1. Kazuo MUTO Tokyo Institute of Technology
核行列要素計算の現状と問題点
Status and problems of nuclear matrix elements calculations
Kazuo MUTO
Tokyo Institute of Technology
Nuclear transition operators
Quasi-particle RPA calculation
Problems of calculations
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2. Part 1
Nuclear transition operators
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3. Two decay modes 2n mode 0n mode Allowed in SM Forbidden in SM
Observed for 10 nuclides
Shortest half-life： T1/2 = 1019 y
0n mode
Forbidden in SM
No observation so far
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4. Momentum transfer 2n mode 0n mode Weak Hamiltonian
1. Nonrelativistic reduction of nucleon current 2. Multipole expansion into spherical tensors
2n mode
0n mode
3. Long wave length limit
Large momentum transfer ～100 MeV/c
Gamow-Teller transitions: momentum transfer ～ 0
→ nucleon form factor
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5. Nuclear transition operators
Sequential (Gamow-Teller) beta decays through virtual intermediate states.
The neutrino exchange with momentum integral generates a neutrino potential, which resembles a Yukawa potential with a “range” of about 20 fm.
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6. Part 2
Quasi-particle RPA calculations
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7. Ground-state correlations
bb decay : 0+gs → 0+gs
Important components of NN interaction:
Pairing between like nucleons →　BCS
Proton-neutron interaction, in particular QQ force →　RPA 　Random-Phase Approximation
BCS
RPA
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8. proton-neutron QRPA BCS RPA Ground state ansatz
Variation with constraints
quasiparticle
RPA
Excitation Operator charge-changing modes
Equation of motion
RPA equation quasi-boson approximation
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9. pnQRPA Calculations 0n mode 2n mode
by hand
0n mode
2n mode
A poor predictive power for 2nbb NME
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10. The best pnQRPA prediction
Experimental half-life of 2nbb decay is used to fix gpp.
Rodin et al., NPA 793 (2007) 213
a) different model spaces
b) different effective NN interactions
c) different QRPA models
d) gA = 1.25 and gA = uncertainties of about 30%
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11. Problems of calculations
Part 3
Problems of calculations
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12. Model dependence of 0nbb NME
Systematic differences between nuclear models
Interacting Boson Model
J. Barea and F. Iachello, Phys. Rev. C79, (2009)
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13. Ratio of MGT and MF
A prediction of MF is expected to be more stable.
gV is free from quenching (no spin operator)
little affected by ground-state correlations
If there is “universality” in the ratio MGT / MF, it can be used for more reliable predictions.
bb decay : nn → pp
A pair of nucleons with T=1, S=0
MGT / MF = -3
QRPA calculations give
MGT / MF = -2 ～ -3
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14. Multipole expansion of 0nbb operators
bb decay :0+ → 0+
Multipole single-particle transition operators
Nuclear intermediate states
all Jp, except 0+
only natural-parity states
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15. Violation of isospin symmetry
Spin-parity of nuclear intermediate states
The large 0+ component of MF is due to violation of isospin symmetry involved in the QRPA models.
Isospin is a good quantum number in shell model.
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16. Classification of beta decays
spin-parity of nuclear intermediate states DL
DJp
allowed
0+, 1+
1st forbidden 1
0-, 1-, 2-
2nd forbidden 2
1+, 2+, 3+
3rd forbidden 3
2-, 3-, 4-
4th forbidden 4
3+, 4+, 5+
5th forbidden 5
4-, 5-, 6-
Fermi, Gamow-Teller
Spin-Dipole type X X X X
Unique : only one transition operator
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17. Quenching of Gamow-Teller strengths
A systematic shell-model calculation：(sd)A-16
Experimental data are well reproduced with a quenching factor of 0.77.
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18. Quenching of M2 strengths?
Unique first-forbidden b decay
quenching factor ?
B. Singh et al., Nuclear Data Sheets 84, 487 (1998)
A number of experimental data, which are, however, uncertain.
Many of them are in heavy nuclei, and structure calculations are difficult.
There are no systematic calculations which allow to discuss quenching.
New data are expected in neutron-rich nuclei.
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19. Single-particle occupation probabilities
J.P. Schiffer, et al. Phys. Rev. Lett. 100, (2008)
B.P. Kay, et al. Phys. Rev. C79, (R) (2009)
Occupation probabilities of QRPA deviate much from experimental data.
g9/2 is most important in 76Ge
Improved occupation probabilities reduce 0nbb NME （76Ge）by 20%.
Data for other nuclei !!
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20. Summary
Predictions of 0nbb nuclear matrix elements by nuclear structure models involve uncertainties of a factor of two For more reliable predictions
Comparison between nuclear structure models
Multipole decomposition of NME
Violation of isospin symmetry
Ratio of MGT and MF
Experimental data and comparison with systematic nuclear structure calculations
Occupation probabilities of single-particle orbits
Spin-dependent transitions, in particular, M2
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