104/21/23 1

Symposium on Astro-Particle and Nuclear Physics

In Honour of 70th Birthday of

Prof. Q.N. Usmani

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Professor M. Z. Rahman Khan

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Energies of multi-strange α-cluster hypernuclei using variational Monte

Carlo Method

MOHAMMAD SHOEB

Department of Physics, Aligarh Muslim University, Aligarh-202 002,

India

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Outline

1.Introduction2. Hamiltonian in α-cluster model3. Potential models4.Variational wavefunctions5. Results and discussion6. Summary

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1. Introduction:Aim of nuclear physics

Complete knowledge of the interaction among octet of baryons in a unified way

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Motivation of studying Strange and multi-strange hypernuclei

• to extract interaction between hyperon-N and hyperon-hyperon

Existence of hypernuclei represent a new state of matter may exhibit new symmetries,

selection rules, etc.

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Presence of hyperon(s) may modify the properties of the core

moment inertia of deformed nucleus rotational and vibrational states structure of

nucleusHypernucleus provides us a opportunity to

investigate properties of hyperon(s) in nuclear medium

Hyperon(s) inside nuclei may be used as probe to study the nuclear structure

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904/21/23 9

It is believed that hyperons matter forming the inner core of neutron stars would have significant effect on their properties.

Schaffner-Bielich [NP 804(2008)309 and ref. their in ] has discussed that hypernuclear potential depths, two-body hyperon- nucleon and hyperon three-body forces as well as hyperon-hyperon interaction would

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10

Therefore, determination of hyperon- nucleon and hyperon-hyperon interaction becomes very important for investigating

the properties of neutron stars.

have a impact on the maximum mass, mass-radius relation, and cooling properties of

neutron stars.

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Hypernuclear physics is likely to play key role in the study: Properties of neutron stars

Equation state of nuclear matter Figure in the next slide shows interdisciplinary

nature of hypernuclei linking particle, nuclear, many-body, astrophysics etc.

[ref. Erni et al arXiv: hep-ex/0903.3905]04/21/23

1204/21/23

1304/21/23 13

Segr tablee

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14

Extension of the nuclear chart in a new dimension, strangeness S

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-Hypernuclear events

, ( ground and excited states), or

(Hida event ) and

He6 Be10

Be11

Be12 B13

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Hypernuclear experiments planned or operative at various (nine) laboratories all over the World

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Experimental facilities for hypernuclear physics program

List of a few leading laboratories where

Hypernuclear physics program to produce and identify hypernuclei with strangeness S= -1 to -3 is being pursued

• TJNAF(Thomas Jefferson National Accelerator Facility) at

Newport news in USA

Electro-production

• FINUDA(FIsica NUclearea DA NE): A special accelerator,

DA NE (Double Annular ring For Nice Experiment), designed at INFN (Instituto Nazionale di Fisica Nucleare)

Kepe

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1804/21/23 18

e e510

MeV510

MeV

(M=1020 MeV, 20

2310 s)

Head on collision

decays

),( 00 KK

),( KKCopious production

A beam of of extremely high intensity and precise low energy is expected to insert “strangeness” inside nucleus to

produce hypernuclei.

K

ZZK AAStop

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• J-PARC(Japan Proton Accelerator Research Complex) at

KEK: Already a rich data related to both the spectroscopy and

decay of hypernuclei at KEK have been measured.

Program for production and unambiguous identification of

hypernuclei and excited states using reaction

( )

Excited states of double-Λ hypernuclei

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KK ,

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20

• ANDA ( ANnihilation at DArmstadt ): A

beam hits primary target to produce

;

Stopping and absorption of in the

secondary target produce hypernucei.

Program to produce S= -3, -hypernuclei

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P P p

pp 0np

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21

MeV 28 p ppIn primary target In secondary target, e.g. Li, Be, B,

Schematic picture describing production of double Λ hypernuclei at PANDA

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Multi-strange hypernuclei

Schaffner et al [Ann. Phys.(NY)235(1994) 35 ] observed that a would become particle stable against the strong decay

if a sufficient number of bound ’s Pauli blocked this decay mode. Thus

is the lightest system suggested to study. At present production of multi-strange hypernuclei seems to be impossible.

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N

)4(7 SHe

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23

However, it will interesting to theoretically study the stability of multi-strange systems. Such a study is likely to have implication on future experimental efforts in producing,

identifying and measuring the properties of multi-strange hypernuclei.

Therefore, we have included in our study multi-strange hypernuclei apart from strange

ones.

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2. Hamiltonian in α-cluster modelHypernuclei studied in the α-cluster model using VMC

s-shell:

p-shell:

Systems within rectangular boxes are the ones whose stability predictions are to be made.

,6He He7

,9Be

,11Be ,13C

,9Be ,10Be),2/5,2/3(*9 Be ),2(*10

Be

,10Be ,13C C14),2(13

C

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Hamiltonian for the five-body system in Ξαα model with α treated as rigid:

Be11

4

5

α

α

Ξ

3

2

1Λ

Λ

5

412

2

1

)()(

)3()(

i

i

rViK

KiKH

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2

1

2

1

5

4

5

44533 )()()()(

i i j iiiji rVrVrVrV

),(2

154

iii rrV

…..(1)

hK K. E. operator , 21hhV potential energy for the

21hh

V; phenomenological

dispersive three-body potential with Yukawa

form factors.

the particle pair

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Hamiltonian for

5

312

2

1

)()()(ii

rViKiKH

C14 in ΛΛααα model

2

1

5

3

5

3

2

1

5

3

5

),()()(i j

jkijkji j ji

ijij rrVrVrV

),( 45,3534 rrrV3

4

5

1

2

….(2) α

αα

Λ

Λ:V phenomenological repulsive

three-body potential with

Gaussian form factors

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3. Potential Models

3.1 Two-body potentialsThree-range Gaussian BB(=ΛΛ, Ξ) potentials in spin state (=s,t)

….(3)

(7.26 MeV)

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Potentials

For =

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Potentials

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…(4)

(4)

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potentials for l th partial waves that fit scattering phase shifts. potential of Chien and Brown has been used for only as the energy is not very sensitive to the choice of the potential.

V

Ali-Bodmer

potentials

Be9

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Two-range Gaussian potentials

•Isle fits of and its weak decay

modes

B He5

•MSA is obtained from Brueckner-Hartee-Fock

Theory and slightly modified to fit

of

[Euro. Phys. J. 16(2003)21]

B He5

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…..(5)

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0V B

V

(4). In the previous slide its graph

is shown by black color line.

WS24 with = 24.0, as suggested by Dover and Gal [Ann.Phys. 146 (1983 )309], has been obtained from a analysis

of old and ambiguous data B

energy = -2.09 MeV for WS24 and Isle potential

= -0.06 MeV for WS14

0V

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3.2 Phenomenological Three-body potential among

and clusters

Microscopic calculations of Bodmer and Usmani

for shows that contribution of dispersive

three-body NN force for the triad

where one nucleon from each is participating

Be9

21NN

1N

2N

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is quite significant, neglecting it among cluster

overbinds and . In cluster model calculation we [Pramana68 (2007)943] have proposed to simulate phenomenologically the dispersive energy in the triad through a simple form

(6)

)()(210 rfrfWV

Be9 Be10

. arar /)exp(

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and .

Phenomenological three-body potential

gives good fit to the binding energy and rms radius of

in the cluster model for AB [ NP 83(1966)66 & phys

Lett B 389(1996)631] potential.

C12

15.0 fma

( 7)

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4.Variational wavefunctions Construction of good trial wavefunction• Physics necessary to describe the ground and excited state• Reasonably efficient to compute Wavefunctions are product of two-body correlation functions and the appropriate spin functions

4.1 Wavefunctions

(i) and model

g.s. ,degenerate doublet and ,

. Replacing by gives w.f. for .

)(rfhh

Be9

)2/1,2/1(),( zJJ 021l

)2/3,2/3(),( zJJ )2/5,2/5(

221l

:)2/5,2/3(*9 Be

Be9

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(ii) and : model

g.s. , excited state ,

(iii) and : model

g.s. , degenerate doublet ,

Be10 )2(*10

Be

)0,0(),( zJJ 021l )2,2(),( zJJ

221l

C13 )2/5,2/3(13

C

),2/1( m 021l

)2/5,2/3( 221l

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Replacing by gives w.f. of . (iv) Wavefunctions for , and

(a) wavefunction for in model:

=

C13

,6He He7 ,10Be

Be11

Be11

JJ z

)(43lmY

m

m

2/1

2

1

2100 )()(

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(b) Wavefunction for : suppress a and a indices

in the wavefunction of in (a) above

(c) Wavefunction for : suppress a index

in the wavefunction of in (a) above

(d) Similarly wavefunction for can be obtained.

He6

Be11

He7

Be11

,10Be

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4.2 Calculation of correlation function

A procedure developed by Urbana group. Solution of the following Schroedinger type equation

etc.

pair .

Potential between particles .

)(rf hh

hh

hh

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5.Procedure for energy calculation

( 8 )

For local operator H the energy can be written in a formsuitable for Monte Carlo calculation. Defining local energy

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and a multivariate probability distribution

( 9)

The variational energy is written as

(10 )

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(11)

General procedure for calculation of energy in VMCmethod:

(12)

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The energy is evaluated using

(i) model of : and

(ii) model of : and

(iii) model of : and (iv) model of : and

Similarly for other hypernuclei

He6 N

He7 N

N

,9Be

Be10 N

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To explore the structure of and

The quadrupole moment ( ) in the cluster model

is calculated using

where runs over coordinate of two s, treated as point particles

and distances are being measured from the cm of two alphas.

Be9 Be10

2efm

i

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5. Result and Discussion

VMC energy: s-Shell Hypernuclei and

NSC97e

,

:ESC00, ND, NSC97b

He6 He7

He6

He7

V

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: Isle and WS

= 2.09 MeV , =3.45 (Isle) and 3.36 (WS) fm

( ) fm for WS24

Isle WS

(12.6, 2.93) (4.8, 2.2)

00

B

00 , ra

0R

0potentials

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Filikhin et al

Faddeev

Faddeev-Yakubovsky

predicted for unbound for Isle soft

replusive potential bound for WS24 potential

Prediction: binding energy

capable of discriminating between radial shape of central

potentials, very unlikely, as this violates shape independence of

low energy data

He7

0

He6

He7

He7

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potential dependent configuration for

stronger potential a configuration speculated

i.e. screened by . configuration for weaker potential

Detailed VMC calculation

:ESC00, ND, NAGSIM, NSC97e, NSC97b

:NSC97e

: Isle , WS24, WS14

He7

0)( 0 )( 0

V

0V

0V

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F and FY [Filikhin et al JPG35(2008)] bold face within round bracket

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Our VMC calculation demonstrates and have

negative energies and is bound for WS24 and Isle

potential ( while is unbound for Isle potential in F-Y

method) .

Gross property such as energy, not good discriminator of the

shapes of the two-body potential.

is unbound for WS14 potential.

Due to strong conversion process ,we will comment

on stability of in the last.

He6 He7

He7

He7

He7

NHe6

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0

Pyramid on triangular isosceles base

Isosceles plane

Two planes

ESC00, NAGSIM, NSC97b

,R ,R R RandArms

in general increase with decrease in

strength of potential

but no change in the configuration

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)()( 76 HeSHeS

)(

being screened by two lambdas

our calculation support not )(

as opposed to speculation of Filikhin et al

configuration whether interaction is weak or strong

.[JPG36(2009)045104]

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p-Shell hypernuclei:

(i) , , , and

Experimental =6.71 MeV

=17.6 0.4 MeV ( 14.5 0.4 MeV assuming

, a ray of about 3.0 MeV must have escaped the identification of decay product from the emulsion ).Excited states:

=3.66 MeV

Demchi-Yanagi event

= MeV

Be9 Be10

C13 C14

)( 9BeB

)( 10BeB *910 BepBe

))2/5,2/3(( 9 BeB

))2(( 10 BeB 35.0

21.033.12

C130

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62

Revised = 11.90 0.13 MeV

=11.69 MeV

))2(( 10 BeB

)(13CB

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and of three-body potential are

adjusted to fit ground state energy

0W

Ali-Bodmer [Chien-Brown]

a0W

potential

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No free potential parameters

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No free potential parameters

Isle+AB

MSA+AB

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No free potential parameters

Demchi-Yanagi event B =12.33 MeV

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-Ve quadrupole moment for : Oblate shape

: Experimental Q =5.3

N replaced by

Prolate Oblate

Not much difference between the calculated energy using AB and

CB alpha-alpha, Isle and MSA lambad-alpha interactions.

Therefore, we consider only AB alpha-alpha and Isle lambda-alpha

interactions.

)2/5,2/3(9 Be

)2/3(9 Be 2fme

)2/3(9 Be )2/5,2/3(9 Be

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= 11.69 MeV First we analysed sub system Ground state energy = -7.26 MeV Excited state energy = -2.84 MeV Energies of ( and ) calculated variationally in cluster

model. i) interaction(AB) very repulsive, gives energy -0.7 MeV.

ii) + parameters and

adjusted to fit the ground state. sitting at the vertices of

equilateral triangle.

)(13CB Experimental

)0(12 C

)2(12 CC12 0 2

MeVW 0.163 fm7.7V

V V

4.17 fm

C12

2.4fm

c.m.04/21/23

6904/21/23 69

Filikhin et al JPG35 (2008)

FY calculation with out

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and in < for (Shrinkage of core )

> in , wanders at the periphery of core

R C13 C12

)3(R

)3( R )3(R C13

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Variational energy VMC energy

(present work) = - 4.99 MeV

in alpha cluster model using

Jacobian-coordinate

MeV

Gaussian-basis function method

( Hiyama et al [Prog.Theo.Phys.97(1997)881] ).

)2(12 C

0-7.17

-4.29

2

C12

Exp. -2.84MeV

Exp. -7.26MeVJ

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Ground and degenerate doublet

Interaction (AB)+ (Isle) + +

(no free

parameter)

Exp. -7.26+(-11.69) MeV

C13 )2/5,2/3(

2/1

)2/5,2/3(

-18.81

-14.98 Prediction

V V

C13 JMeV

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: VMC prediction for G.S. energy in model

fits energy(-7.26 MeV) + Isle + (AB) VMC Energy =-31.29 MeV; Predicted binding of = -7.16- (-31.29)= 24.12MeV

C14

He6

B

VV V

C14

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in model :VMC Prediction

(Isle & WS) + +

C13 0

0 V V

Filikhin et al JPG35(2008)

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(ii) Prediction for the energies of and

Calculated energy of the above systems in and models using combinations of (Isle, WS24, Ws14), (NSC97e) and (NSC97( b,e), NAGSIM) potentials along with dispersive three-body force.

,9Be Be10 Be11

,0

0 0

0 0

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VMC energy for < Faddeev method

Due to interplay between Calculation limited to s-wave

two-body correlations ,

contribution from partial

waves higher than s-wave

is simulated.

energy for Isle and WS24 differ by 3%

for twice of

Separation energy in nearly independent of

strength.

Be9

Be10

B Be10 He6

Be11S

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Do the negative energies of and implies stabilty ?

No

Due to strong conversion

depending on the depth of WS

potential, these systems can decay as:

;

Be10

He6

N0

HeHe 56

BeBe 910

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Calculated

level scheme

for two depths

of

potential.

It seems very

unlikely that

will ever

observed in

future experi-

mental effort.

0

Be10

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81

Note: All the calculations where appears were performed for hyperon. To obtain binding of hypernucleus a coulomb correction 1.5 to 2.0 MeV per alpha particle is to be added to the binding of hypernucleus containing .

04/21/23 81

0

0

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6.Summary• VMC for binding energy of three-, four-, and

five-body alpha cluster s- and p-shell hypernuclei

• First cluster model VMC calculation for predicting the energy of multistrange hypernuclei

• VMC energy for is insensitive to the shape of potential as opposed to Faddeev-Yakubovsky method

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He7

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• is unbound and stability depends on

potential depth• Predicted the energies for and for ground

state • , and are predicted to be stable for

WS14 • Demachi-Yanagi event is interpretted as excited

and exited state energy of degenerate doublet of explained.

)2(13 C

C14

Be90

C130

)2(10 Be

Be110

)2/5,2/3( Be9

Be100

He60

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Thank you

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