2003．9. 2，DUBNA

Effects of the Dirac Sea on Giant Resonance Statesin Relativistic Nuclear Models
Haruki Kurasawa (Chiba Univ.)
Toshio Suzuki (Fukui Univ. / RIKEN)

Purpose
1. Sum Rules require Anti-Nucleons.
2. Effects of the Renormalization should be examined.

Example : Gamow-Teller Sum Rules

K. Ikeda, S. Fujii and J.I. Fujita, Phys. Lett. 3, 271 (1963)

GT Transitions in the Relativistic Mean Field

A2( N-Z )０β+β-ＰNＰN

H. Kurasawa, T. Suzuki and N.V. Giai, Phys. Rev. Lett. 91, 062501(2003);
Phys. Rev. C68, 064311 (2004)

GT Transitions in the Relativistic Mean Field
A+(B +∞)-β-( C + ∞ ) = 2( N-Z )β+ＰN

ＤｉｒａｃNＰ

RPA without Renormalization

S.A. Chin, Ann. of Phys. 108,301(1977)
H. Kurasawa and T. Suzuki, Nucl. Phys.
A445,685 (1985)

RPA I ( No Free Term Approx. )Free TermPauli Blocking Terms

Divergent
Fermi Sea

Fermi Sea

=Dirac Sea

Dirac Sea

Dirac Sea

Current Conservation is satisfied.

H. Kurasawa and T. Suzuki, Phys. Rev. C69, 014306 (2004)

GT Transition in RPA I ( No Free Term Approx. )A’+( - B’ )-( - C’ )= 2( N-Z )Pauli Blocking Terms
β-β+ＰN

ＤｉｒａｃNＰ

Ikeda sum rule is satisfied.

RPA without Renormalization
RPA II ( No Sea Approx. )J.F. Dawson and R.J. Furnstahl,Phys. Rev. C42, 2009 (1990) ;Z. Ma et al., Nucl. Phys. A685,173 (2001)

Transition with Negative Energy

Fermi Sea

Fermi Sea

Dirac Sea

Dirac Sea
Current Conservation is satisfied.

H. Kurasawa and T. Suzuki, Phys. Rev. C69, 014306 (2004)

GT Transition in RPA II ( No Sea Approx. )A’
Ex. E > 0+( C’ )-( B’ )

Ex. E < 0β-= 2( N-Z )Ex. E < 0Eβ+Ｐ

N0ＤｉｒａｃＰN

D.J. Thouless, Nucl. Phys. 22, 78 (1961)

Energy-Weighted Sum
RPA Theorem by Thouless
in Non-Relativistic and No Charge-Exchange Excitations
Energy-Weighted Sum of the Strengths in RPA=< |[F,[H,F]]| >We can also prove the same theorem
for the sum of the β- and β+ transition strengths
in Relativistic RPA I and II .
H. Kurasawa and T. Suzuki, Phys. Rev. C69, 014306 (2004)

Summary of RPA I and II
If we include a part of antinucleon degrees of freedom,
1. Ikeda-Fujii-Fujita Sum Rule is satisfied.
2. RPA theorem also holds for the sum of β- and β+.
H. Kurasawa and T. Suzuki, Phys. Rev. C69, 014306 (2004)

Moreover,3. In the giant resonance region, RPA seems to work.

But there is a serious problemof RPA I and II.

The sum rules are satisfied, owing tothe negative energy-weighted strengths from the anti-nucleons.

Outside of the Giant Resonance RegionStrength

EnergyEnergy-Weighted Strength

RPA I

negative

positive

negative

RPA II

positive

negative

negative

Example :Non Energy-Weighted
Strength in Nucl. Matter
Sum Rule : S(total)/V=0.06863
S(GT) / S(total) = 88%

H. Kurasawa and T. Suzuki,
Phys. Rev. C69, 014306 (2004)

We have unphysical results in RPA I and II,
as a price of Neglecting the Divergence.
We need the renormalization of the Divergence.
How is the Effect of the Renormalization importantfor Observables ?
(For GT states, we do not know the answer yet.)
It depends on Observables in Previous Results.

RPA I, II

Correct

Correct

RPA I, IIS L( q )Qausielastic Electron Scatteringq = (q , q 0 = ω)0(e,e’)
ωHartree＋RPA

Mean Field+RPA

HartreeMean Field

See also H. Kurasawa and T. Suzuki, N. P. A490 (1988) 571 ; P. T. P. 86 (1991) 773.
C. J. Horowitz and J. Piekarewicz, P. R. L. 62 (1989) 391.

GT Transition
How is the renormalizatio important for GT ?
I do not know the answer, But it is important question.
A’+( - B’ )-( - C’ )Pauli Blocking Terms
β-β+ＰNＤｉｒａｃN

Ｐ= 2( N-Z )H. Kurasawa, T. Suzuki and N.V. Giai, Phys. Rev. Lett. 91, 062501(2003)

A’( 6% in finite nuclei, H. Kurasawa et al., Phys. Rev. 68, 064311 (2003);
Z. Ma et al., in press in EPJ A )This is small, but important.

C. Gaarde, Nucl. Phys. A396 (1983) 127c
Old ～ 50 %T. Wakasa et al., Phys. Rev. C55 (1997) 2909
Phys. Lett. B426 (1998) 257

New８４ ～ ９０ ％( ｐ－ｈ )( Δ－ｈ )

StrengthＡｇ’

NΔ>0Ｎ－ＺＥｘ～１０～３００（ＭｅＶ）

Quenchingg’(NΔ)

Pion condensation
critical density/ρ0

<1997

40%

0.75 (quark model)> 8>199710%0.21 (quark model)< 24%0.12 (quark model)( 6% due to the anti-nucleons )～ 1

T. Suzuki, H. Sakai and T. Tatsumi, Proc. of the Int. Workshop on Nucl. Res.
and Medium Effects (Universal Academy Press, Tokyo 1999 ) p77

Conclusions
Conclusions

1. Various laws are satisfied, if a part of anti-nucleon
degrees of freedom are taken into account.
2. As a price of neglecting the divergence, however,
we obtain unphysical results also.
3. We should estimate effects of the renormalization
on Observables.