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. 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.

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

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

Left and Right are eigenstates of (chirality)

U(1) L xU(1) R Chiral symmetry

Kinetic term is invariant under U(1) L x U(1) R

U(1) L xU(1) R Chiral 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

The ground state: For f 2 < 0 The minimum energy configuration

The ground state: For f 2 < 0 The minimum energy configuration

The ground state: For f 2 < 0 The minimum energy configuration Minimum energy (density) is given by

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

The ground state: For f 2 > 0 Minimum energy (density) is given by an any point on the circle This vacuum is not invariant under the chiral transformation Localize Clusterize Translation changes the location of the cluster This corresponds to

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

Particle properties Fluctuations around the vacuum For f 2 < 0 Masses: Degenerate between P=+,- particles

For f 2 < 0 Masses: Fermion acquires a mass, and the pion becomes massless Nambu-Goldstone theorem The Goldberger-Treiman relation

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

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

Nonlinear model Also introduce

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

3. Application Force photon,  weak boson, W, Z gluon graviton u d c s b t e     e Quarks Leptons Matter

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

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

 (1405) 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) Spin, parity; 1/2 – l = 0 l = 1 … … It could be KN (hadron-hadron) molecule, a new form of matter

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

T-matrix = T V V V G V V V G G … m M Two ingredients V and G V: Chiral interaction (Weinberg-Tomozawa) G: 1/(E – H 0 )

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

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.