Relevant degrees of freedom for 0 decay nuclear matrix elements - - PowerPoint PPT Presentation

SLIDE 1

Tomás R. Rodríguez

Interfacing theory and experiment for reliable double-beta decay matrix element calculations Vancouver, May 11-13, 2016

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with energy density functionals

SLIDE 2

Outline

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• J. Menéndez (University of Tokio)
• G. Martínez-Pinedo (TU-Darmstadt)
• A. Poves (UAM-Madrid)

F . Nowacki (IPHC-Strasbourg)

• J. Engel (UNC-Chapel Hill)
• N. Hinohara (MSU-East Lansing)
• N. López-Vaquero (UAM-Madrid)
• J. L. Egido (UAM-Madrid)

SLIDE 3

• 1. EDF method
• 2. Multipole deformation

Outline

• 3. Pairing

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 5. Summary and open questions
• 4. Seniority and SU(4)

SLIDE 4

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)
• Nuclear structure methods for calculating these NME:

Nuclear Matrix Elements

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.
• Difficulties to include collective/single particle degrees of freedom.
• Problems with particle number/isospin conservation.

SLIDE 5

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

Gogny EDF

Effective nucleon-nucleon interaction: Gogny force (D1S/D1M)

V (1, 2) =

2

• i=1

e−(⌥

r1−⌥ r2)2/µ2

i (Wi + BiP ⇥ − HiP ⇤ − MiP ⇥P ⇤)

+iW0(⇥1 + ⇥2)⌥ k × (⌥ r1 − ⌥ r2)⌥ k

+t3(1 + x0P σ)( r1 − r2)⇥α (( r1 + r2)/2) +VCoulomb(⌅ r1,⌅ r2)

2

• b
• d

y p

• t

e n t i a l D e n s i t y d e p e n d e n t t e r m

SLIDE 6

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

Gogny EDF

Effective nucleon-nucleon interaction: Gogny force (D1S/D1M)

V (1, 2) =

2

• i=1

e−(⌥

r1−⌥ r2)2/µ2

i (Wi + BiP ⇥ − HiP ⇤ − MiP ⇥P ⇤)

+iW0(⇥1 + ⇥2)⌥ k × (⌥ r1 − ⌥ r2)⌥ k

+t3(1 + x0P σ)( r1 − r2)⇥α (( r1 + r2)/2) +VCoulomb(⌅ r1,⌅ r2)

2

• b
• d

y p

• t

e n t i a l D e n s i t y d e p e n d e n t t e r m

Next step: Variational method!!!

SLIDE 7

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

(M. Anguiano et al., Nucl. Phys. A 683, 227 (2001))

• Initial intrinsic states: PN-VAP
• M. Anguiano, J. L. Egido, and L. M. Robledo, Nucl. Phys. A 696, 467 (2001).

EN,Z [Φ] = hΦ| ˆ H2b ˆ P N ˆ P Z|Φi hΦ| ˆ P N ˆ P Z|Φi + εN,Z

DD (Φ) λq20hΦ| ˆ

Q20|Φi

EDF axial

SLIDE 8

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

(M. Anguiano et al., Nucl. Phys. A 683, 227 (2001))

• Initial intrinsic states: PN-VAP
• M. Anguiano, J. L. Egido, and L. M. Robledo, Nucl. Phys. A 696, 467 (2001).

EN,Z [Φ] = hΦ| ˆ H2b ˆ P N ˆ P Z|Φi hΦ| ˆ P N ˆ P Z|Φi + εN,Z

DD (Φ) λq20hΦ| ˆ

Q20|Φi

• Intermediate Particle Number and Angular Momentum Projected states

|I; NZ; β2i = 2I + 1 2 Z π dI∗

00(β)e−iβ ˆ Jy ˆ

P N ˆ P Z|Φi dβ

EDF axial

SLIDE 9

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

(M. Anguiano et al., Nucl. Phys. A 683, 227 (2001))

• Initial intrinsic states: PN-VAP
• M. Anguiano, J. L. Egido, and L. M. Robledo, Nucl. Phys. A 696, 467 (2001).

EN,Z [Φ] = hΦ| ˆ H2b ˆ P N ˆ P Z|Φi hΦ| ˆ P N ˆ P Z|Φi + εN,Z

DD (Φ) λq20hΦ| ˆ

Q20|Φi

• Intermediate Particle Number and Angular Momentum Projected states

|I; NZ; β2i = 2I + 1 2 Z π dI∗

00(β)e−iβ ˆ Jy ˆ

P N ˆ P Z|Φi dβ

• Final GCM states

|I; NZ; σi = X

β2

f I;NZ;σ

β2

|I; NZ; β2i

EDF axial

SLIDE 10

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

(M. Anguiano et al., Nucl. Phys. A 683, 227 (2001))

• Initial intrinsic states: PN-VAP
• M. Anguiano, J. L. Egido, and L. M. Robledo, Nucl. Phys. A 696, 467 (2001).

EN,Z [Φ] = hΦ| ˆ H2b ˆ P N ˆ P Z|Φi hΦ| ˆ P N ˆ P Z|Φi + εN,Z

DD (Φ) λq20hΦ| ˆ

Q20|Φi

• Intermediate Particle Number and Angular Momentum Projected states

|I; NZ; β2i = 2I + 1 2 Z π dI∗

00(β)e−iβ ˆ Jy ˆ

P N ˆ P Z|Φi dβ

• Final GCM states

|I; NZ; σi = X

β2

f I;NZ;σ

β2

|I; NZ; β2i

X

β0

2

⇣ HI;NZ

β2,β0

2 − EI;NZ;σN I;NZ

β2,β0

2

⌘ f I;NZ;σ

β0

2

= 0

HI;NZ

β2,β0

2 = hI; NZ; β2| ˆ

H2b|I; NZ; β

2i + εI;NZ DD

⇣ Φ(β2), Φ(β

2)

⌘

N I;NZ

β2,β0

2 = hI; NZ; β2|I; NZ; β

2i

EDF axial

SLIDE 11

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)
• 0.5

0.5 1 β 5 10 15 Enorm (MeV)

PN-VAP

150Sm

• 0.5

0.5 1 β 5 10 15 Enorm (MeV)

PN-VAP

150Nd

Determination of initial and final states (I)

EDF axial

SLIDE 12

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)
• 0.5

0.5 1 β 5 10 15 Enorm (MeV)

J=0 PN-VAP

150Sm

• 0.5

0.5 1 β 5 10 15 Enorm (MeV)

J=0 PN-VAP

150Nd

Determination of initial and final states (II)

EDF axial

SLIDE 13

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)
• 0.5

0.5 1 β 5 10 15 Enorm (MeV)

J=0 PN-VAP

150Sm

• 0.5

0.5 1 β 5 10 15 Enorm (MeV)

J=0 PN-VAP

150Nd

EDF axial

Determination of initial and final states (and III)

SLIDE 14

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)
• 1. Axial states
• 2. Angular momentum
• 3. Quadrupole deformations
• 4. Quadrupole and pairing pp/nn correlations
• 5. Quadrupole and pn correlations
• 6. Quadrupole and octupole deformations

I = 0

K = 0

q = q20

|0; NiZi; σ =

• Λi

G0;NiZi;σ

Λi

|Λ0;NiZi

i

• |0; NfZf; σ

=

• Λf

G0;NfZf ;σ

Λf

|Λ0;NfZf

f

• q = (q20, δ)

q = (q20, q30)

EDF axial

q = (q20, p0)

SLIDE 15

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)
• 1. Axial states
• 2. Angular momentum
• 3. Quadrupole deformations
• 4. Quadrupole and pairing pp/nn correlations
• 5. Quadrupole and pn correlations
• 6. Quadrupole and octupole deformations

I = 0

K = 0

q = q20

|0; NiZi; σ =

• Λi

G0;NiZi;σ

Λi

|Λ0;NiZi

i

• |0; NfZf; σ

=

• Λf

G0;NfZf ;σ

Λf

|Λ0;NfZf

f

• TRANSITIONS:

M 0νββ

ξ

= 0+

f | ˆ

O0νββ

ξ

|0+

i ⇥ = 0; NfZf| ˆ

O0νββ

ξ

|0; NiZi⇥ = ⌥

Λf Λi

• G0;NfZf

Λf

⇥∗ Λ0;NfZf

f

| ˆ O0νββ

ξ

|Λ0;NiZi

i

⇥G0;NiZi

Λi

= ⌥

qiqf ;Λf Λi

⇤ ⇧ u0;NfZf

qf,Λf

• n0;NfZf

Λf

⌅ ⌃

∗

G0;NfZf

Λf

⇥∗ 0; NfZf; qf| ˆ O0νββ

ξ

|0; NiZi; qi⇥

• G0;NiZi

Λi

⇥ ⇤ ⇧ u0;NiZi

qi,Λi

• n0;NiZi

Λi

⌅ ⌃

q = (q20, δ)

q = (q20, q30)

EDF axial

q = (q20, p0)

SLIDE 16

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)
• 1. Axial states
• 2. Angular momentum
• 3. Quadrupole deformations
• 4. Quadrupole and pairing pp/nn correlations
• 5. Quadrupole and pn correlations
• 6. Quadrupole and octupole deformations

I = 0

K = 0

q = q20

|0; NiZi; σ =

• Λi

G0;NiZi;σ

Λi

|Λ0;NiZi

i

• |0; NfZf; σ

=

• Λf

G0;NfZf ;σ

Λf

|Λ0;NfZf

f

• TRANSITIONS:

M 0νββ

ξ

= 0+

f | ˆ

O0νββ

ξ

|0+

i ⇥ = 0; NfZf| ˆ

O0νββ

ξ

|0; NiZi⇥ = ⌥

Λf Λi

• G0;NfZf

Λf

⇥∗ Λ0;NfZf

f

| ˆ O0νββ

ξ

|Λ0;NiZi

i

⇥G0;NiZi

Λi

= ⌥

qiqf ;Λf Λi

⇤ ⇧ u0;NfZf

qf,Λf

• n0;NfZf

Λf

⌅ ⌃

∗

G0;NfZf

Λf

⇥∗ 0; NfZf; qf| ˆ O0νββ

ξ

|0; NiZi; qi⇥

• G0;NiZi

Λi

⇥ ⇤ ⇧ u0;NiZi

qi,Λi

• n0;NiZi

Λi

⌅ ⌃ Matrix elements of the double beta transition operators between particle number and angular momentum projected states

q = (q20, δ)

q = (q20, q30)

EDF axial

q = (q20, p0)

SLIDE 17

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 4. Seniority and SU(4)
• 3. Pairing
• 5. Summary and open questions
• 0.6 -0.4 -0.2

0.2 0.4 0.6 0.8

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

1 2

1.5 0.5 0.5 0.5

• 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| ˆ O0νββ

ξ

|0; NiZi; qi⇥

• 0; NfZf; qf|0; NfZf; qf⇥0; NiZi; qi|0; NiZi; qi⇥
• 0.6 -0.4 -0.2

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

GT

F

β (150Nd) β (150Sm) β (150Nd) β (150Sm)

NME: axial quadrupole deformation

T.R.R., Martínez-Pinedo, PRL 105, 252503 (2010)

A=150

SLIDE 18

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 4. Seniority and SU(4)
• 3. Pairing
• 5. Summary and open questions
• 0.6 -0.4 -0.2

0.2 0.4 0.6 0.8

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

1 2

1.5 0.5 0.5 0.5

• 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| ˆ O0νββ

ξ

|0; NiZi; qi⇥

• 0; NfZf; qf|0; NfZf; qf⇥0; NiZi; qi|0; NiZi; qi⇥
• 0.6 -0.4 -0.2

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

GT

F

β (150Nd) β (150Sm) β (150Nd) β (150Sm)

• 0.5

0.5 β 0.1 0.2 0.3 0.4 |F(β)|2

150Nd (0i +) 150Sm (0f +)

(b)

NME: axial quadrupole deformation

T.R.R., Martínez-Pinedo, PRL 105, 252503 (2010)

• Final result depends on the distribution of probability of the corresponding initial and final collective states within

this plot

A=150

SLIDE 19

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 4. Seniority and SU(4)
• 3. Pairing
• 5. Summary and open questions
• 0.6 -0.4 -0.2

0.2 0.4 0.6 0.8

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

1 2

1.5 0.5 0.5 0.5

• 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| ˆ O0νββ

ξ

|0; NiZi; qi⇥

• 0; NfZf; qf|0; NfZf; qf⇥0; NiZi; qi|0; NiZi; qi⇥
• 0.6 -0.4 -0.2

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

GT

F

β (150Nd) β (150Sm) β (150Nd) β (150Sm)

NME: axial quadrupole deformation

T.R.R., Martínez-Pinedo, PRL 105, 252503 (2010)

• Final result depends on the distribution of probability of the corresponding initial and final collective states within

this plot

A=150

SLIDE 20

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 4. Seniority and SU(4)
• 3. Pairing
• 5. Summary and open questions
• 0.6 -0.4 -0.2

0.2 0.4 0.6 0.8

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

1 2

1.5 0.5 0.5 0.5

• 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| ˆ O0νββ

ξ

|0; NiZi; qi⇥

• 0; NfZf; qf|0; NfZf; qf⇥0; NiZi; qi|0; NiZi; qi⇥
• 0.6 -0.4 -0.2

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

GT

F

β (150Nd) β (150Sm) β (150Nd) β (150Sm)

|M 0νββ

GT

| = 1.28 |M 0νββ

F

| g2

A

= 0.43 |M 0νββ

T OT |

= 1.71

NME: axial quadrupole deformation

T.R.R., Martínez-Pinedo, PRL 105, 252503 (2010)

• Final result depends on the distribution of probability of the corresponding initial and final collective states within

this plot

A=150

SLIDE 21

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 4. Seniority and SU(4)
• 3. Pairing
• 5. Summary and open questions

NME: axial quadrupole plus octupole deformation

• J. M. Yao and J. Engel, arXiv 1604.06297 (2016)

SLIDE 22

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 4. Seniority and SU(4)
• 3. Pairing
• 5. Summary and open questions

NME: axial quadrupole plus octupole deformation

• J. M. Yao and J. Engel, arXiv 1604.06297 (2016)
• FIG. 5: (Color online) The final matrix element M 0ν from

the GCM calculation with and without [46] octupole shape fluctuations (REDF) and those of the QRPA (“QRPA F” [66], “QRPA M” [45], “QRPA T” [47]), the IMB-2 [67], and the non-relativistic GCM, based on the Gogny D1S interaction, with [68] and without [44] pairing fluctuations.

SLIDE 23

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 4. Seniority and SU(4)
• 3. Pairing
• 5. Summary and open questions

10 20 30 40 50 60

γ β

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 10 20 30 40 50 60

γ β

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

1 2 3 4 5 6 7 8 9 10

11 11 9 9 7 5 5 3 3 1 1

10 20 30 40 50 60

γ β2

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Ex (MeV)

01 21 41 01 21 41 61 0+ 2+ 4+ 6+ 8+

Theory Experiment

76Ge

1 2 3 4 5 6 7 8 9 10

11 11 9 9 7 7 5 5 3 3 1 1

10 20 30 40 50 60

γ β2

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Ex (MeV)

01 21 41 61 01 21 41 61 81 0+ 2+ 4+ 6+ 8+

Theory Experiment

76Se

T.R.R., in progress

NME: triaxial quadrupole deformation

PES J=0 g.s. coll. wf. PES J=0 g.s. coll. wf.

A=76

SLIDE 24

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 4. Seniority and SU(4)
• 3. Pairing
• 5. Summary and open questions

10 20 30 40 50 60

γ β

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 10 20 30 40 50 60

γ β

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

1 2 3 4 5 6 7 8 9 10

11 11 9 9 7 5 5 3 3 1 1

10 20 30 40 50 60

γ β2

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Ex (MeV)

01 21 41 01 21 41 61 0+ 2+ 4+ 6+ 8+

Theory Experiment

76Ge

1 2 3 4 5 6 7 8 9 10

11 11 9 9 7 7 5 5 3 3 1 1

10 20 30 40 50 60

γ β2

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Ex (MeV)

01 21 41 61 01 21 41 61 81 0+ 2+ 4+ 6+ 8+

Theory Experiment

76Se

T.R.R., in progress

NME: triaxial quadrupole deformation

PES J=0 g.s. coll. wf. PES J=0 g.s. coll. wf.

N M E ? ? A=76

SLIDE 25

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 4. Seniority and SU(4)
• 3. Pairing
• 5. Summary and open questions
• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

1 2 3 4 5 6 8 1 1 4 1 8 20 2 8

• 15
• 30
• 45
• 60
• 11

0.0 0.2 0.4 0.6 0.8

Exp=418.70 MeV 5DCH=419.671 MeV

HFB-D1S Bruyères-le-Châtel 48Ti 26 22

Binding Energy

• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

2 5 6 8 10 12 14 16 18 20 24 28 35

• 15
• 30
• 45
• 60
• 12

0.0 0.2 0.4 0.6 0.8

Exp=415.99 MeV 5DCH=414.196 MeV

HFB-D1S Bruyères-le-Châtel 48Ca 28 20

Binding Energy

• 0.5

0.0 0.5 10 20

E (MeV)

HFB+ZPE HFB

1 2 3 4 5 6 8 10 12 1 4 16 18 20 24 28 3 5

• 15
• 30
• 45
• 60
• 13

0.0 0.2 0.4 0.6 0.8

Exp=662.07 MeV 5DCH=662.947 MeV

HFB-D1S Bruyères-le-Châtel 76Se 42 34

Binding Energy

• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

1 2 3 4 5 6 8 10 12 14 1 8 20 24 28 3 5

• 15
• 30
• 45
• 60
• 12

0.0 0.2 0.4 0.6 0.8

Exp=661.60 MeV 5DCH=661.878 MeV

HFB-D1S Bruyères-le-Châtel 76Ge 44 32

Binding Energy

• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

1 2 3 4 5 6 8 10 12 1 4 16 18 2 24 28 35

• 15
• 30
• 45
• 60
• 10

0.0 0.2 0.4 0.6 0.8

Exp=714.27 MeV 5DCH=715.266 MeV

HFB-D1S Bruyères-le-Châtel 82Kr 46 36

Binding Energy

• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

2 3 4 5 6 8 10 1 2 14 1 6 1 8 2 24 28 35 45

• 15
• 30
• 45
• 60
• 10

0.0 0.2 0.4 0.6 0.8

Exp=712.84 MeV 5DCH=712.608 MeV

HFB-D1S Bruyères-le-Châtel 82Se 48 34

Binding Energy

• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

1 2 3 4 5 6 8 1 12 16 18 20 24 28 35 45

• 15
• 30
• 45
• 60
• 9

0.0 0.2 0.4 0.6 0.8

Exp=830.78 MeV 5DCH=831.488 MeV

HFB-D1S Bruyères-le-Châtel 96Mo 54 42

Binding Energy

• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

1 2 3 4 5 6 8 10 12 16 18 20 28 35 45

• 15
• 30
• 45
• 60
• 10

0.0 0.2 0.4 0.6 0.8

Exp=814.68 MeV 5DCH=814.098 MeV

HFB-D1S Bruyères-le-Châtel 94Zr 54 40

Binding Energy

NME: triaxial quadrupole deformation

• 0.5

10 20 30

E (MeV)

1 3 4 5 6 8 1 1 4 18 24 28 35 45

• 15
• 30
• 45
• 60
• 11

0.0 0.2 0.4 0.6 0.8

Exp=860.46 MeV 5DCH=860.294 MeV

HFB-D1S Bruyères-le-Châtel 100Mo 58 42

Binding Energy

10 20 30

E (MeV)

2 1 3 4 5 6 8 10 12 1 4 16 1 8 20 24 28 35 45

• 15
• 30
• 45
• 60
• 9

0.0 0.2 0.4 0.6 0.8

Exp=861.93 MeV 5DCH=862.960 MeV

HFB-D1S Bruyères-le-Châtel 100Ru 56 44

Binding Energy

CEA-Bruyeres-le-Chatel data base

HFB-PES

SLIDE 26

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 4. Seniority and SU(4)
• 3. Pairing
• 5. Summary and open questions

NME: triaxial quadrupole deformation

• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

1 3 4 5 6 8 10 1 2 1 4 1 6 2 24 28 35 45

• 15
• 30
• 45
• 60
• 9

0.0 0.2 0.4 0.6 0.8

Exp=988.68 MeV 5DCH=987.956 MeV

HFB-D1S Bruyères-le-Châtel 116Sn 66 50

Binding Energy

• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

2 3 4 5 6 8 10 12 14 1 6 18 2 28 35 45

• 15
• 30
• 45
• 60
• 10

0.0 0.2 0.4 0.6 0.8

Exp=987.44 MeV 5DCH=986.646 MeV

HFB-D1S Bruyères-le-Châtel 116Cd 68 48

Binding Energy

• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

1 2 3 4 5 6 8 1 12 1 4 16 18 20 2 4 2 8 35 45

• 15
• 30
• 45
• 60
• 8

0.0 0.2 0.4 0.6 0.8

Exp=1050.68 MeV 5DCH=1051.015 MeV

HFB-D1S Bruyères-le-Châtel 124Te 72 52

Binding Energy

• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

2 4 5 8 10 1 2 14 16 18 20 24 28 3 5 45

• 15
• 30
• 45
• 60
• 7

0.0 0.2 0.4 0.6 0.8

Exp=1049.96 MeV 5DCH=1048.999 MeV

HFB-D1S Bruyères-le-Châtel 124Sn 74 50

Binding Energy

• 0.5

0.0 0.5 10 20 30

E (MeV)

HFB+ZPE HFB

1 2 3 4 5 6 8 1 1 2 1 4 1 6 2 2 4 2 8 35 45

• 15
• 30
• 45
• 60
• 8

0.0 0.2 0.4 0.6 0.8

Exp=1096.91 MeV 5DCH=1097.034 MeV

HFB-D1S Bruyères-le-Châtel 130Xe 76 54

Binding Energy

• 0.5

0.0 0.5 10 20 30 40

E (MeV)

HFB+ZPE HFB

2 4 5 8 10 1 2 14 16 18 20 24 2 8 3 5 45

• 15
• 30
• 45
• 60
• 6

0.0 0.2 0.4 0.6 0.8

Exp=1095.94 MeV 5DCH=1095.764 MeV

HFB-D1S Bruyères-le-Châtel 130Te 78 52

Binding Energy

• 0.5

0.0 0.5 10 20 30 40

E (MeV)

HFB+ZPE HFB

1 3 4 6 8 1 1 2 14 1 6 18 20 24 28 35 4 5

• 15
• 30
• 45
• 60
• 7

0.0 0.2 0.4 0.6 0.8

Exp=1142.78 MeV 5DCH=1143.040 MeV

HFB-D1S Bruyères-le-Châtel 136Ba 80 56

Binding Energy

• 0.5

0.0 0.5 10 20 30 40

E (MeV)

HFB+ZPE HFB

3 8 12 14 1 6 1 8 2 24 2 8 35 45

• 15
• 30
• 45
• 60
• 8

0.0 0.2 0.4 0.6 0.8

Exp=1141.88 MeV 5DCH=1139.819 MeV

HFB-D1S Bruyères-le-Châtel 136Xe 82 54

Binding Energy

E (MeV)

3 2 4 5 1 6 8 12 1 6 20 24 28 35 4 5

• 15
• 30
• 45
• 60
• 8

0.0 0.2 0.4 0.6 0.8

Exp=1239.25 MeV 5DCH=1238.120 MeV

HFB-D1S Bruyères-le-Châtel 150Sm 88 62

Binding Energy

E (MeV)

5 5 4 6 3 2 8 10 1 2 14 1 8 24 2 8 3 5 4 5

• 15
• 30
• 45
• 60
• 6

0.0 0.2 0.4 0.6 0.8

Exp=1237.45 MeV 5DCH=1235.306 MeV

HFB-D1S Bruyères-le-Châtel 150Nd 90 60

Binding Energy

CEA-Bruyeres-le-Chatel data base

HFB-PES

SLIDE 27

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

1 2 3 4 5 6

• 0.01

0.02 0.03 0.04 0.05 0.06

136Xe

• 0.4

0.4 2 1 2 3 4 5 6

• 0.4

0.4 2 1 2 3 4 5 6

• 0.01

0.02 0.03 0.04 0.05

136Ba

1 2 3 4 5 6

• 2

4 6 8 10 12 14 2 4 4 6 6 8 8 10 10

136Xe

• 0.4

0.4 2 1 2 3 4 5 6

• 0.4

0.4 2 1 2 3 4 5 6

• 2

4 6 8 10 12 14 2 2 4 4 6 6 8 8 8 10 10

136Ba

(a) (b) (c) (d)

MeV MeV

NME: Shape and pp/nn pairing fluctuations

Angular momentum projected potential energy surfaces Collective ground state wave functions

• N. López-Vaquero, T.R.R., J.L. Egido, PRL 111, 142501 (2013)

SLIDE 28

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

1 2 3 4 5 6 (136Xe) 1 2 3 4 5 6 ’ (136Ba)

1 2 3 4 5 5 5 4 4 3 3 2 2 1 1 1

• 0.4

0.4 0.8 2 (136Xe)

• 0.4

0.4 0.8 ’2 (136Ba)

1 2 3 4 5 4 3 3 2 2 2 2 1 1 1 1

(a) (b) GT GT

Dependence on deformation Dependence on pp/nn pairing

• N. López-Vaquero, T.R.R., J.L. Egido, PRL 111, 142501 (2013)

NME: Shape and pp/nn pairing fluctuations

SLIDE 29

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

H = h0

1

X

µ=1

gT =1

µ

S†

µSµ χ

2

2

X

K=2

Q†

2KQ2K

gT =0

1

X

ν=1

P †

ν Pν + gph 1

X

µ,ν=1

F µ†

ν F µ ν ,

(2) where h0 contains spherical single particle energies, Q2K are the components of a quadrupole operator defined in

• Ref. [15], and

S†

µ =

1 p 2 X

l

ˆ l[c†

l c† l ]001 00µ ,

P †

µ =

1 p 2 X

l

ˆ l[c†

l c† l ]010 0µ0 ,

F µ

ν = 1

2 X

i

σµ

i τ ν i =

X

l

ˆ l[c†

l ¯

cl]011

0µν .

(3)

H0 = HλZNZλNNN λQQ20 λP 2 ⇣ P0 + P † ⌘ , (6)

2 4 6 8 10 12 0.1 0.2

φF |Ψ(φF)|2

76Se

0.1 0.2

|Ψ(φI)|2 |Ψ(φF)|2

76Ge

2 4 6 8 10 12

φI

• 15
• 10
• 5

5 10

8 4

• 4
• 4
• 8
• 12

gT=0 6= 0 gT=0 = 0 gT=0 6= 0 gT=0 = 0

FIG. 3. (Color

• nline.)

Bottom right: NφI NφF hφF | PF ˆ M0νPI |φIi for projected quasiparticle vacua with different values of the initial and final isoscalar pairing amplitudes φI and φF , from the SkO0-based interac- tion (see text). Top and bottom left: Square of collective wave functions in 76Ge and 76Se.

• N. Hinohara and J. Engel, PRC 031031(R) (2014)

NME: Shape and pn pairing fluctuations

SLIDE 30

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

NME: pf-shell

Where do the differences between SM and GCM come from?

The “inverted parabola” from initial number of neutrons

20 24 28 32 36 Number of Neutrons (initial) 2 3 4 5 6 M GT

EDFsph (D1S) SMsen=0 (KB3G) SMsen=0 (GXPF1A)

(a) Ca Ti 20 24 28 32 36 Number of Neutrons (initial) (b) Ti Cr 20 24 28 32 36 Number of Neutrons (initial) (c) Cr Fe

• 20

24 28 32 36 Number of initial neutrons Ca,Fe (1) Ti,Cr (2) Ca,Fe (2) Ti,Cr (1)

(b) (c)

GS

• J. Menéndez, T. R. R., A. Poves, G. Martínez-Pinedo, PRC 90, 024311 (2014).

SLIDE 31

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

NME: pf-shell

• Same pattern in spherical EDF

, seniority 0 Shell Model, and Generalized Seniority model (overall scale?)

• What is the effect of including more correlations?

Where do the differences between SM and GCM come from?

The “inverted parabola” from initial number of neutrons

20 24 28 32 36 Number of Neutrons (initial) 2 3 4 5 6 M GT

EDFsph (D1S) SMsen=0 (KB3G) SMsen=0 (GXPF1A)

(a) Ca Ti 20 24 28 32 36 Number of Neutrons (initial) (b) Ti Cr 20 24 28 32 36 Number of Neutrons (initial) (c) Cr Fe

• 20

24 28 32 36 Number of initial neutrons Ca,Fe (1) Ti,Cr (2) Ca,Fe (2) Ti,Cr (1)

(b) (c)

GS

• J. Menéndez, T. R. R., A. Poves, G. Martínez-Pinedo, PRC 90, 024311 (2014).

SLIDE 32

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

NME: pf-shell

EDF sph EDF min EDF full

• 0.5

0.5 2

• 510
• 500
• 490
• 480
• 470
• 460

E (MeV)

58Cr 58Ti

(a) (b) (c) (d) Ca Ti Cr Fe

1 2 3 4 5 6 M0

GT

EDF full EDF min EDF sph 20 24 28 32 36 Number of initial neutrons 1 2 3 4 5 6 M0

GT

20 24 28 32 36 Number of initial neutrons

• J. Menéndez, T. R. R., A. Poves, G. Martínez-Pinedo, PRC 90, 024311 (2014).

SLIDE 33

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

NME: pf-shell

EDF sph EDF min EDF full

• 0.5

0.5 2

• 510
• 500
• 490
• 480
• 470
• 460

E (MeV)

58Cr 58Ti

(a) (b) (c) (d) Ca Ti Cr Fe

1 2 3 4 5 6 M0

GT

EDF full EDF min EDF sph 20 24 28 32 36 Number of initial neutrons 1 2 3 4 5 6 M0

GT

20 24 28 32 36 Number of initial neutrons

• NMEs are reduced with respect to the

spherical value when correlations are included.

• The biggest reduction is produced by angular

momentum restoration and configuration mixing produces an increase of the NME.

• Cross-check nuclei: 42Ca, 50Ca, 56Fe
• J. Menéndez, T. R. R., A. Poves, G. Martínez-Pinedo, PRC 90, 024311 (2014).

SLIDE 34

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

NME: pf-shell

0.5 1 1.5 2 2.5 MGT

EDF sph EDF full

(a) 50Ca 50Ti 2 4 6 8 10 Seniority

• 1
• 0.8
• 0.6
• 0.4
• 0.2

MF (c) 50Ca 50Ti 2 4 6 MGT

SM isosp. proj SM full

(b) 48Ti 48Cr 2 4 6 8 10 Seniority

• 3
• 2
• 1

MF (d) 48Ti 48Cr

• FIG. 5. (Color online) Gamow-Teller [

0ν , panels (a),(b)] and

• J. Menéndez, T. R. R., A. Poves, G. Martínez-Pinedo, PRC 90, 024311 (2014).

SLIDE 35

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

NME: pf-shell

0.5 1 1.5 2 2.5 MGT

EDF sph EDF full

(a) 50Ca 50Ti 2 4 6 8 10 Seniority

• 1
• 0.8
• 0.6
• 0.4
• 0.2

MF (c) 50Ca 50Ti 2 4 6 MGT

SM isosp. proj SM full

(b) 48Ti 48Cr 2 4 6 8 10 Seniority

• 3
• 2
• 1

MF (d) 48Ti 48Cr

• FIG. 5. (Color online) Gamow-Teller [

0ν , panels (a),(b)] and

• The biggest reduction (in Shell

model calculations) is produced by including higher seniority components in the nuclear wave functions.

• Isospin projection is relevant for the

Fermi part of the NME and less important for the Gamow-Teller part.

• EDF does not include properly those

higher seniority components, specially in spherical nuclei.

• p-n pairing effects could also be

important in the reduction of the NME.

• J. Menéndez, T. R. R., A. Poves, G. Martínez-Pinedo, PRC 90, 024311 (2014).

SLIDE 36

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

Hcoll = HM + gT =1

1

• n=−1

S†

nSn + gT =0 1

• m=−1

P †

mPm

+ gph

1

• m,n=−1

: F†

mnFmn : +χ 2

• µ=−2

: Q†

µQµ :

• J. Menéndez, et al., PRC 93, 014305 (2016).

NME: pf-shell

• Increase of the NME when isoscalar

pairing is removed.

• Further increase when spin-isospin is

also removed

SLIDE 37

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

Hcoll = HM + gT =1

1

• n=−1

S†

nSn + gT =0 1

• m=−1

P †

mPm

+ gph

1

• m,n=−1

: F†

mnFmn : +χ 2

• µ=−2

: Q†

µQµ :

• J. Menéndez, et al., PRC 93, 014305 (2016).

NME: pf-shell

• GT operator is SU(4) invariant (neglecting the

neutrino potential)

• GT operator can only connect states

belonging to the same irreducible representation of SU(4)

• SU(4) is more broken when T=0 and spin-

isospin terms are removed from the Hamiltonian⇒ the number of SU(4) irreps present both in the mother and daughter g.s. wave functions are larger⇒ larger NMEs

SLIDE 38

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

Hcoll = HM + gT =1

1

• n=−1

S†

nSn + gT =0 1

• m=−1

P †

mPm

+ gph

1

• m,n=−1

: F†

mnFmn : +χ 2

• µ=−2

: Q†

µQµ :

• J. Menéndez, et al., PRC 93, 014305 (2016).

NME: pf-shell

• SM/GCM comparison with the same

interaction.

• 1D: only pn strength as a generator

coordinate.

• 2D: pn strength and axial quadrupole

deformation as generator coordinates.

EXACT vs. VARIATIONAL!!

SLIDE 39

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

๏ NMEs differ a factor of three between the different methods but we need to understand which are the pros/cons of each method to provide reliable numbers (precision vs. accuracy). ๏ Nuclear physics aspects like deformation, pairing, shell effects, etc., are understood similarly within different approaches. ๏ Systematic comparisons between ISM/EDF methods have been performed but… we need more!

Summary

SLIDE 40

Tomás R. Rodríguez

Relevant degrees of freedom for 0νββ decay nuclear matrix elements with EDF TRIUMF double-beta decay workshop

• 1. EDF method
• 2. Multipole deformation
• 3. Pairing
• 5. Summary and open questions
• 4. Seniority and SU(4)

๏ Isospin mixing and restoration have to be done in the future. Why is it so difficult (perhaps impossible) with the current Gogny EDFs? ๏ Triaxiality has to be taken into account in A=76 and A=100 decays (at least). ๏ How relevant is the proper description of the spectra in 0νββ NMEs? ๏ Occupation numbers with EDF to define physically sound valence spaces. ๏ Odd-odd nuclei is still a major challenge for GCM calculations. ๏ Computational time?!?

Open questions