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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Shell Model Analysis of the Double-Beta Decay Half-Life
Mihai Horoi Department of Physics,
Central Michigan University, Mount Pleasant, Michigan 48859,
USA
Support from NSF grant PHY-0758099 and DOE grant
DF-FC02-09ER41584 is acknowledged
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Plan of the Talk
• Shell Model Approaches to 2 Neutrino DBD – Recent status of
the experimental data – Direct diagonalization approach – Lanczos
strength functions – Comparison with experimental data
• Shell Model Approach to Neutrinoless DBD – Status of present
and future experiments – The anatomy of the 0vββ matrix element –
Analysis of 48Ca, 76Ge and 82Se
• Conclusions and Outlook
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Office of Science Financial Assistance Funding Opportunity
Announcement
DE-PS02-09ER09-24Topical Collaborations in Nuclear Theory
・a. Effective field theory descriptions of nuclear forces ・b.
Properties of nuclei far from stability ・c. Microscopic studies of
nuclear input parameters for astrophysics ・d. Calculations of
electroweak corrections to precision data ・e. Microscopic nuclear
reaction theory ・f. Analysis of the spectrum of excited baryons and
mesons ・g. Studies of the phases of strongly-interacting matter ・h.
Phenomenology of hard probes of hot, dense matter ・i. Phenomenology
of thermal probes of hot matter ・j. Simulations of core collapse
supernovae ・k. Lattice simulations of hadron properties ・l. Lattice
simulations of thermal quantum chromodynamics ・m. Ab initio
many-body calculations ・n. Phenomenology of neutrino oscillations
・o. Dynamics of fission ・p. Calculations of double beta decay
nuclear matrix elements ・q. Extensions of the Standard Model
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Neutrino Masses
EFES-NSCL February 6, 2010
Mihai Horoi CMU
- Tritium decay:
- Cosmic Microwave Background (CMB) power spectrum:
€
Δm122 ≈ 8 ×10−5 eV 2 (solar) Δm23
2 ≈ 2.4 ×10−3 eV 2 (atmospheric)€
c12 ≡ cosθ12 , s12 = sinθ12 , etc
Two neutrino mass hierarchies
€
3H → 3He + e− +ν e
mν e = Uei2mi
i=1
3
∑ < 2.2eV (Mainz exp.)
Katrin exp. (in progress): goal mν e < 0.3eV
€
m1 +m2 +m3 < 0.6eV
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Double Beta Decay Problem
EFES-NSCL February 6, 2010
Mihai Horoi CMU
Adapted from Avignone, Elliot, Engel, Rev. Mod. Phys. 80, 481
(2008) -> RMP08
2-neutrino double beta decay
neutrinoless double beta decay
€
mββ = mkUek2
k∑
€
T1/ 2−1 (0v) =G0v (Qββ ) M
0v (0+)[ ] 2< mββ >me
⎛
⎝ ⎜
⎞
⎠ ⎟
2
€
Qββ
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Two neutrino double-beta decay half-lives: recommended
values
Source: Barabash arXiv:0807.2948
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
2v Double Beta Decay (DBD) of 48Ca
Horoi, Stoica, Bown, PRC 75, 034303 (2007) 48Ca - 250 1+
states
€
f7 / 2€
p1/ 2
€
f5 / 2
€
p3 / 2
€
G
€
T1/ 2−1 =G2v (Qββ ) MGT
2v (0+)[ ] 2
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Comparison with Experiment
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
€
MGT2v = 3 〈σ τ +0 f
1k+〉〈1k
+
H − Egs + E0k∑ στ−0i〉
= 3〈σ τ +0 f1
H − Egs + E0σ τ−0i〉
€
HLN =
α1 β1 0 0β1 α2 β2 0 00 β2α3 β3 0 0 0 βN−1αN
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
€
σ τ−0i〉 = c− dw−〉 = c− L1−〉
σ τ +0 f 〉 = c+ dw+〉 = c+ L1+〉
€
MGT2v ≈ 3c+c− 〈dw +1k
+〉LN 1EL
N (1k+) − Egs + E0k
∑ LN〈1k+ dw−〉
Caurier, Poves, Zuker, PLB 252, 13 (1990)
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
€
HLN =
α1 β1 0 0β1 α2 β2 0 00 β2α3 β3 0 0 0 βN−1αN
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
€
σ τ−0i〉 = c− dw−〉 = c− L1−〉
σ τ +0 f 〉 = c+ dw+〉 = c+ L1+〉
€
MGT2v ≈ 3c+c− 〈dw +1k
+〉LN 1EL
N (1k+) − Egs + E0k
∑ LN〈1k+ dw−〉
€
1HL
N − Egs + E0L1− = g1
− L1− ++ gN
− LN−
Engel, Haxton, Vogel, PRC 46, 2153R (1992)
€
MGT2v ≈ 3c+c− gm
− 〈dw + Lm− 〉
m∑
€
HLNV =V EL
N
1HL
N − Egs + E0L1− =
V1kVm kEL
N (1k+) − Egs + E0k=1
N
∑⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
m=1
N
∑ Lm−
≡ gm−( )
m=1
N
∑ Lm−
48Ca - revisited
€
≈ 3c+c− gm+ 〈Lm
+ dw−〉m∑
€
MGT2v = 3〈σ τ +0 f
1H − Egs + E0
σ τ−0i〉
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Double EC Decay of 58Ni Suhonen, Civitarese, Phys.Rep. 300, 123
(1999) Table 30
€
0.476 MeV −1 0.151 MeV −1
€
Max m− scheme dimension :1.6 ×109€
MGT2v ≈ 3c+c− gm
− 〈dw + Lm− 〉
m∑
€
≈ 3c+c− gm+ 〈Lm
+ dw−〉m∑
€
gm± =
V1k±Vm k
±
ELN (1k
+) − Egs + E0k=1
N
∑
- factor of ~ 10,000 comparing with the sum on intermediate
(exact) states - factor of 2-3 comparing with Caurier et al.
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Results
0
0.05
0.1
0.15
0.2
0.25
48Ca 82Se 128Te 130Te 130Ba 58Ni 64Zn
M (
MeV
^-1
)
Experiment
Theory
€
σ τ → 0.77σ τquenching
pf/GXPF1A: 48Ca(2nββ), 58Ni(2nECEC), 64Zn(2nECEC)
pf5/2g9/2/new int: 82Se(2nββ)
g7/2sdh11/2/PRC 71: 128Te(2nββ), 130Te(2nββ), 130Ba(2nECEC)
PRC 71, 044317 (2005) (0g7/2 1d5/2 )N-t(1d3/22s1/20h11/2 )t
t M Max Dim
0 0.282 3,466,564" 2 0.107 760,338,824" 4 0.103 16,668,221,492"
14 ? 220,000,635,778"
130Ba(2nECEC)
9,332,053,548
€
data for A > 48 suggestsστ →0.5στ quenching
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Neutrinoless Double Beta Decay
€
T1/ 2−1 (0v) =G0v (Qββ ) M
0v (0+)[ ] 2< mββ >me
⎛
⎝ ⎜
⎞
⎠ ⎟
2
i. Contributions from heavy neutrinos neglected
ii. Only contribution from Qββ considered - eliminate majorons
contributions
Effective νββ Mass
RMP08
€
mββ = mkUek2
k∑
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The Effect of the Energy Resolution
EFES-NSCL February 6, 2010
Mihai Horoi CMU
76Ge -> 76Se
F.T. Avignone et al, New J. Phys. 7, 6 (2005)
(2004)
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Planned Experiments
Source: RMP 80, 481(2008)
Homestake
Japan
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CANDLES Projects: Japan
EFES-NSCL February 6, 2010
Mihai Horoi CMU
T. Kishimoto, DBD07
48Ca natural abundance: 0.187% !!!
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Neutrinoless Double Beta Decay
€
T1/ 2−1 (0v) =G0v (Qββ ) M
0v (0+)[ ] 2< mββ >me
⎛
⎝ ⎜
⎞
⎠ ⎟
2
Barea & Iachello, PRC 79 044301 (2009)
Menendez et al arXiv:0906.0179
€
mββ = mkUek2
k∑
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
The Anatomy of the M0v
€
MS0v = Γ( ) 0 f
+ ap+a ʹ′ p
+( )J
˜ a ʹ′ n ˜ a n( )J⎡
⎣ ⎢ ⎤ ⎦ ⎥
0
0i+ p ʹ′ p ;J ˆ S
h(q) jκ (qr)GFS2 fSRC
2
q q+ < E >( )τ1−τ2−
⎡
⎣ ⎢
⎤
⎦ ⎥ n ʹ′ n ;J
a
− closure∑€
M 0v = MGT0v( ) − gVgA
⎛
⎝ ⎜
⎞
⎠ ⎟
2
MF0v + MT
0v
€
p ʹ′ p ;J [ ] n ʹ′ n ;J = ...( )lp 1/2 j pl ʹ′ p 1/2 j ʹ′ p λ s
J
⎧
⎨ ⎪
⎩ ⎪
⎫
⎬ ⎪
⎭ ⎪ sλ
∑ln 1/2 j pl ʹ′ n 1/2 j ʹ′ n λ s J
⎧
⎨ ⎪
⎩ ⎪
⎫
⎬ ⎪
⎭ ⎪
s || ˆ S || s ×
ml NL | nplp n ʹ′ p l ʹ′ p λ ʹ′ m l NL | nnln n ʹ′ n l ʹ′ n λm
ʹ′ m l NL∑ m l h(q) jκ (qr)GFS
2 fSRC2
q q+ < E >( ) ʹ′ m l
€
ˆ S = σ 1⋅ σ 2 GT
ˆ 1 F
⎧ ⎨ ⎩
€
T1/ 2−1 (0v) =G0v (Qββ ) M
0v (0+)[ ] 2< mββ >me
⎛
⎝ ⎜
⎞
⎠ ⎟
2
- Closure approximation
- Includes higher order corrections in the nucleon currents
€
ml h(q) jκ (qr)GFS2 (q) fSRC
2
q q+ < E >( ) ʹ′ m l = drdq jκ (qr)( )∫∫
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Short Range Correlations I
€
Vv (r)→ fSRC (r)Vv (r) fSRC (r)
€
fSRC (r) =1− c e−a r 2 (1− br2)
F. Simkovic et al, PRC 79, 055501 (2009)
a b c
Spencer-Miller 1.10 0.68 1.0 CCM/AV18 1.59 1.45 0.92 CCM/CD-Bonn
1.52 1.88 0.46
€
M 0v = dr12C(r12)0
∞
∫
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Short Range Correlations II Engel & Hagen, PRC 79, 064317
(2009)
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
48Ca: M0v vs and ΛA
Conclusion: about 5-8% variation
€
GΛV ,A (q) =ΛV ,A2
q2 +ΛV ,A2
⎛
⎝ ⎜
⎞
⎠ ⎟
2
€
ΛV = 850 MeVΛA = 1085 MeV⎧ ⎨ ⎩
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
48Ca: M0v vs the Effective Interaction
€
M 0v
0
0.2
0.4
0.6
0.8
1
1.2
GXPF1 GXPF1A KB3 KB3G FPD6
No SRC M-S SRC CD-Bonn SRC AV18 SRC
M. Horoi, S. Stoica, arXiv:0911.3807, accepted at Phys. Rev.
C
€
Prediction : M 0v = 0.85 ± 0.15 T1/2 (0v )≥1026 y⎯ → ⎯ ⎯ ⎯ ⎯ mββ
≤ 0.230± 0.045eV
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
100Mo - QRPA
Dependence on gpp (overall multiplication factor of the pn
effective interaction)
F. Simkovic et al., Phys. Rev. C 77, 045503 (2008).
€
Jπ
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
The Contribution of Intermediate J- States with Closure
Approximation
€
MS0v = Γ( ) 0 f
+ ap+a ʹ′ p
+( )J
˜ a ʹ′ n ˜ a n( )J⎡
⎣ ⎢ ⎤ ⎦ ⎥
0
0i+ p ʹ′ p ;J ˆ S
h(q) jκ (qr)GFS2 fSRC
2
q q+ < E >( )τ1−τ2−
⎡
⎣ ⎢
⎤
⎦ ⎥ n ʹ′ n ;J
a
− closure∑
Closure & 0hω approximation for 48Ca and 48Ti => no
contribution from 1hω states in 48Sc
€
pf
€
sd
€
48Ca
€
0ω
€
48Sc
€
48Ti
€
0ω
€
1ω (J −)
€
???
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
76Ge and 82Se
jj4b: f5/2 p3/2 p1/2 g9/2
€
M 0v
0
0.5
1
1.5
2
2.5
3
3.5
76Ge-A 76Ge-H 82Se-A
No SRC
Spencer-Miller
CD-Bonn SRC
AV18 SRC
A - Brown and Horoi (similar to JUN45 – Honma) H - K. Kaneko, M.
Hasegawa, and T. Mizusaki, Phys. Rev. C 70, 051301(R) (2004). E –
Experimental data: PRL 100, 112501 (2008), PRC 79, 021301
(2009)
0
2
4
6
8
10
12
14
16
18
76Ge N H
76Ge N E
76Se N H
76Se N E
76Ge P H
76Ge P E
76Se P H
76Se P E
0f5/2
1p
0g9/2
0
2
4
6
8
10
12
14
16
18
76Ge N A
76Ge N E
76Se N A
76Se N E
76Ge P A
76Ge P E
76Se P A
76Se P E
0f5/2
1p
0g9/2
Occupation probabilities
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Neutrinoless Double Beta Decay
€
T1/ 2−1 (0v) =G0v (Qββ ) M
0v (0+)[ ] 2< mββ >me
⎛
⎝ ⎜
⎞
⎠ ⎟
2
Barea & Iachello, PRC 79 044301 (2009) Menendez et al
arXiv:0906.0179 Present work
€
Prediction for 76Ge : M 0v = 3 T1/2 (0v )≥1026 y⎯ → ⎯ ⎯ ⎯ ⎯ mββ
≤ 0.220eV
€
mββ = mkUek2
k∑
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Short Range Correlations II
Engel & Hagen, PRC 79, 064317 (2009)
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Two 0vββ decay cases - 76Ge -> 76Se"- fp-g9/2 valence
space"
- p,n: 0f7/2 0f5/2 1p3/2 1p1/2 0g9/2"
- 76Ge: dim 1,296,156,991,047"- 76Se: dim
18,333,463,355,503"
- 150Nd -> 150Sm"- p: 0g7/2 1d5/2 1d3/2 2s1/2 0h11/2"
- n: 1f7/2 1f5/2 0h9/2 2p3/2 2p1/2 0i13/2"
- 150Nd: dim 222,314,413,121,622"- 150Sm: dim
32,199,157,066,956"
150Nd 150Sm
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Summary and Outlook
• 48Ca case suggests that 2ν double-beta decay can be
reasonably well described within the shell model, provided that all
spin partners are included and the quenching factor is well
determined from experiment.
• Efforts must be done to study these effects for the heavier
systems.
• Shell model 0νββ matrix elements seems being not very
sensitive to the effective interaction used, at least for 48Ca.
• The effects of the quenching and the missing spin partners
could be important, and they should be further investigated.
• Prediction for 48Ca: • Prediction for 76Ge:
€
M 0v = 0.85 ± 0.15 T1/2 (0v )≥1026 y⎯ → ⎯ ⎯ ⎯ ⎯ mββ ≤ 0.230±
0.045eV
€
M 0v = 3 T1/2 (0v )≥1026 y⎯ → ⎯ ⎯ ⎯ ⎯ mββ ≤ 0.220eV
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EFES-NSCL February 6, 2010
Mihai Horoi CMU
Collaborators • B. A. Brown - MSU/NSCL • S. Stoica - Bucharest
• A. Neacsu - Bucharest • Z. Gao - CMU/Beijing