving the LS equation SU(3) (flavor) extension of this Lagrangian Coefficients of the WT interaction Large attraction

  26. V G V V G V G V V T-matrix Two ingredients Vand G V: Chiral interaction (Weinberg-Tomozawa) G: 1/(E – H0) m = + T M + … +

  27. Poles on the complex energy plane • Two states near that energy? • Molecular like structure; the new form of hadrons

  28. Summary • The law of the nature is simple. • But spontaneous symmetry breaking must occur, which brings us varieties. • For hadrons it also generates mass. • SSB induces collective (Nambu-Goldstone) mode (Pion) which governs the low energy dynamics of the broken world. => Chiral dynamics • The chiral dynamics can be extended to resonance physics • This might predict new form of hadronic matter.

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