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

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Magnetic Hexadecapole γ Transitions and Neutrino-Nuclear Responses in Medium- Heavy Nuclei Jokiniemi, Lotta; Suhonen, Jouni; Ejiri, Hiroyasu Jokiniemi, L., Suhonen, J., & Ejiri, H. (2016). Magnetic Hexadecapole γ Transitions and Neutrino-Nuclear Responses in Medium-Heavy Nuclei. Advances in High Energy Physics, 2016, Article 8417598. https://doi.org/10.1155/2016/8417598 2016 Research Article Magnetic Hexadecapole 𝛾Transitions and Neutrino-Nuclear Responses in Medium-Heavy Nuclei Lotta Jokiniemi,1Jouni Suhonen,1and Hiroyasu Ejiri2 1Department of Physics, University of Jyvaskyla, P.O. Box 35, 40014 Jyvaskyla, Finland 2Research Center for Nuclear Physics, Osaka University, Osaka 567-0047, Japan Correspondence should be addressed to Jouni Suhonen; [email protected] Received 17 March 2016; Accepted 14 April 2016 Academic Editor: Enrico Lunghi Copyright © 2016 Lotta Jokiniemi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3. Neutrino-nuclear responses in the form of squares of nuclear matrix elements, NMEs, are crucial for studies of neutrino-induced processes in nuclei. In this work we investigate magnetic hexadecapole (M4) NMEs in medium-heavy nuclei. The experimentally derived NMEs, 𝑀EXP(M4), deduced from observed M4 𝛾transition half-lives are compared with the single-quasiparticle (QP) NMEs, 𝑀QP(M4), and the microscopic quasiparticle-phonon model (MQPM) NMEs 𝑀MQPM(M4). The experimentally derived M4 NMEs are found to be reduced by a coefficient 𝑘≈0.29with respect to 𝑀QP(M4) and by 𝑘≈0.33with respect to 𝑀MQPM(M4). The M4 NMEs are reduced a little by the quasiparticle-phonon correlations of the MQPM wave functions but mainly by other nucleonic and nonnucleonic correlations which are not explicitly included in the MQPM. The found reduction rates are of the same order of magnitude as those for magnetic quadrupole 𝛾transitions and Gamow-Teller (GT) and spin-dipole (SD) 𝛽transitions. The impacts of the found reduction coefficients on the magnitudes of the NMEs involved in astroneutrino interactions and neutrinoless double beta decays are discussed. 1. 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]𝛽𝛽) 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 [1–4] and references therein. The present work aims at investigating the magnetic hexadecapole (M4) 𝛾NMEs, 𝑀𝛾(M4), in medium-heavy nuclei to study higher-multipole axial-vector NMEs associated with higher-energy components of astroneutrino reactions and 0]𝛽𝛽 decays.Suchcomponentsareshowntobeimportant for, for example, the 0]𝛽𝛽decays [5]. Neutrino-nuclear responses associated with neutral-cur- rent (NC) and charged-current (CC) interactions are studied by investigating the relevant 𝛾and 𝛽decay transitions or NC and CC scatterings on nuclei. The momenta involved in astroneutrino scatterings and 0]𝛽𝛽 decays are of the order of 50–100 MeV/c. Accordingly, depending on the involved momentum exchanges, the multipoles 𝐽𝜋with angular momenta 𝐽up to around 4-5 are involved (e.g., 0]𝛽𝛽decays mediated by light Majorana neutrinos; see [5]), or even higher multipoles can be engaged (0]𝛽𝛽decays mediated by heavy Majorana neutrinos; see [5]). In some previous works, axial-vector CC resonances of GT(1+)andSD(2 −) NMEs for allowed and first-forbidden 𝛽transitions are shown to be reduced much in comparison with the quasiparticle (QP) and pnQRPA (proton-neutron quasiparticle random-phase approximation) NMEs [6–10] due to spin-isospin (𝜎𝜏) nucleonic and nonnucleonic correlations and nuclear-medium effects. These studies show that exact theoretical evaluations for the astroneutrino and 0]𝛽𝛽NMEs, including possible renormalization of the axialvector coupling constant 𝑔A, are hard. The corresponding NC nuclear responses of magnetic dipole (M1) and quadrupole Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 8417598, 8 pages http://dx.doi.org/10.1155/2016/8417598 2Advances in High Energy Physics (M2) 𝛾transitions are also known to be much reduced with respect to the QP NMEs [11]. Similar studies have been conductedinthecaseofthetwo-neutrinodoublebetadecays 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]𝛽𝛽 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 [15–18]. In the light of the above discussions it is of great interest to investigate the spin-hexadecapole (4−)NMEstoseehow the higher-multipole NMEs are reduced by the nucleonic and nonnucleonic spin-isospin correlations. Actually, there are almost no experimental CC hexadecapole 𝛽NMEs in medium-heavy nuclei since the 𝛽decays 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 𝛽transitions and they can serve as potential testing grounds concerning the quenching effects of the weak vector (𝑔V)andaxialvector (𝑔A) coupling constants [19]. On the other hand, there are many experimental data on NC M4 𝛾NMEs, where the isovector component of the 𝛾NME is related to the analogous 𝛽NME on the basis of the isospin symmetry. Thus we discuss mainly the M4 𝛾transitions in the present report with the aim of helping evaluate/confirm, for example, the 0]𝛽𝛽 NMEs concerning their higher-multipole aspects. 2. Experimental M4 NMEs Here we discuss stretched M4 𝛾transitions with 𝐽𝑖=𝐽 𝑓±𝐽, where 𝐽𝑖and 𝐽𝑓are the initial and final state spins and 𝐽= 4.TheM4𝛾transition rate (per sec) is given in terms of the reduced M4 𝛾strength 𝐵𝛾(M4) as [20] 𝑇(M4)=1.87×10−6𝐸9𝐵𝛾(M4)(1+𝛼)−1 ,(1) where 𝐸is the 𝛾rayenergyinunitsofMeVand𝛼is the conversion-electron coefficient. The reduced strength is expressed in terms of the M4 𝛾NME in units of 𝑒ℏ/(2𝑀𝑐)fm3 as 𝐵𝛾(M4)=(2𝐽 𝑖+1)−1 [𝑀𝛾(M4)]2.(2) The M4 𝛾NME is expressed in terms of the M4 𝛾coupling constants 𝑔(M4) and the M4 matrix element 𝑀(M4) as 𝑀𝛾(M4)=𝑔 p(M4)𝜏p𝑀(M4)+𝑔n(M4)𝜏n𝑀(M4),(3) where the first and the second terms are for the odd-proton (𝜏p=(1−𝜏 3)/2) and odd-neutron (𝜏n=(1+𝜏 3)/2) transition NMEs with 𝜏3being the isospin 𝑧component (𝜏3=1for neutron and 𝜏3=−1for proton). The 𝛾coupling constant is written as 𝑔𝑖(M4)=𝑒ℏ 2𝑀𝑐6(𝜇𝑖−1 5𝑔𝑖),(4) where 𝑖=p for proton and 𝑖=nforneutron,𝜇p= 2.79and 𝜇n= −1.91are the proton and neutron magnetic moments, Table 1: 𝑀(M4) NMEs for M4 𝛾transitions in the mass region of 𝐴=70–120, where the major single-QP transition is 1g9/2-2p1/2. Here p/n stands for the odd-proton/odd-neutron transition. 𝑀EXP, 𝑀QP,and𝑀MQPM are the experimental, single-QP, and MQPM NMEs in units of 103fm3. Nucleus Transition p/n𝑀EXP 𝑀QP 𝑀MQPM 85Kr 2p1/2 →1g9/2 n 0.528 1.57 1.44 89Y2p1/2 →1g9/2 p 0.739 2.12 2.02 89Zr 2p1/2 →1g9/2 n 0.559 1.60 1.47 91Y1g9/2 →2p1/2 p 0.480 2.13 1.83 105In 2p1/2 →1g9/2 p 0.706 2.38 2.09 107In 2p1/2 →1g9/2 p 0.670 2.40 2.13 109In 2p1/2 →1g9/2 p 0.640 2.33 2.10 111In 2p1/2 →1g9/2 p 0.609 2.45 2.03 113In 2p1/2 →1g9/2 p 0.603 2.46 2.05 115In 2p1/2 →1g9/2 p 0.614 2.48 2.03 Table2:ThesameasTable1for𝐴= 130–150, where the major singlequasiparticle transition is 1h11/2-2d3/2. Nucleus Transition p/n𝑀EXP 𝑀QP 𝑀MQPM 135Xe 1h11/2 →2d3/2 n 1.11 3.12 2.87 137Ba 1h11/2 →2d3/2 n 1.03 3.13 2.77 139Ba 1h11/2 →2d3/2 n 0.968 3.12 2.82 141Nd 1h11/2 →2d3/2 n 0.893 3.17 2.79 143Sm 1h11/2 →2d3/2 n 0.878 3.19 2.81 and 𝑔𝑙p=1and 𝑔𝑙n=0are the proton and neutron orbital 𝑔 coefficients. The M4 𝛾matrix element is expressed as 𝑀(M4)=⟨𝑓󵄩 󵄩 󵄩 󵄩 󵄩𝑖3𝑟3[𝜎×𝑌3]4󵄩 󵄩 󵄩 󵄩 󵄩𝑖⟩, (5) where 𝑟is the nuclear radius and 𝑌3is the spherical harmonic for multipole 𝑙=3. The isotopes used for ongoing and/or future 𝛽𝛽 experiments are 76Ge, 82Se, 96Zr, 100Mo, 116Cd, 130Te, and 136Xe [2]. They are in the mass regions of 𝐴=70–120and𝐴=130–150. The single-quasiparticle (single-QP) M4 transitions in these mass regions are uniquely tagged by the pairs 1g9/2-2p1/2 and 1h11/2-2d3/2, 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 𝛾transitions in the mentionedtwomassregionsareanalyzedinTables1and2.The M4 NMEs derived from the experimental half-lives are given in the third column of these tables. The values of 𝑀EXP(M4) are plotted against the mass number in Figure 1. They are around (0.6±0.1)×103fm3and (1.0±0.1)×103fm3for the two mass regions, respectively. They are well expressed as 𝑀EXP (M4)≈6×𝐴fm3,(6) where the mass number 𝐴reflects the 𝑟3dependence of the M4 NME. Advances in High Energy Physics 3 M(M4) 70 80 90 100 110 120 130 140 150 Mass number QP EXP 0.1 1 10 (a) M(M4) 70 80 90 100 110 120 130 140 150 Mass number MQPM EXP 0.1 1 10 (b) Figure 1: (a) EXP: experimental NMEs 𝑀EXP(M4) for odd-neu- tron transitions (light-blue squares) and odd-proton transitions (dark-blue diamonds). QP: quasiparticle NMEs 𝑀QP(M4) for oddneutron transitions (light-blue tip-up triangles) and odd-proton transitions (dark-blue tip-down triangles). (b) EXP: experimental NMEs 𝑀EXP(M4) for odd-neutron transitions (light-blue squares) and odd-proton transitions (dark-blue diamonds). MQPM: NMEs 𝑀MQPM(M4) for odd-neutron transitions (light-blue tip-up triangles) and odd-proton transitions (dark-blue tip-down triangles). 3. Quasiparticle M4 NMEs The M4 𝛾transitions given in Tables 1 and 2 are all, in their simplest description, transitions between single-quasiparticle states.TheNMEsforthesingle-quasiparticletransitionsare written by using the single-particle matrix element 𝑀SP(M4) and the pairing coefficient 𝑃as 𝑀QP (M4)=𝑀 SP (M4)𝑃𝑖𝑗,(7) where the pairing coefficient is given by 𝑃𝑖𝑗 =𝑈 𝑖𝑈𝑓+𝑉 𝑖𝑉𝑓,(8) and 𝑈𝑖(𝑈𝑓)and𝑉𝑖(𝑉𝑓) are the vacancy and occupation amplitudes for the initial (final) state. The single-quasiparticle states discussed here are low-lying states located at the diffusedFermisurface,asshowninFigure2.Thustheoccupation and vacancy probabilities are in the region of 𝑈2=1− 𝑉2=0.5±0.3, and the pairing coefficient is given roughly as 𝑃≈1. In this work the single-quasiparticle NMEs 𝑀QP(M4) arecalculatedbyusingtheBCSwavefunctionswithHO E Ei EfV iV f UiUf 0 1.0 V2 V2 |i⟩ |f⟩ Figure 2: Schematic diagram of the energy (𝐸) and the occupation probabilities, 𝑉2 𝑖and 𝑉2 𝑓,fortheinitialandfinalstates(seebodyof text). The energy levels are shown by the horizontal lines. Vacancy probabilities are given as 𝑈2 𝑖=𝑉 2 𝑖−1and 𝑈2 𝑓=𝑉 2 𝑓−1. The paring coefficient 𝑃𝑖𝑗 for the 𝛾transition is given by 𝑈𝑖𝑈𝑓+𝑉 𝑖𝑉𝑓. 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 𝑀EXP(M4) are uniformly smaller by a coefficient of around 0.29 ±0.05 than the single-quasiparticle ones, 𝑀QP(M4). We introduce a reduction coefficient 𝑘𝑖as in the case of GT(1+)andSD(2 −) [8, 10] transitions. It is defined as 𝑀EXP (M4)=𝑘𝑖(M4)𝑀QP (M4),(9) where 𝑘𝑖,with𝑖=p,n, are the reduction coefficients for single quasi-proton and quasi-neutron M4 𝛾transitions, respectively. The ratios 𝑘𝑖(M4)are 𝑘p≈0.3and 𝑘n≈0.3,asshown in Figure 3(a). The found reductions are consistent with the reductions discussed in [11]. The quasiparticle NMEs 𝑀QP(M4)arecalculatedby assuming a stretched M4 transition between the initial and final nuclear states (see column 2 of Tables 1 and 2) that are assumedtohaveaone-quasiparticlestructure.Thesestates are thus described as |𝛼⟩=𝑎† 𝛼|BCS⟩,(10) where 𝑎† 𝛼creates a quasiparticle on a nuclear mean-field orbital with quantum numbers 𝛼=𝑎,𝑚 𝛼,where𝑎contains the principal quantum number 𝑛, the orbital angular momentum (𝑙), and total angular momentum (𝑗) quantum numbers in the form 𝑛𝑙𝑗asdisplayedincolumn2ofTables1and2. Here 𝑚𝛼is the 𝑧projection of the total angular momentum and |BCS⟩istheBCSvacuum.Thequasiparticlesaredefined by the Bogoliubov-Valatin transformation as 𝑎† 𝛼=𝑈 𝑎𝑐† 𝛼+𝑉 𝑎 𝑐𝛼,  𝑎𝛼=𝑈 𝑎 𝑐𝛼−𝑉 𝑎𝑐† 𝛼,(11) 4Advances in High Energy Physics 70 80 90 100 110 120 130 140 150 Mass number 0 0.2 0.4 0.6 0.8 1 k=M EXP(M4)/MQP(M4) (a) 70 80 90 100 110 120 130 140 150 Mass number 0 0.2 0.4 0.6 0.8 1 MEXP(M4)/MMQPM(M4) (b) Figure 3: Reduction coefficients 𝑘for the M4 NMEs. (a) 𝑘: ratios of experimental NMEs 𝑀EXP(M4) to the quasiparticle NMEs 𝑀QP(M4) for odd-neutron transitions (light-blue squares) and oddproton transitions (dark-blue diamonds). (b) 𝑘: ratios of experimental NMEs 𝑀EXP(M4) to the MQPM NMEs 𝑀MQPM(M4) for odd-neutron transitions (light-blue squares) and for odd-proton transitions (dark-blue diamonds). where 𝑐† 𝛼is the particle creation operator and the timereversed particle annihilation operator  𝑐𝛼is defined by  𝑐𝛼= (−1)𝑗𝑎+𝑚𝛼𝑐−𝛼 with −𝛼=(𝑎,−𝑚𝛼).The𝑈and 𝑉coefficients are the vacancy and occupation amplitudes present also in the quasiparticlematrixelementsof(7)and(8).Thevaluesof 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. 4. 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 randomphaseapproximation)statescalledherephononssincethe lowest ones are usually collective vibrational states. These statescanbeformallywrittenas |𝜔⟩=𝑄† 𝜔|QRPA⟩,(12) where the phonon operator 𝑄† 𝜔creates a nuclear state with quantum numbers 𝜔, containing the angular momentum 𝐽𝜔, parity 𝜋𝜔, and the quantum number 𝑘𝜔which 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 nucleusonehastocoupleaproton(proton-oddnucleus)ora neutron (neutron-odd nucleus) quasiparticle to the phonon operator 𝑄† 𝜔which 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󵄨󵄨󵄨󵄨𝑘𝑗𝑚⟩ = Γ† 𝑘(𝑗𝑚)|QRPA⟩,(13) where a 𝑘th state of angular momentum 𝑗and its 𝑧projection 𝑚is created in an odd-mass nucleus by a creation operator which mixes one-quasiparticle and three-quasiparticle components in the form Γ† 𝑘(𝑗𝑚)=∑ 𝑛𝑋𝑘 𝑛𝑎† 𝑛𝑗𝑚 +∑ 𝑎𝜔𝑋𝑘 𝑎𝜔 [𝑎† 𝑎𝑄† 𝜔]𝑗𝑚 ,(14) where the first term is the one-quasiparticle contribution and the second term is the quasiparticle-phonon contribution. The amplitudes 𝑋𝑘 𝑛and 𝑋𝑘 𝑎𝜔 arecomputedfromtheMQPM equations of motion [22]. In solving these equations special careistobetakentohandletheovercompletenessandthe 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 𝐽𝜋to 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 𝐽𝜋, whenever data existed. When no data was available the bare 𝐺-matrix was used in the calculations. Also the particle-particle part of the multipole interaction was kept as bare 𝐺-matrix interaction. After performing the BCS and QRPA calculations in the reference even-even nuclei, the initial and final nuclear states Advances in High Energy Physics 5 70 80 90 100 110 120 130 140 150 Mass number 0 0.5 1 1.5 MQP(M4)/MMQPM(M4) Figure 4: Ratios of the quasiparticle NMEs 𝑀QP(M4) to the MQPM NMEs 𝑀MQPM(M4) for odd-neutron transitions (light-blue squares) and odd-proton transitions (dark-blue diamonds). 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 𝜔in 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, 𝑀QP(M4), are uniformly smaller, by a coefficient around 0.33, than the MQPM NMEs 𝑀MQPM(M4). Actually, the MQPM NMEs 𝑀MQPM(M4) are 10–20% smaller than the QP NMEs 𝑀QP(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 𝑔coefficients (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 𝜏𝜎correlations 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. 5. Reduction of the Axial-Vector NMEs The 𝛾NMEisdecomposedintotheisovectorandisoscalar ones as [11] 𝑀𝛾(M4)=𝑔 −(M4)𝜏3 2𝑀(M4) +𝑔+(M4)𝜏0 2𝑀(M4),(15) where 𝑀𝛾(M4)is 𝑔p𝑀(M4)and 𝑔n𝑀(M4)for an odd-proton and odd-neutron transition, respectively, and 𝑔−and 𝑔+are the isovector and isoscalar 𝛾coupling constants. They are written as 𝑔±(M4)=𝑒ℏ 2𝑀𝑐6(𝜇±−1 5𝑔𝑙±),(16) with 𝜇±=𝜇 n±𝜇pand 𝑔𝑙± =𝑔 𝑙n±𝑔𝑙p≈∓1. The proton, neutron, isovector, and isoscalar M4 𝛾coupling constants are given as 𝑔p=2.59𝐺, 𝑔n=−1.91𝐺, 𝑔−=−4.50𝐺, 𝑔+=0.68𝐺, (17) where 𝐺=6𝑒ℏ(2𝑀𝑐)−1 is the M4 𝛾coupling coefficient. The M4 NMEs in (15) are rewritten as 𝑘𝑖𝑔𝑖𝑀QP(M4)with 𝑖=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] 𝑘p𝑔p=−𝑔− 2𝑘−+𝑔+ 2𝑘+, 𝑘n𝑔n=𝑔− 2𝑘−+𝑔+ 2𝑘+.(18) After this the isovector and isoscalar reduction coefficients can be derived as 𝑘−=𝑘n−0.567(𝑘n−𝑘p), (19) 𝑘+=𝑘n−3.89(𝑘n−𝑘p). (20) Since the experimentally derived NMEs for the proton and neutron transitions are approximately the same, that is, 𝑘n≈ 𝑘p,wegetfrom(19)𝑘−≈𝑘 n≈𝑘 p≈ 0.3.TheisovectorM4 NMEs are reduced by the coefficient 𝑘−≈0.3,inthesameway as the M4 𝛾transition NMEs. It is notable that the amount of reduction for the isovector 4−𝛾NMEs is the same as that found for the GT(1+)andSD(2 −)𝛽decay NMEs [8, 10]. Axial-vector 𝛽and 𝛾NMEs for low-lying states are much reduced with respect to the QP NMEs due to the strong repulsive 𝜏𝜎 interactions since the axial-vector 𝜏𝜎strengths arepushedupintothe𝜏𝜎 giant-resonance (GR) and the Δ- isobar regions. The M4 reduction coefficient of 𝑘(M4)≈0.29 is nearly the same as the coefficient 𝑘(M2) ≈ 0.24 for M2 𝛾transitions and 𝑘(GT) ≈ 0.235 and 𝑘(SD) ≈ 0.18 in the GT(1+)andSD(2 −)𝛽decayNMEs,respectively[8,10].These reduction coefficients are plotted in Figure 5 with 𝜆denoting the angular momentum content of the transition operator. It seems that the reduction coefficients of the axial-vector NMEsareuniversalforNCandCCNMEsandfortheangular momenta of 𝜆= 1–4. The reduction is considered to be due to such 𝜎𝜏 polarization interactions and nuclear-medium effects that are not explicitly included in the models. Then the 6Advances in High Energy Physics Reduction rate k=M EXP/MQP 0 0.2 0.4 0.6 0.8 1 012345 Multipolarity Figure 5: Reduction coefficients 𝑘(M)=𝑀 EXP(M)/𝑀QP(M)for the multipolarity M =M2(𝜆=2)𝛾and M =M4(𝜆=4)𝛾transitions (dark-blue diamonds) and those for M =GT (𝜆=1)𝛽and M =SD (𝜆=2)𝛽transitions (light-blue squares) in medium-heavy nuclei. Here 𝜆denotes the transition angular momentum. The dotted line shows 𝑘(M)=0.27. reduction rate 𝑘(M)≈0.2–0.3, with respect to the QP NME, is expressed as [8, 10] 𝑘(M)=𝑘𝜎𝜏 (M)×𝑘NM (M),(21) where 𝑘𝜎𝜏(M) ≈ 0.5and 𝑘NM(M) ≈ 0.5stand for the reductions due to the nucleonic 𝜎𝜏polarization effects and nonnucleonic Δ-isobar and nuclear-medium effects, respectively. These universal effects may be represented by the effective weak coupling 𝑔eff A, depending on the model NME, to incorporatetheeffectsthatarenotexplicitlyincludedinthe nuclear-structuremodel.IncaseofthepnQRPAwithexplicit nucleonic 𝜎𝜏correlations, one may use 𝑔A≈0.6, while in the caseoftheQPmodel,withoutany𝜎𝜏correlations, 𝑔A≈ 0.3, both in units of the bare value of 𝑔A=1.26𝑔V. Here we note that the 𝜏𝜎 repulsive interaction concentrates the 𝜏𝜎strength to the highly excited 𝜏𝜎GR, resulting in the reduction of the 𝜏𝜎 NMEs for low-lying states. Such reduction effect is incorporated in the pnQRPA with the strong 𝜏𝜎interaction, so that the reduction factor is improved from 𝑘QP ≈ 0.3 to 𝑘QRPA ≈ 0.6. On the other hand, such strong 𝜏𝜎interaction 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 𝑘QP ≈ 0.29to 𝑘MQPM ≈ 0.33. 6. 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 𝑔Amagnifies since the 0]𝛽𝛽 NME includes the axial-vector NME proportional to the square of 𝑔A.TheM4NC𝛾results 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 𝛽𝛽 NMEsarequenchedroughlybythesameamountby the 𝜎𝜏correlations of the Δregion 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 𝛽𝛽 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 𝑔A derived from the GT 𝛽decays [30]. Actually, the observed single 𝛽GT NMEs are reduced by theeffectivecouplingconstant𝑘eff with respect to the model NME, and thus the observed 2]𝛽𝛽NMEs are well reproduced by using the experimental 𝑘eff , that is, the experimental single 𝛽NMEsforlow-lyingstates[1,2,31].However,itshouldbe kept in mind that these results, important as such, cannot be directly applied to the 0]𝛽𝛽 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 𝑔A= 1.26𝑔Vand be free from the uncertainties introduced bytheeffective(quenched)𝑔eff 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, 7Be, 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 [15–18]. 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 𝑔eff Aand 𝑔eff V. QRPA calculations were made for 92Nb 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 𝑔Aand 𝑔Vhas been conjectured [35–38]. 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 𝑔Aand 𝑔Varegivenbythespectrum-shapemethod 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 Advances in High Energy Physics 7 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 𝜏𝜎 NME, including 𝑔A,isrelatedto the shift of the strength to the higher GR and Δisobar regions. Charge-exchange reactions report about a 50–60% oftheGTsumrule(theIkedasumrule)uptoGTGR, 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 𝜏𝜎strengthsispartlydueto the nonnucleonic (ΔN−)𝜏𝜎correlations [8, 20]. 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