Double Beta Decay and Neutrino Mass

Transcription

1 Double Beta Decay and Neutrino Mass Jenni Kotila Nuclear, Particle, and Astrophysics seminar Yale, May

2 Contents Motivation Phase Space Factors Nuclear Matrix Elements Quenching of g A Half-Life Predictions 0νβ β 0νβ + β + and 0νECβ + Resonantly Enhanced 0νECEC Limits on Average Neutrino Mass Other 0νββ decay scenarios and mechanisms Conclusions

3 Motivation Mass parabola for isobaric nuclei with even A Splits into two parabolas due to the nuclear pair energy: odd-odd, even-even Favored decay: Single β-decay Changes the nuclear charge Z by a value of ±1 Can only occur if the energy of the daughter nucleus is smaller than the energy of the parent nucleus If not, two consecutive β-decays as single process are possible

4 Motivation Nucleus (A,Z) decays to nucleus (A,Z ± 2) by emitting two electrons or positrons + other light particles For processes allowed by the standard model two neutrinos are emitted and the half-life is TWO NEUTRINO p Ν p A2 Ν e e [ τ 2ν ] 1 1/2 = G2νgA m 4 ec 2 M (2ν) 2 G 2ν is the phase space factor, g A is the axial vector coupling constant, and M (2ν) is the nuclear matrix elements Observed in several nuclei with half-lives τ 2ν 1/2 > 1018 y n n A2

5 Motivation For neutrinoless modes the half-life is [ ] 1 τ 0ν 1/2 = G0νgA 4 M(0ν) 2 f(m i,u ei ) 2 f(m i,u ei ) contains the physics beyond standard model Different scenarios and mechanisms: exchange of light neutrino or heavy neutrino, emission of Majoron, exchange of sterile neutrino... For both, 2ν and 0ν, processes, two crucial ingredients are the phase space factors and the nuclear matrix elements!

6 Phase Space Factors Starting from differential decay rate ) dw 0ν = (a (0) + a (1) cosθ 12 w 0ν dǫ 1 d(cosθ 12 ) w 0ν ǫ 1,ǫ 2 a (i) f(m i,u ei ) 2, M (0ν) 2, and f (i) 11 (combination of emitted electron wave functions) And keeping in mind the factorized form [ ] 1 τ1/2 0ν = G0ν ga 4 M(0ν) 2 f(m i,u ei ) 2 PSF can be written as G (i) 0ν = 1 Qββ +m ec 2 f (i) 2R 2 11 ln2 w 0νdǫ 1 m ec 2

7 Phase Space Factors The key ingredient for the evaluation of phase space factors (and thus also double beta decay) are the electron wave functions To simulate realistic situation, we take radial functions that satisfy Dirac equation and potential that takes into account the finite nuclear size and the electron screening Comparison with previous calculations: Considerable difference Example: 150 Nd decay, Z d = 62 at ǫ = 2.0MeV, R(150) = 6.38fm e S 1/2 s (ǫ, r) WF1 = Leading finite size Coulomb (previous studies) WF2 = Exact finite size Coulomb WF3 = Exact finite size Coulomb & electron screening

8 Phase Space Factors G0Ν10 15 yr 1 approx G0ΝG 0Ν Ca Ge Nd Zr Cd Mo Te Xe Se Sn Pd Te Nd Sm Gd approximate this work Th U Pt Mass Number Relative difference: G 0ν/G approx 0ν Ca approximate this work Se Zr Ge Pd Mo Cd Sn Te Nd Te Xe NdSm Gd Pt Current 0νβ β PSFs PRC 85, (2012) (red) compared to previous calculations (blue) Our results have been confirmed by independent calculations of Stoica et al. PRC 88, (2013) Estimate of uncertainties Q-value 3 δq/q R = r 0A 1/3 7% Screening 0.10% Example: decay of 110 Pd precision of Q ββ from 11 to 6.4 gives 1 2 total error for G 0ν D. Fink et al. PRL 108, (2012) Neutron Number

9 Phase Space Factors In addition to phase space factors we also obtain distributions, with normalization N 0ν = g 4 A M(0ν) 2 f 2 single electron spectrum angular correlation double differential rate In case of 2νββ in addition we have summed electron spectrum triple differential rate

10 Nuclear Matrix Elements Previously most used models: QRPA: Results depend on fine-tuning of the interaction, especially near the spherical- deformed transition, for example 150 Nd ISM: Cannot address nuclei with many particles in the valence shells, for example 150 Nd, due to the exploding size of the Hamiltonian matrices (> 10 9 ) Our calculation: Interacting Boson Model (IBM-2) Describes even-even nuclei in terms of correlated pairs of nucleons treated as bosons with L = 0 (s-boson) and L = 2 (d-boson) that are associated with pairs of valence fermions Can be used in any nucleus and thus all nuclei of interest can be calculated within the same model Realistic and well checked wave functions (excitation energies, B(E2) values and quadrupole moments, B(M1) values and magnetic moments, etc. )

11 Nuclear Matrix Elements Example of wave functions: 154 Gd Shape transitional region rapid changes of nuclear deformation Old calculation: No experimental information about 1 + scissors mode New experimental data parameters of Majorana operator can be fitted Little effect on low-lying full-symmetric states BUT significant effect on the mixed symmetry state wave function, like new M0ν (0 + 2 ) = 0.37 (old M 0ν (0 + 2 ) = 0.02) Gd th old th new exp

12 Nuclear Matrix Elements Transition operator for ββ decay: T(p) = H(p)f(m i,u ei ), where H(p) = n,n τ + n τ+ n [ h F (p) + h GT (p) σ n σ n + h T (p)s p nn ] (in momentum space, including higher order corrections) Form factors h F,GT,T (p) v(p) Neutrino potential v(p) different for different scenarios and mechanisms The matrix elements F, GT, and T can be calculated at once using the compact expression And in second quantized form

13 Nuclear Matrix Elements In ββ-decay the fermion transition operator creates a pair of protons (neutrons) and annihilates a pair of neutrons (protons), so we need to map this onto the boson space by using (π j π j )(0) A π(j)s π (π j π j )(2) M Bπ(j, j )d π,m A and B are obtained by equating fermionic matrix elements in the Generalized Seniority basis with bosonic matrix elements (OAI-mapping) The basis is constructed with operators S = j D = Ω j α j 2 j j β jj 1 + δjj ( a j a j ) (0) (a j a j ) (2) where the structure coefficients α j,β jj are obtained by diagonalizing the surface delta interaction, with the strength of the interaction chosen to reproduce the 0-2 separation in the two-particle system The fermion matrix elements are calculated using the commutator method of Frank and Van Isacker and Lipas et al. (PRC (1982), NPA (1990)) OAI mapping : T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A309, 1 (1978).

14 Nuclear Matrix Elements Light neutrino exchange: v(p) = 2 π 1, f = mν p(p+ã) m e M 0Ν ( ) 2 M (0ν) = M (0ν) GT gv M (0ν) F g A Pd Mo Ge Zr SnTe Te Se Cd Xe Xe Gd Nd Sm Ca Nd + M (0ν) T IBM2 QRPATü QRPAJy QRPAdef ISM EDF PHFB Pt U Th Neutron number IBM-2: J. Barea et al., PRC 91, (2015), QRPA-Tu: F. Simkovic et al. PRC 87, (2013), QRPA-Jy: Suhonen et al., PRC (2015),QRPA-def: J:L. Fang et al., PRC (2011), ISM: J. Menendez et al., NPA 818, 139 (2009),PHFB: P.K. Rath et al., PRC 82, (2010), EDF: T.R. Rodriguez et al., PRL 105, (2008)

15 Nuclear Matrix Elements M 0Ν ( ) 2 M (0ν) = M (0ν) GT gv M (0ν) F + M (0ν) T g A Ca IBM2 QRPATü QRPAJy Pd QRPAdef ISM Ge Mo Sn EDF Te Se PHFB Cd TeXe Gd Zr XeNd NdSm Pt U Th Neutron number IBM-2: J. Barea et al., PRC 91, (2015), QRPA-Tu: F. Simkovic et al. PRC 87, (2013), QRPA-Jy: Suhonen et al., PRC (2015),QRPA-def: J:L. Fang et al., PRC (2011), ISM: J. Menendez et al., NPA 818, 139 (2009) Comparison of IBM-2, QRPA, ISM NMEs for light neutrinos IBM-2/QRPA/ISM similar trend Larger values at the middle of the shell than at closed shells Lot of deviation between different QRPA calculations (Ge, Sn) The ISM is a factor of 2 smaller than both the IBM-2 and QRPA in the lighter nuclei and the difference is smaller for heavier Effective value of g A?

16 Nuclear Matrix Elements M 0Ν ( ) 2 M (0ν) = M (0ν) GT gv M (0ν) F + M (0ν) T g A Ca Pd IBM2 Mo QRPATü QRPAJy QRPAdef ISM Zr EDF Ge CdSnTe Te PHFB Se Xe Xe Gd Nd Sm Nd Pt U Th Neutron number Comparison of IBM-2, EDF, and PHFB NMEs for light neutrinos Mean field models EDF/PHFB Somewhat opposite behavior when N > 50 Troubling ratio M (0ν) ( 128 Te)/M (0ν) ( 130 Te): Treatment of shell effects? IBM-2: J. Barea et al., PRC 91, (2015), PHFB: P.K. Rath et al., PRC 82, (2010), EDF: T.R. Rodriguez et al., PRL 105, (2008)

17 Nuclear Matrix Elements ISOSPIN RESTORATION reduces matrix elements χ F = (g V /g A ) 2 M (0ν) F /M (0ν) GT Decay IBM-2 QRPA ISM 48 Ca -0.10(-0.39) -0.32(-0.93) 76 Ge -0.09(-0.37) -0.21(-0.34) Se -0.10(-0.40) -0.23(-0.35) Zr -0.08(-0.08) -0.23(-0.38) 100 Mo -0.08(-0.08) -0.30(-0.30) 110 Pd -0.07(-0.07) -0.27(-0.33) 116 Cd -0.07(-0.07) -0.30(-0.30) 124 Sn -0.12(-0.34) -0.27(-0.40) 128 Te -0.12(-0.33) -0.27(-0.38) Te -0.12(-0.33) -0.27(-0.39) Xe -0.11(-0.32) -0.25(-0.38) Nd -0.12(-0.12) 150 Nd -0.10(-0.10) 154 Sm -0.09(-0.09) 160 Gd -0.07(-0.07) 198 Pt -0.10(-0.10) 232 Th -0.08(-0.08) 238 U -0.08(-0.08) 0νβ β : χ F = ( g V ga ) 2 M (0ν) F GT (old values in parentheses): /M (0ν) Considerable reduction obtained! Isospin restored χ F values very close to the ones obtained from ISM, where isospin is a good quantum number by construction Similar prescription has been used for QRPA (Simkovic et al., PRC (2013) and Suhonen et al., PRC (2015))

18 Nuclear Matrix Elements Estimate of error Sensitivity to input parameter changes Single particle energies: 10% Strengths of interactions: 5% Oscillator parameter (SP wave functions): 5% Closure energy in the neutrino potential: 5% Nuclear radius (If NMEs in dimensionless units): 5% Sensitivity to model assumptions Truncation to S-D space: 1% (spherical) - 10% (deformed) Isospin purity: 2% Sensitivity to operator assumptions Form of the transition operator: 5% Finite nuclear size: 1% Short range correlations (SRC): 5% The total error estimate is 16%

19 Nuclear Matrix Elements In heavy neutrino exchange scenario the transition operator has same form as for light neutrinos, but with f m p m 1 ν h m 1 ν h = k=heavy (U ekh ) 2 1 m kh involves the mass eigenstates m kh of heavy neutrinos and m νh 1GeV The Fourier transform of the neutrino potential is v(p) = 2 1 π m pm e Contact interaction in configuration space strongly influenced by short range correlations

20 Nuclear Matrix Elements Comparison of IBM-2, QRPA, and ISM matrix elements for heavy neutrinos M h 0Ν Ca Ge Se Mo Pd Te Te Cd Zr Sn Xe Gd XeNd Nd Sm IBM2 QRPATü ISM Pt U Th Neutron number IBM-2/QRPA/ISM similar trend, factor of 2 difference between IBM-2 and QRPA-Tü Ratio for QRPA-Jy/QRPA-Tü results varies from 1 up to 2.5, reason for this discrepancy is not clear Note: IBM-2 error estimate in this case is 28% mostly coming from SRC IBM-2: J. Barea et al., PRC 91, (2015), QRPA-Tü: F. Simkovic et al. PRC 87, (2013), QRPA-Jy: Suhonen et al., PRC (2015) ISM: A. Neacsu et al.,ahep 2014,724315(2014)

21 Quenching of g A It is well-known from single β decay/ec and 2νββ that g A is renormalized in nuclei. Reasons: Limited model space Omission of non-nucleonic degrees of freedom(,n,...) The effective value of g A can be defined as M eff 2ν = ( ) ga,eff 2 M 2ν g A M eff β/ec = ( ga,eff g A ) M β/ec and obtained by comparing the calculated and measured half-lives for β/ec and/or for 2νββ J. Fujita and K. Ikeda, Nucl. Phys. 67, 145 (1965), D.H. Wilkinson. Nucl. Phys. A225, 365 (1974)

22 Quenching of g A Maximally quenched value from 2νβ β experiments: Nucleus τ 1/2 2ν y) exp Ca Ge Se 92 ± 7 96 Zr 23 ± Mo 7.1 ± Mo Ru(0 2 ) Cd 28.7 ± Te ± Te 690 ± Xe 2110 ± Nd 8.2 ± Nd Sm(0 2 ) U 2000 ± 600 M eff 2ν 2 is obtained from the measured half-life by M eff 2ν 2 = [ τ 2ν 1/2 G 2ν ] 1 eff M 2Ν Ge Mo MoRu0 2 Cd CA SSD 0.1 Se Zr Nd Te U 0.05 Ca Te Te NdSm0 2 Xe Mass number Smallest M eff 2ν for 136 Xe, the newest one measured! A.S. Barabash, Nucl. Phys. A 935, 52 (2015).

23 Quenching of g A g A,eff = g A M eff 2ν /M 2ν ga, eff 1.4 from experimentalτ 12 ISM g ISM A,eff 1.269A from experimentalτ 12 QRPA g QRPA A,eff 1.269A 0.16 from experimentalτ 12 IBM2 g IBM2 A,eff 1.269A Ca Ge Se ZrMo Cd Te Xe Mass number ISM NMEs from E. Caurier et al., Int. J. Mod. Phys. E 16, 552 (2007). a Yoshida and Iachello, PTEP 2013, 043D01 (2013). b QRPA results from J. Suhonen et al., Phys. Lett. B 725, 153 (2013). Nd Extracted g A,eff: IBM QRPA ISM Similar values found by analyzing β /EC for IBFM-2 a and for QRPA b Assumption: g A,eff is a smooth function of A Parametrization: g A,eff = 1.269A γ IBM-2: γ = 0.18 QRPA: γ = 0.16 ISM: γ = 0.12

24 Quenching of g A Let s return to 0νββ NMEs: M 0ν = g 2 A,eff M(0ν) for IBM-2, QRPA, and ISM M 0Ν M0Ν g A,eff IBM2 ISM QRPAJy Neutron number Taking into account the 16% error estimate for IBM-2: Agreement quite good Looks promising...

25 Quenching of g A Effective value of g A is work in progress, since: Is the renormalization of g A the same in 2νββ as in 0νββ? In 2νββ only the 1 + (GT) multipole contributes. In 0νββ all multipoles 1 +, 2,...; 0 +, 1,... contribute. Some of which could be unquenched. However, also in 0νββ, 1 + intermediate states dominate If not, how to estimate g A,eff? Experiments: by measuring the matrix elements to and from the intermediate odd-odd nucleus in 2νββ decay a Quenching coming from other sources than size of the model space? Theoretical studies by using effective field theory (EFT) to estimate the effect of non- nucleonic degrees of freedom (two-body currents) b Half-life predictions with maximally quenched g A are 6 34 times longer due to the fact that g A enters the equations to the power of 4! a P. Puppe et al., Phys. Rev. C 86, (2012). b J. Menendez, D. Gazit, and A. Schwenk, Phys. Rev. Lett. 107, (2011).

26 Half-Life Predictions: 0νβ β Keep this in mind but for now predictions are calculated with g A =1.269 (and m ν = 1eV) Ν Τ yr Ca Ge Se Zr Mo Pd Cd Sn Xe Te Sm Nd Gd Pt Mass number Judging by the half-life, best candidates 150 Nd, 100 Mo, and 130 Te, where half-lives yr

27 Half-Life Predictions: 0νβ β DECAYS TO FIRST EXCITED 0 + STATES In some cases, the matrix elements to the first excited 0 + state are large Although the PSFs are smaller to the excited state, large matrix elements offer the possibility of a direct detection, by looking at the g-ray de-exciting the 0+ level 0Ν Τ yr Ca Ge Zr Mo Gd Pd Sn Xe Cd Te Nd Mass number Best candidates 100 Mo and 150 Nd, τ (0ν) 1/ νββ-decay observed to excited 0 + state in these nuclei!

28 Half-Life Predictions: 0νβ + β + β + β + available kinetic energy much smaller since Q β + β + = M(A,Z) M(A,Z 2) 4mec2 much smaller phase space much longer half-lives Β Β 0Ν Τ yr Ni Zn Kr Ru Cd Xe Ba Ce Mass number All predicted τ β+ β + 0ν > 10 27, for g A=1.269, m ν = 1eV Compared to 0νβ β hardly detectable

29 Half-Life Predictions:0νECβ + Available T ECβ + = M(A,Z) M(A,Z 2) 2m ec 2 ǫ b The energy of the emitted positron is fixed, no integration G ECβ+ 0ν 0Ν Τ yr = constant R B 2 i, 1 i=0,1 [ (g 1 (ǫ p, R)) 2 + (f 1 (ǫ p,r)) 2] p pcǫ p. ---Β Β ECΒ Ni Zn Kr Ru Cd Xe Ba Ce Mass number Best candidates 124 Xe, 130 Ba, and 136 Ce, where half-lives for g A=1.269, m ν = 1eV Compared to 0νβ β hardly detectable

30 Half-Life Predictions: Resonantly Enhanced 0νECEC For 0νECEC available energy larger Q ECEC = M(A,Z) M(A,Z 2), but since all the energies are fixed, additional requirement that Q-value matches the state energy A, Z1 Q Β A, Z2 HH' Q Ec A, Z 0 Q A, Z2 0 Only bound states, no integration G ECEC 0ν = constant R 2 B 2 n e 1, 1B 2 n e 2, 1

31 Half-Life Predictions: Resonantly Enhanced 0νECEC Resonance enhancement: [ ] 1 τ ECEC 1/2 (0 + ) = g 4 A G ECEC 0ν M 0ν ECEC 2 f(m i,u ei) 2 (m ec 2 )Γ 2 + Γ 2 /4, where = Q B 2h E is the degeneracy parameter, and Γ is the two-hole width So in principle, if 0 and Γ 1eV we could obtain up to 10 6 enhancement Accurate Q-value measurements very important! Many candidates, such as 112 Sn, 130 Ba, and 136 Ce, ruled out by recent high precision Q-value measurements

32 Half-Life Predictions: Resonantly Enhanced 0νECEC Decay G ECEC 0ν M (0ν) Γ (m ec 2 )F τ 1/2 (10 19 yr 1 ) (kev) (kev) (10 27 )yr 124 Xe Gd Dy Er W Best candidates at the moment 152 Gd, and 180 W 0ΝECEC Τ yr Xe Mass number Gd Dy Er W Half-lives > for m ν = 1eV and g A = Compared to 0νβ β hardly detectable

33 Limits on Average Light Neutrino Mass Reminder: [ ] 1 τ1/2 0ν = G0νgA 4 M(0ν) 2 f(m i,u ei ) 2 Light neutrinos: f(m i,u ei ) = mν m e = 1 m e k=light (U ek ) 2 m k The average light neutrino mass is now well constrained by atmospheric, solar, reactor and accelerator neutrino oscillation experiments Obtained information on mass differences and their mixing leaves two possibilities: Normal and inverted hierarchy

34 Limits on Average Light Neutrino Mass The average light neutrino mass is then written as: m ν = c 2 13 c 2 12m 1 +c 2 13s 2 12m 2e iϕ 2 + s 2 13m 3e iϕ 3, c ij = cosϑ ij, s ij = sinϑ ij, ϕ 2,3 = [0,2π], ( m 2 1,m 2 2,m2 3) = m m ) ( δm2 2,+δm2 2,± m2 1 ϑ 12,ϑ 13,ϑ 23 and δm, m fitted to oscillation experiments Phases ϕ 2 and ϕ 3 may vary from 0 to 2π mν in ev sin 2 ϑ12 = 0.312, sin 2 ϑ 13 = 0.016, sin 2 ϑ 23 = INVERTED NORMAL lightest neutrino mass in ev δm 2 = ev 2, m 2 = ev 2 (sin 2 ϑ 13 = ± 0.005)

35 Limits on Average Light Neutrino Mass Current lower half-life limits coming from different experiments Experiment nucleus τ 1/2 m ν IGEX 76 Ge > yr < 0.35eV GERDA 76 Ge > yr < 0.30eV NEMO Mo > yr < 0.56eV CUORE 130 Te > yr < 0.35eV EXO 136 Xe > yr < 0.25eV Kamland-Zen 136 Xe > yr < 0.20eV τ 1/2 m ν < m e τ exp 1/2 G 0νg 2 A M(0ν) IGEX: C. E. Aalseth et al., Phys. Rev. D 65, (2002), GERDA: M. Agostini et al. (GERDA collaboration) Phys. Rev Lett (2013), NEMO-3: R. Arnold, et al., Phys. Rev. D 89, (2014), CUORE: K. Alfonso et al., arxiv: [nucl-ex] (2015), EXO: M. Auger et al., Nature 510, 229 (2014)KamLAND-Zen: A. Gando et al., Phys. Rev. Lett. 110, (2013)

36 Limits on Average Light Neutrino Mass Current limits to m ν from CUORE, IGEX, NEMO-3, KamLAND-Zen, EXO, and GERDA 0νββ experiments NEMO3 CUORE IGEX GERDA EXO KamLANDZen X mν in ev INVERTED NORMAL lightest neutrino mass in ev IGEX: C. E. Aalseth et al., Phys. Rev. D 65, (2002), NEMO-3: R. Arnold, et al., Phys. Rev. D 89, (2014), CUORE: K. Alfonso et al., arxiv: [nucl-ex] (2015), KamLAND-Zen: A. Gando et al., Phys. Rev. Lett. 110, (2013), EXO: M. Auger et al., Nature 510, 229 (2014)GERDA: M. Agostini et al. (GERDA collaboration) Phys. Rev Lett (2013), X: H.V. Klapdor-Kleingrothaus et al., Phys. Lett. B 586, 198 (2004),

37 Limits on Average Heavy Neutrino Mass For heavy neutrino exchange f(m i,u ei ) = η Predicted half-lives for η = Νh Τ yr Ca Ge Se Zr Mo Pd Cd Sn Te Xe Sm Nd Gd Pt Mass number

38 Limits on Average Heavy Neutrino Mass Comparison with experimental limits: η < Decay τ 0ν h 1/2,exp (yr) η (10 6 ) 76 Ge 76 Se τ exp 1/2 G 0ν g2 A M(0ν h ) m νh (GeV) > < > 136 > < > Mo 100 Ru > < > Te 130 Xe > < > Xe 136 Ba > < > 257 > < > 196 The average inverse heavy neutrino mass is not constrained by experiments, but we can set model dependent limits Model by Tello et al. η = M4 W M 4 WR m p m νh mν = mp h η m 4 WR, MW 4 where M W is the mass of the W-boson, M W = (80.41 ± 0.10) GeV, M WR is the mass of an hypothetical W triplet boson, M WR = 1.75TeV Constraints on η can be then converted into constraints on the average heavy neutrino mass V. Tello et al., PRL 106, (2011)

39 Limits on Average Neutrino Mass: Remarks If both light and heavy neutrino exchange contribute, the half-lives are given by [τ1/2 0ν ] 1 = G (0) m 0ν M ν 0ν m e + M 0νh η 2 The two contributions could add or subtract depending on their relative phase The question of effective value of g A is still open. Three suggested scenarios are Free value: Quark value: 1 Maximally quenched value: 1.269A γ if g A is closer to maximally quenched value, then...

40 Limits on Average Neutrino Mass: Maximal quenching Current limits to m ν with maximally quenched g A. In this case even the inverted mass hierarchy is still far away NEMO3 CUORE IGEX GERDA EXO KamLANDZen X mν in ev INVERTED NORMAL lightest neutrino mass in ev Other scenarios like Majoron emitting modes of DBD and new mechanisms like sterile neutrinos?

41 Majoron emitting 0νββ This mechanism requires the emission of one or two additional bosons, Majorons (A,Z) (A,Z+2)+2e +χ 0 or e p e M 0 p A2 (A,Z) (A,Z+2)+2e +2χ 0 Ν Ν If Majorons exist, they could play a significant role in the history of the early Universe and in the evolution of stars Dark matter, cosmological and astrophysical processes... Half-life [ τ 0ν ] 1 1/2 = g 4 A G (0) 2m (m,n) mχ 0 n g χ M ee M 2 n n 0νM A2 m is the number of emitted Majorons and n is the spectral index of the decay g χ M ee is the majoron-neutrino coupling constant

42 Majoron emitting 0νββ Different models proposed to describe Majoron emitting decays Model Decay Mode NG boson L n M (m,n) 0νM IB 0νββχ 0 No 0 1 M GT g2 V g 2 AM F + M T IC 0νββχ 0 Yes 0 1 M GT g2 V g 2 AM F M T ID 0νββχ 0χ 0 No 0 3 IE 0νββχ 0χ 0 Yes 0 3 gv 2 g A 2 gv 2 g A 2 M Fω 2 M GTω 2 M Fω 2 M GTω 2 IIB 0νββχ 0 No -2 1 M GT g2 V g AM 2 F + M T IIC 0νββχ 0 Yes -2 3 M CR IID 0νββχ 0χ 0 No -1 3 IIE 0νββχ 0χ 0 Yes -1 7 gv 2 g A 2 gv 2 g A 2 M Fω 2 M GTω 2 M Fω 2 M GTω 2 IIF 0νββχ 0 Gauge boson -2 3 M CR Bulk 0νββχ 0 Bulk field 0 2 -

43 Majoron emitting 0νββ Differential decay rate has the same from as in light and heavy neutrino exchange ) dw mχ0 n = (a (0) + a (1) cosθ 12 w mχ0 ndǫ 1dǫ 2d(cosθ 12) The PSF depends on m and n Qββ G mχ (i) +m ec 2 Qββ +2m ec 2 ǫ 1 0 n = 2 g 4 A ln2 m ec 2 m ec 2 w mχ0 n q n, majoron energy, and ǫ 1,ǫ 2 Different disributions: the single electron spectrum dw mχ0 n dǫ 1 = N (m,n) dg (0) mχ 0 n 0νM dǫ 1 2m m,n M 2. f (i) 11 wmχ 0ndǫ 1dǫ 2, where N (n,m) 0νM = g4 A ln2 gχ M ee 0νM The summed electron spectrum, which shape makes the different Majoron emitting modes experimentally recognizable dw mχ0 n d(ǫ 1 + ǫ = dg (0) 2) N(n,m) mχ 0 n 0νM d(ǫ 1 + ǫ, 2) and the angular correlation between the two outgoing electrons α(ǫ 1) = dg(1) mχ 0 n/dǫ 1 dg (0). mχ 0 n/dǫ 1

44 Majoron emitting 0νββ single summed α a n n n 1 n=1 dε1 0 dg 1Χ dε1ε2 0 dg 1Χ ΑΕ Ε1mec 2 MeV Ε1Ε22mec 2 MeV Ε12mec 2 MeV n=3 dε1 0 dg 1Χ b n 3 dε1ε2 0 dg 1Χ n 3 ΑΕ n Ε1mec 2 MeV Ε1Ε22mec 2 MeV Ε12mec 2 MeV c n n n 7 n=7 dε1 0 dg 2Χ dε1ε2 0 dg 2Χ ΑΕ Ε1mec 2 MeV Ε1Ε22mec 2 MeV Ε12mec 2 MeV

45 Majoron emitting 0νββ yr 1 0 G 1Χ yr 1 0 G 2Χ Ca Se Ge Nd Zr Cd Te Xe Mo Sn Gd Nd Pd TeXe Sm approximate this work Pt Th Ca Mass Number TeXe Nd Zr Se Cd TeXe Mo Sn Gd Nd Pd Ge Sm approximate this work Pt Th U U Mass Number yr 1 0 G 1Χ yr 1 0 G 2Χ Ca Nd Zr Se Cd TeXe Mo Sn Gd Nd Pd Ge Sm approximate this work Pt Th U 1 Xe Te Ca Mass Number Nd Zr Se Cd Mo Te Xe Sn Gd Nd Pd Ge Sm approximate this work Pt Th U Mass Number

46 Majoron emitting 0νββ Half-lives: Ordinary Majoron decay m = 1,n = 1 If the Majoron couples only to light neutrino, the NME needed to calculate the half-life are the same as for light neutrino exchange τ (0νM) 1/2 for models where n = 1 (IB, IC, IIB) with g M ee = 10 4, g A = ΝM Τ yr Ca Ge Se Zr Mo Pd Sn Cd Te Xe Te Xe Nd Sm Nd Mass number Gd Pt Th U

47 Majoron emitting 0νββ Comparison with experimental limits on τ 0νM 1/2,exp gives information about gee M Decay τ 1/2 0νM (1021 yr) τ 1/2,exp g 0νM (yr) ee M (ev) 48 Ca 48 Ti 8.19 > < Ge 76 Se 39.8 > < Se 82 Kr 7.68 > < Zr 96 Mo 5.32 > < Mo 100 Ru 3.62 > < Pd 110 Cd Cd 116 Sn 7.06 > Sn 124 Te Te 128 Xe 765 > Te 130 Xe 6.82 > Xe 134 Ba Xe 136 Ba 10.1 > Nd 148 Sm Nd 150 Sm 1.74 > Sm 154 Gd Gd 160 Dy Pt 198 Hg Th 232 U U 238 Pu 4.94 < < < > < < Values g M ee or g M ee excluded by the observation of SN 1987A

48 Sterile neutrinos Another scenario, currently being extensively discussed, is the mixing of additional sterile neutrinos The NME for sterile neutrinos of arbitrary mass can be calculated using a transition operator in ν light and ν heavy exchange but with f = mνi m e, v(p) = 2 π p2 + m 2 νi where m νi is the mass of the sterile neutrino The product fv(p) = mνi m e 2 π 1 ( p2 + m 2 νi + Ã 1 ( ) p2 + m 2 νi p2 + m 2 νi + Ã ), has the limits: m νi 0 m νi fv(p) = m νi m e 2 π fv(p) = m νi m e 2 π 1 p(p+ã) 1 = 2 m 2 π νi 1 m em νi

49 Sterile neutrinos The PSF for this scenario is the same as for ν light and ν heavy exchange Several types of sterile neutrinos have been suggested. Light sterile neutrinos Neutrino masses are m νi 1eV These neutrinos account for the reactor anomaly in oscillation experiments and for the gallium anomaly Heavy sterile neutrinos Neutrino masses are m νi 1eV kev mass range, MeV-GeV mass range, TeV mass range Limits on sterile neutrino contributions from DBD are being calculated at the present time... Giunti, NEUTEL2015 conference proceedings

50 Conclusions We have studied several scenarios and mechanisms suggested to describe double beta decay This includes two neutrino and neutrinoless decays, exhange of light and heavy neutrinos, majoron emitting ββ, and decays to ground states as well as to first excited 0 + states Concerning the mass mechanism, based on our results 0νβ β is the likeliest mode to be observed, even if 0νECEC is resonantly enhanced Effective value of g A is work in progress and first results suggest considerable quenching No matter what the mechanism of neutrinoless DBD is, its observation will answer the fundamental questions What is the absolute neutrino mass scale Are neutrinos Dirac or Majorana particles How many neutrino species are there

51 THANK YOU! NEMO3 CUORE IGEX GERDA EXO KamLANDZen X mν in ev INVERTED NORMAL lightest neutrino mass in ev More information: nucleartheory.yale.edu