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 2010.12.17-18 二重ベータ崩壊研究懇談会

Part 1 Nuclear transition operators 2010.12.17-18 二重ベータ崩壊研究懇談会

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 2010.12.17-18 二重ベータ崩壊研究懇談会

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 2010.12.17-18 二重ベータ崩壊研究懇談会

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. 2010.12.17-18 二重ベータ崩壊研究懇談会

Part 2 Quasi-particle RPA calculations 2010.12.17-18 二重ベータ崩壊研究懇談会

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 2010.12.17-18 二重ベータ崩壊研究懇談会

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 2010.12.17-18 二重ベータ崩壊研究懇談会

pnQRPA Calculations 0n mode 2n mode by hand 0n mode 2n mode A poor predictive power for 2nbb NME 2010.12.17-18 二重ベータ崩壊研究懇談会

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 = 1.00 uncertainties of about 30% 2010.12.17-18 二重ベータ崩壊研究懇談会

Problems of calculations Part 3 Problems of calculations 2010.12.17-18 二重ベータ崩壊研究懇談会

Model dependence of 0nbb NME Systematic differences between nuclear models Interacting Boson Model J. Barea and F. Iachello, Phys. Rev. C79, 044301 (2009) 2010.12.17-18 二重ベータ崩壊研究懇談会

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 2010.12.17-18 二重ベータ崩壊研究懇談会

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 2010.12.17-18 二重ベータ崩壊研究懇談会

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. 2010.12.17-18 二重ベータ崩壊研究懇談会

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 2010.12.17-18 二重ベータ崩壊研究懇談会

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. 2010.12.17-18 二重ベータ崩壊研究懇談会

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. 2010.12.17-18 二重ベータ崩壊研究懇談会

Single-particle occupation probabilities J.P. Schiffer, et al. Phys. Rev. Lett. 100, 112501 (2008) B.P. Kay, et al. Phys. Rev. C79, 021301(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 !! 2010.12.17-18 二重ベータ崩壊研究懇談会

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 2010.12.17-18 二重ベータ崩壊研究懇談会