Magnetic Hexadecapole γ Transitions and Neutrino-Nuclear Responses in Medium-Heavy Nuclei

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Introduction
Neutrino interactions in nuclei are studied, for example, by investigating scatterings of astroneutrinos on nuclei and by the attempts to record the neutrinoless double beta (0]) decays.Here the neutrino-nuclear responses can be condensed in the squares of nuclear matrix elements (NMEs) and it is necessary to study through them the neutrino properties and astroneutrino reactions that are of interest to particle physics and astrophysics, as discussed in review articles [1][2][3][4] and references therein.
The present work aims at investigating the magnetic hexadecapole (M4)  NMEs,   (M4), in medium-heavy nuclei to study higher-multipole axial-vector NMEs associated with higher-energy components of astroneutrino reactions and 0] decays.Such components are shown to be important for, for example, the 0] decays [5].
Neutrino-nuclear responses associated with neutral-current (NC) and charged-current (CC) interactions are studied by investigating the relevant  and  decay transitions or NC and CC scatterings on nuclei.The momenta involved in astroneutrino scatterings and 0] decays are of the order of 50-100 MeV/c.Accordingly, depending on the involved momentum exchanges, the multipoles   with angular momenta  up to around 4-5 are involved (e.g., 0] decays mediated by light Majorana neutrinos; see [5]), or even higher multipoles can be engaged (0] decays mediated by heavy Majorana neutrinos; see [5]).
In some previous works, axial-vector CC resonances of GT(1 + ) and SD(2 − ) NMEs for allowed and first-forbidden  transitions are shown to be reduced much in comparison with the quasiparticle (QP) and pnQRPA (proton-neutron quasiparticle random-phase approximation) NMEs [6][7][8][9][10] due to spin-isospin () nucleonic and nonnucleonic correlations and nuclear-medium effects.These studies show that exact theoretical evaluations for the astroneutrino and 0] NMEs, including possible renormalization of the axialvector coupling constant  A , are hard.The corresponding NC nuclear responses of magnetic dipole (M1) and quadrupole 2 Advances in High Energy Physics (M2)  transitions are also known to be much reduced with respect to the QP NMEs [11].Similar studies have been conducted in the case of the two-neutrino double beta decays in [12,13] in the framework of the IBA-2 model.Also the derivation of effective operators has been proposed [14].All these studies bear relevance to the previously mentioned Majorana-neutrino mediated 0] decays, to high-energy astroneutrino reactions, but also to the lower-energy (up to 30 MeV) supernova-neutrino scatterings off nuclei, as shown, for example, in [15][16][17][18].
In the light of the above discussions it is of great interest to investigate the spin-hexadecapole (4 − ) NMEs to see how the higher-multipole NMEs are reduced by the nucleonic and nonnucleonic spin-isospin correlations.Actually, there are almost no experimental CC hexadecapole  NMEs in medium-heavy nuclei since the  decays are very rare thirdforbidden unique transitions.However, it turns out that there are few measurements of the half-lives and electron spectra of the more complex fourth-forbidden nonunique  transitions and they can serve as potential testing grounds concerning the quenching effects of the weak vector ( V ) and axialvector ( A ) coupling constants [19].On the other hand, there are many experimental data on NC M4  NMEs, where the isovector component of the  NME is related to the analogous  NME on the basis of the isospin symmetry.Thus we discuss mainly the M4  transitions in the present report with the aim of helping evaluate/confirm, for example, the 0] NMEs concerning their higher-multipole aspects.

Experimental M4 NMEs
Here we discuss stretched M4  transitions with   =   ± , where   and   are the initial and final state spins and  = 4.The M4  transition rate (per sec) is given in terms of the reduced M4  strength   (M4) as [20] where  is the  ray energy in units of MeV and  is the conversion-electron coefficient.The reduced strength is expressed in terms of the M4  NME in units of ℏ/(2) fm 3  as The M4  NME is expressed in terms of the M4  coupling constants (M4) and the M4 matrix element (M4) as where the first and the second terms are for the odd-proton ( p = (1− 3 )/2) and odd-neutron ( n = (1+ 3 )/2) transition NMEs with  3 being the isospin  component ( 3 = 1 for neutron and  3 = −1 for proton).The  coupling constant is written as where  = p for proton and  = n for neutron,  p = 2.79 and  n = −1.91 are the proton and neutron magnetic moments, and  p = 1 and  n = 0 are the proton and neutron orbital  coefficients.The M4  matrix element is expressed as where  is the nuclear radius and  3 is the spherical harmonic for multipole  = 3.
The isotopes used for ongoing and/or future  experiments are 76 Ge, 82 Se, 96 Zr, 100 Mo, 116 Cd, 130 Te, and 136 Xe [2].They are in the mass regions of  = 70-120 and  = 130-150.The single-quasiparticle (single-QP) M4 transitions in these mass regions are uniquely tagged by the pairs 1g 9/2 -2p 1/2 and 1h 11/2 -2d 3/2 , respectively.Here the higher spin state is the intruder one from the higher major shell with opposite parity.The single-particle M4 NMEs corresponding to these tagging transitions are quite large because of the large radial and angular overlap integrals.
The single-quasiparticle M4  transitions in the mentioned two mass regions are analyzed in Tables 1 and 2. The M4 NMEs derived from the experimental half-lives are given in the third column of these tables.
The values of  EXP (M4) are plotted against the mass number in Figure 1.They are around (0.6 ± 0.1) × 10 3 fm 3 and (1.0 ± 0.1) × 10 3 fm 3 for the two mass regions, respectively.They are well expressed as where the mass number  reflects the  3 dependence of the M4 NME.

Quasiparticle M4 NMEs
The M4  transitions given in Tables 1 and 2 are all, in their simplest description, transitions between single-quasiparticle states.The NMEs for the single-quasiparticle transitions are written by using the single-particle matrix element  SP (M4) and the pairing coefficient  as where the pairing coefficient is given by and   (  ) and   (  ) are the vacancy and occupation amplitudes for the initial (final) state.The single-quasiparticle states discussed here are low-lying states located at the diffused Fermi surface, as shown in Figure 2. Thus the occupation and vacancy probabilities are in the region of  2 = 1 −  2 = 0.5 ± 0.3, and the pairing coefficient is given roughly as  ≈ 1.In this work the single-quasiparticle NMEs  QP (M4) are calculated by using the BCS wave functions with HO single-particle states.They are given in the fifth column of Tables 1 and 2 and are plotted in Figure 1(a).
The experimental M4 matrix elements  EXP (M4) are uniformly smaller by a coefficient of around 0.29 ± 0.05 than the single-quasiparticle ones,  QP (M4).We introduce a reduction coefficient   as in the case of GT(1 + ) and SD(2 − ) [8,10] transitions.It is defined as where   , with  = p, n, are the reduction coefficients for single quasi-proton and quasi-neutron M4  transitions, respectively.The ratios   (M4) are  p ≈ 0.3 and  n ≈ 0.3, as shown in Figure 3(a).The found reductions are consistent with the reductions discussed in [11].
The quasiparticle NMEs  QP (M4) are calculated by assuming a stretched M4 transition between the initial and final nuclear states (see column 2 of Tables 1 and 2) that are assumed to have a one-quasiparticle structure.These states are thus described as where  †  creates a quasiparticle on a nuclear mean-field orbital with quantum numbers  = ,   , where  contains the principal quantum number , the orbital angular momentum (), and total angular momentum () quantum numbers in the form   as displayed in column 2 of Tables 1 and 2.
Here   is the  projection of the total angular momentum and |BCS⟩ is the BCS vacuum.The quasiparticles are defined by the Bogoliubov-Valatin transformation as where  †  is the particle creation operator and the timereversed particle annihilation operator c is defined by c = (−1)   +   − with − = (, −  ).The  and  coefficients are the vacancy and occupation amplitudes present also in the quasiparticle matrix elements of ( 7) and (8).The values of these amplitudes are obtained in BCS calculations (for details see, e.g., [21]) and they are used also in the subsequent nuclear-structure calculations of the odd-mass nuclei and their neighboring even-even-mass reference nuclei.

Microscopic Quasiparticle-Phonon Model for M4 NMEs
The microscopic quasiparticle-phonon model (MQPM) takes the structure of the nuclear states beyond the approximation (10).In the MQPM this extension is done in the traditional way of starting from an even-even reference nucleus where the states are described as QRPA (quasiparticle randomphase approximation) states called here phonons since the lowest ones are usually collective vibrational states.These states can be formally written as where the phonon operator  †  creates a nuclear state with quantum numbers , containing the angular momentum   , parity   , and the quantum number   which enumerates states with the same angular momentum and parity.The state ( 12) is a linear combination of two-quasiparticle states as explicitly written in [22] where the MQPM was first introduced.To arrive at a state in the neighboring odd-mass nucleus one has to couple a proton (proton-odd nucleus) or a neutron (neutron-odd nucleus) quasiparticle to the phonon operator  †  which is a two-quasiparticle operator.In this way one creates three-quasiparticle states in the traditional quasiparticle-phonon coupling scheme and these states are then mixed with the one-quasiparticle states by the residual nuclear Hamiltonian (for details see [22]).Hence we obtain the MQPM states where a th state of angular momentum  and its  projection  is created in an odd-mass nucleus by a creation operator which mixes one-quasiparticle and three-quasiparticle components in the form where the first term is the one-quasiparticle contribution and the second term is the quasiparticle-phonon contribution.
The amplitudes    and    are computed from the MQPM equations of motion [22].In solving these equations special care is to be taken to handle the overcompleteness and the nonorthogonality of the quasiparticle-phonon basis, as described in detail in [22].
In the actual calculations we used slightly modified Woods-Saxon single-particle energies to improve the quality of the computed energy spectra of the odd-mass nuclei involved in the present work.This resulted in a good correspondence between the computed and experimental lowenergy spectra of these nuclei.We adopted a residual Hamiltonian with realistic effective two-nucleon interactions derived from the Bonn-A one-boson-exchange potential [23].The free parameters of the interaction were fixed in the BCS and QRPA phases of the calculations as explained in [24,25].The two-body monopole matrix elements were multiplied by one parameter for protons and one for neutrons to scale phenomenologically the proton and neutron pairing strengths separately.This was done by fitting the computed pairing gaps to the phenomenological ones, derived from the measured proton and neutron separation energies [26].The QRPA step contained two parameters for each multipole   to control the energies of the even-even excited states.These were the strengths of the particle-hole and particle-particle parts of the two-nucleon interaction.The particle-hole interaction controls the energies of collective states and thus it was fitted to reproduce the experimental excitation energy of the lowest state of a given multipolarity   , whenever data existed.When no data was available the bare -matrix was used in the calculations.Also the particle-particle part of the multipole interaction was kept as bare -matrix interaction.
After performing the BCS and QRPA calculations in the reference even-even nuclei, the initial and final nuclear states in the neighboring odd-mass nuclei, of interest in the present work, were formed by first creating the quasiparticle-phonon components of the operator (14).The convergence of the MQPM results for the energies of the involved states and the M4 transition amplitudes between them were monitored by adding more and more QRPA phonons of different multipolarities  in the diagonalization of the residual Hamiltonian.
In terms of the cut-off energy of the added phonons the convergence was achieved at around 14 MeV.
The converged M4 results are given in the sixth column of Tables 1 and 2 and are plotted in Figures 1(b) and 3(b).It is seen that the experimental M4 matrix elements,  QP (M4), are uniformly smaller, by a coefficient around 0.33, than the MQPM NMEs  MQPM (M4).Actually, the MQPM NMEs  MQPM (M4) are 10-20% smaller than the QP NMEs  QP (M4), as shown in Figure 4.The admixtures of the quasiparticle-phonon components in the wave functions (14) of the M4 initial and final states reduce the M4 NMEs a little, but not nearly enough to bring the MQPM NMEs close to the corresponding experimental ones, at least by using the bare  coefficients (4) adopted in the present work.Hence, the major part of reduction from the MQPM NME to the experimental one (a reduction coefficient of 0.3) is considered to be due to such nucleonic and nonnucleonic  correlations and nuclear-medium effects that are not explicitly included in the (traditional) quasiparticle-phonon coupling scheme that MQPM uses, that is, the even-even nucleus serving as a reference for the odd-mass one.
The M4 NMEs in (15) are rewritten as      QP (M4) with  = p, n, −, +, the symbols standing for the proton, neutron, isovector, and isoscalar components, respectively.Then the reduction coefficients and the weak couplings for proton and neutron transitions are expressed in terms of those for the isovector and isoscalar components as [11] After this the isovector and isoscalar reduction coefficients can be derived as Since the experimentally derived NMEs for the proton and neutron transitions are approximately the same, that is,  n ≈  p , we get from (19)  − ≈  n ≈  p ≈ 0.3.The isovector M4 NMEs are reduced by the coefficient  − ≈ 0.3, in the same way as the M4  transition NMEs.It is notable that the amount of reduction for the isovector 4 −  NMEs is the same as that found for the GT(1 + ) and SD(2 − )  decay NMEs [8,10].Axial-vector  and  NMEs for low-lying states are much reduced with respect to the QP NMEs due to the strong repulsive  interactions since the axial-vector  strengths are pushed up into the  giant-resonance (GR) and the Δisobar regions.The M4 reduction coefficient of (M4) ≈ 0.29 is nearly the same as the coefficient (M2) ≈ 0.24 for M2  transitions and (GT) ≈ 0.235 and (SD) ≈ 0.18 in the GT(1 + ) and SD(2 − )  decay NMEs, respectively [8,10].These reduction coefficients are plotted in Figure 5 with  denoting the angular momentum content of the transition operator.
It seems that the reduction coefficients of the axial-vector NMEs are universal for NC and CC NMEs and for the angular momenta of  = 1-4.The reduction is considered to be due to such  polarization interactions and nuclear-medium effects that are not explicitly included in the models.Then the reduction rate (M) ≈ 0.2-0.3, with respect to the QP NME, is expressed as [8,10] where   (M) ≈ 0.5 and  NM (M) ≈ 0.5 stand for the reductions due to the nucleonic  polarization effects and nonnucleonic Δ-isobar and nuclear-medium effects, respectively.These universal effects may be represented by the effective weak coupling  eff A , depending on the model NME, to incorporate the effects that are not explicitly included in the nuclear-structure model.In case of the pnQRPA with explicit nucleonic  correlations, one may use  A ≈ 0.6, while in the case of the QP model, without any  correlations,  A ≈ 0.3, both in units of the bare value of  A = 1.26V .
Here we note that the  repulsive interaction concentrates the  strength to the highly excited  GR, resulting in the reduction of the  NMEs for low-lying states.Such reduction effect is incorporated in the pnQRPA with the strong  interaction, so that the reduction factor is improved from  QP ≈ 0.3 to  QRPA ≈ 0.6.On the other hand, such strong  interaction to give rise to the possible M4 GR is not explicitly incorporated in the MQPM, and thus the reduction factor is only a little improved from  QP ≈ 0.29 to  MQPM ≈ 0.33.

Discussion and Conclusions
Among other models, the QRPA-based models are used to compute the NMEs of double beta decays [4,27].In these calculations the importance of the quenching of  A magnifies since the 0] NME includes the axial-vector NME proportional to the square of  A .The M4 NC  results of the present investigation, together with the earlier M2 NC and GT and SD CC results, suggest that many of the leading multipoles in the decomposition of the neutrinoless  NMEs are quenched roughly by the same amount by the  correlations of the Δ region as well as other nuclearmedium effects.Since the corresponding investigations have been done using data on low-lying nuclear states it is safe to say that the quenching applies at least to the low-lying states in nuclei.These observations are very relevant for the twoneutrino  decays since they are low-energy phenomena and usually involve only one or few lowest states in the intermediate nucleus [1,28,29].A consistent description of these decays can be achieved by using the quenched  A derived from the GT  decays [30].
Actually, the observed single  GT NMEs are reduced by the effective coupling constant  eff with respect to the model NME, and thus the observed 2] NMEs are well reproduced by using the experimental  eff , that is, the experimental single  NMEs for low-lying states [1,2,31].However, it should be kept in mind that these results, important as such, cannot be directly applied to the 0] processes since there large momentum exchanges are involved and also vector-type of NMEs and higher excited states contribute.In this case it is a big challenge to develop such nuclear models for the axialvector weak processes that include explicitly appropriate nucleonic and nonnucleonic correlations.If successful, then in such models one could use the axial weak coupling of  A = 1.26V and be free from the uncertainties introduced by the effective (quenched)  eff A .Neutrino-nucleus scatterings are important to probe many astrophysical phenomena, like the solar and supernova neutrinos [1,32,33].The GT NMEs bring in most of the contributions for solar neutrinos and low-energy supernova neutrinos for neutrino energies below 15 MeV (see, e.g., [16,17]).The SD NMEs play a role for medium-energy neutrinos above 15 MeV.For the low-energy solar neutrinos reliable calculations of the GT NMEs are needed to evaluate the SNU values for the pp, 7 Be, CNO, and other neutrinos.Experimental GT strengths can also be used, if available [1].The SD contributions can be important for the CC supernova antineutrino scatterings off nuclei even at low energies, as shown in [15][16][17][18].
The supernova-neutrino nucleosyntheses are sensitive to the neutrino CC and NC interactions, as discussed in a review article [33].Here SD and higher-multipole NMEs are involved in the high-energy components of the supernova neutrinos.It is important for accurate evaluations of the isotope distributions to use appropriate NMEs and effective weak couplings of  eff A and  eff V .QRPA calculations were made for 92 Nb nuclei in [34].
The involved GT and SD NMEs can be studied via beta decays in nuclei where beta decay data is available.In some of these studies a strong quenching of both  A and  V has been conjectured [35][36][37][38].Such quenching for the higher multipoles is extremely hard to study, the present study being a rather unique one in this respect.Quenching of the higher multipoles can also be studied via high-forbidden beta decays [19] but the available data is extremely scarce at the moment.Perspectives for the studies of the quenching of both  A and  V are given by the spectrum-shape method introduced in [19].There the shape of the beta spectrum of the high-forbidden nonunique beta decays has been studied for the determination of the possible quenching of the weak constants.Use of this method can be boosted by future highsensitive measurements of electron spectra in underground laboratories.
Finally, it is worth pointing out that the universal reduction/quenching of the  NME, including  A , is related to the shift of the strength to the higher GR and Δ isobar regions.Charge-exchange reactions report about a 50-60% of the GT sum rule (the Ikeda sum rule) up to GT GR, while the (p, n) reactions claim that around 90% of the GT sum-rule strength is seen by including the strength beyond the GT GR up to 50 MeV [39].Very careful investigations of the GT, SD, and higher-multipole strength distributions, by using charge-exchange reactions, are called for to see if the reduction/quenching of the  strengths is partly due to the nonnucleonic (ΔN − )  correlations [8,20].These investigations not only are interesting from the point of view of the double beta decay but also touch the projected double chargeexchange reactions [40] where the high-momentum response of nuclei is also probed.

|f⟩
Figure 2: Schematic diagram of the energy () and the occupation probabilities, 2  and  2  , for the initial and final states (see body of text).The energy levels are shown by the horizontal lines.Vacancy probabilities are given as  2  =  2  − 1 and  2  =  2  − 1.The paring coefficient   for the  transition is given by     +     .