Critical view on double-beta decay matrix elements within Quasi Random Phase Approximation-based methods

Nuclear Physics A 694 (2001) 269–294 www.elsevier.com/locate/npe

Critical view on double-beta decay matrix elements within Quasi Random Phase Approximation-based methods S. Stoica a , H.V. Klapdor-Kleingrothaus b a Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, PO Box MG-6,

76900-Bucharest, Romania b Max-Planck-Institut fur ¨ Kernphysik, W-6900 Heidelberg, Germany

Received 11 January 2001; revised 12 March 2001; accepted 23 March 2001

Abstract A systematic study of the two-neutrino and neutrinoless double-beta decay matrix elements (ME) for the nuclei with A = 76, 82, 96, 100, 116, 128, 130 and 136 is done. The calculations are performed with four different quasi random phase approximation (QRPA)-based methods, i.e. the proton–neutron QRPA (pnQRPA), the renormalized proton–neutron QRPA (pnRQRPA), the fullRQRPA and the second-QRPA (SQRPA). First we checked the conservation of the Ikeda sum rule (ISR) and found that it is fulfilled with a good accuracy for the SQRPA, while for the pnRQRPA and full-RQRPA the deviations are up to 17%. Then, we studied the dependence of the ME on the single-particle (s.p.) basis. For that we performed the calculations using the same set of parameters and two different s.p. basis. For the two-neutrino decay mode the ME manifest generally the largest sensitivity to the choice of the basis when they are calculated with the pnQRPA, while the smallest sensitivity is got with the SQRPA. For all the methods the largest differences between the results were found for 128,130Te and 136 Xe. For the neutrinoless decay mode the ME display generally a stronger dependence on the basis than for the two-neutrino decay mode, when they are calculated with the pnQRPA, RQRPA and full-RQRPA, while for SQRPA differences in the results are within 30%. A better stability against the change of the s.p. basis used and a good fulfillment of the ISR allow to reduce the uncertainties in the values of the neutrinoless ME predicted by the QRPA-based methods to about 50% from their magnitude. Further, we fixed the values of gpp from the two-neutrino calculations and according to recent experimental data, and then we used them to compute the ME for the neutrinoless decay mode. Taking the most recent experimental limits for the neutrinoless half-lives, we deduce new upper limits for the neutrino mass parameter. Finally, there are estimated, for each nucleus, scales for the neutrinoless double-beta decay half-lives that the experiments should reach for exploring neutrino masses around 0.1 eV. This might guide the experimentalists in planning their setups.  2001 Elsevier Science B.V. All rights reserved. PACS: 21.60.Jz; 23.40.Hc; 23.40.Bw

E-mail address: [email protected] (S. Stoica). 0375-9474/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 0 9 8 8 - 5

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1. Introduction Investigations on the nuclear double-beta decay are still actual due to the large potential of this process to provide us with information about challenging issues like the lepton number nonconservation, massive neutrinos, the existence of right-handed currents in the weak interaction, etc., which appear as basic hypothesis in the theories beyond the Standard Model (SM). In the theoretical estimations of the ββ decay half-lives the evaluation of the involved nuclear ME represents one source of uncertainty. Since the nuclei undergoing such a decay are generally open-shell nuclei, for their calculation the QRPA-based methods [1–29] have been the most employed. The pnQRPA [1] was the first method widely used for calculations of nuclear chargeexchanging processes. However, its success in explaining the suppression mechanism of the two-neutrino double beta (2νββ) decay ME was achieved later on, by the inclusion of the particle–particle correlations [6–11]. However, this inclusion leads to a strong sensitivity of the ME on the particle–particle component of the pn residual interaction. Namely, the 2νββ decay ME as functions of the particle–particle interaction strength (usually denoted by gpp ) decrease rapidly and change sign, within a very narrow interval of values of gpp and this causes difficulties for fixing this parameter adequately. Trying to overcome this drawback several improvements of this method have been proposed during the last decade: the double commutator method [14,17], the appropriate treatment of the particle-number nonconservation [15,16], the inclusion of the pn pairing [22,23], etc. However, more successful have been the extensions of the pnQRPA beyond the quasiboson approximation (QBA) developed in Refs. [13,20,24–26,29]. Their main achievement is that the ME become more stable against gpp and the instability of the pnQRPA is shifted towards the region of unphysical values of this parameter. The first method including higher-order corrections to pnQRPA was developed in Ref. [13] and applied to the evaluation of the 2νββ decay ME of 82 Se. In this approach the pnQRPA phonon operator and the transition β± operators were expressed as boson expansions of appropriate pair operators and there were kept the next-order terms from these series beyond the QBA. Then, this method, called SQRPA, has been employed, with some extensions, for similar calculations of other isotopes, for deformed nuclei, as well as for transitions to excited states in Refs. [19,21,30]. An alternative approach for extending pnQRPA is based on the idea of partial restoration of the Pauli exclusion principle by taking into account the next terms in the commutator of the like-nucleon operators involved in the derivation of the QRPA equations. The commutator is replaced by its expectation value in the RPA (correlated) g.s. and this leads to a renormalization of the relevant operators and of the forward- and backwardgoing QRPA amplitudes as well. This method (called pnRQRPA) was first developed in Refs. [31,32] for the standard QRPA and adapted later on for charge-exchanging processes in Ref. [24]. Then, it has been extensively used for both 2ν- and 0νββ decay modes and for transitions to g.s. and excited states and for different nuclei in Refs. [26–29,33]. The extension of this method when the proton–neutron pairing interactions, besides the proton–

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proton and neutron–neutron ones, are also included was called the full-RQRPA [28]. However, the RQRPA-type methods face an undesirable drawback namely, a significant degree of nonconservation of the Ikeda Sum Rule (ISR) [34]. Although refinements in the way of calculating the averages of the quasiparticle number operator are proposed [26] the result was a rather small reduction of the violation. Another challenging issue of this method is the dependence of the calculated ME on the size of the s.p. basis, especially for the neutrinoless mode. Analyzing the calculations of the ββ decay ME existent in literature one still observes discrepancies between the values of the same ME, which may differ up to a factor three. On the other hand, it is difficult to compare results obtained with different versions of the QRPA-based methods and using different parameters and codes. In order to reduce such discrepancies a systematic study becomes necessary . In this paper we make a study of both two-neutrino and neutrinoless ββ decay ME for the experimentally interesting nuclei 82 Se, 96 Zr, 100 Mo, 116Cd, 128,130Te, and 136 Xe, with the pnQRPA, pnRQRPA, full-RQRPA and SQRPA methods. For each method are used two different s.p. basis in order to see the dependence of the results on the size of the Hilbert space. For completeness some results for 76 Ge, partially published very recently in [46], are also included. For a better comparison between the results the calculations are performed for each method with the same set of parameters regarding all steps of the QRPA codes (the construction of the s.p. basis, of the BCS and QRPA wave functions, as well as the renormalization of the G-matrix elements). To our knowledge such a systematic study, with several methods, and for eight nuclei, for both ββ decay modes, has not yet been done, but rather there were compared values of the ββ decay ME obtained under different conditions of calculation. On the other side, in the literature do not exist complete calculations for all the nuclei and with all the four methods presented here, and we also intend to cover this lack. In addition, matrix elements for the neutrinoless decay mode have not been performed, using the SQRPA method, until now. With our study we aim to reduce the discrepancies existent in the literature in the values of the ββ ME predicted by the QRPAbased methods and we hope this will be useful for both theoreticians and experimentalists. First, we analyzed the degree of conservation of the ISR within each method and advance possible explanations for the different deviations. Then, we computed the nuclear ME involved in the 2νββ decay mode and we fixed the gpp parameter according to the recent experimental half-lives reported for this mode. Further, we calculated 0νββ decay ME and using the best presently available half-lives for this decay mode we deduced new upper limits for the neutrino mass parameter. With our calculated ME we also estimated time scales for neutrinoless half-lives that experiments should be able to measure for probing neutrino masses in the mass region of 0.1 eV. The paper is organized as follows. In Section 2 we give a short theoretical description of the QRPA-based methods used in our calculations. Details of the calculations, results and discussions of them are presented in Section 3 which is divided into three subsections. Subsection 3.1 refers to the ISR, 3.2 to the results for the two- neutrino decay mode, while 3.3 is dedicated to the neutrinoless decay mode. In Section 4 we end up with conclusions derived from our work.

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2. Theoretical framework In this section we give a brief theoretical presentation of the QRPA-based methods that we use in our calculation. A detailed description of each method can be found in the corresponding references. In the QRPA-based methods one assumes that the nuclear motion is harmonic and the excitation QRPA operator can be written in the following general form:
   † m π Xµµ Aµµ (k, l, J, M) ΓJm+  k, l, J Mπ = k,l,µ
µ

   m π   +Yµµ Aµµ (k, l, J, M) ,  k, l, J

(1)

where the summation is taken with k
l if µ = µ . X and Y are the forwardand backward-going QRPA amplitudes and A, A† the bifermion quasiparticle operators coupled to angular momentum J and projection M:
† † CjJkM A†µµ (k, l, J, M) = N (kµ, lµ ) mk jl ml aµkmk aµ lml , mk ,ml

µµ (k, l, J, M) = (−) A

J −M

Aµµ (k, l, J, −M).

(2)

N is a normalization constant, which differs from unity only in case when both quasiparticles are in the same shell [26], µ, µ = 1, 2 and 1 ≡ protons, 2 ≡ neutrons. Using for instance the equation of motion method one can derive the pnQRPA equations which, in the matrix representation, may be written as follows:    m    m X U 0 X A B m = ΩJ π , (3) m 0 −U J π Y m B A Jπ Y where the matrices A, B and U have the following expressions:

        +

0

 † AJ µk, νl; µ k  , ν  l  = 0+ RPA Aµν (k, l, J, M), H , Aµ ν  k , l , J, M RPA ,

     

 +

      µ ν  k , l , J, M , H  0+ , (4) BJ µk, νl; µ k ; ν l = 0RPA Aµν (k, l, J, M), A RPA

  †     +

0

U = 0+ (5) RPA Aµν (k, l, J, M), Aµ ν  k , l , J, M RPA . Here the ΩJmπ are the QRPA excitation energies for the mode J π . The simplest way to calculate the A, B and U matrices is to adopt the so-called quasiboson approximation (QBA), as is done within the pnQRPA method, i.e. the quasiparticle operators A, A† are assumed to behave like bosons and satisfy thus exactly the boson commutation relations:    Aµν (k, l, J, M), A†µ ν  k  , l  , J, M    = N (kµ, lν)N k  µ , l  ν  δµµ δνν  δkk  δll  − δµν  δνµ δlk  δkl  (−)jk +jl −J . (6) But, proceeding in this way the Pauli principle is violated and this represents a serious drawback of this method. To improve the situation, within the RQRPA method the above commutator is calculated more precisely by adding in expression (6), besides the scalar term, the next terms which are just the proton and neutron number operators. The value of

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this commutator is replaced by its expectation values in the RPA g.s. Further, one observes that one can mimic the boson behavior of the A, A† operators if one renormalizes them as follows [24,26]: µµ (k, l, J, M) = D −1/2 π Aµµ (k, l, J, M), A µkνk J

(7)

where the Dµkνk  matrices are defined as follows:    Dµkνk  J π = N (kµ, lν)N k  µ , l  ν  δµµ δνν  δkk  δll  − δµν  δνµ δlk  δkl  (−)jk +jl −J



  



 × 1 − j −1 0+ a † aνl  0+ − j −1 0+ a † aµk  0+RPA . l

RPA

νl

00

RPA

k

RPA

µk

00

(8) If one renormalizes further the QRPA amplitudes, A, B matrices and the QRPA phonon operator, m = D −1/2 AD −1/2 , m = D 1/2 Xm , m = D 1/2 Y m , X Y A m = D −1/2 BD −1/2 , (9) B
    †   m+ m π m π  k, l, J A  k, l, J A  (k, l, J, M) + Y µµ (k, l, J, M) , Γ π= X µµ

JM

k,l,µ
µ

µµ

µµ

(10) one observes that the RQRPA equations have the form as in the ordinary QRPA, but now the quantities in the Eq. (3) are replaced by the renormalized ones. and B we need to determine the renormalization matrices D. This is To calculate A done by solving numerically a system of nonlinear equations by an iterative procedure. As input values one can use their expressions in which the averages of the number operators are replaced by the backward-going amplitudes obtained as initial solutions of the QRPA equation. In QRPA-type methods, before starting the RPA procedure, we need the occupation amplitudes (u, v) and the quasiparticle energies, in order to get the quasiparticle representation of the RPA operators. This is done by solving the HFB equations, in which one may include, in the general case, both like- and unlike-nucleon pairing interactions. When one includes only like-nucleon pairing in these equations, the QRPA procedure described above was named pnRQRPA [24,26]. Later on this method was extended by the inclusion of both types of pairing interaction [27] and this version was named full-RQRPA. In this work by the full-RQRPA method we also understand this extension. In the SQRPA method [13,19] the way of including higher-order corrections beyond pnQRPA is different from that of the RQRPA methods. Here, the two-quasiparticle and the quasiparticle-density dipole operators are expanded into series of boson pair operators from which one retains the next terms beyond the QBA:
 (1,0)  (0,1) + + Ak1 Γ1µ (k) + Ak1 Γ1µ (k) , (11) A†1µ (pn) = k

† (pn) B1µ

=


 k1 k2

where

     Bk(2,0) (pn) Γ1† (k1 )Γ2† (k2 ) 1µ + Bk(0,2) (pn) Γ1 (k1 )Γ2 (k2 ) 1µ , (12) 1 k2 1 k2

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S. Stoica, H.V. Klapdor-Kleingrothaus / Nuclear Physics A 694 (2001) 269–294 † B1µ (pn) =




† CjJpM mp jn mn ajp mp ajn mn ,

1µ (pn) = (−)J −M B1µ (pn). B

(13)

mk ,ml

The boson expansion coefficients A(1,0), A(0,1) , B (2,0) , B (0,2) are determined so that the Eqs. (11)–(12) are also valid for the corresponding ME in the boson basis. Further, in the quasiparticle representation, the transition β± operators can also be expressed in terms of the dipole operators A1µ and B1µ : † †  ¯  β− µ (k) = θk A1µ (k) + θk A1µ + ηk B1µ (k) + η¯ k B1µ ,   †   ¯ † β+ µ (k) = − θk A1µ (k) + θk A1µ + η¯ k B1µ (k) + ηk B1µ ,

(14)

where

jˆp jˆp θ¯k = √ jp ||σ ||jn Un Vp , jˆ = 2j + 1, θk = √ jp ||σ ||jnUp Vn , 3 3 ˆ ˆ jp jp η¯ k = √ jp ||σ ||jn Vp Vn . (15) ηk = √ jp ||σ ||jnUp Un , 3 3 For consistency, by using (11)–(12) their expressions are also obtained in the same order beyond QBA. Their complete expressions can be found in Ref. [13]. It is important to stress that these expressions contain, besides the one boson terms present in the QBA, higherorder contributions which are proportional to products of two boson operators. Thus, in the SQRPA method, the higher-order corrections to the pnQRPA are consistently introduced both in the wave functions (through the phonon operators), and in the expressions of the β± operators. For the 2ν- and 0ν-ββ decay half-lives we used in the calculations the following known factorized forms:

2ν 2  2ν −1

, T1/2 = F 2ν MGT (16) where F 2ν is the lepton phase-space integral and

+ + +

− − +




0+ f ||σ τ ||1+k  1k 1l 1l |σ τ | 0i 2ν MGT = . El + Qββ /2 + me − E0

(17)

l,k

In (17), l, k denote the two different sets of 1+ states in the odd–odd nucleus obtained with two separate RPA procedures applied onto the g.s. of the initial and final nuclei participating in the ββ decay. El is the energy of the lth intermediate 1+ state, and E0 is the initial g.s. energy.    0ν −1 mν  2 = Cmm , (18) T1/2 me where mν  is the effective neutrino mass and  2  2 gv 0ν 0ν 0ν Cmm = F1 MGT − MF . gA

(19)

0ν F10ν is the phase-space integral and MGT and MF0ν are Gamow–Teller and Fermi ME. In (18) we neglected the effects of right-handed weak currents (see, e.g., [8]).

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3. Numerical calculations and results We performed calculations of the nuclear ME involved in the ββ decay of 82 Se, 96 Zr, and 136Xe with the pnQRPA, pnRQRPA, full-RQRPA and SQRPA methods. For the Hilbert space used to generate the s.p. basis needed in the calculations we made two choices. For the nuclei with A
100 we included: (i) the full ¯ oscillator shells and (ii) the full (2–4)hω ¯ oscillator shells. For the nuclei with (3–4)hω ¯ oscillator shells and (ii) the full A > 100 the two s.p. basis include: (i) the full (3–5)hω ¯ (2–5)hω oscillator shells. From here on we will call (s) the small basis (i) and (l) the large basis (ii) and these indices are used in tables and figures to distinguish between calculations performed with the two basis. The s.p. energies were obtained by solving the Schrödinger equation with a Coulombcorrected Woods–Saxon potential. The λ-pole nucleon–nucleon residual interactions were taken as Brueckner G-matrix derived from the Bonn-A one-pion-exchange potential. The quasiparticle energies and the BCS occupation amplitudes were calculated by solving the HFB equations. For the pnQRPA, pnRQRPA and SQRPA only proton–proton (pp) and neutron–neutron (nn) pairing interactions were included, while for the full-RQRPA also the pn pairing interactions were added. The calculations were performed separately for the initial and final nuclei participating in the ββ decay and for the two basis sets. Also, we included in the model space the states with all the multipolarities J π . The renormalization constants were chosen as follows: gpp = 1.0 for all the multipolarities, except the 1+ channel for which it was left as a free parameter. This fixed value for gpp represents a choice our model calculations and it is justified by the weak dependence of the results on this parameter for the other multipolarities than the 1+ . gph = 1.0 for all the multipolarities except the 2+ channel where it was fixed to 0.8, since for larger values the p–h interaction in this channel is too strong producing the collapse of the RPA procedure. 100 Mo, 116 Cd, 128 Te, 130 Te

3.1. The Ikeda sum rule First we checked the conservation of the ISR [34]:










0+ |β− | 1+ 2 − 1+

β+

0+ 2 S− − S+ = g.s.

m

=




m

m

g.s.

m

g.s.

 + − 

(−)m 0+ g.s. βm , βm |0g.s.  = 3(N − Z).

(20)

m

In Table 1, we present the degree of deviations from the ISR, in percentages, for the three extensions of pnQRPA used in our calculations, for the two basis sets. The first values in the rows represent the results obtained with the large basis, while the second values are obtained with the small one. In the pnRQRPA and full-RQRPA we found deviations from the right side up to 17%. In previous papers [25,35,36] it was shown that within RQRPA methods the ISR is violated even if all the spin–orbit partners are included in the s.p. basis, and deviations over 10–12% may shed doubts on the values of the calculated ME. This violation is due mainly to the omission of scattering terms both in the definition of the RQRPA phonon operator

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Table 1 The numbers represent the deviations (in percent) from the ISR calculated within the specified methods RQRPA 82 Se 96 Zr 100 Mo 116 Cd 128 Te 130 Te 136 Xe

16.36 15.14 12.06 14.03 13.86 12.05 11.76

16.64 15.87 12.82 14.75 14.02 12.73 12.24

full-RQRPA 16.76 16.12 12.97 14.12 14.28 12.92 12.55

17.01 16.81 13.09 15.93 14.44 13.31 12.89

SQRPA 2.3 2.83 2.17 3.41 2.20 2.03 1.91

2.7 3.08 2.67 4.08 2.60 2.11 2.11

The first (second) numbers in the rows represent the calculation with (l) and (s) s.p. basis. The calculations were performed with the gpp values fixed by comparison with experimental 2νββ decay half-lives.

and in the transition β± operators, which leads to an unbalanced calculation of the two sides of Eq. (20). From Table 1 one can also observe that for the larger basis the degree of nonconservation of the ISR is bigger. This feature could be related to one of the approximations made in the derivation of the RQRPA equations, namely the overlap matrix U which appears in the RQRPA equation is approximated by its diagonal form. However, this approximation is well justified if the s.p. basis does not exceed two oscillator full shells. Thus, a too large enlargement of the s.p. basis makes this approximation questionable. There are also some other inconsistencies of the RQRPA related to the neglection of the next-order terms, besides the scalar one, also in the expression of the B, B † operator commutator, and to the treatment of the BCS and RQRPA vacua [29], which might also generate deviations from the ISR. Such deviations may shed doubts on the reliability of the ME values since an important part of the β strength is lost. On the other side, one observes that in the SQRPA method the deviations are within a few percents, confirming the result reported earlier in Refs. [19,20] where the calculations were performed with smaller s.p. basis. An explanation of this would be the presence in the expressions of the transition operators of additional terms as it was shown in Section 2. These higherorder terms cooperate with the improved expression of the SQRPA wave function in the calculation of the left-hand side of Eq. (20) and give positive contributions to the fulfillment of the ISR. This may give more confidence in the calculation of the ME performed with this method. We should mention that the good degree of conservation of the ISR within SQRPA is valid only when contributions from the three boson states are neglected in the boson expansion of the pair operators, as it has been done in the present calculations. These contributions also introduce undesirable spurious states in the QRPA space and may deteriorate the degree of conservation of the ISR up to 17%.

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3.2. Two-neutrino double beta decay 2ν versus g calculated with pnQRPA and SQRPA methIn Figs. 1–14 we plotted the MGT pp ods (Figs. 1–7) and with pnRQRPA and full-RQRPA (Figs. 8–14) for the ββ decays mentioned in the captions. For each method the calculations are performed with the two different s.p. basis. In the figures also drawn is the line representing the value of the ME corresponding to the most recent reported 2νββ decay half-lives for each case. By inspection, one can see a dependence of the results on the size of the s.p. basis for most of the calculations, whose magnitude varies from case to case. The pnQRPA manifests the largest dependence, except for the nuclei 96 Zr and 130 Te, as compared with the other methods, while SQRPA gives generally the smallest differences. One also observes that these differences are bigger for the heaviest isotopes, i.e. 128,130Te and 136 Xe where the experimental ME are small (or expected to be small, as in the case of 136 Xe). One may estimate that the largest uncertainty in the prediction of the 2ν ME, including these last nuclei, is about a factor two, if one uses one basis or another. Fortunately, this does not affect much the fixing of gpp needed for the calculations of the neutrinoless decay mode. Indeed, within this uncertainty for the ME this parameter has a rather small variation and

2ν matrix elements versus g calculated with pnQRPA and SQRPA methods for the Fig. 1. The MGT pp 82 ββ decay of Se. The curves correspond to the calculations performed in the following conditions: (a) SQRPA (s); (b) SQRPA (l); (c) pnQRPA (s); (d) pnQRPA (l). (l) and (s) means that calculations are performed with the large and small s.p. basis, respectively. The horizontal line represents the experimental ME, i.e. the value calculated with (16) and using the experimental 2νββ decay half-life reported in Ref. [42].

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Fig. 2. The same as in Fig. 1 but for 96 Zr. Experimental half-life is taken from Ref. [43]. (a) SQRPA (s); (b) SQRPA (l); (c) pnQRPA (s); (d) pnQRPA (l).

Fig. 3. The same as in Fig. 1 but for 100 Mo. Experimental half-life is taken from Ref. [38]. (a) SQRPA (s); (b) SQRPA (l); (c) pnQRPA (s); (d) pnQRPA (l).

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Fig. 4. The same as in Fig. 1 but for 116 Cd. Experimental half-life is taken from Ref. [45]. (a) SQRPA (s); (b) SQRPA (l); (c) pnQRPA (s); (d) pnQRPA (l).

Fig. 5. The same as in Fig. 1 but for 128 Te. Experimental half-life is taken from Ref. [37]. (a) SQRPA (s); (b) SQRPA (l); (c) pnQRPA (s); (d) pnQRPA (l).

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Fig. 6. The same as in Fig. 1 but for 130 Te. Experimental half-life is taken from Ref. [40]. (a) SQRPA (s); (b) SQRPA (l); (c) pnQRPA (s); (d) pnQRPA (l).

Fig. 7. The same as in Fig. 1 but for 136 Xe. Experimental half-life is taken from Ref. [39]. (a) SQRPA (s); (b) SQRPA (l); (c) pnQRPA (s); (d) pnQRPA (l).

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2ν matrix elements versus g calculated with pnRQRPA and full-RQRPA methods Fig. 8. The MGT pp for the ββ decay of 82 Se. The curves correspond to the calculations performed in the following conditions: (a) full-RQRPA (s); (b) full-RQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l). (l) and (s) means that calculations are performed with the large and small s.p. basis, respectively. The horizontal line representing the experimental ME is also drawn.

Fig. 9. The same as in Fig. 8 but for 96 Zr. (a) full-RQRPA (s); (b) full-RQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l).

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Fig. 10. The same as in Fig. 8 but for 100 Mo. (a) full-RQRPA (s); (b) full-RQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l).

Fig. 11. The same as in Fig. 8 but for 116 Cd. (a) full-RQRPA (s); (b) full-RQRPA (l); (c) pnRQRPA (s); pnRQRPA (l).

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Fig. 12. The same as in Fig. 8 but for 128 Te. (a) SRQRPA (s); (b) SRQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l).

Fig. 13. The same as in Fig. 8 but for 130 Te. (a) full-RQRPA (s); (b) full-RQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l).

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Fig. 14. The same as in Fig. 8 but for 136 Xe. (a) full-RQRPA (s); (b) full-RQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l).

can be fixed rather precisely according to the experimental 2νββ decay half-lives. That, together with a weaker dependence of the neutrinoless ME on this parameter as compared with the 2ν decay mode, leads to an insignificant error in the calculation of the 0νββ decay ME, which could appear from this way of fixing gpp . As another general feature one confirms previous calculations that within the higher-order QRPA methods the collapse of the QRPA solutions is pushed towards higher values of gpp . Also, one observes that 2ν versus g calculated with the larger s.p. bafor all the used methods the functions MGT pp sis cross the zero axis faster than the ones calculated with the smaller basis. This could be understood by the fact that a larger basis generates a larger amount of g.s. correlations which may cause a faster break-down of the QRPA with the increase of the particle–particle strength. Using the most recent 2νββ decay data reported in the literature we deduce from Eq. (16) the experimental ME for this mode. The phase-space factors were taken from Ref. [33]. The values of these factors as well as the experimental half-lives used, are given in Table 2. Then, using our calculated ME we fixed the gpp parameters for all the methods and nuclei in order to use their values for neutrinoless decay mode calculations. Since in the case of 136 Xe there is not yet settled an experimental half-life for the 2ν decay mode, 2ν = but only a lowest limit of 3.6 × 1020 yr, we fixed in this case the value of gpp for a T1/2 21 1.0 × 10 yr.

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Table 2 The integrated phase space factors F12ν and F10ν [25], the recent experimental 2ν- and 0ν-ββ decay half-lives and the estimated by the SQRPA 0νββ half-lives for mν = 0.1 eV Nucleus F12ν [yr−1 ] 76 Ge 82 Se 96 Zr 100 Mo 116 Cd 128 Te 130 Te 136 Xe

1.3 × 10−19 4.3 × 10−18 1.8 × 10−17 8.9 × 10−18 7.4 × 10−18 8.5 × 10−22 4.8 × 10−18 4.9 × 10−18

F10ν [yr−1 ]

T 2ν [yr]

6.31 × 10−15 1.55 × 1021 [44] 2.73 × 10−14 8.3 × 1019 [42] 5.7 × 10−14 2.1 × 1019 [43] 1.13 × 10−13 0.95 × 1019 [38] 4.68 × 10−14 2.6 × 1019 [45] 1.66 × 10−15 7.7 × 1024 [37] 4.14 × 10−14 2.6 × 1021 [40] 4.37 × 10−14 > 3.6 × 1020 [39]

0ν T0.1 eV [yr]

T 0ν [yr] > 1.9 × 1025 > 9.5 × 1021 > 1.0 × 1021 > 5.2 × 1022 > 0.7 × 1023 > 7.7 × 1024 > 5.6 × 1022 > 4.4 × 1023

[44] [42] [43] [38] [45] [37] [41] [39]

2.83−4 × 1026 5.6−7.6 × 1025 0.63−1.02 × 1026 1.14−1.29 × 1025 0.78−1.06 × 1026 1.38−1.90 × 1027 0.98−1.08 × 1026 5.63−6.22 × 1026

The experimental half-lives are taken from the references indicated in parenthesis.

3.3. Neutrinoless double-beta decay In Figs. 16–25 we display the ME for the neutrinoless decay mode calculated with pnQRPA, pnRQRPA full-RQRPA and SQRPA, for the two choices of the s.p. basis. From the Figs. 16–21 one observes that the sensitivity of the ME to the choice of the s.p.

0ν matrix elements versus g Fig. 15. The MGT pp calculated with pnQRPA, pnRQRPA and full-RQRPA methods for the ββ decay of 82 Se. The curves correspond to the calculations performed in the following conditions: (a) pnQRPA (s); (b) pnQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l); (e) full-RQRPA (s); f) full-RQRPA (l).

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Fig. 16. The same as Ibidem like in Fig. 15 but for 96 Zr. (a) pnQRPA (s); (b) pnQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l); (e) full-RQRPA (s); (f) full-RQRPA (l).

Fig. 17. The same as Ibidem like in Fig. 15 but for 100 Mo. (a) pnQRPA (s); (b) pnQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l); (e) full-RQRPA (s); (f) full-RQRPA (l).

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Fig. 18. The same as Ibidem like in Fig. 15 but for 116 Cd. (a) pnQRPA (s); (b) pnQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l); (e) full-RQRPA (s); (f) full-RQRPA (l).

Fig. 19. The same as Ibidem like in Fig. 15 but for 128 Te. (a) pnQRPA (s); (b) pnQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l); (e) full-RQRPA (s); (f) full-RQRPA (l).

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Fig. 20. The same as in Fig. 15 but for 130 Te. (a) pnQRPA (s); (b) pnQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l); (e) full-RQRPA (s); (f) full-RQRPA (l).

Fig. 21. The same as in Fig. 15 but for 136 Xe. (a) pnQRPA (s); (b) pnQRPA (l); (c) pnRQRPA (s); (d) pnRQRPA (l); (e) full-RQRPA (s); (f) full-RQRPA (l).

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0ν matrix elements versus g calculated with the SQRPA method for the ββ decay Fig. 22. The MGT pp 76 82 of Ge and Se. The curves correspond to the calculations performed in the following conditions: (a) 76 Ge (s); (b) 76 Ge (l); (c) 82 Se (s); (d) 82 Se (l).

Fig. 23. The same as in Fig. 22 but for 96 Zr and 100 Mo. (a) 96 Zr (s); (b) 96 Zr (l); (c) 100 Mo (s); (d) 100 Mo (l).

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Fig. 24. The same as in Fig. 22 but for 116 Cd and 136 Xe. (a) 116 Cd (s); (b) 116 Cd (l); (c) 136 Xe (s); (d) 136 Xe (l).

Fig. 25. The same as in Fig. 22 but for 128 Te and 130 Te. (a) 128 Te (s); (b) 128 Te (l); (c) 130 Te (s); (d) 130 Te (l).

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basis used is generally bigger than in the case of the two-neutrino mode for the pnQRPA, pnRQRPA and full-RQRPA as also it was found in previous calculations [28,29]. However, it was shown by the authors of Ref. [28] that by increasing the dimension of the s.p. basis such dependence becomes smaller. On the other side, as we discussed in Section 3.1, a too large increase of the basis (e.g. over two full oscillator shells) may shed doubts on one of the basic approximations made in the derivation of the RQRPA, and consequently on the results. Distinctively from this, one can see from the Figs. 22–25 that when they are calculated with the SQRPA the ME display a weak dependence on the s.p. basis used, the differences between the results being not more than 30%. A better stability against the change of the basis and a good fulfillment of the ISR give more confidence in the results obtained with the SQRPA. Further, in order to have a more complete image on the calculations, we display in Table 3 the results of the neutrinoless ME calculated with all the four methods and for the two basis sets, at the values of gpp fixed, for each method, as it was explained previously. For completeness we also show the results for 76 Ge obtained in Ref. [46]. By inspection, one observes that in all cases the differences between the results obtained with the same method but with different basis are not so big around the physical values of gpp . With a few exceptions one would say that, the uncertainty in the calculations of the neutrinoless ME performed with all these methods, coming from the use of different Table 3 The neutrinoless matrix elements and upper limits for the neutrino mass parameter calculated in this paper with pnQRPA, pnRQRPA, full-RQRPA and SQRPA methods using the experimental limits given in Table 2 Nucleus 76 Ge 82 Se 96 Zr 100 Mo 116 Cd 128 Te 130 Te 136 Xe

pnQRPA

pnRQRPA

full-RQRPA

SQRPA

1.71(l); 4.45(s) 0.84(l); 0.33(s) 4.71(l); 5.60(s) 6.75(l); 5.67(s) 2.75(l); 4.16(s) 24.61(l); 16.27(s) 3.81(l); 5.37(s) 1.75(l); 1.24(s) 2.85(l); 3.99(s) 3.13(l); 2.24(s) 3.43(l); 4.84(s) 1.31(l); 0.94(s) 3.77(l); 4.73(s) 2.81(l); 2.24(s) 1.35(l); 1.69(s) 2.17(l); 2.73(s)

1.87(l); 3.74(s) 0.79(l); 0.40(s) 2.70(l); 4.30(s) 11.71(l); 7.38(s) 2.72(l); 3.01(s) 24.88(l); 22.48(s) 3.40(l); 4.36(s) 1.96(l); 1.53(s) 3.39(l); 3.61(s) 2.63(l); 2.47(s) 2.83(l); 4.29(s) 1.60(l); 1.05(s) 3.00(l); 4.55(s) 3.54(l); 2.33(s) 1.02(l); 1.57(s) 3.6(l); 2.35(s)

2.40(l); 3.68(s) 0.62(l); 0.41(s) 2.63(l); 4.15(s) 12.04(l); 7.64(s) 2.42(l); 2.99(s) 27.97(l); 22.64(s) 3.35(l); 4.11(s) 1.63(l); 1.53(s) 2.35(l); 2.62(s) 3.80(l); 3.41(s) 2.85(l); 3.75(s) 1.59(l); 1.21(s) 2.61(l); 3.49(s) 4.07(l); 3.04(s) 0.89(l); 0.99(s) 4.14(l); 3.72(s)

3.21(l); 3.82(s) 0.46(l); 0.39(s) 3.54(l); 4.13(s) 8.96(l); 7.68(s) 2.12(l); 2.70(s) 31.92(l); 25.06(s) 4.23(l); 4.51(s) 1.58(l); 1.48(s) 2.29(l); 2.67(s) 3.90(l); 3.34(s) 2.85(l); 3.38(s) 1.58(l); 1.34(s) 2.42(l); 2.53(s) 4.39(l); 4.19(s) 0.98(l); 1.03(s) 3.76(l); 3.57(s)

The numbers in the first row of each nucleus are the matrix elements, while the numbers in the second row are the the neutrino mass parameters. The calculations are performed with the large basis (l) or with the small (s) at gpp values fixed by comparison with experimental 2νββ decay half-lives.

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s.p. basis, is within a factor of two. The largest discrepancies between the results occur for pnQRPA and RQRPA and possible explanations for this may be related to the shortcomings of these methods which were already discussed previously. One also observes that the fullRQRPA and SQRPA values do not differ from each other by more that 50%. Having in view that the SQRPA displays both a better stability against the change of the basis and a good fulfillment of the ISR, we also may reduce the uncertainties in the predictions of the ME, related to these two shortcomings in the QRPA-based methods, to this percentage. Further, using the corresponding phase-space factors and the most recent reported limits for the neutrinoless half-lives given in Table 1, we also deduced new limits for the neutrino mass parameter for all the methods and for each nuclei. These values are also given in Table 3 below the corresponding ME, in the second rows of the boxes. Since the ME and mν  enter the half-lives formulae at the same power, the same degree of uncertainty as for the ME, applies to the effective neutrino mass. As a general feature, one should mention that with a larger s.p. basis one obtains smaller values for the ME and consequently bigger values for the neutrino mass limits. This could also be understood by the fact that a larger Hilbert space introduces a larger amount of g.s. correlations which may produce a more rapid collapse of the QRPA procedure. Finally, using our calculated ME we also estimated for each nucleus the half-lives that experiments should reach for exploring the neutrino mass in the region of 0.1 eV. These values are given in the last column and they may serve as a guide for future planned experiments.

4. Conclusions We have performed a systematic study of the ββ decay ME for the two-neutrino and neutrinoless decay modes with the pnQRPA, pnRQRPA, full-RQRPA and SQRPA methods, using two different s.p. basis sets. The calculations are extended to the following experimentally interesting nuclei: 76 Ge, 82 Se, 96 Zr, 100 Mo, 116Cd, 128Te, 130Te and 136 Xe. First, we analyzed the degree of conservation of the ISR. It was found that it is well fulfilled within pnQRPA, as expected, and with a good approximation within SQRPA method (deviations are within 2–4%), while within RQRPA and full-RQRPA the ISR is underestimated up to 17%. This result is not much dependent on the size of the s.p. basis used. Possible explanations for this behavior were advanced. In case of the RQRPA methods the deviations could be mainly related to the omission of scattering terms both in the definition of the RQRPA phonon operator and in the transition β± operators. Concerning SQRPA, the better fulfillment of the ISR obtained with this method may be attributed to the inclusion also in the expressions of the transition operators of additional higher-order terms beyond the QBA. These terms seem to cooperate properly with those from the improved expressions of the wave functions in the calculation of the expectation values of the transition operators and do not deteriorate significantly the conservation of the ISR. This result gives more confidence in the calculation of the neutrinoless ME with this method.

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Then we studied the dependence of the results on the size of the s.p. basis used and found the following main results: (i) For the two-neutrino mode the ME calculated with the pnQRPA method display the largest dependence, as compared with the other methods, while SQRPA gives generally the smallest differences in their values. The differences between calculations corresponding to the two basis sets are bigger for the heaviest isotopes, i.e. 128,130Te and 136 Xe. For these nuclei the experimental 2νββ decay ME are the smallest and the predictive power of all the methods is the worst, as compared with the other nuclei. However, the uncertainty in their prediction, associated to the use of different s.p. basis is within a factor two. (ii) Within this uncertainty gpp has only a small variation and one can fix it with reasonable precision from the experimental 2νββ decay half-lives. This small variation, together with the weaker sensitivity of the ME for the neutrinoless decay mode on this parameter, as compared with the 2ν mode, produces only a quite small error in the calculation of the 0ν ME. 2ν versus g calculated with the larger basis (iii) For all methods used the functions MGT pp cross the zero axis faster than the ones obtained with the smaller one. This could be understood by the fact that a larger basis generates a larger amount of g.s. correlations which may cause a faster break-down of the QRPA with the increase of the particle–particle strength. (iv) For the neutrinoless decay mode the sensitivity of the ME on the size of the s.p. basis is generally bigger than for the 2ν decay mode. However, the differences between the results obtained with the same method, but with different basis, are smaller (within a factor up to two) in the region around the fixed gpp , which is the physical one. Moreover, one can reduce these uncertainties to 50% if one compares the closer values of the ME obtained with the full-RQRPA and SQRPA methods. This conclusion is also strengthened by the better stability against the change of the s.p. basis and by the good degree of fulfillment of the ISR, that the SQRPA method displays. (v) Using our calculated ME and the most recent experimental limits for the 0νββ decay half-lives found in the literature we deduced new limits for the neutrino mass parameter for each case. (vi) We estimated the half-lives that experiments should be able to measure for exploring neutrino masses in the region of 0.1 eV. These values may help experimentalists in planning their setups. One remarks that the Heidelberg– Moscow ββ decay experiment on 76 Ge is the closest, as compared with other experiments, to the scaled half-life.

Acknowledgements One of the authors (S.S.) would like to thank the Max Plank Institut für Kernphysik for the hospitality extended to him during his stay in Heidelberg.

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