0 νββ nuclear matrix elements within QRPA and its variants

1. 0νββ nuclear matrix elements within QRPAand its variants W. A. Kamiński1, A. Bobyk1 A. Faessler2 F. Šimkovic2,3, P. Beneš4 1Dept. of Theor. Phys., Maria Curie-Skłodowska University, Lublin, Poland 2Inst. of Theor. Phys., University of Tuebingen, Germany 3Dept. of Nucl. Phys., Comenius University, Bratislava, Slovakia 4IEAP, Czech Technical University, Prague, Czech Republic

2. Experiment gives upper bound on • must come from the theory Motivation • Upper bound on the neutrino mass:

3. The operators: do not fulfil the bosonic commutation relations • BCS state is not a QRPA ground state: QRPA drawbacks

4. The QRPA built naively on the BCS would be a pure TDA: • QRPA ground statehas a non-vanishing quasiparticle content: QRPA drawbacks (cont.)

5. What should we learn? • The description of the ground state should be made consistent with that of the excited states • One should go beyond QBA and not neglect the scattering terms

6. The mapping: • The renormalized operators and amplitudes: Formalism – RQRPA

7. The linear equations for q.p. densities: with Formalism – RQRPA (cont.) • Solve the RQRPA iteratively, i.e. na(out)=na(in).

8. The modified BCS tensors: Formalism – SQRRPA • Computational procedure: iterate between BCS and RQRPA untill the convergence is achieved, i.e. na(out)=na(in).

9. 0.4 0.3 0.2 0.1 0.0 -0.1 ] -1 -0.2 [MeV -0.3 GT n -0.4 2 M -0.5 -0.6 SRQRPA -0.7 RQRPA QRPA -0.8 -0.9 10 13 16 20 26 13 16 19 23 76Ge →76Se: dependence on the s.p. basis 16 no core O core SRQRPA RQRPA QRPA number of levels

10. 0.4 0.3 0.2 0.1 0.0 -0.1 ] -0.2 -1 [MeV -0.3 -0.4 GT n 2 M -0.5 -0.6 -0.7 QRPA SRQRPA RQRPA RQRPA -0.8 SRQRPA QRPA -0.9 -1.0 13 16 19 13 16 19 23 number of levels 100Mo →100Ru: dependence on the s.p. basis 16 no core O core

11. 116Cd →116Sn 0.2 gph=1.0 QRPA gph=1.0 RQRPA gph=1.0 SRQRPA gph=0.8 QRPA gph=0.8 RQRPA 0.1 gph=0.8 SRQRPA ] -1 [MeV GT n 0.0 2 M -0.1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 g pp

12. 0.5 0.4 0.3 ] -1 0.2 [MeV 0.1 GT n 2 M 0.0 -0.1 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 g pp 128Te →128Xe g =1.0 QRPA ph g =1.0 RQRPA ph g =1.0 SRQRPA ph g =0.8 QRPA ph g =0.8 RQRPA ph g =0.8 SRQRPA ph

13. 0.3 0.2 0.1 0.0 -0.1 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 g pp 130Te →130Xe 0.5 g =1.0 QRPA ph g =1.0 RQRPA 0.4 ph g =1.0 SRQRPA ph g =0.8 QRPA ph g =0.8 RQRPA ph g =0.8 SRQRPA ] ph -1 [MeV GT n 2 M

14. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 g pp 136Xe →136Ba 0.9 g =1.0 QRPA ph 0.8 g =1.0 RQRPA ph g =1.0 SRQRPA 0.7 ph g =0.8 QRPA ph g =0.8 RQRPA 0.6 ph g =0.8 SRQRPA ] ph 0.5 -1 0.4 [MeV 0.3 GT n 2 0.2 M 0.1 0.0 -0.1

15. 146Nd →146Sm 0.5 g =1.0 QRPA ph g =1.0 RQRPA 0.4 ph g =1.0 SRQRPA ph g =0.8 QRPA ph 0.3 g =0.8 RQRPA ph g =0.8 SRQRPA ph ] 0.2 -1 [MeV 0.1 n GT 2 M 0.0 -0.1 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 g pp

16. 0.7 0.6 0.5 0.4 0.3 0.2 n 2 M 0.1 0.0 -0.1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 g pp 150Nd →150Sm g =1.0 QRPA ph g =1.0 RQRPA ph g =1.0 SRQRPA ph g =0.8 QRPA ph g =0.8 RQRPA ph g =0.8 SRQRPA ] ph -1 [MeV GT

17. Summary

18. Conclusions • The RQRPA and the SRQRPA are more stable with growing dimension of the single-particle model space • The RQRPA reproduces the experimental data for higher values of the particle-particle force • The SRQRPA behaves like QRPA, but the collapse is pushed forward towards higher gpp values

19. Conclusions (cont.) • 0νββ nuclear matrix elements can beaccuratelyreproduced within QRPA, RQRPA and SQRPA by fixing the gpp value using 2νββ experimental data • For the closed and partially closed shell nuclei (48Ca, 116Sn, 136Xe) a further improvement in the description of pairing interaction is necessary

20. References • S. M. Bilenky, A. Faessler, F. Šimkovic, Phys. Rev. D 70, 033003 (2004) • V. A. Rodin, A. Faessler, F. Šimkovic, P. Vogel, Phys. Rev. C 68, 044302 (2003) • A. Bobyk, W. A. Kamiński, F. Šimkovic, Phys. Rev. C 63, 051301(R) (2001)

21. Thank you for attention! W. A. Kamiński, A. Bobyk A. Faessler F. Šimkovic, P. Beneš