Introduction
Model
B̄ -nucleus interaction
Results
Spin symmetry
Conclusions
Calculations of antibaryon nuclear bound states
Jaroslava Hrtánková
Ji°í Mare²
Nuclear Physics Institute, eº, Czech Republic
ECT* workshop, Trento, October 27 - 31, 2014
Introduction
Model
B̄ -nucleus interaction
Results
Spin symmetry
Conclusions
Introduction
study of antibaryon (p̄ , Λ̄, Σ̄, Ξ̄) bound states in selected nuclei:
behavior of antibaryons in the nuclear medium
changes of the binding energy, single particle energies and density
distributions in the nuclear core
p̄ absorption in a nucleus
knowledge of p̄ nucleus and Ȳ nucleus interaction for future
experiments (PANDA@FAIR)
testing models of (anti)hadronhadron interactions
possibility of long living p̄ in the nuclear medium due to phase space
suppression? (I.N. Mishustin et al, Phys. Rev. C 71 (2005))
2
Introduction
Model
B̄ -nucleus interaction
Results
Spin symmetry
Conclusions
RMF approach
Baryons treated as Dirac elds interacting via the exchange of meson
elds
isoscalar-scalar eld σ , isoscalar-vector eld ωµ , isovector-vector eld
ρ
~µ , and massless vector eld Aµ .
The standard RMF models TM and density dependent model TW99
giN (ρVN ) = giN (ρsat )fi (x ) , i = σ, ω, ρ ,
where ρVN is a vector baryon density and
(S.
x = ρVN /ρsat
Typel, H.H. Wolter, Nucl. Phys. A 656 (1999) 331 )
3
Introduction
B̄ -nucleus interaction
Model
Results
Spin symmetry
Conclusions
RMF approach
Dirac equation for nucleons and antibaryon
~ + β(mj + Sj ) + Vj ]ψ α = α ψ α ,
[−i α
~∇
j
j j
S
j = N , B̄
,
1 + τ3
A0 + Σ R ,
2
∂ gρN
∂ gσN
∂ gωN
ρVN ω0 +
ρ IN ρ 0 −
ρSN σ .
ΣR =
∂ρVN
∂ρVN
∂ρVN
= gσj σ,
Vj
= gωj ω0 + gρj ρ0 τ3 + ej
Klein-Gordon equations for meson elds
(−4 + mσ2 )σ = −gσN (ρVN )ρS −gσB̄ (ρVN )ρS B̄
(−4 + mω2 )ω0 = gωN (ρVN )ρV +gωB̄ (ρVN )ρV B̄
(−4 + mρ2 )ρ0 = gρN (ρVN )ρI +gρB̄ (ρVN )ρI B̄
−4A0 = e ρp +eB̄ ρB̄ .
4
Introduction
Model
B̄ -nucleus interaction
Results
Spin symmetry
Conclusions
B -nucleus interaction
Nucleon-meson couplings obtained by tting nuclear matter and nite
nuclei properties
Hyperon-meson coupling constants:
for ω and ρ eld obtained from SU(6) symmetries,
for σ eld obtained from ts to experimental data (Λ hypernuclei, Σ
atoms, Ξ production in (K + , K − ) reactions)
gσΛ = 0.621gσN ,
gσΣ = 0.5gσN ,
gσΞ = 0.299gσN ,
gωΛ = 2/3gωN ,
gωΣ = 2/3gωN ,
gωΞ = 1/3gωN ,
gρΛ = 0 ,
gρΣ = 2/3gρN
gρΞ = gρN
,
5
Introduction
B̄ -nucleus interaction
Model
Results
Spin symmetry
Conclusions
B̄ -nucleus interaction
NN → N̄N interaction G-parity + optical potential
B -nucleus → B -nucleus interaction G-parity transformation
gσB̄ = gσB , gωB̄ = −gωB , gρB̄ = gρB
400
400
repulsive
200
200
0
U [MeV]
U [MeV]
S+V ~ -750 MeV
0
16
-200
attractive
16
O
Op
S+V ~ -50 MeV
-400
0
1
2
3
4
r [fm]
5
-200
S
V
6
1
2
3
4
r [fm]
Fig.1: The scalar and vector potential acting on nucleon in
(right), calculated statically in the TM2 model.
16
5
O (left) and
-400
6
p̄
in
16
Op̄
6
Introduction
Model
B̄ -nucleus interaction
Results
Spin symmetry
Conclusions
p̄-nucleus interaction
Antiprotonic atoms and p̄ scattering o nuclei at low energies
→ the depth of ReVp̄ ∼ 100 − 300 MeV
Reduced
p̄ coupling constants
gσp̄ = ξ gσN , gωp̄ = −ξ gωN , gρp̄ = ξ gρN
,
where parameter ξ is from h0, 1i
large polarization eects conrmed
(I.N. Mishustin et al, Phys. Rev. C 71 (2005))
7
Introduction
Model
The issue of the
B̄ -nucleus interaction
Results
Spin symmetry
Conclusions
p̄ self-interaction
KG equations for a meson
eld acting on nucleons:
ξ=0.25
ξ=0.5
ξ=0.75
ξ=1
1
-3
ρp [fm ]
2
(−4 + mM
)ΦN = gMN ρMN
+ gM p̄ ρM p̄
1.2
0.8
208
208
Pbp TM1
1.2
1
0.8
Pbp TM1
0.6
no self-interaction 0.6
acting on p̄ :
0.4
0.4
2
)Φp̄ = gMN ρMN
(−4 + mM
h
(
(p̄(
+h
gMh
ρM(
h
(
p̄
h
(
h
0.2
0.2
0
0.5
1
r [fm]
1.5
0
0.5
1
r [fm]
1.5
2
Fig.2: The p̄ density in 208 Pbp̄ , calculated
with and without the p̄ self-interaction.
8
Introduction
B̄ -nucleus interaction
Model
Results
Spin symmetry
Conclusions
Density-dependent vs. standard RMF model
0.1
0
0.08
-0.01
0.06
Pbp TM1
-3
ρp - ρn [fm ]
208
0.04
0.02
208
ξ=0.25
ξ=0.5
ξ=0.75
ξ=1
208
Pbp TW99
-0.02
-0.03
-0.04
Pb
0
-0.05
-0.02
-0.06
G [-]
12
TM1
4
0
1
2
r [fm]
Fig.3:The isovector density in
208
12
Gσ
Gω
Gρ
8
3
8
TW99
ξ=1
0
1
2
r [fm]
4
3
4
Pbp̄ , calculated within the TM1 and TW99 model.
9
Introduction
B̄ -nucleus interaction
Model
Results
Spin symmetry
Conclusions
Antibaryon potential
0
20
-50
S+V [MeV]
0
ξ=0.2
-20
TM2
16
-40
16
16
-60
0
Fig.4: The
16
1
2
3 4
r [fm]
B nucleus
TM2
16
Op
16
OΛ
16
OΣ0
16
OΞ0
5
(left) and
0
1
2
B̄ nucleus
3 4
r [fm]
-100
-150
Op
OΛ
OΣ0
-200
-250
OΞ0
5
6
(right) potentials in
16
O.
Introduction
B̄ -nucleus interaction
Model
Results
Spin symmetry
Conclusions
Antibaryon spectrum
0
Ξ
+
0
Ξ
+
Σ
EΒ [MeV]
-50
0
Σ
−
-100
Σ
Λ
TM
40
-150
6
16
Li
12
90
Ca
ξ=0.2
Zr
208
Pb
p
O
C
-200
0
10
20
[-]
30
40
2/3
A
Fig.5: The A dependence of
ξ = 0.2.
B̄
single particle energies, calculated in TM model for
11
Introduction
B̄ -nucleus interaction
Model
Results
Spin symmetry
Conclusions
Baryon vs. antibaryon s.p. energies
0
-5
6
Li
16
12
O
+
Ξ ξ=0.2
−
Ξ
C
E [MeV]
-10
TM
40
-15
90
Ca
Zr
208
-20
Pb
-25
-30
-35
-40
0
5
10
15
20
2/3
A
25
30
35
40
Fig.6: Single particle energies of Ξ− and Ξ̄+ for ξ = 0.2 in various nuclei, calculated
dynamically in the TM model.
Introduction
The
Model
B̄ -nucleus interaction
Results
Spin symmetry
Conclusions
p̄ absorption
p̄nucleus optical
1.0
potential:
ξ = 0.2, Imb0 = 1.9 fm
√
s = 2mN − Bp̄ − BN
2π
fs [-]
ReVp̄ =ξ VRMF ,
X
ImVp̄ =
fs Br Imb0 ρRMF ,
channel
0.8
πω
0.6
ηω
πρ
0.4
ρω
2ω
3π
2πρ
2πω
0.2
2πη
4π
0
0.5
s
1/2
1
[GeV]
5π
6π
1.5
7π
2
Fig.7: The phase space suppression factor f√
s
as a function of the center-of-mass energy s .
13
Introduction
The
B̄ -nucleus interaction
Model
Results
Spin symmetry
Conclusions
p̄ absorption
Table 1: The 1s single particle energies Ep̄ and widths Γp̄ (in MeV) in
16
Op̄ , calculated dynamically (Dyn) and statically (Stat) with the real,
complex and complex with fs potentials (TM2 model), consistent with
p̄ atom data.
Ep̄
Γp̄
Real
Dyn
Stat
193.7 137.1
-
Complex
Dyn
Stat
175.6 134.6
552.3 293.3
Complex + fs
Dyn
Stat
190.2 136.1
232.5 165.0
14
Introduction
Model
B̄ -nucleus interaction
Results
Spin symmetry
Conclusions
CMS vs LAB frame
p̄ absorption in a nucleus → non-negligible contribution from the
momentum dependent term in
s = (EN + Ep̄ )2 − (~pN + ~pp̄ )2 , ~pN + ~pp̄ 6= 0,
where Ei = mi − Bi for i = N , p̄ .
(A.
Cieply et al., Phys. Lett. B 702, 402 (2011) )
Table 2: The 1s single particle energies Ep̄ and widths Γp̄ (in MeV) in
16
√Op̄ , calculated dynamically in TM2 model with dierent approach to
s , consistent with p̄ atom data.
Ep̄
Γp̄
CMS
190.2
232.5
LAB
191.6
179.9
Introduction
B̄ -nucleus interaction
Model
Spin symmetry in
Results
Spin symmetry
Conclusions
p̄ spectrum
relativistic symmetry of Dirac Hamiltonian when V=S+const
J.N. Ginocchio, Phys. Rep. 414, 165 - 261 (2005)
spin doublets - states with j = ` ±
upper components of
1
2
are degenerate
wave function are equal gnr ,`+1/2 (r ) = gnr ,`−1/2 (r )
lower components are related by the equation
∂
`+2
∂
`−1
+
fnr ,`+1/2 (r ) =
−
fnr ,`−1/2 (r )
∂r
r
∂r
r
p̄
spin symmetry is well preserved in antinucleon spectra
X.T. He, S.G. Zhou, J. Meng, E.G. Zhao and W. Scheid, Eur. Phys. J. A 28, 265
- 269 (2006)
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Introduction
B̄ -nucleus interaction
Model
Results
Spin symmetry
Conclusions
Spin symmetry
0.003
0
0.002
-0.0004
-0.0006
0.001
-0.0008
f p1/2
f p3/2
]
-0.0002
0.0005
16
p in O TM2
-0.0004
ξ=0.2 dyn
with absorption
16
p in O TM2
-0.001
ξ=0.2 dyn
0
1
2
r [fm]
3
40
1
2
r [fm]
3
0
-3/2
wave function [fm
wave function [fm
-3/2
]
-0.0002
g p1/2
g p3/2
0.001
differential relation
f p1/2
f p3/2
differential relation
g p1/2
g p3/2
0
4
Fig.8: Upper (g) and lower (f)
components of the p̄ wave function
in 16 Op̄ TM2, calculated dynamically.
0
1
2
r [fm]
3
40
1
2
r [fm]
3
4
Fig.9: Upper (g) and lower (f)
components of the p̄ wave function
in 16 Op̄ TM2, calculated dynamically
with p̄ absorption in the nucleus.
17
Introduction
Model
B̄ -nucleus interaction
Results
Spin symmetry
Conclusions
Conclusions
antibaryons are deeply bound in the nuclear medium
large polarization eects of the nuclear core due to
B̄
conrmed
p̄ absorption in a nucleus p̄ widths are suppressed due to the phase
space reduction, but still remain large for potentials consistent with
p̄atom data
signicant contribution from ~pp̄ and ~pN to Γp̄
spin symmetry in p̄ spectrum is preserved even if the polarization
eects in the nucleus and the p̄ absorption are taken into account
18