Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with energy density functionals Tomás R. Rodríguez Interfacing theory and experiment for reliable double-beta decay matrix element calculations Vancouver, May 11-13, 2016
Outline 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) Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
Outline 1. EDF method 2. Multipole deformation 3. Pairing 4. Seniority and SU(4) 5. Summary and open questions Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
Nuclear Matrix Elements 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation • Nuclear structure methods for calculating these NME: Di ff erent 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. - Di ffi culties to include collective/single particle degrees of freedom. - Problems with particle number/isospin conservation. Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
Gogny EDF 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation Effective nucleon-nucleon interaction: Gogny force (D1S/D1M) 2 l r 2 ) 2 /µ 2 a � i ( W i + B i P ⇥ − H i P ⇤ − M i P ⇥ P ⇤ ) e − ( ⌥ r 1 − ⌥ i V (1 , 2) = t n e t o p y i =1 d o b - 2 + iW 0 ( ⇥ 1 + ⇥ 2 ) ⌥ r 2 ) ⌥ k × � ( ⌥ + V Coulomb ( ⌅ r 2 ) r 1 − ⌥ k r 1 , ⌅ m r e t t n e d n e p e d r 2 ) ⇥ α (( � y t + t 3 (1 + x 0 P σ ) � ( � r 1 − � r 1 + � r 2 ) / 2) i s n e D Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
Gogny EDF 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation Effective nucleon-nucleon interaction: Gogny force (D1S/D1M) 2 l r 2 ) 2 /µ 2 a � i ( W i + B i P ⇥ − H i P ⇤ − M i P ⇥ P ⇤ ) e − ( ⌥ r 1 − ⌥ i V (1 , 2) = t n e t o p y i =1 d o b - 2 + iW 0 ( ⇥ 1 + ⇥ 2 ) ⌥ r 2 ) ⌥ k × � ( ⌥ + V Coulomb ( ⌅ r 2 ) r 1 − ⌥ k r 1 , ⌅ m r e t t n e d n e p e d r 2 ) ⇥ α (( � y t + t 3 (1 + x 0 P σ ) � ( � r 1 − � r 1 + � r 2 ) / 2) i s n e D Next step: Variational method!!! Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
EDF axial 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation • Initial intrinsic states: PN-VAP ( M. Anguiano et al., Nucl. Phys. A 683, 227 (2001)) M. Anguiano, J. L. Egido, and L. M. Robledo, Nucl. Phys. A 696, 467 (2001). P N ˆ E N,Z [ Φ ] = h Φ | ˆ H 2b ˆ P Z | Φ i DD ( Φ ) � λ q 20 h Φ | ˆ + ε N,Z Q 20 | Φ i P N ˆ h Φ | ˆ P Z | Φ i Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
EDF axial 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation • Initial intrinsic states: PN-VAP ( M. Anguiano et al., Nucl. Phys. A 683, 227 (2001)) M. Anguiano, J. L. Egido, and L. M. Robledo, Nucl. Phys. A 696, 467 (2001). P N ˆ E N,Z [ Φ ] = h Φ | ˆ H 2b ˆ P Z | Φ i DD ( Φ ) � λ q 20 h Φ | ˆ + ε N,Z Q 20 | Φ i P N ˆ h Φ | ˆ P Z | Φ i • Intermediate Particle Number and Angular Momentum Projected states Z π | I ; NZ ; β 2 i = 2 I + 1 P N ˆ 00 ( β ) e − i β ˆ J y ˆ d I ∗ P Z | Φ i d β 2 0 Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
EDF axial 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation • Initial intrinsic states: PN-VAP ( M. Anguiano et al., Nucl. Phys. A 683, 227 (2001)) M. Anguiano, J. L. Egido, and L. M. Robledo, Nucl. Phys. A 696, 467 (2001). P N ˆ E N,Z [ Φ ] = h Φ | ˆ H 2b ˆ P Z | Φ i DD ( Φ ) � λ q 20 h Φ | ˆ + ε N,Z Q 20 | Φ i P N ˆ h Φ | ˆ P Z | Φ i • Intermediate Particle Number and Angular Momentum Projected states Z π | I ; NZ ; β 2 i = 2 I + 1 P N ˆ 00 ( β ) e − i β ˆ J y ˆ d I ∗ P Z | Φ i d β 2 0 • Final GCM states f I ; NZ ; σ X | I ; NZ ; σ i = | I ; NZ ; β 2 i β 2 β 2 Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
EDF axial 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation • Initial intrinsic states: PN-VAP ( M. Anguiano et al., Nucl. Phys. A 683, 227 (2001)) M. Anguiano, J. L. Egido, and L. M. Robledo, Nucl. Phys. A 696, 467 (2001). P N ˆ E N,Z [ Φ ] = h Φ | ˆ H 2b ˆ P Z | Φ i DD ( Φ ) � λ q 20 h Φ | ˆ + ε N,Z Q 20 | Φ i P N ˆ h Φ | ˆ P Z | Φ i • Intermediate Particle Number and Angular Momentum Projected states Z π | I ; NZ ; β 2 i = 2 I + 1 P N ˆ 00 ( β ) e − i β ˆ J y ˆ d I ∗ P Z | Φ i d β 2 0 • Final GCM states f I ; NZ ; σ X | I ; NZ ; σ i = | I ; NZ ; β 2 i β 2 β 2 ⇣ ⌘ X f I ; NZ ; σ H I ; NZ 2 − E I ; NZ ; σ N I ; NZ = 0 β 2 , β 0 β 2 , β 0 β 0 2 2 β 0 2 0 N I ; NZ 2 = h I ; NZ ; β 2 | I ; NZ ; β 2 i β 2 , β 0 ⇣ ⌘ 2 = h I ; NZ ; β 2 | ˆ 0 0 H I ; NZ 2 i + ε I ; NZ H 2b | I ; NZ ; β Φ ( β 2 ) , Φ ( β 2 ) β 2 , β 0 DD Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
EDF axial 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation Determination of initial and final states (I) 15 15 10 10 E norm (MeV) E norm (MeV) 5 5 150 Nd 150 Sm PN-VAP PN-VAP 0 0 -0.5 0 0.5 1 -0.5 0 0.5 1 β β Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
EDF axial 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation Determination of initial and final states (II) 15 15 10 10 E norm (MeV) E norm (MeV) 5 5 150 Nd 150 Sm PN-VAP PN-VAP J =0 J =0 0 0 -0.5 0 0.5 1 -0.5 0 0.5 1 β β Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
EDF axial 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation Determination of initial and final states (and III) 15 15 10 10 E norm (MeV) E norm (MeV) 5 5 150 Nd 150 Sm PN-VAP PN-VAP J =0 J =0 0 0 -0.5 0 0.5 1 -0.5 0 0.5 1 β β Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
EDF axial 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation � G 0; N i Z i ; σ | Λ 0; N i Z i 1. Axial states | 0; N i Z i ; σ � � = K = 0 Λ i i 2. Angular momentum I = 0 Λ i 3. Quadrupole deformations G 0; N f Z f ; σ | Λ 0; N f Z f q = q 20 � | 0; N f Z f ; σ � � = q = ( q 20 , δ ) 4. Quadrupole and pairing pp/nn correlations Λ f f Λ f 5. Quadrupole and pn correlations q = ( q 20 , p 0 ) 6. Quadrupole and octupole deformations q = ( q 20 , q 30 ) Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
EDF axial 5. Summary and open questions 1. EDF method 3. Pairing 4. Seniority and SU(4) 2. Multipole deformation � G 0; N i Z i ; σ | Λ 0; N i Z i 1. Axial states | 0; N i Z i ; σ � � = K = 0 Λ i i 2. Angular momentum I = 0 Λ i 3. Quadrupole deformations G 0; N f Z f ; σ | Λ 0; N f Z f q = q 20 � | 0; N f Z f ; σ � � = q = ( q 20 , δ ) 4. Quadrupole and pairing pp/nn correlations Λ f f Λ f 5. Quadrupole and pn correlations q = ( q 20 , p 0 ) 6. Quadrupole and octupole deformations q = ( q 20 , q 30 ) M 0 νββ O 0 νββ O 0 νββ = � 0 + f | ˆ | 0 + i ⇥ = � 0; N f Z f | ˆ | 0; N i Z i ⇥ = TRANSITIONS: ξ ξ ξ ⇥ ∗ � ⌥ G 0; N f Z f � Λ 0; N f Z f ⌥ O 0 νββ | Λ 0; N i Z i ⇥ G 0; N i Z i | ˆ = Λ f f ξ i Λ i Λ f Λ i q i q f ; Λ f Λ i ∗ � ⇤ ⌅ ⇤ ⌅ ⇧ u 0; N f Z f ⇧ u 0; N i Z i ⇥ ∗ � ⇥ q f , Λ f G 0; N f Z f q i , Λ i O 0 νββ G 0; N i Z i � 0; N f Z f ; q f | ˆ | 0; N i Z i ; q i ⇥ ⌃ ⌃ Λ f Λ i ξ � � n 0; N f Z f n 0; N i Z i Λ f Λ i Tomás R. Rodríguez TRIUMF double-beta decay workshop Relevant degrees of freedom for 0 νββ decay nuclear matrix elements with EDF
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