Neutrinoless double beta decay. Introduction, what its observation would prove, and its nuclear matrix elements.

Transcription

1 Neutrinoless double beta decay. Introduction, what its observation would prove, and its nuclear matrix elements. Petr Vogel Caltech ECT,Trento, 7/31/2009

2 ββ decay can exist in two modes. The two-neutrino (2νββ) decay is an allowed but slow process, while the neutrinoless (0νββ) mode would violate the total lepton number conservation law and thus would be a sign of new physics (so far) virtual state of the intermediate nucleus virtual state of the intermediate nucleus and of the exchanged neutrino

3 ν masses are much smaller than the masses of other fermions D E S E R T Is that a possible Hint of a new mass-generating mechanism?

4 The most popular theory of why neutrinos are so light is the See-Saw Mechanism (Gell-Mann, Ramond, Slansky (1979), Yanagida(1979), Mohapatra, Senjanovic(1980)) ν { Familiar light neutrino Very heavy neutrino } N R It assumes that the very heavy neutrinos N R exist. Their large mass explains why the observed neutrinos are so light, m ν ~ v 2 /M N (v is the Higgs vacuum expectation value, v ~ 250 GeV). Both the light and heavy neutrinos are then Majorana fermions.

5 Study of the 0νββ-decay is one of the highest priority issues in particle and nuclear physics APS Joint Study on the Future of Neutrino Physics (2004) (physics/ ) We recommend, as a high priority, a phased program of sensitive searches for neutrinoless double beta decay (first on the list of recommendations) The answer to the question whether neutrinos are their own antiparticles is of central importance, not only to our understanding of neutrinos, but also to our understanding of the origin of mass.

6 Candidate Nuclei for Double Beta Decay 48 Ca 48 Ti 76 Ge 76 Se 82 Se 82 Kr 96 Zr 96 Mo 100 Mo 100 Ru 110 Pd 110 Cd 116 Cd 116 Sn 124 Sn 124 Te 130 Te 130 Xe 136 Xe 136 Ba 150 Nd 150 Sm Q (MeV) Abund.(%) All candidate nuclei on this list have Q > 2MeV. The nuclei with an arrow are used in the present or planned large mass experiments. For most of the nuclei in this list the 2νββ decay has been observed

7 If (or when) the 0νββ decay is observed two theoretical problems must be resolved: a)what is the mechanism of the decay, i.e., what kind of virtual particle is exchanged between the affected nucleons (or quarks)? b) How to relate the observed decay rate to the fundamental parameters, i.e., what is the value of the corresponding nuclear matrix elements?

8 Two basic categories are long-range and short-range contributions to the 0νββ decay. The long-range category involves two pointlike vertices and the exchange of a light Majorana neutrino between them. The standard (plain vanilla) type of that category is when 1/T 1/2 0ν = G 0ν (Q,Z) M 0ν 2 <m ββ > 2, <m ββ >=Σ i U ei 2 m i, which represents simple relation between the decay rate and the parameters of the neutrino mass matrix. The short-range category involves only a single pointlike vertex (six fermions, four hadrons and two leptons), i.e. a dimension 9 operator. The relation between the decay rate and neutrino mass is not simple in that case.

9 It is well known that the amplitude for the light neutrino exchange scales as <m ββ >. On the other hand, if heavy particles of scale Λ are involved the amplitude scales as 1/Λ 5 The relative size of the heavy (A H ) vs. light particle (A L ) exchange to the decay amplitude is (a crude estimate, due originaly to Mohapatra) (dimension 9 operator). A L ~ G F 2 m ββ /<q 2 >, A H ~ G F 2 M W 4 /Λ 5, where Λ is the heavy scale and q ~ 100 MeV is the virtual neutrino momentum. For Λ ~ 1 TeV and m ββ ~ ev A L /A H ~ 1, hence both mechanisms contribute equally.

10 Note in passing that less attention has been devoted in the past to the evaluation of the nuclear matrix elements for the case of heavy particle exchange (short-range contribution to 0νββ decay). Proper treatment of the nucleon-nucleon repulsion in that case is obviously crucial; it is traditionally treated crudely using nucleon form factors. Including pion exchange avoids this problem and seems to lead to larger and more consistently evaluated matrix elements. (Vergados 82, Faessler et al. 97, Prezeau et al. 03) 0νββ amplitude is contained in the ππee vertex

11 Lets restrict our further considerations to the simplest long-range mechanism involving the virtual exchange of a light Majorana neutrino. As long as the mass eigenstates ν i that are the components of the flavor neutrinos ν e, ν µ, and ν τ are really Majorana neutrinos, the 0νββ decay will occur, with the rate 1/T 1/2 = G(E tot,z) (M 0ν ) 2 <m ββ > 2, where G(E tot,z) is easily calculable phase space factor, M 0ν is the nuclear matrix element, calculable with difficulties (and discussed later), and <m ββ > = Σ i U ei 2 exp(iα i ) m i, where α i are unknown Majorana phases (only two of them are relevant). Using the formula above we can relate <m ββ > to other observables related to the absolute neutrino mass.

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

13 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. It is therefore necessary to evaluate, with a sufficient accuracy, the ground state wave functions of both nuclei, and evaluate the matrix element of the 0νββdecay operator connecting them. This cannot be done exactly; some approximation and/or truncation is always needed. Moreover, unfortunately, there is no other analogous observable that can be used to judge the quality of the result.

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

15 To obtain the 0νββ operator, we need to integrate neutrino propagator first over dq 0 (the energy of the virtual neutrino), and then Fourier transform the corresponding second order perturbation expression (this is now the integral over the three-momentum of the virtual neutrino): This is the `neutrino potential (A m is the total energy in the virtual nuclear state with respect to (M i + M f )/2). The constants are added for future convenience. r is the distance between the two neutrons that are transformed into protons. This H(r, E), together with spin and isospin operators will appear in the nuclear matrix elements, In compact form H(r ) = R/r Φ(Εr) where Φ (Εr) 1 is a slowly varying function. Since r < R the potential is 1 (but less than 5-10).

16 Without short range correlations How good is the closure approximation? 130 Te 76 Ge 100 Mo With short range correlations Comparison between the QRPA M 0ν with the proper energies of the virtual intermediate states (symbols with arrows) and the closure approximation (lines) with different <E n - E i >. 76 Ge 130 Te Note the mild dependence on <E n - E i > and the fact that the exact results are reasonably close to the closure approximation results for <E n - E i > < 20 MeV. 100 Mo Graph by F. Simkovic

17 Two complementary procedures are commonly used: a) Nuclear shell model (NSM) b) 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 0νββ 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. Relatively simple, thus more 0νββ calculations.

18 Evaluation of M 0ν involves transformation to the relative coordinates of the nucleons (the operators O K depend on r ij ) 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 π refers to the virtual state in odd-odd nucleus, while J refers to the angular momentum of the neutron pair transformed into proton pair.

19 The vectors X and Y are obtained by solving the equations of motion: with Eigenvalue equation for ω 2, unphysical solutions with ω 2 < 0 possible particle-hole particle-particle

20 Evaluate the M 2ν is relatively simple overlap But the two `vacua 0 + QRPA > are not identical, hence the Overlap is included (this is an approximation). But more importantly, how does one choose g pp? The usual practice in QRPA is to give up on the predictability of the 2νββ decay, instead to choose g pp such that the M 2ν has the correct value ( ~ ±20% deviation from the nominal g pp = 1)

21 2νββ matrix elements for 76 Ge as a function of g pp in QRPA and RQRPA, calculation performed with 9,12, and 21 orbits. Note the crossing of zero and approach to collapse (infinite slope) 9 9 experimental M 2ν

22 Important bonus: Fixing g pp in order to reproduce M 2ν correctly also essentially removes the dependence on the size of the s.p. basis. experimental value of M 2ν M 0ν full lines, M 2ν dashed lines. By fixing g pp to the exp. value of M 2ν we get the same M 0ν with 9 and 21 levels, but with different g pp for the two cases, 1.05 vs. 0.85

23 from isovector p-p and n-n pairing interaction from isoscalar n-p interaction 82 Se 130 Te 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. J The same restricted s.p. space is used for QRPA and NSM. There is a reasonable qualitative agreement between these two methods

24 The opposite signs, and similar magnitudes of the J = 0 and J 0 parts is universal. Here for three nuclei with coupling constant g pp adjusted so that the 2νββ rate is correctly reproduced. Now two oscillator shells are included.

25 J + J - Contributions to the 0νββ-decay NME in 100 Mo from different J π multipole states in the virtual intermediate odd-odd 100 Tc. The strength g pp of the particleparticle neutron-proton interaction is varied by ± 5% with respect to the value g pp = that gives good agreement with the known 2νββ decay rate. Note the rapid variation with g pp of the 1 + multipole. The other multipoles (as well as the negative parity ones not shown) change only slowly with g pp. It is therefore reasonable to adjust g pp so that the contribution of the 1 + is fixed to the known 2νββ where only that multipole contributes.this is what we do.

26 How good is QRPA? Can we check its validity? To do that (approximately) we use a two-level model that can be solved exactly using the algebra based on SO(5)xSO(5). It has many features analogous to real nuclei. The hamiltonian is number operator distance between the two shells pair operators strength of n-p interaction From Engel & Vogel, PRC69,034304(2004)

27 Comparison of exact (solid lines) and QRPA (dashed) M 2ν and M 0ν, for different level spacings ε. In this model QRPA works perfectly.

28 Dependence of the M 0ν on the distance r between the two neutrons that are transformed into the two protons. The neutrino potential is H(r)= R/r Φ(ωr) where Φ(ωr) is rather slowly varying function. This is a long range potential, more or less like a Coulomb potential. Thus, naively, one expect that the matrix element will get its main contribution from r ~ R, i.e. the mean distance between the nucleons in a nucleus. This is not so. Due to the pairing and broken pairs competition, only distances r < 2-3 fm contribute, i.e., only nearest neighbors.

29 C(r) C J (r) total Full matrix element pairing part broken pairs part The radial dependence of M 0ν for the three indicated nuclei. The contributions summed over all components is 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 or large values of q are quite important. M 0ν = C(r)dr

30 The radial dependence of M 0ν for the indicated nuclei, evaluated in the nuclear shell model. (Menendes et al, arxiv: ). Note the similarity to the QRPA evaluation of the same function.

31 The radial dependence of M 0ν evaluated in the exactly solvable model described earlier. Note that the cancellation at r > 2-3 fm is most pronounced near g pp = 1.

32 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: a) Short range repulsion b) Nucleon finite size c) Induced weak currents (Pseudoscalar and weak magnetism) Each of these, with the present treatment, causes correction (or uncertainty) of ~20% in the 0νββ matrix element. There is a consensus now that these effects must be included but only emerging consensus how to treat them, in particular a).

33 Two-nucleon probability distribution, with and without correlations, MC with realistic interaction. O. Benhar et al. RMP65,817(1993) no s.r.c. = nuclear matter, saturation density = nuclear matter, half of the saturation density only protons See also Bisconti et al., Phys. Rev. C73, (2006) for the more modern version of this

34 Dependence on the distance between the two transformed nucleons and the effect of different treatments of short range correlations. This causes changes of M 0ν by ~ 20%. (5.3) (4.0) (4.1) (5.0) C(r) Graph by F. Simkovic

35 Figure from Engel & Hagen, arxiv: The GT part of M 0ν is shown. Full line for the bare operator, s.r.c not included. Dashed line is obtained when the effective operator generated by summing over high energy ladder diagrams is used. The dotted line is from Simkovic et al. ( ), obtained by fits to the coupled-cluster equations at the one and two-body level. These results, together with the consistent treatment of the UCOM method suggest that the effect of s.r.c is quite small.

36 Dependence of the 0νββ matrix element on the Μ Α = M V = Λ cut parameter in the usual dipole nucleon form factor. When correction for short range correlations is included the M 0ν changes little for Λ cut 1000 MeV. f(q 2 ) ~1/(1+q 2 /Λ 2 ) 2 Graph by F. Simkovic

37 Hadronic current expressed in terms of nucleon fields Ψ: Vector g V (q 2 ) = g V /(1 + q 2 /M V2 ) 2, g V = 1, M V = 0.85 GeV Axial vector g A (q 2 ) = g A /(1 + q 2 /M A2 ) 2, g A = 1.25, M A = 1.09 GeV Weak Magnetism g M (q 2 ) = (µ p - µ n ) g V (q 2) Induced pseudoscalar g P (q 2 ) = 2m p g A (q 2 )/(q 2 + m π2 ) After the nonrelativistic reduction the space part of the current is

38 Contributions of different parts of the nucleon current. Note that the AP (axial-pseudoscalar interference) contains q 2 /(q 2 + m π2 ), and MM contains q 2 /4M p2. 76 Ge 76 Se

39 Full estimated range of M 0ν within QRPA framework and comparison with NSM (higher order currents now included in NSM)

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

41 0νββ nuclear matrix elements calculated very recently with the Interacting Boson Model-2, see Barea and Iachello, Phys. Rev. C79, (2009).

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

43 Assumed occupancies of individual valence orbits might be different Neutron orbit occupancies, original Woods-Saxon vs. adjusted effective mean field. For 76 Ge -> 76 Se experiment p f 5/2 g 9/2 P f 5/2 g 9/2 P 0.5 f 5/2 0.8 g 9/2 0.7 Experiment from J.P.Schiffer et al, Phys.Rev.Lett. 100, (2008), used (d,p),(p,d),( 3 He,α),(α, 3 He) to derive occupancies of neutron orbits

44 Proton orbit occupancies, original Woods-Saxon vs. adjusted effective mean field. experiment P 1.8 ± 0.15 f 5/2 2.0 ± 0.25 g 9/2 0.2 ± 0.25 P 2.1 ± 0.15 f 5/2 3.2 ± 0.25 g 9/2 0.8 ± 0.25 P 0.3 f 5/2 1.2 g 9/2 0.6 Experiment from B.P.Kay et al, Phys.Rev.C79,021301(2009), based on (d, 3 He)

45 Full estimated range of M 0ν 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

46 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 = which is only ~12% of the total matrix element 3.27 (The total matrix element is made of = 3.27.) In the figure all entries are, however, normalized so that their sum is unity..

47 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 150 Nd -> 150 Sm very difficult.

48 Summary on matrix elements 1) There is, as of now, agreement of all practitioners on what needs to be included in the evaluation of the 0νββ nuclear matrix elements, even though there is no complete agreement how to do it (e.g. for the short range correlations). 2) 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. 3) There are still noticeable differences between these two methods, and several possible causes have been identified. 4) Both methods predict that the 0νββ nuclear matrix elements should vary slowly and rather smoothly with A and Z, unlike the 2νββ matrix elements. That makes the comparison of experiments with different sources easier.

49 Moore s law of 0νββ decay: There is a steady progress in the sensitivity of the searches for 0νββ decay. Several experiments that are funded and almost ready to go will reach sensitivity to ~0.1 ev. There is one (so far unconfirmed) claim that the 0νββ decay of 76 Ge was actually observed. The deduced mass <m ββ > would be then ev.

50 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

51 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).