Some Microscopic Aspects of Double Beta Decay Nuclear Matrix Elements in IBM-2

Transcript

1 Some Microscopic Aspects of Double Beta Decay Nuclear Matrix Elements in IBM- J. Barea 1 J. Kotila,3 F. Iachello 3 1 Departamento de Física, Universidad de Concepción, Chile Department of Physics, University of Jyväskylä, Finland 3 Sloane Physics Laboratory, Yale University, USA TRIUMF Double-Beta Decay Workshop May 11-13, 016, Vancouver, Canada

2 Contents 1 Double beta decay theory The interacting boson model 3 Results 4 The role of the single particle levels 5 Concluding remarks

3 Contents 1 Double beta decay theory The interacting boson model 3 Results 4 The role of the single particle levels 5 Concluding remarks

4 Halflives and NME [ τ (0ν) 1/ ] 1 G0ν M 0ν f (m i, U ei ), [ τ (ν) 1/ ] 1 Gν mec M ν, M 0ν M ν F ; J F h F 0ν + h GT 0ν F ; J F h F ν + h GT ν + h0ν T I ; I ; F,GT,T hx = 1 4 α πα π α ν α ν ( 1) J F,GT,T ( GX απα πα να ν; J ) J 1 + ( 1) J δ αν α ν 1 + ( 1) J δ απα π ( ) (J) π α π π ) (J) α ( ναν ν α π ν α ρ = (n ρl ρj ρ), ρ = ν, π, X = 0ν, 0ν h, ν Source: J. Barea and F. Iachello, PRC 79, (009)

5 Two body matrix elements I F,GT,T ( GX απ α πα ν α ν; J ) = α π α π; JM h F,GT,T X α ν α ν; JM F,GT,T h X h (s 1,s,λ) X = 1 τ n + τ + n n,n ( Σ (s 1) n Σ (s ) n ) (λ) HX (r nn )C (λ) (Ω nn ), h F h (0,0,0) X, h GT h (1,1,0) X, h T h (1,1,) 0ν Σ (0) n = 1, Σ (1) n = σ n, C (λ) (Ω) = 4π/ (λ + 1)Y (λ) (Ω)

6 Two body matrix elements II F,GT,T GX (α π α πα ν α ν; J; J) G (s 1,s,λ) X (α π α πα ν α ν; J; J) = lπ+lν l π +l ν kmax k 1 = l π l ν k = l π l ν k=k min i k 1 k +λˆk 1 ˆk k 10 k 0 λ0 { } { ( 1) s +k 1 k1 s 1 k ( 1) j π+j ν+j jπ j π } J ˆk jˆ π ĵ ν 1 1 s k λ 1 l π j π 1 l ν j ν s 1 k 1 k ˆk ˆ j πĵ ν j ν j ν k 1 l π j π 1 l ν j ν s k k Σ (s 1) 1 ( 1) k 1 lˆ π l π 0 k 1 0 l ν 0 ( 1) k l ˆ π l π0 k 0 l ν0 Σ (s ) 1 R (s 1,s,λ) X,k 1,k (n π, l π, n π, l π, n ν, l ν, n ν, l ν),

7 Radial Integrals 0 h (s 1,s,λ) X (p) p dp R (s 1,s,λ) X,k 1,k (n π, l π, n π, l π, n ν, l ν, n ν, l ν) = 0 0 R nπl π (r 1 ) R nνl ν (r 1 ) j k1 (pr 1 ) r 1 dr 1 R n π l π (r ) R n ν l ν (r ) j k (pr ) r dr h (s 1,s,λ) X (p) = v X (p) h (s 1,s,λ) (p) }{{} +HOC+FNS+SRC δ(p) for ν p 1 v X (p) = for light 0ν π p(p+ã) 1 π m em p for heavy 0ν ; HOC source: F. Šimkovic et al, PRC 60, (1999)

8 Contents 1 Double beta decay theory The interacting boson model 3 Results 4 The role of the single particle levels 5 Concluding remarks

9 The DBD transition operator in IBM- F,GT,T h = X α πα π α ν α ν J 1 4 ( 1)J F,GT,T G (α X πα ) π αν α ν ; J 1 + ( 1) J δ απ α 1 + ( 1) J δ π αν α ( ) ν (J) ) π απ (J) π α ( ν αν ν α π ν ( ) π απ (0) ( π Aαπ s απ π + A απ s π d π d ) (0) π +... ( ) () ( π απ π α B απ α π π d π + C απ α π s π s π d ) () π +... ) ( ( ναν ν (0) αν Ã αν s ν + Ã αν sν d ν d ) (0) ν +... ) () ( ) ( ν αν ν α B ν αν α d ν + C ν αν α d ν ν () sν sν +... Source: E. Caurier et al, PRL 100, (008) F,GT,T h X F,GT,T h s X,AA π,gt,t sν + hf d X,BB π d ν, where F,GT,T h X,AA F,GT,T h X,BB = α π α ν F,GT,T G (α X πα πα ν α ν ; 0) A απ Ã αν = δαπ α α π, α 1 + δαν π α ν π α ν, α ν F,GT,T ( G X απα π αν α ν ; ) B απ α B π αν α ν 0 Ν M GT AA A 76 BB BC CB CC BD DB CD DC DD

10 Isospin correction of the transition operator Monopole term removed R (s 1,s,λ) X,k 1,k R (s 1,s,λ) X,k 1,k δ k1 0δ k 0δ λ0 δ απαν δ α π α ν R(s 1,s,0) X,0,0 Consequences: M F ν 0 and M F 0ν are strongly reduced M GT ν and M GT,T 0ν do not change

11 Contents 1 Double beta decay theory The interacting boson model 3 Results 4 The role of the single particle levels 5 Concluding remarks

12 NMEs for light neutrino exchange Source: J. Barea, J. Kotila and F. Iachello, PRC 91, (015)

13 NMEs for heavy neutrino exchange and β + β + /β + EC/ECEC decays

14 NMEs for sterile neutrino exchange [ τ (0ν) 1/ ] 1 G0ν (U en ) M 0ν (m N ) f (m N ) ; N v (p) = π p + mn f (m N ) = m N m e lim f (m N)v (p) = m N m N 0 m e ( lim f (m N)v (p) = mp m N m N π 1 ( p + mn + Ã ) ; 1 ( ) π p p + Ã ) 1 m em p J. Barea, J. Kotila and F. Iachello, PRD 9, (015)

15 Exclusion zones

16 Contents 1 Double beta decay theory The interacting boson model 3 Results 4 The role of the single particle levels 5 Concluding remarks

17 Occupations for 130 Te Energies from a Woods-Saxon potential for different radii

18 Sensitivity of the NMEs for A = 130

19 Occupations for 150 Nd

20 Sensitivity of the NMEs for A = 150

21 Realistic occupancies in 130 Te Orbital Neutron S.P.E. [MeV] Neutron Occupancies (calculated) Neutron Occupancies (experiment) 3s 1/ (0) d 3/ d 5/ (0) 1g 7/ h 11/ (3)

22 Contents 1 Double beta decay theory The interacting boson model 3 Results 4 The role of the single particle levels 5 Concluding remarks

23 Summary The formalism of the DBD in IBM- has been explained. The results obtained have been shown. The influence of the single particle levels has been investigated.

24 Summary Thank you for your attention!

25 Woods-Saxon potential parametrization The single particle energies were obtained diagonalizing the Woods-Saxon Hamiltonian H = T + V (r) + V so (r) + 1 (1 + τ 3) V C (r), where ( V V (r) = 1 + e r/a ; V = V 0 1 ± κ N Z ) ; N + Z ( ) V so (r) = λ V 1 + e r/a (σ p) ; We used the Blomqvist and Wahlborn parametrization for V 0, κ, a, λ, etc. The values of N and Z correspond to those of the most stable beta decay isobars for A = 100, 110, 10, 130, 140, 150, 160 Computer Code WSBETA, S. Cwiok et al, Comp. Phys. Commun. 46 (1987) 379.

26 High Order Corrections / Finite Nucleon Size New terms in GT and a Tensor contribution h F 0ν = h F VV h GT 0ν = h GT AA + hgt AP + hgt PP + hgt MM h T 0ν = h T AP + ht PP + ht MM Coupling constants become momentum dependent g V (p g ( V ) = ( ), MV = 0.71 GeV/c ) 1 + p MV g A (p ) = g A ( ), M A = 1, 09 GeV/c 1 + p MA

27 HOC + FNS Terms HOC term h(p) h F VV h GT AA h GT AP h GT PP h MM GT h AP T h PP T h MM T g A g A [ [ ga gv /g A (1+p /MV ) 4 ga 1 (1+p /MA) 4 1 p 3 (1+p /MA) 4 p +mπ p +m π p (1+p /M [ A) g A 3 g V g A 1 ( ( κ β p (1+p /MV ) 4 4mp hap GT h GT PP 1 hgt MM ) ] 1 m π MA ) ] 1 m π MA ]

28 Short Range Correlations Jastrow function in coordinate space ψ SRC = f (r) ψ ψ SRC = f (r) ψ } ψ SRC H ψ SRC = ψf (r) H f (r) ψ H F,GT,T (r) H F,GT,T (r)f (r) f (r) = 1 Ce Ar ( 1 Br ) H F,GT (r) = H T (r) = 0 0 j 0 (pr)h F,GT,T (p)p dp j (pr)h T (p)p dp

29 SRC Parametrizations Name A (fm ) B (fm ) C Miller-Spencer Argonne CD-Bonn Source: F. Šimkovic et al, Phys. Rev. C 79, (009)

30 IBM- and isospin For heavy nuclei the valence protons occupy orbits full of neutrons T ψ = 0 T = M T max = 1 N Z For nuclei where protons and neutrons are in the same major shell, but with different character as particles or holes, isospin symmetry is violated only of order For lighter nuclei IBM-3 and IBM-4 are required to produce states with definite isospin Source: J. P. Elliot, Prog. Part. Nucl. Phys. 5, 35 (1990) 1 Ω