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

0 νββ decay NMEs with the generator coordinate method Changfeng Jiao Department of Physics Central Michigan University Feb. 3 rd @ UMASS

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 Current results with EDF-based GCM GCM (REDF) 8 GCM (NREDF) ISM 6 0 n M 4 2 0 136 Xe 48 Ca 76 Ge 82 Se 124 Sn 130 Te

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 Current results with EDF-based GCM Both the shell model and the EDF- GCM (REDF) 8 GCM (NREDF) based GCM could be missing ISM important physics. 6 0 n The discrepancy may be M 4 because: • The GCM omits correlations. 2 • The shell model omits many single-particle levels 0 136 Xe 48 Ca 76 Ge 82 Se 124 Sn 130 Te 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.

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. GCM 2. Correlations 3. Multi-shell GCM 4. Summary ① Using a shell-model Hamiltonian ② HFB states with multipole constraints q . | Φ ( q ) i We are trying to include all possible collective correlations. ③ Angular momentum and particle number projection MK ˆ P N ˆ | JMK ; NZ ; q i = ˆ P J P Z | Φ ( q ) i ④ Configuration mixing within GCM: X | Ψ J f JK NZ σ i = ( q ) | JMK ; NZ ; q i σ K,q 0 f JK X {H J KK 0 ( q ; q 0 ) − E J σ N J KK 0 ( q ; q 0 ) } f JK ( q 0 ) = 0 ( q ) σ σ K 0 ,q 0 M 0 νββ O 0 νββ = h Ψ J =0 N f Z f | ˆ | Ψ J =0 N i Z i i ξ ξ

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 0 = H − λ Z N Z − λ N N N − λ 0 Q 20 − λ P 2 ( P 0 + P † H 0 ) isoscalar pn pairing constrained g T = 0 = 0 0.2 g T = 0 = 0 is the isoscalar pairing amplitude φ | Ψ ( φ I ) | 2 76 Ge 0.1 φ = h P 0 + P † 0 i / 2 12 10 g T = 0 = 0 -12 | Ψ ( φ F ) | 2 1 ˆ P † l [ c † l c † X l ] L =0 ,S =1 ,T =0 g T = 0 = 0 10 0 = -4 5 M S =0 √ -8 2 8 l 0 -4 76 Se φ F 6 0 -5 The wave functions are pushed into 4 4 a region with large isoscalar pairing -10 2 8 amplitude. 0 -15 0 0.1 0.2 2 4 6 8 10 12 reduce the 0 νββ NMEs. | Ψ ( φ F ) | 2 φ I N. Hinohara and J. Engel, PRC 90, 031301(R) (2014)

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. 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 use the KB3G interaction for the pf shell 2.5 0.15 w/o isoscalar pairing 54 Ti w/ isoscalar pairing exact solution expt. 2.0 2.0 0.12 1.5 0.09 1.5 M GT 1.0 0.06 1.0 0.5 0.03 0.5 0.0 0.00 -0.2 -0.1 0.0 0.1 0.2 0.0 Deformation 48 48 Ti 54 54 Cr 54 54 Fe 2 collective wave function shows peaks Reduction of NME at nonzero isoscalar-pairing amplitude.

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 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary 0 = H − λ Z N Z − λ N N N − λ 0 Q 20 − λ P 2 ( P 0 + P † H 0 ) − λ 2 Q 22 triaxial deformation constrained 60 -81.5 76 Se GCN2850 60 (deg) 76 Ge GCN2850 -65.0 (deg) -82.0 45 -82.5 -65.5 45 -83.0 -66.0 -83.5 30 30 -84.0 -66.5 -84.5 -67.0 -85.0 15 15 -85.5 -67.5 -86.0 -68.0 -86.5 -87.0 -68.5 0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 2 2 With GCN2850 or JUN45 interaction, projected potential energy surfaces for 76 Ge and 76 Se 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 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary 0 = H − λ Z N Z − λ N N N − λ 0 Q 20 − λ P 2 ( P 0 + P † H 0 ) − λ 2 Q 22 triaxial deformation constrained 60 -81.5 76 Se GCN2850 60 (deg) 76 Ge GCN2850 -65.0 (deg) -82.0 45 -82.5 -65.5 45 -83.0 -66.0 -83.5 30 30 -84.0 -66.5 How does triaxial shape affect NMEs? -84.5 -67.0 -85.0 15 15 -85.5 -67.5 -86.0 -68.0 -86.5 -87.0 -68.5 0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 2 2 With GCN2850 or JUN45 interaction, projected potential energy surfaces for 76 Ge and 76 Se 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 1. GCM 2. Correlations 3. Multi-shell GCM 4. Summary 5 M GT w/o triaxial M GT w/ triaxial 15%~20% reduction 4 M F w/o triaxial for both GT and Fermi M F w/ triaxial part of NME if triaxial 3 0 n shape fluctuation is M included. 2 1 0 76 Ge JUN45 82 Se GCN2850 76 Ge GCN2850

Benchmarking: 0 νββ 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 6 SM 5 GCM 4 JUN45 GCN2850 GCN2850 M GT 3 2 SDPFMU-DB KB3G 1 0 48 Ca 76 Ge 76 Ge 82 Se 48 Ca

Benchmarking: 0 νββ 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 6 SM 5 GCM 4 JUN45 GCN2850 GCN2850 M GT 3 2 SDPFMU-DB KB3G A full sdpf -shell 1 GCM calculation 0 48 Ca 76 Ge 76 Ge 82 Se 48 Ca The NMEs given by SM and GCM are in good agreement, indicating that the GCM captures most important valence-shell correlations.

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

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

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