Nuclear structure model for double-charge-exchange processes

A new model, based on the BCS approach, is especially designed to describe nuclear phenomena ( A , Z ) → ( A , Z ± 2) of double-charge exchange (DCE). Although it was proposed and applied in the particle-hole limit, by one of the authors [Krmpoti´c, Fizika B 14 , 139 (2005)], it has not yet been applied within the BCS mean-ﬁeld framework, nor has its ability to describe DCE processes been thoroughly explored. It is a natural extension of the pn -QRPA model, developed by Halbleib and Sorensen [Nucl. Phys. A 98 , 542 (1967)] to describe the single β decays ( A , Z ) → ( A , Z ± 1), to the DCE processes. As such, it exhibits several advantages over the pn -QRPA model when used in the evaluation of the double beta decay (DBD) rates. For instance, (i) the extreme sensitivity of the nuclear matrix elements (NMEs) on the model parametrization does not occur; (ii) it allows us to study the NMEs, not only for the ground state in daughter nuclei, as the pn -QRPA model does, but also for all ﬁnal 0 + and 2 + states, accounting at the same time for their excitation energies and the corresponding DBD Q values; (iii) together with the DBD-NMEs it also provides the energy spectra of Fermi and Gamow-Teller DCE transition strengths, as well as the locations of the corresponding resonances and their sum rules; (iv) the latter are relevant for both the DBD and the DCE reactions, since the underlying nuclear structure is the same; this correlation does not exist within the pn -QRPA model. As an example, detailed numerical calculations are presented for the ( A , Z ) → ( A , Z + 2) process in 48 Ca → 48 Ti and the ( A , Z ) → ( A , Z − 2) process in 96 Ru → 96 Mo, involving all ﬁnal 0 + states and 2 + states.


I. INTRODUCTION
The double-charge-exchange (DCE) processes relate the (A, Z ) nuclei with the (A, Z + 2) and (A, Z − 2) nuclei and will be labeled as {+2} and {−2} processes respectively.
The most studied DCE process is the double beta decay (DBD).It is the slowest physical process observed so far, and can be used to learn about neutrino physics, provided we know how to deal with the nuclear structure.According to the number and type of leptons we may have the following DBD modes: (i) double-electron decay (2β − ), (ii) double-positron decay, (iii) electron capture-positron emitting decay (eβ + ), and (iv) double electron capture decay (ee).Each one of these decays occurs either with the emission of two neutrinos (2ν decay) or they are neutrinoless (0ν decay).To simplify the notation and avoid confusion, we will designate the first process as DBD − and the remaining three as DBD + .
The 0ν-decay rates depend on several unknown parameters such as neutrino mass, Majoron coupling, the coupling constants of the right-handed components of the weak Hamiltonian, etc., and the only way to put these in evidence is by having sufficient command over the nuclear structure.It is precisely at this point that the 2ν2β ∓ , 2νeβ + , and 2νee decay modes are important.A comparison between experiment and theory for them provides a measure of the confidence that one may have in the nuclear wave functions employed for extracting the unknown parameters from 0ν-lifetime measurements.
The number of possible candidates for 2β − decay is quite large: there are 35 nuclei.In addition, 34 nuclei can undergo 2e-electron capture, while 22 and 6 nuclei can undergo eβ + and 2β + decays respectively [1].The discovery of the massiveness of the neutrino, through the observation of oscillations, boosted the importance of the 0ν DBD, since they are the only observables capable of providing the magnitude of the effective neutrino mass.
It is well known that the involved nuclear structure in 2ν and 0ν DBD is the same one that describes the DCE reactions.This fact reignited recently the interest in the measurements of heavy-ion-induced DCE reactions, such as the NUMEN project [2][3][4] involving the 40 Ca( 18 O, 18 Ne) 40 Ar process.Simultaneously, the interest in the theoretical study of the DCE reactions has been renewed [5,6].Moreover, Shimizu, Menéndez, and Yako [7] have also noted recently a correlation between the DCE and 0ν2β − decay.In all the theoretical studies mentioned, the calculations were made within the framework of the shell model (SM).
The neutrinoless DBDs occur in medium-mass nuclei that are often far from closed shells and, as a consequence, the calculations are mostly made in the proton-neutron quasiparticle random phase approximation (pn-QRPA), since this tool is computationally much simpler than the SM.As discussed in Ref. [8], the kind of correlations that these two methods  (2.15), and indicates that the first β − decay is switched on in the initial state and the second in the intermediate state.The 2β + NMEs are represented in the same way after making the substitution O ∓ J ↔ O ± J .The pn and nn + pp nuclear interactions between protons and neutrons are indicated by red and blue dashed lines, respectively.The five vertices of the diagram (b) correspond to five of six angular momentum coupling in the symbol 9 j in Eq. (2.47).The sixth coupling (JJ )J corresponds to the three unconnected lines in this figure .include are not the same.The pn-QRPA deals with a large fraction of nucleons in a large single-particle space, but within a modest configuration space.The SM, by contrast, deals with a small fraction of nucleons in a limited single-particle space, but allows them to correlate in arbitrary ways within a large configuration space.
There is another important difference.The standard pn-QRPA only allows us to calculate the double-charge-exchange transitions from the ground state of the decaying (A, Z ) nucleus to the ground state in the final (A, Z ± 2) nuclei.In fact, to evaluate the transitions going to the excited states, a second (pp + nn)-QRPA must be performed [9,10], which introduces additional free parameters and it is limited to one and two quadrupole phonon states.
To deal with the DCE processes we will resort here to a Tamm-Dancoff approximation (TDA), which has been suggested and discussed on its ph limit (where ph means particle-hole) 1 for 48 Ca more than a decade ago in Ref. [11].In this model it is assumed that the initial, intermediate, and final nuclei are the BCS vacuum, pn excitations, and 2p2n excitations respectively.The resulting model will be labeled as (pn, 2p2n)-QTDA.The main differences with the standard pn-QRPA model are illustrated in Fig. 1.
The present model is a natural extension to double-chargeexchange processes of the pn-QRPA model, originally proposed by Halbleib and Sorensen (HS) in 1967 to describe single β decays [12].As such, it permits the evaluation of the NMEs, not only for the ground states but also for all final 0 + and 2 + states, as well as the Q values for the 2β − decay (Q 2β − ), and for the 2e capture (Q 2e ).It yields as well the DCE energy strength spectra and their sum rules, which are relevant for associated reaction processes and resonances.Detailed numerical calculations are performed in the present work for the 48 Ca → 48 Ti and 96 Ru → 96 Mo processes, involving their final 0 + and 2 + states.

A. Nuclear matrix elements and double-charge-exchange excitations
Independently of the nuclear model used, and when only the allowed transitions, i.e., the Gamow-Teller (GT) and Fermi (F) transitions are considered, the NMEs for the 2ν2β ± decay, from the ground state 0 + i in the initial nucleus (A, Z ) to one of the states 0 where g 0 ≡ g V and g 1 ≡ g A are the vector and axial-vector weak coupling constants respectively, and the summation goes over all intermediate virtual states The one-body operators are with O 0 = 1, and O 1 = σ for F and GT transitions, respectively, c † k ≡ c † j k ,m k and ck ≡ (−) j k −m k c j k ,−m k are the single-particle creation and annihilation operators, and Ĵ = √ 2J + 1.The energy denominator in Eq. (2.1) is3 where are the energies of the decaying (A, Z ) nucleus in its ground state, the intermediate (A, Z ± 1) nuclei in the state J + α , and final (A, Z ± 2) nuclei in the state J + f , respectively.The matrix elements hJ + α ||O ± J ||0 + i i can be expressed as i.e., as a product of model dependent one-body densities, and the single-particle NME, which is a purely geometric (angular) factor, with the property O J (np) = (−) j n − j p O J (pn). 4imilarly, where are the corresponding density matrices for transitions from the intermediate states |J α i to the final states |J + f i, with J = 0, 2. The corresponding matrix elements are where the GT-NMEs to 2 + f states have also been included [17][18][19].
All the information about the nuclear structure is contained in the one-body density matrices (2.4) and (2.8) or, more precisely, in the two-body density matrices, The NME for the 0ν2β ± decays to the 0 + f final states can be easily evaluated from these densities.In fact, after making the replacement in Ref. [[20], Eqs.(2.20)] we can express them as where X = V, A, P, M stands for vector (V ), axial-vector (A), pseudoscalar (P), and weak-magnetism (M) terms.With the NME M 0ν ± (2 + f ) we proceed in the same way.It is well known that the single β-decay processes to the states J α in (A, Z + 1) and (A, Z − 1) nuclei are related to the following single charge-exchange transition strengths: (2.13) When |J + α i is a complete set of excited states that can be reached by operating with O ± J on the initial state |0 + i i, they satisfy the single-charge exchange (SCE) or Ikeda sum rule, for both the F and GT transitions, Similarly, both M 2ν2β ± (J + f ) and M 0ν2β ± (J + f ) are related to the double-charge-exchange operators (O ± J O ± J ) J and to their spectral distributions in (A, Z ± 2) nuclei given by (2.15) When both |J + α i and |J + f i are a complete set of excited states that can be reached by operating with O ± J , and (O ± J O ± J ) J on the initial state |0 + i i, their differences obey the double-charge-exchange sum rules (DCESRs), which were evaluated in Refs.[5,6,[21][22][23] with the following results: 19) where C is a relatively small quantity given by [ [22], Eq. ( 4)].These equations agree with Eq. (8) in Ref. [6] except for a factor of 3 and the omission of Combining Eqs.(2.18) and (2.19), one obtains the sum rule for the total GT strength, which is independent of the structure of the ground-state wave function [22].The relationship between the DBD and DCE reactions has been discussed recently in Refs.[2,7].

B. (pn, 2p2n)-QTDA model
The pn-QRPA evaluations of the DBDs are generally limited to the ground state of the final nuclei, i.e., to the calculation of M 2ν (0 + 1 ) and M 0ν (0 + 1 ).Moreover, the SCESRs (2.14) are fulfilled within this model, but it does not allow us to evaluate the strengths S 2β ± JJ given by (2.16) and to discuss the corresponding DCESRs listed in Eqs.(2.17)- (2.20).
To describe the intermediate states J + α in both pn-QRPA and (pn, 2p2n)-QTDA models a nuclear Hamiltonian of the type is used, where is the independent-quasiparticle Hamiltonian, with a † α and a ᾱ being the single-quasiparticle creation and annihilation operators, defined by the Bogoliubov transformation [ [24], Eqs.(13.10)] (2.23) The proton and neutron pairing interactions are contained in the transformation coefficients u a and v a , and in the quasiparticle energy where e k is the shell-model single-particle energy (spe), and λ is the chemical potential or Fermi level.The energy gap parameters 1 k and the pairing coupling constants are determined to reproduce the experimental odd-mass difference for each nucleus.Finally, H pn is the quasiproton-quasineutron interaction.
Within the pn-QRPA, the states |J α i with excitation energy ω J α are created from the correlated initial and final 0 + ground states by proton-neutron phonon creation operators Q J α , which are defined as a linear superposition of creation and annihilation proton-neutron quasiparticle pair operators (2.25)That is, and Frequently, this is done for both initial |0 + I i and final |0 + F i ground states, obtaining in this way two different sets of intermediate states J α and J α 0 in the (N − 1, Z + 1) nucleus.Therefore, in the evaluation of the 2β-NME, it is necessary to consider their overlap, which is indicated in Fig. 1, and corresponds to the substitution (2.28) The ground state defined in Eq. (2.27) is more accurate than the BCS ground state (a k |BCSi = 0) since it contains terms with 0, 4, 8, . . .quasiparticles [8].Nevertheless, in the present model we approximate the initial ground state in the (A, Z ) nucleus by the BCS vacuum and the states |J α i in the intermediate (A, Z ∓ 1) nuclei as This disadvantage of the present model is counteracted by the description that we make of the final states in the (A, Z ∓ 2) nuclei; that is, instead of the correlated |0 + F i state defined in Eq. (2.26), we have where J + = 0 + , 2 + and are antisymmetrized and normalized two-proton-two-neutron quasiparticle states, where are normalized two-quasiparticle states.The amplitudes X p 1 p 2 J 12 ,n 1 n 2 J 0 12 ;J + f are obtained by diagonalizing the Hamiltonian [25,26] in the basis (2.31) where H nn and H pp are neutron-neutron and proton-proton interactions.Details on the evaluation of matrix elements of H 2 can be found in Refs.[25,26].However, our final results are different.The matrix element of H pn for the odd-odd nucleus reads where the functions G and F are defined in the standard way [25,26].They differ from those of Baranger [27] by a factor of −2, as can be seen, for example, from the comparison of [ [27], Eq. ( 20)] with [ [28], Eq. ( 4)] (or with [[29], Eq. (3.2)]).Our G and F functions are also different from those of Suhonen [24].The matrix elements of H pn in the basis (2.31) are derived by employing the relation (1A-25) from [30].We get where the operator exchanges the particles p 1 and p 2 .
Finally, the matrix element of the neutron-neutron Hamiltonian H nn in the same basis is and analogously for the proton-proton Hamiltonian H pp .The energies in the denominator D 2ν ± J α ,J f , defined by (2.3), are where λ p and λ n are the proton and neutron chemical potentials.Therefore, for both 2β + and 2β − decays we have (2.39) The lowest energies ) and for the 2e capture (Q 2e ), defined as 5 where the M's are the atomic masses.Namely, (2.41) 5 The 2β + and β + e Q values are To evaluate the one-body densities (2.4) and (2.8) we make use of [ [24], Eqs.(15.4)] to get which, from (2.2) and (2.5), immediately yields and with
Making use of orthogonality and completeness of both basis |J α i and A † (pnJ )|BCSi in Eq. (2.29), the relation (2.15) can be expressed in a more compact form, namely as In this way, the transition strength turns out to be independent of the intermediate states.
It is important to emphasize that the permutation operators in the last two equations only act on the right side.The physical meaning of these permutations can be inferred from Fig. 1(b), where the DCE matrix element graphically.This quantity is used in the evaluation of both the DBD NME (2.9) and the DCE transition strengths (2.15), but the Eq.(2.47) is applicable only in the latter case.
Together with the NMEs M 2ν ± (J + f ) given by (2.1) and (2.9) with where the NMEs are given in natural units ( h = m e = c = 1), and the leptonic kinematics factors, G α 2ν (J + f ), are in yr −1 .These factors can be found in Ref. [ [31], Table II] for several nuclei of interest.(For the most recent computations of phase space factors see Refs.[32,33].) The excitation energies in the final nuclei are calculated from (2.49) It should be noted that, just as the pn-QRPA model predicts identical excitation energies for the (Z, A ± 1) nuclei, the current model yields the same the excitation energies in the (Z, A ± 2) nuclei.
Finally, the centroid energies of the DCE transition strengths are defined as (2.50)

III. NUMERICAL RESULTS AND DISCUSSION
The residual interaction is described by the δ force (in units of MeV fm 3 ) where v s and v t are the spin-singlet and spin-triplet parameters.

A. Mean field
The mean field is defined by the unperturbed Hamiltonian H 0 and the spe that are specified below.As usual, the pairing strengths for protons and neutrons, v pair s (p) and v pair s (n), are obtained from the fitting of the experimental pairing gaps.
The DBD − 48 Ca → 48 Ti is a rather unique case, since 48 Ca is a double closed nuclei, and we can make use of the experimental spe, labeled as e expt j , which comprises the energies of both the single-particle and single-hole states.All of them were taken from the binding and excitation energies, weighted with spectroscopic factors, of odd-mass neighboring nuclei: 47 Ca and 49 Ca for neutrons, and 47 K and 49 Sc for protons.They are listed in Fig. 2 and are those from [ [34], Table II], except for the proton f 5/2 spe, which is estimated from the proton f 5/2f 7/2 splitting given in Ref. [35].We need this level to saturate both the SCE and DCE sum rules.Once this has been done, the spe e expt j have been used in two different ways: (1) The following steps are done in handling the BCS equations [13,36,37]: (a) The BCS energies relative to the Fermi level λ, are introduced, where the positive (negative) sign is adopted if the corresponding single-particle state is a particle (hole) state.(b) It is assumed that neutron and proton Fermi levels λ n and λ p lay between j n = 2p 3/2 -1 f 7/2 and j p = 1 f 7/2 -1d 3/2 states respectively, and that all states above λ are pure quasiparticle excitations E (+) j and all states below λ are pure quasihole excitations E (−) j .(c) Starting from a set of harmonic oscillator energies e HO j , the energies E (±) j are adjusted to the experimental spectra e expt j by means of a χ 2 search varying the strengths v pair s and the bare spe e j which that appear in the BCS gap equations (2.24).All energies involved in this procedure are illustrated in Fig. 2.
(2) For the sake of completeness, the pairing parameters v pair s (p) and v pair s (n) were fixed in the standard manner [29].That is, by fitting the experimental pairing gaps to the calculated pairing gaps 1 j , given by [ [30], Eq. (2.96)], with j = 1 f 7/2 for neutrons and j = 2s 1/2 for protons.
The most relevant difference between the spe e expt j and e j is the disappearance of the energy gap between the holes and the particles in the last case.The resulting parameters v pair s , FIG. 2. Mean field energies (in units of MeV) for neutrons (left panel) and for protons (right panel).In both cases we show (i) harmonic oscillator energies e HO j , (ii) adjusted single-particle energies e j , (iii) BCS energies relative to the Fermi level (E (±) j ), and (iv) experimental energies (e expt j ).
and λ are given in the Table I for the two sets of spe e j and e expt j .The quasiparticle energies E (±) j with e expt j are obviously slightly different from those shown in Fig. 2.
In the 96 Ru nucleus, the neutron and proton shells are open and its energy spectrum is clearly rotational.This fact gives rise to a strong interplay between collective and singleparticle degrees of freedom in the low energy spectra of the neighboring odd-mass nuclei 95 Ru, 97 Ru, 95 Tc, and 97 Rh.For instance, it is very likely that the ground state 5/2 + in 95 Ru is a consequence of the j − 1 anomaly in the three neutron cluster (1g 7/2 ) 3 [38,39].Such a complex nuclear structure makes it to difficult determine the appropriate spe spectra and the pairing interaction strengths from experimental energy spectra, as it was done in the case of 48 Ca.
Instead, we used the spe provided by Paar [40], which were calculated in the relativistic Hartree-Bogoliubov model, as outlined in Ref. [41].They are shown in Table II, together with the resulting quasiparticle energies (3.3), which were obtained following procedure 2 in the case of 48 Ca.This suggests fitting the calculated pairing gaps 1 j , with j = 1g 7/2 for neutrons and j = 1g 9/2 for protons, to the experimental ones.The similarity between the spe e j and the quasiparticle energies E (±) j is remarkable.The corresponding pairing parameters v pair s and chemical potentials λ are listed in Table I.It is important to note that in 48 Ca it is λ n > λ p while in 76 Ru this difference is λ n < λ p .We will soon see that this fact is decisive with respect to the Q values of DBD.More precisely, this will explain why DBD − occurs in 48 Ca and DBD + occurs in 96 Ru.
Our method of calculation is similar in several aspects to that used in the SM framework in Refs.[5,6].In fact, the illustration of their calculations, made in Fig. 1 of the last reference, is also valid in our case.The biggest difference between the two models, in addition to the residual interactions that were used, arises from the size of the configuration spaces of the final states J + f .We have 664 0 + states and 2.470 2 + states, while Auerbach and Bui Minh Loc [6] have, in their evaluation of double-charge-exchange GT strength 48 Ca → 48 Ti, the quantity of 14.177 and 61.953 final states in the f p space, respectively.In the case of the 96 Ru → 96 Mo decay, we also have 664 0 + states, but 2.583 2 + states.6

B. Residual interaction
The same coupling constants v pp s , v pp t , v ph s , and v ph t were used in the numerical evaluations of the matrix elements of the Hamiltonians H pn , H pp , H nn , given by Eqs.(2.34) and (2.37).
That is, we assume that the residual interaction is the same for identical and nonidentical particles, which is a very strong constraint on the model parameters.
In all our previous works within the pn-QRPA model [20,28,[42][43][44][45][46][47] the values of the interaction coupling constants, both in ph and pp channels, were set from the partial restoration of the spin-isospin SU(4) symmetry (PSU4SR) induced by the residual interaction.The corresponding physical motivations for this, and the procedure are discussed in those references in detail.However, to make the present paper selfcontained, it might be appropriate to use some paragraphs to explain the philosophy of the PSU4SR approach.In doing this, we necessarily have to partially repeat what has been said previously, in particular in Ref. [47].
(1) Particle-hole channel.The SU(4) symmetry is dynamically broken by the spin-orbit coupling and the supermultiplet destroying residual interactions, but these two effects have a tendency to cancel each other.Indeed, it is an experimental fact that the energy difference between the GT resonance and the isobaric analog state (IAS) behaves as where the first and second terms are interpreted as arising from the mean spin-orbit splitting and the difference in the corresponding residual interactions, when described within the TDA [48][49][50], respectively.Moreover, for the residual interaction (3.1), the last term in Eq. (3.3) implies that and we say that the PSU4SR takes place within the ph channel when this condition is met.(2) Particle-particle channel.For a system with N 6 = Z, the isospin and spin-isospin symmetries are violated in the mean field approximation, since the neutron excess automatically yields isovector mean fields, i.e., different Hartree-Fock (HF) potentials for protons and neutrons.This occurs even if the nuclear Hamiltonian commutes with the corresponding excitation operators O ± J .However, it was very early realized that when a nondynamical violation occurs in the HF solution, the RPA-induced ground state correlations (GSCs) can be invoked to restore the symmetry [51][52][53][54].These correlations are not explicitly evidenced, but only implicitly through their effects on the matrix elements and the corresponding decay rates [see, for instance, Eqs.For the F excitations, and when the isospin nonconserving forces are absent, a self-consistent inclusion of the GSCs (s = 1) leads to the following: (i) the β − strength S + IAS ≡ S {+1} 0 is fully concentrated in the IAS, (ii) the β + strength S {−1} 0 , which in pn-QRPA can be viewed as an extension of the β − spectrum to negative energies, is totally quenched, and (iii) the NME M 2ν − F (0 + 1 ) is null for all practical purposes.All this can be seen, for instance, from [ [20], Figs.1(b) and 1(c)], and [ [47], Fig. 1].The extent to which the above conditions are fulfilled at s = s sym = 1 may be taken as a measure of the isospin symmetry restoration.
Thus, in pn-QRPA model the SU(4) symmetry is partially restored within the pp channel at minima of the strengths S {+1} 0 , and S {+1} 1 .For the residual interaction (3.1), this occurs at s = s sym = 1, and t = t sym ∼ = 1.2. (3.6) Briefly, the spin-isospin symmetry is partially restored within the pn-QRPA when the conditions (3.4) and (3.6) are met, and this restoration is the prerequisite for a physically adequate evaluation of the observables.
It has also been shown that in the vicinity of the place where the restoration is performed, the F and GT NMEs behave á la Padé [20,[44][45][46][47], i.e., with s 1 > 1 being the value of s where the pn-QRPA collapses for J = 0, and with t 1 > t 0 , where t 0 and t 1 denote the zero and pole of this NME, respectively.In fact, for J = 1 the pn-QRPA collapses at t = t 1 .Such behaviors can be visualized, for instance, from [ [20], Figs.1(b) and 1(b 0 )].As noted before [47], in the pn-TDA, where the GSCs are neglected, the results for the observables are quite different, even if the self-consistence s = 1 is imposed.In fact, in this case, (i) the S {+1} 0 strength is always fragmented, (ii) the perturbed S {−1} 0 , and S {−1}  II], which are M 2ν − F (0 + 1 ) = −0.004,and M 2ν − GT (0 + 1 ) = 0.022.From the previous discussion it is clear that for the model developed here, we can use only the ph part (3.4) of the PSU4SR, namely (i) v ph s = 27 and v ph t = 64 for 48 Ca, and (ii) v ph s = 55 and v ph t = 92 for 96 Ru.With respect to the pp parameters, we observe that the first relationship in Eq. (3.6) is always valid (both in the QRPA and QTDA models), since it is a consequence of the self-consistency between the pairing interaction v pair s , and the isovector pp interaction v pp s .On the other hand, in the (pn, 2p2n)-QTDA there is no physical restriction on the isoscalar pp interaction v pp t , therefore we will basically use 1 states in the final nuclei 48 Ti and 96 Mo, respectively.Results from Refs.[18,58] for 48 Ca and from Refs.[9,14,[59][60][61] for 96 Ru, as well as the experimental value for 48 Ca [1], are also shown in same units.Suhonen's results [9] correspond to those listed in his Table II for the channel α = 2e. 48Ca  [9] 415-1437 492-1554 0.1-8.4[14] 251 [59] 101 [60] 5 4 s = 1, while the value of t, given in Eq. (3.6), will be taken only as a guide.Moreover, instead of using the bare value g A = 1.27 for the axial-vector coupling constant [55], we use an effective value g A = 1.0. 7Still smaller values for g A have been used in the literature [57].

C. Nuclear matrix elements
We calculate simultaneously the NMEs M 2ν , and M 2ν GT (2 + f ) for all above mentioned J + f final states with the following three sets of pp parameters: T 1 : s = 1.00, t = 0.80, T 2 : s = 1.00, t = 1.00, (3.9) In the upper part of Table III are shown the results for the decays 48 Ca → 48 Ti (0 + 1 , 0 + 2 , 2 + 1 ), evaluated with the two sets of spe drawn in Fig. 2. The results obtained by other authors [18,58], as well as the experimental value [1] of the NMEs, are shown below.The agreement between the calculated and measured results for M 2ν (0 + 1 ) can be considered satisfactory (in particular with spe e j ), taking into account that all nuclear parameters in the pp and ph channels are basically fixed, both for identical particles in H nn and H pp , and for nonidentical particles in H pn .We hope that in the near future the NMEs M 2ν (0 + 2 ) and M 2ν GT (2 + 1 ) will also be measured.Our results are consistent with the previous ones for all three decays.The NME M 2ν (0 + 1 ) has been calculated many times, but there are only a very few theoretical studies of M 2ν (0 + 2 ) and M 2ν GT (2 + 1 ).As far as we know, the first one of these has been evaluated only in Ref. [58], and the second one in Refs.[18,58] (see also Ref. [62]).
Table III shows that our F NMEs are not null, which is understandable since the current model does not consider the GSCs.To make them null, one has to go from (pn, 2p2n)-QTDA to (pn, 2p2n)-QRPA.This is a formidable task that we are planning to do next.Certainly, GSCs will also affect the results for other observables, whose scope is difficult to estimate at this time.
The present F NMEs are relatively small for both 0 + levels, when compared with the corresponding GT ones, but in no way are they negligible.As seen in Eq. (2.1), these two NMEs always interfere destructively.It should be pointed out that the contributions of M 2ν F (0 + f =1,2 ) have not been considered in the just mentioned studies [18,58], therefore, strictly speaking, their results for |M 2ν (0 It is interesting to note that the F NMEs are often omitted in the calculations, simply invoking isospin conservation.But, as explained above, this is a necessary but not sufficient condition for these NMEs to be null (see also the work of Satula et al. [63]).
An exception is the work of Šimkovic, Rodin, and Faessler [64], where partial restoration of isospin symmetry is achieved in the pn-QRPA, which leads to the disappearance of M 2ν F .This is accomplished by separating the renormalization parameter g pp into isovector and isoscalar parts.The isovector parameter g T =1 pp is chosen to be essentially equal to the pairing constant g pair , which is equivalent to our condition for selfconsistency s = 1 [see also the Eq.(3.7)].
In the lower part of Table III the results for the 96 Ru → 96 Mo (0 + 1 , 0 + 2 , 2 + 1 ) decays are shown.The outcomes of previous calculations [9,14,[59][60][61] are also presented.It is noticeable that the differences between our three calculations are much smaller than the differences with all the other works.
The discrepancies between the results for the NMEs presented in Table III come from the use of different nuclear models.For instance, in the study of the DBD + in 96 Ru, done by Suhonen [9], was used the multiple-commutator model (MCM), designed to connect states of the odd-odd nuclei to excited states of the neighboring even-even final nuclei [65].In this model, the states in the odd-odd nucleus, as well as the final J + f = 0 + 1 state, are described in the standard way, i.e., by the pn-QRPA in the form (2.26), while excited states of the even-even nuclei are generated by the charge-conserving QRPA (cc-QRPA), described in detail in Ref. [24].That is, GT , and M 2ν for the ground 0 + state (in natural units ×10 3 ), as a function of the pp parameters t and s for 48 Ti for (a) 48 Ca with spe e expt j , (b) 48 Ca with spe e j , and (c) 96 Ru.The experimental value for 48 Ti |M 2ν (0 + 1 )| = (38 ± 3) × 10 −3 is also indicated by the red line, whose width represents the experimental error.
the final states 3 , and 0 + 3 are assumed to be basic cc excitations (one-phonon cc-QRPA states), and the J + f = 0 + 2 and 2 + 2 states are considered as consisting of two 2 + 1 cc-QRPA phonons.Therefore, there is no physical similarity between the model used by Suhonen [9] and ours.The strong dependence of the NME within the pn-QRPA model with respect to the isoscalar pp parameter t is well known and is often discussed, therefore, it could be interesting to analyze that dependence in the current model.This is done in Fig. 3 showing the NMEs M 2ν F , M 2ν GT , and M 2ν for the ground 0 + state in 48 Ca and 96 Ru, as a function of the pp parameters t and s.The experimental value of |M 2ν (0 + 1 )| is also drawn.It can be concluded that within the present model such dependence is only moderate.The same statement is valid also for all remaining 2ν NMEs.

D. Half-lives
Knowing the NMEs and the leptonic kinematic factors G α 2ν (J + f ), the corresponding half-lives τ α 2ν (J + f ) are evaluated trivially from (2.48).Despite this, we present some of them only for the sake of completeness.
Our results for τ 2β − 2ν (J 48 Ca, evaluated with spe e j and the three sets of the pp parameters are compared with the previous calculations and with the data in Table IV.For the state 0 + 1 we agree both with the experiment [1] and with the SM calculation [58].On the other hand, our half-life for the first 2 + 1 is shorter by a factor of about 2 than the same evaluated with SM [58], and by a factor of about 4 than the one obtained by Raduta et al. [18]. in units of yr for the 2ν-DBD 96 Ru → 96 Mo, with the pp parameter set T 2 and g A = 1 are shown and confronted with previous calculations.In Eq. (2.48) we use the G α 2ν (J + f ) factors from Ref. [17] for the channels 2β + (=1.080 × 10 −26 yr −1 ), eβ + (=0.454 × 10 −21 yr −1 ), and 2e(=2.740× 10 −21 yr −1 ).In the case of 96 Ru, our results for the half-lives τ α 2ν (0 + 1 ), for different decay modes (α = 2β + , eβ + , 2e), are shown in Table V.All of them are larger by one to three orders of magnitude when compared with previous theoretical studies, which are shown in the same table.The biggest discrepancies are in the pn-QRPA calculation made in Ref. [14].The origin of what is difficult to elucidate given the difference between the two models used.
The experimental limits for the 96 Ru half-lives are τ 2β + 2ν (0 + 1 ) > 1.4 × 10 20 yr and τ eβ + 2ν (0 + 1 ) > 0.8 × 10 20 yr [68].Thus, from Table V we see that our predictions are nine orders of magnitude larger for the 2β + mode, and four orders of magnitude larger for the eβ + mode.This, in turn, implies that the accuracy of the measurements would have to be significantly improved to observe these processes.
We have not evaluated the half-lives of the 96 Ru final states J + f = 0 + 2 , 2 + 1 , since their kinematic factors G α are not available.

E. Q values and energy spectra
Before starting with the discussion of Q values, it is convenient to remember that a physical phenomenon is allowed only when this quantity is positive.
In Table VI the experimental data are confronted with our results for the Q ββ and Q 2e values in the DCE processes: (a) 48 Ca → 48 Ti and 48 Ca → 48 Ar, and (b) 96 Ru → 96 Mo and 96 Ru → 96 Pd.One sees that the model is capable of reproducing fairly well not only the signs of the Q-values, but also their magnitudes, without having to modify the parameters of the model.This is very encouraging!In addition, it seems that the model "knows" what type of DCE decay can occur in a given nucleus.
The nature of Q value is predominantly determined by the proton and neutron pairing mean fields, as seen from (2.41) or, more precisely, from the relation The dependence on the residual interaction is rather weak and takes place through the ground state energy ω 0 + 1 in the residual nuclei, as More details on how the Qvalues depend on the pp coupling constants are shown in the Fig. 4.
As we stated before, in the same way that the pn-QRPA model [12] predicts identical energy spectra for odd-odd nuclei (A, Z ± 1), the present model predicts identical excitation energies in even-even nuclei (A, Z ± 2).This is obviously not realistic due to the large neutron excess.
It is pertinent to mention here that the use of particlenumber-projection can become very important when working with the BCS mean-field [42].Without a doubt, with this method different energy spectra are obtained in nuclei with the same mass number but different numbers of protons and neutrons.
But, despite the above-mentioned disadvantage, the predicted 0 + and 2 + excitation energies in 48 Ti and 48 Ar are consistent with the data, as shown in Fig. 5 for the spes e expt j , e j , and pp parametrization T 2.
Because of the size of the Q 2β − value (= 4.268 MeV), the 2β − decays are energetically possible for all states, except for the 0 + 4 .We have evaluated the NMEs for all these states, but we do not consider it necessary to present them here.We have  not found in the literature any detailed calculation of the low energy 48 Ti spectrum to compare with the ones obtained for us and shown in Fig. 5.
The experimental energy spectra of 96 Pd and 96 Mo are poorly explained by our calculations.However, for the sake of completeness, the comparison between theory and data is presented in Fig. 5(b).These two nuclei exhibit very different spectra from each other, and most likely no nuclear model explains their properties simultaneously, without dramatically varying the parameters of the model, which we do not do here.It is possible that GSCs significantly modify the energy spectra of final even-even nuclei, that is, when switching from (pn, 2p2n)-QTDA to (pn, 2p2n)-QRPA, as happens with the spectra of odd-odd nuclei when changing from pn-QTDA to pn-QRPA [24].
Excitation energies of 96 Mo are not evaluated in the MCM [9] as we do.What is done in the cc-QRPA model is to adjust the theoretical energies to the experimental ones, obtaining in this way the particle-hole channel parameters.This is carried out for each one-phonon state [65].

F. Double-charge-exchange strengths and their sum rules
The results for the DCE transition strengths S {±2} JJ given by (2.15), both for Fermi (J = 0; J = 0) and Gamow-Teller (J = 1; J = 0, 2) are displayed in Table VII.The corresponding sum rules S {2} JJ calculated from (2.16) are also shown, and confronted with the predicted sum rules S {2} JJ given by (2.17)- (2.19).In addition, to know the locations of the DCE resonances, we have evaluated the energy centroid (2.50).
To obtain an idea of the magnitudes of the DBD, the strengths B {+2} JJ 1 (B {−2} JJ 1 ) going to the levels 0 + 1 and 2 + 1 in final 48 Ti ( 96 Mo) nucleus are explicitly given.
All calculations related to DCE transition strengths were performed for the three sets of pp parameters (3.9), finding that all produce identical results.We have also found that, at least in the case of 48 Ca, there is a certain dependence of the results with respect to the used spe spaces.
The results derived for 48 Ca within the SM for the p f space8 by other authors are given in the last three bins of the upper part of Table VII-specifically, by (i) Sagawa and Uesaka [5] with GXFF1A interaction, and (ii) by Auerbach and Minh Loc [6] and Shimizu, Menéndez, and Yako [7], both with KB3G interaction.In fact, the values of strengths S {+2} 1J =0,2 attributed to the last authors [7] have been extracted from their Fig.1(b).
Several observations are in order regarding the results shown in Table VII: (1) The strengths S {−2} JJ are always small in comparison with the strengths S {+2} JJ and, as a consequence, JJ .This is clearly due to the relatively large neutron excess.
(2) Although small, the strengths S {−2} JJ are significant in relation to the DBD.They are proportionally higher in 96 Ru, which decays by 2β + , than in 48  JJ are dimensionless, and the energies are in MeV.The SM results, obtained by Shimizu et al. [7] with the KB3G interaction, are also shown.
As seen in Fig. 7, the DIAS in 96 Pd is located at around 21 MeV, while both J = 0 + and J = 2 + DGTGRs are at about 23 MeV.These resonances are not directly related to the DBD of 96 Ru, but their locations in 96 Pd can be searched through heavy ion reactions.
The smallness of B {−2} 00 and its energy distribution, shown in Fig. 8, are fully consistent with the small value of M 2ν F (0 + 1 ) in The role played by the residual interaction in generating DIAS is illustrated in Fig. 9, where a few results for the F distribution B {+2} 00 in 48 Ti are exhibited.Panel (a) shows how the unperturbed (BCS) strength moves upwards in energy by about 15 MeV, due to the residual interaction in the ph channel.In panel (b), the F distributions are shown when the full residual interaction acts, with t = 1, and several different values of s.The maximum concentration of strength occurs for s = 1 at around 22 MeV, thus generating the double isobaric analogous state (DIAS).This, in turn, clearly shows the importance of the self-consistency (s = 1) between the pairing force and the pp isovector interaction.
But, as seen from the insert in the panel (b), not all the double F strength is contained in a single state.If this would happen after moving from the QTDA to the QRPA, and thus completely reset the spin-isospin symmetry, it remains an open question.

H. Comparison between ground state 2ν-DBD NME and DCE strengths
In the so-called closure approximation is used the completeness character of the intermediate states |J α i in Eqs.(2.1) and (2.9), after replacing the energies E {∓1} J α in Eq. (2.3) by some average Ē {∓1} J [67].Thus, except for the constant energy denominator, the ground state DCE densities B {±2} J0 1 are the closure approximations of the squares of the NMEs.In view of this, to know how reasonable the closure approximation is, it may be interesting to compare the behaviors of these two quantities as a function of the pp parameters.
As an example, in Fig. 10   JJ are dimensionless, and the energies are in MeV. 48Ca with the spe e j .The squares of the NMEs are in natural units, while the strengths are dimensionless.The proportionality between these two observables suggests that the closure approximation in the case of 48 Ca is reasonable.However, there is no guarantee that this result will be valid i n general.

IV. FINAL REMARKS
We have developed a nuclear structure model that involves (pn, 2p2n)-QTDA excitations on the BCS mean field, which is capable of simultaneously describing the DBD and the DCE transition strengths.So far, this has been done only in the context of SM, where these two problems are generally treated separately, although it is well known that they are intimately related to each other.This is the case, for instance, of 48 Ca, where the DBD − s are described in Refs.[18,58], while the transition strength distributions B {+2} JJ and the corresponding total strengths S {+2} JJ were evaluated in Refs.[5][6][7].The (pn, 2p2n)-QTDA model has additional advantages over the standard pn-QRPA model, namely: (1) Together with the NMEs of the ground state, the NMEs of all the excited states 0 + and 2 + are calculated simultaneously.In order to do the same in the pn-QRPA model, it is necessary to resort to supplementary calculations through several charge-conserving QRPAs, thus introducing several new model parameters.(2) It allows the evaluation of the Q values for DBDs, which play a very important role in these types of processes.
The (pn, 2p2n)-QTDA aims to describe the DCE processes and is the natural extension of pn-QRPA, developed by Halbleib and Sorensen [12] to describe the SCE processes.However, the first model, unlike the second, does not include ground state correlations, which play a central role within the pn-QRPA to reduce NMEs.To include this type of correlation at the level of 2p2n excitations, it is necessary to go from (pn, 2p2n)-QTDA to (pn, 2p2n)-QRPA.
It should be borne in mind that, in the same way as the pn-QRPA put into play the seniority-4 configurations, the (pn, 2p2n)-QRPA will induced the seniority-6 and -8 configurations, which could be relevant in the evaluation of NMEs, such as are in the shell model calculations [69,70].
Our next aim is to evaluate and discuss the 0ν-NMEs (2.12) making use of the replacement (2.11) in our previous work [20].One expects that the relationship between DCE nuclear reactions and DBD will be more clearly visible at 0ν decays than at 2ν decays, due to a lower dependence on the NME of their energy denominators than in the first case.
During the development of the present study, Santopinto et al. [71], based on a previous work of Bertulani [72], have reported that, in the low-momentum-transfer limit, the heavy ion 40 Ca( 18 O, 18 Ne) 40  where P and T stand for projectile and target nuclei respectively.
The correspondence with our notation is as follows: (1) For the matrix elements10 Therefore, the present model possesses all the necessary ingredients to evaluate the heavy-ion cross section in the low-momentum-transfer limit.Of course, now it is necessary to solve two eigenvalue problems, one for the target nucleus 40 Ca, and one for the projectile nucleus 18 O.
In summary, we have developed a new model, based on the BCS approach, to describe the double-charge-exchange nuclear phenomena (A, Z ) → (A, Z ± 2).It is a natural extension of the Halbleib and Sorensen [12] model, aimed to describe the single-charge exchange processes (A, Z ) → (A, Z ± 1).As an example, detailed numerical calculations are presented for the (A, Z ) → (A, Z + 2) process in 48 Ca → 48 Ti and the (A, Z ) → (A, Z − 2) process in 96 Ru → 96 Mo, involving several final 0 + and 2 + states.At the moment we are extending this study in three directions: (1) A thorough evaluation of all 2ν-DBD ± , together with the associated nuclear reaction strengths will be performed.
(3.7)  and(3.8)below].Here it is convenient to define the ratios s (p) + v pair s (n)]/2.In the pp channel the F (GT) excitations depend only on s(t ).

FIG. 4 .
FIG. 4. Calculated Q 2β − and Q 2e values in 48 Ca with spe e expt j (a), and with spe e j (b), and 96 Ru (c), as a function of pp parameters t and s.The experimental Q values are also shown.

FIG. 5 .
FIG. 5. (a)Experimental excitation energies of the final states J π = 0 + and 2 + are compared with calculations, performed with the pp parametrization T 2, for (a)48 Ti and48 Ar and (b)96 Pd and 96 Mo.
FIG. 6. DCE strength distributions B {+2}JJfor the transition48 Ca → 48 Ti with spe e j , and e expt j , and pp strengths T 2, as a function of the excitation energy E in48 Ti.The B {+2} JJ are dimensionless, and the energies are in MeV.The SM results, obtained by Shimizu et al.[7] with the KB3G interaction, are also shown.

FIG. 7 .
FIG. 7. DCE strength distributions B {+2} JJ for the transition 96 Ru → 96 Pd with pp strengths T 2 as a function of the excitation energy E in 96 Pd.The B {+2} JJ are dimensionless, and the energies are in MeV.

FIG. 8 .
FIG. 8. DCE strength distributions B {−2} JJ for the transition 96 Ru → 96 Mo. with pp strengths T 2 as a function of the excitation energy E in 96 Mo.The B {−2} JJ are dimensionless, and the energies are in MeV.

FIG. 9 .
FIG. 9. DCE strength distributions B {+2} 00 for the transition 48 Ca → 48 Ti with spe e expt j , for several parametrizations of the residual interaction.The inserted figure in (b) shows the splitting of the double F strength within the DIAS.See text for details.

TABLE I .
Results for the BCS coupling constants and Fermi levels.All notation is explained in the text.The λ's are given in units of MeV, and the couplings v pair s is in units of MeV fm 3 .

TABLE II .
Neutron and proton spe for 96 Ru, which were obtained in the manner explained in the text, together with the resulting quasiparticle energies(3.3).All notation is explained in the text.The energies are given in units of MeV.

TABLE VI .
Calculated Q ββ and Q 2e values (in units of MeV) with the three sets of pp parameters T 1, T 2, and T 3 and for the DCE processes: (a)48Ca → 48 Ti and48Ca →48Ar and (b)96Ru → 96 Mo and 96 Ru → 96 Pd are confronted with experiments.
Table III and of B {−2} 00 1 in Table VII.Moreover, the distributions of the B {−2} 1J =0,2 clearly indicate that the DBD + of 96 Ru will be very slow.