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LANL April 15, 2009

Double beta decay theory, in particular an “anatomy” of the nuclear matrix elements. Petr Vogel Caltech. LANL April 15, 2009. Outline: Introduction - 0 nbb as a test of the total lepton number conservation.

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LANL April 15, 2009

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  1. Double beta decay theory, in particular an “anatomy” of the nuclear matrix elements Petr Vogel Caltech LANL April 15, 2009

  2. Outline: Introduction - 0nbb as a test of the total lepton number conservation. Mechanism of the decay and how can one tell when the 0nbb is actually observed. Calculating the nuclear matrix elements and why is it difficult. Similarities and differences in the quasiparticle random phase (QRPA) and nuclear shell model (NSM) approaches.

  3. Experimental discovery of neutrino oscillations opened a `window to physics beyond the Standard Model’. In the next stage of experiments effort will be made to refine our knowledge of oscillation phenomenology (`precision neutrino physics ’ ), and b) to search for qualitatively new phenomena that should help making sense out of the things that we have seen in that window. Study of the neutrinoless double beta decay, i.e., tests of the total lepton number conservation, is a primary example of this effort.

  4. How can we tell whether the total lepton number is conserved? A partial list of processes where the lepton number would be violated: Neutrinoless bb decay: (Z,A) -> (Z2,A) + 2e(), T1/2 > ~1025 y Muon conversion: m- + (Z,A) -> e+ + (Z-2,A), BR < 10-12 Anomalous kaon decays: K+ -> p-m+m+ , BR < 10-9 Flux of ne from the Sun: BR < 10-4 Flux of ne from a nuclear reactor: BR < ? Production at LHC of pair of same charge leptons with no missing energy: BR <? Observing any of these processes would mean that the lepton number is not conserved, and that neutrinos are massive Majorana particles. It turns out that the study of the 0nbb decay is by far the most sensitive test of the total lepton number conservation, so we restrict further discussion to this process.

  5. e– e– ()R L 0 u d d u W W Whatever processes cause 0, its observation would imply the existence of a Majorana mass term: Schechter and Valle,82 By adding only Standard model interactions we obtain (n)R  (n)L Majorana mass term Hence observing the0nbb decay guaranties that n are massive Majorana particles.

  6. What is the nature of the `black box’? In other words, what is the mechanism of the 0nbb decay? All these diagrams can contribute to the 0nbb decay amplitude d u d u WL WR Light or heavy Majorana neutrino. Model extended to include right-handed WR. Mixing extended between the left and right-handed neutrinos. e- Light Majorana neutrino, only Standard Model weak interactions e- n n e- e- WL WL d u d u d u d u WR e (selectron) Supersymmetry with R-parity violation. Many new particles invoked. Light Majorana neutrinos exist also. e- Heavy Majorana neutrino interacting with WR. Model extended to include right-handed current interactions. e-  (neutralino) nheavy e- e- WR e (selectron) d u d u

  7. The relative size of heavy (AH) vs. light particle (AL) exchange to the 0nbb decay amplitude is(a crude estimate): AL ~ GF2 mbb/<k2>, AH ~ GF2 MW4/L5 , where L is the heavy scale and k ~ 100 MeV is the virtual neutrino momentum. For L ~ 1 TeV and mbb ~ 0.1 – 0.5 eV AL/AH ~ 1, hence both mechanism would contribute equally. Note that if L >> 1 TeV, the heavy particle exchange would result in unobservably small 0nbb rate. From the observation of the 0nbb decay it is, in general, impossible to decide which of the possible graphs is relevant. A diagnostic tool in deciding which mechanism dominates might be in linking lepton number violation (LNV) with lepton flavor violation (LFV) .

  8. Linking LNV to LFV Summary: G(m- +(Z,A)  e- + (Z,A)) Bmeg = G(meg)/G(menmne) Bme = G(m- +(Z,A)  nm + (Z,A)) • SM extensions with low ( TeV) scale LNV** Left-right symmetric model, R-parity violating SUSY, etc. possibly G0nbb unrelated tombb2 R = Bme/Bmeg» 10-2 • SM extensions with high (GUT) scale LNV [G0nbb~ mbb2] R~ O(a/p) ~10-3 - 10-2 ** In absence of fine-tuning or hierarchies in flavor couplings. Important caveat! See: V. Cirigliano et al., PRL93,231802(2004)

  9. As long as the mass eigenstates ni, which are components of the flavor neutrinos ne, nm and nt, are Majorana neutrinos, the 0nbb decay will occur for sure, with the rate 1/T1/2= G(Etot,Z) (M0n)2|<mbb>|2, where G(Etot,Z) is easily calculable phase space factor, M0n is the nuclear matrix element, calculable with difficulties (and discussed later), and <mbb> = Si |Uei|2 exp(iai) mi, where ai are unknown Majorana phases (only two of them are relevant). The <mbb>, the effective neutrino Majorana mass, is the quatity that we would like to extract from experiments. We can relate |<mbb>| to other observables related to the absolute neutrino mass.

  10. Usual representation of that relation. It shows that the <m> axis can be divided into three distinct regions. However, it creates the impression (false) that determining <m> would decide between the two competing hierarchies. degenerate inverted normal

  11. <mbb> vs. the absolute mass scale minimum mass, not observable from observational cosmology, M = m1+m2+m3 from b decay blue shading: normal hierarchy, Dm231 > 0. red shading: inverted hierarchy Dm231 < 0 shading:best fit parameters, lines 95% CL errors. Thanks to A. Piepke

  12. Nuclear Matrix Elements: In double beta decay two neutrons bound in the ground state of an initial even-even nucleus are simultaneously transformed into two protons that again are bound in the ground state of the final nucleus. The nuclear structure problem is therefore to evaluate, with a sufficient accuracy, the ground state wave functions of both nuclei, and evaluate the matrix element of the 0nbb-decay operator connecting them. This cannot be done exactly; some approximation and/or truncation is always necessary. Moreover, there is no other analogous observable that can be used to judge the quality of the result.

  13. Can one use the 2nbb-decay matris elements for that? What are the similarities and differences? Both 2nbb and 0nbb operators connect the same states. Both change two neutrons into two protons. However, in 2nbb the momentum transfer q < few MeV; thus eiqr ~ 1, long wavelength approximation is valid, only the GT operator st need to be considered. In 0nbb q ~ 100-200 MeV, eiqr = 1 + many terms, there is no natural cutoff in that expansion. Explaining 2nbb-decay rate is necessary but not sufficient

  14. Basic procedures: Define the valence space 2) Derive the effective hamiltonian Heff using the nucleon-nucleon interaction plus some empirical nuclear data. Solve the equations of motion to obtain the ground state wave functions

  15. Two complementary procedures are commonly used: Nuclear shell model (NSM) Quasiparticle random phase approximation (QRPA) In NSM a limited valence space is used but all configurations of valence nucleons are included. Describes well properties of low-lying nuclear states. Technically difficult, thus only few 0nbb calculations. In QRPA a large valence space is used, but only a class of configurations is included. Describes collective states, but not details of dominantly few-particle states. Rather simple, thus many 0nbb calculations.

  16. QRPA proceeds in two steps. First pairing between like nucleons is included in a simple fashion: Bogoliubov transformation, proton and neutron Fermi levels are smeared. However, particle numbers are conserved only in average. particles quasiparticles 2) Then the proton-neutron interaction is included two quasiparticle creation operator two quasiparticle annihilation operator correlated ground state, includes zero-point motion

  17. Evaluation of M0n involves transformation to the relative coordinates of the nucleons (the operators OK depend on rij) unsymmetrized two-body radial integral involves `neutrino potentials’ From QRPA for final nucleus overlap From QRPA for initial nucleus Note the two separate multipole decompositions. J refers to the virtual state in odd-odd nucleus, while J refers to the angular momentum of the neutron pair transformed into proton pair.

  18. Why it is difficult to calculate the matrix elements accurately? Contributions of different angular momenta J of the neutron pair that is transformed in the decay into the proton pair with the same J. Note the opposite signs, and thus tendency to cancel, between the J = 0 (pairing) and the J 0 (ground state correlations) parts. The same restricted s.p. space is used for QRPA and NSM. There is a reasonable agreement between the two methods 82Se 130Te Sorry, this should be the script J

  19. The opposite signs, and similar magnitudes of the J = 0 and J  0 parts is universal in QRPA. Here for three nuclei with coupling constant gpp adjusted so that the 2nbb rate is correctly reproduced. Now two oscillator shells are included.

  20. Dependence on the relative distance, nucleon structure, short range repulsion, higher order currents, etc. The neutrino propagator connecting the two participating nucleons introduces dependence on the relative distance r( or equivalently momentum transfer q ) between them. If small values of r (or large values of q) are important, we have to worry about induced weak currents, nucleon finite size, and the short range nucleon-nucleon repulsion.

  21. Graphs representing the elementary 0nbb amplitude. The neutrino propagator causes dependence of the corresponding transition operator on the momentum transfer q and, in the Fourier transform on the distance r between the participating nucleons. (n) (p) neutrino propagator (Z+2,A) (Z,A) (p) (n) (p) (n) The ``neutrino potential” is H(r) = R/r F(wr), where F(wr) is rather slowly varying function. Thus, naively, one expects that the typical distance is r ~ R. (Z,A) (Z+2,A) (n) (p)

  22. The radial dependence of M0n for the three indicated nuclei. The contributions summed over all components ss shown in the upper panel. The `pairing’ J = 0 and `broken pairs’ J  0 parts are shown separately below. Note that these two parts essentially cancel each other for r > 2-3 fm. This is a generic behavior. Hence the treatment of small values of r and large values of q are quite important. C(r) total Full matrix element pairing part CJ(r) broken pairs part M0n = C(r)dr

  23. The radial dependence of M0n for the indicated nuclei, evaluated in the nuclear shell model. (Menendes et al, arXiv:0801.3760). Note the similarity to the QRPA evaluation of the same function.

  24. The finding that the relative distances r < 2- 3 fm, and correspondingly that the momentum transfer q > ~100 MeV means that one needs to consider a number of effects that typically play a minor role in the structure of nuclear ground states: Short range repulsion Nucleon finite size Induced weak currents (Pseudoscalar and weak magnetism) Each of these, with the present treatment, causes correction (or uncertainty) of ~20% in the 0nbb matrix element. There is a consensus now that these effects must be included but no consensus how to treat them, in particular a).

  25. Dependence on the distance between the two transformed nucleons and the effect of different treatments of short range correlations. This causes changes of M0n by ~ 20%. (5.3) (4.0) (4.1) (5.0) C(r) Graph by F. Simkovic

  26. Contributions of different parts of the nucleon current. Note that the AP (axial-pseudoscalar interference) contains q2/(q2 + m2), and MM contains q2/4Mp2. 76Ge76Se

  27. Full estimated range of M0n within QRPA framework and comparison with NSM (higher order currents now included in NSM)

  28. The 2 matrix elements, unlike the 0 ones, exhibit pronounced shell effects. They vary fast as a function of Z or A.

  29. 0nbb nuclear matrix elements calculated very recently with the Interacting Boson Model-2, see Barea and Iachello, Phys. Rev. C79, 044301(2009).

  30. Why are the QRPA and NSM matrix elements different? Various possible explanations: Assumed occupancies of individual valence orbits might be different In QRPA more single particle states are included In NSM all configurations (seniorities) are included In NSM the deformation effects are included All of the above

  31. Assumed occupancies of individual valence orbits might be different Neutron orbit occupancies, original Woods-Saxon vs. adjusted effective mean field. For 76Ge -> 76Se experiment p f5/2 g9/2 P f5/2 g9/2 P 0.5 f5/2 0.8 g9/2 0.7 Experiment from J.P.Schiffer et al, Phys.Rev.Lett. 100, 1120501(2008), used (d,p),(p,d),(3He,),(,3He) to derive occupancies of neutron orbits

  32. Proton orbit occupancies, original Woods-Saxon vs. adjusted effective mean field. experiment P 1.8  0.15 f5/22.0  0.25 g9/2 0.2  0.25 P 2.1  0.15 f5/2 3.2  0.25 g9/2 0.8  0.25 P 0.3 f5/2 1.2 g9/2 0.6 Experiment from B.P.Kay et al, Phys.Rev.C79,021301(2009), based on (d,3He)

  33. Full estimated range of M0n within QRPA framework and comparison with NSM (higher order currents now included in NSM) New QRPA value with adjusted mean field so that experimental occupancies are reproduced  New NSM value with adjusted mean field (monopole) where experimental occupancies are better reproduced 

  34. In QRPA more single particle states are included Contribution of initial neutron orbit pairs against the final proton pairs. The nonvalence orbits are labeled as r. Adding all parts with r-type orbits gives +2.83 - 3.22 = -0.39 which is only ~12% of the total matrix element 3.27 In the figure all entries are, however, normalized so that their sum is unity..

  35. In NSM the 0 matrix elements converge slowly as higher and higher seniorities (more complicated configurations) are included. In QRPA the sm = 4 should be well described, but higher sm might not be. (from the PhD thesis of J. Menendez)

  36. NSM evaluation of the (hypothetical) mirror decay 66Ge -> 66Se. The deformation of 66Se is artificially changed, while 66Ge is not changed. The matrix element is reduced significantly if the deformations of the initial and final nuclei are different. 0 matrix element From the PhD thesis of J. Menendez

  37. It appears, therefore, that all of these effects, possible differences in the assumed occupancies of valence orbits, additional single particle states included in QRPA but not in NSM, inclusion of complicated configurations (higher seniority and/or deformation) in NSM but only crudely in QRPA, can and probably do affect the resulting nuclear matrix elements, and might explain the different outcomes of the two methods. In particular, the difference in deformation of the initial and final nuclei makes the evaluation of the matrix element for 150Nd -> 150Sm very difficult.

  38. Summary There is, as of now, agreement of all practitioners on what needs to be included in the evaluation of the 0 nuclear matrix elements, even though there is no complete agreement how to do it (e.g. for the short range correlations). The NSM and QRPA have both many basic features in common, in particular the (sometimes severe) cancellation between the effect of pairing and `broken pairs’ configurations and in the radial distance dependence. There are still noticeable differences between the two methods, and several possible causes have been identified. Both methods predict that the 0 nuclear matrix elements should vary slowly and rather smoothly with A and Z, unlike the 2 matrix elements. That makes the comparison of experiments with different sources easier.

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