Strength of J π = 1 + Gamow-Teller and isovector spin monopole transitions in double-β-decay triplets

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I. INTRODUCTION
Theoretical study of the nuclear-structure properties of double-beta-decay emitters is one of the most challenging activities in the field of nuclear physics, and it is of utmost relevance for the analysis and design of experiments [1,2].Analyses of the related nuclear matrix elements (NMEs) have mostly been performed within the framework of the protonneutron quasiparticle random-phase approximation (pnQRPA) or some higher order variant of it (see, e.g., [3]).However, many other models have also been used earlier [2] and new models have joined recently [4].Among the various nuclear-structure aspects related to double-beta-decay studies, some attention has been recently paid to detailed comparisons between experimental results of single-beta decay and chargeexchange-reaction data and theoretical calculations of the energy distribution of the intensity for allowed Gamow-Teller transitions (see, e.g., [5]).
Low-energy and high-energy Gamow-Teller (GT) strength distributions are a very useful tool to access the validity of various theoretical assumptions and approximations, such as the sequences of single-particle states, the values of the level occupancies, and the sizes of the model spaces used to perform calculations (see, e.g., [6][7][8]).Here we shall focus on the systematics of the excitation of the isovector spin monopole (IVSM) mode.In a first investigation on the subject [9] we addressed the question of the interplay between the IVSM and Gamow-Teller (GT) modes in the context of a schematic model, and tested its results by performing realistic pnQRPA calculations.In this work we extend our search to some of the most favored double-beta-decay emitters in order to extract some conclusions about the energetics, strength distributions, and mass dependence of total intensities of the IVSM ∓ modes in the (p,n) and (n,p) charge-exchange directions.
In Sec.II we give a brief introduction to the underlying formalism of the GT and IVSM modes of excitation and in Sec.III we display and discuss the obtained results.The final conclusions are drawn in Sec.IV.

II. BRIEF REVIEW OF THE FORMALISM
The formalism developed in [9] is now applied to calculate the nuclear response to the excitation of Gamow-Teller (GT) and isovector spin monopole (IVSM) modes, starting from the ground states of few selected even-even nuclei.For the present study we have chosen the double-beta-decay emitters in the A = 76, A = 82, A = 100, A = 116, A = 128, A = 130, and A = 136 triplets of isobars, in order to perform a case-bycase analysis of the transitions.In particular, we aim at the identification of the IVSM strength at low energies because of the potentially important implications upon nuclear doublebeta-decay studies [10][11][12][13][14][15][16].The main steps of the calculations are the following: (i) Single-particle bases: We build the single-particle bases for each involved nucleus by solving the radial Schrödinger equation for a Coulomb-corrected Woods-Saxon potential by starting from the nl j = 0s 1/2 state (no-core basis), for both protons and neutrons.We use the Woods-Saxon parameters given in [17].We deal with bound states and quasibound states only.Extra care has to be taken when dealing with the quasibound states near the top of the angular-momentum and centrifugal barriers.These cases can be dealt with by very carefully choosing the iteration increment of the solver for unbound states.The sequence of levels is adjusted, when possible, to correspond to the observed sequence of single-particle states and/or the known experimental occupancies of the key orbits of calculation.These adjustments affect the values of monopole pairing strengths, pairing gaps, quasiparticle energies, occupancies of orbitals, etc., as explained next.(ii) Pairing gaps, quasiparticle energies, and BCS occupancy factors: The two-body potential used in the calculations was derived from the Bonn-A oneboson-exchange potential [18].The monopole pairing strength was fixed by fitting the BCS gaps to the observed odd-even mass differences [19][20][21].We tabulate the values of the pairing strength constants and the obtained proton and neutron gaps in the next section.(iii) Spectra of the 1 + excitations in odd-odd nuclei: The wave functions and energies, for the complete set of 1 + excitations in the odd-odd nuclei, were obtained by performing a pnQRPA diagonalization in the space of unperturbed quasiproton-quasineutron pairs coupled to J π = 1 + .The formalism is well known and we shall skip the details here (see, e.g., [19,21]).In the tables of the next section we give for each mass system the values of the relevant parameters, i.e., strengths of the particle-hole and particle-particle interactions used in the calculations.(iv) Transition operators and strength distributions for the GT and IVSM excitations: The transition operators are written in terms of quasiproton-quasineutron pairs and further transformed to the pnQRPA basis by expressing the transition densities in terms of onephonon variables.
The final expressions for the relevant transition matrix elements, connecting the initial ground state with the kth 1 + state, read [9,19] 1 where X m pn and Y m pn are the forward-and backward-going amplitudes of the state with energy E m .Furthermore, u and v are the BCS vacancy and occupancy factors, respectively.The GT and IVSM operators are defined as and the strength S, for each operator, is given by the sum over all 1 + states belonging to the spectrum of the daughter odd-odd mass nucleus, such that For the GT operator the difference between the total intensities of the two branches of excitation gives the standard Ikeda sum rule, S − (GT) − S + (GT) = 3(N − Z), which is model independent and which is fulfilled if both members of the spin-orbit pairs corresponding to a given value of the single-particle orbital angular momentum are included in the basis [19].This sum rule is preserved quite accurately by all the present pnQRPA calculations.For the IVSM mode there is no model-independent sum rule due to the radial dependence of the operator, but it is possible to extract a limit for it, in the chosen model space for calculations, by shifting the IVSM operator with the excess square radius r 2 excess [22], defined as the average over the excess neutron orbits.By this procedure the appropriate IVSM strength is obtained by the use of the We use this prescription to obtain the IVSM − and IVSM + strength distributions discussed in Sec.III.

III. RESULTS AND DISCUSSION
In this section we present and discuss the results of the calculations.For the sake of completeness we shall introduce each of the elements entering the calculations, referring to the already published material for details.

A. Single-particle bases and energies
The single-particle energies are the eigenstates of the Woods-Saxon potential (adding Coulomb force for protons) and we keep bound and quasibound states, e.g., eigenstates with very small decay width.The parameters of the central, orbital, and spin-orbit terms of the potential, as well as the radius and surface thickness parameters for each term, are taken from [17].Since these are no-core calculations, we are taking all orbits from the N = 0 oscillator shell up to two oscillator major shells above the respective Fermi surfaces for protons and neutrons in each system.For the A = 82,116,130,136 isobars the Woods-Saxon single-particle energies work well, but for the A = 100 isobars we have to resort to the "EXPWS" energies of Ref. [23] and for the A = 128 isobars we use the basis introduced in [5].
For the A = 76 isobars we can use two prescriptions.In Table I we show for the mass A = 76 system the values of single-particle energies of protons and neutrons for the TABLE III.Renormalization factors for the particle-hole and particle-particle interactions in the J π = 1 + channel, and the energy of the first pnQRPA eigenvalue (in MeV) relative to the ground state of the mother nucleus.Also are given the excess radii used to compute the IVSM − and IVSM + strength distributions by the prescription (4).For the A = 76 nuclei "WS + BCS" denotes the Woods-Saxon-based and "exp" the experiment-based occupancies adopted in the calculations.6) and (8), in the transition 76 Ge → 76 As.The excitation energies are measured from the ground state of 76 Ge.The IVSM strength is given in units of fm 4 .The occupancies of the single-particle orbitals are based on either the WS + BCS model (left panels) or on experiment (right panels).

Mother nucleus
Woods-Saxon potential, adopted as input for the calculation of the occupancies and quasiparticle spectra in this work.Furthermore, we compare the results emerging from these occupancies and quasiparticle energies with those emerging from the experimentally measured proton and neutron occupancies [24,25] at the respective Fermi surfaces.These experimental occupancies were already used in the works [6][7][8]26].The information about the single-particle levels and their energies for the mass A = 100, A = 116, and A = 128 systems has been given in Ref. [5], in Tables 2 and 3.
As already mentioned, we can adopt for the mass A = 76 case the measured occupancies of Refs.[24,25].To cope with the experimental occupancies one can take as a starting point the Woods-Saxon energies of Table I and perform the usual BCS calculation, after which one subsequently replaces manually the computed occupancies with the measured ones close to the proton and neutron Fermi surfaces.These partly modified occupancies are not quite consistent with the underlying BCS calculation, but it was shown in [6] that the resulting average proton and neutron numbers correspond quite accurately to the  7) and (9), in the transition 76 Se → 76 As.The excitation energies are measured from the ground state of 76 Se.The IVSM strength is given in units of fm 4 .The occupancies of the single-particle orbitals are based on either the WS + BCS model (left panels) or on experiment (right panels).actual numbers of protons and neutrons in the 76 Ge and 76 Se nuclei.The occupancies then determine the theoretical pairing gaps and quasiparticle energies, and the quasiparticle energies in turn, together with the occupancies, determine uniquely the results of the pnQRPA equations of motion.

B. Pairing gaps, quasiparticle spectra, and BCS occupation factors
The monopole two-body matrix elements generated from the Bonn-A interaction were adapted to finite nuclei by an overall renormalization of the strength of the monopole proton and neutron channels, separately.The corresponding scaling factor for neutrons (protons) is denoted by g (n)  pair (g pair ), and its adopted values are tabulated in column two (column three) of Table II.With the tabulated scaling factors the BCS-calculated pairing gaps reproduce the empirically deduced gaps that are tabulated in columns four and five of Table II.Using the tabulated scaling factors we have performed BCS calculations .The excitation energies are measured from the ground state of 82 Se for GT − and IVSM − , and from the ground state of 82 Kr for GT + and IVSM + .The IVSM strength is given in units of fm 4 .
to obtain the one-quasiparticle energies and occupation factors needed to perform the subsequent pnQRPA calculations.
The occupancies, for the cases in which they have been determined experimentally, as in the case of the A = 76 system, are valuable indicators of the reliability of the adopted single-particle energies [6,26].In addition to the systematics on the observed odd-even mass differences, the adjustment of the single-particle energies [23,[27][28][29][30] reduces the tension with the data concerning low-lying quasiparticle excitations in the adjacent odd-mass nuclei.The different adjustment schemes and the resulting occupation amplitudes have been tabulated extensively for 76 Ge, 76 Se, 130 Te, and 130 Xe in Tables 1-4 of [6].In the present work we exploit the "WS + BCS" and experimental schemes of these tables for A = 76 and the Woods-Saxon scheme for A = 130.For the A = 100, A = 116, and A = 128 systems the occupancies (occupation amplitudes squared times the degeneracies of the j orbitals) at the respective Fermi surfaces have been tabulated in Ref. [5], in Tables 5, 6, and 7. .The excitation energies are measured from the ground state of 100 Mo for GT − and IVSM − , and from the ground state of 100 Ru for GT + and IVSM + .The IVSM strength is given in units of fm 4 .

C. Spectra of the 1 + excitations in odd-odd nuclei and the GT and IVSM transitions
For each of the isobaric systems with mass number A we have taken the triplet to be composed of "left-hand-side" eveneven (A,N,Z), "right-hand-side" even-even (A,N − 2,Z + 2), and "intermediate" odd-odd (A,N − 1,Z + 1) nuclei.The spectra of 1 + excitations, belonging to the odd-odd mass nuclei, were constructed by applying the pnQRPA formalism [20,21,31], including particle-hole and particleparticle channels in the residual (proton-neutron) two-body interactions.The linearization procedure yields, therefore, two sets of energies and wave functions of 1 + states which are reached by the action of the t − and t + components of the GT and IVSM operators from the ground states of the left-hand-side and right-hand-side nuclei, respectively.Table III shows the values of the particle-hole renormalization factors g ph which roughly reproduce the energetics of the GT − giant resonance (GTGR).In the table we list also the values of the particle-particle renormalization factor g pp and the energies of the first excited 1 + state obtained by applying the pnQRPA formalism starting from the left and right ground states of the double-even mass nuclei of each mass system.
While the value of g ph is determined by the properties of the GTGR, the value of the particle-particle parameter g pp is not so easily pinned down.Typically its value has been fixed either by the half-lives of 2νββ decays [32][33][34][35][36] or the comparative half-lives (log f t values) of beta decays [27,37].In [5,38] a novel approach was introduced where both beta-decay and 2νββ-decay data was used to determine both the value of g pp and the (effective) value of the axial-vector coupling constant g A (β) simultaneously.From the analysis of [5] we can extract the following average values of these quantities: g pp = 0.63 ± 0.17, g A (β) = 0.57 ± 0.21. (5) We adopt the above value of the particle-particle coupling strength to our further analyses of the GT and IVSM properties of the discussed triplets of isobars listed in Table III.6), ( 7), ( 8) and ( 9), in the transitions 128 Te → 128 I [panels (a) and (c)] and 128 Xe → 128 I [panels (b) and (d)].The excitation energies are measured from the ground state of 128 Te for GT − and IVSM − , and from the ground state of 128 Xe for GT + and IVSM + .The IVSM strength is given in units of fm 4 .

B(IVSM)
where 0 + L (0 + R ) is the ground state of the left-hand-side (righthand-side) even-even nucleus and 1 + m is the mth 1 + state in the intermediate nucleus.These strength functions are illustrated in Figs.1-8.In Figs. 1 and 2 the left panels show the results based on the WS + BCS occupancies, whereas the right panels show the results based on BCS occupancies supplemented with the experimental occupancies for the key orbitals at the proton and neutron Fermi surfaces.
There is a possibility to compare the presently computed GT − and GT + distributions with the corresponding experimental ones [13] for the A = 76 case.The experimental GT − distribution is conveniently displayed in the uppermost panel  6), ( 7), ( 8) and ( 9), in the transitions 136 Xe → 136 Cs [panels (a) and (c)] and 136 Ba → 136 Cs [panels (b) and (d)].The excitation energies are measured from the ground state of 136 Xe for GT − and IVSM − , and from the ground state of 136 Ba for GT + and IVSM + .The IVSM strength is given in units of fm 4 . of Fig. 1 of [39] and the GT + distribution in the uppermost panel of Fig. 3 of the same article.When comparing the left and right upper panels of our present Fig. 1 with the experimental GT − distribution [39], one notices a great similarity of the experimental and computed distributions: there are two strong peaks below 10 MeV of relative excitation energy (i.e., relative to the energy of the first 1 + state) both in the calculated and in the experimental distributions, in addition to the strong GTGR peak at around 11-12 MeV of relative excitation.It seems that both the "WS" and "exp" based calculations reproduce the gross features of the experimental GT − distribution rather well.Comparing the experimental and theoretical GT + distributions one notices that the "WS" based calculation reproduces the strength of the experimental distribution but is twice as broad as it.The "exp" based calculation has too much GT + strength at low energies.Also for the A = 100 and A = 128 cases the computed GT − distributions agree nicely with the experiments as seen in Tables 10 and 12 of [5].
At this point it is worth noting that, with respect to the twoneutrino double-beta decays, the decay of 116 Cd shows singlestate dominance [40,41], as clearly shown in Fig. 3 of [5].Instead, the decay 100 Mo is not quite single-state dominated (see Fig. 3 of [5]) and the decay of 128 Te not at all as seen in Fig. 4 of [5].The decays of 76 Ge, 82 Se, 130 Te, and 136 Xe are not single-state dominated, as seen from the sizable overlap of the GT − and GT + strengths in Figs.1-3, 7, and 8.

E. Energy centroids and total and peak intensities of the GT and IVSM modes
In Tables IV-VII we show the results for the calculated energy centroids, both for GT ± and IVSM ± transitions, as well as the energies and intensities of the states which carry the largest strength for each type of operator.The overall feature which emerges from these results is pretty consistent with the model estimates of our previous work [9]; that is, (a) the fraction of the total intensity carried by the state with the largest intensity is roughly the same for GT − and IVSM − modes (which is approximately 20-30 % of the strength), and (b) the energy difference between the two energy centroids (as also between the states with the largest intensities) is of the order of 2 ω.Concerning the mass dependence of the right-hand side of the transitions, the results for the IVSM + mode are very much concentrated around energies of the order of 20-25 MeV, while the intensities of the GT + tend to be dominated by few states in the low-energy part of the spectrum.
As seen in Figs.1-8, there is also a visible mismatch between the branches of the GT − and GT + excitations at the GTGR energies, indicating that there are practically no contributions to the 2νββ NME that stem from the GTGR region and beyond.This was shown tangibly for the A = 100, A = 116, and A = 128 double-beta systems in [5] by recording cumulative sums of the 2νββ NMEs.The other interesting feature concerning the IVSM modes is the appearance of significant strength, for the IVSM − side of the transitions, at energies of the order of 10-20 MeV where also the GT − giant resonance appears.This happens consistently for all the considered triplets of isobars.For example, for 76 Ge there is an exact match of the IVSM − and GT − strengths at the largest (two largest) GT − peaks for the "WS + BCS" occupancies ("exp" occupancies) in the left (right) panel of Fig. 1.For 100 Mo there is an exact match at the GT − peak energy (Fig. 4) and for 116 Cd a number of matching strengths for the GT − peaks between 14 and 17 MeV, as seen in Fig. 5.In these cases the interaction between both modes may be possible.
The matching of the GT − peaks and IVSM − peaks around the GTGR region is further studied systematically in Table VIII, where we show the amplitudes of transitions to 1 + states around the GTGR, for which the magnitude of the GT − amplitude is larger than unity.These transitions constitute the most important contributions to the GT − and IVSM − strengths around the GTGR region.As seen from the table, the ratio between the IVSM − amplitudes (expressed in fm 2 ) and the GT − amplitudes is of the order of 7 to 10.They are, of course, in phase, as it is expected from the structure of both transition operators.Hence, interaction between the GT − and IVSM − TABLE IV.Energetics of the GT − and IVSM − modes.The quantities E max and E centroid are the energy of the state with the largest intensity and the energy centroid of the intensity distribution of each mode.All values are expressed in units of MeV.The two adopted occupancy schemes for 76 Ge have been indicated by "WS + BCS" and "exp".VIII.Amplitudes of the GT − and IVSM − transitions, given in the last two columns, from the ground states of the nuclei of column 1 to the J π = 1 + states of the nuclei listed in column 2. Listed are the amplitudes of transitions to states around the GTGR with a magnitude of the GT − amplitude larger than unity.The third column gives the pnQRPA energy.For 76 Ge both the "WS + BCS" occupancies and the "exp" occupancies are used in the calculations.modes is likely and it may induce a shift to lower energies of some of the spin-isospin strength carried by the IVSM − mode, as predicted by the perturbative analysis of Ref. [9], performed in the context of a schematic model.This could have significant effects for the experimental analysis of the GT − strength at the GTGR energies.Concerning the differences caused by the two occupation schemes used for the A = 76 triplet of isobars we notice from Fig. 1 and Tables IV and V that no drastic differences are observed with the resulting GT − and IVSM − distributions of strength.The GT − distributions are quite similar, as is also indicated by the cumulative sum of Fig. 9, but there is some strength redistribution for the IVSM − strength when going from the "WS + BCS" to the "exp" occupancies.For the GT + and IVSM + strength distributions a stronger effect appears, as noticed from Fig. 2 and Tables VI and VII.In particular the total and peak GT + strengths are drastically increased when going from the "WS + BCS" to the "exp" occupancies.The total IVSM + strength is not much affected by going from one occupancy scheme to the next but there is some redistribution of strength at the main peak.

Nucleus
Finally, in Fig. 10 we show the mass dependence of the total effective GT − and GT + strengths (left panel) and the total effective strengths of the IVSM ± transitions (right panel) taken from Tables V and VII.The GT results of Fig. 10 agree with the Ikeda 3(N − Z) sum rule.The interesting thing about the IVSM modes is that the total IVSM − strength has a much steeper slope than the total IVSM + strength as a function of the mass number A. This makes the difference of the strengths an increasing function of A. As already noted earlier, for the A = 76 isobars the results with the BCS and experimental occupancies do not deviate notably from each other in the case of the IVSM strengths.For the GT + strength the relative difference is notable.  7Ge → 76 As calculated by using the "WS + BCS" and the "exp" occupancies.The excitation energies are measured from the ground state of 76 Ge.The IVSM strength is normalized by dividing by R 4 where R = 5.08 fm is the nuclear radius.

IV. CONCLUSIONS
In this work we have performed realistic pnQRPA calculations of the GT and IVSM excitations in double-odd mass nuclei, which belong to double-beta-decay triplets with A = 76, A = 82, A = 100, A = 116, A = 128, A = 130, and A = 136.The calculations were performed in large single-particle bases with realistic Bonn-A based two-body interactions.We have fitted the observed odd-even mass differences by adjusting the couplings of the pairing monopole channels separately for protons and neutrons.Similarly, we have fitted the energy of the GTGR centroids by varying the parameters entering the proton-neutron particle-hole channel of the pnQRPA.The systematics shows that the IVSM and GT modes may be correlated, and that the presence of strength, due to the t − side of the IVSM excitations at energies near that of the GT − giant resonance may be significant for double-beta-decay and (p,n)-type of reaction studies.6) and ( 7), as functions of the mass number.Right panel: The total effective IVSM − (upper curve) and IVSM + (lower curve) strengths, ( 8) and ( 9), as functions of the mass number in units of fm 4 .The small circles indicate the results obtained with the "exp" occupancies for the A = 76 nuclei.For the IVSM the circles coincide in the scale of the figure (see also Tables V and VII).
The overall picture which emerges from these calculations seems to stress the need of a detailed exploration of the region of excitations between 10 to 20 MeV in the odd-odd nuclei which are members of double-beta-decay triplets.There the presence of (p,n) strength may be partly due to excitations of the type σ r 2 t − .

FIG. 1 .
FIG. 1. Distribution of the GT − [panels (a) and (b)] and IVSM − [panels (c) and (d)] strength, (6) and (8), in the transition76 Ge → 76 As.The excitation energies are measured from the ground state of 76 Ge.The IVSM strength is given in units of fm4 .The occupancies of the single-particle orbitals are based on either the WS + BCS model (left panels) or on experiment (right panels).

FIG. 2 .
FIG.2.Distribution of the GT + [panels (a) and (b)] and IVSM + [panels (c) and (d)] strength,(7) and(9), in the transition76 Se → 76 As.The excitation energies are measured from the ground state of 76 Se.The IVSM strength is given in units of fm4 .The occupancies of the single-particle orbitals are based on either the WS + BCS model (left panels) or on experiment (right panels).

FIG. 9 .
FIG. 9. Cumulative sums of the effective GT − strength (6) [left panel] and IVSM − strength (8) [right panel] for the transition76 Ge → 76 As calculated by using the "WS + BCS" and the "exp" occupancies.The excitation energies are measured from the ground state of 76 Ge.The IVSM strength is normalized by dividing by R 4 where R = 5.08 fm is the nuclear radius.

FIG. 10 .
FIG.10.Left panel: The total effective GT − (upper curve) and GT + (lower curve) strengths, (6) and(7), as functions of the mass number.Right panel: The total effective IVSM − (upper curve) and IVSM + (lower curve) strengths, (8) and (9), as functions of the mass number in units of fm4 .The small circles indicate the results obtained with the "exp" occupancies for the A = 76 nuclei.For the IVSM the circles coincide in the scale of the figure (see also TablesV and VII).

TABLE I .
Adopted neutron and proton single-particle states and their energies for the mass A = 76 system.The energies are given in units of MeV.

TABLE II .
Pairing scaling factors and the resulting pairing gaps, given in units of MeV.For the A = 76 nuclei "WS + BCS" denotes the Woods-Saxon based and "exp" the experiment-based occupancies.For 116 Sn ( 136 Xe) the proton (neutron) pairing strength is adopted from 116 Cd ( 136 Ba) since there is no proton (neutron) pairing gap for 116 Sn ( 136 Xe) due to its proton (neutron) magicity.

TABLE V .
E max (GT − ) E centroid (GT − ) E max (IVSM − ) E centroid (IVSM −) Intensity of the GT − and IVSM − modes.The quantitity S max is the intensity carried by the state of energy E max , and S total is the total intensity of each mode.The intensities corresponding to the IVSM mode are given in units of fm4.The two adopted occupancy schemes for 76 Ge have been indicated by "WS + BCS" and "exp".

TABLE VI .
Energetics of the GT + and IVSM + modes.The quantities E max and E centroid are the energy of the state with the largest intensity and the energy centroid of the intensity distribution of each mode.All values are expressed in units of MeV.The two adopted occupancy schemes for 76 Se have been indicated by "WS + BCS" and "exp".

TABLE VII .
Intensity of the GT + and IVSM + modes.The quantitity S max is the intensity carried by the state of energy E max , and S total is the total intensity of each mode.The intensities corresponding to the IVSM mode are given in units of fm4.The two adopted occupancy schemes for 76 Se have been indicated by "WS + BCS" and "exp".