70zn, 86kr, 94zr, 104ru, 110pd and 124sn

8/17/2019 70Zn, 86Kr, 94Zr, 104Ru, 110Pd and 124Sn

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Nuclear Physics A 864 (2011) 63–90

www.elsevier.com/locate/nuclphysa

On the double-beta decays of 70Zn, 86Kr, 94Zr,104Ru, 110Pd and 124Sn

Jouni Suhonen

Department of Physics, P.O. Box 35 (YFL), FI-40014 University of Jyväskylä, Finland

Received 23 February 2011; received in revised form 3 May 2011; accepted 8 June 2011

Available online 12 June 2011

Abstract

Double-beta decays of 70Zn, 86Kr, 94Zr, 104Ru, 110Pd and 124Sn to the final ground states and the decaysof 110Pd and 124Sn to the excited states in 110Cd and 124Te are studied in the framework of the proton–neutron quasiparticle random-phase approximation (pnQRPA) combined with the multiple-commutator

model (MCM) for description of decays to the excited final states. Reasonably large single-particle modelspaces and G-matrix-based effective nuclear forces are used to compute the relevant nuclear matrix elementsand decay half-lives. The present study is among the very few that have been dedicated to double-beta de-cays of these nuclei, although the associated double-beta Q values exceed 1.0 MeV for the ground-statedecays.© 2011 Elsevier B.V. All rights reserved.

Keywords: Quasiparticle random-phase approximation; Multiple-commutator model; Two-neutrino double-beta decay;Neutrinoless double-beta decay

1. Introduction

Double-beta decays are among the most interesting disintegration processes occurring inatomic nuclei. The two-neutrino double-beta (2νββ) decay is a second-order weak process oc-curring in the standard model of electro-weak interactions. Several measured values of the 2 νββhalf-lives are available at the present [1]. The neutrinoless mode of double-beta decay (0νββ de-cay) is the more interesting channel of disintegration involving necessarily a massive Majorana

E-mail address: [email protected].

0375-9474/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2011.06.021


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J. Suhonen / Nuclear Physics A 864 (2011) 63–90 65

This article is organized as follows. In Section 2 the basic theoretical framework is brieflyreviewed. In Section 3 the methods of determination of the values of the model parameters areoutlined. In Section 4 the results for the single beta decays, 2νββ and 0νββ decays are given in

terms of decay log f t values, NMEs and decay half-lives. Finally in Section 5 a summary of thecalculations is provided.

2. Outline of the theoretical framework for double-beta decays

In this section a brief outline of the basic formalism of double-beta decays is presented to helpthe reader to tie the obtained results with the underlying weak-interaction and nuclear physics.For a more comprehensive account of the theory the reader is referred to the reviews [2–4].

2.1. Two-neutrino double-beta decay

The 2νββ decay is a second-order weak process in the standard model of electroweak inter-actions. The decay proceeds through a complete set of virtual states in the nucleus intermediateto the mother and daughter nuclei of the decay. Due to the small momenta involved the decaytransitions are of the allowed Gamow–Teller type and thus the set of intermediate states consistsof all the 1+ states. The decay starts always from the 0+ ground state of the even–even mothernucleus denoted by the initial state, 0+i , in this work. The final states in the daughter even–evennucleus can be either the ground state 0+gs or an excited 0

+ or 2+ state. All these states can becalled generically as the final J +f

= 0+, 2+ states. The associated 2νββ-decay half-life, t (2ν)1/2 , can

be written ast

(2ν)1/2

0+i → J +f

−1 = G(2ν) (J f )M (2ν) (J f )2, (1)where G(2ν) (J f ) is an integral over the phase space of the leptonic variables [2]. The nucleardouble Gamow–Teller matrix element, M (2ν) (J f ), can be written as

M (2ν) (J f ) =1√

1 + 2δJ 2k1k2

(J f

m σ mt −m 1+k1 )1

+k1

|1+k2(1+k2

n σ nt −n 0+i )( 12 Qββ + Ek1 − M i )/me + 1

, (2)

where the transition operators are the usual Gamow–Teller operators for β − transitions, Qββ isthe 2νββ Q value, Ek is the mass energy of the k-th intermediate state, M i is the mass energy of the initial nucleus, and me is the rest-mass energy of the electron. The various transition matrixelements of (2) are given in Section 2.3 and the overlap factor in (2) is given by

J πk1

J πk2 =pn

XJ

π k2pn

X̄J π k1pn − Y J π k2

pn Ȳ J π k1pn

(3)

and it takes care of the matching of the corresponding states in the two sets of states based on theinitial and final even–even reference nuclei.

In practical calculations Ek is taken to be the average of the pnQRPA energies for the initial (i)and final (f ) even–even reference nuclei, i.e. Ek = (Eik + E f k )/2. The energy Ek is furthernormalized such that the energy difference E1 −M i corresponds to the experimental mass energydifference of the intermediate and initial nuclei plus the measured excitation energy of the first1+ state in the intermediate nucleus.


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66 J. Suhonen / Nuclear Physics A 864 (2011) 63–90

2.2. Neutrinoless double-beta decay to 0+ states

In the present work it is assumed that the 0νββ decay proceeds dominantly via the exchange

of light Majorana neutrinos and that the final states have the multipolarity 0+. In this case thedecay half-life can be written ast

(0ν)1/2

0+i → 0+f

−1 = G(0ν)M (0ν) 2mν[eV]2, (4)where the effective neutrino mass (given above in units of eV) is defined as

mν =

j

λCPj mj |U ej |2. (5)

The coefficients λCPj are the Majorana CP phases and U ej are the components of the electron

row of the neutrino-mixing matrix. The value of the phase-space integral G(0ν)

should alwaysbe calculated for the value gA = 1.25 of the axial-vector coupling constant due to the specificdefinition of the NME M (0ν)

given below. The values of G(0ν) for the discussed nuclei and

transitions are given later in Section 4.5. The nuclear matrix element of (4) can be written as alinear combination of three terms, i.e.

M (0ν) =

gA

gbA

2M

(0ν)GT −

gV

gA

2M

(0ν)F + M (0ν)T

, (6)

where gbA = 1.25 is the bare-nucleon value of the axial-vector coupling constant.

The exchange of a light Majorana neutrino generates a neutrino potential that is useful toexpand in multipoles to facilitate a pnQRPA calculation of the decay rate. Due to the high mo-mentum of the exchanged neutrino (of the order of hundred MeV/c) all the multipoles contributeto the decay rate and thus it is relevant to sum over all multipolarities J π of the virtual states of the intermediate nucleus. Hence, the double Fermi and Gamow–Teller nuclear matrix elementsare defined through

M (0ν)F =

a

0+f

mn

hF(rmn, Ea)t −m t

−n

0+i , rmn = |rm − rn|, (7)M

(0ν)GT

=a 0+f

mn

hGT(rmn, Ea )(σ m

·σ n)t

−m t

−n 0+i , (8)

where the summation over a ≡ kJ π in Eqs. (7) and (8) runs over all the states of the intermediateodd–odd nucleus, k denoting the k-th state of multipolarity J π . The quantity rmn is the relativedistance between the two decaying neutrons labeled m and n. The ground state of the initial even–even nucleus is denoted, as before, by 0+i and the ground state or the excited 0

+ state of the finaleven–even nucleus is denoted by 0+f . The quantity Ea is the mass energy of the nuclear state a

of the intermediate nucleus. The tensor matrix element M (0ν)T can be dropped as its magnitude isquite small [25,31,32]. The neutrino potentials hK (rmn, Ea), K = F, GT, are given by

hK (rmn, Ea) = 2π

RA

dq qhK (q2

)q + Ea − (M i + Ef )/2

j 0(qrmn), (9)

where RA = 1.2A1/3 fm is the nuclear radius, M i is the ground-state mass energy of the initialnucleus, Ef the (ground-state or excited-state) mass energy of the final nucleus and j 0 is the


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spherical Bessel function. The term hK (q2) in (9) includes the contributions arising from theinduced currents and the finite nucleon size [45].

Again, in practical calculations the lowest pnQRPA energy, Emin, is normalized in such a

way that the energy difference Emin − (M i + Ef )/2 corresponds to the experimentally deter-mined mass energy difference Einterm − (M i + Ef )/2 where Einterm is the mass energy of theintermediate nucleus

The nuclear matrix elements can be written in the pnQRPA framework as

M (0ν)K =

J π ,k1,k2,J

ppnn

(−1)j n+j p+J +J √

2J + 1

×

j p j n J

j n j p J

pp : J

OK

nn : J

× 0+f c†p ̃cnJ J πk1J πk1 J πk2 J πk2c†p c̃nJ 0+i , (10)where k1 and k2 label the different pnQRPA solutions for a given multipole J π . The operatorsOK inside the two-particle matrix element derive from (7) and (8) and they can be written as

OF = hF(r,Ea ), OGT = hGT(r,Ea )σ 1 · σ 2, r = |r1 − r2|, (11)where Ea is the average of the energies of the corresponding states a of the two pnQRPA calcu-lations based on the initial and final nuclei of the decay.

2.3. Basic transition densities

The pnQRPA states of the intermediate nucleus are written asJ πk M =pn

XJ

π kpn

a†pa

†n

J M

− Y J π kpn

a†pa†n

†J M

|QRPA, (12)where |QRPA is the QRPA vacuum. The operator a†p (a†n) creates a proton (neutron) quasipar-ticle in the orbital p (n). The sum runs over all proton–neutron configurations in the chosenvalence space. In the case 0+f = 0+gs the form (12) of the pnQRPA state leads to the transitiondensities

0+f c†

p ̃cn

J

J πk1= √ 2J + 1v̄p ̄un X̄J π k1pn + ūp ̄vn Ȳ J π k1pn , (13)J πk2

c†p c̃nJ 0+i = √ 2J + 1upvnXJ π k2pn + vpunY J π k2pn , (14)where v (v̄) and u (ū) correspond to the BCS occupation and unoccupation amplitudes of theinitial (final) even–even nucleus. The amplitudes X and Y ( X̄ and Ȳ ) come from the pnQRPAcalculation starting from the initial (final) nucleus of the double-beta decay. In the case 0+f =0+1 ≈ 0+2-ph, i.e. the final state is a two-phonon-like state, the transition density (13) is replacedby an other one given later in this section. The overlap factor in (10) is given in (3).

The multiple-commutator model (MCM) [12,46] is designed to connect excited states of aneven–even reference nucleus to states of the neighboring odd–odd nucleus. The states of theodd–odd nucleus are given by the pnQRPA in the form (12). The excited states of the even–evennucleus are generated by the (charge conserving) quasiparticle random-phase approximation (cc-QRPA) described in detail in [47]. Here the symmetrized form of the phonon amplitudes is


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Fig. 1. Experimental low-energy spectra of 104Pd and 110Cd. The numbers to the right of the energy levels are excitationenergies in units of MeV.

adopted contrary to Ref. [47] so that the first excited 2+ state can be written as a ccQRPA phononin the form

2+1 M

= Q†

2+1 , M

|QRPA =

ab Z

2+1ab

a†a a

†b

2M − W

2+1ab

a†a a

†b

†2M

|QRPA, (15)

where the amplitudes Z and W are obtained by solving the ccQRPA equations of motion [47].Above |QRPA denotes the QRPA vacuum.

From the above 2+ phonons (15) one can build ideal two-phonon J + states of the formJ +2-ph= 1√ 2

Q†

2+1

Q†2+1

J =0,2,4|QRPA. (16)

An ideal two-phonon triplet consists of partner states J π = 0+, 2+, 4+ that are degenerate inenergy, and exactly at an energy twice the excitation energy of the 2+1 state. In practice thisdegeneracy is always lifted by the residual interaction between the one- and two-phonon states

[48]. In nuclei there are many candidates for two-phonon states [48,49] and examples of suchnuclei are 104Pd and 110Cd which are double-beta-decay daughter nuclei of 104Ru and 110Pd.The low-energy spectra of 104Pd and 110Cd are depicted in Fig. 1. In these cases the paradigm of a two-phonon triplet seems to work fine in terms of the small splitting of the triplet. However, thecentroids of the triplets are slightly higher than twice the excitation energy of the 2+1 state. Basedon these considerations the wave functions of the triplets of states in these nuclei are presumedto be reasonably well described by the ansatz wave function (16). Some further examples will bediscussed in Section 4.1.

Having defined the form of the one-phonon excitation (15) and the structure of the pnQRPAstate (12) one can connect them by the MCM procedure [12,46]. The corresponding general

expression for the decay amplitude can be written asI +kc†

p ̃cn

L

J πk1= 2

(2I + 1)(2L + 1)(2J + 1)(−1)I +L+J


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×

p1

ūp ̄un X̄

J πk1p1n

Z̄I +k

pp1 − v̄p ̄vn Ȳ J πk1

p1n W̄ I

+k

pp1

J I Lj p j n j p1

+n1 (−1)I

+j n

+j n1

v̄p ̄vn X̄J πk1

pn1 Z̄

I +knn1 − ūp ̄un Ȳ

J πk1

pn1 W̄

I +knn1 J I Lj n j p j n1 (17)

instead of the expression (13) for the ground-state transition. This decay amplitude has been usedin (2) to evaluate the transition amplitude

2+1

m

σ mt −m

1+k1= 1√ 3pn

pσ

n2+1 c†p ̃cn11+k1. (18)After defining the form of the two-phonon excitation (16) the MCM procedure can be used toderive the following general expression for the transition density

I +2-phc†

p ̃cn

L

J πk1= 40√

2

(2I + 1)(2L + 1)(2J + 1)(−1)J +L+1

×p1n1

ūp ̄vn X̄J

π k1p1n1

Z̄2+1pp1 Z̄

2+1nn1 + v̄p ̄un Ȳ

J π k1p1n1

W̄ 2+1pp1 W̄

2+1nn1

j p j p1 2j n j n1 2L J I

(19)

instead of the expression (13) for the 0νββ ground-state transition. In the case of the 2νββ decaythis transition density has been used to compute the amplitude

I +2-ph

m

σ mt −m

1+k1= 1√ 3pn

pσ

nI +2-phc†p ̃cn11+k1. (20)3. Model spaces and parameters

The calculations of this work were performed in reasonably large valence spaces includingseveral single-particle orbitals below and above the proton and neutron Fermi surfaces. Thesingle-particle orbitals of the present calculations belong to two categories. Firstly, most of theorbitals are bound, i.e. their single-particle energies are negative. Secondly, there are positive-

energy resonant states in the continuum, their energies determined by the angular-momentumand Coulomb barriers [47]. The size of the model space is limited from above by the appear-ance of large positive energies of the computed single-particle resonances in the positive-energycontinuum. This is why for some model spaces the 0h9/2 orbital is missing in the calculationsthus introducing a small violation of the Ikeda 3(N − Z) sum rule. However, the 0h9/2 orbitalis quite far in energy from the active valence space so that its effect on the double-beta-decayhalf-lives can be neglected. Having said this, the adopted model spaces in this work are the fol-lowing: for the mass A = 70, 86 nuclei the single-particle model space 0f–1p–0g–1d–2s–0h11/2was used and for the masses A = 94, 104 the space 0f–1p–0g–1d–2s–0h was adopted, both forprotons and neutrons. In the case of the A

= 110 nuclei the proton single-particle space was

chosen to be 0f–1p–0g–1d–2s–0h11/2 and the one for neutrons was 0f–1p–0g–1d–2s–0h–2p–1f.For the A = 124 nuclei both the proton and neutron single-particle spaces consisted of the0f–1p–0g–1d–2s–0h–2p–1f orbitals. The asymmetry of the proton and neutron models spacesfor the A = 110 nuclei stems from the fact that the proton orbitals are pushed up by the Coulomb


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term and thus their energies enter the positive-energy continuum more easily than the ones forneutrons.

At this point it should be noted that the values of the single-particle energies close to the

proton and neutron Fermi surfaces play a role in many-body calculations since they determinethe occupancies of the single-particle orbitals in the active valence space. This effect is wellknown in many-body calculations based on the BCS approach [38,39,41,43,47] and in other typeof calculations like the interacting shell-model [40].

The single-particle energies of the adopted model spaces were generated by the use of aspherical Coulomb-corrected Woods–Saxon (WS) potential with a standard parametrization [50],optimized for nuclei near the line of beta stability. The resulting single-particle energies are givenin Tables 1 and 2 for neutrons and protons respectively. Only the relevant orbitals close to theneutron and proton Fermi surfaces are listed. The resulting WS bases were used for all nucleidiscussed in this work.

Modifications of the WS energies were done for all nuclei to allow a better reproduction of the spectra of their odd-A neighbors. These bases are the adjusted bases, denoted from hereon as “Adj”. The changes of the neutron single-particle energies for the A = 70–110 nuclei areindicated in Table 1 and the changes of the proton energies are given in Table 2. Only the changedvalues of energies are listed in the rows referring to the adjusted basis (Adj). The adjusted single-particle energies of 124Sn and 124Te were taken from Table 8 of Ref. [15].

The Bonn-A G-matrix has been used as the starting point and it has been renormalized inthe standard way [9,12]: The quasiparticles are treated in the BCS formalism and the pairingmatrix elements are scaled by a common factor, separately for protons and neutrons. In practicethese factors are fitted such that the lowest quasiparticle energies obtained from the BCS match

the experimental pairing gaps for protons and neutrons respectively. The experimental excitationenergies of the low-lying states of the proton- and neutron-odd nuclei, adjacent to the even–evenreference nuclei of this work, are compared with the relative single-quasiparticle energies of theBCS in Tables 3 and 4 for the neutron-odd and proton-odd nuclei respectively. The comparisonsfor the odd A = 125 nuclei have been done in Ref. [15] and the values of the experimental andcomputed quasiparticle energies can be seen in Table 9 of that reference.

It should be noted that a pure single-quasiparticle structure has been assumed for the exper-imental states. In most of the listed cases this may be a valid assumption but for some statesthe mixing with the three-quasiparticle degrees of freedom could be expected. In particular, for105Rh the 7/2+ ground state and for 105Ag and 111Ag the first excited 7/2+ states are not de-scribable as one-quasiparticle states and most likely they are three-quasiparticle states so thattheir energies are not given in Table 4. Due to the uncertainties in the three-quasiparticle con-tributions the BCS quasiparticle energies were adjusted only very qualitatively. In fact, the finalaim of these rough adjustments was to see how the resulting increased densities of single-particlestates at the proton and neutron Fermi surfaces influence the final observables of the calculations,i.e. the beta-decay and ββ-decay rates. This also helps in placing some kind of error estimatesfor the 0νββ NMEs and half-lives. Finally, it is worth pointing out that the adjustments donein this way may produce rather strong differences in spin–orbit splittings between two adjacenteven–even nuclei. This might be considered either as an artifact of the procedure or as some kindof effective way to take account of sudden changes of nuclear deformation along isobaric chainsof nuclei.

The particle–hole and particle–particle parts of the proton–neutron two-body interaction areseparately scaled by the particle–hole parameter gph and particle–particle parameter gpp [9]. Theparticle–hole parameter affects the position of the Gamow–Teller giant resonance (GTGR) and


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Table 1Changes made to the Woods–Saxon energies (WS) of neutrons in the adjusted basis (Adj) for the nuclei listed in column one and fo3–11. All the energies are given in MeV.

Nucleus Basis Neutron orbital

1p1/2

1p3/2

2s1/2

1d3/2

1d5/2

0g7/2

70Zn WS −8.877 −10.694 −1.526 −0.026 −2.444 1.243 70Ge WS −10.116 −12.017 −2.275 −0.743 −3.463 0.215

Adj 86Kr WS −11.564 −13.161 −3.533 −2.430 −4.972 −2.683

Adj −4.5086Sr WS −12.663 −14.310 −4.336 −3.243 −5.933 −3.668

Adj −10.20 −11.2094Zr WS −13.252 −14.770 −4.946 −4.041 −6.594 −4.863 94Mo WS −14.295 −15.853 −5.758 −4.864 −7.527 −5.816

Adj −4.10 −6.90104Ru WS −14.664 −16.070 −6.270 −5.576 −8.036 −6.871

Adj −5.00 −6.40 −5.05104Pd WS −15.637 −17.074 −7.060 −6.377 −8.920 −7.773

Adj −6.00 −7.50 −6.30110Pd WS −15.255 −16.594 −6.879 −6.294 −8.675 −7.794

Adj −7.80 −8.10 −8.60110Cd WS

−16.188

−17.554

−7.650

−7.075

−9.527

−8.665

Adj −8.60 −9.00 −9.50124Sn WS −16.229 −17.424 −7.983 −7.611 −9.796 −9.458

Adj −7.37 −8.05 124Te WS −17.078 −18.294 −8.704 −8.342 −10.580 −10.260

Adj −9.08 −8.77 −10.32 −10.41


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Table 2Changes made to the Woods–Saxon energies (WS) of protons in the adjusted basis (Adj) for the nuclei listed in columnone and for the single-particle orbitals listed in columns 3–10. All the energies are given in MeV.

Nucleus Basis Proton orbital

1p1/2 1p3/2 0f 5/2 2s1/2 1d3/2 1d5/2 0g7/2 0g9/270Zn WS −5.491 −7.758 −6.542 2.520 4.181 0.974 4.942 −3.732

Adj −5.5070Ge WS −3.563 −5.685 −4.529 3.781 5.512 2.608 6.616 −1.748

Adj −3.3086Kr WS −8.292 −10.225 −9.975 0.520 1.676 −1.483 0.794 −6.89886Sr WS −6.499 −8.444 −8.175 1.932 3.104 −0.030 2.410 −5.13694Zr WS

−8.542

−10.337

−10.441 0.269 1.210

−1.836

−0.126

−7.278

94Mo WS −6.836 −8.661 −8.734 1.657 2.617 −0.415 1.428 −5.606104Ru WS −9.521 −11.193 −11.664 −0.673 0.121 −2.916 −1.806 −8.415

Adj −9.90104Pd WS −7.915 −9.592 −10.054 0.693 1.409 −1.523 −0.320 −6.841

Adj −8.30110Pd WS −10.330 −11.937 −12.593 −1.457 −0.885 −3.772 −2.965 −9.303

Adj −10.50110Cd WS −8.735 −10.382 −11.035 −0.107 0.478 −2.327 −1.525 −7.779

124Sn WS −12.418 −13.887 −14.899 −3.467 −3.183 −5.960 −5.781 −11.535Adj −4.87 −4.58 −5.98 −6.56 −9.15

124Te WS −10.980 −12.433 −13.446 −2.230 −1.908 −4.596 −4.408 −10.117Adj −4.86 −4.77 −5.82 −5.61

its value was fixed roughly by the available systematics [47] on the location of the giant state.While it is well known that the position of the GTGR is rather sensitive to the strength of theparticle–hole interaction, the amount of low-lying strength is rather sensitive to correlations that

go beyond the pnQRPA approach. Nevertheless, the value of gph still affects to some extent thelow-lying Gamow–Teller strength distribution that is relevant for the value of the 2νββ NME.Through this the effects of gph are mediated also to the final values of the 0νββ NMEs. Thecomputed GT strength distributions in the presently discussed intermediate nuclei are given inSection 4.2.

After fixing the value of the particle–hole parameter one needs to access in some way thephysical values of the particle–particle parameter gpp of the proton–neutron two-body interac-tion. Typically this has been done by fitting the value of gpp in such a way that the experimental2νββ half-lives are reproduced in the interval gA = 1.00–1.25 of the axial-vector coupling con-stant [30–33,51] where gA

= 1.00 corresponds to the quenched value and gA

= 1.25 to the

bare-nucleon value of gA. The experimental error and the uncertainty in the value of gA theninduce an interval of acceptable values of gpp, the minimum value of gpp related to gA = 1.00and the maximum value to gA = 1.25. This is because the magnitude of the calculated 2νββNME, M (2ν) , decreases with increasing value of gpp in a pnQRPA calculation [7–9] and this


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Table 3Comparison of the experimental and BCS-calculated neutron quasiparticle energies for the nuclei listed in column oneand for the single-particle orbitals listed in columns 3–10. All the energies are given in MeV. Comparisons for theneutron-odd A

= 125 nuclei have been done in Table 9 of Ref. [15].

Nucleus Source Neutron orbital

1p1/2 1p3/2 2s1/2 1d3/2 1d5/2 0g7/2 0g9/2 0h11/271Zn Exp 0.000 ? 0.157

WS 0.000 1.746 0.110

71Ge Exp 0.000 0.500 0.198WS 0.000 1.110 1.052Adj 0.000 0.854 0.328

87Kr Exp ? ? 0.532 0.000 ? ?WS 3.160 4.756 1.439 0.000 2.289 0.603

Adj 3.545 5.142 0.472 0.000 2.289 0.98987Sr Exp 0.388 0.873 ? 1.229 ? 0.000

WS 2.219 3.752 3.484 1.950 4.332 0.000Adj 0.369 1.079 3.269 1.754 4.075 0.000

95Zr Exp 0.954 ? 0.000 ? ?WS 0.881 1.821 0.000 0.971 3.137

95Mo Exp 0.821 ? 0.000 0.766 1.938WS 1.359 2.313 0.000 1.352 3.470Adj 0.862 2.511 0.000 0.863 2.056

105Ru Exp 0.159 0.000 0.021 0.230 0.207WS 0.057 0.539 0.337 0.000 1.509Adj 0.093 0.000 0.065 0.273 0.136

105Pd Exp ? 0.281 0.000 0.306 0.489WS 0.285 0.918 0.105 0.000 1.821Adj 0.284 0.196 0.000 0.226 0.442

111Pd Exp 0.072 ? 0.000 ? 0.172WS 0.000 0.158 0.782 0.225 0.848Adj 0.000 0.414 0.158 0.452 0.596

111Cd Exp 0.000 0.342 0.246 0.417 0.396

WS 0.040 0.394 0.452 0.000 1.140Adj 0.000 0.787 0.070 0.286 0.944

magnitude is compared with the magnitude of the experimental NME, M (2ν) (exp) ∝ (gA)−2[see Eq. (1)], deduced from the experimental 2νββ half-life.

For the presently discussed nuclei there are no experimental data for the 2νββ half-livescorresponding to ground-state transitions. Then a conservative upper limit for gpp is obtainedby requiring the computed value of the ground-state-to-ground-state 2νββ NME to go to zero,M (2ν)

→ 0, leading to infinite 2νββ half-life. This corresponds to the minimum magnitude of the

NME and thus to the maximum value of gpp and hence to the value gA = 1.25 of the axial-vectorcoupling constant. All the deduced upper limits of gpp are summarized in Table 5. The decay of 94Zr is a special case since the associated NME has a minimum value for gpp = 1.09 but neverreaches zero. In this case then the value gpp = 1.09 is adopted as the maximum value of gpp


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Table 4Comparison of the experimental and BCS-calculated proton quasiparticle energies for the nuclei listed in column one andfor the single-particle orbitals listed in columns 3–6. All the energies are given in MeV. Comparisons for the proton-oddA

= 125 nuclei have been done in Table 9 of Ref. [15].

Nucleus Source Proton orbital

1p1/2 1p3/2 0f 5/2 0g9/271Ga Exp 0.390 0.000 0.487 1.495

WS 1.729 0.000 0.639 3.255Adj 1.792 0.000 0.745 1.615

71As Exp ? ? 0.000 1.001WS 0.949 0.000 0.082 2.394Adj 1.111 0.000 0.295 1.106

87Rb Exp 0.845 0.000 0.403 ?

WS 0.437 0.238 0.000 1.43487Y Exp 0.000 0.980 0.793 0.381

WS 0.000 0.637 0.420 0.554

95Nb Exp 0.236 ? ? 0.000WS 0.000 1.162 1.274 0.137

95Tc Exp 0.039 0.646 0.667 0.000WS 0.373 1.607 1.798 0.000

105Rh Exp 0.130 0.392 ? 0.149WS 0.646 1.830 2.287 0.000

Adj 0.299 0.971 1.433 0.000

105Ag Exp 0.000 0.347 ? 0.053WS 0.822 2.159 2.711 0.000Adj 0.131 1.131 1.580 0.000

111In Exp 0.536 0.803 ? 0.000WS 0.883 2.338 3.045 0.000

111Ag Exp 0.000 0.290 ? 0.130WS 0.807 2.097 2.830 0.000Adj 0.221 1.232 1.871 0.000

and the associated minimum values of the NMEs in the two adopted bases are given in the lastcolumn of Table 5 along with the deduced upper limits of gpp. In Table 5 the first column givesthe mother nucleus, the second column the source of the single-particle energies, the third thefitted values of the particle–hole parameter of the ccQRPA and the fourth column the values of the particle–hole parameter of the pnQRPA extracted from the fit to the excitation energy of theGTGR. The fifth column lists the minimum values of gpp and in the parenthesis the source of derivation of this value is given. Column six of the table lists the values of the 2 νββ NMEs thatcorrespond to the minimum values of gpp.

The minimum values of gpp are obtained either by fitting roughly the experimental log f t value of an allowed β− transition 1+1 → 0+gs (decays of 70Zn, 104Ru, 110Pd) or from a reasonableconservative estimate (decays of 94Zr, 86Kr and 124Sn). As discussed earlier in each case theminimum value of gpp corresponds to gA = 1.00. The reasonable conservative estimate refers to


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Table 5Values of the model parameters adopted for the double-beta decays discussed in this article. The first column gives themother nucleus, the second the source of the single-particle energies, the third the value of the particle–hole parameterof the ccQRPA (determined from the experimental location of the 2+1 state in the ββ -decay daughter nucleus) and the

fourth the value of the particle–hole parameter of the pnQRPA. The fifth and sixth columns list the minimum values of the parameter gpp of the pnQRPA (with their sources in parenthesis) and the resulting value of the 2νββ NME. The lastcolumn lists the maximum values of gpp defined by the zero of the 2νββ NME. For more information see the text.

Nucleus Basis gph(2+) gph gminpp (source) M (2ν) (gminpp ) gmaxpp (M (2ν) ≈ 0.00)

70Zn WS 0.746 1.10 0.91 (log f t ) 0.447 1.086Adj 0.667 1.10 0.93 (log f t ) 0.509 1.084

86Kr WS 0.813 1.00 0.80 (ad hoc) 0.110 1.03Adj 0.738 1.00 0.80 (ad hoc) 0.127 0.97

94Zr WS 0.782 1.00 0.80 (ad hoc) 0.373 1.09 (M (2ν) = 0.270)

Adj 0.750 1.00 0.80 (ad hoc) 0.347 1.09 (M (2ν)

= 0.170)104Ru WS 0.675 1.10 1.00 (log f t ) 0.474 1.069

Adj 0.581 1.00 1.23 (log f t ) 0.282 1.272

110Pd WS 0.745 1.10 1.05 (log f t ) 0.245 1.097Adj 0.745 1.10 0.93 (log f t ) 0.271 1.060

124Sn WS 0.730 1.00 0.80 (ad hoc) 0.110 0.92Adj 0.540 1.00 0.80 (ad hoc) 0.192 0.907

situations where there are no experimental observables by which to estimate the minimum value

of gpp. In these cases (denoted by “ad hoc” in Table 5) a conservative lower limit gpp = 0.80has been arbitrarily chosen. The minimum values of gpp correspond to the physically allowedmaximum values of the NMEs due to the earlier mentioned smooth increase in the magnitudeof the ground-state-to-ground-state NMEs of (2) with decreasing value of gpp in a pnQRPAcalculation.

Concerning the ccQRPA calculation, the value of the particle–hole parameter gph(2+) of theeffective Hamiltonian [12] is determined by the energy of the first 2+ state, 2+1 , in the ββ -decaydaughter nucleus. The corresponding wave function is given in (15). In the present calculationsthe experimental energy of the 2+1 state is reproduced by varying the magnitude of gph(2

+), andthe resulting fitted values of gph(2+) are listed in column three of Table 5. A striking differencecan be seen between the values of the two gph parameters in Table 5. The difference stems fromthe fact that in the ccQRPA the 2+1 state of an even–even nucleus collects easily collectivity [47]and thus even a modest gph(2+) strength is enough to reproduce the experimental energy of thisstate. Contrary to this the value of the gph parameter of the pnQRPA relates to the location of the1+ GTGR in an odd–odd nucleus and subsequently has a totally different origin. Furthermore, nosuch strong collectivity of the 1+ states in odd–odd nuclei as the one of 2+ states in even–evennuclei has been observed in general.

4. Results and discussion

In this section the discussion is divided in four parts: first the properties of the single betadecays are discussed since they determine the minimum values of the gpp parameters for part of the double-beta decays. At the same time comparisons of the computed log f t values with theavailable data serve as checks of the plausibility of the present calculations. Second, the 2νββ


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Fig. 2. Experimental low-energy spectra of 70 Ge, 94 Mo and 124 Te. The numbers to the right of the energy levels areexcitation energies in units of MeV.

decays are discussed and estimates of the associated decay half-lives are derived. Third, thevalues of the 0νββ NMEs are quoted and analyzed in terms of the magnitude of the axial-vectorcoupling constant gA and in terms of the adopted forms of the nucleon–nucleon short-rangecorrelations. Finally, the adopted values of the 0νββ NMEs are quoted and the corresponding

decay half-lives are given in a form that enables easy extraction of the half-life for any neutrinomass of the reader’s choice.

4.1. Single beta decays

The available data on β−-decay rates allow for studies of the lateral beta-decay feedings of the 0+ ground states and low-lying excited 0+, 2+ and 4+ states in the final nuclei of double-betadecays considered in this article. These states are shown in Figs. 1 and 2. From these figures onesees that for 94Mo, 104Pd, 110Cd and 124Te the picture of a two-phonon triplet works rather well sothat the wave functions of the triplets of states in these nuclei are presumed to be reasonably well

described by the ansatz wave function (16). For 70Ge the two-phonon picture is not realized andthat is why only the 2+1 state is treated in the MCM calculation in the subsequent determinationof the theoretical log f t values of the lateral β− decay feeding. For 86Sr only the ground stateand the 2+1 state are accessible due to the small β

− decay Q value. Also it has to be pointed outthat here the only 0+1 states accessible via 0νββ-decay transitions are the ones in

110Cd and 124Tedue to the too small 0νββ Q values of all the other nuclei considered in this work.

In Table 6 the calculated log f t values, based on the above-described pnQRPA and MCMwave functions, have been quoted in columns six and seven. They are calculated for gA = 1.00and hence correspond to the minimum values of the gpp parameter of the pnQRPA. Columnfive lists the corresponding experimental log f t values. The transitions concern β

− or β

+/EC

decays from the J π ground states of the mother nuclei (listed in column three) to the groundor excited states of the daughter nuclei (listed in column four). The excited states are denotedas 0+1 (the first excited 0

+ state) or 2+1 and 2+2 (the first and second 2

+ states) or 4+1 (the firstexcited 4+ state). The values of the particle–hole parameter gph(2+) of the ccQRPA (see Table 5)


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Table 6Experimental and calculated log f t values (columns 5–7) for allowed and first-forbidden β− and β +EC (indicated incolumn two) transitions between the nuclei indicated in the first column. The spin-parities of the initial (ground) statesare indicated in the third column and column four lists the spin-parities of the final states of the transitions.

Transition Mode J πi J πf

log f t

Exp. WS basis Adj basis

70Ga → 70Ge β− 1+ 0+gs 5.1 5.11 5.102+1 6.0 5.53 5.89

70Ga → 70Zn β+/EC 1+ 0+gs 5.0 4.07 4.0586Rb → 86Sr β− 2− 0+gs 9.4 9.13 9.31

2+1 7.9 7.17 7.20

86

Rb → 86

Kr β+/EC 2− 0+gs 9.8 12.3 13.594Nb → 94Mo β− 3+ 2+1 6.5 6.79 6.63

4+1 7.4 6.63 6.952+2 6.8 6.25 6.47

104Rh → 104Pd β− 1+ 0+gs 4.5 4.32 4.332+1 5.8 6.50 6.040+1 7.4 5.08 5.952+2 8.7 6.85 8.43

104Rh →

104Ru β+

/EC 1+

0+gs

4.4 3.73 3.95

2+1 5.4 5.35 5.500+1 5.1 4.63 4.87

110Ag → 110Cd β− 1+ 0+gs 4.7 4.84 4.722+1 5.5 5.46 5.570+1 6.8 5.02 5.222+2 7.2 7.52 7.28

110Ag → 110Pd β+/EC 1+ 0+gs 4.1 3.72 3.79124Sb

→ 124Te β− 3− 2+1 10.3 7.18 7.73

4+1 10.7 8.01 8.432+2 10.2 8.25 8.50

are obtained by fitting the experimental energy of the first 2+ state in the beta-decay daughternuclei. For the transitions 104Rh → 104Ru the values of the particle–hole parameters in 104Ru aregph(2+) = 0.665 (WS) and gph(2+) = 0.555 (Adj). The resulting wave function of the 2+1 stateis then used to construct the two-phonon states (16) and the related transition amplitudes in theMCM formalism.

All the transitions considered in Table 6 pertain to allowed or first-forbidden decays. For theallowed decays of this work the log f t value is defined as [47]

log f t = log(f 0t 1/2) = log

6147

BGT + BF

, (21)


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where the reduced Gamow–Teller and Fermi transition probabilities are defined as

BGT =g2A

2J i +

1J +f σ t

±

J +i

2, BF =

g2V

2J i +

1J +f 1t

±

J +i

2(22)

for the initial J +i and final J +

f states. It should be noted that for the Fermi transitions we alwayshave J f = J i . For the first-forbidden unique (FFU) transitions 2− → 0+ (the transition 86 Rb →86Sr being the unique example in this work) we can define [47]

log f t = log(f 1ut 1/2) = log

61471

12 B1u

, B1u =

g2A

2J i + 1M

21u, (23)

where

M1u =

mec2

√ 4π0+[σ r]2t ±2− (24)

for the initial 2− and final 0+ states. For the first-forbidden non-unique transitions J −i → J +f ,|J i − J f | 1, of this work we can define [47]

log f t = log(f 0t 1/2) = log

f 06147

S (−)1

, (25)

where the shape function S (−)1 can be inferred from [12].Interesting conclusions can be drawn based on Table 6: it is well visible that the adjustments

in the single-particle energies (see Section 3) improve the description of the beta-decay rates,sometimes considerably. This is a nontrivial consequence of the adjustments since they weredone to improve the quality of the energy spectra of the odd-A nuclei adjacent to the even–evenreference nuclei, a procedure having little to do with beta decays between an odd–odd and aneven–even nucleus in the region next to the reference nucleus. The computed log f t values for theβ− decays are mostly in fair agreement with the experimental ones, except for the decay of 124Sbwhere the predicted partial decay rates are clearly too fast. In all the other nuclei the β− decayto the 2+1 one-ccQRPA-phonon state is quite well described by the MCM in the adjusted basis.Also the decays to the 2+2 members of the two-phonon triplets are reasonably well reproducedby the adjusted-basis MCM calculations. Computed β− decay rates to the 0+1 members of the

two-phonon triplets are systematically too fast in both bases. This could have some consequencesfor the double-beta-decay feeding of the 0+1 state in

110Cd.The calculation of the β+/EC decay of 70Ga to the ground state of 70Zn produces too strong a

transition rate as seen in Table 6. This is a typical feature of the pnQRPA calculations as discussedin [44]. For the β+/EC decay of 86Rb the calculations in both bases produce a far too suppresseddecay rate. This, in turn, could have an effect on the 0νββ decay rate. For the β +/EC decays of 104Rh and 110Ag the adjusted basis works reasonably well, a bit better than the WS basis.

4.2. Gamow–Teller strength distributions in the intermediate nuclei

After fixing the model parameters gph and gpp of the pnQRPA one can calculate the Gamow–Teller strength distributions in the intermediate nuclei of the ββ decays. The GT strength in atransition from the 0+ ground state of an even–even nucleus to the m-th 1+ state in the neigh-boring odd–odd nucleus is given by


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GT−(m) =1+mσ t −

0+gs2 β− strength, (26)GT+(m) =

1+m

σ t +

0+gs

2

β+ strength

. (27)

The GT−(m) strength corresponds to transitions from the ground state of the ββ-decay parentnucleus to the intermediate nucleus and the GT+(m) strength relates to transitions from theground state of the ββ -decay daughter nucleus to the intermediate nucleus.

In Table 7 the GT− and GT+ strengths are given as summed over energy bins of 2 MeV.For example, the 70Ga nucleus is fed by GT− from 70Zn (ββ parent) and by GT+ from 70Ge(ββ daughter). The final column lists the total GT− and GT+ strengths. It should be noted herethat the total strengths do not satisfy the Ikeda 3(N − Z) sum rule since the involved initialground states are not the one and the same ground state as required by the sum rule. In thetable the energy of the first 1+ state in the intermediate nucleus is scaled to zero. Further-more, the value gA = 1.25 of the axial-vector coupling constant has been used to derive thestrength distributions. As explained before the gpp parameter is connected to the value of gA.Thus the strength distributions with gA = 1.25 and gA = 1.00 are slightly different. However,Table 7 gives a good idea how the pnQRPA-computed strengths look like, at least qualita-tively.

From Table 7 it is obvious that in the GTGR region the GT− strength is redistributed whengoing from the WS to the adjusted basis, the adjustment typically increasing the fragmentation of the strength. For the low-energy 1+ states the redistribution is more gentle. It may be noted thatfor 94Nb the GT− strengths are the same for the WS and Adjusted bases since for the involved ββparent nucleus 94Zr actually only the WS basis has been used in the calculations. The total GT+strength is small but sometimes quite different for the two basis sets used. Also here redistribution

of strength takes place when going from the WS to the adjusted basis.For 124Sb one can compare the present results with the ones computed by the use of the

energy-density-functional (EDF) method of [28]. There the sum strengths S +(GT) = 1.63 andS −(GT) = 40.65 were obtained by the use of the quenching factor (0.74)2 for the Gamow–Teller operator. Adopting the same quenching factor produces the values S +(GT) = 0.26–0.38and S −(GT) = 39.6 in the present calculations. The S −(GT) strengths of the present and theEDF calculations agree well but for the S +(GT) strength the present calculations yield a muchsmaller value.

4.3. Two-neutrino double-beta decays

For the double-beta-decaying nuclei of this work not many previous calculations of the 2νββ-decay rates exist. In particular the studies of decays to excited final states are scarce. The availableprevious calculations have been listed in the last column of Table 8. They are based on thepnQRPA [15,52], higher-RPA theory [15,21], boson expansion methods [22,53], the single-state-dominance hypothesis [54] or the interacting shell model [55].

In Table 8 are given the presently calculated minimum half-lives in years (second last column)and the corresponding NMEs (third last column) for the 2νββ decays of the nuclei listed in thefirst column. For the ground-state transitions the NME is the maximum NME and it correspondsto the minimum value gmin

pp of the g

pp parameter obtained for g

A = 1.00 and listed in Table 5.

Also given are the spin-parity of the final state (second column), the decay Q values in MeV(third column) and the corresponding 2νββ phase-space factor in units of inverse years (fourthcolumn). The phase-space factor and the corresponding half-life have been calculated for thevalue of gA indicated in parenthesis below the value of the phase-space factor. As mentioned


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Table 7Calculated strength distributions GT− and GT+ in the nuclei listed in the first column of the table. Both the WS and the adjusted strength distribution is given in bins of 2 MeV width for the value gA = 1.25 of the axial-vector coupling constant. The total summeenergies of the first 1+ states in the listed nuclei have been scaled to zero.

Nucl. Basis Mode Energy interval in MeV

0–2 2–4 4–6 6–8 8–10 10–12 12–14 1470Ga WS GT− 2.05 1.90 3.06 0.00 1.99 20.70 0.40

GT+ 0.05 0.09 1.01 0.00 0.08 0.00 0.12 Adj GT− 2.18 1.72 3.16 0.00 15.50 7.12 0.47

GT+ 0.07 0.09 1.31 0.00 0.16 0.00 0.26 86Rb WS GT− 1.85 3.80 5.14 0.03 0.00 29.26 2.04

GT+ 0.00 0.38 0.00 0.13 0.04 0.00 0.00 Adj GT− 1.75 0.99 7.94 0.03 0.00 0.00 31.44

GT+ 0.10 0.06 0.34 0.01 0.11 0.01 0.02 94Nb WS GT

− 1.31 0.00 0.79 5.89 2.12 2.52 0.08 2

GT+ 1.00 0.13 0.53 0.07 0.01 0.03 0.01 Adj GT− 1.31 0.00 0.79 5.89 2.12 2.52 0.08 2

GT+ 0.56 0.00 0.61 0.07 0.03 0.02 0.03 104Rh WS GT− 4.78 0.00 0.69 2.24 2.56 4.49 0.10 24

GT+ 0.00 0.00 0.22 0.37 0.22 0.02 0.00 Adj GT− 2.46 0.00 0.00 2.78 4.04 7.18 0.11

GT+ 0.03 0.00 0.30 1.08 0.51 0.10 0.01 110Ag WS GT− 3.94 0.00 1.50 2.71 6.80 0.39 9.55 24

GT+ 0.01 0.00 0.11 0.07 0.22 0.02 0.05 Adj GT

− 3.58 0.00 2.05 1.72 6.97 0.52 15.29 2

GT+ 0.01 0.00 0.08 0.20 0.00 0.02 0.04 124Sb WS GT− 3.18 4.97 5.38 10.15 1.73 46.14 0.16

GT+ 0.00 0.17 0.06 0.00 0.01 0.18 0.00 Adj GT− 3.99 1.53 5.08 4.75 11.01 0.05 29.86 14

GT+ 0.09 0.25 0.01 0.06 0.17 0.01 0.06


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Table 8Calculated minimum half-lives (the second last column) and the corresponding NMEs (third last column) for the 2νββdecays of the nuclei listed in the first column. Also given are the spin-parity of the final state (second column), thedecay Q values (third column) and the corresponding 2νββ phase-space factor (fourth column). The phase space and

the corresponding half-life have been calculated for the value of gA indicated in parenthesis below the value of thephase-space factor. Column five lists the adopted single-particle bases and the last column the available results of otherworks.

Nucleus J πf Qββ (MeV) G(2ν) (yr−1) Basis M (2ν) t (2ν)1/2 (min) (yr) t

(2ν)1/2 (yr) [Ref.]

70Zn 0+gs 1.001 1.24 × 10−22 WS 0.447 4.0 × 1022 (0.25–64) × 1022 [21](gA = 1.00) Adj 0.509 3.1 × 1022 70 × 1022 [54]

86Kr 0+gs 1.256 1.29 × 10−21 WS 0.110 6.4 × 1022 (22–990) × 1022 [21](gA = 1.00) Adj 0.127 4.8 × 1022

2+1 0.179 1.38 × 10−32 WS −0.00170 2.5 × 1037(g

A = 1.00) Adj

−0.00382 5.0

×1036

94Zr 0+gs 1.144 8.82 × 10−22 WS 0.373 8.2 × 1021 (31–6600) × 1021 [21](gA = 1.00) Adj 0.347 9.4 × 1021

2+1 0.273 5.81 × 10−30 WS 0.0170 6.0 × 1032(gA = 1.25) Adj 0.0155 7.2 × 1032

104Ru 0+gs 1.299 3.47 × 10−21 WS 0.474 1.3 × 1021 6.4 × 1021 [54](gA = 1.00) Adj 0.282 3.6 × 1021

2+1 0.743 8.52 × 10−25 WS 0.00792 1.9 × 1028 18 × 1028 [54](gA = 1.25) Adj 0.00811 1.8 × 1028 6.2 × 1028 [53]

110Pd 0+gs 2.013 1.50×

10−19 WS 0.245 1.1×

1020 (1.2–1.8)×

1020 [52](gA = 1.00) Adj 0.271 0.91 × 1020 1.6 × 1020 [22]

1.2 × 1020 [54]2+1 1.355 1.30 × 10−21 WS −0.0112 0.62 × 1025 8.4 × 1025 [22]

(gA = 1.25) Adj 0.00766 1.3 × 1025 4.4 × 1025 [54]1.5 × 1025 [53]

0+1 0.540 1.19 × 10−23 WS −0.447 4.2 × 1023 2400 × 1023 [54](gA = 1.25) Adj −0.304 9.1 × 1023

2+2 0.537 2.36 × 10−26 WS 0.00195 11 × 1030 38 × 1030 [54](gA = 1.25) Adj 0.00240 7.4 × 1030

124Sn 0+gs 2.287 6.26×

10−19 WS 0.110 1.3×

1020 0.78

×1020 [15]

(gA = 1.00) Adj 0.192 0.43 × 1020 2.9 × 1020 [55]2+1 1.684 2.61 × 10−20 WS 0.00890 4.8 × 1023 6500 × 1023 [15]

(gA = 1.25) Adj −0.00294 44 × 10232+2 0.961 3.10 × 10−23 WS 0.00360 2.5 × 1027 1.7 × 1027 [15]

(gA = 1.25) Adj −0.00294 1.4 × 10270+1 0.630 5.49 × 10−23 WS −0.455 8.8 × 1022 55 × 1022 [15]

(gA = 1.25) Adj −0.441 9.4 × 1022

before, for the ground-state transitions gA =

1.00 but usually not for the excited-state transitionssince the NMEs corresponding to excited-state transitions can either increase or decrease as func-tions of gpp, as shown in Ref. [15]. In fact most of the excited-state transitions are the strongestfor gA = 1.25, mainly because the high value of gA enhances the decay rate since the 2νββphase-space factor scales as G(2ν) ∝ g4A. Column five lists the adopted sets of single-particle en-


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ergies used to calculate the NMEs of column six. The last column lists the available results of other works.

From Table 8 one notices that the results of other calculations do not contradict the lower lim-

its of half-lives obtained in this work, although in some cases they are quite close to the presentlyobtained lower limits. For the transitions to excited states no results of other calculations couldbe found for the decays of 86Kr and 94Zr. The excited states fed by the 2νββ transitions from70Zn, 94Zr, 104Ru, 110Pd and 124Sn are depicted in Figs. 1 and 2. Concerning the ground-statedecays Table 8 indicates that the presently computed lower limits for the 2νββ half-lives of 110Pdand 124Sn are of the order of 1020 years whereas for the rest of the nuclei they are in the range(1.3–64) × 1021 years.

It is worth pointing out that the gpp dependence of the NMEs corresponding to 2νββ decaysto excited 0+ and 2+ states is rather weak in the MCM formalism [15] so that the increase of

the magnitude of gpp from gmin

pp does not affect the magnitudes of the computed NMEs dras-tically. The reason for this difference from the ground-state transitions is seen from the decayamplitudes (13), (17) and (19). In the case of the ground-state transitions, Eq. (13), the ground-state correlations, mediated by the (backward-going) Y amplitudes, can affect strongly the decayamplitude through the favorable combination up vn of the BCS amplitudes, as first discussedin [7]. The magnitudes of the Y amplitudes are always much smaller than the magnitudes of the(forward-going) X amplitudes due to the small-oscillations nature of the RPA theory [47]. Themagnitudes of the Y amplitudes increase with the increasing value of the gpp parameter and thuswith the increasing ground-state correlations. Hence with favorable BCS occupations the Y termin (13) can even cancel the X term leading to an infinite 2νββ half-life. For the decays to theone-phonon and two-phonon states, Eqs. (17) and (19), the conglomerates of the BCS occupa-tions are unfavorable for the Y amplitude terms, and in addition, the Y terms are of the formY 2 or Y 3 for the one-phonon and two-phonon amplitudes respectively, thus further suppressingthe Y terms. Due to these reasons the effect of the increasing ground-state correlations, with theincreasing value of gpp, is weak for the excited-state decays.

For the decays of 110Pd and 124Sn to the 2+1 states the computed lower limits of half-livesrange between (4.8–130) × 1023 years, where the suppression comes mainly from the very smallNMEs. For the 2+1 -decays of the other nuclei the predicted half-lives are extremely long, above1.8

× 1028 years, the suppression being mostly due to the tiny Q values. For decays to the 0+1

states the computed range of lower limits of the half-life is (8.8–91)×1022 years, the suppressionstemming from the smallness of the involved Q values. The decays to the 2+2 states are heavilysuppressed by the tiny NMEs and small Q values: t (2ν)1/2 (min) > 1.4 × 1027 years.

From the experimental side the decay transitions of Table 8 are rather scarcely studied and thefew obtained lower limits are not very sensitive. For the decay of 70Zn a lower limit of 1.3× 1016years was obtained in [56]. For the decay of 86Kr no experimental data could be found. For 104Ruthere is a measured lower limit 3.5 × 1019 years for the decay to the 2+1 state [57]. For the 2νββdecays of 94Zr the measured lower limits are weak, 1.3 × 1019 years for the 2+1 transition [58].For the decays of 110Pd no recent measurements have been done but for the decays of 124Sn

some recent measurements could be found, like [59,60]. The following best lower limits for thecombined 2νββ and 0νββ decays of 124Sn were obtained in [60]: 9.1× 1020 years (decay to 2+1 ),1.1×1021 years (decay to 0+1 ) and 9.4×1020 years (decay to 2+2 ). As can be seen, all these limitsare still far from the predicted theoretical ones.


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Table 9Calculated values of the NMEs of Eq. (6) with the Jastrow (columns 4–6) and UCOM (columns 7–9) short-range cor-relations for the ground-state-to-ground-state 0νββ decays of the nuclei listed in the first column. The second columngives the sets of adopted single-particle energies and the third column the values of the axial-vector coupling constant.

Nucleus Basis gA Jastrow UCOM

M (0ν)F M

(0ν)GT M

(0ν) M (0ν)F M (0ν)GT M

(0ν)

70Zn WS 1.00 −1.244 3.338 2.933 −1.467 4.007 3.5041.25 −0.767 1.260 1.751 −0.975 1.850 2.474

Adj 1.00 −1.167 3.438 2.947 −1.413 4.180 3.5801.25 −0.558 1.079 1.437 −0.789 1.732 2.237

86Kr WS 1.00 −1.182 2.689 2.477 −1.382 3.305 3.0001.25 −0.819 1.679 2.203 −1.008 2.241 2.886

Adj 1.00 −1.210 2.857 2.603 −1.411 3.470 3.1241.25 −0.898 1.670 2.245 −1.087 2.229 2.925

94Zr WS 1.00 −1.356 3.752 3.269 −1.632 4.598 3.9881.25 −0.934 2.281 2.879 −1.192 3.039 3.802

Adj 1.00 −1.299 3.541 3.098 −1.571 4.372 3.8041.25 −0.884 1.774 2.340 −1.137 2.517 3.245

104Ru WS 1.00 −1.789 4.448 3.992 −2.212 5.738 5.0881.25 −1.617 2.291 3.325 −2.028 3.533 4.831

Adj 1.00 −1.265 1.950 2.057 −1.690 3.161 3.1041.25 −1.139 0.478 1.207 −1.555 1.630 2.625

110Pd WS 1.00 −1.783 3.886 3.628 −2.178 5.085 4.6491.25 −1.691 2.777 3.859 −2.080 3.951 5.282

Adj 1.00 −2.146 4.899 4.508 −2.576 6.220 5.6301.25 −1.897 3.232 4.446 −2.314 4.488 5.969

124Sn WS 1.00 −1.278 2.759 2.583 −1.553 3.607 3.3021.25 −1.073 1.968 2.655 −1.339 2.774 3.631

Adj 1.00 −2.154 4.372 4.177 −2.592 5.732 5.3271.25 −1.839 2.775 3.952 −2.263 4.064 5.513

4.4. General features of the discussed neutrinoless double-beta decays

In Table 9 are listed the calculated NMEs of Eq. (6) that correspond to 0νββ decays toground states of the final nuclei. The Jastrow-correlated NMEs are listed in columns 4–6 andthe UCOM-correlated ones in columns 7–9. The second column lists the origins of the single-particle energies and the third column gives the values of the axial-vector coupling constant.

From Table 9 one can see that the UCOM correlations have a much softer effect on the NMEsM (0ν)

than the Jastrow correlations (less reduction from the uncorrelated NMEs), as first pointed

out in [30]. The differences between the results of the Woods–Saxon (WS) and adjusted (Adj)single-particle energies stem from the differences in the occupancies of the individual orbitsclose to the proton and neutron Fermi surfaces. These occupancy effects have been studied indetail in [41] for the ground-state transitions of a set of more frequently discussed 0νββ decays.In the present calculations the differences between the WS- and Adj-based NMEs are large for104Ru and 124Sn, the WS-calculated NMEs being much larger (some 40–60%) for 104Ru and


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Fig. 3. Calculated NMEs M (0ν) for the discussed 0νββ decays. The upper panel shows the NMEs separately for gA =

1.25 and gA = 1.00, whereas the lower panel displays the NMEs separately for the UCOM and Jastrow short-rangecorrelations. For 110Pd and 124Sn the ground-state (gs) and excited-state (0+1 ) NMEs are shown separately.

some 50–60% smaller for 124Sn than those of the adjusted basis. For 94Zr the WS-calculatedNMEs are some 5–20% larger than the ones calculated in the adjusted basis and for 110Pd the

NMEs of the adjusted basis are some 10–25% larger than the WS-calculated ones. For the rest of the cases the magnitudes of the NMEs do not depend much on the adopted set of single-particleenergies.

The dependence of the magnitudes of the NMEs M (0ν) on the value of the axial-vector cou-

pling constant gA is analyzed in the upper panel of Fig. 3. There one can see that only for 70Zn themagnitudes of the ground-state NMEs depend drastically on the value of gA. As explained be-fore the gA dependence of the NMEs arises directly from the gA dependence of M (0ν)

in (6) and

indirectly since the values of the particle–particle parameter gpp and gA are related as explainedin Section 3.

In the lower panel of Fig. 3 the magnitudes of the NMEs M (0ν) are given separately for the

UCOM and Jastrow short-range correlations. There the drastic difference between these two setsof NMEs for the ground-state decays is highlighted. The effects of the UCOM correlations areclose to the ones obtained in a self-consistent coupled-cluster method [35] and thus only theUCOM-correlated NMEs are used in the further analyzes pursued in this paper.


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Table 10Calculated values of the NMEs of Eq. (6) with the Jastrow (columns 4–6) and UCOM (columns 7–9) short-range corre-lations for the 0νββ decays of 110Pd and 124Sn to the 0+1 states in

110Cd and 124Te. The second column gives the setsof adopted single-particle energies and the third column the values of the axial-vector coupling constant.

Nucleus Basis gA Jastrow UCOM

M (0ν)F M

(0ν)GT M

(0ν) M (0ν)F M (0ν)GT M

(0ν)

110Pd WS 1.00 −0.309 1.302 1.031 −0.322 1.340 1.0641.25 −0.306 1.538 1.733 −0.319 1.577 1.781

Adj 1.00 −0.336 1.161 0.958 −0.350 1.201 0.9931.25 −0.331 1.229 1.441 −0.345 1.268 1.488

124Sn WS 1.00 −1.709 4.480 3.961 −1.778 4.699 4.1451.25 −1.713 4.512 5.608 −1.781 4.732 5.872

Adj 1.00 −1.864 4.606 4.141 −1.930 4.816 4.3181.25

−1.858 4.688 5.877

−1.924 4.898 6.130

In Table 10 the same information as in Table 9 is given for the 0νββ decays of 110Pd and124Sn to the two-phonon 0+1 states in

110 Cd and 124Te. Here the differences between the resultsof the Woods–Saxon (WS) and adjusted (Adj) bases are rather small, less than 20% for 110Pdand less than 5% for 124Sn. In these cases the effect of the different gA values is considerable butthe effects caused by the different short-range correlations are very small as also well evident inFig. 3. These drastically different behaviors of the ground-state decays and excited-state decayswith respect to the influence of the short-range correlations were already pointed out in the stud-ies of Refs. [42] and [43]. In [42,43] it was noticed that the relative reductions by the UCOM

and Jastrow correlations are roughly the same for both types of transition, whereas the absolutereductions are much less and more uniform for the excited-state transitions. This observation isconfirmed by the present study.

4.5. Final results for the 0νββ NMEs and decay half-lives

In this section the final values of the calculated NMEs M (0ν) and the associated predicted

half-lives are summarized. Here only the UCOM-correlated NMEs are considered since theyare expected to be more realistic than the Jastrow-correlated ones [30,35]. The final values of

the NMEs are listed in Table 11 in columns two and three. There the values of the NMEs areseparately given for the two extreme values of gA = 1.00 (quenched) and gA = 1.254 (barenucleon) by taking the arithmetic means

M̄ = 1N

k

M (0ν)k

(28)

of the NMEs over the different single-particle sets (WS and Adj) used in the calculations. Alsothe corresponding standard deviations

M

= 1

N − 1

k ¯M − M

(0ν)k

2 (29)are given. In the above expressions N denotes the number of the NMEs.

The presently computed NMEs are compared with the available PHFB NMEs of Ref. [27] incolumns four and five of Table 11, as also with the available interacting-shell-model NMEs of


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Table 11Mean values and standard deviations of the computed NMEs M (0ν)

for the two extreme values of the axial-vector

coupling constant and for the UCOM short-range correlations (columns two and three). The PHFB results of [27], theISM results of [25] and the EDF results of [28] have been included for comparison.

Nucleus M̄ ± M (UCOM) M̄ ±M [27] M (ISM) M (EDF)gA = 1.00 gA = 1.25 gA = 1.00 gA = 1.254 gA = 1.25 gA = 1.25

70Zn 3.54 ± 0.05 2.36 ± 0.1786Kr 3.06 ± 0.09 2.91 ± 0.0394Zr 3.90 ± 0.13 3.52 ± 0.39 3.10 ± 0.18 4.45 ± 0.25104Ru 4.10 ± 1.40 3.73 ± 1.56110Pd (0+gs) 5.14 ± 0.69 5.63 ± 0.49 5.75 ± 0.45 8.23 ± 0.62110Pd (0+1 ) 1.03 ± 0.05 1.63 ± 0.21124Sn (0+gs) 4.31

±1.43 4.57

±1.33 2.10 4.81

124Sn (0+1 ) 4.23 ± 0.12 6.00 ± 0.18 0.80

Ref. [25] (column six) and the recently published NMEs computed by using the energy densityfunctional (EDF) approach in closure approximation [28] (column seven). Here the original re-sults of [27] have been scaled to correspond to the definition of the NME M (0ν)

in (6). As can

be seen the present results and the results of [27] are quite nicely correlated and the NMEs haveroughly the same magnitude in both calculations for the ground-state-to-ground-state decays of 94Zr and 110Pd. For 124Sn the EDF results of [28] are consistent with the present calculations butthe ISM-computed NMEs deviate notably from the presently computed, in particular for the de-

cay to the first excited 0+ state. It may also be pointed out that for 70Zn and 86Kr the uncertaintiesin M (0ν)

stemming from the use of different bases is quite small whereas for 104Ru and 124Sn

(for the ground-state transition) the uncertainties are large. The bases effects on the ground-stateNME of 110Pd are moderate. For all the excited-state NMEs the uncertainties induced by the useof different bases are quite modest.

One can further combine the result of (4) with the values of the NMEs listed in Table 11 togive a useful summary of the computed half-lives in the form

t (0ν)1/2 =

C(0ν)

(|mν|[eV])2 × 1024 yr. (30)

The computed ranges of the factor C(0ν) are listed in the second last column of Table 12 andthe effective neutrino mass should be given in units of eV in the above equation. In Table 12the mother nuclei are listed in the first column, the spin-parities of the final states in the secondcolumn and the calculated values of the NMEs (NMEs of columns two and three of Table 11combined) with the UCOM short-range correlations in the third column. The phase-space factorspresent in (4) are given in column four in units of inverse years. They are calculated for gA = 1.25as required by the specific definition of the NME M (0ν)

in (6).

In the last column of Table 12 there are listed the values of C(0ν) emerging from the avail-able numerical results of other works. It can be seen that for the ground-state decays of 94Zr and110

Pd the present values of C (0ν)

coincide nicely with those of [27]. For the decays of

124

Sn thepresent results and the other two available results [16,20] are roughly compatible. At this point itshould be noted that in [16,20] the computational scheme was based on the relativistic harmonicconfinement model (RHCM) of quarks and the resulting nucleon form factors [61–63]. In thisframework a folding of the free nucleon current with the confined quark degrees of freedom was


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Table 12Calculated values of the NMEs M (0ν)

with the UCOM short-range correlations (third column) and the corresponding

auxiliary factors of Eq. (30) (second last column). The mother nuclei are listed in column one and the spin-parities of thefinal states in column two. Also the associated phase-space factors, calculated for gA

= 1.25, are given in units of inverse

years in column four. The final column lists available results of other works.

Nucleus Final state M̄ ±M (UCOM) G(0ν) (yr−1) C(0ν) C(0ν) [Ref.]70Zn 0+gs 2.95 ± 0.69 2.34 × 10−27 49.1+34.6−16.886Kr 0+gs 2.98 ± 0.10 6.37 × 10−27 17.7 ± 1.294Zr 0+gs 3.71 ± 0.32 6.37 × 10−27 11.4+2.3−1.7 11.0

+6.7−3.5 [27]

104Ru 0+gs 3.91 ± 1.23 1.20 × 10−26 5.45+6.15−2.30110Pd 0+gs 5.38 ± 0.56 5.38 × 10−26 0.64+0.16−0.11 0.38

+1.24−0.12 [27]

0+1 1.33 ± 0.37 1.40 × 10−27 404+371−157

124Sn 0+gs 4.44 ± 1.14 1.05 × 10−25 0.48+0.39−

0.18 0.46–1.14 [16]

0+1 5.12 ± 1.03 2.65 × 10−27 14.4+8.2−4.4 70.2 [20]

done resulting in a nucleonic current that differed from that of the standard formulation [3,64]. Inaddition, no short-range correlations were taken into account beyond the RHCM-predicted nu-cleon form factors. This is why direct comparison of the present results, obtained in the standardformulation, and the results of Refs. [16,20] is not straightforward.

From Table 12 one observes that the shortest half-lives are expected for the ground-statedecays of 110Pd and 124Sn, of the order of 1026 years for |mν| = 0.1 eV. For the rest of thedecays at least an order of magnitude longer half-lives are expected, the longest being related tothe decays of 70Zn and 110Pd to the ground state and 0+1 state, respectively.

From the experimental side not much is known about the presently discussed 0νββ-decaytransitions. The only available recent measurement concerns the decay of 124Sn [65] with thehalf-life limit t (0ν)1/2 > 2.0 × 1019 years for the ground-state transition.

5. Summary

In this work the double-beta decays, both 2νββ and 0νββ, of the nuclei 70Zn, 86Kr, 94Zr,104Ru, 110Pd and 124Sn are investigated in a fully microscopic pnQRPA

+ MCM many-body

formalism suitable for description of decay transitions to both the final ground state and ex-cited states. The chosen nuclei are very seldom studied but still have double-beta Q valuesabove 1.0 MeV. The value of the proton–neutron particle–particle interaction constant gpp wasdetermined by available beta-decay data, zeros of the 2νββ nuclear matrix elements or by rea-sonable estimates. Lower limits of the 2νββ-decay half-lives were derived. For the 0νββ decaysan easy-to-use expression for the half-life estimates is provided. The results for the nuclear ma-trix elements and decay half-lives were compared with the results of other works where everpossible.

Acknowledgements

This work was supported by the Academy of Finland under the Finnish Center of ExcellenceProgram 2006–2011 (Nuclear and Accelerator Based Program at JYFL). The contribution of Dr. Mika Mustonen in drawing the figures of this article is gratefully acknowledged.


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