Lecture IX: Ab Initio Nuclear Structure for Double-Beta Decay J. Engel November 1, 2017
Ab Initio Shell Model Partition of Full Hilbert Space P Q P = valence space Q = the rest ^ PH ^ ^ PH ^ P P Q Task: Find unitary transformation to make H block-diagonal in P and Q , with H eff in P reproducing d most important eigenvalues. QH ^ ^ QH ^ ^ Q P Q Shell model done here.
Ab Initio Shell Model Partition of Full Hilbert Space P Q P = valence space Q = the rest H eff P Task: Find unitary transformation to make H block-diagonal in P and Q , with H eff in P reproducing d most important eigenvalues. H eff-Q Q Shell model done here.
Ab Initio Shell Model Partition of Full Hilbert Space P Q P = valence space Q = the rest H eff P Task: Find unitary transformation to make H block-diagonal in P and Q , with H eff in P reproducing d most important eigenvalues. H eff-Q Q For transition operator ^ M , must apply same transformation to get ^ M eff . Shell model done here.
Ab Initio Shell Model Partition of Full Hilbert Space P Q P = valence space Q = the rest H eff P Task: Find unitary transformation to make H block-diagonal in P and Q , with H eff in P reproducing d most important eigenvalues. H eff-Q Q For transition operator ^ M , must apply same transformation to get ^ M eff . As difficult as solving full problem. But idea is that N-body ef- fective operators may not be important for N > 2 or 3. Shell model done here.
Method 1: Coupled-Cluster Theory Ground state in closed-shell nucleus: 1 � � | Ψ 0 � = e T | ϕ 0 � t m i a † 4 t mn ij a † m a † T = m a i + n a i a j + . . . i , m ij , mn Slater determinant m , n > F i , j < F States in closed-shell + a few constructed in similar way.
Method 1: Coupled-Cluster Theory Ground state in closed-shell nucleus: 1 � � | Ψ 0 � = e T | ϕ 0 � t m i a † 4 t mn ij a † m a † T = m a i + n a i a j + . . . i , m ij , mn Slater determinant m , n > F i , j < F States in closed-shell + a few constructed in similar way. Construction of Unitary Transformation to Shell Model for 76 Ge: 1. Calculate low-lying spectra of 56 Ni + 1 and 2 nucleons (and 3 nucleons in some approximation), where full calculation feasible. 2. Do Lee-Suzuki mapping of lowest eigenstates onto f 5 / 2 pg 9 / 2 shell, determine effective Hamiltonian and decay operator. Lee-Suzuki maps d lowest eigenvectors to orthogonal vectors in shell model space in way that minimizes difference between mapped and original vectors. 3. Use these operators in shell-model calculation of matrix element for 76 Ge (with analogous plans for other elements).
Option 2: In-Medium Similarity Renormalization Group Flow equation for effective Hamiltonian. Asymptotically decouples shell-model space. d dsH ( s ) = [ η ( s ) , H ( s )] , η ( s ) = [ H d ( s ) , H od ( s )] , H ( ∞ ) = H eff ✛ ✲ V [ MeV fm 3 ] pp hh 10 hh ✻ 5 0 pp -5 -10 -15 ❄ -20 s = 0 . 0 s = 1 . 2 s = 2 . 0 s = 18 . 3 Hergert et al. Trick is to keep all 1- and 2-body terms in H at each step after normal ordering . Like truncation of coupled-clusters expansion. If shell-model space contains just a single state, approach yields ground-state energy. If it is a typical valence space, result is effective interaction and operators.
Ab Initio Calculations of Spectra + 8 4 6 7 22 O 23 O 24 O + 1 + 1 + + ) 7 4 3 + (4 6 2 + + 4 + 1 + ) 2 + 5 + 2 (2 2 + + + 4 2 3 + 2 1 6 + 2 + ) + 5 2 ( 3 + + 3 2 + 4 2 0 5 + ) 2 E x [ MeV] + + 3 (0 0 + 0 + 3 4 3 + + ) + + ( 5 Neutron-rich 4 3 3 5 2 2 5 + + 5 3 + 2 + 2 2 2 3 + 2 + oxygen isotopes 2 2 2 2 1 1 1 + + + + + + + + 0 1 1 1 1 + + + + 0 0 0 0 0 0 0 0 0 0 2 2 2 2 G B . G B . G B . I p I p I p E R D x E R D x E R D x C E C E C E S S S S S S C - U C - U C - U M M M I I I 14 20 Ne 24 Mg + 8 13 + 8 + 12 8 + 8 + 8 + 8 + 8 11 10 9 + + 6 6 + + 6 E x [ MeV] 6 6 + 6 + + 8 6 + 6 7 6 Deformed nuclei 5 4 + + 4 + + + 4 + 4 4 4 4 + + 4 4 3 2 + + + + 2 2 + + 2 + 2 2 + 2 2 2 1 + + + + + + + + 0 0 0 0 0 0 0 0 0 . . I G B p I G B p E E D D R x R x C C S E S E S S C C - U - U M M I I
Coupled Cluster Test in Shell-Model Space: 48 Ca − → 48 Ti No Shell-Model Mapping From G. Hagen 3 + 3 + 2 + 0 + 4 + 0 + y 2 + 4 + r 1 + 2 + 1 + a 2 + 4 + 3 + 3 + n 4 + 4 + 2 + i 4 + 2 + m 2 + 2 + i l e 0 + 0 + r P EOM CCSDT-1 Exact 48 Ti Spectrum
Coupled Cluster Test in Shell-Model Space: 48 Ca − → 48 Ti No Shell-Model Mapping From G. Hagen 3 + 3 + 2 + 0 + 4 + 0 + y 2 + 4 + r 1 + 2 + 1 + a 2 + 4 + 3 + 3 + n 4 + 4 + 2 + i 4 + 2 + m 2 + 2 + i l e 0 + 0 + r P EOM CCSDT-1 Exact 48 Ti Spectrum ββ 0 ν Matrix Element GT F T Exact .85 .15 -.06 CCSDT-1 .86 .17 -.08
Full Chiral NN + NNN Calculation (Preliminary) From G. Hagen y M 0 ν Method E 3 max r a CC-EOM (2p2h) 0 1 . 23 n CC-EOM (3p3h) 10 0 . 33 i m CC-EOM (3p3h) 12 0 . 45 CC-EOM (3p3h) 14 0 . 37 i l CC-EOM (3p3h) 16 0 . 36 e r SDPFMU-DB - 1 . 12 P SDPFMU - 1 . 00 Last two are two-shell shell-model calculations with effective interactions.
Complementary Ideas: Density Functionals and GCM Construct set of mean fields by constraining coordinate(s), e.g. quadrupole moment � Q 0 � . Then diagonalize H in space of symmetry-restored quasiparticle vacua with different � Q 0 � . Collective wave functions 0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.4 (b) 76 Ge (0 i + ) � 76 Se (0 f + ) β 2 β 2 = deformation Rodriguez and Martinez-Pinedo: Wave functions peaked at β 2 ≈ ± . 2 Robledo et al.: Minima at β 2 ≈ ± . 15 We’re now including crucial isoscalar pairing amplitude as collective coordinate... � � �
Capturing Collectivity with Generator Coordinates How Important are Collective Degrees of Freedom? Can extract collective separable interaction —— monopole + pairing + isoscalar pairing + spin-isospin + quadrupole —— from shell model interaction, see how well it mimics full interaction for ββ matrix elements in light pf -shell nuclei. 4 KB3G H coll . 3 . 5 H coll . (no T = 0 pairing) Good news for collective models! 3 2 . 5 M GT 2 1 . 5 Ca → Ti 1 0 . 5 0 22 24 26 28 30 32 34 36 38 40 N mother
GCM Example: Proton-Neutron (pn) Pairing Can build possibility of pn correlations into mean field. They are frozen out in mean-field minimum, but included in GCM. 0 νββ matrix element Collective pn-pairing wave functions 0.2 15 pn-GCM 0.1 10 Ordinary GCM 76 Ge | Ψ ( φ ) | 2 M 0 ν 5 0.2 0 0.1 76 Se − 5 0 0 0 . 5 1 1 . 5 2 2 . 5 3 0 2 4 6 8 10 φ = pn pairing amplitude g pp g pn Proton-neutron pairing significantly reduces matrix element.
GCM in Shell-Model Spaces 76 Ge 76 Se 4 Excitation energy (MeV) + 0 + 3 3 0 3 + 0 2 + + 0 0 3 + 3 2 0 2 ββ Matrix Elements in 1 and 2 Shells + 0 2 + 0 1 2 + 2 + 1 2 + 2 + 2 1 1 1 + + + 0 + 0 0 0 0 1 1 1 1 GCM Exp. Exp. GCM GCM Spectrum in 2 Shells
Combining DFT-like and Ab Initio Methods GCM incorporates some correlations that are hard to capture automatically (e.g. shape coexistence). So use it to construct initial “reference” state, let IMSRG, do the rest. Test in single shell for “simple” nucleus. In progress: Improving GCM-based flow. Coding IMSRG-evolved ββ transition operator. To do: applying with DFT-based GCM.
Improving RPA/QRPA 16 O (a) 0.04 RPA 0.02 Fraction E0 EWSR/MeV 0 RPA produces states in (b) intermediate nucleus, but 0.04 DFT-Corrected SSRPA_F form is restricted to 1p-1h Second RPA excitations of ground 0.02 state. Second RPA adds 0 2p-2h states. (c) Exp 0.04 0.02 0 5 10 15 20 25 30 35 40 E (MeV)
Issue Facing All Models: “ g A ” 40-Year-Old Problem: Effective g A needed for single-beta and two-neutrino double-beta decay in shell model and QRPA. 1.4 from experimental τ 1 2 ISM ISM 1.269 A 0.12 g A ,eff 1.2 from experimental τ 1 2 IBM 2 CA SSD IBM 2 1.269 A 0.18 1.0 g A ,eff 0.8 g A , eff Ca 0.6 Ge Se ZrMo Cd Te Xe Nd 0.4 0.2 0.0 40 60 80 100 120 140 160 Mass number from F. Iachello If 0 ν matrix elements quenched by same amount as 2 ν matrix elements, ex- periments will be much less sensitive; rates go like fourth power of g A .
Arguments Suggesting Strong Quenching of 0 ν Both β and 2 νββ rates are strongly quenched, by consistent factors. Forbidden (2 − ) decay among low-lying states appears to exhibit similar quenching. Quenching due to correlations shows weak momentum dependence in low-order perturbation theory.
Arguments Suggesting Weak Quenching of 0 ν Many-body currents seem to suppress 2 ν more than 0 ν . Enlarging shell model space to include some effects of high- j spin-orbit partners reduces 2 ν more than 0 ν . Neutron-proton pairing, related to spin-orbit partners and investigated pretty carefully, suppresses 2 ν more than 0 ν . g pp =0.0 Ca → Ti g pp =0.8 3 2 g pp =1.0 2 g pp =1.2 -1 ) 1 0 ν 0 P(r) (fm M GT 0 full no T = 0 pairing 3 -2 2 1 2 ν -4 0 4 8 12 0 r (fm) 22 24 26 28 30 32 34 36 Large r contributes more to 2 ν . N mother
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