Description of 0νββ and 2νββ decay using interacting boson model with isospin restoration. Jenni Kotila

Transcription

1 Description of 0νββ and 2νββ decay using interacting boson model with isospin restoration Jenni Kotila FIDIPRO-HIP workshop on Nuclear Isospin Properties Helsinki Institute of Physics, Helsinki

2 Contents Introduction Isospin restored IBM-2 nuclear matrix elements for 2νββ Mapping fermionic operators into bosonic operators Isospin restoration Comparison with other models Using isospin restored 2νββ NMEs to estimate maximally quenched values of g A Few words about Phase Space Factors Combining the results: Half-Life Predictions Comparison with experiments: Limits on Average Neutrino Mass Conclusions & Outlook

3 Introduction Mass parabola for isobaric nuclei with even A Splits into two parabolas due to the nuclear pair energy: odd-odd, even-even Favoured 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 Introduction 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 the half-life is [ ] 1 τ1/2 2ν = G2ν ga 4 m ec 2 M (2ν) 2 and for neutrinoless modes [ ] 1 τ1/2 0ν = G0ν ga 4 M(0ν) 2 f(m i,u ei ) 2 m ν 2 and/or m 1 ν h 2 G 2ν and G 0ν are the phase space factors g A is the axial vector coupling constant (effective value essentially model dependent!) M (2ν) and M (0ν) are the nuclear matrix elements f(m i,u ei ) contains the physics beyond standard model For both processes, two crucial ingredients are the nuclear matrix elements and the phase space factors!

5 Nuclear Matrix Elements Evaluation of the nuclear matrix elements in IBM-2

6 Nuclear Matrix Elements The interacting boson model / The interacting boson fermion model

7 Nuclear Matrix element Transition operators

8 Nuclear Matrix element Mapping onto boson space

9 Nuclear Matrix element Mapping onto boson space

10 Nuclear Matrix Elements Isospin restoration Problem: We obtain M (2ν) F 0 Due to: Truncation of high orders in the boson expansion of the transition operator, truncation of high multipoles in the transition operator, IBM-2 is not a isospin invariant model... But: From the second quantized form we remember that V (λ) s 1,s 2 R (k 1,k 2,λ) (n 1,l 1,n 2,l 2,n 1,l 1,n 2,l 2) = 0 vλ(p)p2 dp 0 Rn 1l 1 (r 1)R n 1 l 1 (r1)jk 1(pr 1)r 2 1dr 1 0 Rn 2l 2 (r 1)R n 2 l 2 (r1)jk 2(pr 2)r2dr 2 1 The offending isospin violating NME is the Fermi NME in 2νββ, for which v λ (p) = δp, λ = 0, s p 2 1 = s 2 = 0 The radial integral in the definition of the two-body matrix element between two fermion states is just overlap integral between initial and final states R (0,0,0) 2ν (n 1,l 1,n 2,l 2,n 1,l 1,n 2,l 2) δ n1,n 1 δ l 1,l 1 δ n 2,n 2 δ l 2,l 2

11 Nuclear Matrix Elements Isospin restoration Isospin restoration is obtained by using 2νββ : R (k 1,k 2,λ) (n 1,l 1,n 2,l 2,n 1,l 1,n 2,l 2) δ k1,0δ k2,0δ k,0δ λ,0δ j1,j 1 δ j 2,j 2 δ n 1,n 1 δ l 1,l 1 δ n 2,n 2 δ l 2,l 2 Fermi NME vanish for 2νββ 0νββ : R (k 1,k 2,λ) (n 1,l 1,n 2,l 2,n 1,l 1,n 2,l 2) δ k1,0δ k2,0δ k,0δ λ,0δ j1,j 1 δ j 2,j 2 δ n 1,n 1 δ l 1,l 1 δ n 2,n 2 δ l 2,l 2 R (0,0,0) 0ν (n 1,l 1,n 2,l 2,n 1,l 1,n 2,l 2) Fermi NME for 0νββ is reduced by subtraction of the monopole term in the expansion of the matrix element intomultipoles Isospin restoration reduces NMEs!

12 Nuclear Matrix Elements ISOSPIN RESTORATION χ F = (g V /g A ) 2 M (0ν) F /M (0ν) GT Decay IBM-2 QRPA ISM ISO(old) ISO(old) 48 Ca -0.10(-0.42) -0.32(-0.93) 76 Ge -0.09(-0.38) -0.21(-0.34) Se -0.10(-0.42) -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(-0.35) -0.27(-0.40) 128 Te -0.12(-0.34) -0.27(-0.38) Te -0.12(-0.34) -0.27(-0.39) Xe -0.11(-0.33) -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) 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))

13 Nuclear Matrix Elements Full matrix element: M 0Ν ( ) 2 M (0ν) = M (0ν) GT gv M (0ν) F + M (0ν) T g A Ca Ge Mo Pd Se Te Cd Te Xe Gd Sn Zr Xe Nd NdSm IBM2 QRPATü ISM Pt U Th Neutron number Comparison of IBM-2, QRPA, and ISM matrix elements for light neutrinos with Argonne/UCOM SRC IBM-2/QRPA/ISM similar trend Shell effects: NMEs smaller at the closed shells than in the middle of the shell Deformation effects: Deformation reduces NMEs IBM-2 NMEs in (surprisingly) good agreement with QRPA NMEs The ISM is a factor of (approximately) two smaller than both the IBM-2 and QRPA in the lighter nuclei and the difference is smaller for heavier nuclei Effective value of g A?

14 Nuclear Matrix Elements M 0Ν ( ) 2 M (0ν) = M (0ν) GT gv M (0ν) F + M (0ν) T g A 7 Pd IBM2 6 QRPATü Ge Zr Mo SnTe Te QRPAJy U 5 Se Cd QRPAdef Xe Xe GdISM Th 4 EDF PHFB 3 Ca Nd NdSm Pt Neutron number Comparison of IBM-2, QRPA, ISM, EDF, and PHFB matrix elements for light neutrinos with Argonne/UCOM SRC EDF and PHFB: rather different behavior than IBM-2/QRPA/ISM, when N > Zr Ratio 128 Te/ 130 Te Treatment of shell effects?

15 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: 1% Sensitivity to operator assumptions Form of the transition operator: 5% Finite nuclear size: 1% Short range correlations (SRC): 5% Total error estimate is 16%

16 Nuclear Matrix Elements Decay M (0ν) h IBM-2 QRPA-Tü ISM 48 Ca 48 Ti Ge 76 Se Se 82 Kr Zr 96 Mo Mo 100 Ru Pd 110 Cd Cd 116 Sn Sn 124 Te Te 128 Xe Te 130 Xe Xe 136 Ba Nd 148 Sm Nd 150 Sm Sm 154 Gd Gd 160 Dy Pt 198 Hg Th 232 U U 238 Pu 189 For completenes: Comparison of IBM-2, QRPA, and ISM matrix elements for heavy neutrinos with Argonne/UCOM SRC Other available calculations very limited BUT: IBM-2/QRPA/ISM seem to have similar trend Note: Error estimate in this case is 28% mostly coming from SRC The neutrino potential for heavy neutrino exchange is a contact interaction in configuration space and thus strongly influenced by SRC

17 Effective value of g A Maximally quenched value from 2νβ β experiments: Nucleus τ1/2 2ν y) exp Ca Ge 1500 ± Se 92 ± 7 96 Zr 23 ± Mo 7.1 ± Mo Ru(0 2 ) Cd 28 ± Te ± Te Xe 2110 ± Nd 8.2 ± Nd- 150 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 A.S. Barabash, Phys. Rev. C 81, (2010) M eff 2Ν CA Mo M eff 2Ν SSD 0.2 U MoRu Ge Cd 0.1 Se Zr Nd Te 0.05 Ca Te NdSm0 2 Xe Mass number Extracted dimensionless quantity M eff 2ν 2 = g 4 A (m ec 2 )M (2ν) 2 Smallest M eff 2ν for 136 Xe, the newest one measured!

18 Effective value of g A Now, if we add to the same figure the theoretical IBM-2 isospin restored nuclear matrix elements ( gv F (m ec 2 )M (2ν) = (m ec 2 ) M(2ν) GT Ã GT ) 2 (2ν) M g A Ã F include the factor g 2 A but they are still much larger than M eff 2ν g A,eff < 1.0, at least in the case of 2νβ β! which DO NOT 1.2 Te M eff 2Ν CA M eff 2Ν SSD 1. Mo m e c 2 M 2Ν CA 0.8 m e c 2 M 2Ν SSD Cd 0.6 Zr Nd 0.4 Ge Se U 0.2 Ca Te Xe Mass number

19 Effective value of g A g A is renormalized in nuclei renormalization depends on the size of the model space IBM-2: small model space ISM: large model space g A,eff can be extracted comparing M eff 2ν and (m e c 2 )M (2ν) ga, eff Ca from experimentalτ 12 ISM from experimentalτ 12 IBM2 CASSD Ge Se ZrMo Cd Te Xe Nd Mass number ISM NMEs from E. Caurier et al., Int. J. Mod. Phys. E 16, 552 (2007).

20 Effective value of g A g A is renormalized in nuclei renormalization depends on the size of the model space IBM-2: small model space ISM: large model space g A,eff can be extracted comparing M eff 2ν and M 2ν Assumption: g A,eff is a smooth function of A ga, eff Ca from experimentalτ 12 ISM g ISM A,eff 1.269A 0.12 from experimentalτ 12 IBM2 CASSD g IBM2 A,eff 1.269A 0.18 Ge Se ZrMo Cd Te Xe Nd Mass number ISM NMEs from E. Caurier et al., Int. J. Mod. Phys. E 16, 552 (2007). Parametrization: g A,eff = 1.269A γ IBM-2: γ = 0.18 ISM: γ = 0.12 Similar values found for IBM-2 by analyzing β /EC, Yoshida and Iachello, PTEP 2013, 043D01 (2013) Similar values found for QRPA by J. Engel et al., PRC 89, (2014).

21 Effective value of g A Let s return to 0νββ: IBM-2 and ISM 0νβ β NMEs multiplied by g 2 A,eff 2.0 M 0Ν g A,eff IBM2 ISM Neutron number Taking into account the 16% error estimate for IBM-2: Agreement very good Looks promising...

22 Effective value of g A Effective value of g A is work in progress, since: The closure approximation may not be good for 2νβ β and one needs to evaluate explicitly the matrix elements to and from the individual intermediate odd-odd nucleus If this is the case phase space factors can not be exactly separated from the nuclear matrix elements Is the renormalization of g A the same in 2νββ as in 0νββ? If not, how to estimate g A,eff? Quenching coming from other sources than size of the model space? Half-life predictions with maximally quenched g A are 40 times longer due to the fact that g A enters the equations to the power of 4!

23 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) m ν 2 and/or m 1 ν h 2, M (0ν) 2, and f (i) 11 (combination of emitted electron wave functions) And using 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

24 Phase Space Factors The key ingredient for the evaluation of phase space factors (and thus double beta decay) are the electron wave functions Depending on mode, we use positive/negative energy Dirac central field scattering wave functions (β /β + decay), or positive energy central field bound state wave functions (EC), consisting on radial functions g κ and f κ, and spherical spinors χ κ In all cases g κ and f κ satisfy the radial Dirac equations: dg κ (r) = κ r g κ(r) + ǫ V + m ec 2 f κ (r) dr df κ (r) dr c = ǫ V m ec 2 g κ (r) + κ c r f κ(r) Numerical solution by Salvat et al., Comput. Phys. Commun. 90, 151 (1995)

25 Phase Space Factors To simulate realistic situation, we take into account the finite nuclear size and the electron screening Thomas-Fermi screening function, ϕ(r) = Z eff /Z d, obtained by Majorana solution of Thomas Fermi equation (Esposito, Am. J. Phys. 70, 852 (2002)) Boundary conditions take into account the fact that final atom is charged ion or neutral atom depending on mode Comparison with previous calculations: Considerable difference Example of obtained radial wave functions: 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

26 Phase Space Factors G0Ν10 15 yr Ca Ge Nd Zr Cd Mo Te Xe Se Sn Pd Te Nd Sm Gd approximate this work Th U Pt Current 0νβ β PSFs (red) compared to previous calculations (blue) approx G0ΝG 0Ν Mass Number Ca approximate this work Se Zr Ge Pd Mo Cd Sn Te Nd Te Xe NdSm Gd Pt Neutron Number Relative difference: G 0ν/G approx 0ν Difference: few % for Ca, 30% for Nd, and already 60% for Pt Does affect half-life and average neutrino mass predictions!

27 Phase Space Factors These results have been confirmed by independent calculations of Stoica et al. PRC 88, (2013) Their conclusion: An inadequate numerical treatment can change significantly the results Now: Very detailed (1keV/100eV increments) calculations using Yale supercomputer to 1. Minimize uncertainty coming from numerical integration 2. Offer adequate numerical accuracy for single electron, summed energy and two-dimensional spectra, and angular correlations to be used in the analysis of experimental data 0 dg 0ΝdΕ Ε 1 m e c 2 MeV Ε 1 m e c 2 MeV Example 76 Ge 76 Se decay: (left) single electron spectrum, dw 0ν dg dǫ 1 = N (0) 0ν 0ν dǫ 1, (right) angular correlations, α(ǫ 1) = dg(1) 0ν /dǫ 1 ΑΕ dg (0) 0ν /dǫ 1

28 Phase Space Factors Estimate of uncertainties introduced to PSF 0ν Q-value 3 δq/q Radius 7% Screening 0.10% Most Q-values very accurately known Error coming from radius, R = r 0 A 1/3, can be significantly reduced by adjusting r 0 for each nucleus instead of using r 0 = 1.2 fm: 3 5 r2 0 A2/3 = r 2 exp where r 2 exp is obtained from electron scattering and/or muonic x-rays. The screening error is estimated to be 10% of the Thomas-Fermi contribution, known to overestimate the electron density at the nucleus

29 Half-Life Predictions: 0νβ β For now predictions are calculated with g A =1.269 (and m ν = 1eV) Ν Τ yr Ca Ge Se Zr Mo Pd Sn Xe Cd 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

30 Half-Life Predictions: 0νβ + β + and 0νECβ + β + β +, ECβ + available kinetic energy much smaller much smaller phase space much longer half-lives 0Ν Τ yr Β Β ECΒ Ni Zn Kr Ru Cd Xe Ba Ce Mass number Best candidates 0νECβ + in 124 Xe, 130 Ba, and 136 Ce, where half-lives for g A =1.269, m ν = 1eV Compared to 0νβ β hardly detectable BUT 130 Ba nucleus where 2νECEC observed in geochemical experiments

31 Half-Life Predictions: Resonantly Enhanced 0νECEC 0νECEC available energy larger, but since all the energies are fixed, additional requirement that Q-value matches the state energy Resonance enhancement: A, Z2 HH' A, Z2 [ ] 1 τ ECEC 1/2 (0 + ) = g 4 A G ECEC 0ν M 0ν 0 ECEC Q Ec A, Z1 Q Β 0 A, Z 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 Many candidates, such as 112 Sn, 130 Ba, and 136 Ce, ruled out by recent high precision Q-value measurements Q

32 Half-Life Predictions: Resonantly Enhanced 0νECEC Best candidates at the moment 152 Gd, and 180 W Decay (kev) Γ(keV) (m ec 2 )F τ 1/2 (10 27 )yr 124 Xe Gd Dy Er W For 152 Gd, 164 Er and 180 W decay resonance state is ground state, for 156 Dy and 124 Xe resonance state is excited 0 + -state importance of reliable wave functions! Half-lives > for m ν = 1eV and g A = Compared to 0νβ β hardly detectable

33 Limits on Average Neutrino Mass Factorized half-life: [ ] 1 τ1/2 0ν = G0ν ga 4 M(0ν) 2 f(m i,u ei ) 2 Light neutrinos: f(m i,u ei) = mν = 1 (U ek) 2 m k m e m e Advance: The average light neutrino mass is now well constrained by atmospheric, solar, reactor and accelerator neutrino oscillation experiments mν in ev k=light INVERTED NORMAL lightest neutrino mass in ev Heavy neutrinos: f(m i,u ei) = η = m p m 1 ν h = m p (U ekh ) 2 1 k=heavy m kh

34 Limits on Average Neutrino Mass Current limits to m ν from CUORICINO, IGEX, NEMO-3, KamLAND-Zen, EXO, and GERDA 0νββ experiments NEMO3 CUORICINO IGEX KamLANDZen GERDA EXO 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., Nucl. Phys. A 765, 483 (2006), CUORICINO: C. Arnaboldi et al., Phys. Rev. C 78, (2008), KamLAND-Zen: A. Gando et al., Phys. Rev. C 85, (2012), EXO: M. Auger et al., Phys. Rev. Lett. 109, (2012), GERDA: M. Agostini et al. (GERDA collaboration) arxiv: v1 [nucl-ex] (2013), X: H.V. Klapdor-Kleingrothaus et al., Phys. Lett. B 586, 198 (2004),

35 Limits on Average Neutrino Mass Same figure as the previous one with maximally quenched g A. In this case even the inverted mass hierarchy is still far away NEMO3 CUORICINO IGEX KamLANDZen GERDA EXO 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., Nucl. Phys. A 765, 483 (2006), CUORICINO: C. Arnaboldi et al., Phys. Rev. C 78, (2008), KamLAND-Zen: A. Gando et al., Phys. Rev. C 85, (2012), EXO: M. Auger et al., Phys. Rev. Lett. 109, (2012), GERDA: M. Agostini et al. (GERDA collaboration) arxiv: v1 [nucl-ex] (2013), X: H.V. Klapdor-Kleingrothaus et al., Phys. Lett. B 586, 198 (2004),

36 0νββ: Remarks The renormalization of g A is not known for 0νββ This is very important issue since g A enters the calculation to the fourth power! 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 There is a possibility of interference between light and heavy neutrino exchange

37 Conclusions & Outlook Conclusions We have calculated NMEs and phase space factors needed for the description of double beta decay and competing modes and analyzed the results This includes two neutrino modes, as well as the exchange of heavy neutrinos Effective value of g A is work in progress and first results suggest considerable quenching Based on our results 0νβ β is the likeliest mode to be observed, even if 0νECEC is resonantly enhanced Outlook Deeper analysis about the similarities and differences of IBM-2 and QRPA results more reliable NMEs Additional improvements may be included to PSFs if needed (P-wave contribution, finite extent of nuclear surface, etc.) Investigation of Majoron emission and right-handed current modes of DBD

38 Thanks to my collaborators F. Iachello and J. Barea and THANK YOU FOR your attention! NEMO3 CUORICINO IGEX KamLANDZen GERDA EXO X mν in ev INVERTED NORMAL lightest neutrino mass in ev