0 decay in medium- and heavy-mass nuclei within the ... · 0 decay in medium- and heavy-mass nuclei...

0νββ decay in medium- and heavy-massnuclei within the realistic shell model

Luigi Coraggio

Istituto Nazionale di Fisica Nucleare - Sezione di Napoli

Fundamental Physics withElectroweak Probes of Light Nuclei

July 3rd, 2018Institute for Nuclear Theory, Seattle

Luigi Coraggio INT program INT-18-2a

Acknowledgements

L. De Angelis (INFN-NA)

T. Fukui (INFN-NA)

A. Gargano (INFN-NA)

N. Itaco (UNICAMPANIA and INFN-NA)

F. Nowacki (IPHC Strasbourg and UNICAMPANIA)

L. C. (INFN-NA)


Outline

The neutrinoless double-β decay

The calculation of the nuclear matrix element (NME) of 0νββdecay

The realistic nuclear shell model (RSM)

Present work:

Testing the RSM: calculation of the GT strengths and thenuclear matrix element of 2νββ decayRSM calculation of 0νββ nuclear matrix element M0ν andcomparison with other SM resultsPerturbative properties of the 0νββ effective operatorEvaluation of the blocking effect

Outlook


The detection of the 0νββ decay is nowadays one of the maintargets in many laboratories all around the world, triggered bythe search of ”new physics” beyond the Standard Model.

Its detection

would correspond to a violation of the conservation of theleptonic number,

may provide more informations on the nature of theneutrinos (the neutrino as a Majorana particle,determination of its effective mass, ..).


The neutrinoless double β-decay

The inverse of the 0νββ-decay half-life is proportional to thesquared nuclear matrix element M0ν .This property evidences the relevance to calculate M0ν

[T 0ν

1/2

]−1= G0ν

∣∣∣M0ν∣∣∣2 〈mν〉2

G0ν is the so-called phase-spacefactor, which can be accuratelyevaluated by atomic physicscalculations;〈mν〉 =|

∑k mkU2

ek | effectivemass of the Majorana neutrino(light-neutrino exchange)


The detection of the 0νββ-decay

It is necessary to locate the nuclei that are the best candidates todetect the 0νββ-decay

The main factors to be taken into account are:

the Q-value;

the phase-space factor G0ν ;

the isotopic abundance

First group: 76Ge, 130Te, and136Xe.

Second group: 82Se,100Mo,and 116Cd.

Third group: 48Ca, 96Zr, and150Nd.


Our aim is to compute the 0νββ-decay NME for 48Ca, 76Ge,82Se,130Te, and 136Xe.


The calculation of the NMEThe NME is given by

M0ν = M0νGT −

(gV

gA

)2

M0νF + M0ν

T ,

The matrix elements M0να are defined, within the closure approxima-

tion, as follows:

M0να =

∑jn jn′ jp jp′Jπ

TBTD (jnjn′ , jp jp′ ; JiJf )⟨jnjn′ ; Jπ | τ−1 τ

−2 Oα

12 | jp jp′ ; Jπ⟩

with α = (GT , F , T )

The TBTD are the two-body transition-density matrix elements, andthe Gamow-Teller (GT ), Fermi (F ), and tensor (T ) operators:

OGT12 = ~σ1·~σ2HGT

OF12 = HF

OT12 = [3 (~σ1 · r) (~σ1 · r)− ~σ1 · ~σ2] HT

Hα(r) =2Rπ

∫ ∞0

jα(qr)hα(q2)qdqq+ < E >


The calculation of the NME

To describe the nuclear properties detected in the experiments,one needs to resort to nuclear structure models.

Every model is characterized by a certain number ofparameters.

The calculated value of the NME may depend upon thechosen nuclear structure model.

All models may present advantages and/or shortcomings tocalculate the NME


Nuclear structure calculations

The spread of nuclear structure calculations evidencesinconsistencies among results obtained with differentmodels


The problem of geffA ,g

effV

Model dependence of calculated NME is due to different wavefunctions, and also because of different geff

A ,geffV

The renormalization of gA,gV

There are some arguments to employ geffA ,geff

V .Effective coupling constants are necessary to take into account:

the degrees of freedom that have been excludedbecause of the truncation of the Hilbert space;

contributions to the free values from meson exchangecurrents;the short-range correlations excluded to soften the NNforce, when starting from realistic potentials.

In this study we tackle the first issue deriving effective-decayoperators by way of the many-body perturbation theory


The realistic shell model

The derivation of the shell-model hamiltonian, starting from arealistic nuclear potential VNN and using the many-body theory,may provide a reliable approach to the study of the 0νββ decay

The model space may be “shaped” according to thecomputational needs of the diagonalization of the shell-modelhamiltonian

In such a case, the effects of the neglected degrees of freedomare taken into account by the effective hamiltonian Heff and theeffective transition operators Oeff via the many-body theory


An example: 19F

16O

p3/2p1/2

s1/2

19F

protons neutrons

s1/2

d5/2d3/2

s1/2

p3/2p1/2

s1/2d5/2d3/2

model space

9 protons & 10 neutronsinteracting

spherically symmetric meanfield (e.g. harmonic oscillator)

1 valence proton & 2 valenceneutrons interacting in atruncated model space

The degrees of freedom of the core nucleons and the excitations ofthe valence ones above the model space are not considered explicitly.


Effective shell-model hamiltonian

The shell-model hamiltonian has to take into account in an effectiveway all the degrees of freedom not explicitly considered

Two alternative approaches

phenomenological

microscopic

VNN (+VNNN)⇒ many-body theory⇒ Heff

DefinitionThe eigenvalues of Heff belong to the set of eigenvalues of thefull nuclear hamiltonian.This may be provided by a similarity transformation Ω of thefull Hilbert-space hamiltonian H


Workflow for a realistic shell-model calculation

1 Choose a realistic NN potential (NNN)

2 Renormalize its short range correlations

3 Identify the model space better tailored to study the physicsproblem

4 Derive the effective shell-model hamiltonian and consistentlyeffective transition operators, by way of the many-bodyperturbation theory

5 Calculate the observables (energies, e.m. transitionprobabilities, β-decay amplitudes...), using only theoretical SPenergies, two-body matrix elements, and effective operators.


Realistic nucleon-nucleon potential: VNN

Several realistic potentials χ2/datum ' 1:CD-Bonn, Argonne V18, Nijmegen, ...

Strong short-rangerepulsion

How to handle the short-range repulsion ?

Brueckner G matrix

EFT inspired approaches

Vlow−k, our choiceSRG R0

R1

R2

k’

k


The shell-model effective hamiltonianWe start from the many-body hamiltonian H defined in the full Hilbertspace:

H = H0 + H1 =A∑

i=1

(Ti + Ui ) +∑i<j

(V NNij − Ui )

PHP PHQ

QHP QHQ

H = Ω−1HΩ

⇒QHP = 0

PHP PHQ

0 QHQ

Heff = PHP

Suzuki & Lee⇒ Ω = eω with ω =

(0 0

QωP 0

)Heff

1 (ω) = PH1P + PH1Q1

ε−QHQQH1P−

−PH1Q1

ε−QHQωHeff

1 (ω)


The perturbative approach to the shell-model Heff

The Q-box vertex function

Q(ε) = PH1P + PH1Q1

ε−QHQQH1P

Exact calculation of the Q-box is computationally prohibitive for many-body system⇒ we perform a perturbative expansion

1ε−QHQ

=∞∑

n=0

(QH1Q)n

(ε−QH0Q)n+1

The diagrammatic expansion of the Q-box

1 2 3 4 5

6 7 8 9

a b b ba a a

c c c cd d d

a

aa b

b

c c

b

hp

hp hpp

a ab b

c c cd d d

p

ph h

c d

2h1

1 2


The shell-model effective operators

Consistently, any shell-model effective operator may be calculated

It has been demonstrated that, for any bare operator Θ, a non-Hermitianeffective operator Θeff can be written in the following form:

Θeff = (P + Q1 + Q1Q1 + Q2Q + QQ2 + · · · )(χ0 +

+χ1 + χ2 + · · · ) ,

where

Qm =1

m!

dmQ(ε)

dεm

∣∣∣∣ε=ε0

,

ε0 being the model-space eigenvalue of the unperturbed hamiltonianH0

K. Suzuki and R. Okamoto, Prog. Theor. Phys. 93 , 905 (1995)



The χn operators are defined as follows:

χ0 = (Θ0 + h.c.) + Θ00 ,

χ1 = (Θ1Q + h.c.) + (Θ01Q + h.c.) ,

χ2 = (Θ1Q1Q + h.c.) + (Θ2QQ + h.c.) +

(Θ02QQ + h.c.) + QΘ11Q ,

· · ·

and

Θ(ε) = PΘP+PΘQ1

ε−QHQQH1P

Θ(ε1; ε2) = PH1Q1

ε1 −QHQ×

QΘQ1

ε2 −QHQQH1P

Θm =1

m!

dmΘ(ε)

dεm

∣∣∣∣ε=ε0

Θnm =1

n!m!

dn

dεn1

dm

dεm2Θ(ε1; ε2)

∣∣∣∣ε1,2=ε0



We arrest the χ series at the leading term χ0, and then expandit perturbatively:

One-body operator

*

**

*a

b

a

b

a

b

a

b

hpp

h

b

=X

a

Two-body operatora

=X

a b

c d

h

p

b a b a a a ab b bb

c c c c c cdd d d d d

h

p

hp p

211

h2

see also J. D. Holt and J. Engel, Phys. Rev. C 87, 064315 (2013)


The choice of the cutoff Λ = 2.6 fm−1

L. C., A. Gargano, and N. Itaco, JPS Conf. Proc. 6, 020046 (2015)


Nuclear models and predictive power

Realistic SM calculations for 48Ca, 76Ge, 82Se, 130Te, and 136Xe⇓

Check this approach calculating observables related to the GTstrengths and 2νββ decay and compare the results with data.[

T 2ν1/2

]−1= G2ν

∣∣∣M2νGT

∣∣∣2


Model spaces

48Ca: four proton and neutron orbitals outsidedoubly-closed 40Ca0f7/2,0f5/2,1p3/2,1p1/2

76Ge,82Se: four proton and neutron orbitals outsidedoubly-closed 56Ni0f5/2,1p3/2,1p1/2,0g9/2

130Te,136Xe: five proton and neutron orbitals outsidedoubly-closed 100Sn0g7/2,1d5/2,1d3/2,2s1/2,0h11/2


Spectroscopic properties

0

1

2

3

4

5E

(MeV

)

Exp

48Ca

Th

3+

4+

2+

3+

0+ 0+

2+

4+2.7

18

14

2.7

0

1

2

3

E (M

eV)

Exp

76Ge

Th

3+

4+

2+

4+

0+

0+

0+

2+

2+

2+4+3+4+0+

540

730

720

70

560

750

730

100

0

1

2

3

E (M

eV)

Exp

82Se

Th

4+

2+

4+

0+

0+

0+

2+

2+

2+

0+

390360

550

400

111

106

210

78

0

1

2

E (M

eV)

Exp

130Te

Th

6+

4+

2+

4+

0+

2+0+

0+

2+

2+

2+4+6+

4+ 0+2+

580

430

240220

0

1

2

E (M

eV)

Exp

136Xe

Th

6+4+

2+

3+,4+

0+ 0+

2+

4+6+

4+

420 300

53 9


GT− running sums

0 1 2 3Ex (MeV)

0

1

2

3

Σ B

(GT)

bare

effective

exp(3He,t)

76Ge GT strength

0 1 2 3Ex (MeV)

0

0,5

1

1,5

2

2,5

3

Σ B

(GT)

bare

effective

exp(p,n)

82Se GT strength

0 500 1000 1500 2000 2500 3000Ex (keV)

0

0,5

1

1,5

2

Σ B

(GT)

130Te GT strength

bare

effective

exp(3He,t)

0 1000 2000 3000 4000Ex (keV)

0

1

2

3

4

Σ B

(GT)

136Xe GT strengthbare

effective

exp(3He,t)


2νββ nuclear matrix elements

0 0,1 0,2 0,3Theory

0

0,1

0,2

0,3

Experiment

136Xe

130Te

82Se

76Ge

48Ca

Blue dots: bare GT operator

Decay Expt. Bare

48Ca→48 Ti 0.038± 0.003 0.02076Ge→76 Se 0.113± 0.006 0.29782Se→82 Kr 0.083± 0.004 0.336

130Te→130 Xe 0.031± 0.004 0.142136Xe→136 Ba 0.0181± 0.0007 0.0975

Experimental data from A. S. Barabash, Nucl. Phys.A 935, 52 (2015)


2νββ nuclear matrix elements

0 0,1 0,2 0,3Theory

0

0,1

0,2

0,3

Experiment

136Xe

130Te

82Se

76Ge

48Ca

76Ge82Se

48Ca136Xe

130Te

Blue dots: bare GT operatorBlack triangles: effective GT

operator

Decay Expt. Eff.

48Ca→48 Ti 0.038± 0.003 0.01276Ge→76 Se 0.113± 0.006 0.10782Se→82 Kr 0.083± 0.004 0.103

130Te→130 Xe 0.031± 0.004 0.044136Xe→136 Ba 0.0181± 0.0007 0.0285

Experimental data from A. S. Barabash, Nucl. Phys.A 935, 52 (2015)


Matrix elements of the neutron-proton effective GT− operator

na la ja nb lb jb 3rd order GT−eff quenching na la ja nb lb jb 3rd order GT−eff quenching

0f5/2 0f5/2 -0.977 0.37 0g7/2 0g7/2 -1.239 0.500f5/2 1p3/2 -0.143 0g7/2 1d5/2 -0.0191p3/2 0f5/2 0.046 1d5/2 0g7/2 0.1311p3/2 1p3/2 2.030 0.62 1d5/2 1d5/2 1.864 0.641p3/2 1p1/2 -1.621 0.55 1d5/2 1d3/2 -1.891 0.611p1/2 1p3/2 1.713 0.58 1d3/2 1d5/2 1.794 0.581p1/2 1p1/2 -0.697 0.67 1d3/2 1d3/2 -1.023 0.660g9/2 0g9/2 3.121 0.70 1d3/2 2s1/2 -0.093

2s1/2 1d3/2 0.1172s1/2 2s1/2 1.598 0.65

0h11/2 0h11/2 2.597 0.69

Perturbative properties of the effective operator

Decay 1st ord M2νGT 2nd ord M2ν

GT 3rd ord M2νGT Expt.

130Te→130 Xe 0.142 0.042 0.044 0.034± 0.003136Xe→136 Ba 0.0975 0.0272 0.0285 0.0218± 0.0003

More than 70% from 1st→ 2nd orderLess than 5% from 2nd→ 3rd order

L.C., L. De Angelis, T. Fukui, A. Gargano, and N. Itaco, Phys. Rev. C 95,064324 (2017)


0νββ nuclear matrix elementsDecay M0ν

GT M0νF M0ν

48Ca→48 Ti1st order 0.32 -0.13 0.412nd order 0.27 -0.12 0.353rd order 0.44 -0.18 0.55

76Ge→76 Se1st order 2.76 -0.52 3.092nd order 1.42 -0.51 1.753rd order 3.03 -0.80 3.54

82Se→82 Kr1st order 2.72 -0.51 3.042nd order 1.38 -0.51 1.703rd order 2.98 -0.78 3.47

130Te→130 Xe1st order 2.51 -0.57 2.872nd order 1.57 -0.63 1.973rd order 2.71 -0.73 3.11

136Xe→136 Ba1st order 1.82 -0.41 2.082nd order 1.13 -0.45 1.423rd order 1.96 -0.53 2.29

The perturbative behavioris not so satisfactory:third-order contribution israther large compared tothe second order one


The perturbative behavior

Decay M0ν (I) M0ν (II)

76Ge→76 Se1st order 3.09 3.122nd order 2.22 2.203rd order 4.54 4.14

82Se→82 Kr1st order 3.04 2.732nd order 2.16 2.403rd order 4.45 3.92

(I) Present calculations: Vlow−k, cutoff Λ = 2.6 fm−1, effective0νββ operator without folding

(II) Calculations by Engel and Holt∗: Vlow−k, cutoff Λ = 2.1 fm−1,effective 0νββ operator without folding

(∗) J. D. Holt and J. Engel, Phys. Rev. C 87, 064315 (2013)


The perturbative behaviorThe issue of the perturbative behavior of the 0νββ effective operatorneeds to be framed within the approach of the renormalization of theNN potential

Vlow−k: the configurations of VNN(k , k ′) are restricted tothose with k , k ′ < kcutoff = Λ

The Vlow−k is obtained via a similarity transformation Ω

Hlow−k = T + Vlow−k(k , k ′) = Ω−1HNN(k , k ′)Ω = T + Ω−1VNN(k , k ′)Ω

Consistently, the 0νββ operator O has to be transformed by way ofthe similarity transformation Ω

⇓The high-momentum (short-range) components will be effectivelytaken into account, and the perturbative properties should improve(as it happens for the NN potential)


The perturbative behavior

The matrix elements M0να should be defined, without the closure ap-

proximation, as follows:

M0να =

∑k

∑jp jp′ jn jn′Jπ

〈f |a†pan|k〉〈k |a†p′an′ |i〉⟨jnjn′ ; Jπ | τ−1 τ

−2 Oα

12 | jp jp′ ; Jπ⟩

with α = (GT , F , T )

The neutrino potentials Hα should depend from the energy of theinitial, final, and intermediate states:

Hα(r) =2Rπ

∫ ∞0

jα(qr)hα(q2)qdqq + Ek − (Ei + Ef )/2


The perturbative behaviorActually, because of the computational complexity, the energies of theintermediate states are replaced by an average value:

Ek − (Ei + Ef )/2→ 〈E〉∑k

〈f |a†pan|k〉〈k |a†p′an′ |i〉 = 〈f |a†pana†p′an′ |i〉

This approximation is reasonable since the values of q that domina-te the matrix elements are of the order of 100-200 MeV, while therelevant excitation energies in the model space are about 10 MeV

However this might not be true for the intermediate-state configura-tions that appear in the perturbative expansion, that are relevant atleast up to 8~ω excitation energy

⇓〈E〉 may underestimate the real energy denominator for those

configurations


Shell model calculations of M0ν

0

2

4

6

M0ν

136Xe

130Te82Se

76Ge

48Ca

Blue dots:Madrid-Strasbourggroup, bare 0νββoperator

Red dots: Horoi et al.,bare 0νββ operator

Black dots: RSM,bare 0νββ operator


Shell model calculations of M0ν

0

2

4

6

M0ν

136Xe

130Te82Se

76Ge

48Ca

Blue dots:Madrid-Strasbourggroup, bare 0νββoperator

Red dots: Horoi et al.,bare 0νββ operator

Black dots: RSM,effective 0νββoperator


The blocking effect

Blocking (Pauli) effect: the filling of the model-space orbitals bythe valence nucleons affects the calculation of the effective 0νββoperator:

822 P. J. Ellis and E. Osnes: Effective operators in nuclei

TABLE X. Comparison of important second-order effective charge diagrams with the corre-sponding third-order effective interaction diagrams.

Class of diagrams

Effective chargeEgg / 2—GEg / p

N P

Effective interaction"&a,'» v = o, T = I ~u. ~ a,'~, z= o, r = I&

(in MeV)

Verte~TDALadderNumber conservingTotal

—0.0380.0810.0560.0120.111

-0.0600.1290.012—0.0600.021

0.478-0.241-0.1850.1180.170

Ellis and Siegel, 1971 and unpublished work.Barrett and Kirson, 1970, 1972; Barrett, 1972.

understood by reference to Table X. Here we comparesecond-order diagrams for the effective charge with thecorresponding third-order diagrams for the effectiveinteraction. The vertex;correction diagram is seen tobe relatively much bigger for the effective interactionthan for the neutron effective charge, and thus in theformer case the TDA is much too large, whereas it isfairly reasonable in the latter. Now comparing neutronand proton effective charges from the matrix calculationwith the first-order results, we find an increase in mag-nitude for neutrons, whereas the proton case shows amuch smaller increase or even a reduction. From TableX we see that this behavior arises firstly from the lad-der diagram, which is much smaller for protons. Sec-ondly, the number-conserving sets give a large nega-tive value for protons. Much of this comes from thefolded diagrams which normalize the initial and finalwave functions (see Sec. II.D.). The importance of nor-malization effects has been stressed by Qoode et a1.(1972). Incidentally, this reference contains an interest-ing decomposition of the matrix approach, which showsthe sequence of approximations needed to obtain theTDAIt should be noted that the 2P-1h and 3P-2h. bases used

here will contain spurious components which involveexcitations of the center-of-mass of the whole nucleus,as well as the intrinsic excitations that are physicallyrelevant. Ando eg ~E. (1977) have found that the approxi-mate elimination of spurious states leads to a moreattractive effective interaction, although the effectivecharge is little changed. Perhaps more attention shouldbe paid to this difficult question of spurious effects intruncated bases, but we shall not discuss it further here.Finally we mention that a matrix calculation has been

performed with a basis consisting of the (sd )' valencestates plus all 2A&u excitations (Watt et al. , 1974). Thisenormous calculation is only possible using the power-ful Glasgow shell-model technique. Unfortunately, thereis no known way to obtain just the linked two-body effec-tive interaction with this technique. It is therefore diffi-cult to know what weight to attach to the results, whichsuggest that the correction to the bare effective inter-action is somewhat overestimated by second-order per-turbation theory. Note, however, that this is in agree-ment with the trend shown in Table VIII.Summary. In making comparison between perturba-

tion calculations and the "exact" results from matrix

diagonalizations, great care is needed to eliminate ex-traneous effects, in particular, unlinked diagrams.Third-(second-)order perturbation theory for the effec-tive interaction (charge) often gives good results, butits accuracy cannot be relied on. The TDA approxima-tion is fairly reasonable for the neutron effective charge,but is strongly cut back by vertex corrections for theeffective interaction and proton effective charge. In thelatter case normalization effects are also important.

G. Many-body effective operatorsWe have in this review restricted our attention to the

calculation of the two-body effective interaction and theone-body effective charge appropriate to the nuclearshell model. These quantities are frequently applied tostudies of nuclei involving many particles. However,we have pointed out in Sec. II that in many-particle sys-tems the effective operators in question will have many-body components. Note that we are not talking hereabout many-body nuclear forces. Even if we start froma basic two-nucleon interaction, many-body effectiveforces will arise from using a truncated valence space.Examples of three-body effective forces are shown inFig. 53. Diagram (a) arises as one particle is excited

(b)

jmP

(c)FIG. 53. Second-order contributions to the three-body effec-tive interaction. Diagrams (c) and (d) are particular cases ofdiagrams (a) and (b) and arise from corrections for violationof the Pauli exclusion principle, as shown.

Rev. Mod. Phys. , Vol. 49, No. 4, October 1977

Many-body correlations need to be takent into account: we cal-culate three-body correlations diagram and sum over one of theincoming/outcoming nucleons

⇓We obtain a density-dependent effective operator


The blocking effectGamow-Teller two-body matrix elements

Decay jajb jc jd ; J = 0+ ladder 3b (a) 3p-1h 3b (b)

48Ca→48 Ti0f7/20f7/20f7/20f7/2 -0.013 0.003 0.243 -0.016

76Ge→76 Se0g9/20g9/20f5/20f5/2 0.138 -0.210 -1.052 0.2290g9/20g9/21p3/21p3/2 0.178 -0.264 -0.204 0.046

82Se→82 Kr0g9/20g9/20f5/20f5/2 0.141 -0.303 -1.073 0.3300g9/20g9/21p3/21p3/2 0.182 -0.281 -0.209 0.056

130Te→130 Xe0h11/20h11/20g7/20g7/2 0.177 -0.176 -1.008 0.306

136Xe→136 Ba0h11/20h11/20g7/20g7/2 0.183 -0.226 -1.051 0.388

As we expect:3-body (a) diagram reduces the contribution of the 2-body ladderdiagram

3-body (b) diagram reduces the contribution of the 2-body 3p-1h(core polarization) diagram


The blocking effect

Decay M0νGT M0ν

F M0ν

48Ca→48 Ti2nd order 0.27 -0.12 0.352nd order + blocking 0.27 -0.12 0.35 0%

76Ge→76 Se2nd order 1.42 -0.51 1.752nd order + blocking 1.60 -0.54 1.94 10%

82Se→82 Kr2nd order 1.38 -0.51 1.702nd order + blocking 1.58 -0.55 1.93 14%

130Te→130 Xe2nd order 1.57 -0.63 1.972nd order + blocking 1.85 -0.68 2.29 16%

136Xe→136 Ba2nd order 1.13 -0.45 1.422nd order + blocking 1.34 -0.50 1.66 17%

Obviously, the blocking effect is stronger for decays involving nucleiwith a larger number of valence nucleons


Outlook

Tackle the problem of the perturbative behavior of theexpansion of the 0νββ effective operator:

Calculation of 0νββ matrix elements consistently withVlow−k renormalization procedureAvoid the closure approximation

Derivation of Heff from chiral two- and three-body potentialsEvaluation of the effects of chiral two-body currents (forboth 2νββ and 0νββ decays)


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