09.12.16toki@osaka-groningen1 relativistic chiral mean field model for nuclear physics (ii) hiroshi...

09.12.16 toki@osaka-groningen 1

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 & nucleiConfinement, Mass generation

QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄÇ™Ç±ÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

Potential energy surface of the vacuum

Chiral order parameter

Yoichiro Nambu

Hosaka


QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ

Ç™Ç±ÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

He was motivated by the BCS theory.

€

E p = ±(ε p2 + Δ2)1/ 2

€

E p = ±( p2 + m2)1/ 2

Nobel prize (2008)

€

εiψ i + Δψ ̃ i = E iψ i

−ε iψ ̃ i + Δ*ψ i = E iψ ̃ i

€

− r

σ ⋅ r

p ψ L + mψ R = E pψ Lr σ ⋅

r p ψ R + mψ L = E pψ R

€

Δ is the order parameter is the order parameter

€

m

Particle number Chiral symmetry


Nambu-Jona-Lasinio Lagrangian

Mean field approximation; Hartree approximation

Fermion gets mass.

The chiral symmetry is spontaneously broken.

€

ψ ψ → ψ ψ cos(2α ) +ψ iγ 5ψsin(2α )

ψ iγ 5ψ →ψ iγ 5ψ cos(2α ) −ψ ψsin(2α )

€

ψ → e iαγ 5ψChiral 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

Gc

€

1 =GcΛ

2

π 2


The particle-hole excitation (pion channel): RPA

T =K

+ T

KJ(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, butthey are generated dynamically.Massless boson appears (Nambu-Goldstone boson) with pseudo-scalar quantum number.


Bosonization (Eguchi:1974)

€

Lσ =1

2(∂ μσ )2 + (∂ μπ a )2

( ) −λ

4(σ 2 + π a 2) − v 2

( )2

Fermion field is quark

€

Z = Dψ Dψ exp i d4∫ x LNJL[ ]∫€

LNJL = iψ γ μ∂μψ + G ψ ψ( )

2+ ψ iγ 5τ

aψ( )2

[ ]

Auxiliary fields

€

σ =ψ ψ

€

π a =ψ iγ 5τaψ

€

Z = Dψ DψDσDπ a exp∫ i d4∫ x[ψ iγ μ∂ μψ

−ψ g0(σ + iγ 5τaπ a )ψ −

1

2μ0

2(σ 2 + π a 2)]


Nuclear physics with NJL model

Auxiliary fields

€

σ =ψ ψ

€

π a =ψ iγ 5τaψ

SU2 chiral transformation

exp (aqa g5 )

€

σ → σ cos2θ + ˆ θ aπ asin2θ

π a → π a cos2θ − ˆ θ aσ sin2θ

€

LNJL = iψ γ μ∂μψ + G ψ ψ( )

2+ ψ iγ 5τ

aψ( )2

[ ]

Confinement (Polyakov NJL Mode)

€

Lσ =ψ iγ μ∂μ − g(σ + iτ aπ a )( )ψ

+1

2(∂ μσ )2 + (∂ μπ a )2

( ) −λ

4(σ 2 + π a2) − v 2

( )2

SU2c is doneSU3c is not yet done.


Chiral sigma modelChiral sigma model

Linear Sigma Model Lagrangian

Polar coordinate

Weinberg transformation

Pion is the Nambu boson of chiral symmetry


Non-linear sigma modelNon-linear sigma model

Lagrangian = fπ +

whereM = gσfπ M* = M + gσ

mπ2 = 2 + fπ mσ

2 = 2 +3 fπ

m = gfπ m

= m + g

~ ~

Free parameters are and

€

mσ

€

gω (Two parameters)

N


Relativistic mean field model(standard)

Mean field approximation:

€

σ = σ + ′ σ

Then take only the mean field part, which is just a number.

€

σ =σ

=δ0ω

€

π a = 0

The pion mean field is zero. Hence, the pion contributionis zero in the standard mean field approximation.






Relativistic mean field model(pion condensation)

€

π a = π a Ogawa, Toki, et al.Brown, Migdal..

Since the pion has pseudo-scalar (0-) nature, the parityand charge symmetry are broken.

In finite nuclei, we have to project out spin and isospin,which involves a complicated projection.

Dirac equation

€

mπ2π a = gπ∇ i ψ iγ 5γ iψ


€

Ψ=Ψ(σ ,ω)⊗Ψ(N)Mean field approximation for mesons.

€

Ψ(N) = C0 RMF + Ci

i

∑ 2p − 2hi

Nucleons are moving in the mean field and occasionally broughtup to high momentum states

due to pion exchange interaction

€

σ

€

σ

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?

€

σ

€

σ

G.S.

Spin-saturated

The spin flipped states are alreadyoccupied by other nucleons.

Pauli forbidden


Energy minimization with respect tomeson and nucleon fields

€

δΨ H Ψ

Ψ Ψ= 0

€

δE

δσ= 0

δE

δω= 0

(Mean field equation)

€

δE

δψ i(x)= 0

δE

δCi

= 0



Hartree-Fock

G-matrix component


Numerical results (1)

4He12C16O

Ogawa TokiNP 2009

Adjust binding energyand size

Tensor

Spin-spin

Pion

Total

12C


Numerical results 2

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

The difference comes from low pion spin states (J<3).This is the Pauli blocking effect.

P3/2

P1/2

C

O

S1/2

Pion energy Pion tensor provides large attraction to 12C

OC

Cumulative Individual contribution


Chiral symmetry

Nucleon mass is reducedby 20% due to sigma.

We want to work out heavier nuclei for magic number.Spin-orbit splitting should be worked out systematically.

Ogawa TokiNP(2009)

Not 45% as discussedin RMF model.

N

€

One half is from sigma meson and the other half isfrom the pion.


Nuclear matterHu Ogawa TokiPhys. Rev. 2009

€

ψ ψ

E/A

Total

Pion

€

Σ ~ 50MeV

Total

€

ψ ψ

€

σ


Deeply bound pionic atom

Toki Yamazaki, PL(1988)

Predicted to exist

Found by (d,3He) @ GSIItahashi, Hayano, Yamazaki..Z. Phys.(1996), PRL(2004)

Findings: isovector s-wave

€

b1

b1(ρ )=1− 0.37

ρ

ρ 0

€

fπ2mπ

2 = −2mq ψ ψ

€

ψ ψ

ψ ψ=1− 0.37

ρ

ρ 0

€

b1 ∝1

fπ2




Suzuki, Hayano, Yamazaki..PRL(2004)

Optical model analysisfor 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 Physics30 Sep 09:00- CEST/16:00- JST Atsushi HOSAKA07 Oct 09:00- CEST/16:00- JST Nuclear Structure21 Oct 09:00- CEST/16:00- JST Nasser KALANTAR-NAYESTANAKI28 Oct 09:00- CET/17:00- JST Low-energy tests of the Standard Model25 Nov 09:00- CET/17:00- JST Rob TIMMERMANS02 Dec 09:00- CET/17:00- JST Relativistic chiral mean field model description of finite nuclei09 Dec 09:00- CET/17:00- JST Hiroshi TOKI16 Dec 09:00- CET/17:00- JST + WRAP-UP/DISCUSSION

09.12.16toki@osaka-groningen1 relativistic chiral mean field model for nuclear physics (ii) hiroshi...

Documents