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Forbidden Beta Transitions in Neutrinoless Double Beta Decay

Forbidden Beta Transitions in Neutrinoless Double Beta Decay. Kazuo Muto Department of Physics, Tokyo Institute of Technology. 1. Quenching of spin-dependent transitions Violation of isospin symmetry 3 . Nuclear monopole interaction. V - A. V - A. A • A. V • V. A • A.

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Forbidden Beta Transitions in Neutrinoless Double Beta Decay

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  1. Forbidden Beta Transitionsin Neutrinoless Double Beta Decay Kazuo Muto Department of Physics, Tokyo Institute of Technology • 1.Quenching of spin-dependent transitions • Violation of isospin symmetry • 3. Nuclear monopole interaction July 29-30, 2010, Dresden

  2. V-A V-A A • A V • V A • A The mass term of 0nbbdecay The momentum integral of the virtual neutrino gives rise to a neutrino potential, which acts on the nuclear wave functions, being a long-range Yukawa-type (“range” ~ 20 fm). There appear three nuclear matrix elements, with two-body nuclear transition operators. July 29-30, 2010, Dresden

  3. Multipole expansion of NME(QRPA) spin-parities of nuclear intermediate states July 29-30, 2010, Dresden

  4. Part 1 Quenching of spin-dependent transitions July 29-30, 2010, Dresden

  5. Renormalization of operators due to model space truncation • Nuclear structure calculations (QRPA and shell models) in a finite model space • Renormalization of effects of coupling: model space and outside the model space • NN interaction (eg. G-matrix) • Transition operators First-order approximation by effective coupling constant July 29-30, 2010, Dresden

  6. Quenching of GT Strength Systematic analysis of GT beta decays in sd-shell nuclei The experimental data are well reproduced with a quenching factor of 0.77, in(sd)A-18calculation. July 29-30, 2010, Dresden

  7. outside the model space GT-strength Distribution The strength distribution, deduced from the charge-exchange (p,n) reaction, extends to high-excitation energy region, far beyond the giant resonance. T. Wakasa et al., Phys. Rev. C55, 2909 (1997) July 29-30, 2010, Dresden

  8. Magnetic Stretched States(Jp = 4-,Jp = 6-,Jp = 8- ) • Transitions between single-particle orbits with the largest in respective major shells • Unique configuration within excitation • The observed strengths are quenched considerably, compared with the s.p. strength, probably due to coupling with higher excitations: July 29-30, 2010, Dresden

  9. Quenching of M4 strengths (1) A perturbative calculation of M4 transition strength in 16O with G-matrix. July 29-30, 2010, Dresden

  10. second order first order Quenching of M4 strengths (2) July 29-30, 2010, Dresden

  11. Reductions in amplitude (%): at q = qpeak at q = 100 MeV/c Quenching of M4 strengths (3) July 29-30, 2010, Dresden

  12. Part 2 Violation of isospin symmetry July 29-30, 2010, Dresden

  13. Multipole expansion of NME(QRPA) The large 0+ contribution in QRPA calculationsis due to isospin symmetry breaking. July 29-30, 2010, Dresden

  14. with for the modified Hamiltonian with constraints for expectation values of the nucleon numbers BCS formalism (1) Ansatz for the BCS ground state Variation with respect to the occupation amplitudes July 29-30, 2010, Dresden

  15. The variation gives BCS formalism (2) the BCS equations pairing interaction s.p.e. for thecore nucleus two-body interactionbetween valence nucleons July 29-30, 2010, Dresden

  16. Isospin symmetry in BCS The proton and neutron systems are coupled through the proton-neutron interaction. • Isospin symmetry is conserved, if • the s.p.e. spectra of the proton and neutron systems are the same (or a constant shift) for the N = Z core nucleus, • s.p.e. are calculated with the two-body interaction. July 29-30, 2010, Dresden

  17. In QRPA calculations, we usually replace the s.p.e. by energy eigenvalues of a nucleon in a Woods-Saxon potential. This introduces a violation of isospin symmetry. Isospin violation in BCS Shell model and self-consistent HF(B) calculations conserve the isospin symmetry, or a small violation. July 29-30, 2010, Dresden

  18. Part 3 Nuclear monopole interaction July 29-30, 2010, Dresden

  19. Definition of single-particle energies (1) Prescription by Baranger Nucl. Phys. A149 (1970) 225 July 29-30, 2010, Dresden

  20. interaction with the core nucleons interaction with the valence nucleons monopole Definition of single-particle energies (2) the same form as the BCS formalism July 29-30, 2010, Dresden

  21. exchange : exactly the same quantity that appears in s.p. energies. Monopole interaction The monopole interaction is defined as the lowest-rank term of multipole expansion of two-body NN interaction. Proton-neutron interaction monopole interaction Like-nucleon interaction monopole interaction July 29-30, 2010, Dresden

  22. particle-particle particle-hole Universality of pn interactionwhen normalized by the monopole J.P. Schiffer and W.M. True,Rev. Mod. Phys. 48, 191 (1976) July 29-30, 2010, Dresden

  23. j’< j’> j< j> Roles of C, T，LS interactions When both spin-orbit parners, j< and j>, are filled with nucleons, + LS central + tensor For s.p. energies of j’< and j’>， (1) Central forces give the same gain, (2) Tensor forces give no change, (3) Spin-orbit forces enlarge the splitting. July 29-30, 2010, Dresden

  24. G-matrix is not good! USB: filled symbols G-matrix: open symbols “G-matrix is good except the monopole” The monopole strengths are accumulated in s.p.e., especially in a calculation with a large model space. July 29-30, 2010, Dresden

  25. Conclusions • Spin-dependent transitions are quenched by a factor of about 0.75 in amplitudes in a truncated model space due to coupling to higher-lying configurations. The quenching factor seems to be independent of the multipoles. • Approximations in the commonly used QRPA model violate the isospin symmetry, which overestimates the 0+ component of the 0nbb NME to a large extent. • Improvement is necessary in the monopole component of effective NN interactions. • A more reliable prediction of the 0nbb NME requires detailed comparison between results of QRPA, shell- model and IBM calculations. July 29-30, 2010, Dresden

  26. July 29-30, 2010, Dresden

  27. 0nbbdecay transition operators Double Gamow-Teller ME (magnetic type) Double Fermi ME (electric type) July 29-30, 2010, Dresden

  28. Jp = 1-Component • Isovector Electric Dipole Transitions • E1 excitation strengths in the same nucleus well satisfy the TRK sum rule. Highly collective • No renomalization of the coupling constant July 29-30, 2010, Dresden

  29. Jp = 2+Component • Electric Quadrupole Transitions • Systematic analyses of E2 transitions have shown that • Isoscalar transitions are enhanced, • Isovector strengths have no renormalization. July 29-30, 2010, Dresden

  30. Jp = 0+Component • The largest component of the double Fermi ME about 1/3 of • A shell-model calculation(for 48Ca) gives almost 0. • This large value is possibly due to violation of isospin symmetry in QRPA calculations. July 29-30, 2010, Dresden

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