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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
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