Isovector and isoscalar spin-multipole giant resonances in the parent and daughter nuclei of double-β -decay triplets

The strength distributions, including giant resonances, of isovector and isoscalar spin-multipole transitions in the commonly studied double-β -decay triplets are computed in the framework of the quasiparticle random-phase approximation (QRPA) using the Bonn-A two-body interaction in no-core single-particle valence spaces. The studied nuclei include the double-β parent and daughter pairs ( 76 Ge, 76 Se), ( 82 Se, 82 Kr), ( 96 Zr, 96 Mo), ( 100 Mo, 100 Ru), ( 116 Cd, 116 Sn), ( 128 Te, 128 Xe), ( 130 Te, 130 Xe), and ( 136 Xe, 136 Ba). The studied transitions proceed from the ground states to the J π = 0 − , 1 − , 2 − (spin-dipole transitions) and J π = 1 + , 2 + , 3 + (spin-quadrupole transitions) excited states in these nuclei. Comparison of the present results with potential future data may, indirectly, shed light on the reliability of QRPA-based nuclear-structure frameworks in description of the wave functions of nuclear states relevant for the two-neutrino and neutrinoless double β decays in the studied triplets.


I. INTRODUCTION
Two-neutrino double beta (2νββ) and neutrinoless double beta (0νββ) decays keep attracting keen attention of both the nuclear-and particle-physics communities.The 0νββ decay is of particular interest since, if detected, it has strong implications to physics beyond the standard model [1][2][3][4][5][6][7].The 2νββ decay proceed through the 1 + virtual states [1] and the 0νββ decay proceeds through virtual states of all multipolarities J π [8,9] in the intermediate nuclei of the double-β-decay triplets consisting of an even-even mother, an odd-odd intermediate, and an even-even daughter nucleus.The wave functions of these virtual states can be probed by calculations and experiments on β − -type (from the 0 + ground state of the ββ mother nucleus) and β + -type (from the 0 + ground state of the ββ daughter nucleus) isovector spin-multipole transitions to the J π = 0 − , 1 − , 2 − (L = 1 spin-dipole transitions) and J π = 1 + , 2 + , 3 + (L = 2 spin-quadrupole transitions) excited states in the ββ intermediate nuclei.Here, L refers to the orbital angular momentum of the operator mediating the multipole transition.The corresponding theoretical studies have been conducted in [10] as an extension of the studies conducted in [11][12][13] for the 1 + isovector spin-monopole resonances.Experimentally, these transitions have typically been probed by the partial-wave L = 0 charge-exchange reactions (CXRs) by using the β − type of (p, n) or ( 3 He,t) reactions and β + type of (n, p), (d, 2 He), or (t, 3 He) reactions [6,[14][15][16].Recently, the partial-wave L = 1 CXRs to 2 − states have attracted interest by development of improved experimental methods and facilities, e.g., at the RCNP in Osaka, Japan [6,17].
In [10] the β − -type and β + -type isovector spin-dipole (IVSD) and isovector spin-quadrupole (IVSQ) strength distributions and giant resonances in the ββ-decay triplets 76 Ge-76 As-76 Se, 82 Se-82 Br-82 Kr, 96 Zr-96 Nb-96 Mo, 100 Mo-100 Tc − 100 Ru, 116 Cd-116 In-116 Sn, 128 Te-128 I-100 Xe, 130 Te-130 I-130 Xe, and 136 Xe-136 Cs-136 Ba were studied using the proton-neutron quasiparticle random-phase approximation (pnQRPA).In the present work we extend these studies to non-CXR isoscalar and isovector spin-dipole (ISSD and IVSD) and isoscalar and isovector spin-quadrupole (ISSQ and IVSQ) strength distributions and giant resonances in the mother and daughter even-even nuclei of the listed ββ-decay triplets.The corresponding transitions start from the 0 + ground states and lead to the J π = 0 − , 1 − , 2 − (L = 1 spin-dipole transitions), and J π = 1 + , 2 + , 3 + (L = 2 spin-quadrupole transitions) states in the same even-even nuclei.The L = 1 and L = 2 isovector transitions have been studied long ago by Auerbach and Klein for non-superfluid nuclei in the random-phase approximation (RPA) framework in [18].For the description of spin-multipole transitions in superfluid nuclei, like in the present case, we have to use the charge-conserving quasiparticle RPA (QRPA) [19] instead of the above-mentioned charge-changing pnQRPA and charge-conserving RPA.Theoretical and experimental studies of the CXR and non-CXR type of transitions can probe indirectly the ββ decays by testing the ability of QRPA-based nuclear-theory frameworks (QRPA and pnQRPA) to yield reliable wave functions in both channels of transitions.In particular, in the pnQRPA and QRPA calculations the quantities to be probed are the single-particle valence spaces, the single-particle energies, and the resulting Bardeen-Cooper-Schrieffer (BCS) ground states as foundations of the correlated pnQRPA and QRPA ground states [19,20].The relevant experiments include (p, p ), (e, e ), (α, α ), (d, d ), ( 3 He, 3 He ), etc., experiments [21].

II. THEORY
The formalism used in this work is the quasiparticle random phase approximation (QRPA).The formalism is explicitly introduced, for example, in [19].Excitation energies of the observed nuclei are calculated as quasiparticle excitations of the QRPA ground state.The excitation operator in the QRPA case is where ω includes quantum numbers n, J π , and M, and X ω ab and Y ω ab are the amplitudes that describe the probability of creating or annihilating a quasiparticle pair in the QRPA ground state.A † ab (JM ) is the quasiparticle-pair creation operator and Ãab (JM ), the corresponding annihilation operator, is of the form Above the N ab (J ) is the normalization factor.
The QRPA equations can be written in a matrix form as Matrices A and B are written explicitly in [19].
The transition operator for an isovector excitation can be written as and the isoscalar transition operator as where t 0 is the third component of the isospin operator.With the use of the transition operators ( 5) and ( 6) the reduced single-particle nuclear matrix elements (NMEs) can be calculated.For these operators the reduced single-particle NMEs can be written as where R (L)  ab is a radial integral and ρ equals 's' or 'v' such that η s ab = 1 and η v ab = 1 if a and b denote neutron orbitals and η v ab = −1 if a and b denote proton orbitals.Now the reduced transition NMEs can be written as TABLE I. Pairing gap scaling parameters g (n)  pair and g (p)  pair for each nucleus are listed in columns 2 and 3. Also the particle-hole parameters for each J π excitation are tabulated in columns 4-9.Transition strengths are calculated as the square of the transition NME

III. RESULTS AND DISCUSSION
The lowest quasiparticle energies for protons and neutrons are fitted to the proton and neutron pairing gaps.Pairing gaps for protons and neutrons are calculated by using the threepoint formulas [19] where S p and S n are the separation energies for protons and neutrons, respectively.Also the lowest energies for each multipole are adjusted to the corresponding experimental ones by using the particlehole parameter g ph .The only exception are the 1 − states where the lowest one is spurious and has been removed from the calculations.Here, instead, the second 1 − state has been fitted to the lowest experimental 1 − energy.If the energy is not known experimentally (this is exclusively the case for the 0 − states) or the computed energy is not sensitive to the value g ph (this is the case for most 1 + and 3 + states), the default value g A = 1.00 is adopted.The resulting values of g ph for each multipole state and each nucleus are also tabulated in Table I.The pnQRPA calculations are sensitive to the value of the particle-particle parameter g pp [1,6,20], and its value can be fixed, e.g., by comparison with the measured half-life of a two-neutrino ββ transition, as was done in [10].For QRPA the g pp parameter plays a negligible role since the results are essentially independent of its value.This is why we adopted the default value g pp = 1.00.
We calculate the strength ( 9) for each excitation energy E and multipolarity J π .The corresponding discrete strength functions are then folded with a Lorentzian in order to make the strengths easier to compare with the potential future experimental data.For the Lorentzian we choose as used also in [10].In the Lorentzian fit (11) E is the excitation energy, E 0 is the energy of the peak, and W is the peak width which is set to 0.5 MeV following the convention of [10].
The computed strength functions are plotted as functions of E for a representative set of cases in Figs.1-3, and the average energies and total strengths are given in Tables II and III.In Fig. 1, the strength functions are plotted for the 76 Ge nucleus for different transition types.Panel (a) in that figure presents the isovector strength for spin-dipole (L = 1) transitions, panel (b) presents the isovector strength for spinquadrupole (L = 2) transitions, and panels (c) and (d) present the isoscalar strengths for the L = 1 and L = 2 transitions.In Fig. 2 the isoscalar strength functions for spin-quadrupole transitions for (a) 82 Se, (b) 82 Kr, (c) 96 Zr, and (d) 96 Mo nuclei are plotted.Finally, in Fig. 3 isovector and isoscalar strength functions for 136 Xe are shown the same way as in Fig. 1.
In spin-dipole transitions, the isovector strength is located almost only in one peak, as shown in Figs. 1 and 3.The dominant peak consists primarily of a 0 − state, and all other J π states are almost negligible.This excitation is located at approximately 20 MeV for all the nuclei.
The isoscalar spin-dipole strengths are more spread than the isovector ones.In this case, the 0 − state is also the most significant contributor to the strength function but also the 1 − state has a notable contribution.The largest strengths are observed in approximately around 10 MeV.From Figs. 1 and  3 and Tables II and III, it can be seen that the 2 − state has the lowest overall strength compared to the other multipoles, so it does not contribute to the total dipole strength functions very much.
Isovector spin-quadrupole strength functions are more spread in energy E than the spin-dipole ones.The strength function has its largest peak for a 1 + state at about 30 MeV, as can be seen in Figs. 1 and 3.There is also a correlation between the mass number A and the ratio of the highest peak and other peaks: When A increases the largest peak gets more dominant.This correlation can also be seen in Figs. 1 and 2.
Lastly, the isoscalar spin-quadrupole strength is divided between all J π states, which is different from the previously analyzed strength functions.The 2 + and 1 + states have the largest strength peaks, but the 3 + state has a considerably large peak for some nuclei at approximately 15 MeV.These peaks are seen in Fig. 2. The strength is spread up to 30 MeV and is considerable at all energies.The total L = 2 strength is by far larger than the L = 1 strength, as seen in Tables II and III.For all nuclei the 1 + strength is the largest and the 2 − spin-dipole strength the smallest.On average, the spin-quadrupole strength clearly increases with increasing mass number.
The presently obtained results for the isovector strength functions can also be compared with the corresponding strength functions for the charge-changing modes, calculated in Ref. [10].In [10] the (p, n) β − -type and (n, p) β + -type isovector strength functions were treated.For both the spin-dipole and spin-quadrupole transitions the strength distributions of the charge-changing modes are more widely spread.As discussed earlier, the isovector excitations in this work, as also in [10], tend to be concentrated in few TABLE II.Average energies (columns 4 and 6) and total strengths (columns 5 and 7) for isovector (superscript v) and isoscalar (superscript s) spin-dipole and spin-quadrupole excitations for the nuclei with mass numbers A = 76, 82, 96, and 100.The average isovector energies E v ave are compared with those of [10] (column 3), corrected for nuclear mass differences between the ββ intermediate nucleus and the ββ mother nucleus [E (GR) − ] or the ββ daughter nucleus [E (GR) + ].The energy E (GR) − is listed for the ββ mother (β − branch) and E (GR) + for the ββ daughter (β + branch) in order to have the same initial nucleus for the charge-changing transitions of [10] and the charge-conserving ones of the present work.The strengths are given in units of fm 2 for J π = 0 − , 1 − , 2 − and in fm 4 for peak-like structures rather than spreading over the whole excitation range, as in isoscalar excitations.
In Tables II and III the third column displays energies "E (GR) − " and "E (GR) + " from Table III of [10], corrected for mass differences between the ββ intermediate nucleus and the ββ mother and daughter nuclei, respectively.This is done because in [10] the average energies were referred to the ground state of the double-β intermediate nucleus, and in order to compare the general trends of average isovector energies with those of Ref. [18].Since in [10] the feeding of the intermediate nuclei was discussed we have to adopt the "E (GR) − " for the mother nuclei ( 76 Ge, 82 Se, 96 Zr, 100 Mo, 116 Cd, 128 Te, 130 Te, and 136 Xe) and "E (GR) + " for daughter nuclei ( 76 Se, 82 Kr, 96 Mo, 100 Ru, 116 Sn, 128 Xe, 130 Xe, and 136 Ba) of ββ decays.The average energies in Tables II and III reveal a common pattern, coincident with that found in [18]: The energies E (GR) − , E v ave , and E (GR) + form on average a hierarchy, with E (GR) − being the largest, E (GR) + the smallest, and E v ave falling in the middle.This hierarchy is particularly distinct for the zirconium region of A = 96, as also was the case for 90 Zr in [18].

IV. CONCLUSIONS
The strength distributions of isovector and isoscalar spin-multipole transitions were studied in this work.The FIG. 3. Same as in Fig. 1 but for 136 Xe. excitation energies and transition strengths were calculated using the QRPA formalism.The strength distributions were calculated for a set of double-β decaying nuclei and the corresponding daughter nuclei.The same double-β decaying nuclei were studied in [10] using the pnQRPA formal-ism for charge-changing isovector spin-multipole strength distributions.
In our QRPA calculations the quasiparticles were handled by using the BCS formalism, fitting the phenomenological proton and neutron pairing gaps with the help of the corresponding pairing-strength parameters.Particle-hole parameters were adjusted so that the lowest proton and neutron quasiparticle energies were fitted to the proton and neutron pairing gaps.We also adjusted the energy of the lowest state for each multipole J π to fit the data, if available.For this we used the particle-hole parameter, available in the QRPA formalism.
The results show that the isoscalar strengths are more spread in energy than the isovector ones.The isovector spindipole strength is located almost only in one dominant peak, and for the spin-quadrupole strength, the most significant peak becomes more dominant when the mass number increases.The isoscalar transition-strength distributions show notable strength for all excitation states, contrary to isovector transitions.
Similarly to the previously made study [10], where the isovector multipole-strength distributions of β − and β + types of excitations were investigated, the presently studied isovector transition strengths tend to locate in one peak.Most of the nuclei had larger centroid energies in β − -type excitations than those calculated in this study.The situation was the opposite for the β + -type strength; the isovector excitation energies were lower on average than those calculated in this work.
When the experimental data are available, the results calculated in this work could be compared to the experimentally observed ones.Such comparison could shed light on the potency of a BCS-based random-phase-approximation framework to access high-lying excited states and their wave functions.Since the presently used QRPA and the charge-changing pnQRPA frameworks are based on the same quasiparticle mean field, the present study also serves as an indirect method to check the reliability of the pnQRPA framework in producing nuclear wave functions of high-lying states relevant for the studies of the nuclear matrix elements of the neutrinoless double β decay.

TABLE III .
Same as inTable II but for the nuclei with mass numbers A = 116, 128, 130, and 136.