Variational Procedure Find best Slater det. | ψ � by minimizing H ≡ � ψ ′ | H | φ ′ � / � φ ′ | ψ ′ � : In coordinate space, resulting equations are � − ∇ 2 � d � r − � j ( � r ′ ) φ j ( � φ ∗ r ′ V ( | r ′ | ) r ′ ) 2 m φ a ( � r ) + � φ a ( � r ) � �� � j � F ρ ( � r ′ ) � � � � d � r − � j ( � r ′ ) φ a ( � r ′ | ) φ ∗ r ′ V ( | r ′ ) − � φ j ( � r ) = ǫ a φ a ( � r ) . j � F First potential term involves the “direct” (intuitive) potential � d � r − � r ′ | ) ρ ( � � r ′ V ( | � r ′ ) . U d ( r ) ≡
Variational Procedure Find best Slater det. | ψ � by minimizing H ≡ � ψ ′ | H | φ ′ � / � φ ′ | ψ ′ � : In coordinate space, resulting equations are � − ∇ 2 � d � r − � j ( � r ′ ) φ j ( � φ ∗ r ′ V ( | r ′ | ) r ′ ) 2 m φ a ( � r ) + � φ a ( � r ) � �� � j � F ρ ( � r ′ ) � � � � d � r − � j ( � r ′ ) φ a ( � r ′ | ) φ ∗ r ′ V ( | r ′ ) − � φ j ( � r ) = ǫ a φ a ( � r ) . j � F First potential term involves the “direct” (intuitive) potential � d � r − � r ′ | ) ρ ( � � r ′ V ( | � r ′ ) . U d ( r ) ≡ Second term contains the nonlocal “exchange potential” � r , � r − � j ( � r ′ | ) φ ∗ � r ′ ) ≡ � r ′ ) φ j ( � U e ( V ( | r ) . j � F
Self Consistency Note that in potential-energy terms U d and U e depend on all the occupied levels. So do the eigenvalues ǫ a , therefore, and Solutions are “self-consistent” To solve equations: 1. Start with a set of basis states φ a , φ b , φ c ...and calculate construct U d and U e . 2. Solve the HF Schr¨ odinger equation to obtain a new set of basis states φ a ′ , φ b ′ . . . 3. Repeat steps 1 and 2 until you get essentially the same basis out of step 2 as you put into step 1.
Second-Quantization Version Theorem (Thouless) Suppose | φ � ≡ a † 1 · · · a † F | 0 � is a Slater determinant. The most general Slater determinant not orthogonal to | φ � can be written as � � C mi a † C mi a † | φ ′ � = exp ( m a i + O ( C 2 )] | φ � m a i ) | φ � = [ 1 + m > F , i < F m , i Minimizing E = � ψ | H | ψ � : ∂ H = � φ | Ha † n a j | φ � = 0 ∀ n > F , j � F ∂ C nj � = ⇒ h nj ≡ T nj + V jk , nk = 0 ∀ n > F , j � F k < F where T ab = � a | p 2 2 m | b � and V ab , cd = � ab | V 12 | cd � − � ab | V 12 | dc � . This will be true if ∃ a single particle basis in which h is diagonal, � h ab ≡ T ab + V ak , bk = δ ab ǫ a ∀ a , b . k � F Another version of the HF equations.
Brief History of Mean-Field Theory 1. Big problem early: Doesn’t work with realistic NN potentials because of hard core, which causes strong correlations.
Brief History of Mean-Field Theory 1. Big problem early: Doesn’t work with realistic NN potentials because of hard core, which causes strong correlations. 2. Hard core included implicitly through effective interaction: Brueckner G matrix Still didn’t work perfectly; three-body interations neglected. G V = V + + + . . . V
Brief History of Mean-Field Theory 1. Big problem early: Doesn’t work with realistic NN potentials because of hard core, which causes strong correlations. 2. Hard core included implicitly through effective interaction: Brueckner G matrix Still didn’t work perfectly; three-body interations neglected. G V = V + + + . . . V 3. Three-body interaction included approximately as density-dependent two-body interaction, in the same way as two-body interaction is approximated by density-dependent mean field. Results better, and a convenient “zero-range” approximation used.
Brief History (Cont.) 4. Phenomenology successfully evolved toward zero-range density-dependent (Skyrme) interactions, with H = t 0 ( 1 + x 0 ^ P σ ) δ ( � r 1 − � r 2 ) + 1 � � 2 t 1 ( 1 + x 1 ^ ( � ∇ 1 − � ∇ 2 ) 2 δ ( P σ ) � r 1 − � r 2 ) + h . c . + t 2 ( 1 + x 2 ^ P σ ) ( � ∇ 1 − � r 2 )( � ∇ 1 − � ∇ 2 ) · δ ( � r 1 − � ∇ 2 ) + 1 6 t 3 ( 1 + x 3 ^ r 2 ) ρ α ([ P σ ) δ ( � r 1 − � � r 1 + � r 2 ] / 2 ) σ 2 ) · ( � ∇ 1 − � r 2 )( � ∇ 1 − � + iW 0 ( � σ 1 + � ∇ 2 ) × δ ( � r 1 − � ∇ 2 ) , where P σ = 1 + σ 1 · σ 2 ^ , 2 and t i , x i , W 0 , and α are adjustable parameters.
Brief History (Cont.) 4. Phenomenology successfully evolved toward zero-range density-dependent (Skyrme) interactions, with H = t 0 ( 1 + x 0 ^ P σ ) δ ( � r 1 − � r 2 ) + 1 � � 2 t 1 ( 1 + x 1 ^ ( � ∇ 1 − � ∇ 2 ) 2 δ ( P σ ) � r 1 − � r 2 ) + h . c . + t 2 ( 1 + x 2 ^ P σ ) ( � ∇ 1 − � r 2 )( � ∇ 1 − � ∇ 2 ) · δ ( � r 1 − � ∇ 2 ) + 1 6 t 3 ( 1 + x 3 ^ r 2 ) ρ α ([ P σ ) δ ( � r 1 − � � r 1 + � r 2 ] / 2 ) σ 2 ) · ( � ∇ 1 − � r 2 )( � ∇ 1 − � + iW 0 ( � σ 1 + � ∇ 2 ) × δ ( � r 1 − � ∇ 2 ) , where P σ = 1 + σ 1 · σ 2 ^ , 2 and t i , x i , W 0 , and α are adjustable parameters. Abandoning first principles leads to still better accuracy.
Brief History (Cont.) 5. Convenient because exchange potential is local; easy to solve. Also, variational principal can be reformulated in terms of a local energy-density functional.
Brief History (Cont.) 5. Convenient because exchange potential is local; easy to solve. Also, variational principal can be reformulated in terms of a local energy-density functional. Defining � � r , s ) | 2 � � ρ ab = � b | φ i �� φ i | a � , ρ ( r ) = | φ i ( i � F i � F , s � � r , s ) | 2 , � r , s ′ ) × � τ ( � r ) = | ∇ φ i ( � J ( � r ) = − i φ i ( � r , s )[ ∇ φ i ( � σ ss ′ ] i � F , s , s ′ i � F , s and � h 2 � 2 n τ + 3 8 t 0 ρ 2 + 1 16 ρ 3 + 1 d � E = � φ | H | φ � = r [ 16 ( 3 t 1 + 5 t 2 ) ρτ + 1 64 ( 9 t 1 − 5 t 2 )( ∇ ρ ) 2 + 3 J + 1 ∇ · � � 4 W 0 ρ � J 2 ] 32 ( t 1 − t 2 )
Brief History (Cont.) 5. Convenient because exchange potential is local; easy to solve. Also, variational principal can be reformulated in terms of a local energy-density functional. Defining � � r , s ) | 2 � � ρ ab = � b | φ i �� φ i | a � , ρ ( r ) = | φ i ( i � F i � F , s � � r , s ) | 2 , � r , s ′ ) × � τ ( � r ) = | ∇ φ i ( � J ( � r ) = − i φ i ( � r , s )[ ∇ φ i ( � σ ss ′ ] i � F , s , s ′ i � F , s and � h 2 � 2 n τ + 3 8 t 0 ρ 2 + 1 16 ρ 3 + 1 d � E = � φ | H | φ � = r [ 16 ( 3 t 1 + 5 t 2 ) ρτ + 1 64 ( 9 t 1 − 5 t 2 )( ∇ ρ ) 2 + 3 J + 1 ∇ · � � 4 W 0 ρ � J 2 ] 32 ( t 1 − t 2 ) you find ∂ ( E − � i ǫ i ρ ii ) = h ab − ǫ a δ ab = 0 , ∀ a , b ∂ρ ab i.e. the Hartree-Fock equations, with Hamiltonian h ab = ∂ E /∂ρ ab .
Brief History (Cont.) 6. “Shoot, we can include more correlations, get back to first principles, if we mess with the density functional via:”
Brief History (Cont.) 6. “Shoot, we can include more correlations, get back to first principles, if we mess with the density functional via:” Theorem (Hohenberg-Kohn and Kohn-Sham, vulgarized) ∃ universal functional of the density that, together with a simple one depending only on external potentials, gives the exact ground-state energy and density when minimized through Hartree-like equations. (Finding the functional is up to you!) There is some work to construct functionals form first principles, but they are determined largely by fitting Skyrme parameters. Results are pretty good, but it’s hard to quantify systematic error.
Densities Near Drip Lines This and next 2 slides from J. Dobacewski -3 ) 0.10 100 Sn 100 Zn (n) (n) Particle density (fm (p) (p) 0.05 0.00 0 2 4 6 8 2 4 6 8 10 R (fm)
Two-Neutron Separation Energies Experiment Theory
Deformation
Collective Excited States Can do time-dependent Hartree-Fock in an external potential r ) e − i ω t + f † ( r ) e i ω t . TDHF equation is: f ( � r , t ) = f ( � � − id ρ ab = ∂ E [ ρ ] + f ab ( t ) dt ∂ρ ab
Collective Excited States Can do time-dependent Hartree-Fock in an external potential r ) e − i ω t + f † ( r ) e i ω t . TDHF equation is: f ( � r , t ) = f ( � � − id ρ ab = ∂ E [ ρ ] + f ab ( t ) dt ∂ρ ab Assuming small amplitude oscillations ρ = ρ 0 + δρ e − i ω t + δρ † e − i ω t gives � ∂ h mi δρ nj + ∂ h mi i ωδρ mi = δρ jn + f mi ∂ρ nj ∂ρ jn n > F , j � F
Collective Excited States Can do time-dependent Hartree-Fock in an external potential r ) e − i ω t + f † ( r ) e i ω t . TDHF equation is: f ( � r , t ) = f ( � � − id ρ ab = ∂ E [ ρ ] + f ab ( t ) dt ∂ρ ab Assuming small amplitude oscillations ρ = ρ 0 + δρ e − i ω t + δρ † e − i ω t gives � ∂ h mi δρ nj + ∂ h mi i ωδρ mi = δρ jn + f mi ∂ρ nj ∂ρ jn n > F , j � F Resulting δρ ( ω ) , is the transition density. “Transition strength” to hω is roughly R = � excited state with energy E = � mi f mi δρ mi ( ω ) . This is the “random phase approximation” (RPA).
Evolution of the IV dip ole strength in Sn isotop es IV dip ole strength & ' Isovector Dipole in RPA Strength Distribution Transition Densities 8 neutrons E=14.04 MeV 0.20 protons 132 Sn 14.04 MeV 0.00 6 -0.20 E=11.71 MeV ] 0.10 1 2 ] - m 2 fm f 0.00 [ 4 R [e ρ 11.71 MeV δ -0.10 2 r 7.60 MeV E=7.60 MeV 0.20 2 0.00 -0.20 0 0 2 4 6 8 10 12 0 10 20 30 5 15 25 r [fm] E [MeV] % $
Generalization to Include Pairing HFB (Hartree-Fock-Bogoliubov) is the most general “mean-field” theory in these kinds of operators: � � � � � � ac a † α † U ac a † U ∗ ac a c + V ∗ α a = , a = c + V ac a c , c c c Ground state is the “vacuum” for these operators. In addition to having density matrix ρ ab , one also has “pairing density:” � κ ab ≡ � 0 | a b a a | 0 � , κ ( � r ) = ϕ a ( � r ) ϕ b ( � r ) κ ab . ab Quasiparticle vacuum violates particle-number conservation, but includes physics of correlated pairs. Energy functional E [ ρ ] replaced by E [ ρ, κ ] . Minimizing leads to HFB equations for U and V . Generalization to linear response is called the quasiparticle random phase approximation (QRPA).
Gamow-Teller Strength Gamow-Teller strength from 208 Pb 10 B(GT +) B(GT -) 8 6 Transition operators B (GT) are those that generate allowed β decay: � στ ± . 4 2 0 0 5 10 15 20 25 30 EQRPA
QRPA Calculations of 0 νββ Decay These very different in spirit from shell-model calculations, which involve many Slater determinants restricted to a few single-particle shells. QRPA involves small oscillations around a single determinant, but can involve many shells (20 or more). Recall that the 0 ν operator has terms that look like � ^ M = H ( r ij ) σ i · σ j . ij where i and j label the particles. QRPA evaluates this by expanding in multipoles, and inserting set of intermediate-nucleus states: � � F | ^ � F | ^ O i , JM | N � � N | ^ M | I � = O j , JM | I � , ij , JM , N and uses calculated transition densities to evaluate the matrix elements.
More on QRPA Strength of neutron-proton pairing in effective interaction is not well determined by data, often fit to reproduce 2 ν lifetime. 0.25 3.90 9 levels -1 ) 2 ν (MeV 0 ν M M 0.00 0.00 21 levels -0.25 -3.90 0.7 0.8 0.9 1.0 1.1 1.2 1.3 g pp Problem: Computation of transition densities for initial and final nuclei are completely separate. No way to match the states N computed in initial-nucleus and final-nucleus QRPA.” Must cheat.
Beyond Mean-Field Theory: Generator Coordinates Sometime called “EDF” Sometimes a single mean field won’t do, even with density functionals that includes the effects of many correlations. Basic idea: Construct set of mean fields by constraining coordinate(s), e.g. Collective wave functions quadrupole moment: � r 2 i Y 2 , 0 0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.4 � Q 0 � ≡ � � . i i (b) Minimize 76 Ge (0 i + ) � 76 Se (0 f + ) � H ′ � = � H � − λ � Q 0 � for a whole range of the coordinate β 2 � Q 0 � . Then diagonalize H in space of quasiparticle vacua (projected onto Wave functions peaked at β 2 ≈ ± . 2 good particle number and angular momentum) with different � Q 0 � . � � �
Calculating ββ Decay with Generator Coordinates -0.4 -0.2 0 0.2 0.4 0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.4 (a) (b) (c) 48 Ca (0 i + ) 76 Ge (0 i + ) 150 Nd (0 i + ) 0.3 | F ( � )| 2 0.2 48 Ti (0 f + ) 76 Se (0 f + ) 150 Sm (0 f + ) 0.1 0 0.6 6 (d) (e) (f) 2.5 2.5 5 0.4 0.5 0.5 0.5 0.5 0.5 2.5 4 2.5 0.2 0.5 150 Sm 3 76 Se β 0 48 Ti 4.5 4.5 2 2.5 2.5 0.5 2.5 -0.2 2.5 0.5 0.5 1 0.5 2.5 0.5 0.5 -0.4 48 Ca 2.5 76 Ge 150 Nd Rodr´ ıguez and 0 0.6 6 Martinez-Pinedo (g) 2.5 (h) (i) 2.5 4.5 5 0.4 0.5 2.5 0.5 0.5 0.5 0.5 2.5 0.5 4 0.2 150 Sm 2.5 76 Se 48 Ti 3 2.5 β 0 4.5 4.5 4.5 2.5 2 4.5 0.5 -0.2 2.5 2.5 0.5 0.5 0.5 1 2.5 0.5 0.5 2.5 -0.4 48 Ca 76 Ge 150 Nd 0 30 48 Ca 76 Ge (l) 150 Nd 25 (j) (k) -E pp (MeV) 76 Se 48 Ti 150 Sm 20 15 10 5 -0.4 -0.2 0 0.2 0.4 0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.4 -0.2 0 0.2 0.4 0.6 � � �
Level of Agreement So Far 8 NR-EDF 7 R-EDF QRPA Jy 6 QRPA Tu QRPA CH 5 IBM-2 SM Mi M 0 ν 4 SM St-M,Tk 3 2 Significant spread. And all 1 the models could be missing 0 important physics. 10 31 Uncertainty hard to quantify. 2 [y meV 2 ] 10 30 0 ν m ββ T 1/2 10 29 10 28 48 76 82 96 100 116 124 130 136 150 A
Matrix Elements for Double-Beta Decay II. Hadronic Physics
Relating Theory to Experiment d u u u LQCD d d n n n p p p p ab-initio methods n e ~ ? Many-body methods e
u d e - W − How do we get to ν ¯ nuclear scales? ν ¯ e - W − u d
u d u d e - e - W − ν ¯ Λ ≪ Λ BSM ν ¯ e - e - W − u d u d
Prezeau, Ramsey-Musolf, Vogel (2003) u d u d e - e - W − ν ¯ Λ ≪ Λ BSM ν ¯ e - e - W − u d u d
Prezeau, Ramsey-Musolf, Vogel (2003) p n u d u d e - e - Λ ≪ Λ QCD e - e - Λ ≪ Λ QCD p u d n u d Λ ≪ Λ QCD p n p n π + e - π - e - e - π + e - p n p n
Prezeau, Ramsey-Musolf, Vogel (2003) p n u d u d e - e - 𝜓 PT: NNLO Λ ≪ Λ QCD e - e - Λ ≪ Λ QCD p u d n u d Λ ≪ Λ QCD p n p n π + 𝜓 PT: NNLO 𝜓 PT: LO e - π - e - e - π + e - p n p n
p n e - π - e - π + p n
p n e - π - e - π + p n ~g A
This is the matrix element we need to calculate using LQCD _ p n u d π - e - e - π - e - π + p e - n _ π + u d ~g A
What Do You Do With These Amplitudes? Chiral effective field theory! In QCD vacuum � � m q ¯ qq � � = 0 q = u , d which spontaneously breaks a chiral (left-right) symmetry. Like spontaneous magnetization, which gives rise to massless “magnons” (spin waves). Pions are the analog of magnons for chiral symmetry. If the u and d quarks had the same mass, pions would be massless. In the real world they have mass, but much less than other hadronic objects. Chiral perturbation theory is the “effective theory” for interacting pions. It has infinitely many parameters but only a finite number at each order of λ χ = q /Λ or m π /Λ , the expansion parameter ( q is a typical momentum and Λ is the scale at which other hadrons can exist, about 1 GeV.) The theory breaks down if λ χ gets close to Λ .
Chiral Effective Field Theory with Nucleons Here you try to add nucleons to the mix. There is no problem with adding a single nucleon, but with two or more, things get a little tricky. Proceeding naively, the terms in the nucler interaction have effect only at increasingly large powers of λ χ . 2N Force 3N Force 4N Force LO ( Q/ Λ χ ) 0 NLO ( Q/ Λ χ ) 2 NNLO ( Q/ Λ χ ) 3 +... N 3 LO ( Q/ Λ χ ) 4 +... +... +...
Chiral Effective Field Theory with Nucleons Here you try to add nucleons to the mix. There is no problem with adding a single nucleon, but with two or more, things get a little tricky. Proceeding naively, the terms in the nucler interaction have effect only at increasingly large powers of λ χ . 2N Force 3N Force 4N Force LO ( Q/ Λ χ ) 0 NLO ( Q/ Λ χ ) 2 NNLO ( Q/ Λ χ ) 3 +... N 3 LO ( Q/ Λ χ ) 4 +... +... +...
Comes with Consistent Weak Current Pions are axial, just like the part of the weak current important for ββ decay. The leading piece of the axial current is π c , c just the usual one-body current, more or less. At next order, you get π c 3 , c 4 c D with the constants fixed by the three-body interaction:
Operators for Heavy Particle Exchange Subleading diagrams n p Leading diagrams for heavy particle exchange n p π e π e e n p e π n p n p e e n p
How Useful? In principle, this is exactly what you’d need for a controlled calculation of weak processes with controlled error bars. In practice... 1. Extension of “power counting” to nonperturbative nuclear-structure calculations not fully rigorous. 2. Arguments about how best to determine parameters 3. You also need a many-body calculation with quantifiable errors (we’ll get to that next).
Matrix Elements for Double-Beta Decay III. Ab Initio Nuclear Structure
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 + 1 + r 2 + 1 + a 2 + 4 + 3 + 3 + 4 + n 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 + 1 + r 2 + 1 + a 2 + 4 + 3 + 3 + 4 + n 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 � � �
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) 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
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
Effects of Closure on Quenching Two-level model: | 1 M � | 1 I � | 1 F � E 1 Shell-model | 0 M � E 0 | 0 I � | 0 F � space Initial Intermediate Final Assume Lower levels: � 0 M | β | 0 I � = � 0 F | β | 0 M � ≡ M β Upper levels: � 1 M | β | 1 I � = � 1 F | β | 1 M � = − α M β Operator doesn’t connect lower and upper levels. “Shell-model” calculation gets M 2 β M cl ββ = M 2 M ββ = β E 0
Effects of Closure on Quenching (Cont.) In full calculation, low and high-energy states mix: | 0 ′ � = cos θ | 0 � + sin θ | 1 � | 1 ′ � = − sin θ | 0 � + cos θ | 1 � in all three nuclei. Then we get β = M β ( cos 2 θ − α sin 2 θ ) 2 M ′ + ( α + 1 ) 2 sin 2 θ cos 2 θ � � 1 M ′ 2 ν = M ′ 2 β E 0 E 1 cl = M ′ 2 � 1 + ( α + 1 ) 2 sin 2 θ cos 2 θ � M ′ 2 ν β
Effects of Closure on Quenching (Cont.) In full calculation, low and high-energy states mix: | 0 ′ � = cos θ | 0 � + sin θ | 1 � | 1 ′ � = − sin θ | 0 � + cos θ | 1 � in all three nuclei. Then we get β = M β ( cos 2 θ − α sin 2 θ ) 2 M ′ M β < + ( α + 1 ) 2 sin 2 θ cos 2 θ � � 1 M ′ 2 ν = M ′ 2 β E 0 E 1 cl = M ′ 2 � 1 + ( α + 1 ) 2 sin 2 θ cos 2 θ � M ′ 2 ν β
Effects of Closure on Quenching (Cont.) In full calculation, low and high-energy states mix: | 0 ′ � = cos θ | 0 � + sin θ | 1 � | 1 ′ � = − sin θ | 0 � + cos θ | 1 � in all three nuclei. Then we get β = M β ( cos 2 θ − α sin 2 θ ) 2 M ′ M β < E 0 ≪ E 1 + ( α + 1 ) 2 sin 2 θ cos 2 θ � � M ′ 2 1 β M ′ 2 ν = M ′ 2 ≈ β E 0 E 1 E 0 cl = M ′ 2 � 1 + ( α + 1 ) 2 sin 2 θ cos 2 θ � M ′ 2 ν β
Effects of Closure on Quenching (Cont.) In full calculation, low and high-energy states mix: | 0 ′ � = cos θ | 0 � + sin θ | 1 � | 1 ′ � = − sin θ | 0 � + cos θ | 1 � in all three nuclei. Then we get β = M β ( cos 2 θ − α sin 2 θ ) 2 M ′ M β < E 0 ≪ E 1 + ( α + 1 ) 2 sin 2 θ cos 2 θ � � M ′ 2 1 β M ′ 2 ν = M ′ 2 ≈ β E 0 E 1 E 0 cl = M ′ 2 � 1 + ( α + 1 ) 2 sin 2 θ cos 2 θ � M ′ M ′ 2 > 2 ν β β M cl = α = 1 2 ν , So if α = 1, the closure matrix element is not suppressed at all.
Effects of Closure on Quenching (Cont.) In full calculation, low and high-energy states mix: | 0 ′ � = cos θ | 0 � + sin θ | 1 � | 1 ′ � = − sin θ | 0 � + cos θ | 1 � in all three nuclei. Then we get β = M β ( cos 2 θ − α sin 2 θ ) 2 M ′ M β < E 0 ≪ E 1 + ( α + 1 ) 2 sin 2 θ cos 2 θ � � M ′ 2 1 β M ′ 2 ν = M ′ 2 ≈ β E 0 E 1 E 0 cl = M ′ 2 � 1 + ( α + 1 ) 2 sin 2 θ cos 2 θ � M ′ M ′ 2 > 2 ν β β M cl = α = 1 2 ν , So if α = 1, the closure matrix element is not suppressed at all. If α = 0, it’s suppressed as much as the single- β matrix element, but still less than the non-closure ββ matrix element.
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