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Advances in the theory of 0νββ decay
Nuclear Physics B (Proc. Suppl.) 217 (2011) 5–8 www.elsevier.com/locate/npbps
Advances in the theory of 0νββ decay F. Iachelloa and J. Bareab a
Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, CT 06520-8120, USA b
Departamento de F´ısica, Universidad de Concepci´on, Casilla 160-C, Concepci´ on, Chile Recent advances in the theory of 0νββ decay are briefly discussed. New (2010) results for the nuclear matrix elements within the framework of the Interacting Boson Model (IBM-2) are presented.
1. THEORY OF 0νββ DECAY The process 0νββ in which a nucleus X is transformed into a nucleus Y with the emission of two electrons and no neutrinos, A Z XN → A − Y + 2e , is of fundamental importance N −2 Z+2 for measuring the neutrino mass and determining whether or not there is physics beyond the standard model. The half-life for the process can be written as
(0ν)
τ1/2
−1
2 2 = G0ν M (0ν) |fb (m, η)|
(1)
where G0ν is a phase-space factor (atomic physics), M (0ν) are the matrix elements (nuclear physics), and fb contains physics beyond the standard model (particle physics). When written in this form, the calculation of τ1/2 splits into three different parts. In the particle physics part, one starts from the assumed weak Lagrangean, L, and derives the transition operator inducing the decay, which under certain circumstances, can be factorized as T (p) = H(p)fb (m, η), where m and η are masses and coupling constants of the neutrino or other hypothetical particle beyond the standard model. In the nuclear physics part, one computes the matrix elements of H(p) between the initial and final states, M (0ν) = f |H(p)| i, and in the atomic physics part one computes the phase space factor which depends on the Qvalue of the process, Qββ = Ei − Ef − 2me c2 , and the charge Zd of the daughter nucleus Y, G0ν = G0ν (Qββ , Zd ). 0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.04.055
1.1. Particle physics The transition operator T (p) depends on the model of 0νββ decay. Three scenarios have been considered [1], [2]: (i) the emission and reabsorption of a light (mlight < 1MeV) neutrino; (ii) the emission and reabsorption of a heavy (mheavy > 1GeV) neutrino and (iii) the emission and absorption of a light neutrino together with emission of an hypothetical particle (Majoron). After the discovery of neutrino oscillations, attention has been focused on the first scenario, which will be considered henceforth. In order to construct the transition operator T (p), one starts from the weak interaction Hamiltonian GF (2) H β = √ [¯ eγμ (1 − γ5 ) νeL ] JLμ† + h.c. 2 ¯ + gV (q 2 )γ μ − igM (q 2 ) qν σ μν JLμ† = Ψτ 2mp μ q 2 μ 2 5 − gA (q )γ γ5 + gP (q ) γ Ψ, (3) 2mp where in addition to the usual V-A terms there are weak-magnetism and pseudo-scalar terms [2]. Here q μ is the momentum transferred from hadrons to leptons. From (2), after nonrelativistic reduction, one finds the transition operator, T (p), which, for scenario (i), can be written as mν 2 , mν = (Uνk ) mk , (4) T (p) = H(p) me k=light
where U is the neutrino mixing matrix and p = |q|. The transition operator H(p) has the form
6
F. Iachello, J. Barea / Nuclear Physics B (Proc. Suppl.) 217 (2011) 5–8
sion for the fermion operator inducing the process (λ) 1 + + (s1 ) (s2 ) Vs(λ) Σ = τ τ × Σ n n n n 1 ,s2 2
[2] H(p)
τn+ τn+ −hF (p) + hGT (p)σn · σn p + hT (p)Snn .
=
(5)
[In a more general approach [1], one considers also additional terms in H β (R-L couplings) and a recoil contribution, giving rise to nine terms in all, 3GT, 3F, 1T, one pseudoscalar (P) and one recoil (R).] The form factors hF,GT,T are given ˜ F,GT,T (p), with v(p) = by hF,GT,T (p) = v(p)h 2 1 ˜ .This form assumes the closure approxπ p(p+A) imation, (a good approximation for 0νββ since the virtual neutrino momentum is of the order of 100MeV/c and the energy scale of the nuclear excited states is 1 MeV), with closure energy A˜ = 1.12A1/2 (MeV). The quantity v(p) is called ˜ ”neutrino potential”. The form factors h(p) are listed in [2]. The finite nucleon size (FNS) is taken into account by taking the coupling constant momentum dependent and short range correlation (SRC) are taken into account by convoluting the potential v(p) with the correlation function j(p) usually taken as a Jastrow function. [It should be noted that Tomoda’s form factors are slightly difˇ ferent from Simkovic. His formulation is in coordinate space, i.e. the form factors are the Fourier transforms of those given above.] 1.2. Nuclear physics The ”nuclear matrix elements”, i.e. the matrix elements of H(p), can be written as M (0ν)
=
2 ˜ (0ν) gA M
≡
MGT −
(0ν)
gV gA
2
(0ν)
MF
(0ν)
+ MT
. (6)
Calculations up to 2008 were done within the framework of: (i) the quasi-particle random phase approximation (QRPA) [3] and (ii) the shell model (SM) [4]. A recent advance is (iii) the development of a program to compute 0νββ (and 2νββ) nuclear matrix elements in the closure approximation with the framework of the microscopic Interacting Boson Model (IBM-2) [5]. All matrix elements, F, GT and T, in this model can be calculated at once using the compact expres-
n,n
·
V (rnn )C (λ) (Ωnn ).
(7)
Here λ = 0, s1 = s2 = 0(F), λ = 0, s1 = s2 = 1(GT), λ = 2, s1 = s2 = 1(T). In second quantized form (−1)J+1 G(λ) s1 s2 (j1 j2 j1 j2 ; J) = Vs(λ) 1 ,s2 4 j1 j2 j1 j2 J × 1 + (−1)J δj1 j2 1 + (−1)J δj1 j2
(J) (J) × πj†1 × πj†2 · ν˜j1 × ν˜j2 . (8) Seen from the nucleus, the process amounts to the annihilation of a pair of neutrons with angular momentum J and the creation of a pair of protons with the same angular momentum J. The fermion operator V is then mapped
(0) into the boson space by using πj† × πj† →
(2) → Bπ (j, j )d†π,M , and simAπ (j)s†π , πj† × πj† M ilar for neutrons. The coefficients A, B are obtained by the so-called OAI mapping procedure [6]. Matrix elements of mapped operators are then evaluated with realistic wave functions of the initial and final nuclei either taken from the literature, when available, or obtained from a fit to the observed energies and other properties. Results for the matrix elements in dimensionless units are shown in figure 1. In IBM-2 these are obtained from the matrix elements of V in fm−1 by multipliplication by 2R, where R = 1.2A1/3 fm, the factor of 2 arising from the definition of V in Eq.(6). An error analysis of the IBM-2 results has been performed. Our estimated sensitivity to input parameter changes is: 1. Single-particle energies 10%. 2. Strength of the interaction that generates the collective pairs (surface delta) 5%. 3. Oscillator size parameter of the basis functions 5% . 4. Closure energy 5%. Our estimated sensitivity to model assumption is: 1. Truncation to S,D space 1% (spherical nuclei)-10% (deformed nuclei). 2. Isospin purity 1% (GT)20%(F)-1%(T). Our estimated sensitivity to operator assumptions is: 1. Form of the operator
7
F. Iachello, J. Barea / Nuclear Physics B (Proc. Suppl.) 217 (2011) 5–8 6 IBM2 QRPA SM
5
M 0Ν
4
3
2
0
40
1
60
80
Neutron number
100
120
Figure 1. Neutrinoless double beta decay matrix elements for IBM-2 [5], QRPA with gA equal to 1.25 and Jastrow SRC [3], and SM [4].
approximation may not be good for 2νββ, since only a selected number of states contributes to the decay. The average neutrino momentum is of the order of 2 MeV/c. Also, renormalization effects could be different in 0νββ than in 2νββ. This remains an important unsolved problem. 1.3. Atomic physics For an extraction of the neutrino mass and for estimates of the half-life one needs also the phase space factor G0ν . A general relativistic formulation was given by Tomoda [1] and results for selected cases tabulated. The phase space factor is given in terms of the quantity (0) (0) F11 = F11 (Qββ , Zd ) shown in Figure 2.
100
0Ν
5%. 2. Finite nuclear size (FNS) 1%. 3. Short range correlations (SRC) 2%. Estimated overall sensitivity 30%. We also note some simple features of the IBM2 calculations: (i) The mass dependence is very mild, approximately M (0ν) ∼ = 89A−2/3 . (ii) Shell effects are large. The matrix elements are small at the closed shells. (iii) Deformation effects appear to always decrease the matrix elements. We estimate: 76 Ge (-19%),128 Te (-26%),154 Sm (-32%). The nuclear matrix elements contain the axial vector coupling constant gA . In free space gA = 1.254. However, it is well known that in single beta decay in nuclei gA is renormalized to gA,ef f ≈ 0.7gA . This is a crucial problem for extraction of the neutrino mass since gA appears to the fourth power in the half-life. There are two main sources of renormalization: 1. Limited model space. 2. Missing hadronic degrees of freedom, Δ,... The determination of gA in 0νββ is a difficult problem to solve. For case 1, we are limited by the size of the matrices to diagonalize (> 109 ). For case 2, we are limited by a detailed knowledge of the decay process. An indirect solution is provided by the study of 2νββ. We have done a calculation of 2νββ in the closure approximation and find a renormalization of gA,ef f ∼ 0.7gA . The same result has been obtained in SM and in QRPA. However, the closure
F11 1013y 1fm2
500
50
130
136
10 5
A 76
82
100
150
(0)
Figure 2. The phase space factor F11 as calculated by Tomoda [1].
(0)
The relation between G0ν and F11 is interconnected with the definition of M (0ν) . Using To(0) moda definition, we have G0ν = F11 /4R2 with (0) R = 1.2A1/3 fm. The theory of F11 is by no (0) means simple. F11 is proportional to the scattering electron wave functions at the nucleus, (0) 2 F11 ∝ |ψe1 (0)ψe2 (0)| . If the wave functions are 2 (Zα) non-relativistic |ψ(0)| = 1−e2πy −2πy , y = (v/c) . If 2
the wave function are relativistic |ψ(0)| diverges. To regularize it, Tomoda solved the Dirac equation numerically for a uniform charge distribution with R = 1.2A1/3 fm. Because of the large depen-
8
F. Iachello, J. Barea / Nuclear Physics B (Proc. Suppl.) 217 (2011) 5–8
dence on Zd , i.e. Zdβ (β ∼ 3), the phase space factor favors heavy nuclei. For 150 Nd decay, the wave function is highly relativistic (Zd α ∼ 0.45). Because of the complex nature of the calculation and the resulting strong dependence on Zd we are planning to do a new and independent calculation (0) of F11 . Also, the consistency between the defini(0) tion of F11 and M (0ν) needs to be checked. 1.4. Results for half-life −1 (0ν) = Results for half-life, using τ1/2
(0) 2 F11 (0ν) 2 mν and the dimensionless 4R2 M me IBM-2 matrix elements, are shown in figure 3 for mν = 1eV and gA = 1.25. For other values, −2 −4 they scale as mν and (gA ) . These results are shown for relative comparison. The absolute scale must await a check of Tomoda’s formula, in particular the factor of 4 in the denominator.
1.50
0Ν
76
Ge → 76 Se 82 Se → 82 Kr 96 Zr → 96 Mo 100 Mo → 100 Ru 110 Pd → 110 Cd 116 Cd → 116 Sn 128 Te → 128 Xe 130 Te → 130 Xe 136 Xe → 136 Ba 148 Nd → 148 Sm 150 Nd → 150 Sm 154 Sm → 154 Gd
IBM-2 5.46 4.41 2.53 3.73 3.62 2.78 4.52 4.06 3.35 1.98 2.32 2.51
QRPA 4.68 4.17 1.34 3.53 2.93 3.77 3.38 2.22
SM 2.22 2.11
2.26 2.04 1.70
3. ACKNOWLEDGEMENTS
This work was performed in part under the US DOE Grant DE-FG02-91ER-40608.
1.00
Τ12 1024y
Table 1 Neutrinoless double beta decay matrix elements, M (0ν) , calculated in the IBM-2 [5], the QRPA [3], and the SM [4]. All matrix elements in dimensionless units.
0.70 0.50
REFERENCES
0.30
0.20 A 76
82
100
130
136
150
Figure 3. Half-lives for 0νββ calculated using IBM-2 matrix elements [5] and Tomoda phase space factors [1] for mν = 1eV and gA = 1.25.
2. TABLE OF NUCLEAR MATRIX ELEMENTS Since the publication of the IBM-2 results of [5], additional IBM-2 calculations have been done. In Table 1 and figure 1 we show, for future reference, our current (September 2010) results.
1. T. Tomoda, Rep. Prog. Phys. 54 (1991) 53. ˇ 2. F. Simkovic, G. Pantis, J.D. Vergados, and A. Faessler, Phys. Rev. C60 (1999) 055502. ˇ 3. F. Simkovic, A. Faessler, V. Rodin, P. Vogel, and J. Engel, Phys. Rev. C77 (2008) 045503. 4. E. Caurier, J. Men´endez, F. Nowacki, and A. Poves, Phys. Rev. Lett. 100 (2008) 052503. 5. J. Barea and F. Iachello, Phys. Rev. C79 (2009) 044301. 6. T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A309 (1978) 1.
Advances in the theory of 0νββ decay F. Iachelloa and J. Bareab a
Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, CT 06520-8120, USA b
Departamento de F´ısica, Universidad de Concepci´on, Casilla 160-C, Concepci´ on, Chile Recent advances in the theory of 0νββ decay are briefly discussed. New (2010) results for the nuclear matrix elements within the framework of the Interacting Boson Model (IBM-2) are presented.
1. THEORY OF 0νββ DECAY The process 0νββ in which a nucleus X is transformed into a nucleus Y with the emission of two electrons and no neutrinos, A Z XN → A − Y + 2e , is of fundamental importance N −2 Z+2 for measuring the neutrino mass and determining whether or not there is physics beyond the standard model. The half-life for the process can be written as
(0ν)
τ1/2
−1
2 2 = G0ν M (0ν) |fb (m, η)|
(1)
where G0ν is a phase-space factor (atomic physics), M (0ν) are the matrix elements (nuclear physics), and fb contains physics beyond the standard model (particle physics). When written in this form, the calculation of τ1/2 splits into three different parts. In the particle physics part, one starts from the assumed weak Lagrangean, L, and derives the transition operator inducing the decay, which under certain circumstances, can be factorized as T (p) = H(p)fb (m, η), where m and η are masses and coupling constants of the neutrino or other hypothetical particle beyond the standard model. In the nuclear physics part, one computes the matrix elements of H(p) between the initial and final states, M (0ν) = f |H(p)| i, and in the atomic physics part one computes the phase space factor which depends on the Qvalue of the process, Qββ = Ei − Ef − 2me c2 , and the charge Zd of the daughter nucleus Y, G0ν = G0ν (Qββ , Zd ). 0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.04.055
1.1. Particle physics The transition operator T (p) depends on the model of 0νββ decay. Three scenarios have been considered [1], [2]: (i) the emission and reabsorption of a light (mlight < 1MeV) neutrino; (ii) the emission and reabsorption of a heavy (mheavy > 1GeV) neutrino and (iii) the emission and absorption of a light neutrino together with emission of an hypothetical particle (Majoron). After the discovery of neutrino oscillations, attention has been focused on the first scenario, which will be considered henceforth. In order to construct the transition operator T (p), one starts from the weak interaction Hamiltonian GF (2) H β = √ [¯ eγμ (1 − γ5 ) νeL ] JLμ† + h.c. 2 ¯ + gV (q 2 )γ μ − igM (q 2 ) qν σ μν JLμ† = Ψτ 2mp μ q 2 μ 2 5 − gA (q )γ γ5 + gP (q ) γ Ψ, (3) 2mp where in addition to the usual V-A terms there are weak-magnetism and pseudo-scalar terms [2]. Here q μ is the momentum transferred from hadrons to leptons. From (2), after nonrelativistic reduction, one finds the transition operator, T (p), which, for scenario (i), can be written as mν 2 , mν = (Uνk ) mk , (4) T (p) = H(p) me k=light
where U is the neutrino mixing matrix and p = |q|. The transition operator H(p) has the form
6
F. Iachello, J. Barea / Nuclear Physics B (Proc. Suppl.) 217 (2011) 5–8
sion for the fermion operator inducing the process (λ) 1 + + (s1 ) (s2 ) Vs(λ) Σ = τ τ × Σ n n n n 1 ,s2 2
[2] H(p)
τn+ τn+ −hF (p) + hGT (p)σn · σn p + hT (p)Snn .
=
(5)
[In a more general approach [1], one considers also additional terms in H β (R-L couplings) and a recoil contribution, giving rise to nine terms in all, 3GT, 3F, 1T, one pseudoscalar (P) and one recoil (R).] The form factors hF,GT,T are given ˜ F,GT,T (p), with v(p) = by hF,GT,T (p) = v(p)h 2 1 ˜ .This form assumes the closure approxπ p(p+A) imation, (a good approximation for 0νββ since the virtual neutrino momentum is of the order of 100MeV/c and the energy scale of the nuclear excited states is 1 MeV), with closure energy A˜ = 1.12A1/2 (MeV). The quantity v(p) is called ˜ ”neutrino potential”. The form factors h(p) are listed in [2]. The finite nucleon size (FNS) is taken into account by taking the coupling constant momentum dependent and short range correlation (SRC) are taken into account by convoluting the potential v(p) with the correlation function j(p) usually taken as a Jastrow function. [It should be noted that Tomoda’s form factors are slightly difˇ ferent from Simkovic. His formulation is in coordinate space, i.e. the form factors are the Fourier transforms of those given above.] 1.2. Nuclear physics The ”nuclear matrix elements”, i.e. the matrix elements of H(p), can be written as M (0ν)
=
2 ˜ (0ν) gA M
≡
MGT −
(0ν)
gV gA
2
(0ν)
MF
(0ν)
+ MT
. (6)
Calculations up to 2008 were done within the framework of: (i) the quasi-particle random phase approximation (QRPA) [3] and (ii) the shell model (SM) [4]. A recent advance is (iii) the development of a program to compute 0νββ (and 2νββ) nuclear matrix elements in the closure approximation with the framework of the microscopic Interacting Boson Model (IBM-2) [5]. All matrix elements, F, GT and T, in this model can be calculated at once using the compact expres-
n,n
·
V (rnn )C (λ) (Ωnn ).
(7)
Here λ = 0, s1 = s2 = 0(F), λ = 0, s1 = s2 = 1(GT), λ = 2, s1 = s2 = 1(T). In second quantized form (−1)J+1 G(λ) s1 s2 (j1 j2 j1 j2 ; J) = Vs(λ) 1 ,s2 4 j1 j2 j1 j2 J × 1 + (−1)J δj1 j2 1 + (−1)J δj1 j2
(J) (J) × πj†1 × πj†2 · ν˜j1 × ν˜j2 . (8) Seen from the nucleus, the process amounts to the annihilation of a pair of neutrons with angular momentum J and the creation of a pair of protons with the same angular momentum J. The fermion operator V is then mapped
(0) into the boson space by using πj† × πj† →
(2) → Bπ (j, j )d†π,M , and simAπ (j)s†π , πj† × πj† M ilar for neutrons. The coefficients A, B are obtained by the so-called OAI mapping procedure [6]. Matrix elements of mapped operators are then evaluated with realistic wave functions of the initial and final nuclei either taken from the literature, when available, or obtained from a fit to the observed energies and other properties. Results for the matrix elements in dimensionless units are shown in figure 1. In IBM-2 these are obtained from the matrix elements of V in fm−1 by multipliplication by 2R, where R = 1.2A1/3 fm, the factor of 2 arising from the definition of V in Eq.(6). An error analysis of the IBM-2 results has been performed. Our estimated sensitivity to input parameter changes is: 1. Single-particle energies 10%. 2. Strength of the interaction that generates the collective pairs (surface delta) 5%. 3. Oscillator size parameter of the basis functions 5% . 4. Closure energy 5%. Our estimated sensitivity to model assumption is: 1. Truncation to S,D space 1% (spherical nuclei)-10% (deformed nuclei). 2. Isospin purity 1% (GT)20%(F)-1%(T). Our estimated sensitivity to operator assumptions is: 1. Form of the operator
7
F. Iachello, J. Barea / Nuclear Physics B (Proc. Suppl.) 217 (2011) 5–8 6 IBM2 QRPA SM
5
M 0Ν
4
3
2
0
40
1
60
80
Neutron number
100
120
Figure 1. Neutrinoless double beta decay matrix elements for IBM-2 [5], QRPA with gA equal to 1.25 and Jastrow SRC [3], and SM [4].
approximation may not be good for 2νββ, since only a selected number of states contributes to the decay. The average neutrino momentum is of the order of 2 MeV/c. Also, renormalization effects could be different in 0νββ than in 2νββ. This remains an important unsolved problem. 1.3. Atomic physics For an extraction of the neutrino mass and for estimates of the half-life one needs also the phase space factor G0ν . A general relativistic formulation was given by Tomoda [1] and results for selected cases tabulated. The phase space factor is given in terms of the quantity (0) (0) F11 = F11 (Qββ , Zd ) shown in Figure 2.
100
0Ν
5%. 2. Finite nuclear size (FNS) 1%. 3. Short range correlations (SRC) 2%. Estimated overall sensitivity 30%. We also note some simple features of the IBM2 calculations: (i) The mass dependence is very mild, approximately M (0ν) ∼ = 89A−2/3 . (ii) Shell effects are large. The matrix elements are small at the closed shells. (iii) Deformation effects appear to always decrease the matrix elements. We estimate: 76 Ge (-19%),128 Te (-26%),154 Sm (-32%). The nuclear matrix elements contain the axial vector coupling constant gA . In free space gA = 1.254. However, it is well known that in single beta decay in nuclei gA is renormalized to gA,ef f ≈ 0.7gA . This is a crucial problem for extraction of the neutrino mass since gA appears to the fourth power in the half-life. There are two main sources of renormalization: 1. Limited model space. 2. Missing hadronic degrees of freedom, Δ,... The determination of gA in 0νββ is a difficult problem to solve. For case 1, we are limited by the size of the matrices to diagonalize (> 109 ). For case 2, we are limited by a detailed knowledge of the decay process. An indirect solution is provided by the study of 2νββ. We have done a calculation of 2νββ in the closure approximation and find a renormalization of gA,ef f ∼ 0.7gA . The same result has been obtained in SM and in QRPA. However, the closure
F11 1013y 1fm2
500
50
130
136
10 5
A 76
82
100
150
(0)
Figure 2. The phase space factor F11 as calculated by Tomoda [1].
(0)
The relation between G0ν and F11 is interconnected with the definition of M (0ν) . Using To(0) moda definition, we have G0ν = F11 /4R2 with (0) R = 1.2A1/3 fm. The theory of F11 is by no (0) means simple. F11 is proportional to the scattering electron wave functions at the nucleus, (0) 2 F11 ∝ |ψe1 (0)ψe2 (0)| . If the wave functions are 2 (Zα) non-relativistic |ψ(0)| = 1−e2πy −2πy , y = (v/c) . If 2
the wave function are relativistic |ψ(0)| diverges. To regularize it, Tomoda solved the Dirac equation numerically for a uniform charge distribution with R = 1.2A1/3 fm. Because of the large depen-
8
F. Iachello, J. Barea / Nuclear Physics B (Proc. Suppl.) 217 (2011) 5–8
dence on Zd , i.e. Zdβ (β ∼ 3), the phase space factor favors heavy nuclei. For 150 Nd decay, the wave function is highly relativistic (Zd α ∼ 0.45). Because of the complex nature of the calculation and the resulting strong dependence on Zd we are planning to do a new and independent calculation (0) of F11 . Also, the consistency between the defini(0) tion of F11 and M (0ν) needs to be checked. 1.4. Results for half-life −1 (0ν) = Results for half-life, using τ1/2
(0) 2 F11 (0ν) 2 mν and the dimensionless 4R2 M me IBM-2 matrix elements, are shown in figure 3 for mν = 1eV and gA = 1.25. For other values, −2 −4 they scale as mν and (gA ) . These results are shown for relative comparison. The absolute scale must await a check of Tomoda’s formula, in particular the factor of 4 in the denominator.
1.50
0Ν
76
Ge → 76 Se 82 Se → 82 Kr 96 Zr → 96 Mo 100 Mo → 100 Ru 110 Pd → 110 Cd 116 Cd → 116 Sn 128 Te → 128 Xe 130 Te → 130 Xe 136 Xe → 136 Ba 148 Nd → 148 Sm 150 Nd → 150 Sm 154 Sm → 154 Gd
IBM-2 5.46 4.41 2.53 3.73 3.62 2.78 4.52 4.06 3.35 1.98 2.32 2.51
QRPA 4.68 4.17 1.34 3.53 2.93 3.77 3.38 2.22
SM 2.22 2.11
2.26 2.04 1.70
3. ACKNOWLEDGEMENTS
This work was performed in part under the US DOE Grant DE-FG02-91ER-40608.
1.00
Τ12 1024y
Table 1 Neutrinoless double beta decay matrix elements, M (0ν) , calculated in the IBM-2 [5], the QRPA [3], and the SM [4]. All matrix elements in dimensionless units.
0.70 0.50
REFERENCES
0.30
0.20 A 76
82
100
130
136
150
Figure 3. Half-lives for 0νββ calculated using IBM-2 matrix elements [5] and Tomoda phase space factors [1] for mν = 1eV and gA = 1.25.
2. TABLE OF NUCLEAR MATRIX ELEMENTS Since the publication of the IBM-2 results of [5], additional IBM-2 calculations have been done. In Table 1 and figure 1 we show, for future reference, our current (September 2010) results.
1. T. Tomoda, Rep. Prog. Phys. 54 (1991) 53. ˇ 2. F. Simkovic, G. Pantis, J.D. Vergados, and A. Faessler, Phys. Rev. C60 (1999) 055502. ˇ 3. F. Simkovic, A. Faessler, V. Rodin, P. Vogel, and J. Engel, Phys. Rev. C77 (2008) 045503. 4. E. Caurier, J. Men´endez, F. Nowacki, and A. Poves, Phys. Rev. Lett. 100 (2008) 052503. 5. J. Barea and F. Iachello, Phys. Rev. C79 (2009) 044301. 6. T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A309 (1978) 1.