0 decay NMEs with the generator coordinate method Changfeng Jiao - - PowerPoint PPT Presentation

Changfeng Jiao

Department of Physics Central Michigan University

• Feb. 3rd @ UMASS

0νββ decay NMEs with the generator coordinate method

Generator Coordinate Method (GCM)

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

Generator Coordinate Method: an approach that treats large-amplitude fluctuations, which is essential for nuclei that cannot be approximated by a single mean field.

How it works: Construct a set of mean-field states by constraining coordinates, e.g., quadrupole moment. Then diagonalize Hamiltonian in space of symmetry-restored nonorthogonal vacua with different amounts of quadrupole deformation. GCM based on EDF has been applied to double-beta decay, however…

Comparison between GCM and SM

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

2 4 6 8

136Xe 130Te 124Sn 82Se 76Ge

GCM (REDF) GCM (NREDF) ISM

M

0n

48Ca

Current results with EDF-based GCM

Comparison between GCM and SM

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

2 4 6 8

136Xe 130Te 124Sn 82Se 76Ge

GCM (REDF) GCM (NREDF) ISM

M

0n

48Ca

Our long-term goal is to combine the virtues of both frameworks through an EDF-based or ab-initio GCM that includes all the important shell model correlations and a large single-particle space.

Current results with EDF-based GCM

The discrepancy may be because:

• The GCM omits correlations.
• The shell model omits many

single-particle levels

Both the shell model and the EDF- based GCM could be missing important physics.

To get closer to the ultimate goal:

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

We can use SM Hamiltonian in the GCM Our short-term goal is more modest: a shell-model Hamiltonian-based GCM in one and two (and possibly more) shells.

At a minimum, we can use these as a first step in the MR-IMSRG (see J. M. Yao’s talk).

Our Current Procedure

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary

① Using a shell-model Hamiltonian ② HFB states with multipole constraints q. We are trying to include all possible collective correlations. ③ Angular momentum and particle number projection ④ Configuration mixing within GCM:

• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

|Φ(q)i

|JMK; NZ; qi = ˆ P J

MK ˆ

P N ˆ P Z|Φ(q)i

|ΨJ

NZσi =

X

K,q

f JK

σ

(q)|JMK; NZ; qi

X

K0,q0

{HJ

KK0(q; q0) − EJ σ N J KK0(q; q0)}f JK σ

(q0) = 0

M 0νββ

ξ

= hΨJ=0

Nf Zf | ˆ

O0νββ

ξ

|ΨJ=0

NiZii

f JK

σ

(q)

Level 1 GCM: Axial shape and pn pairing fluctuation

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

H

0 = H − λZNZ − λNNN − λ0Q20 −λP

2 (P0 + P †

0 )

isoscalar pn pairing constrained is the isoscalar pairing amplitude

φ = hP0 + P †

0 i/2

φ

The wave functions are pushed into a region with large isoscalar pairing amplitude. reduce the 0νββ NMEs.

• N. Hinohara and J. Engel, PRC 90, 031301(R) (2014)

2 4 6 8 10 12 0.1 0.2

φF |Ψ(φF)|2

76Se

0.1 0.2

|Ψ(φI)|2 |Ψ(φF)|2

76Ge

2 4 6 8 10 12

φI

• 15
• 10
• 5

5 10

8 4

• 4
• 4
• 8
• 12

gT=0 = 0 gT=0 = 0 gT=0 = 0 gT=0 = 0

P †

0 =

1 √ 2 X

l

ˆ l[c†

l c† l ]L=0,S=1,T =0 MS=0

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary

Level 1 GCM: Axial shape and pn pairing fluctuation

• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

We use the KB3G interaction for the pf shell

• 0.2
• 0.1

0.0 0.1 0.2 0.0 0.5 1.0 1.5 2.0 2.5

54Ti

Deformation

2

0.00 0.03 0.06 0.09 0.12 0.15

0.0 0.5 1.0 1.5 2.0

48 48Ti 54 54Fe 54 54Cr

MGT

w/o isoscalar pairing w/ isoscalar pairing expt.

collective wave function shows peaks at nonzero isoscalar-pairing amplitude.

exact solution

Reduction of NME

With GCN2850 or JUN45 interaction, projected potential energy surfaces for 76Ge and 76Se give minima with triaxial deformation.

Level 2 GCM: Triaxial deformation

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary

H

0 = H − λZNZ − λNNN − λ0Q20 −λP

2 (P0 + P †

0 ) −λ2Q22

triaxial deformation constrained

• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

15 30 45 60 0.0 0.1 0.2 0.3 0.4 0.5

(deg)

76Ge GCN2850

2

• 68.5
• 68.0
• 67.5
• 67.0
• 66.5
• 66.0
• 65.5
• 65.0

15 30 45 60 0.0 0.1 0.2 0.3 0.4 0.5

(deg)

76Se GCN2850

2

• 87.0
• 86.5
• 86.0
• 85.5
• 85.0
• 84.5
• 84.0
• 83.5
• 83.0
• 82.5
• 82.0
• 81.5

With GCN2850 or JUN45 interaction, projected potential energy surfaces for 76Ge and 76Se give minima with triaxial deformation.

Level 2 GCM: Triaxial deformation

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary

H

0 = H − λZNZ − λNNN − λ0Q20 −λP

2 (P0 + P †

0 ) −λ2Q22

triaxial deformation constrained

• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

15 30 45 60 0.0 0.1 0.2 0.3 0.4 0.5

(deg)

76Ge GCN2850

2

• 68.5
• 68.0
• 67.5
• 67.0
• 66.5
• 66.0
• 65.5
• 65.0

15 30 45 60 0.0 0.1 0.2 0.3 0.4 0.5

(deg)

76Se GCN2850

2

• 87.0
• 86.5
• 86.0
• 85.5
• 85.0
• 84.5
• 84.0
• 83.5
• 83.0
• 82.5
• 82.0
• 81.5

How does triaxial shape affect NMEs?

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary

Level 2 GCM: triaxial deformation

• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

1 2 3 4 5

M

0n

82Se GCN2850 76Ge JUN45

MGT w/o triaxial MGT w/ triaxial MF w/o triaxial MF w/ triaxial

76Ge GCN2850

15%~20% reduction for both GT and Fermi part of NME if triaxial shape fluctuation is included.

Benchmarking: 0νββ NMEs given by GCM and SM

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

1 2 3 4 5 6

GCN2850 SDPFMU-DB

76Ge 48Ca

SM GCM MGT

48Ca 82Se 76Ge

KB3G JUN45 GCN2850

Benchmarking: 0νββ NMEs given by GCM and SM

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

1 2 3 4 5 6

GCN2850 SDPFMU-DB

76Ge 48Ca

SM GCM MGT

48Ca 82Se 76Ge

KB3G JUN45 GCN2850

The NMEs given by SM and GCM are in good agreement, indicating that the GCM captures most important valence-shell correlations.

A full sdpf-shell GCM calculation

Multi-shell GCM

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

In principle, effective pfsdg-shell interaction based on chiral EFT can be calculated by many-body perturbation theory (MBPT), similarity renormalization group (SRG) or couple cluster (CC). We employ two effective pfsdg-shell interactions calculated by MBPT, which are provided by J. D. Holt.

Computing Usage:

• Our calculation within pf5g9 shell used about 15K CPU hours,

including axial shape, triaxial shape, and isoscalar pairing as coordinates.

• Extension to pfsdg shell will increase time by a factor of 25,

because of the increased number of orbits.

pfsdg-1: 3N forces normal ordered with respect to 40Ca pfsdg-2: 3N forces normal ordered with respect to 56Ni

Multi-shell GCM: SPEs optimization

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

We optimize the single-particle energies for pfsdg-shell interactions by fitting the measured occupancies of valence neutron and proton orbits.

Neutron-orbit occupancies Proton-orbit occupancies

Multi-shell GCM: low-lying spectra

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

GCM GCM

Multi-shell GCM: collective wave function

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary
• 0.4
• 0.2

0.0 0.2 0.4 1 2 3 76Se: GCN2850

Deformation

2

0.00 0.04 0.08 0.12 0.16 0.20

• 0.4
• 0.2

0.0 0.2 0.4 1 2 3 4 5 76Se: pfsdg

Deformation

2

0.00 0.02 0.04 0.06 0.08 0.10

• Larger model space: larger isoscalar pairing in pfsdg-shell calculation
• How does triaxial shape influence NMEs?

Multi-shell GCM: triaxial deformation

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

0.0 0.2 0.4 1 2 3 4 5

Deformation

2

0.00 0.02 0.04 0.06 0.08 0.10 0.12

= 60

0.0 0.2 0.4 1 2 3 4 5

= 30

With triaxially deformed configurations, the wave functions: ① are pushed to the region with larger isoscalar pn pairing. ② spread widely to the region with larger deformation

76Se 76Se

Multi-shell GCM: triaxial deformation

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

0.0 0.2 0.4 1 2 3 4 5

Deformation

2

0.00 0.02 0.04 0.06 0.08 0.10 0.12

= 60

0.0 0.2 0.4 1 2 3 4 5

= 30

With triaxially deformed configurations, the wave functions: ① are pushed to the region with larger isoscalar pn pairing. ② spread widely to the region with larger deformation

76Se 76Se

Multi-shell GCM

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. Introduction 2. GCM based on shell-model Hamiltonian 3. Calculations and results 4. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

full pfsdg-shell GCM calculations

A relatively simple strategy for stochastic basis selection

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

Start axial for initial nucleus

HJ(q, q0)

axial for final nucleus

HJ(q, q0)

axial

GCM given by axially- deformed basis states

MJ=0(qf, q0

i)

M 0ν

stochastically selection from rest of |Φ(qs)i GCM updated

M 0ν

If , keep

• therwise, throw it
• away. Loop over all the

initial and final calculate and with

M

H

|Φ(qs)i

∆M 0ν

ξ

> cξ |Φ(qs)i |Φ(qs)i

We still want to add additional coordinates, e.g., pp/nn pairing, quasiparticle excitation, etc.

Summary

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary
• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary

We are trying to combine the virtues of the shell model and EDF calculations by including all collective correlations in the GCM. Tests against exact solutions in one shell indicate that we indeed have all important valence-space correlations. Calculation has been extended to two major shell (e.g., pfsdg shell) model space, which is out of scope of the conventional

• SM. Including triaxially deformed configurations significantly

affect the calculated NMEs. To speed up the two-shell calculation, stochastic selection of basis states is under construction, and we are looking for more efficient methods.

• 1. Introduction 2. GCM method 3. Shell-model interaction 4. Calculations and results 5. Summary

Collaborators:

• Jonathan Engel, UNC
• Jiangming Yao, UNC
• Mihai Horoi, CMU
• Jason Holt, TRIUMF
• Javier Menendez, University of Tokyo
• Nobuo Hinohara, University of Tsukuba

Summary

Thank you for your attention!

• 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary