Double-Beta Decay from First Principles

J. Engel

April 23, 2020 Goal is set of matrix elements with real error bars by May, 2021

DBD Topical Theory Collaboration

Lattice QCD Data

Chiral EFT Similarity Renormalization Group

Ab-Initio Many-Body Methods

Harmonic No-Core Quantum Oscillator Basis Shell Model Monte Carlo Effective Theory Light Nuclei (benchmarking)

DFT-Inspired Coupled Multi-reference In-Medium SRG Clusters In-Medium SRG for Shell Model Heavy Nuclei

DFT Statistical Model Averaging (for EDMs) Shell Model Goal is set of matrix elements with real error bars by May, 2021

DBD Topical Theory Collaboration

Papenbrock LQCD = Latice QCD Hagen EFT = Effective Field Theory Bogner QMC = Quantum Monte Carlo Morris Hergert DFT = Densty Functional Theory Sun Nazarewicz Novario SRG = Similarity Renormilazation Group More IM-SRG IM-SRG = In-Medium SRG Coupled Clusters DFT HOBET = Harmonic-Oscillator-Based Statistics Effective Theory NC-SM = No-Core Shell Model SM = Shell Model DBD Topical Theory Collaboration

Haxton HOBET Walker-Loud McIlvain LQCD Brantley Monge- Johnson Camacho Ramsey- SM Musolf Horoi Engel EFT SM Cirigliano Nicholson Mereghetti "DFT" EFT LQCD QMC Carlson Jiao Quaglioni Vary Goal is set of matrixSRG elementsNC-SM with Yao real error bars by May, 2021 Papenbrock LQCD = Latice QCD Hagen EFT = Effective Field Theory Bogner QMC = Quantum Monte Carlo Morris Hergert DFT = Densty Functional Theory Sun Nazarewicz Novario SRG = Similarity Renormilazation Group More IM-SRG IM-SRG = In-Medium SRG Coupled Clusters DFT HOBET = Harmonic-Oscillator-Based Statistics Effective Theory NC-SM = No-Core Shell Model SM = Shell Model Part 0 0νββ Decay and neutrinos are their own antiparticles... n Example: 136Xe p can observe two neutrons turning Others: 76Ge, 130Te, 150Nd ... w into protons, emitting two electrons l e and nothing else.

Different from already observed e two-neutrino process. wl n p

Neutrinoless Double-Beta Decay

. . If energetics are right (ordinary beta decay forbidden)...

Z+1, N-1 Z, N Z+2, N-2 and neutrinos are their own antiparticles... n p can observe two neutrons turning w into protons, emitting two electrons l e and nothing else.

Different from already observed e two-neutrino process. wl n p

Neutrinoless Double-Beta Decay

. . If energetics are right (ordinary beta decay forbidden)...

Z+1, N-1 Z, N Z+2, N-2

Example: 136Xe Others: 76Ge, 130Te, 150Nd ... n Example: 136Xe p can observe two neutrons turning Others: 76Ge, 130Te, 150Nd ... w into protons, emitting two electrons l e and nothing else.

Different from already observed e two-neutrino process. wl n p

Neutrinoless Double-Beta Decay

. . If energetics are right (ordinary beta decay forbidden)...

Z+1, N-1 Z, N and neutrinos are their own antiparticles... Z+2, N-2 Example: 136Xe Others: 76Ge, 130Te, 150Nd ...

Different from already observed two-neutrino process.

Neutrinoless Double-Beta Decay

. . If energetics are right (ordinary beta decay forbidden)...

Z+1, N-1 Z, N and neutrinos are their own antiparticles... Z+2, N-2 n p can observe two neutrons turning w into protons, emitting two electrons l e and nothing else. ν

e wl n p Example: 136Xe Others: 76Ge, 130Te, 150Nd ...

Neutrinoless Double-Beta Decay

. . If energetics are right (ordinary beta decay forbidden)...

Z+1, N-1 Z, N and neutrinos are their own antiparticles... Z+2, N-2 n p can observe two neutrons turning w into protons, emitting two electrons l e and nothing else. ν ν Different from already observed e two-neutrino process. wl n p 4

1 ] -2

[eV] a) NO, QRPA b) IO, QRPA β

β m 10−1 − 1 10−2 probability density [eV

− 10−1 − − 10 3

−4 −2 10 − − − − 10 10 5 10−4 10 3 10−2 10−1 1 10 5 10−4 10 3 10−2 10−1 1 m [eV] m [eV] − l l

FIG. 1. Marginalized posterior distributionsAgostini, for mββ and Benato,ml for NO (a) Detweiler and IO (b). The solid lines show the allowed parameter Whateverspace assuming 3σ intervals the hierarchy, of the neutrino oscillation the Majorana observables from nu-ﬁt mass [12]. The must plot is producedcome assuming from QRPA NMEs and the absence of mechanisms that drive ml or mββ to zero. The probability density is normalized by the logarithm of mββ somewhere,and of ml. and the Standard Model by itself doesn’t allow it. ]

Its-1 presenceIO IO & Planck impliesIO & m =0 new particles, whichposterior distribution could make is slightly theshifted low to smaller values l for IO, and the discovery probability of future experi- 10 NO NO & Planck NO & ml=0 masses of neutrinos natural, and couldments remains also verychange high. In 0 NO,νββmββ israte. pushed below the reach of future experiments, and the discovery probabilities are driven to be very small as shown in FIG.2. 1 Using ml in the ﬁt basis with a log-ﬂat scale invariant

probability density [eV prior would provide the same results as long as the cutoﬀ on ml, required to have normalizable posterior distributions, is set low enough to make the result independent 10−1 of the choice of cutoﬀ. 1

0.8 III. EXPERIMENTAL SENSITIVITY 0.6

cumulative probability 0.4 The experimental search for 0νββ decay is a very ac- tive field. There is a number of isotopes that can undergo 0.2 0νββ decay and many detection techniques have been 0 developed and tested in recent years [49, 50]. Examples −3 −2 −1 10 10 10 are: high-purity Ge detectors [51, 52], cryogenic bolome- mββ [eV] ters [53, 54], loaded organic liquid scintillators [27], time- projection chambers [55, 56], and tracking chambers [57]. FIG. 2. Top: marginalized posterior distributions of mββ Various larger-scale experiments with the sensitivity to (solid line) for NO and IO, normalized by the logarithm of probe the full IO parameter space are being mounted m . Bottom: complementary cumulative distribution func- ββ or proposed for the near or far future. This work fo- tions for mββ . The band shows the deformation of the pos- 1terior distribution due to different assumptions on the NME. cuses on those projects considered recently by the U.S. The data from cosmology provide a somewhat stronger con- DOE/NSF Nuclear Science Advisory Committee’s Sub- straint on mββ than the current 0νββ decay experiments. committee on Neutrinoless Double Beta Decay [58]: CU- 3 The sharp peaks visible in the mββ distributions are due to a PID [59, 60], KamLAND-Zen [61], LEGEND [62, 63], volume effect dominated by Σ and the Majorana phases. For nEXO [64], NEXT [65], PandaX-III [66], SNO+ [67, 68], 2 NO with m = 0 there is negligible variation due to the NME. l and SuperNEMO [69, 70]. Most of these projects follow a staged-approach in which the target mass will be progres- sively increased. The various phases and parameters of change by only tens of percent. each project are summarized in TABLEI and discussed 4 When the fit is performed with Σ fixed to its mini- in AppendixC. We would like to caution the reader, how- mum allowed value (corresponding to ml = 0), the mββ ever, that many of these experiments are under rapid de- posterior distribution is constrained to lie within the hor- velopment, and the parameters publicly available during izontal bands that extend to m 0 in FIG.1. The m the snapshot of time in which this manuscript was pre- 10101010 1l → ββ

Neutrino Physics in Neutrinoless Double-Beta Decay Diagram is proportional to n p effective “Majorana mass” WL e of light neutrinos,

Õ 2 ν mββ = Uei mi , i e WL so if 0νββ decay is seen neutrinos are their own n p antiparticles!

And if mass hierarchy is inverted, or maybe even if it’s normal, next generation of experiments should be able to see the decay. #it{m_{#beta#beta}} [eV] Whatever the hierarchy, the Majorana mass must come from somewhere, and the Standard Model by itself doesn’t allow it. Its presence implies new particles, which could make the low masses of neutrinos natural, and could also change 0νββ rate.

Neutrino Physics in Neutrinoless Double-Beta Decay Diagram is proportional to n p effective “Majorana mass” WL e of light neutrinos, Õ 2 4 ν mββ = Uei mi ,

1 ]

i -2

[eV] a) NO, QRPA b) IO, QRPA β

β m 10−1 e νββ 1 − WL so if 0 decay is seen neutrinos are their own 10−2

n p probability density [eV − antiparticles! 10−1 − − 10 3 And if mass hierarchy is inverted, or maybe even if it’s normal, next generation of

−4 −2 10 − − − − 10 experiments10 5 10−4 should10 3 be able10−2 to10 see−1 the1 decay.10 5 10−4 10 3 10−2 10−1 1 m [eV] m [eV] − l l

FIG. 1. Marginalized posterior distributionsAgostini, for mββ and Benato,ml for NO (a) Detweiler and IO (b). The solid lines show the allowed parameter space assuming 3σ intervals of the neutrino oscillation observables from nu-ﬁt [12]. The plot is produced assuming QRPA NMEs and the absence of mechanisms that drive ml or mββ to zero. The probability density is normalized by the logarithm of mββ and of ml. ]

-1 IO IO & Planck IO & m =0 posterior distribution is slightly shifted to smaller values l for IO, and the discovery probability of future experi- 10 NO NO & Planck NO & ml=0 ments remains very high. In NO, mββ is pushed below the reach of future experiments, and the discovery probabilities are driven to be very small as shown in FIG.2. 1 Using ml in the ﬁt basis with a log-ﬂat scale invariant

0.8 III. EXPERIMENTAL SENSITIVITY 0.6

1 ] -2

[eV] a) NO, QRPA b) IO, QRPA β

β m 10−1 − 1 10−2 probability density [eV

− 10−1 − − 10 3

−4 −2 10 − − − − 10 10 5 10−4 10 3 10−2 10−1 1 10 5 10−4 10 3 10−2 10−1 1 m [eV] m [eV] − l l

0.8 III. EXPERIMENTAL SENSITIVITY 0.6

Neutrino Physics in Neutrinoless Double-Beta Decay Diagram is proportional to n p effective “Majorana mass” WL e of light neutrinos,

Õ 2 ν mββ = Uei mi , i e WL so if 0νββ decay is seen neutrinos are their own n p antiparticles!

And if mass hierarchy is inverted, or maybe even if it’s normal, next generation of experiments should be able to see the decay.

Whatever the hierarchy, the Majorana mass must come from somewhere, and the Standard Model by itself doesn’t allow it. Its presence implies new particles, which could make the low masses of neutrinos natural, and could also change 0νββ rate. #it{m_{#beta#beta}} [eV] hHi

X NR νL ...and the left-handed neutrino gets a Majorana mass from the concatenation of two such diagrams. 2 νL is light (≈ me/mR) because of the two electron-mass-like Higgs interactions and the heavy-neutrino propagator.

But if there are heavy right-handed neutrinos, then an off-diagonal version of the same diagram operates: hHi

νL NR

Natural Masses through ’Seesaw’ hHi This is the way charged fermions get mass in the Standard Model. Neutrinos cannot because the right-handed states are weak SU(2) singlets. eL eR hHi

Natural Masses through ’Seesaw’ hHi This is the way charged fermions get mass in the Standard Model. Neutrinos cannot because the right-handed states are weak SU(2) singlets. eL eR

But if there are heavy right-handed neutrinos, then an off-diagonal version of the same diagram operates: hHi

νL NR hHi

X NR νL

2 νL is light (≈ me/mR) because of the two electron-mass-like Higgs interactions and the heavy-neutrino propagator.

Natural Masses through ’Seesaw’ hHi This is the way charged fermions get mass in the Standard Model. Neutrinos cannot because the right-handed states are weak SU(2) singlets. eL eR

But if there are heavy right-handed neutrinos, then an off-diagonal version of the same diagram operates: hHi hHi

X νL NR NR νL ...and the left-handed neutrino gets a Majorana mass from the concatenation of two such diagrams. hHi

X NR νL

Natural Masses through ’Seesaw’ hHi This is the way charged fermions get mass in the Standard Model. Neutrinos cannot because the right-handed states are weak SU(2) singlets. eL eR

But if there are heavy right-handed neutrinos, then an off-diagonal version of the same diagram operates: hHi hHi

X νL NR NR νL ...and the left-handed neutrino gets a Majorana mass from the concatenation of two such diagrams. 2 νL is light (≈ me/mR) because of the two electron-mass-like Higgs interactions and the heavy-neutrino propagator. Heavy-particle exchange can occur at the same rate as light-ν exchange (or even a

larger rate) if mN ≈ mWR ≈ 1 TeV.

New Physics Can Contribute Directly to ββ Decay

If neutrinoless decay occurs then ν’s are Majorana, no matter what,

but high-scale physics can contribute directly alongside light-neutrino exchange:

n p

WL e

e WL n p Heavy-particle exchange can occur at the same rate as light-ν exchange (or even a

larger rate) if mN ≈ mWR ≈ 1 TeV.

New Physics Can Contribute Directly to ββ Decay

If neutrinoless decay occurs then ν’s are Majorana, no matter what,

but high-scale physics can contribute directly alongside light-neutrino exchange:

n p

WR e Exchange of heavy right-handed neutrino in left-right symmetric model. N

e WR n p New Physics Can Contribute Directly to ββ Decay

If neutrinoless decay occurs then ν’s are Majorana, no matter what,

but high-scale physics can contribute directly alongside light-neutrino exchange:

n p

WR e Exchange of heavy right-handed neutrino in left-right symmetric model. N Heavy-particle exchange can occur at the W e R same rate as light-ν exchange (or even a n p larger rate) if mN ≈ mWR ≈ 1 TeV. This is the matrix element we need to calculate using LQCD

_ n p d π- u e- π- e-

- π+ e - _ e n p + ~gA d π u

From A. Nicholson

New Physics at Hadronic Level

New-physics operators Light-ν exchange: p n p n π e WL e π ν e n p e WL n p n p π e e Effective field theory lists pion- n p nucleon-level operators and deter- mines their importance. n p e Lattice QCD can then compute dependence of blobs on new particle e masses and couplings. n p

New physics inside blobs New Physics at Hadronic Level

New-physics operators Light-ν exchange: p n p n π e WL e π ν e This is the matrix element we n p need to ecalculate using LQCD WL n p n p π e _ e n p d π- u Effective field theory lists pion- e-n p nucleon-levelπ- operatorse- and deter- p mines their importance.- n π+ e - e Lattice QCD can then compute de- _ e n p + e pendence of~g blobsA on new particle d π u masses and couplings. n p From A. Nicholson New physics inside blobs Light-ν Exchange in a Nucleus

[ 0ν ]−1 ( ) | |2 2 T1/2 = G Z, N M0ν mββ

Phase-space factor Nuclear matrix element, the focus of this talk

g2 GT − V F M0ν = M0ν 2 M0ν + ... gA with Õ GT h | ( ) · + + | i M0ν = F H rij σi σj τi τj I + ... i, j F Õ + + M0ν =hF | H(rij) τi τj |I i + ... i, j 2R ¹ ∞ sin qr H(r) ≈ dq roughly ∝ 1/r π r 0 q + E − (Ei + Ef)/2 Corrections are from “forbidden” terms, weak nucleon form factors, many-body currents, other effects of high-energy physics that depend on framework. Need “first-principles” calculations.

Nuclear Matrix Elements Pre-Topical Collaboration

8 NR-EDF 7 R-EDF QRPA Jy 6 QRPA Tu QRPA CH 5 IBM-2

ν SM Mi Significant spread. And all the 0 4

M SM St-M,Tk models may miss important 3 physics. 2 1

Uncertainty can’t be 1031

quantified. ] 2

1030 [y meV 2 ββ m

ν 29 0 1/2 10 T

1028 48 76 82 96 100 116 124 130 136 150 A Nuclear Matrix Elements Pre-Topical Collaboration

8 NR-EDF 7 R-EDF QRPA Jy 6 QRPA Tu QRPA CH 5 IBM-2

ν SM Mi Significant spread. And all the 0 4

M SM St-M,Tk models may miss important 3 physics. 2 1

Uncertainty can’t be 1031

quantified. ] 2

1030 [y meV

Need “first-principles” 2 ββ m

ν 29 0 calculations. 1/2 10 T

1028 48 76 82 96 100 116 124 130 136 150 A Part I Ab-Initio Many-Body Methods Ab Initio Nuclear Structure Often starts with chiral effective-field theory Nucleons, pions sufficient below chiral-symmetry breaking scale. Expansion of operators in powers of Q/Λχ .

Q = mπ or typical nucleon momentum.

2N Force 3N Force 4N Force LO 0 (Q/⇤) At each “order,” only a finite number of operators exist. NLO 2 (Q/⇤)

NNLO 3 (Q/⇤) +...

N3LO 4 (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,and4,respectively.Furtherexplanationsare given in the text.

The reason why we talk of a hierarchy of nuclear forces is that two- and many-nucleon forces are created on an equal footing and emerge in increasing number as we go to higher and higher orders. At NNLO, the first set of nonvanishing three-nucleon forces (3NF) occur [70, 71], cf. column ‘3N Force’ of Fig. 1. In fact, at the previous order, NLO, irreducible 3N graphs appear already, however, it has been shown by Weinberg [52] and others [70, 127, 128] that these diagrams all cancel. Since nonvanishing 3NF contributions happen first 3 0 at order (Q/⇤) , they are very weak as compared to 2NF which start at (Q/⇤) . More 2PE is produced at ⌫ = 4, next-to-next-to-next-to-leading order (N3LO), of which we show only a few symbolic diagrams in Fig. 1. Two-loop 2PE graphs show up for the first time and so does three-pion exchange (3PE) which necessarily involves two loops. 3PE was found to be negligible at this order [57, 58]. Most importantly, 15 new contact terms Q4 arise and are represented by the four-nucleon-leg graph with a solid diamond. They include a quadratic⇠ spin-orbit term and contribute up to D-waves. Mainly due to the increased number of contact terms, a quantitative description of the two-nucleon interaction up to about 300 MeV lab. energy is possible, at N3LO (for details, see below). Besides further 3NF, four-nucleon forces (4NF) start at this order. Since the leading 4NF come into existence one order higher than the leading 3NF, 4NF are weaker than 3NF. Thus, ChPT provides a straightforward explanation for the empirically known fact that 2NF 3NF 4NF . . . .

4. Two-nucleon interactions The last section was just an overview. In this section, we will ﬁll in all the details involved in the ChPT development of the NN interaction; and 3NF and 4NF will be discussed in Section 5. We start by talking 19 As difficult as solving original problem. But many-body effective operators (beyond Must must apply same unitary 2- or 3-body) can be treated approximately. transformation to transition operator.

Ab Initio Many-Body Methods

Partition of Full Hilbert Space

P Q P = subspace you want ˆ P PHˆ Pˆ PHˆ Q Q = the rest

Task: Find unitary transformation to make H block-diagonal in P and Q, with Heﬀ in P reproducing most important eigenvalues. Q QHˆ Pˆ QHˆ Qˆ

Simpler calculation done here. As difficult as solving original problem. But many-body effective operators (beyond Must must apply same unitary 2- or 3-body) can be treated approximately. transformation to transition operator.

Ab Initio Many-Body Methods

Partition of Full Hilbert Space

P Q P = subspace you want P Heﬀ Q = the rest

Task: Find unitary transformation to make H block-diagonal in P and Q, with Heﬀ in P reproducing most important eigenvalues. Q Heﬀ-Q

Simpler calculation done here. As difficult as solving original problem. But many-body effective operators (beyond 2- or 3-body) can be treated approximately.

Ab Initio Many-Body Methods

Partition of Full Hilbert Space

P Q P = subspace you want P Heﬀ Q = the rest

Task: Find unitary transformation to make H block-diagonal in P and Q, with Heﬀ in P reproducing most important eigenvalues. Q Heﬀ-Q Must must apply same unitary transformation to transition operator.

Simpler calculation done here. Ab Initio Many-Body Methods

Partition of Full Hilbert Space

P Q P = subspace you want P Heﬀ Q = the rest

Task: Find unitary transformation to make H block-diagonal in P and Q, with Heﬀ in P reproducing most As difficult as solving original problem.important eigenvalues. Q Heﬀ-Q But many-body effective operators (beyond Must must apply same unitary 2- or 3-body) can be treated approximately. transformation to transition operator.

Simpler calculation done here. In-Medium Similarity Renormalization Group One way to determine the transformation

Flow equation for effective Hamiltonian. Gradually decouples selected set of states.

hh✛ pp ✲ V [MeVfm3] hh 10 ✻ 5 0 pp -5 -10 -15 ❄ -20 s = 0.0 s = 1.2 s = 2.0 s = 18.3

Figure 7: Decoupling for the Whitefrom generator, H. Hergert Eq. (41), in the J π = 0+ neutron- 40 neutron interaction matrix elements of Ca (emax =8, ~ω = 20 MeV, Entem-Machleidt 3 1 N dLO(500)H(s) = evolved[η(s), toH(λs)]=, 2.0 fm−η().s) Only= [Hhhhh,(s), hhpp,H ( pphh,s)] , and ppppH(∞)blocks= H of the matrixds are shown. d od eff Trick is to keep all 1- and 2-body terms in H at each step after normal mechanism. A likely explanation is that the truncation of the commutator (49) to one- orderingand(IMSRG-2 two-body contributions, includes only most (Eqs. important (50), (51)) causes parts an imba oflance 3, 4-body in the inﬁnite- ...terms). order re-summation of the many-body perturbation series. For the time being, we have to If selectedadvise against set the is use a single of the Wegner state, generator end upin IM-SRG with calculatio g.s. energy.ns with (comparably)If it is a “hard” interactions that exhibit poor order-by-order convergence of the perturbation valenceseries. space, get effective shell-model interaction and operators.

5.4. Decoupling As discussed in Sec. 4.1, the IM-SRG is built around the concept of decoupling the reference state from excitations, and thereby mapping it onto the fully interacting ground state of the many-body system within truncation errors. Let us now demonstrate that the decoupling occurs as intended in a sample calculation for 40Ca with our standard 3 1 chiral N LO interaction at λ = 2.0 fm− . Figure 7 shows the rapid suppression of the oﬀ-diagonal matrix elements in the J π = 0+ neutron-neutron matrix elements as we integrate the IM-SRG(2) ﬂow equations. At s = 2.0, after only 20–30 integration steps

with the White generator, the Γpp′hh′ (s) have been weakened significantly, and when we reach the stopping criterion for the flow at s = 18.3, these matrix elements have vanished to the desired accuracy. While the details depend on the specific choice of generator, the decoupling seen in Fig. 7 is representative for other cases. With the suppression of the off-diagonal matrix elements, the many-body Hamiltonian is driven to the simplified form first indicated in Fig. 2. The IM-SRG evolution not only decouples the ground state from excitations, but reduces the coupling between excitations as well. This coupling is an indicator of strong correlations in the many-body system, which usually require high- or even infinite-order treatments in approaches based on the Goldstone expansion. As we have discussed in Sec. 3, the IM-SRG can be understood as a non-perturbative, infinite-order re-summation of the many-body perturbation series, which builds the effects of correlations into the flowing Hamiltonian. To illustrate this, we show results from using the final IM-SRG Hamiltonian H( ) in Hartree-Fock and post-HF methods in Fig. 8. ∞ After the same 20–30 integration steps that lead to a strong suppression of the off- diagonal matrix elements (cf. Fig. 14), the energies of all methods collapse to the same result, which is the IM-SRG(2) ground-state energy. By construction, this is the result 29 Coupled-Cluster Theory

Ground state in closed-shell nucleus:

Õ Õ 1 |Ψ i = eT |ϕ i T = tma† a + tmna† a†a a + ... 0 0 i m i 4 ij m n i j i,m ij,mn Slater determinant m,n>F i,j

Here the Hamiltonian is transformed in a non-unitary way:

H −→ H˜ ≡ e−THeT

so that |ϕ0i is its ground state. Must solve algebraic equations for the t’s. Excited states, states in closed-shell + a few nucleons, constructed from simple excitations of |ϕ0i. Ab Initio Calculations of Spectra 8 4+ 6 7 22 23 1+ 24 O O + O + + 3+ 1 7 4 (4 ) + + + 2 6 2 1 + 4 + 5 + + 2 + (2 ) 3+ 2 4 2 + + 2 1 6 2 + 3+ 5 2 + ( ) + + 3 2 2 0 4 2 5 + + + 0 3+ (0 ) 0 3+ 3+ 4 + 4 3+ 3 5+ (5 ) + 2 2 [ MeV] 5 5+ Neutron-rich

x + + 2 2 3 + 2 2

E 3 + 2 2 2 2 oxygen isotopes 2 1 1 1

+ + + + 1+ 1+ 1+ 1+ + + + + 0 0 0 0 0 0 2 2 2 2 0 0 0 0 0

Exp. Exp. Exp. CCEI USDB CCEI USDB CCEI USDB IM-SRG IM-SRG IM-SRG 14 20 + 24 13 Ne 8 Mg + 8 + 12 8+ 8+ 8 8+ + 11 8 10 + 9 + 6 + + 6 + 6 + 6 6 6 6+ 8 6+ 7 [ MeV]

x 6

Deformed nuclei E 5 + + + 4 + 4+ 4 + 4 4+ 4 4 4+ 4 3

2 + + + 2+ 2 2 2+ + 2+ 2 2+ 1 2 0 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0+

Exp. Exp. CCEI USDB CCEI USDB IM-SRG IM-SRG December 5, 2018 16:59 WSPC/INSTRUCTION FILE Ring

Modern applications of covariant density functional theory 7

GCM in DFT 2.0 GCM EXP X(5) + 1.8 10 1 + 10 + 1 10 1.6 1 + 322 4 1.4 2 + 204(12) 8 + 300 1 + + 1.2 + 8 42 8 4 1 1 138 2 + 2 1.0 2 277 + 170(51) + 208 278(25) 2 + 2 2 261 6 2 + + + 91 0.8 1 + 6 + 6 02 1 0 114(23) 1 02 147 2

Energy (MeV) s=2 0.6 226 + + 210(2) + 228 4 4 4 0.4 1 1 1

+ + + 178 2 182(2) 2 182 0.2 21 1 1 + + + 0 113 150 0 115(2) 01 115 0.0 1 Nd 1 s=1

P. Ring et al, 2011

Fig. 4. The particle-number projected GCM spectrum of 150Nd (left), compared with the and the X(5)-symmetry predictions (right) for the excitation energies, and B(E2) values (in Weis- skopf units) of the ground-state and the β-band. The theoretical spectra are normalized to the + experimental energy of the state 21 , and the X(5) transition strengths are normalized to the experimental B(E2;2+ 0+). 1 → 1

were derived microscopically form the covariant density functional PC-F1. The resulting spectrum and the electromagnetic transition probabilities are more or less the same as in the 1D-AMP GCM calculation shown in Fig. 4.

4. Summary and outlook. Among the microscopic approaches to the nuclear many-body problem, the framework of nuclear density functional theory provides the most complete description of ground-state properties and collective excitations over the whole nuclide chart. Here we have seen that the self-consistent relativistic mean-field model based on a universal density functional provides an excellent description of the inner fission barriers in actinide nuclei with an average deviation between theory and exper- iment of 0.76 MeV. For transitional nuclei relativistic models not only describe general features of shape transitions but also particular properties of spectra and transition rates at the critical point of the QPT. However, to calculate excitation spectra and transition probabilities, the self-consistent mean-field approach has to be extended to include correlations related to restoration of broken symmetries and fluctuations of collective variables. This can be done either by performing GCM

MICROSCOPIC DESCRIPTION OF SPHERICAL TO ... PHYSICAL REVIEW C 81, 034316 (2010) New Idea: Reference State with Collective Correlations Background: Generator Coordinate Method

Construct set of mean fields by constraining coordinate(s), e.g. quadrupole moment hQ0i. Then diagonalize H in space of symmetry-restored quasiparticle vacua with different hQ0i.

Potential energy surface Collective squared wave functions FIG. 2. (Color online) Same as Fig. 1, but for the isotopes 134-128Xe. -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) 48Ca (0 +) 76Ge (0 +) 150Nd (0 +) 2 0.3 i i i )|

` 48 + 76 + 150 + ( 0.2 Ti (0f ) Se (0f ) Sm (0f ) F | 0.1 0 0.6 β 6 (d) (e) 2 2.5 (f) 2.5 0.5 0.4 0.5 5 Rodr´ıguez and Mart´ınez-Pinedo:0.5 Wave 0.5 0.5 130 functions in 76Ge,Se peaked2.5 at two Li et al.: Potential energy surface2.5 for Xe 0.2barriers are concentrated around β ≈ 0, and0.5 therefore,different rather deformedbinding shapes. energy surfaces in Figs. 1 and 2, are used to calculate 4 than two separate minima, the potentials display continuous the parameters that determine the collective Hamiltonian [21],

γ -soft minima that extend from prolate to oblate shapes. 3

ˆ = ˆ + Sm ˆ + β 0Of particularTi interest in the present analysis are theSe nuclei H Tvib Trot Vcoll, 4.5 (3) that have48 been identified as possible candidates for76 a shape 4.5 150 2.5 2.5 2 0.5 with the vibrational kinetic energy, phase transition that can be characterized by the E(5) dynami- 2.5 -0.2 2.5 cal symmetry [7]. The experimental realization of this critical- 2 h¯ 1 ∂ r ∂0.5 0.5 0.5 Tˆ =− √ β4B point symmetry,2.5 associated with a second-order quantum vib 4 0.5 γγ 1 0.5 2 wr β ∂β w ∂β -0.4phase transition between48Ca spherical and γ -soft potential2.5 shapes, 76Ge 150Nd 134 ∂ r ∂ 1 ∂ r was first identified in Ba [8]. E(5) is the symmetry of a − 3 + − 0 0.6 β Bβγ sin 3γ five-dimensional (intrinsic variables β and γ and the three ∂β w ∂γ β sin 3γ ∂γ w 6 2.5 Euler angles)(g) infinite well in the axial deformation(h) variable β (i) 2.5 ∂4.5 1 ∂ r ∂ [V (β) = 0for|β| βW , and V (β) =∞for |β| >βW ], and × B + sin 3γB , (4) 0.4 0.5 βγ 2.5 ββ 5 the potential is completely γ independent. The microscopic 0.5 ∂β β ∂γ w ∂γ 0.5 0.5 binding energy curve E(β)of134Ba (Fig. 1) displays a shape 2.5 0.5 0.5 0.2that is almost symmetric with respect to β = 0. One notes 4 ≈− ≈ a relatively flat bottom between β2.5 0.1 and β 0.1(the

≈ Ti oblate conﬁguration is only 0.5 MeV above the prolate Sm Se 2.5 3 48 β 0 | 76 | 4.5 minimum), and the potential is rather stiff for β > 0.15. 150 4.5 4.5 The dependence on the triaxial2.5 deformation parameter γ is 4.5 2 shown0.5 in the corresponding three-dimensional energy map -0.2 2.5 and, even more clearly, in Fig. 3, where we plot the binding 2.5 0.5 0.5 0.5 energy2.5 curves as functions of γ for several values of axial 0.5 1 0.5 = 2.5 -0.4deformation: β 480.05, 0.1, 0.15, and 0.2. In the region of 76 150 Ca 134 Ge Nd the ﬂat bottom |β| 0.1 the binding energy of Ba is 0 30 indeed almost independent of γ , and even for somewhat larger | | 48 76Ge 150 deformations,25 (j) 0.1 β 0.2, only aCa weak dependence(k) on γ 76 (l) Nd 48Ti Se 150Sm is20 predicted by the calculation based on the PC-F1 functional.

A very similar energy surface is calculated for the isotone (MeV) 13215 pp Xe (Figs. 2 and 4). -E 10In the next step the constrained self-consistent solutions FIG. 3. (Color online) Self-consistent RMF + BCS binding en- of5 the relativistic mean-ﬁeld plus BCS equations, that is, the ergy curves of the 134Ba nucleus, as functions of the deformation single-particle-0.4 -0.2 wave functions, 0 0.2 occupation 0.4 probabilities, 0.6 -0.4and -0.2parameter 0 0.2γ , for 0.4 four values 0.6 of-0.4 axial deformation, -0.2 0β = 0.20.05, 0. 0.41, 0.6 quasiparticle energies that` correspond to each point on the 0.15,` and 0.2. `

034316-3 Figure 1: (a)-(c) Collective wave functions, GT intensity with, (d)-(f) full and, (g)-(i) constant spatial dependence and (j)-(l) pairing energies for (left) A = 48, (middle) A = 76 and (right) A = 150 decays. Shaded areas corresponds to regions explored by the collective wave functions.

different deformations (β +0.40 and β +0.25, respectively). According to Eq. 6, the final results depend on the convolution≈ of the collective≈ wave functions with the 0νββ matrix elements as a function of deformation. In Fig. 1(d)-(f) we show schematically -shaded circles- the areas of the GT intensity explored by the collective wave functions. We observe, on the one hand, that configuration mixing is very important in the final result because several shapes can contribute to the value of NME, especially in A = 48 and 76. On the other hand, we see that the regions with largest values of the GT intensity are excluded by the collective wave functions. For example, calculations assuming spherical symmetry give systematically larger NME -except for A = 96- as we show in Figure 2. To summarize, we have presented a method for calculating 0νββ nuclear matrix elements based on Gogny D1S Energy Density Functional including beyond mean field effects such as symmetry restoration

5 December 5, 2018 16:59 WSPC/INSTRUCTION FILE Ring

MICROSCOPIC DESCRIPTION OF SPHERICAL TO ... PHYSICAL REVIEW C 81, 034316 (2010)

New Idea: Reference StateModern applications with Collective of covariant density Correlations functional theory 7 Background: Generator Coordinate Method

GCM in DFT Construct2.0 set of mean fields by constraining coordinate(s), e.g. GCM h i EXP X(5) quadrupole moment+ Q0 . Then diagonalize H in space of 1.8 10 1 + 10 + symmetry-restored quasiparticle1 vacua with10 different hQ i. 1.6 1 0 + 322 4 1.4 2 + 204(12) 8 + 300 1 + + 1.2 + 8 42 8 Potential energy surface4 1 Collective1 squared wave138 functions FIG. 2. (Color online) Same as 2 + 2 134-128 1.0 2 Fig. 1, but for the isotopes Xe. 277 + 170(51) -0.4 -0.2 0+ 0.2208 0.4278(25) 0.62 -0.4 -0.2 0 0.2 0.4 0.6 -0.4 -0.2 0 0.2 0.4 0.6 + 2 2 261 6 2 + + + 91 0.8 1 + 6 + 6 02 1 0 114(23) 1 0.4 02 147 2 (a) Energy (MeV) (b) s=2 (c) 0.6 226 48Ca (0 +) 76Ge (0 +) 150Nd (0 +) 2 0.3 i i i + + 210(2) + 228 )| 41 4 4

1 1 ` 0.4 48 + 76 + 150 + ( 0.2 Ti (0f ) Se (0f ) Sm (0f ) F | + 178 + 182(2) + 2 2 2 182 0.1 0.2 1 1 1 + + + 0 113 150 0 115(2) 01 115 0 0.0 1 Nd 1 s=1 0.6 β 6 (d) (e) 2 2.5 (f) P. Ring et al, 2011 2.5 0.5 0.4 0.5 5 Fig. 4. The particle-number projected GCM spectrumRodr´ıguez of 150Nd (left),and Mart´ınez-Pinedo: compared0.5 with the and Wave 0.5 0.5 the X(5)-symmetry predictions (right)130 for the excitationfunctions energies, in and76Ge,Se B(E2) peaked values2.5 (in at Weis- two Li et al.: Potential energy surface2.5 for Xe skopf units) of the ground-state and≈ the β-band.0.5 The theoretical spectra are normalized to the 4 0.2barriers are concentrated around β+ 0, and therefore,different rather deformedbinding shapes. energy surfaces in Figs. 1 and 2, are used to calculate experimental energy of the state 21 , and the X(5) transition strengths are normalized to the thanexperimental two separate B(E2;2 minima,+ 0+ the). potentials display continuous the parameters that determine the collective Hamiltonian [21], 1 → 1

γ -soft minima that extend from prolate to oblate shapes. 3

ˆ = ˆ + Sm ˆ + β 0Of particularTi interest in the present analysis are theSe nuclei H Tvib Trot Vcoll, 4.5 (3) thatwere have48 derivedbeen identified microscopically as possible form candidates the covariant for76 a density shape 4.5 functional PC-F1. The 150 2.5 2.5 2 resulting0.5 spectrum and the electromagnetic transition probabilitieswithare the more vibrational or less kinetic energy, phase transition that can be characterized by the E(5) dynami- 2.5 -0.2 2.5 cal symmetrythe same [ as7]. in The the experimental 1D-AMP GCM realization calculation of this shown critical- in Fig. 4. 2 h¯ 1 ∂ r ∂0.5 0.5 0.5 Tˆ =− √ β4B point symmetry,2.5 associated with a second-order quantum vib 4 0.5 γγ 1 0.5 2 wr β ∂β w ∂β -0.4phase4. transition Summary between48Ca and outlook.spherical and γ -soft potential2.5 shapes, 76Ge 150Nd 134 ∂ r ∂ 1 ∂ r was first identified in Ba [8]. E(5) is the symmetry of a − 3 + − 0 0.6 Among the microscopic approaches to the nuclear many-body problem, the frame-β Bβγ sin 3γ five-dimensional (intrinsic variables β and γ and the three ∂β w ∂γ β sin 3γ ∂γ w 6 work of nuclear density functional2.5 theory provides the most complete description Euler angles)(g) infinite well in the axial deformation(h) variable β (i) 2.5 of ground-state properties and collective excitations over the whole nuclide chart.∂4.5 1 ∂ r ∂ [V (β) = 0for|β| βW , and V (β) =∞for |β| >βW ], and × B + sin 3γB , (4) 0.4 Here we have seen that0.5 the self-consistent relativistic mean-field model basedβγ on 2.5 ββ 5 the potential is completely γ independent. The microscopic 0.5 ∂β β ∂γ w ∂γ 0.5 0.5 a universal density functional provides an excellent description of the inner fission binding energy curve E(β)of134Ba (Fig. 1) displays a shape 2.5 0.5 barriers in actinide nuclei with an average0.5 deviation between theory and exper- 0.2that is almost symmetric with respect to β = 0. One notes 4 iment of 0.76 MeV. For transitional≈− nuclei relativistic≈ models not only describe a relatively flat bottom between β2.5 0.1 and β 0.1(the general features of shape transitions but also particular properties of spectra and

≈ Ti oblate conﬁguration is only 0.5 MeV above the prolate Sm Se 2.5 3

transition48 rates at the critical point of the QPT. However, to calculate excitation β 0 | 76 | 4.5 minimum), and the potential is rather stiff for β > 0.15. 150 spectra and transition probabilities, the self-consistent mean-fie4.5 ld approach has to 4.5 The dependence on the triaxial2.5 deformation parameter γ is be extended to include correlations related to restoration of brok4.5en symmetries and 2 -0.2shown0.5 in the corresponding three-dimensional energy map fluctuations of collective variables. This can be done either2.5 by performing GCM and, even more clearly, in Fig. 3, where we plot the binding 2.5 0.5 0.5 0.5 energy2.5 curves as functions of γ for several values of axial 0.5 1 0.5 = 2.5 -0.4deformation: β 480.05, 0.1, 0.15, and 0.2. In the region of 76 150 Ca 134 Ge Nd the flat bottom |β| 0.1 the binding energy of Ba is 0 30 indeed almost independent of γ , and even for somewhat larger | | 48 76Ge 150 deformations,25 (j) 0.1 β 0.2, only aCa weak dependence(k) on γ 76 (l) Nd 48Ti Se 150Sm is20 predicted by the calculation based on the PC-F1 functional.

5 1. Use bare Hamiltonian to construct approximate GCM ground states in initial and final nuclei and the statistical ensemble: ρ = a |IihI| + b |FihF| 2. Use flow equation to construct Hamiltonian H¯ so that Tr(ρH¯ ) is minimized, apply resulting transformation to decay operator M to get M¯ . 3. Neither initial nor final state is an exact eigenvector of H¯ , even without truncation of flow equations, so do a second GCM calculation with H¯ in both nuclei. 4. Compute hF|M¯ |Ii.

Procedure for 0νββ decay:

In-Medium GCM J.M. Yao, B. Bally, JE, R. Wirth, T.R. Rodr´ıguez,H. Hergert Uses GCM reference state to capture collective multi-particle multi-hole correlations that escape truncated IMSRG. \ complications for transitions, which involve two separate states. 2. Use flow equation to construct Hamiltonian H¯ so that Tr(ρH¯ ) is minimized, apply resulting transformation to decay operator M to get M¯ . 3. Neither initial nor final state is an exact eigenvector of H¯ , even without truncation of flow equations, so do a second GCM calculation with H¯ in both nuclei. 4. Compute hF|M¯ |Ii.

Procedure for 0νββ decay: 1. Use bare Hamiltonian to construct approximate GCM ground states in initial and final nuclei and the statistical ensemble: ρ = a |IihI| + b |FihF| 3. Neither initial nor final state is an exact eigenvector of H¯ , even without truncation of flow equations, so do a second GCM calculation with H¯ in both nuclei. 4. Compute hF|M¯ |Ii.

Procedure for 0νββ decay: 1. Use bare Hamiltonian to construct approximate GCM ground states in initial and final nuclei and the statistical ensemble: ρ = a |IihI| + b |FihF| 2. Use flow equation to construct Hamiltonian H¯ so that Tr(ρH¯ ) is minimized, apply resulting transformation to decay operator M to get M¯ . 3. Neither initial nor final state is an exact eigenvector of H¯ , even without truncation of flow equations, so do a second GCM calculation with H¯ in both nuclei. In-Medium GCM J.M. Yao, B. Bally, JE, R. Wirth, T.R. Rodr´ıguez,H. Hergert Uses GCM reference state to capture collective multi-particle multi-hole correlations that escape truncated IMSRG. \ complications for transitions, which involve two separate states.

Procedure for 0νββ decay: 1. Use bare Hamiltonian to construct approximate GCM ground states in initial and final nuclei and the statistical ensemble: ρ = a |IihI| + b |FihF| 2. Use flow equation to construct Hamiltonian H¯ so that Tr(ρH¯ ) is minimized, apply resulting transformation to decay operator M to get M¯ . 3. Neither initial nor final state is an exact eigenvector of H¯ , even without truncation of flow equations, so do a second GCM calculation with H¯ in both nuclei. 4. Compute hF|M¯ |Ii. In-Medium GCM for Decay of 48Ca GCM Reference States for IMSRG

Potential Energy Surfaces

48Ca is spherical and 48Ti is weakly deformed. β2 only

Smaller ħω gives a bit more collectivity.

Spectrum in 48Ti

In 9 shells

5 48 Ti(2.0/2.0, h¯Ω = 16 MeV) 4 4+ 3 4+ 4+ 2 [MeV]

x +

E 2 + + 1 2 2 + + + 0 0 0 0

GCM (β2) GCM(β2,ω) EXP Spectrum in 48Ti

In 9 shells

5 48 48 4+ Ti(2.0/2.0, h¯Ω = 16 MeV) Ti 4 4+ + 3 152 + 1404 + 4 2 193 4 +

[MeV] 2 201(18)

[MeV] 2

x + x

E + 2

E 2 + + 1 108 1012 136 125(5)2 0+ 0+ 0 + + + 0 0 0 0 2.0/2.0(16) 1.8/2.0(16) 1.8/2.0(12) EXP GCM (β2) GCM(β2,ω) EXP

β2 only

Smaller ħω gives a bit more collectivity. Variation with BE2 and Summary of Results

1.4 Raman Pritychenko EM1.8/2.0(12) emax = 6 EM1.8/2.0(16) emax = 8 1.2 EM2.0/2.0(16) emax = 10

ν 0 1.0 M

0.8 Extrap. 0.6

75 100 125 150 B(E2 : 2+ 0+) [e2fm4] →

The green band is our best guess for the matrix element. IMGCM Matrix Element Vs. Others

From S. Novario

3.0 Spherical Deformed 2.5

2.0 ν 0 1.5 Coupled M Clusters (prelim.) 1.0

0.5

0.0 CCSD CCSDT-1 IMSRG+GCM QRPA IBM EDF SM (pf) SM (MBPT) SM (sdpf) Part 2 Currents and EFT NUCLEAR SHELL MODEL 43 () parameters can be empirically extracted as the residuals between a set of experimental values and the values of the matrix el ements calculated with the free-nucleon operators. Our results are discussed in Sections 3.2 and 3.3 for the GT and MI operators, respectively. Values for the () parameters in the effective operator can al so be calculated from fun damental considerations. Our empirical results are compared with such calculations in Section 3.4.

3.2 Gamow-Teller Results The relationships between experimental GT matrix elements from sd-shell beta decays and the predictions of the W interaction have been studied comprehensively in (57). This study incorporated a compilation of extant beta decay in A = 17-39 nuclei together with shell-model calculations for all the initial and final states concerned. The essential conclusions drawn in (57) can be inferred from the comparisons of experimental and theoretical matrix elements presented in Figure 6. The values of the

matrix elements are normalized to reflect the 3(N - Z) sum rule, such

that R(GT) = M(GT)/W, where W = 19A/9vl[(2Jj+ 1)3(Nj _Zj)]1/2 for

Ni i= Zi and W = 19A/9vl[(2Jr+ 1)3(Nr - Zr)]1/2 for Ni = Zi' The matrix

elements M(GT) are obtained from it = 6170j[B(F)+B(GT)], where

B(GT) = M(GT)2j(2Ji + I). B(GT) is the GT transition probability (which depends on the transition direction). M(GT) is the GT reduced matrix Prelude:element Gamow-Teller (which is independentβ ofDecay the transition direction). It is evident from inspection of the left side of Figure 6 that the exper Leadingimental order values decay of GT operator matrix el ements is στì + .in the sd shell are systematically smaller than the predictions of the W-interaction wave functions coupled 50-Year-Oldwith the free-nucleon Problem: operator, Effective by ag factorneeded of about in all 0.77 calculations (indicated by of the lower line on the left side of Figure 6).A The same wave functions combined shell-modelwith the effective (or related) operator type. account for most of the data extremely well.

Brown & WildenthalR(GT)

FREE-NUCLEON EFFECTIVE

Annu. Rev. Nucl. Part. Sci. 1988.38:29-66. Downloaded from www.annualreviews.org 0.8 by University of North Carolina - Chapel Hill/ACQ SRVCS on 09/29/13. For personal use only.

0.2

o. 0 __----' �_____.L�---L..�--L-�___ ...... L �_L.�--'--�L-..� o 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 THEORY Figure 6 Theoretical vs experimental R(GT) matrix elements (see Sections 3 and 3.2). Many suggestions about the cause but, until recently, no consensus. p p n/p p p n/p ν π ν ν

n n n/p n n n/p (a) (b) (c)

n/p

cp dp

ν in jp an bn (a) (b) (c) (d)

Consider very simple wave function

(e) (f) (g) (h)

Higher order:

p n/p p p n/p p p n/p ν p p p p ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 c3, c4

p p n/p p p n/p ν π ν ν n n n/p n n n/p (a) (b) (c)

n/p

cp dp

ν in jp an bn (a) (b) (c) (d)

Axial Weak Current in Chiral EFT (e) (f) (g) (h) β Decay (simplified) with electron lines omitted Leading order:

p n/p p p n/p p p n/p ν p p p p ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 c3, c4 Usual β-decay current p p n/p p p n/p ν π ν ν

n n n/p n n n/p (a) (b) (c)

n/p

cp dp

ν in jp an bn (a) (b) (c) (d)

(e) (f) (g) (h)

Higher order:

p n/p p p n/p p p n/p ν p p p p ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 c3, c4

p p n/p p p n/p ν π ν ν n n n/p n n n/p (a) (b) (c)

n/p

cp dp

ν in jp an bn (a) (b) (c) (d)

Axial Weak Current in Chiral EFT (e) (f) (g) (h) β Decay (simplified) with electron lines omitted Leading order: Consider very simple wave function

p n/p p p n/p p p n/p ν p p p p ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 c3, c4 protons neutrons Usual β-decay current p p n/p p p n/p ν π ν ν

n n n/p n n n/p (a) (b) (c)

n/p

cp dp

ν in jp an bn (a) (b) (c) (d)

(e) (f) (g) (h)

Higher order:

p n/p p p n/p p p n/p ν p p p p ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 c3, c4

p p n/p p p n/p ν π ν ν n n n/p n n n/p (a) (b) (c)

n/p

cp dp

ν in jp an bn (a) (b) (c) (d)

Axial Weak Current in Chiral EFT (e) (f) (g) (h) β Decay (simplified) with electron lines omitted Leading order: Consider very simple wave function

p n/p p p n/p p p n/p ν p p p p ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 c3, c4 protons neutrons Usual β-decay current p p n/p p p n/p ν π ν ν

n n n/p n n n/p (a) (b) (c)

n/p

cp dp

ν in jp an bn (a) (b) (c) (d)

(e) (f) (g) (h)

Higher order:

p n/p p p n/p p p n/p ν p p p p ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 c3, c4

p p n/p p p n/p ν π ν ν n n n/p n n n/p (a) (b) (c)

n/p

cp dp

ν in jp an bn (a) (b) (c) (d)

Axial Weak Current in Chiral EFT (e) (f) (g) (h) β Decay (simplified) with electron lines omitted Leading order: Consider very simple wave function

p n/p p p n/p p p n/p ν p p p p ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 c3, c4 protons neutrons Usual β-decay current p p n/p p p n/p ν π ν ν n n n/p n n n/p (a) (b) (c)

p p n/p p p n/p n/p ν π ν ν cp dp n n n/p n n n/p (a) (b) ν in (c) jp an bn (a) (b) (c) (d)

n/p c Axial Weak Current in Chiral EFT p dp (e) (f) (g) (h) β Decay (simplified) with electron lines omitted ν in Leadingjp order: Consider very simple wave function a b n n p n/p p p n/p p p n/p ν (a) (b)p p (c)p p (d)ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 c3, c4 protons neutrons Usual β-decay current (e) (f) (g) (h)

Higher order: p n/p p p n/p p p n/p ν p p p p ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 c3, c4 p p n/p p p n/p ν π ν ν n n n/p n n n/p (a) (b) (c)

p p n/p p p n/p n/p ν π ν ν cp dp n n n/p n n n/p (a) (b) ν in (c) jp an bn (a) (b) (c) (d)

Higher order: p n/p p p n/p p p n/p ν p p p p ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 protons c3, c4neutrons p p n/p p p n/p ν π ν ν n n n/p n n n/p (a) (b) (c)

p p n/p p p n/p n/p ν π ν ν cp dp n n n/p n n n/p (a) (b) ν in (c) jp an bn (a) (b) (c) (d)

Higher order: p n/p p p n/p p p n/p ν p p p p ν π ν π π π ∆ + ... ν n n/p n ≈ n/p n n/p n n n n n n gA c3, c4 protons c3, c4neutrons Resolving the quenching puzzle of ! decays: medium-mass nuclei Quenching in the sd and pf Shells

IMSRG calculation, Holt et al

Some quenching from correlations omitted by the shell model.

But a lot comes from the two-body current. What about in ββ Decay

Usual leading-order process

p p n p p ν π ν

n n n n n protons neutrons What about in ββ Decay

Usual leading-order process

p p n p p ν π ν

n n n n n protons neutrons What about in ββ Decay

Usual leading-order process

p p n p p ν π ν

n n n n n protons neutrons e p p n p p e ν ν π n n n n n