Relativistic chiral mean field model for nuclear physics (II)

1
Relativistic chiral mean field model for nuclear
physics (II)

• Hiroshi Toki
• Research Center for Nuclear Physics
• Osaka University

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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)

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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.

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Spontaneous breaking of chiral symmetry
Hosaka
Potential energy surface of the vacuum
Yoichiro Nambu
Chiral order parameter
Quarks gluons
Confinement, Mass generation
Hadrons nuclei
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Nobel prize (2008)
He was motivated by the BCS theory.
is the order parameter
is the order parameter
Chiral symmetry
Particle number
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Nambu-Jona-Lasinio Lagrangian
Chiral transformation
Mean field approximation Hartree approximation
Fermion gets mass.
The chiral symmetry is spontaneously broken.
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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.
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The particle-hole excitation (pion channel) RPA
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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.
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Bosonization (Eguchi1974)
Fermion field is quark
Auxiliary fields
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Nuclear physics with NJL model
Auxiliary fields
SU2 chiral transformation
Confinement
(Polyakov NJL Mode)
SU2c is done SU3c is not yet done.
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Chiral sigma model
Pion is the Nambu boson of chiral symmetry

• Linear Sigma Model Lagrangian

Polar coordinate
Weinberg transformation
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Non-linear sigma model
N
r fp j
Lagrangian
Free parameters are and
(Two parameters)
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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.
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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.
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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
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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
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Energy minimization with respect to meson and
nucleon fields
(Mean field equation)
Hartree-Fock
G-matrix component
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Numerical results (1)
Ogawa Toki NP 2009
12C
Spin-spin
Pion
Tensor
Total
4He 12C 16O
Adjust binding energy and size
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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
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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.
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Nuclear matter
Hu Ogawa Toki Phys. Rev. 2009
E/A
Total
Total
Pion
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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
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Suzuki, Hayano, Yamazaki.. PRL(2004)
Optical model analysis for the deeply bound state.
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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.

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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.

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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