Relativistic chiral mean field model for nuclear physics (II) Hiroshi Toki Research Center for Nuclear Physics Osaka University

Pion is important !! Yukawa introduced pion as a mediator of nuclear interaction (1934) Meyer-Jensen introduced shell model for finite nuclei (1949) Nambu-Jona-Lasinio introduced chiral symmetry and its breaking for mass and pion generation (1961)

Motivation for the second stage Pion is important in nuclear physics. Pion appears due to chiral symmetry. Particles as nucleon, rho mesons,.. may change their properties in medium. Chiral symmetry may be recovered partially in nucleus. Unification of QCD physics and nuclear physics.

Spontaneous breaking of chiral symmetry Quarks & gluons Hadrons & nuclei Confinement, Mass generation Potential energy surface of the vacuum Chiral order parameter Yoichiro Nambu Hosaka

He was motivated by the BCS theory. Nobel prize (2008) is the order parameter Particle number Chiral symmetry

Nambu-Jona-Lasinio Lagrangian Mean field approximation; Hartree approximation Fermion gets mass. The chiral symmetry is spontaneously broken. Chiral transformation

Chiral condensate is The fermion mass is The mass is similar to the pairing gap in the BCS formalism. The mass generation mechanism for a fermion. m G GcGc

The particle-hole excitation (pion channel): RPA T = K + T K J(q)

The pion mass is zero. Nambu-Goldstone mode has a zero mass. The nucleon gets mass by chiral condensation. There appears a massless boson; pseudo-scalar meson. All the masses of particles are zero at the beginning, but they are generated dynamically. Massless boson appears (Nambu-Goldstone boson) with pseudo-scalar quantum number.

Bosonization (Eguchi:1974) Fermion field is quark Auxiliary fields

Nuclear physics with NJL model Auxiliary fields SU2 chiral transformation Confinement (Polyakov NJL Mode) SU2 c is done SU3 c is not yet done.

Chiral sigma model Linear Sigma Model Lagrangian Polar coordinate Weinberg transformation Pion is the Nambu boson of chiral symmetry

Non-linear sigma model Lagrangian  = f  +  where M = g  f  M* = M + g   m  2 =  2 + f  m  2 =  2 +3 f  m  = g  f  m   =  m  + g   ~~ Free parameters are and (Two parameters) N

Relativistic mean field model (standard) Mean field approximation: Then take only the mean field part, which is just a number. The pion mean field is zero. Hence, the pion contribution is zero in the standard mean field approximation.

Relativistic mean field model (pion condensation) Ogawa, Toki, et al. Brown, Migdal.. Since the pion has pseudo-scalar (0 - ) nature, the parity and charge symmetry are broken. In finite nuclei, we have to project out spin and isospin, which involves a complicated projection. Dirac equation

Mean field approximation for mesons. Nucleons are moving in the mean field and occasionally brought up to high momentum states due to pion exchange interaction h h p p Brueckner argument Relativistic Chiral Mean Field Model (powerful method) Wave function for mesons and nucleons

Why 2p-2h states are necessary for pion (tensor) interaction? G.S. Spin-saturated The spin flipped states are already occupied by other nucleons. Pauli forbidden

Energy minimization with respect to meson and nucleon fields (Mean field equation) Hartree-Fock G-matrix component

Numerical results (1) 4 He 12 C 16 O Ogawa Toki NP 2009 Adjust binding energy and size Tensor Spin-spin Pion Total 12 C

Numerical results 2 The difference between 12 C and 16 O is 3 MeV/N. The difference comes from low pion spin states (J<3). This is the Pauli blocking effect. P 3/2 P 1/2 C O S 1/2 Pion energy Pion tensor provides large attraction to 12 C O C Cumulative Individual contribution

Chiral symmetry Nucleon mass is reduced by 20% due to sigma. We want to work out heavier nuclei for magic number. Spin-orbit splitting should be worked out systematically. Ogawa Toki NP(2009) Not 45% as discussed in RMF model. N One half is from sigma meson and the other half is from the pion.

Nuclear matter Hu Ogawa Toki Phys. Rev E/A Total Pion Total

Deeply bound pionic atom Toki Yamazaki, PL(1988) Predicted to exist Found by GSI Itahashi, Hayano, Yamazaki.. Z. Phys.(1996), PRL(2004) Findings: isovector s-wave

Suzuki, Hayano, Yamazaki.. PRL(2004) Optical model analysis for the deeply bound state.

Summary-2 NJL model provides the linear sigma model. Pion (tensor) is treated within the relativistic chiral mean field model. JJ-magic is produced by pion. Nucleon mass is reduced by 20% Deeply bound pionic atom seems to verify partial recovery of chiral symmetry.

Summary Pion is important in Nuclear Physics. Pion is a Goldstone-Nambu boson of chiral symmetry breaking. By integrating out the quark field with confinement, we can get sigma model Lagrangian. Relativistic chiral mean field model is able to work out the sigma model Lagrangian. We have now a tool to unify the quark picture with the hadron picture and describe nucleus from quarks.

Joint Lecture Groningen-Osaka Spontaneous Breaking of Chiral Symmetry in Hadron Physics 30 Sep 09:00- CEST/16:00- JST Atsushi HOSAKA 07 Oct 09:00- CEST/16:00- JST Nuclear Structure 21 Oct 09:00- CEST/16:00- JST Nasser KALANTAR-NAYESTANAKI 28 Oct 09:00- CET/17:00- JST Low-energy tests of the Standard Model 25 Nov 09:00- CET/17:00- JST Rob TIMMERMANS 02 Dec 09:00- CET/17:00- JST Relativistic chiral mean field model description of finite nuclei 09 Dec 09:00- CET/17:00- JST Hiroshi TOKI 16 Dec 09:00- CET/17:00- JST + WRAP-UP/DISCUSSION