Double Beta Decay matrix elements, remarks and perspectives

Transcription

1 Double Beta Decay matrix elements, remarks and perspectives Petr Vogel, Caltech NNR05 Workshop CAST/SPring-8 Dec. 4, 2005

2 Thanks to the discoveries of the recent past we know a lot about neutrinos. But, there are still many things we do not know and would like to know: Are neutrinos Majorana particles? What is the pattern of neutrino masses? What is the absolute mass scale? Is CP invariance violated in the lepton sector? Is there a relation between all of this and the baryon excess in the Universe? Double beta decay is a process that can help in answering some of these questions

3 What are the questions that we should ask and try to answer? 1. What is the physics of the fundamental dd -> uuee lepton number violating process? How can we tell? 2. How to relate the 0νββ decay rate to the fundamental parameters? In other words, how can we evaluate the nuclear matrix elements? And how uncertain their values are? 3. What is the relation of the deduced fundamental parameters and the neutrino mixing matrix? Or, in other words, what is the relation between the 0νββ decay rate and the absolute neutrino mass.

4 Assume that indeed 1/τ ββ ~ <m ββ > 2. What is the relation between this quantity and the absolute neutrino mass? <m ββ > = Σ U ei2 m i. This quatity depends on the oscillation parameters, on the unknown phases of U ei, and on the unknown absolute neutrino mass. The dependence on the Majorana phases can be eliminated by considering the maximum and minimum values of <m ββ >. Only the dependence on the absolute mass remains.

5 minimum mass, not observable from observational cosmology, M = m 1 +m 2 +m 3 from β decay <m ββ > vs. the absolute mass scale blue shading: normal hierarchy, m 2 31 > 0. red shading: inverted hierarchy m 2 31 < 0 shading:best fit parameters, lines 95% CL errors. Thanks to A. Piepke

6 <m ββ > (ev) The degenerate mass region can be explored by ways independent on the Majorana nature of neutrinos Planck +SDSS sensitivity Katrin sensitivity

7 Nuclear matrix elements A provocative question: Do we know at all how large the matrix elements really are? Or, in other words, why there is so much variation among the published results? from Bahcall et al hep-ph/ , spread of published values of squared nuclear matrix element for 76 Ge This suggests an uncertainty of as much as a factor of 5. Is it really so bad?

8 Remark (still provocative): If the uncertainty in the M.E. is as large as a factor of 3 (or more), i.e., an order of magnitude (or more) in halflife, perhaps we need not perform sophisticated calculations at all. Remember the classification of ordinary β-decays: allowed β decays have log(ft) = and we need to know just spins and parities of the corresponding states. However, with such uncertainty our predictions would be much less useful for planning and interpreting 0νββ decay experiments.

9 In contrast, Rodin et al, nucl-th/ suggest that the uncertainty is much less, perhaps only ~ 30% (within QRPA and its generalizations, naturally). So, who is right? Slowly and smoothly decreasing (except 96 Zr) with A

10 Two examples of alternative calculations. Notice that these are 2-3 times larger, essentially (but not quite) constant with A.

11 Where are the differences of these two examples coming from? The bulk of the differences is understood, but there is no consensus which approach is correct (or at least more correct). The main effect (by a factor of ~2) comes from taking into account (for Rodin et al.) and not taking into account (Civitarese and Suhonen) the short-range repulsion O -> fof. Another effect comes (by ~30%) from including (or not) the induced pseudoscalar coupling. It would be helpful (to put it mildly) if we can agree here and now how to treat these effects.

12 More on nuclear matrix elements The most important operator is Σ h(r ij )[σ i σ j (g V /g A ) 2 ]τ + τ + where the sum is over all nucleon pairs, and r ij is the distance between the nucleons. h(r) is the `neutrino potential, the Fourier transform of the neutrino propagator h(r)~e -1.5Er /r. (Often Rh(r) is used to make the m.e. dimensionless, with R being the nuclear radius; see previous plots.) Tests show that it is OK to treat this two-body operator in closure; the result depends only weakly on the assumed average nuclear excitation energy. It is customary to expand the operators in terms of multipole states of the intermediate odd-odd nucleus. As a consequence of the high momentum of the virtual neutrino many multipoles contribute significantly, unlike for the case of the 2ν decay where only 1 + contributes.

13 Many multipoles contribute in each case. Most of them, with the exception of 1 +, have the same sign. This is from Rodin et al, other calculations get a similar pattern.

14 76 Ge 100 Mo Note the reduction caused by s.r.c and higher order effects Figure by F. Simkovic

15 from Civitarese & Suhonen, Phys. Lett. B626,80(2005), (same Figs. in Nucl.Phys. A761,313(2005)

16 Here the contribution of the 1+ multipole, and of all other ones is plotted against the parameter g pp that signifies the strength of the neutron-proton interaction. The nominal value is g pp =1.0. The dots denote the adjusted value that reproduces the 2ν decay rate in each nucleus. Note the steep slope of the 1 + curve and the relatively gentle slope of the dashed lines. This suggests that adjusting properly the contribution of the 1 + is important.

17 Nuclear matrix elements for the 2ν decay deduced from measured halflives. Note the pronounced shell dependence. 1/T 1/2 = G(E,Z) (M GT 2ν ) 2 easily calculable phase space factor Geoch QuickTime and a TIFF (LZW) decompressor are needed to see this picture. NEMO-2 Cuoricino Geoch

18 The 2ν matrix element contains products of the β - (right leg) and EC (left leg) amplitudes, weighed by the energy denominators. Single state dominance is often invoked. Energies, and squares of the amplitudes (but not signs) of the 1 + states could be determined by charge exchange reactions, testing the SSD idea. However, for contribution of the 1 + multipole to 0ν m.e. the energy denominator represents a wrong weight. It looks more like the closure m.e.

19 QRPA calculation for 100 Mo. The closure m.e. and M 2ν are not proportional to each other. But M closure and M 0ν (1+) essentially are. M closure M 2ν (x3.5) m.e. M 0ν (1 + ) adjusted g pp g pp

20 QRPA calculation for 76 Ge. The closure m.e. and M 2ν are not really proportional to each other. But M closure and M 0ν (1+) essentially are. M closure m.e. M 0ν (1+) adjusted g pp M 2ν (x7.5) g pp

21 Everybody agrees that within QRPA, the most popular method of evaluating the 0νββ matrix elements, the crucial parameter is g pp the strength of the particleparticle (neutron-proton) interaction. (There are other, implicit, parameters within QRPA that affect the results as well, more on them later) Consequently, the choice of g pp is important. Since the 1 + channel is the most sensitive one to the variation of g pp it is almost obvious that it should be used for the adjustment. The contribution of all other multipoles to 0νββ m.e. is less sensitive to the exact value of g pp, Hopefully that contribution is correctly reproduced by QRPA. There is no agreement on the proper way of g pp adjustment. Rodin et al. use the M 2ν matrix element.

22 Rodin et al. Phys. Rev. C68, (2003), and Rodin et al. nucl-th/ have shown that their method of adjustment results in very little sensitivity to: a) how many s.p. states are included? b) which parametrization of the G-matrix is used? c) whether the g A is quenched? d) whether one of the generalization of QRPA is used? It is likely that a substantial part of the spread in the QRPA results is caused by different choices a) - d) without corresponding renormalization. There is a heated debate currently whether this is indeed the answer to the problem. The other cause of the spread, already stressed, is the inclusion (or not) of the s.r.c. and induced currents.

23 The outliers predict wrong 2ν halflife. The matrix elements of SM and Rodin et al. are quite close. Bobyk ν wrong 20x SM 96 Caurier QRPA Rodin 2003 Tomoda 86 proj.m.f. 2ν wrong 6x

24 Comparison of M 0ν of Rodin et al. (RQRPA) and Nowacki et al. (SM, private comm., preliminary 2004) and older published (Caurier et al. 1996) Nucleus RQRPA SM 76 Ge Se Zr Mo Cd Te (1.0) 136 Xe (0.6) Except for 100 Mo the agreement between these very different calculations is reasonably good. Note that the SM calculations include the reduction caused by the s.r.c. and induced currents.

25 F.Nowacki, preliminary, private comm. 2004

26 0νββ half-lives for <m ββ > = 50 mev based on the matrix elements of Rodin et al. 76 Ge ( ) x y 82 Se ( ) x y 100 Mo ( ) x y 130 Te ( ) x y 136 Xe ( ) x Y (no 2ν observed yet) Note: Calculated matrix elements decrease with increasing A, but the phase-space factors usually increase, particularly the Coulomb factor, hence relatively little variation of T 1/2 with A. Note: The sensitivity to <m ββ > scales as 1/(T 1/2 ) 1/2

27 Summary and/or Conclusions Study of 0νββ decay entered a new era. No longer is the aim just to push the sensitivity higher and the background lower, but to explore specific regions of the <m ββ > values. In agreement with the `phased program the plan is to explore the `degenerate region (0.1-1 ev) first, with ~100 kg sources, and prepare the study of `inverted hierarchy ( eV) region with ~ ton sources that should follow later. In this context it is important to keep in mind the questions I discussed: a) Relation of <m ββ > and the absolute mass (rather clear already, becoming less uncertain with better oscillation results). b) Mechanism of the decay (exploring LFV, models of LNV, running of LHC to explore the ~TeV mass particles). c) Nuclear matrix elements (exploring better, and agreeing on, the reasons for the spread of calculated values, and deciding on the optimum way of performing the calculations, while pursuing vigorously also the application of the shell model).