A Monte Carlo approach to the study of medium-light...

A Monte Carlo approach to the studyof medium-light hypernuclei properties

Diego Lonardoni

University of Trento - Physics DepartmentINFN - Gruppo Collegato di Trento

December 16, 2011

1. The method• DMC• DMC for nuclei: AFDMC

2. Hypernuclei• hyperon-nucleon interaction• AFDMC for hypernuclei

3. Preliminary results

4. Conclusions and perspectives

Outline

Diffusion Monte Carlo

� =it

� � ⇤

⇤�⇥(R, �) = H⇥(R, �)

⇥(R, �) = e�(H�E0)�⇥(R, 0)

=⇥�

n=0

cne�(En�E0)�⇤n(R)

�(R, 0) =��

n=0

cn⇥n(R)

lim��⇥

⇥(R, �) = c0⇤0(R)

DMC = projection method

Diffusion Monte Carlo

⇥(R, � + d�) = �R|⇥(� + d�)⇥

=

⌅�R| e�(H�E0)d�

⇤⇤R⇥⇥⌥ ⌃⇧ �

G(R,R�,d�)

�R⇥|⇥(�)

⇥dR⇥

walkers

potential term

d� d�

kinetic term

d�

Nucleon-nucleon interaction

2 body

3 body

Nucleon-nucleon interaction

Vij =n�

p=1

vp(r)Opij

2 body

3 body

Op=1,8ij = {1,�i · �j , Sij ,Lij · Sij}� {1, ⇥i · ⇥j}

Op=9,14ij =

⇤L2ij , L

2ij (�i · �j) , (Lij · Sij)

2⌅� {1, ⇥i · ⇥j}

Op=15,18ij =

�Tij , (�i · �j)Tij , Sij Tij , �

zi + �zj

⇥

Vijk = CSW2⇡ O2⇡,SW

ijk + CPW2⇡ O2⇡,PW

ijk + C�R3⇡ O3⇡,�R

ijk + CRORijk

Auxiliary Field Diffusion Monte CarloProblem:

P � e�12�d⇥O

2

e�12�d⇥O

2

=1�2�

�dx e�

x2

2 +⇥��d⇥xO

Idea: Hubbard-Stratonovich transformation

� �= A�⇤

i

ci · ⇥i

⇥

auxiliary fieldMC calculation

� A!

Z!(A� Z)!4A components

Auxiliary Field Diffusion Monte CarloAV6: Op=1,6

ij = {1,�i · �j , Sij}� {1, ⇥i · ⇥j}⇤ ⇥� ⌅VSD+VSI

VSD =1

2

A�

n=1

3�

�=1

�(⌅)n O(⌅)2

n� A(⌅)i,j : A�A

+1

2

3A�

n=1

�(⇤)n O(⇤)2

n A(⇤)i�,j⇥ : 3A� 3A

+1

2

3A�

n=1

3�

�=1

�(⇤⌅)n O(⇤⌅)2

n� A(⇤⌅)i�,j⇥ : 3A� 3A

rotation of spin-isospin states

15A auxiliary fieldsgood for Hubbard-Stratonovich

Hypernuclei

single �hypernucleus

double �hypernucleus

� = u+ d+ s

HypernucleiPANDA @ FAIR- anti-proton beam- double �-hypernuclei- �-ray spectroscopy

STAR @ RHIC- HI collider- anti �-hypernuclei- exotica?

JLAB- electron-production- single �-hypernuclei- �-wavefunction

SPHERE @ JINR- heavy ion beams- single �-hypernuclei- weak decay

BNL- heavy ions beams- anti-hypernuclei- single �-hypernuclei- double �-hypernuclei

ALICE @ LHC- URHIC collider- anti �-hypernuclei- exotica?

KEK � J-PARC- intense K� beams- single and double �-hypernuclei- �-ray spectroscopy for single �

KAOS @ MAMI C- electron-production- single �-hypernuclei- �-wavefunction FINUDA @ DAFNE

- e+e� collider- stopped-K� reaction- single �-hypernuclei- �-ray spectroscopy

HypHI @ GSI- heavy ion beams- single �-hypernuclei

at extreme isospins- magnetic moments

BNL JLABKEKJ-PARC

2010 2020

ALICEPANDASTAR

SPHEREFINUDAHypHI

2000

KAOS

Hypernuclei

H. Ðapo, B.-J. Schaefer, and J. Wambach. Appearance of hyperons in neutron stars. Phys. Rev. C, 81(3):035803, Mar 2010

Hyperon-nucleon interaction

2 body

3 body

[1]

Hyperon-nucleon interaction

V⇤i

(r) = v0(r) + v0(r)"(Px

� 1) +1

4v�

T 2⇡

(m⇡

r)�⇤ · �i

v0(r) = vc(r)� v2�(r)

vc(r) = Wc

�1 + e

r�r̄a

⇥�1

v2�(r) = v̄ T 2� (m�r)

2 body

3 body HS ok!only 2-body operators

HS ok!

V⇤ij = CSW2⇡ O2⇡,SW

⇤ij + CPW2⇡ O2⇡,PW

⇤ij + CDOD⇤ij

[1]

The idea: behind the scene

Startingpoint:

good wave function

robust method for nuclei +

different AFDMC approaches

different single particle orbitals

work in progress

present and future collaborations

-33.0

-32.9

-32.8

-32.7

-32.6

-32.5

-32.4

-32.3

-32.2

-32.1

-32.0

-31.9

-31.8

-31.7

-31.6

0e+00 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 8e-05 9e-05 1e-04

ener

gy [M

eV]

do [1/MeV]

4He binding energy

V4’ potential

Code PsiT-weight Skyrme.orbCode PsiT-weight HF-B1.orbCode Elocal-weight HF-B1.orbCode PsiT-weight HF-V4p.orb


GFMCPhys.Rev.Lett.89182501-1:4(2002)


-34.0

-33.0

-32.0

-31.0

-30.0

-29.0

-28.0

-27.0

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-25.0

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-22.0

-21.0

-20.0

-19.0

-18.0

0e+00 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 8e-05 9e-05 1e-04

en

erg

y [

Me

V]

d! [1/MeV]

4He binding energy

V6’ potential

Code PsiT-weight HF-B1.orb

Code Elocal-weight HF-B1.orb

Code Elocal-weight Skyrme.orb

Code Elocal-weight HF-V6p.orb

GFMCPhys.Rev.Lett.89182501-1:4(2002)

Hypernuclei preliminary results

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

6.50

7.00

0e+00 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 8e-05 9e-05 1e-04

ener

gy [M

eV]

do [1/MeV]

5RHe R-separation energy

V4’+VRN HF-B1.orb

V6’+VRN HF-B1.orb

V4’+VRN+VRNN HF-B1.orb

V6’+VRN+VRNN HF-B1.orb

3 body⇤NN

exp

[2] Journal of Physics G: Nuclear and Particle Physics, 32(3):363-373, 2006

[2]


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

dens

ity [1

/fm3 ]

r [fm]

5RHe density

N-density V4’+VRN HF-B1.orbR-density V4’+VRN HF-B1.orb


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

dens

ity [1

/fm3 ]

r [fm]

5RHe density

N-density V4’+VRN+VRNN HF-B1.orbR-density V4’+VRN+VRNN HF-B1.orb

3 body⇤NN


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

dens

ity [1

/fm3 ]

r [fm]

5RHe density

N-density V6’+VRN HF-B1.orbR-density V6’+VRN HF-B1.orb


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

dens

ity [1

/fm3 ]

r [fm]

5RHe density

N-density V6’+VRN+VRNN HF-B1.orbR-density V6’+VRN+VRNN HF-B1.orb

3 body⇤NN

Hyp: ok!

Conclusions and perspectives

AFDMC algorithm can be easily extended to the study of hypernuclear systems

development of an accurate hyperon-nucleon potential

development of an accurate hyperon-hyperon potential

study of heavier -hypernuclei: , � 17�O7

�He

study of -hypernuclei: 6��He��

Conclusions and perspectives

To do(high priority)

deeper investigation in the AFDMC algorithm

better trial wave function

single particle orbitals

correlation functions

Thank you

[1] Hypernuclear potentials:

A. A. Usmani, Steven C. Pieper, and Q. N. Usmani. Variational calculations of the –separation energy of the hypernucleus. Phys. Rev. C, 51(5):2347–2355, May 1995.

A. A. Usmani and S. Murtaza. Variational Monte Carlo calculations of hypernucleus. Phys. Rev. C, 68(2):024001, Aug 2003.

A. A. Usmani. N space–exchange correlation effects in the hypernucleus. Phys. Rev. C, 73(1):011302, Jan 2006.

5⇤He

5⇤He

⇤

⇤17⇤O

A Monte Carlo approach to the study of medium-light...

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