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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

Hosaka

Potential energy surface of the vacuum

Yoichiro Nambu

Chiral order parameter

Quarks gluons

Confinement, Mass generation

Hadrons nuclei

Nobel prize (2008)

He was motivated by the BCS theory.

is the order parameter

is the order parameter

Chiral symmetry

Particle number

Nambu-Jona-Lasinio Lagrangian

Chiral transformation

Mean field approximation Hartree approximation

Fermion gets mass.

The chiral symmetry is spontaneously broken.

Chiral condensate is

The fermion mass is

m

G

Gc

The mass is similar to the pairing gap in the BCS

formalism. The mass generation mechanism for a

fermion.

The particle-hole excitation (pion channel) RPA

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 (Eguchi1974)

Fermion field is quark

Auxiliary fields

Nuclear physics with NJL model

Auxiliary fields

SU2 chiral transformation

Confinement

(Polyakov NJL Mode)

SU2c is done SU3c is not yet done.

Chiral sigma model

Pion is the Nambu boson of chiral symmetry

- Linear Sigma Model Lagrangian

Polar coordinate

Weinberg transformation

Non-linear sigma model

N

r fp j

Lagrangian

Free parameters are and

(Two parameters)

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.

Dirac equation

In finite nuclei, we have to project out spin and

isospin, which involves a complicated projection.

Relativistic Chiral Mean Field Model (powerful

method)

Wave function for mesons and nucleons

p

p

Mean field approximation for mesons.

h

h

Nucleons are moving in the mean field and

occasionally brought up to high momentum

states due to pion exchange interaction

Brueckner argument

Why 2p-2h states are necessary for pion

(tensor) interaction?

The spin flipped states are already occupied by

other nucleons.

Pauli forbidden

G.S.

Spin-saturated

Energy minimization with respect to meson and

nucleon fields

(Mean field equation)

Hartree-Fock

G-matrix component

Numerical results (1)

Ogawa Toki NP 2009

12C

Spin-spin

Pion

Tensor

Total

4He 12C 16O

Adjust binding energy and size

Numerical results 2

Individual contribution

Cumulative

O

C

O

The difference between 12C and 16O is 3 MeV/N.

P1/2

C

The difference comes from low pion spin states

(Jlt3). This is the Pauli blocking effect.

P3/2

S1/2

Pion tensor provides large attraction to 12C

Pion energy

Chiral symmetry

Ogawa Toki NP(2009)

Nucleon mass is reduced by 20 due to sigma.

N

Not 45 as discussed in RMF model.

We want to work out heavier nuclei for magic

number. Spin-orbit splitting should be worked out

systematically.

One half is from sigma meson and the other half

is from the pion.

Nuclear matter

Hu Ogawa Toki Phys. Rev. 2009

E/A

Total

Total

Pion

Deeply bound pionic atom

Predicted to exist

Toki Yamazaki, PL(1988)

Found by (d,3He) _at_ 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 0900- CEST/1600- JST

Atsushi HOSAKA 07 Oct 0900- CEST/1600- JST

Nuclear Structure 21 Oct 0900- CEST/1600- JST

Nasser KALANTAR-NAYESTANAKI 28 Oct 0900-

CET/1700- JST Low-energy tests of the Standard

Model 25 Nov 0900- CET/1700- JST Rob

TIMMERMANS 02 Dec 0900- CET/1700- JST

Relativistic chiral mean field model description

of finite nuclei 09 Dec 0900- CET/1700- JST

Hiroshi TOKI 16 Dec 0900- CET/1700- JST

WRAP-UP/DISCUSSION