Shell Model Analysis of the Double-Beta Decay Half-Lifebrown/EFES-2010/pdf/horoi.pdfDouble-Beta Decay Half-Life Mihai Horoi Department of Physics, Central Michigan University, Mount

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


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


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

Neutrino Masses


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

Double Beta Decay Problem


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


Mihai Horoi CMU

Two neutrino double-beta decay half-lives: recommended values

Source: Barabash arXiv:0807.2948


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


Mihai Horoi CMU

Comparison with Experiment


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)


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〉


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.


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


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∑

The Effect of the Energy Resolution


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76Ge -> 76Se

F.T. Avignone et al, New J. Phys. 7, 6 (2005)

(2004)


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Planned Experiments

Source: RMP 80, 481(2008)

Homestake

Japan

CANDLES Projects: Japan


Mihai Horoi CMU

T. Kishimoto, DBD07

48Ca natural abundance: 0.187% !!!


Mihai Horoi CMU


€

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∑


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)( )∫∫


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

∞

∫


Mihai Horoi CMU

Short Range Correlations II Engel & Hagen, PRC 79, 064317 (2009)


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⎧ ⎨ ⎩


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


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


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 −)

€

???


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


Mihai Horoi CMU


Mihai Horoi CMU


€

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∑


Mihai Horoi CMU

Short Range Correlations II

Engel & Hagen, PRC 79, 064317 (2009)


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


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


Mihai Horoi CMU

Collaborators •  B. A. Brown - MSU/NSCL •  S. Stoica - Bucharest •  A. Neacsu - Bucharest •  Z. Gao - CMU/Beijing

Documents

Shell Model Analysis of the Double-Beta Decay Half-Lifebrown/EFES-2010/pdf/horoi.pdfDouble-Beta Decay Half-Life Mihai Horoi Department of Physics, Central Michigan University, Mount