Spectra of positive- and negative-energy nucleons in finite nuclei

G. Mao1,2, H. Stöcker2, and W. Greiner2

1)Institute of High Energy Physics Chinese Academy of Sciences2)Frankfurt University, Germany

I. IntroductionII. RHA for Finite NucleiIII. Numerical ResultsIV. Summary and Outlook

1. G.Mao, H. Stöcker, and W. Greiner, Int. J. Mod. Phys. E8, 389 (1999); AIP Conf. Proc. 597, 112 (2001).2. G. Mao, Phys. Rev. C67, 044318 (2003); High Ene. Phys. Nucl. Phys. 27, 692 (2003).

× × × × × × × × × ×

× × × × × × × × × × × × ×

× ×

. . . .. . 1p1s

p 1

s 1

NM

NM

0

E

. nucleon

× nucleon–anti-nucleon pair

shell model states

vacuum

r

(1) potential of nucleons

)(~

MeV 50~~

LS SVdr

dU

SVcenU

MeV 700~~ SVcenU

(2) potential of anti-nucleons

due to G-parity,vectorfields change signs

estimation based on no-sea approximation, param. dep.

)(~ SVdr

dU LS

1. Auerbach et al., PLB182, 221 (1986). 2. Reinhard et al., ZPA323, 13 (1986).

× × × × × × × × × ×× × × × × × × × × ×

Investigate the properties of quantum vacuum in the medium.A verification for the application of relativistic Quantum Field Theory in a many-body system.

Determine the individual scalar and vector potential

SVcenUSVcenU ~ ,~

Build a basis for the study of anti-matter and anti-nuclei.

.

.

.

Relativistic Hartree Approach

tixadtixbtx ee E E ,

seaval000

nucleon anti-nucleon

.

.valence-nucleon contribution

Dirac-sea contribution

describing bound states of nucleons and anti-nucleons consistently

other densitiessimilar

II. RHA for Finite Nuclei

Quantum Hadrodynamics

B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16, 1(1986)

AARRm

RRm

UMi NF

IF

4

1

2

1

4

1

2

1

4

1

2

L

LLL

4322

!4

1

!3

1

2

1 cbmU

A

RM

fRg

M

fgg

e

N

NI

12

1

82

1

4

0

L

,2

i

here

and

Tensor Couplings

VS

dr

dTLULS 2~

0 ψσgMωγgiγ σNμ

μωμ

μ

Dirac equation

In static nuclear matter

ψσgMβgαiψt

i σN

0

particle, posi. ene.

)()(0 pUσgMβPαpUgEσN

particle, neg. ene.

)()(0 pVgMβPαpVgE N

*E

*E

ωgE

ωgE

ω

/

m*P

ω

/

m*P

0

212

2

0

212

2

*

.* mpE

p

NpV

mpE

p.σ

χ

NpU**

σ

s

pEixpisp

pEixpisp ee

tSpVd

tSpUb

pE

mpdtx

.,

.,

21

2/3

3

,,)2(

),(

and are probability amplitudesspb ,

spd ,

The wave packet can be expanded as

antiparticle, posi. ene.

antiparticle, neg. ene.

pUσgMβpαpUωgE σNω

0

ωgmPE

ωgmPE

ω

/

*

ω

/

*

0

212

2

0

212

2

EEEE

s

tpEixpisp

tpEixpisp ee SpVdSpUb

pE

mpdtx

.

,.

,

21

2/3

3

,,)2(

),(

and are the annihilation and creation operators for the particles and antiparticles

spb ,

spd ,

One can expand the wave packet of antiparticles analogous to that of particles. In quantum field

theory:

pVσgMβpαpVωgE σNω

0

In finite nuclei, the Dirac equation can be written as

tx

xAexRiM

fxRg

xiM

fxgxgNMitx

ti

N

N

,

]12

1

42

1

2[,

000,000,00

00

The field operator can be expanded according tonucleons and anti-nucleons

tiexadtiexbtx E E ,

: quantum numberSpherical Nuclei rArRrr 00,00 , , ,

^^^

SLJ

and 0

^^

PeP i commute with^

H

a and are eigenfunctions of

^^^

, , PJH··

11-

1-

l

r

r

r

rGi

r

r

r

r

r

rF

x

l

r

r

r

r

r

rF

r

r

r

rGi

x

jlm

jlm

a

jlm

jlm

P

smm sm

mlmsmmjljlm

2

1Y2

1

spherical spinor

Inserting into the Dirac equation, one gets coupled equations for

FG , GF ,and

Nucleons

rFrAerRgrgrgM

rGrRM

fr

M

f

rdr

drF

rGrAerRgrgrgM

rFrRM

fr

M

f

rdr

drG

N

rN

rN

N

rN

rN

000,000

0,000

000,000

0,000

12

1

2

1

42E

12

1

2

1

42E

Anti-nucleons

rGrAerRgrgrgM

rFrRrM

frrM

f

rdr

drG

rFrAerRgrgrgM

rGrRrM

frrM

f

rdr

drF

N

NN

N

NN

000,000

0,000

000,000

0,000

12

1

2

1

42E

12

1

2

1

42E

2

1for

2

1for 1

ljl

ljl

where

In numerical solutionsNucleons:

rRM

fr

M

f

rrW

rAerRgrgrgMU

rAerRgrgrgMM

rGrWdr

dMrF

rGUrGrWdr

dMrW

dr

drG

rN

rN

Neff

Neff

eff

effeff

0,000

000,000

000,000

1

1

42

12

1

2

1

12

1

2

1E

E

Anti-nucleons:

rGrWdr

dMrF

rGUrGrWdr

dMrW

dr

drG

eff

effeff

1

1E

rRM

fr

M

f

rrW

rAerRgrgrgMU

rAerRgrgrgMM

rN

rN

Neff

Neff

0,000

000,000

000,000

42

12

1

2

1

12

1

2

1E

Vector fields change signs

G-parity

Orthonormalization of wave functions

yxtyatxatytx

tytxtytx

yxtytx

txadtxbtx

, ,, ,

0, , , , , ,

, , ,

,,,

matrix equationFrom the Dirac equation one can have

0 3

if 0 3

if 0 3

xaxxd

xaxaxd

xxxd

From above equations one obtains

0

0

3 3

rFrFrGrGdr

rFrFrGrGdr

yayaydyyyd

rerA

rM

frgrRm

rM

frgrm

rcrbrgrm

rp

T

N

T

N

s

0,02

0,00,00,022

00022

222

42

1

2

!3

1

2

1

Meson-field equations

seaval000

valence-nucleon contribution

Dirac-sea contribution

other densitiessimilar

1

1224

1

1

rFrGjr

rvals

1

1224

1 0

rFrGj

rrval

21

2g

1111

zFVBVgrsea

s

20212

2226

g

2ln2

224

221

g

1

*0

*

*

*

Am

e

m

M

mg

m

gz

N

020

*ln

3

N

sea

M

mgr

rFeffMrFrGeffUrG

rGrWdr

drFrFrW

dr

drGdr

rFeffMrFrGeffUrG

rGrWdr

drFrFrW

dr

drGdr

E

0E

E 0

E

ljn , ,

eff. pot. deri. term.

total derivativebaryon number is conserved

EE

Param:, , ,

, , , , ,

, NNMfMfmm

cbggg

Set: MeV763 MeV938 mM N

7 9~

RMF RHA

----- -----

s.o. splitting in o16 shell fluc. Pb208

Tensor couplings enlarge SLU by a factor of 2Binding Energy

are improved

Dirac-sea effects are enhanced

BE

Charge densities

Vacuum contributions to the scalar density and baryon density

RHAT

Relative amplitude to the baryon density

16O: < 4.0 % 40Ca: < 2.3 % 208Pb: < 0.6 %

RHA1

Scalar and Vector potentials

for S and VRHAT largerthan RHA1 about 20 MeV

ameliorated evidently

deepened20~30 MeV

single particle spectraof

protons and antiprotons

single particle spectraof

neutrons and antineutrons

Proton and anti-proton potentials in Pb208

Proton anti-protonNL1 54.1 750.0RHA1 42.6 362.0RHAT 46.6 396.8 at 0.9 fm

IV. Summary and Outlook

1. RHA including tensor couplings describing bound states of positive- and negative-energy nucleons in finite nuclei consistently.2. Parameters fitted to the properties of spherical nuclei

78.0* NMm

from RHA is about half of RMFE

RHAT: effect of tensor couplings

SLU is increased by a factor of 2

E is deepened 20~30 MeV

···

1. N. Auerbach, A.S. Goldhaber, M.B. Johnson, L.D. Miller and A. Picklesimer, PLB 182, 221(1986)2. Y. Jin and D.S. Onley, PRC 38, 813(1988)

o

nucleon o anti-nucleon × nucleon–anti-nucleon pair

NM

0

NM

E

r