aye aye min, khin swe myint, j. esmaili & yoshinori akaishi august 23, 2011 by theoretical...
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Aye Aye Min, Khin Swe Myint, J. Esmaili &
Yoshinori AKAISHI
August 23, 2011
By
Theoretical Investigation for Production of Double- Hypernuclei
from Stopped Hyperon on
APFB2011
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Abstract
Investigation of the formation ratio of to for various
absorptions from 2S, 2P and 3D orbitals of atom by
assuming a d- cluster model for
Two kinds of d- relative wave function namely 1s d- relative wave
function with phenomenological One Range Gaussian (ORG) potential
and that with Orthogonality Condition Model (OCM ) were used in our
calculations.
We have also investigated differential cross section for single-
hypernuclei, and .
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(K. Nakazawa , Nucl. Phys. A 835 (2010))
It is worthwhile to measure the masses of double- hypernuclei for several nuclear species to determine - interaction without ambiguities.
t
p
Emulsion Experiment
B : interaction energy
B= B(AZ) - 2B(
A-1Z)
Weakly attractive Interaction !
H4
(T. Fukuda et. al., Phys. Rev. Lett. 87 (2001))
BNL BeonreactionK,K 9
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S = 0 sector S = -1 sector S = -2 sector
NN
N
N
N
N
~ 300 MeV
~ 80 MeV
~ 28 MeV
Although coupling effect is not significant in non-strangeness sector, coupling effect plays an important role in strangeness sector.
(K.S. Myint, S. Shinmura and Y. Akaishi, Nucl. Phys. A 721 (2003) 21)
N coupling effect in
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Xa,ΞLi AΛΛstopped
6
ptpnd
d- cluster structure
Production of Double- Hypernuclei
In order to produce and , the reaction is
Target ( )
Pt
dPn
P
t
dPn
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two hyperons and ordinary nucleus
H-dibaryon and ordinary nucleus
atomLi6
SP
D
Elementary process for the reaction
p
28.33MeV
Single-hypernucleusand
-hyperon
Two single-hypernuclei
Double-hypernucleus
Absorption of - in atom and Production of hypernuclei
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No. Reactions Q-value(MeV)
1
2 31.88
3 27.75
4 9.08
5 27.93
6 6.86
7 6.44
8 7.17
9 0.92
10 1.05
11 9.21
12 24.63
B04.7dHLi 56
nHeLi 66
HeLi 66
nHeLi 56
dHLi 46
np�HLi 46
n2HeLi 46
tHLi 36
ndHLi 36
nHHLi 336
HHLi 346
nHeLi 42
6
Table 1. Possible reactions for the stopped hyperon on
MeV0.5~B
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d
t
q~
Q~
q
q tq
dk
q
P
tpq
tq
q~ d
dq
d~q
p~q
q~
K
0K
Transition matrix , int
Linlmcm
dHfi 65 ,0T,,T
kK
Transition matrix in terms of relevant momenta ,
qqQ,qqqqkK '' ~t~~~d~d~dL
2T Htd
29
fi 5
int
Lidpnlm 6~,~~ qqq
internal wave function of
sub-systems
relative wave functions
Triton(t), deuteron(d)
proton-triton (p-t)
deuteron-alpha (d-)
n
n p t
p
n d
p
d
t
q~
Q~
q
q tq
dk
K
dHLi 56
wave function for target
Formation from stopped on
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qqqq ~gT~g~t~0
2q
q ~~g
2
2
12
fm5.2c
mc
andparameterstrengthT0
mesonK,particleexchangeofmassmc2
Interaction for elementary process, is described by separable potential.
p
where
Interaction for elementary process
1~g q 43
a22
1~g~~~d
qqq '
By assumption the interaction is zero range ,
p
K
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d
22
0d22
3
HˆdiTka2
15 k
2t
a12
~5
43
t
23
t fm309.0a,ea3
10
2
1~~t
2Q
Q
2a3
~24
32
3
ppt fm521.0a,ea3
16
2
1~~p
2q
q
We will discuss later!
Decay width to and deutron
Decay width ( ) is Γ
atomLioffunctionwaveradialrR 6n
,rRrq~jdrrq~F n0
2n
2
ddpptnmttt2t
2 ~~~~q~Fq~Y
~~ˆddqq~̂dq~dq~ qqQqq
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P
n
q~
p~q
q~q
~
q pq
nq
q
q~
Q~
q
q q
n nk
K
n
22
0n22
3
He6ˆdiTka2
1k
2
ddppnnm22 ~~~~
q~Fq~Y~~ˆddqq~̂dq~dq~ qqQqq
2q
qp
~
da
143
d
23
ppn ea
8
2
1~~
2Q
Q
~
a8
343
23
ea
3
2
1~~
Formation
2fm309.0a
2d fm199.0a
p
nnn
p
p
0K
1. GBWF (one range
phenomenonlogiacl
Gaussian potential)
2. GBWF (OCM model)
q~
Q~
q
q q
n nk
K
nHeLi 66
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Construction of relative d- wave function by using one range phenomenological Gaussian potential
Gaussian basis radial wave function for d- cluster is
2j
b/r1
jj ercrU
Gaussian one range potential
bj= range parameter and
cj= the expasion coefficient
we adjusted the potential strength( -85.42MeV) to give energy eigen value of 1s state(-1.48 MeV ) and eigen function corresponding to this 1s bound state .
By applying Fourier transform,
2dq
q~
2
b
3j
jjdd
2
j
ebc32
1~~
2.0 fm
2
b/r0d eVrVrV
-85.42
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The Gaussian potential between and x particle
Where,
For our system, case, x is deuteron and . Li6 0
The potential strengths and range parameters for -d system
1kk maxmax 2P
kk fm2.0
MeV21.64Vk
MeV21.10VPk
Construction of relative d- wavefunction with OCM
max 2Pk
2k
max k
1k
rpk
rk
1kkx eV1eVrVrV
x
k
1k
rp,sk
rk
1k
sk SeV1eV
max 2Pk
2k
max
(E. Hiyama et.al., arXiv:nucl-th 24 (2002) 0204059.)
xS = the spin of x
= relative angular momentum between and x
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The Pauli principle between nucleons belonging to and x (x = n, p, d, t )
clusters is taken into account by the Pauli projection operator or OCM projection operator
The forbidden states for d- cluster are 0s and 0p states.
2ar
2
14/13
s0 era
2rU
f
dfdfPauli limV rr
2dq
q~
2
b
3j
jjdd
2
j
ebc32
1~~
MeV105
d
22
0d22
3
HˆdiTka2
15 k
2
ddpptnmttt2t
2 ~~~~q~Fq~Y
~~ˆddqq~̂dq~dq~ qqQqq
n
22
0n22
3
HeˆdiTka2
16 k
2
ddppnnm22 ~~~~
q~Fq~Y~~ˆddqq~̂dq~dq~ qqQqq
Monte Carlo integration
Method
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nHeLi 56 dHLi 46
22
2/12
2
nHe
2
n2n
220
n
He
215
5
kkH2
xkm
ddkiT4dk
d
2
ddppnnmB~~~~
q~Fq̂~Y~~~dd qqqqq
22
2/12
2
dH
2
d2d
220
d
H
214
4
kkH2
xkm
ddkiT4dk
d
2
ddpptnmBt~~~~
q~Fq̂~Y~~~dd qqqqq
Models of single -hypernuclei
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d- density distribution ofIn coordinate space
Results and Discussions
HeH 6ΛΛ
5ΛΛ
Γ/Γ)10|x|T(Γ 42
0 )10|x|T(Γ 42
0
H5ΛΛ He6
ΛΛ
B.E ( ) = 5.0 MeV
Table 2. Formation ratio of to from stopped hyperon on
d- wave function
types
Atomic absorption (arbitary
unit)(arbitary
unit)
GBWF(1s)
(one range pot.)
2S 4.46 1.18 3.78
2P 0.14 0.13 1.08
3D 85.63 77.13 1.11
OCM
2S 8.18 1.29 6.34
2P 0.24 0.12 2.00
3D 153.29 76.40 2.01
H5K.S. Myint, S. Shinmura and Y. Akaishi, Eur. Phys. J. A 16 (2003) 21.
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MeV04.12dHLi 56
MeV88.31nHeLi 66
effect of low and high momenta component of d-
relative motion ???
to clarify this argument more
profoundly!
qd (MeV/c)
d- density distribution of (in momentum space)
Li6
This wave function ( 0s′ ) is obtained by reducing the strength of one range Gaussian potential (-19.152MeV) to give the ground state energy, E = -1.48 MeV.
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Significance of d- relative momentum contribution
Wave function
types
atomic
absorption
GBWF( 1s ) 3D 85.63 77.13
GBWF( ) 3D 239.36 6.37
formation is enhanced and formation is
dropped off significantly!It is important to understand the structure
of a target to propose a feasible reaction topopulate double- hypernuclei from
hyperon captured at rest.
s0
MeV)10x(Γ 4
He6ΛΛH5
ΛΛ
MeV)10x(Γ 4
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For single- hypernuclei case,
234.0
He6
192.0
H5
He5 H4
and are at rest!
150 MeV/c113.82 MeV/c
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It may be deduced the significance of - coupling effect from this experiment.
Formation of is more dominant than that of for all absorption orbitals; 2S, 2P and 3D states from this reaction ( 1.1 for ORG and
2.0 for OCM for the major 3D absorption case).
Concluding remarks
Binding energy of can be measured without ambiguities.
Thus, we have proposed a feasible reaction which can produce , , and with comparable branching ratios.
Low momentum component of d- relative wave function favors the formation.
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Thank you for your
kind attention!
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0.0 MeV
23.21 MeV
28.33 MeV
+ + t
8.0 MeV+
+ n t
+ p + t+ p + t
Pauli Suppressioneffect
N coupling effect in
Coupling effect enhancement
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Strength ()
BE of d- cluster(MeV)
remark
0 -33.3-1.48
Unphysical forbidden state
1 -32.32-1.45
Unphysical forbidden state
101 -23.35-1.45
Unphysical forbidden state
102 -1.50-1.48
Unphysical forbidden state
103 -1.48
104 -1.48 Allowed state
105 -1.48 Allowed state
106 -1.48 Allowed state
Binding Energy of d- cluster by changing the strength of value
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-85.42
E =-1.48 MeV
-85.42
-19.152
-85.42
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P
t
d
Pn
Pt
d Pn
Proton speration ~ 19.81 MeV energy
B.E(d ) =2.224 MeV
n
MeV33.28p
formation formation
t
d
n
(6He)
2S absorption(
6He)2P absorption
(6He)
3D absorption
BE(2.224 MeV) 1.29 0.12 76.4
BE(3.5MeV) 1.32 0.13 78.1
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MeV04.12dHLi 56
Pd=191.80 MeV/c KEd=9 MeV KE(
H)= 3.04 MeV Q=12.04 MeV
MeV88.31nHeLi 66
Pn= 232.47 MeV KEn=28 MeVKE(
He)= 3.88 MeV Q=31.88 MeV
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2Bq
Bq
~
a8
54/32/3
ea
5
2
1~~
mm
mm
hyper_
hyper_
/
H2
xkm
xkm
k2
22
nhyper_
2
emittedhyper_
2
i2
emitted22
hyper
2emitted
2
Ecmcmcm2
H
k
The required data are;
emittedemittedemittedemitted sinsincoscossinsincoscosx
22
2/12
2
nHe
2
n2n
220
n
He
215
5
kkH2
xkm
ddkiT4dk
d
2
ddppnnmB~~~~
q~Fq̂~Y~~~dd qqqqq
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P
t
d
Pn
Pt
d Pn
Proton speration ~ 19.81 MeV energy
B.E(d ) =2.224 MeV
n
MeV33.28p
formation formation
t
d
n
(6He)
2S absorption(
6He)2P absorption
(6He)
3D absorption
BE(2.224 MeV) 1.29 0.12 76.4
BE(3.5MeV) 1.32 0.13 78.1
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Abundant of Lithium
7%
93%
MeV04.12dHLi 56
MeV06.11tHLi 57
Pt
d
6Li
-1.48 MeV
P
t
t
7Li
-2.5 MeV
LiΓ2~LiΓ 7
H5ΛΛ
6
H5ΛΛ
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d- wave function
types
Atomic absorption (arbitary
unit)(arbitary
unit)
GBWF(1s)
(one range pot.)
2S 4.46 1.21 3.69
2P 0.14 0.13 1.08
3D 85.63 79.19 1.08
OCM
2S 8.18 1.32 6.20
2P 0.24 0.13 1.85
3D 153.29 78.27 1.96
(arbitary unit)
(arbitary unit)
4.46 1.18 3.78
0.14 0.13 1.08
85.63 77.13 1.11
8.18 1.29 6.34
0.24 0.12 2.00
153.29 76.40 2.01
H5ΛΛ
)10|x|T(Γ 420
He6ΛΛ
)10|x|T(Γ 420
H5ΛΛ
)10|x|T(Γ 420
He6ΛΛ
)10|x|T(Γ 420
HeH 6ΛΛ
5ΛΛ
Γ/ΓHeH 6
ΛΛ5
ΛΛΓ/Γ
Old data from Nagara_paper(BE(LLHe6))
New data from Nagazawa Sensei(BE(LLHe6))
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Wave function types
Probabilities of
low momentum
component
Probabilities of
high momentum
component
GBWF(1s) 0.74 0.26
OCM 0.71 0.29
Table 3. Probabilities of momentum components of d- relative wave unction of
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Introduction
hyperon
can stay in the nucleus deeply without obeying Pauli exclusion principle
hypernucleus
probes a deep interior of the nucleus and investigates the nuclear structure
gives a new dimension to the traditional world of nuclei
provides the rich information on the baryon dynamic involving the strange particles
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Strangeness-exchange process
Combination of strangeness exchange and associated production of strangeness process
Associated production of strange-hadrons process
Possible production of hypernuclei
nK -
pK - etc.
Kn
Kp etc.
KpK
0KnK etc.
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participant
Spectator -projectile fragment
Spectator -target fragment
coalescence of hyperons to
projectile fragnent
theoretical model (Wakai, Bando, Sano)
High energy heavy-ion collisions
From Professor Dr T. Fukuda’s Presentation
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High energy heavy-ion collisions
Coalescence of strange particles with a nuclear fragment produced in projectile nuclear fragmentation
Coalescence of strange particles and nucleons both produced in the participant part
Secondary process by and K mesons produced in the primary nuclear collisions
p
n
KF
F
Conversion of hypernucleus into single and double-hypernucleus
XNaF)p(Ne
XK
( at 2.1 GeV/nucleon )
XCC ( at 3.7 GeV/nucleon )
XFTO16 ( at 2.1 GeV/nucleon )
HAuSi 3 ( at 14.5 GeV/nucleon ) etc.
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In order to produce a hypernucleus,
where, q = momentum transfer to the hyperon
The hyperon emerging from the reaction must remain in the nucleus.
Formation probability of the hypernucleus
Momentum transfer to the hyperon
Sticking probability, kS
2
HOnk
HOnYYNNk NNYY
qrjn,n;qS
n , = principal quantum number and orbital angular momentum for nucleon and hyperon state qrjk = bessel function with the orbital angular momentum transfer
( initial and final states are Harmonic Oscillator wave functions )
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Direct Process
Via atom
KEK-E 176
P P
H
K-
K+
K- 0
K+
K+
K-
-
K- K+
- or H (?)
K-K+
KEK-E 176 -E 224 BNL-E 813 -E 836 -E 885
KEK-E 176 -E 224 BNL-E 885
KEK-E 224 - atom
K+
K-
-
A
A
orH
Prowse (?), Danysz et al.KEK- E 176, E373BNL- E906
KEK- E 176 E373
KEK- E 176 E 224BNL- E 885
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H. Takahashi, “PhD Thesis”, Kyoto University (2003)
Possible Candidates of double- hypernuclei in emulsion experiments
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KEK-PS E176
MeV7.09.4B
MeV7.09.4B
or interaction energy
attractive or repulsive ???
Double hyper event from E-176 experiment
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Double hyper event from E-373 experiment
t
p
B : interaction energy
B= B(AZ) - 2B(
A-1Z)
Weakly attractive Interaction !
MeV54.082.5951Hem 6
MeV19.025.7B
MeV20.001.1B
MeV13.0Bassumed
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Nakazawa Sensei, 2003 Presentation
(at J-Lab)
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Nakazawa Sensei, 2003 Presentation
(at J-Lab)
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KEK-PS E176
MeV7.09.4B
MeV7.09.4B
or
(Possibility of excited state was not considered!)
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KEK-PS E373
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Construction of relative d- wave function by using one range phenomenological Gaussian potential
Gaussian basis radial wave function for d- cluster is
2j
b/r1
jj ercrU
Gaussian one range potential 2
b/r0d eVrVrV
bj= range parameter and
cj= the expasion coefficient Hamiltonian operator is
we adjusted the potential strength( -85.42MeV) to give energy eigen value of 1s state(-1.48 MeV ) and eigen function corresponding to this 1s bound .
By applying Fourier transform,
2dq
q~
2
b
3j
jjdd
2
j
ebc32
1~~
2.0 fm rVr
1
2dr
d
2H
2
2
2
22
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approximate value of an integral
Pick n randomly distributed points x1, x2, x3,…, xn in the interval [ a ,b ].
n
1iixf
n
1f̂Average value of
the function
f̂abdxxfb
a
Approximate value of an integral
n
f̂f̂abError
22 Estimation
for the error
n
1ii
22 xfn
1f̂
Monte Carlo Integration Method
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Bindingenergy of
Atomic absorption
3.59 MeV
2S 1.98
2P 0.62
3D 0.57
5.0 MeV2S 3.78
2P 1.08
3D 1.12
7.25 MeV
2S 6.37
2P 1.69
3D 2.01
Bindingenergy of
Atomic absorption
3.59 MeV
2S 3.99
2P 1.25
3D 1.19
5.0 MeV2S 6.34
2P 2.00
3D 2.01
7.25 MeV
2S 9.29
2P 2.83
3D 3.17
HeH 6ΛΛ
5ΛΛ
Γ/Γ HeH 6ΛΛ
5ΛΛ
Γ/Γ
GBWF (1s) OCM (1s)
Binding energy effect of
Ms. Hla Hla win (Ph D thesis, private communication) Binding energy of 6He ( NAGARA event data )
K.S. Myint et.al., Eur. Phys. J. A 16 (2003) 21
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Wave function types
Probabilities of low
momentum component
Probabilities of high
momentum component
GBWF(1s) 0.74 0.26
OCM(1s) 0.71 0.29
GBWF(0s’) 0.90 0.10
Table 2. Probabilities of momentum components of d- relative wave function of
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Kepe
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3Li6
no: of proton 3no: of neutron 3P 0s(2-1/2),0p(1-3/2)n 0s(2-1/2),0p(1-3/2)J=J(p)+J(n) =3/2+3/2 =3,2,1,0 ( 2 is impossible)=(-1)**(l_p+l_n) =(-1)**(1+1) =+J_=3+,1+,0+
Iso_spin
Transition matrix ,
qqqqqkkK B~t~~~dd
L
2T Hen
6
fi 5
intLidpnlm 6
~,~~ qqq
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MeV39.13dHLi 56
MeV88.31nHeLi 66
effect of low and high momenta component of d-
relative motion ???
to clarify this argument more
profoundly!
This wave function (0s’) is obtained by reducing the strength of one range Gaussian potential (-19.152MeV) to give the ground state energy E = -1.48 MeV.
d- density distribution of (in momentum space)
Li6
qd (MeV/c)
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Our University will be held the International Conference on February, 2011.