Double-Beta Decay Matrix Elements in the Next Five Years

Transcription

1 Double-Beta Decay Matrix Elements in the Next Five Years J. Engel September 8, 2016

2 Primary Focus: 0ν ββ Decay If energetics are right (ordinary beta decay forbidden)..... Z+1, N-1 Z, N and neutrinos are their own antiparticles... n p Z+2, N-2 can observe two neutrons turning into protons, emitting two electrons and nothing else, e.g. via W! " x e n W p e

3 Significance n p normal inverted W e! " x n W p e In usual scenario, rate depends on effective neutrino mass: m eff m i U 2 ei i If lightest neutrino is light: m eff m 2 sol normal m eff m 2 atm inverted!! New expts.

4 Significance n p normal inverted W e! " x n W p e In usual scenario, But rate also rate depends on ona nuclear matrix element. effective neutrino mass: m eff m i U 2 ei i If lightest neutrino is light: m eff m 2 sol normal m eff m 2 atm inverted!! New expts.

5 How Effective Mass Gets into Rate [T1/2 0ν ] 1 = Z 0ν 2 δ(e e1 + E e2 Q) d3 p 1 d 3 p 2 2π 3 2π 3 spins

6 How Effective Mass Gets into Rate [T1/2 0ν ] 1 = Z 0ν 2 δ(e e1 + E e2 Q) d3 p 1 d 3 p 2 2π 3 2π 3 spins Z 0ν contains lepton part e(x)γ µ (1 γ 5 )U ek ν k (x) ν c k (y)γ ν(1 + γ 5 )U ek e c (y), k where ν s are Majorana mass eigenstates.

7 How Effective Mass Gets into Rate [T1/2 0ν ] 1 = Z 0ν 2 δ(e e1 + E e2 Q) d3 p 1 d 3 p 2 2π 3 2π 3 spins Z 0ν contains lepton part e(x)γ µ (1 γ 5 )U ek ν k (x) ν c k (y)γ ν(1 + γ 5 )U ek e c (y), k where ν s are Majorana mass eigenstates. Contraction gives neutrino propagator: k e(x)γ µ (1 γ 5 ) qρ γ ρ + m k q 2 m 2 k γ ν (1 + γ 5 )e c (y) U 2 ek,

8 How Effective Mass Gets into Rate [T1/2 0ν ] 1 = Z 0ν 2 δ(e e1 + E e2 Q) d3 p 1 d 3 p 2 2π 3 2π 3 spins Z 0ν contains lepton part e(x)γ µ (1 γ 5 )U ek ν k (x) ν c k (y)γ ν(1 + γ 5 )U ek e c (y), k where ν s are Majorana mass eigenstates. Contraction gives neutrino propagator: k e(x)γ µ (1 γ 5 ) qρ γ ρ + m k q 2 m 2 k γ ν (1 + γ 5 )e c (y) U 2 ek, The q ρ γ ρ part vanishes in trace, leaving a factor m eff k m k U 2 ek.

9 What About Hadronic Part? Integral over times produces a factor n f J µ L ( x) n n Jν L ( y) i q 0 (E n + q 0 + E e2 E i ) + ( x, µ y, ν), with q 0 the virtual-neutrino energy and the J the weak current.

10 What About Hadronic Part? Integral over times produces a factor n f J µ L ( x) n n Jν L ( y) i q 0 (E n + q 0 + E e2 E i ) + ( x, µ y, ν), with q 0 the virtual-neutrino energy and the J the weak current. In impulse approximation: ( p J µ (x) p = e iqx u(p) g V (q 2 )γ µ g A (q 2 )γ 5 γ µ ) ig M (q 2 ) σµν q ν + g P (q 2 )γ 5 q µ u(p ). 2m p

11 What About Hadronic Part? Integral over times produces a factor n f J µ L ( x) n n Jν L ( y) i q 0 (E n + q 0 + E e2 E i ) + ( x, µ y, ν), with q 0 the virtual-neutrino energy and the J the weak current. In impulse approximation: ( May not be adequate. p J µ (x) p = e iqx u(p) g V (q 2 )γ µ g A (q 2 )γ 5 γ µ ) ig M (q 2 ) σµν q ν + g P (q 2 )γ 5 q µ u(p ). 2m p

12 What About Hadronic Part? Integral over times produces a factor n f J µ L ( x) n n Jν L ( y) i q 0 (E n + q 0 + E e2 E i ) + ( x, µ y, ν), with q 0 the virtual-neutrino energy and the J the weak current. In impulse approximation: ( May not be adequate. p J µ (x) p = e iqx u(p) g V (q 2 )γ µ g A (q 2 )γ 5 γ µ ) ig M (q 2 ) σµν q ν + g P (q 2 )γ 5 q µ u(p ). 2m p q 0 typically of order inverse inter-nucleon distance, 100 MeV, so denominator can be taken constant and sum done in closure.

13 Final Form of Nuclear Part M 0ν = M GT 0ν g2 V ga 2 M F 0ν +... with M GT 0ν = F i, j H(r ij ) σ i σ j τ + i τ + j I +... M F 0ν = F i, j H(r ij ) τ + i τ + j I +... H(r) 2R πr 0 sin qr dq q + E (E i + E f )/2 roughly 1/r Contribution to integral peaks at q 100 MeV inside nucleus. Corrections are from forbidden terms, weak nucleon form factors, many-body currents...

14 ne, Elliott, and Engel: Double beta decay, Majorana neutrinos, and Totally New Physics Could Contribute If neutrinoless decay 41 occurs then ν s are Majorana, no matter what: re are no W R s in the torbut is roughly light neutrinos propor- may not FIG. drive 3. the Majorana decay: propagator resulting from MeV is a typical Schechter and Valle, Exchange of heavy right-handed neutrino n, instead of Eq. in left-right 41, symmetric model. or more zero-mass bosons, called Majorons, be emitted along with the two electrons in do 42 decay 0, Chikashige et al., 1981; Ge Roncadelli, 1981; Georgi et al., Appare e approximately equal ever, it is difficult for such a process to have a amplitude. Second, even if some exotic lepto Mohapatra 1999; Cirviolating physics exists and light neutrino ex not responsible for the decay, the occurrence still implies that neutrinos are Majorana par 43 nonzero mass Schechter and Valle, The that any diagram contributing to the decay c _ (ν) R W e p n p ββ(0ν) n e W ν L

15 ne, Elliott, and Engel: Double beta decay, Majorana neutrinos, and Totally New Physics Could Contribute If neutrinoless decay 41 occurs then ν s are Majorana, no matter what: re are no W R s in the torbut is roughly light neutrinos propor- may not FIG. drive 3. the Majorana decay: propagator resulting from MeV is a typical Schechter and Valle, Exchange of heavy right-handed neutrino n, instead of Eq. in left-right 41, symmetric model. or more zero-mass bosons, called Majorons, Amplitude of exotic mechanism: be emitted along with the two electrons in do Z heavy 42 ( decay ) 4 ( 0, Chikashige et al., 1981; Ge 0ν MWL q Roncadelli, 2 ) 1981; Georgi et al., Appare Z light M WR m e approximately equal ever, it eff m q MeV 2 is difficult N 0ν for such a process to have a amplitude. Second, even if some exotic lepto Mohapatra 1999; Cir- 1 if m violating N 1 TeV and m physics exists eff m and light 2 atm neutrino ex not responsible for the decay, the occurrence still implies that neutrinos are Majorana par 43 nonzero mass Schechter and Valle, The that any diagram contributing to the decay c _ (ν) R W e p n p ββ(0ν) n e W ν L

16 ne, Elliott, and Engel: Double beta decay, Majorana neutrinos, and Totally New Physics Could Contribute If neutrinoless decay 41 occurs then ν s are Majorana, no matter what: re are no W R s in the torbut is roughly light neutrinos propor- may not FIG. drive 3. the Majorana decay: propagator resulting from MeV is a typical Schechter and Valle, Exchange of heavy right-handed neutrino n, instead of Eq. in left-right 41, symmetric model. or more zero-mass bosons, called Majorons, Amplitude of exotic mechanism: be emitted along with the two electrons in do Z heavy 42 ( decay ) 4 ( 0, Chikashige et al., 1981; Ge 0ν MWL q Roncadelli, 2 ) 1981; Georgi et al., Appare Z light M WR m e approximately equal ever, it eff m q MeV 2 is difficult N 0ν for such a process to have a amplitude. Second, even if some exotic lepto Mohapatra 1999; Cir- 1 if m violating N 1 TeV and m physics exists eff m and light 2 atm neutrino ex not responsible for the decay, the occurrence Diagrams with decay still implies that neutrinos are Majorana par at ππ vertex usually 43 nonzero mass Schechter and Valle, The contribute even more. that any diagram contributing to the decay c _ (ν) R W e p n p ββ(0ν) n e W ν L

17 ne, Elliott, and Engel: Double beta decay, Majorana neutrinos, and Totally New Physics Could Contribute If neutrinoless decay 41 occurs then ν s are Majorana, no matter what: re are no W R s in the torbut is roughly light neutrinos propor- may not FIG. drive 3. the Majorana decay: propagator resulting from MeV is a typical Schechter and Valle, Exchange of heavy right-handed neutrino n, instead of Eq. in left-right 41, symmetric model. or more zero-mass bosons, called Majorons, Amplitude of exotic mechanism: be emitted along with the two electrons in do Z heavy 42 ( decay ) 4 ( 0, Chikashige et al., 1981; Ge 0ν MWL q Roncadelli, 2 ) 1981; Georgi et al., Appare Z light M WR m e approximately equal ever, it eff m q MeV 2 is difficult N 0ν for such a process to have a amplitude. Second, even if some exotic lepto Mohapatra 1999; Cir- 1 if m violating N 1 TeV and m physics exists eff m and light 2 atm neutrino ex So exotic stuff can occur not responsible for the decay, the occurrence Diagrams with with roughly decay the same rate as lightν exchange. Untangling still implies that neutrinos are Majorana par at ππmay vertex require usually several experiments and accurate nuclear 43 matrixnonzero mass Schechter and Valle, The contribute elements even formore. all processes. that any diagram contributing to the decay c _ (ν) R W e p n p ββ(0ν) n e W ν L

18 Nuclear-Structure Methods in One Slide Density Functional Theory & Related Techniques: Mean-field-like theory plus relatively simple (e.g. RPA or GCM) corrections in very large single-particle space with phenomenological interaction. Shell Model: Partly phenomenological interaction in a small valence single-particle space a few orbitals near nuclear Fermi surface but with arbitrarily complex correlations. Interacting Boson Model: Truncation of shell model to collective pairs followed by replacement of pairs by bosons, with phenomenological boson interaction. Ab Initio Calculations: Start from a well justified two-nucleon + three-nucleon Hamiltonian, then solve full many-body Schrödinger equation to good accuracy in space large enough to include all important correlations. At present, works pretty well in with A up to about 50.. New!

19 Nuclear-Structure Methods in One Slide Density Functional Theory & Related Techniques: Mean-field-like theory plus relatively simple (e.g. RPA or GCM) corrections in very large single-particle space with phenomenological interaction. Shell Model: Partly phenomenological interaction in a small valence single-particle space a few orbitals near nuclear Fermi surface but with arbitrarily complex correlations. Interacting Boson Model: Truncation of shell model to collective pairs followed by replacement of pairs by bosons, with phenomenological boson interaction. Ab Initio Calculations: Start from a well justified two-nucleon + three-nucleon Hamiltonian, then solve full many-body Schrödinger equation to good accuracy in space large enough to include all important correlations. At present, works pretty well in with A up to about 50. Has potential to combine. and ground virtues of shell model and density functional theory. New!

20 Contrasting the Approaches QRPA Shell Model protons neutrons

21 Contrasting the Approaches pn QRPA QRPA/DFT: Large single-particle spaces in arbitrary Shell single mean field; relatively simple correlations Modeland excitations within the space. protons neutrons

22 Contrasting the Approaches QRPA Shell Model: Small single-particle space in simple spherical mean field; arbitrarily Shell complex Model correlations within the space. protons neutrons

23 Contrasting the Approaches QRPA Shell Model IBM has additional mapping: nucleon pairs to bosons. protons neutrons

24 Contrasting the Approaches Can we combine the virtues QRPA of these methods? Can we avoid fitting parameters to data directly in heavy nuclei? That s not a bad thing, but makes it hard to estimate accuracy whenshell calculating something different from anything ever measured! Model IBM has additional mapping: nucleon pairs to bosons. protons neutrons

25 Level of Agreement So Far Significant spread. And all the models could be missing important physics. Uncertainty hard to quantify.

26 Level of Agreement So Far Significant spread. And all the models could be missing important physics. Uncertainty hard to quantify. More computing power, new many-body methods responsible for major progress in DFT and ab initio theory. Should take advantage of it.

27 DOE Topical Collaboration

28 Ab Initio Nuclear Structure Typically starts with chiral effective field theory. Nucleons, pions sufficient below chiral-symmetry breaking scale. LO (Q/ ) 0 2N Force 3N Force 4N Force NLO (Q/ ) 2 NNLO (Q/ ) N 3 LO (Q/ ) Figure 1: Hierarchy of nuclear forces in ChPT. Solid lines represent nucleons and dashed lines pions. Small dots, large solid dots, solid squares, and solid diamonds denote vertices of index = 0, 1, 2, and 4, respectively. Further explanations are

29 Ab Initio Nuclear Structure Typically starts with chiral effective field theory. Nucleons, pions sufficient below chiral-symmetry breaking scale. LO (Q/ ) 0 2N Force 3N Force 4N Force NLO (Q/ ) 2 NNLO (Q/ ) 3 π c 3, c c D π c 3, c 4 c D Comes with consistent weak current. N 3 LO (Q/ ) Figure 1: Hierarchy of nuclear forces in ChPT. Solid lines represent nucleons and dashed lines pions. Small dots, large solid dots, solid squares, and solid diamonds denote vertices of index = 0, 1, 2, and 4, respectively. Further explanations are

30 Ab Initio Shell Model Partition of Full Hilbert Space P P ^PH^P Q ^PH^Q P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with H eff in P reproducing d most important eigenvalues. Q ^QH^P ^QH^Q Shell model done here.

31 Ab Initio Shell Model Partition of Full Hilbert Space P P H eff Q P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with H eff in P reproducing d most important eigenvalues. Q H eff-q Shell model done here.

32 Ab Initio Shell Model Partition of Full Hilbert Space P P H eff Q P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with H eff in P reproducing d most important eigenvalues. Q H eff-q For transition operator ^M, must apply same transformation to get ^M eff. Shell model done here.

33 Ab Initio Shell Model Partition of Full Hilbert Space P P H eff Q P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with H eff in P reproducing d most important eigenvalues. Q H eff-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 effective operators may not be important for N >2 or 3. Shell model done here.

34 Method 1: Coupled-Cluster Theory Ground state in closed-shell nucleus: Ψ 0 = e T ϕ 0 T = i,m t m i a ma i + ij,mn 1 4 tmn ij a ma na i a j +... Slater determinant m,n>f i,j<f States in closed-shell + a few constructed in similar way.

35 Method 1: Coupled-Cluster Theory Ground state in closed-shell nucleus: Ψ 0 = e T ϕ 0 T = i,m t m i a ma i + ij,mn 1 4 tmn ij a ma na i a j +... 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).

36 Option 2: In-Medium Similarity Renormalization Group Flow equation for effective Hamiltonian. Asymptotically decouples shell-model space. d ds H(s) = [η(s), H(s)], η(s) = [H d(s), H od (s)], H( ) = H eff hh pp hh pp s = 0.0 s = 1.2 s = 2.0 s = 18.3 V [ MeV fm 3 ] 10 Figure 7: Decoupling for the White Hergert generator, et al. Eq. (41), in the J π = 0 + neutronneutron interaction matrix elements of 40 Ca (e max = 8, ω = 20 MeV, Entem-Machleidt N 3 LO(500) evolved to λ = 2.0 fm 1 ). Only hhhh, hhpp, pphh, and pppp blocks of the matrix are shown. Trick is to keep all 1- and 2-body terms in H at each step after normal ordering. Like truncation of coupled-clusters expansion. mechanism. A likely explanation is that the truncation of the commutator (49) to oneand two-body contributions only (Eqs. (50), (51)) causes an imbalance in the infiniteorder re-summation of the many-body perturbation series. For the time being, we have to advise against the use of the Wegner generator in IM-SRG calculations with (comparably) hard interactions that exhibit poor order-by-order convergence of the perturbation 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

37 Ab Initio Calculations of Spectra Ex [ MeV] O 23 O ( O + ) 2 + ( ) ( ) (0 + ) ( 5 2+ ) Neutron-rich oxygen isotopes 0 CCEI 0 + IM-SRG USDB Exp CCEI IM-SRG Deformed nuclei 1+ 2 USDB 1+ 2 Exp. Ex [ MeV] CCEI IM-SRG USDB Exp Ne Mg CCEI IM-SRG USDB Exp. CCEI IM-SRG USDB Exp.

38 Coupled Cluster Test in Shell-Model Space: 48 Ca 48 Ti No Shell-Model Mapping From G. Hagen EOM CCSDT-1 48 Ti Spectrum Exact Preliminary

39 Coupled Cluster Test in Shell-Model Space: 48 Ca 48 Ti No Shell-Model Mapping From G. Hagen EOM CCSDT-1 48 Ti Spectrum ββ0ν Matrix Element GT F T Exact Preliminary Exact CCSDT

40 Full Chiral NN + NNN Calculation (Preliminary) From G. Hagen Method E3 max M 0ν CC-EOM (2p2h) CC-EOM (3p3h) CC-EOM (3p3h) CC-EOM (3p3h) CC-EOM (3p3h) SDPFMU-DB SDPFMU Preliminary Last two are two-shell shell-model calculations with effective interactions.

41 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. F( ) (a) (d) 0.4 β 2 = deformation 0.5 Robledo et al.: Minima at β 2 ± Collective wave functions Ca (0 + i ) 48 Ti (0 + f ) (b) 76 Ge (0 + i ) 76 Se (0 + f ) β (e) 2 Rodriguez and Martinez-Pinedo: Wave functions peaked at β 2 ± (c (f β 0 Ti Se 0 Sm

42 Complementary Ideas: Density Functionals and GCM F( ) 2 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 β 2 = deformation 0.5 Robledo et al.: Minima at β 2 ±.15 β (a) (d) Ti 48 Ca (0 i + ) 48 Ti (0 f + ) Se Collective wave functions (b) 76 Ge (0 i + ) 76 Se (0 f + ) β (e) 2 Rodriguez and Martinez-Pinedo: Wave functions peaked at β 2 ±.2 We re now including crucial isoscalar pairing amplitude as collective coordinate Sm (c (f

43 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 KB3G H coll. H coll. (no T = 0 pairing) 2.5 M GT Ca Ti N mother

44 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 KB3G H coll. H coll. (no T = 0 pairing) Good news for collective models! M GT Ca Ti N mother

45 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. 15 0νββ matrix element Collective pn-pairing wave functions 0.2 M 0ν pn-gcm Ordinary GCM g pp pn Ψ(φ) Ge Se φ = pn pairing amplitude Proton-neutron pairing significantly reduces matrix element.

46 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. 15 0νββ matrix element Collective pn-pairing wave functions 0.2 M 0ν pn-gcm Ordinary GCM g pp pn Ψ(φ) Ge Se φ = pn pairing amplitude Proton-neutron pairing significantly reduces matrix element. Can include pn pairing in IBM as well, with spin-1 isoscalar bosons.

47 GCM in pf Shell 3 2 H coll. GCM M GT Ti Cr Cr Fe N mother

48 Combining DFT-like and Ab Initio Methods Numerical calculations: comparison with Shell-Model 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. MR-ImSRG for 48 Ca-Ti 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. Numerical D RDM λ calculat NN Inte SM: E( MR-Im

49 Improving RPA/QRPA 16 O 0.04 (a) 0.02 RPA RPA produces states in intermediate nucleus, but form is restricted to 1p-1h excitations of ground state. Second RPA adds 2p-2h states. Fraction E0 EWSR/MeV (b) (c) DFT-Corrected Second SSRPA_F RPA Exp E (MeV)

50 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. Effective axial vector coupling constant in nuclei from 2νββ F. One obtains g IBM-2 Iachello, MEDEX 13 meeting A,eff ~ IBM The extracted values can be parametrized as gaeff, = 1.269A If 0ν matrix elements quenched by same amount as 2ν matrix elements, experiments A similar will beanalysis much lesscan sensitive; be done rates for gothe likeism fourth power ISM of g A for which g ISM ~ gaeff, = 1.269A

51 Arguments Suggesting Strong Quenching of 0ν Both β and 2νββ 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.

52 Arguments Suggesting Weak Quenching of 0ν Many-body currents seem to suppress 2ν more than 0ν. Enlarging shell model space to include some effects of high-j spin-orbit partners reduces 2ν more than 0ν. Neutron-proton pairing, related to spin-orbit partners and investigated pretty carefully, suppresses 2ν more than 0ν. M GT ν Ca Ti full no T = 0 pairing P(r) (fm -1 ) g pp =0.0 g pp =0.8 g pp =1.0 g pp = ν N mother r (fm) Large r contributes more to 2ν.

53 Effects of Closure on Quenching Two-level model: 1 I 1 M 1 F E 1 E 0 0 I 0 M 0 F Shell-model space Initial Intermediate Final Assume Lower levels: 0 M β 0 I = 0 F β 0 M Upper levels: 1 M β 1 I = 1 F β 1 M = α 0 M β 0 I Operator doesn t connect lower and upper levels. Shell-model calculation gets M ββ = M2 β E 0 M cl ββ = M2 β

54 Effects of Closure on Quenching (Cont.) In full calculation, low and high-energy states mix: in all three nuclei. Then we get 0 = cos θ 0 + sin θ 1 1 = sin θ 0 + cos θ 1 M β = M β(cos 2 θ α sin 2 θ) 2 ( ) M 2ν = M β (α + 1)2 sin 2 θ cos 2 θ E 0 E 1 ( ) M 2ν cl = M β (α + 1) 2 sin 2 θ cos 2 θ

55 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 M β = M β(cos 2 θ α sin 2 θ) 2 < M β ( ) M 2ν = M β (α + 1)2 sin 2 θ cos 2 θ E 0 E 1 ( ) M 2ν cl = M β (α + 1) 2 sin 2 θ cos 2 θ

56 Effects of Closure on Quenching (Cont.) In full calculation, low and high-energy states mix: in all three nuclei. Then we get 0 = cos θ 0 + sin θ 1 1 = sin θ 0 + cos θ 1 M β = M β(cos 2 θ α sin 2 θ) 2 < M β ( ) M 2ν = M β (α + 1)2 sin 2 θ cos 2 E 0 E 1 θ M β 2 E 0 E 1 E 0 ( ) M 2ν cl = M β (α + 1) 2 sin 2 θ cos 2 θ

57 Effects of Closure on Quenching (Cont.) In full calculation, low and high-energy states mix: in all three nuclei. Then we get 0 = cos θ 0 + sin θ 1 1 = sin θ 0 + cos θ 1 M β = M β(cos 2 θ α sin 2 θ) 2 < M β ( ) M 2ν = M β (α + 1)2 sin 2 θ cos 2 E 0 E 1 θ M β 2 E 0 E 1 E 0 ( ) M 2ν cl = M β (α + 1) 2 sin 2 θ cos 2 θ > M β 2 = M cl 2ν, α = 1 So if α = 1, the closure matrix element is not suppressed at all.

58 Effects of Closure on Quenching (Cont.) In full calculation, low and high-energy states mix: in all three nuclei. Then we get 0 = cos θ 0 + sin θ 1 1 = sin θ 0 + cos θ 1 M β = M β(cos 2 θ α sin 2 θ) 2 < M β ( ) M 2ν = M β (α + 1)2 sin 2 θ cos 2 E 0 E 1 θ M β 2 E 0 E 1 E 0 ( ) M 2ν cl = M β (α + 1) 2 sin 2 θ cos 2 θ > M β 2 = M cl 2ν, α = 1 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.

59 We Hope to Resolve the Issue Soon Problem must be due to some combination of: 1. Truncation of model space. Should be fixable in ab-initio shell model, which compensates effects of truncation via effective operators. 2. Many-body weak currents. Size still not clear, particularly for 0νββ decay, where current is needed at finite momentum transfer q. Leading terms in chiral EFT for finite q only recently worked out. Careful fits and use in decay computations will happen in next year or two.

60 Benchmarking and Error Estimation Systematic Error: 1. Calculate and benchmark spectra and transition rates (including β decay) with all good methods. 2. Calculate β, 2νββ and 0νββ matrix elements in light nuclei 6 He, 8 He, 22 O, 24 O with methods discussed here plus no-core shell model and quantum Monte Carlo. 3. Do the same in 48 Ca. 4. Test effects of next order in EFT Hamilton, coupled-cluster truncation, restrictions to N-body operators, etc. 5. Benchmark methods against spectra and electromagnetic transitions in A = 76, 82, 100, 130, 136, 150.

61 Benchmarking and Error Estimation Systematic Error: 1. Calculate and benchmark spectra and transition rates (including β decay) with all good methods. 2. Calculate β, 2νββ and 0νββ matrix elements in light nuclei 6 He, 8 He, 22 O, 24 O with methods discussed here plus no-core shell model and quantum Monte Carlo. 3. Do the same in 48 Ca. 4. Test effects of next order in EFT Hamilton, coupled-cluster truncation, restrictions to N-body operators, etc. 5. Benchmark methods against spectra and electromagnetic transitions in A = 76, 82, 100, 130, 136, 150. Statistical Error: Chiral-EFT Hamiltonians contain many parameters, fit to data. Posterior distributions (for Bayesian analysis) or covariance matrices (for linear regression) developed to quantify statistical errors for ββ matrix elements.

62 Finally... Existence of topical collaboration will speed progress in next few years. Goal is accurate matrix elements with quantified uncertainty by end of collaboration (5 years from now).

63 Finally... Existence of topical collaboration will speed progress in next few years. Goal is accurate matrix elements with quantified uncertainty by end of collaboration (5 years from now). That s all; thanks for listening.