Recent advances in neutrinoless double beta decay with energy density functional methods · 2012....

Recent advances in neutrinoless double beta decay with energy

density functional methods

Tomás R. Rodríguez

Recent advances in neutrinoless double beta decay with energy

density functional methods

Tomás R. Rodríguez

Congratulations to Alfredo, Andrés, Etienne and all the people from Strasbourg-Madrid (SM) collaboration!!

2. Method: GCM+PNAMP

3. Results

1. Introduction: 0νββ decay

4. Summary and Conclusions

Outline


3. Results



Outline

Perrot’s

and

Menéndez’s talk!


3. Results



Outline

Perrot’s

and

Menéndez’s talk!

Robledo’s talk!

Tomás R. RodríguezRecent advances in neutrinoless double beta decay with EDFStrasbourg, October 2012

taken from J. Menéndez

Process mediated by the weak interaction which occurs in those even-even nuclei where

the single beta decay is energetically forbidden.

Double beta decay2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions


Two neutrino double beta decay 2⇥��

Neutrinoless double beta decay 0⇥��

AZXN �A

Z+2 YN�2 + 2e� + 2�e

AZXN �A

Z+2 YN�2 + 2e�

• Conserves the leptonic number• Compatible with massive or massless Dirac/Majorana neutrinos• Experimentally observed (T1/2~1019-21 y)• Within the Standard Model

• Violates the leptonic number conservation• Neutrinos are massive Majorana particles • Experimentally not observed (yet?) (T1/2 >1025 y)• Beyond the Standard Model

2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions

Double beta decay


Half-life neutrinoless double beta decay (Doi et al (1985))

• Kinematic phase space factor:

• Effective neutrino mass: �m�⇥ =�

j

U2ejmj

G01 =(GgA(0))4m4

e

64⇥5 ln 2

�F0(Z, ⇤1)F0(Z, ⇤2)

⇤p1p2�(⇤1 + ⇤2 � Ef � Ei)d⇤1d⇤2d(�p1 · �p1)

• Nuclear Matrix Element (NME):

M0⇥�� = ��

gV (0)gA(0)

⇥2

M0⇥��F + M0⇥��

GT �M0⇥��T

Fermi Gamow-Teller Tensor

⇥T 0⇥��

1/2 (0+ � 0+)⇤�1

= G01

��M0⇥��2

⌅⇥m⇥⇤me

⇧2

light-neutrino exchange mechanism

Double beta decay2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions


• Energy Density Functional

M0⇥�� = ��

gV (0)gA(0)

⇥2

M0⇥��F + M0⇥��

GT �M0⇥��T

• Each term can be written as the expectation value of a transition operator acting on the initial al final states:

M0⇥��⇤ = �0+

f |O0⇥��⇤ |0+

i ⇥

• Nuclear structure methods for calculating these NME:

• Quasiparticle Random Phase Approximation in different versions: QRPA, RQRPA, SRQRPA. (Tübingen group, Jyüvaskylä group)

• Interacting Shell Model -ISM- (Strasbourg-Madrid collaboration, Michigan)

• Interacting Boson Model -IBM- (Yale group)

• Projected Hartree-Fock-Bogoliubov -PHFB- (Lucknow-UNAM group)

Nuclear Matrix Elements2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions


• Energy Density Functional

M0⇥�� = ��

gV (0)gA(0)

⇥2

M0⇥��F + M0⇥��

GT �M0⇥��T

• Each term can be written as the expectation value of a transition operator acting on the initial al final states:

M0⇥��⇤ = �0+

f |O0⇥��⇤ |0+

i ⇥

• Nuclear structure methods for calculating these NME:

• Quasiparticle Random Phase Approximation in different versions: QRPA, RQRPA, SRQRPA. (Tübingen group, Jyüvaskylä group)

• Interacting Shell Model -ISM- (Strasbourg-Madrid collaboration, Michigan)

• Interacting Boson Model -IBM- (Yale group)

• Projected Hartree-Fock-Bogoliubov -PHFB- (Lucknow-UNAM group)

Different ways to deal with: - Finding the best initial and final ground states. - Handling the transition operator (inclusion of most relevant terms, corrections, approximations, etc.).

Some remarks about these methods: - Calculations with limited single particle bases. - Interactions fitted to the specific region (ISM) or to each nucleus individually (rest). - Difficulties to include collective degrees of freedom. - Problems with particle number conservation.

Nuclear Matrix Elements2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions


Method: GCM+PNAMP

• Effective nucleon-nucleon interaction: Gogny force (D1S-D1M) that is able to describe properly many phenomena along the whole nuclear chart.

First step: Particle Number Projection (before the variation) of HFB-type wave functions.

• Method of solving the many-body problem:

Second step: Simultaneous Particle Number and Angular Momentum Projection (after the variation).

Third step: Configuration mixing within the framework of the Generator Coordinate Method (GCM).

V (1, 2) =2X

i=1

e�(~r1�~r2)2/µ2

i (Wi +BiP� �HiP

⌧ �MiP�P ⌧ )

+t3(1 + x0P�)�(~r1 � ~r2)⇢

↵ ((~r1 + ~r2)/2)+iW0(�1 + �2)~k ⇥ �(~r1 � ~r2)~k

+VCoulomb

(~r1

,~r2

)



Determination of initial and final states (I)Intrinsic state: Solve the PN-VAPequations with the Gogny D1S/D1M interaction

��EN,Z

⇤|�(q)�

⌅⇥|��=|�� = 0

EN,Z[|�⇤] =⇥�|HPN PZ |�⇤⇥�|PN PZ |�⇤

+ ⇥N,ZDD (|�⇤) � �q⇥�|Q|�⇤

|��HFB states

|IMK; NZ; q⇥ =2I + 18�2

�DI�

MK(⇥)R(⇥)PN PZ |�(q)⇥d⇥Particle number and angular momentum projected state:

|IM ; NZ�� =�

Kq

f I;NZ,�Kq |IMK; NZ; q�General form (GCM state):

Hill-Wheeler-Griffin equation (GCM)

⇤

K�q�

�HI;NZ

KqK�q� � EI;NZ;�N I;NZKqK�q�

⇥f I;NZ;�

K�q� = 0

N I;NZKqK�q� � ⌅IMK; NZ; q|IMK �; NZ; q�⇧

HI;NZKqK�q� � ⌅IMK; NZ; q|H|IMK �; NZ; q�⇧ + �IKK�;NZ

DD [|�(q)⇧, |�(q�)⇧]

generalized eigenvalue problem

Particle number projection2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions


-0.5 0 0.5 1`

0

5

10

15

E norm

(MeV

)

PN-VAP150Nd

Determination of initial and final states (I)

��EN,Z

⇤|�(q)�

⌅⇥|��=|�� = 0 EN,Z[|�⇤] =

⇥�|HPN PZ |�⇤⇥�|PN PZ |�⇤

+ ⇥N,ZDD (|�⇤) � �q⇥�|Q|�⇤|��HFB states



-0.5 0 0.5 1`

0

5

10

15

E norm

(MeV

)

PN-VAP150Sm


��EN,Z

⇤|�(q)�

⌅⇥|��=|�� = 0 EN,Z[|�⇤] =

⇥�|HPN PZ |�⇤⇥�|PN PZ |�⇤

+ ⇥N,ZDD (|�⇤) � �q⇥�|Q|�⇤|��HFB states



-0.5 0 0.5 1`

0

5

10

15

E norm

(MeV

)PN-VAP

150Sm

-0.5 0 0.5 1`

0

5

10

15

E norm

(MeV

)

PN-VAP150Nd




Determination of initial and final states (II)Intrinsic state: Solve the PN-VAPequations with the Gogny D1S interaction

��EN,Z

⇤|�(q)�

⌅⇥|��=|�� = 0

EN,Z[|�⇤] =⇥�|HPN PZ |�⇤⇥�|PN PZ |�⇤

+ ⇥N,ZDD (|�⇤) � �q⇥�|Q|�⇤

|��HFB states

|IMK; NZ; q⇥ =2I + 18�2

�DI�


|IM ; NZ�� =�

Kq



⇤

K�q�

�HI;NZ


⇥f I;NZ;�

K�q� = 0



DD [|�(q)⇧, |�(q�)⇧]


Particle number and angular momentum projection



-0.5 0 0.5 1`

0

5

10

15

E norm

(MeV

)J=0PN-VAP

150Sm

-0.5 0 0.5 1`

0

5

10

15

E norm

(MeV

)

J=0PN-VAP

150Nd

Determination of initial and final states (II)

Particle number and angular momentum projection



|IM ; NZ�� =�

Kq



⇤

K�q�

�HI;NZ


⇥f I;NZ;�

K�q� = 0



DD [|�(q)⇧, |�(q�)⇧]


Configuration (shape) mixing

Determination of initial and final states (& III)Intrinsic state: Solve the PN-VAPequations with the Gogny D1S interaction

��EN,Z

⇤|�(q)�

⌅⇥|��=|�� = 0

EN,Z[|�⇤] =⇥�|HPN PZ |�⇤⇥�|PN PZ |�⇤

+ ⇥N,ZDD (|�⇤) � �q⇥�|Q|�⇤

|��HFB states

|IMK; NZ; q⇥ =2I + 18�2

�DI�




|IM ; NZ�� =�

Kq


�

K�q�

N I;NZKqK�q�u

I;NZK�q�;� = nI;NZ

� uI;NZKq;�

Solving HWG equation:

1. Diagonalization of the norm overlap:

2. Natural basis: |�IM ;NZ� =�

Kq

uI;NZKq;�⇥nI;NZ

�

|IMK; NZ; q� ; nI;NZ� /nI,NZ

max > �

3. Normal eigenvalue problem:

�

��

��I;NZ |H |��I;NZ⇥GI;NZ;�� = EI;NZ;�GI;NZ;�

�



��EN,Z

⇤|�(q)�

⌅⇥|��=|�� = 0

EN,Z[|�⇤] =⇥�|HPN PZ |�⇤⇥�|PN PZ |�⇤

+ ⇥N,ZDD (|�⇤) � �q⇥�|Q|�⇤

|��HFB states

|IMK; NZ; q⇥ =2I + 18�2

�DI�




General form (GCM state): |IM ; NZ�� =�

�

GI;NZ;�� |�I;NZ�

�

K�q�

N I;NZKqK�q�u

I;NZK�q�;� = nI;NZ

� uI;NZKq;�

Solving HWG equation:

1. Diagonalization of the norm overlap:

2. Natural basis: |�IM ;NZ� =�

Kq

uI;NZKq;�⇥nI;NZ

�

|IMK; NZ; q� ; nI;NZ� /nI,NZ

max > �

3. Normal eigenvalue problem:

�

��

��I;NZ |H |��I;NZ⇥GI;NZ;�� = EI;NZ;�GI;NZ;�

�



��EN,Z

⇤|�(q)�

⌅⇥|��=|�� = 0

EN,Z[|�⇤] =⇥�|HPN PZ |�⇤⇥�|PN PZ |�⇤

+ ⇥N,ZDD (|�⇤) � �q⇥�|Q|�⇤

|��HFB states

|IMK; NZ; q⇥ =2I + 18�2

�DI�




-0.5 0 0.5 1`

0

5

10

15

E norm

(MeV

)J=0PN-VAP

150Sm

-0.5 0 0.5 1`

0

5

10

15

E norm

(MeV

)

J=0PN-VAP

150Nd

Determination of initial and final states (& III)

Configuration (shape) mixing2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions


1. Axial states 2. Angular momentum3. Quadrupole deformations

I = 0K = 0

q = q20

|0; NiZi; �� =�

�i

G0;NiZi;��i

|�0;NiZii �

|0; NfZf ; �� =�

�f

G0;Nf Zf ;��f

|�0;Nf Zf

f �

Transitions2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions


Good agreement between experimental and theoretical Q-values, radii and total strength (quenched)

440 T.R. Rodríguez, G. Martinez-Pinedo / Progress in Particle and Nuclear Physics 66 (2011) 436–440

Table 1Masses, rms charge radii and total Gamow–Teller strengths S�(+) for mother (granddaughter) calculated with Gogny D1S GCM+PNAMP functionalcompared to experimental values. Theoretical values for S+/� are quenched by a factor (0.74)2.

Isotope BEth (MeV) BEexp (MeV) [27] Rth (fm) Rexp(fm) [28] Stheo�/+ Sexp�/+48Ca 420.623 415.991 3.465 3.473 13.55 (14.4 ± 2.2 [29])48Ti 423.597 418.699 3.557 3.591 1.99 (1.9 ± 0.5 [29])76Ge 664.204 661.598 4.024 4.081 20.97 (19.89 [30])76Se 664.949 662.072 4.074 4.139 1.49 (1.45

± 0.07 [31])82Se 716.794 712.842 4.100 4.139 23.56 (21.91 [30])82Kr 717.859 714.273 4.130 4.192 1.2496Zr 829.432 828.995 4.298 4.349 27.6396Mo 833.793 830.778 4.319 4.384 2.56 (0.29

± 0.08 [32])100Mo 861.526 860.457 4.372 4.445 27.87 (26.69 [30])100Ru 864.875 861.927 4.388 4.453 2.48116Cd 988.469 987.440 4.556 4.628 34.30 (32.70 [30])116Sn 991.079 988.684 4.567 4.626 2.61 (1.09+0.13

�0.57 [33])124Sn 1051.668 1049.96 4.622 4.675 40.65124Te 1051.562 1050.69 4.664 4.717 1.63128Te 1082.257 1081.44 4.686 4.735 40.48 (40.08 [30])128Xe 1080.996 1080.74 4.723 4.775 1.45130Te 1096.627 1095.94 4.695 4.742 43.57 (45.90 [30])130Xe 1097.245 1096.91 4.732 4.783 1.19136Xe 1143.333 1141.88 4.756 4.799 46.71136Ba 1143.202 1142.77 4.786 4.832 0.96150Nd 1234.512 1237.45 5.034 5.041 50.32150Sm 1235.936 1239.25 5.041 5.040 1.45

Acknowledgements

We thank A. Poves, J. Menéndez, J.L. Egido, K. Langanke, T. Duguet and F. Nowacki for fruitful discussions. TRR is supportedby the Programa de Ayudas para Estancias de Movilidad Posdoctoral 2008 and FPA2009-13377-C02-01 (MICINN). GMPis partly supported by the DFG through contract SFB 634, by the ExtreMe Matter Institute EMMI and by the HelmholtzInternational Center for FAIR.

References

[1] F.T. Avignone, S.R. Elliott, J. Engel, Rev. Modern Phys. 80 (2008) 481.[2] H.V. Klapdor-Kleingrothaus, et al., Phys. Lett. B 586 (2004) 198.[3] H. Ejiri, Prog. Part. Nucl. Phys. 64 (2010) 249.[4] E. Caurier, et al., Phys. Rev. Lett. 100 (2008) 052503.[5] J. Menendez, et al., Nuclear Phys. A 818 (2009) 139.[6] F. Simkovic, et al., Phys. Rev. C 60 (1999) 055502.[7] F. Simkovic, et al., Phys. Rev. C 77 (2008) 045503.[8] M. Kortelainen, J. Suhonen, Phys. Rev. C 75 (2007) 051303.[9] J. Suhonen, O. Civitarese, Nuclear Phys. A 847 (2010) 207.

[10] K. Chaturvedi, et al., Phys. Rev. C 78 (2008) 054302.[11] P.K. Rath, et al., Phys. Rev. C 80 (2009) 044303.[12] J. Barea, F. Iachello, Phys. Rev. C 79 (2009) 044301.[13] T.R. Rodríguez, G. Martinez-Pinedo, Phys. Rev. Lett. 105 (2010) 252503. arXiv:1008.5260.[14] S. Umehara, et al., J. Phys. Conf. Ser. 39 (2006) 356.[15] H. Simgen, Nuclear Phys. B 143 (2005) 567.[16] K. Zuber, et al., AIP Conf. Proc. 942 (2007) 101.[17] M. Bender, P.-H. Heenen, P.-G. Reinhard, Rev. Modern Phys. 75 (2003) 121.[18] J.F. Berger, M. Girod, D. Gogny, Nuclear Phys. A 428 (1984) 23.[19] H. Feldmeier, et al., Nuclear Phys. A 632 (1998) 61.[20] R. Rodríguez-Guzmán, J.L. Egido, L.M. Robledo, Nuclear Phys. A 709 (2002) 201.[21] D. Lacroix, T. Duguet, M. Bender, Phys. Rev. C 79 (2009) 044318.[22] P. Ring, P. Schuck, The Nuclear Many Body Problem, Springer-Verlag, Berlin, 1980.[23] M. Anguiano, J.L. Egido, L.M. Robledo, Nuclear Phys. A 696 (2001) 467.[24] T.R. Rodríguez, J.L. Egido, Phys. Rev. Lett. 99 (2007) 062501.[25] E. Caurier, et al., Rev. Modern Phys. 77 (2005) 427.[26] R. Álvarez-Rodríguez, et al., Phys. Rev. C 70 (2004) 064309.[27] G. Audi, A.H. Wapstra, C. Thibault, Nuclear Phys. A 729 (2003) 337.[28] I. Angeli, At. Data Nucl. Data Tables 87 (2004) 185.[29] K. Yako, et al., Phys. Rev. Lett. 103 (2009) 012503.[30] R. Madey, et al., Phys. Rev. C 40 (1989) 540.[31] R.L. Helmer, et al., Phys. Rev. C 55 (1997) 2802.[32] H. Dohmann, et al., Phys. Rev. C 78 (2008) 041602(R).[33] S. Rakers, et al., Phys. Rev. C 71 (2005) 054313.[34] J. Menendez, et al., 2008. arXiv:0809.2183.

Nuclear structure properties

Neutrinoless double beta decay candidates



-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0

1

2

1.5

0.5

0.5

0.5

A=150

- GT strength greater than Fermi.- Similar deformation between mother and granddaughter is favored by the transition operators- Maxima are found close to sphericity although some other local maxima are found

�0; NfZf ; qf |O0⇥��⇤ |0; NiZi; qi⇥�

�0; NfZf ; qf |0; NfZf ; qf ⇥�0; NiZi; qi|0; NiZi; qi⇥

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

4.5

2.5

0.5

0.5

0.5

0.5

0.0

1.0

2.0

3.0

4.0

5.0

6.0

GTF

� (150Nd)

�(1

50Sm

)

� (150Nd)�

(150Sm

)

NME: deformation and mixing2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions

T.R.R., Martínez-Pinedo, PRL 2010


-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0

1

2

1.5

0.5

0.5

0.5

A=150


�0; NfZf ; qf |O0⇥��⇤ |0; NiZi; qi⇥�


- Final result depends on the distribution of probability of the corresponding initial and final collective states within this plot

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

4.5

2.5

0.5

0.5

0.5

0.5

0.0

1.0

2.0

3.0

4.0

5.0

6.0

GTF

� (150Nd)

�(1

50Sm

)

� (150Nd)�

(150Sm

)

-0.5 0 0.5`

0

0.1

0.2

0.3

0.4|F

(`)|2 150Nd (0i

+)

150Sm (0f+)

(b)




-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0

1

2

1.5

0.5

0.5

0.5

A=150


�0; NfZf ; qf |O0⇥��⇤ |0; NiZi; qi⇥�



-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

4.5

2.5

0.5

0.5

0.5

0.5

0.0

1.0

2.0

3.0

4.0

5.0

6.0

GTF

� (150Nd)

�(1

50Sm

)

� (150Nd)�

(150Sm

)




-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0

1

2

1.5

0.5

0.5

0.5

A=150


�0; NfZf ; qf |O0⇥��⇤ |0; NiZi; qi⇥�



-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

4.5

2.5

0.5

0.5

0.5

0.5

0.0

1.0

2.0

3.0

4.0

5.0

6.0

GTF

� (150Nd)

�(1

50Sm

)

� (150Nd)�

(150Sm

)

|M0⇥��GT | = 1.28

|M0⇥��F |g2

A

= 0.43

|M0⇥��TOT | = 1.71




0.0

1.0

2.0

3.0

4.0

5.0

6.0

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

4.5

2.5

2.5

2.5

0.5

0.50.5

0.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0

1

2

0.5

0.5

0.5

GT

�(7

6Se

)

� (76Ge)�

(76Se

)� (76Ge)


F

NME: deformation and mixing

�0; NfZf ; qf |O0⇥��⇤ |0; NiZi; qi⇥�

�0; NfZf ; qf |0; NfZf ; qf ⇥�0; NiZi; qi|0; NiZi; qi⇥ A=762. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions


0.0

1.0

2.0

3.0

4.0

5.0

6.0

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

4.5

2.5

2.5

2.5

0.5

0.50.5

0.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0

1

2

0.5

0.5

0.5

GT

�(7

6Se

)

� (76Ge)�

(76Se

)� (76Ge)


F


-0.5 0 0.5`

0

0.1

0.2

0.3

0.4|F

(`)|2 76Ge (0i

+)

76Se (0f+)


�0; NfZf ; qf |O0⇥��⇤ |0; NiZi; qi⇥�



0.0

1.0

2.0

3.0

4.0

5.0

6.0

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

4.5

2.5

2.5

2.5

0.5

0.50.5

0.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0

1

2

0.5

0.5

0.5

GT

�(7

6Se

)

� (76Ge)�

(76Se

)� (76Ge)


F



�0; NfZf ; qf |O0⇥��⇤ |0; NiZi; qi⇥�



0.0

1.0

2.0

3.0

4.0

5.0

6.0

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

4.5

2.5

2.5

2.5

0.5

0.50.5

0.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0

1

2

0.5

0.5

0.5

GT

�(7

6Se

)

� (76Ge)�

(76Se

)� (76Ge)


F



�0; NfZf ; qf |O0⇥��⇤ |0; NiZi; qi⇥�

�0; NfZf ; qf |0; NfZf ; qf ⇥�0; NiZi; qi|0; NiZi; qi⇥ A=76

|M0⇥��GT | = 3.72

|M0⇥��F |g2

A

= 0.88

|M0⇥��TOT | = 4.60



Noticeable difference in the NME if only intrinsic spherical configurations are considered without configuration mixing.

T.R. Rodríguez, G. Martinez-Pinedo / Progress in Particle and Nuclear Physics 66 (2011) 436–440 439

Fig. 1. (a)–(c) Collectivewave functions, GT intensity with, (d)–(f) full and, (g)–(i) constant spatial dependence and (j)–(l) pairing energies for (left) A = 48,(middle) A = 76 and (right) A = 150 decays. Shaded areas correspond to regions explored by the collective wave functions.

Fig. 2. 0⇥�� matrix elements evaluated with full configuration mixing (circles) and assuming spherical symmetry (squares).



-0.4

-0.2

0

0.2

0.4

0.6

0

1

2

3

4

5

6

4.5

2.52.5

0.5

0.50.5

0.5

-0.4 -0.2 0 0.2 0.4 0.6`

51015202530

-Epp

(MeV

) 150Sm150Nd

- The structure is related to the pairing energy (particle-particle) of the nuclei involved in the transition

- Maxima of the strength correspond to maxima in pairing energy

- In agreement with seniority arguments (increasing seniority decreases NME)

- Pairing energy and NME involve similar Wick theorem’s contractions in this formalism

NME: Pairing2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions


-0.4

-0.2

0

0.2

0.4

0.6

0

1

2

3

4

5

6

4.5

2.5

2.5

2.5

2.5 0.5

0.50.5

0.5

-0.4 -0.2 0 0.2 0.4 0.6`

51015202530

-Epp

(MeV

) 76Se76Ge







-0.4

-0.2

0

0.2

0.4

0.6

0

1

2

3

4

5

6

2.5

2.5

2.5

2.5 0.5

0.5

0.5

0.5

-0.4 -0.2 0 0.2 0.4 0.6`

51015202530

-Epp

(MeV

) 48Ti48Ti







-0.4

-0.2

0

0.2

0.4

0.6

0

1

2

3

4

5

6

4.5

2.5

2.5

2.5

2.5

0.5

0.5

0.5

0.5

-0.4 -0.2 0 0.2 0.4 0.6`

51015202530

-Epp

(MeV

) 82Kr82Se







-0.4

-0.2

0

0.2

0.4

0.6

0

1

2

3

4

5

6

4.5

4.5

2.5

2.5

2.5

0.5

0.50.5

0.5

-0.4 -0.2 0 0.2 0.4 0.6`

51015202530

-Epp

(MeV

) 100Ru100Mo







-0.4

-0.2

0

0.2

0.4

0.6

0

1

2

3

4

5

6

4.5

2.5

2.5

2.5

2.5

0.5

0.50.5

0.5

-0.4 -0.2 0 0.2 0.4 0.6`

51015202530

-Epp

(MeV

) 124Te124Sn







-0.4

-0.2

0

0.2

0.4

0.6

0

1

2

3

4

5

6

4.52.5

2.5

0.5

0.50.5

0.5

-0.4 -0.2 0 0.2 0.4 0.6`

51015202530

-Epp

(MeV

) 128Xe128Te







-0.4

-0.2

0

0.2

0.4

0.6

0

1

2

3

4

5

6

6.54.5

2.5

2.5

2.5

0.5

0.50.5

0.5

-0.4 -0.2 0 0.2 0.4 0.6`

51015202530

-Epp

(MeV

) 130Xe130Te







-0.4

-0.2

0

0.2

0.4

0.6

0

1

2

3

4

5

6

4.5

2.5

2.5

2.5

0.5

0.50.5

0.5

-0.4 -0.2 0 0.2 0.4 0.6`

51015202530

-Epp

(MeV

) 136Ba136Xe







-0.4 -0.2 0 0.2 0.4 0.6`

-10

-5

-10

-5

82

50

(a) neutrons

(b) protons

164Er164Dy

82

f7/2

h9/2

p3/2

i13/2 f5/2

p1/2

g7/2

d5/2

d3/2

h11/2 s1/2

s.p.

e. (M

eV)

s.p.

e. (M

eV)

- The minima (maxima) of the NME as a function of deformation are related to areas with low (high) level density around the Fermi level of the initial and final states.

-0.4

-0.2

0

0.2

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0

0.2

0.4

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1

1.2

1.4

1.30.9

0.50.5

0.1

0.1

0.1

0.1

Fermi

164Er (0i+)

164Dy (0f+)

-0.4 -0.2 0 0.2 0.4 0.6`

510152025303540

-Epp

(MeV

)

164Er

164Dy

β2 164Er

β 2 16

4 Dy


T.R.R., G. Martiínez-Pinedo, PRC 2012


A 48 76 82 96 100 116 124 128 130 136 150 152 164 180

M0ν 2.37 4.60 4.22 5.65 5.08 4.72 4.81 4.11 5.13 4.20 1.71 1.07 0.64 0.58

T1/2(y)

28.5 x1023

76.9 x1023

20.8x1023

5.48 x1023

8.64 x1023

9.24 x1023

16.2 x1023

343.1 x1023

8.84x1023

12.7 x1023

16.5 x1023

4.2 x1031

1.3 x1036

1.6 x1034

A 48 76 82 96 100 116 124 128 130 136 150 152 164 180

M0ν 2.43 4.64 4.28 5.70 5.19 4.83 4.71 3.98 5.07 4.29 1.36 0.89 0.50 0.38

T1/2(y)

27.1 x1023

75.6 x1023

20.2x1023

5.38 x1023

8.28 x1023

8.82 x1023

16.9 x1023

365.8 x1023

9.05x1023

12.2 x1023

26.1x1023

6.2 x1031

2.1 x1036

3.8 x1034

NME: Summary of the results

double electron capturedouble beta decay

Gogny D1S parametrization

Gogny D1M parametrization



- Higher values than the ones predicted by ISM calculations (larger valence space, lower seniority components). - For A=76, 82, 128, 150 we predict smaller values than the ones given by QRPA and/or IBM while for A=96, 100, 116, 124, 130, 136 larger values are obtained.- Consistent results with the rest of the models. Notice that we are using the same interaction for all the nuclei.- Further studies are needed to understand what is missing in the different models.

QRPA (Jy): )M. Kortelainen, J. Suhonen, PRC 75, 051303(R) (2007) and PRC 76, 024315 (2007)

QRPA(Tu): F. Simkovic et al., PRC 77, 045503 (2008)

ISM: J. Menendez et al., PRL 100, 52503 (2008)

IBM-2:J. Barea, F. Iachello, PRC 77, 045503 (2008)

PHFB: K. Chaturvedi et al. PRC 78, 054302 (2008)0

1

2

3

4

5

6

7

M0i

``

GCM+PNAMPISM PHFB

IBM-2QRPA(Jy)QRPA(Tu)

48Ca 76Ge 82Se 96Zr 100Mo116Cd 124Sn 128Te 130Te 136Xe 150Nd





- Higher values than the ones predicted by ISM calculations (larger valence space, lower seniority components). - For A=76, 82, 128, 150 we predict smaller values than the ones given by QRPA and/or IBM while for A=96, 100, 116, 124, 130, 136 larger values are obtained.- Consistent results with the rest of the models. Notice that we are using the same interaction for all the nuclei.- Further studies are needed to understand what is missing in the different models.

QRPA (Jy): )M. Kortelainen, J. Suhonen, PRC 75, 051303(R) (2007) and PRC 76, 024315 (2007)

QRPA(Tu): F. Simkovic et al., PRC 77, 045503 (2008)

ISM: J. Menendez et al., PRL 100, 52503 (2008)

IBM-2:J. Barea, F. Iachello, PRC 77, 045503 (2008)

PHFB: K. Chaturvedi et al. PRC 78, 054302 (2008)0

1

2

3

4

5

6

7

M0i

``

GCM+PNAMPISM PHFB

IBM-2QRPA(Jy)QRPA(Tu)

48Ca 76Ge 82Se 96Zr 100Mo116Cd 124Sn 128Te 130Te 136Xe 150Nd

THEORETICAL UNCERTAINTY IS

A

FACTOR TWO IN MOST

OF T

HE CASES





• Energy density functional methods allow the inclusion of deformation of the initial and final states in the calculation of NME for neutrinoless double beta decay and double electron capture.

• Equal deformation of initial and final states is favored by the transition operator. Spherical configurations are more favored than deformed ones.

• NME values are strongly correlated to the pairing energies of initial and final states.

• NMEs are much larger for the double beta decay candidates than for double electron capture ones (deformation).

Summary and conclusions2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions


• Calculations in the pf-shell.

• Other degrees of freedom should be also explored (pairing vibrations, octupole deformations, triaxiality, explicit quasiparticle excitations...)

• Isospin symmetry breaking and restoration.

• Occupation numbers.

Outlook2. Method: GCM+PNAMP 3. Results: GCM+PNAMP 1. Introduction 4. Summary and Conclusions


Acknowledgments

G. Martínez-Pinedo (TU-Darmstadt)

K. Blaum (Universität Heidelberg)T. Duguet (CEA-Saclay)J. L. Egido (UAM-Madrid)K. Langanke (GSI-Darmstadt)N. López-Vaquero (UAM-Madrid)J. Menéndez (TU Darmstadt)F. Nowacki (Strasbourg)A. Poves (UAM-Madrid)C. Smorra (Universität Heidelberg)

Recent advances in neutrinoless double beta decay with energy density functional methods · 2012....

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