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

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 .

(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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

For single- hypernuclei case,

234.0

He6

192.0

H5

He5 H4

and are at rest!

150 MeV/c113.82 MeV/c

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.

Thank you for your

kind attention!

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

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

-85.42

E =-1.48 MeV

-85.42

-19.152

-85.42

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

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

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

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

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

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))

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

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

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.

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

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.

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 )

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

H. Takahashi, “PhD Thesis”, Kyoto University (2003)

Possible Candidates of double- hypernuclei in emulsion experiments

KEK-PS E176

MeV7.09.4B

MeV7.09.4B

or interaction energy

attractive or repulsive ???

Double hyper event from E-176 experiment

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

Nakazawa Sensei, 2003 Presentation

(at J-Lab)

Nakazawa Sensei, 2003 Presentation

(at J-Lab)

KEK-PS E176

MeV7.09.4B

MeV7.09.4B

or

(Possibility of excited state was not considered!)

KEK-PS E373

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

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

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

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

Kepe

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

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)

Our University will be held the International Conference on February, 2011.