Spin-dipole nuclear matrix elements for double beta decays and astro-neutrinos

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Physics Letters B
www.elsevier.com/locate/physletb Spin-dipole nuclear matrix elements for double beta decays and astro-neutrinos Spin-dipole (SD) nuclear matrix elements (NMEs) M ± (SD2) for unique first forbidden β ± 2 − → 0 + ground-state-to-ground-state transitions are studied by using effective microscopic two-nucleon interactions in realistic single-particle model spaces. The observed values of the NMEs M ± exp (SD2) are compared with the values of the single-quasiparticle NMEs M ± qp (SD2) without nucleon spin-isospin (σ τ ) correlation and the QRPA NMEs M ± QRPA (SD2) with the σ τ correlation. The observed SD matrix elements are found to be reduced by the factor k ≈ 0.2 with respect to M ± qp (SD2) and by the factor k NM ≈ 0.5 with respect to M ± QRPA (SD2). We then infer that the SD NME is reduced considerably partly by the nucleon σ τ correlations and partly by other non-nucleonic and nucleonic correlations which are not explicitly included in the QRPA. Impact of the found reduction factors on the magnitudes of the NMEs involved in neutrino-less double beta decays and astro-neutrino interactions are discussed. Fundamental properties of neutrinos and their interactions can be studied by investigating neutrino-less double beta (0νββ) decays and astro-neutrino-nuclear processes. Here the associated nuclear matrix elements (NMEs) are crucial to extract quantitative neutrino properties that are of interest to particle and astrophysics, as discussed in reviews [1][2][3][4][5][6] and refs. therein.
The present Letter aims to analyze the spin-dipole (SD) β ± NMEs for 2 − → 0 + ground-state-to-ground-state transitions and to compare them with current model calculations. These NMEs are associated with the J π = 2 − component of the 0νββ matrix element and the cross sections of the medium-energy neutrinonuclear interactions. We show that the SD matrix elements are considerably reduced due to nucleonic spin(σ )-isospin(τ ) and other correlations as well as possible non-nucleonic nuclear medium effects.
The 0νββ decays of several isotopes are under extensive experimental and theoretical study to access the Majorana properties of neutrinos and their absolute mass scales [2][3][4][5]. Then a reliable value of the 0νββ NME, M 0ν , is required to design an optimum 0νββ detector and to deduce the neutrino mass from the measured 0νββ decay rate in case it is observed. In fact, exact theoretical evaluations for M 0ν are hard [2,6,7], and thus comprehensive model calculations using appropriate nuclear interactions and the renormalized weak coupling constant g A are necessary to get accurate values for M 0ν [7,8].
In case of the light-neutrino mediated 0νββ decay a virtual Majorana neutrino is exchanged between two nucleons of the mother nucleus. Then the momentum of the virtual neutrino is around q ≈ 100-50 MeV/c and the angular momenta involved in the process are mainly lh = 1h − 3h and the spin and parity of the intermediate states are J ± = 1 ± − 4 ± . Among them one of the largest components is J π = 2 − associated with the p-wave neutrino axial-vector (AV) interaction [7]. The NMEs relevant to this virtual-neutrino interaction are the β − and β + matrix elements of Thus the single β NME M(SD2) is of great interest due to its intimate relation with M 0ν , as discussed in review articles [2,4,9] and also in a workshop proceedings [10].
Astro-neutrino charged-current (CC) interaction rates are given by inverse β ± decay rates. Then the β ± NME is needed to derive the neutrino flux from the interaction rate and vice versa. The supernova neutrino energy extends up to a few tens of MeV, and then the NME M ± (SD2) associated with the p-wave neutrino AV interaction gets relatively important [11][12][13].  The 2 − single β transition strength is given as where J i is the initial state spin and M ± (SD2) is the reduced SD matrix element. It is expressed as where τ ± are the isospin operators for β ± and σ is the Pauli spin operator. The SD 2 − matrix elements have been studied by investigating forbidden β and M2 γ decays as discussed in Refs. [14][15][16][17][18].
The SD matrix elements are shown to be reduced by the repulsive τ σ interaction which pushes the 2 − strength up to the possible SD 2 − giant resonance (SDGR). The SDGRs are expressed as coherent sums of nucleon-particle-nucleon-hole SD states [18]. The observed SD NME is suggested to be reduced further since the strength is pushed up to the possible isobar GR, which is expressed as a coherent sum of nucleon-hole--particle SD states at the 300 MeV region [19]. SD strengths spread over other states as well. The reduction is a common feature of τ σ weak and electro-magnetic transitions [18]. The SD strength distributions are schematically shown in Fig. 1.
The Gamow-Teller (GT) 1 + NME involved in the s-wave τ σ interaction is the major player in the low-energy s-wave neutrino processes such as the two-neutrino double beta (2νββ) decays and low-energy astro-neutrino-nuclear interactions. The single β GT strengths have been studied by measuring allowed β decays and charge exchange reactions (CERs), such as (p, n), (n, p), (d, 2 He), ( 3 He, t) (t, 3 He) and others. They were studied at IUCF, KVI, LAMPF, NSCL, RCNP, RIKEN, TRIUMP, UC-Davis, Uppsala Univ., and many other labs. as given in the references of reviews [1,2,4].
where D contains the weak coupling strength and g V /g A = The major SD transitions in these mass regions are 1g 9/2 -1f 5/2 , 2d 5/2 -2p 1/2 and 1h 11/2 -1g 7/2 , respectively. Their single-particle SD NMEs are quite large because of the large radial and angular overlap integrals. The single β ± 2 − → 0 + SD NMEs for nuclei in the three mass regions are derived from the f 1 t values [32] as given in the third column of Tables 1, 2  In the simplest picture the β decays discussed here proceed between single-quasiparticle states. Then the NMEs are written by The pairing factor P is given as P = U n V p and P = U p V n for β − and β + decays respectively, where U p (U n ) and The k coefficients are scattered around k ≈ 0.2 and they may be considered to stand for the reduction due to the τ σ correlation and nuclear medium effects and others, if any, which are not explicitly included in the QP model. The realistic SD 2 − NMEs were calculated in terms of the quasiparticle RPA (QRPA) model with the τ σ correlation (the protonneutron QRPA, i.e. pnQRPA). The corresponding NMEs are shown in the fifth column of Tables 1, 2 and 3. The adopted singleparticle spaces are large enough to stabilize the value of the SD 2 − NME for the ground-state-to-ground-state transition. The adopted proton-neutron interaction is based on the realistic Gmatrix, where the pairing and multipole channels are scaled by few constants as explained in [34][35][36]. For the multipole channels the particle-hole interaction is scaled by a strength constant g ph such that the centroid of the observed GTGR is reproduced. It should be noted that the low-lying SD 2 − strength is quite insensitive to the particle-particle interaction constant g pp in contrast to the low-lying GT 1 + strength [37]. The observed SD NMEs are found to be significantly reduced with respect to the QRPA values by the reduction factor k NM with where the reduction factor of k NM ≈ 0.5 suggests that the SD axial weak coupling is reduced in nuclei due to some nuclear medium effects which are not explicitly included in the QRPA.
The reduction factor for the SD matrix element may be expressed as where k ≈ 0.2, k τ σ ≈ 0.4 and k NM ≈ 0.5. These values depend little on the nuclear parameters used for the present quasiparticle and QRPA models.
Actually, the values of the SD matrix elements depend on the nucleon configurations at the nuclear Fermi surface, i.e. on the occupation and vacancy amplitudes of V and U . The reduction due to the nucleon σ τ correlations and other non-nucleonic renormalization effects such as the isobar correlations are nuclear core/medium effects. In order to elucidate such nuclear core/medium effects, we consider the geometric mean of the β + and β − NMEs for the available cases where both of them have been measured. The mean value is expressed as where M ± (SD2) is the reduced matrix element for the β ± transition A(Z , N) ↔ A(Z ∓ 1, N ± 1). In fact there are two types of such decays, one common odd-odd nucleus decays to two neighboring even-even nuclei, and two neighboring odd-odd nuclei decay to one common even-even nucleus. In terms of the quasiparticle expression, the mean matrix element is expressed as where V p , U n and V n , U p are the pairing factors for the β + and β − final nuclei. Noting that U i ≈ 1 − V 2 i and V i ≈ 1 − U 2 i , with i being p or n, the geometric mean of the pairing factor is given as (V p U n V n U p ) 1/2 ≈ 0.43 ± 0.05 in a wide range of U and V . In other words, the nuclear surface effects due to the valence nucleon configurations are averaged in the mean NME. Thus the geometric mean is very insensitive to the nucleon configuration at the Fermi surface, but rather reflects the nuclear core/medium effects. The mean values are obtained for 6 β ± decays as shown in Table 4. Table 4 Mean SD 2 − NMEs derived from the β ± matrix elements given in Tables 1, 2 We note that these effective reduction factors are rather independent of nuclear parameters used in the quasiparticle and QRPA models.
In short, the present analyses show that the SD 2 − weak NMEs, which play important roles for the 0νββ NMEs and the astroneutrino inverse-β NMEs, are much reduced by reduction factors of k ≈ 0.2 and k NM ≈ 0.5 with respect to the single-quasiparticle values and the standard QRPA calculations, respectively, in medium heavy nuclei. The reductions are considered to be due to the τ σ correlation effect of k τ σ ≈ 0.4 and the other nuclear medium and correlation effect of k NM ≈ 0.5. The latter is associated with the non-nucleonic (isobar) and other correlations, which are not included in the standard QRPA. Then this may be incorporated by using an effective g eff A ≈ 0.5g A in the standard QRPA.
Let us discuss the present results and their impacts on the 0νββ NMEs and astro-ν interactions.
The 2 − ground states discussed here are all simple quasiparticle states with the unique configuration of (l + 1/2, l − 1/2) with l = l − 1. The β transitions are single-quasiparticle ones of (l + 1/2) ↔ (l − 1/2) with a large single-particle NME. The quasiparticle NME depends on the quasiparticle U and V factors, but the effects are minimized in the geometrical mean of the There are no other strong SD states nearby, and thus couplings with other states are not appreciable. They are included in the QRPA calculations. The present QRPA calculations are based on Woods-Saxon single-particle energies with the global Bohr-Mottelson parametrization [33]. These parameters are fitted to reproduce the properties of nuclei close to the stability line of the nuclear chart and thus suitable for the present calculations. The Woods-Saxon energies vary smoothly and so do the computed NMEs in general. Deviations from the smooth trend occur in some cases where the computed NME is very small (the A = 72, 122(β − ), 126 cases) but then also the observed NME M exp (SD2) is anomalously small.
The smooth trends of the Woods-Saxon energies can be broken by manual adjustments of the single-particle energies close to the proton and/or neutron Fermi surfaces. The resulting basis can be coined "adjusted basis" and it has been used on several occasions in the past calculations to better reproduce the experimentally available single-quasiparticle energies in the neighboring odd-A nuclei. Such adjustments along with adoption of the measured orbital occupancies of the mother and daughter nuclei were done in the case of the 76 Ge → 76 Se(0 + gs ) double beta transition in [38]. These adjustments resulted in a great improvement of the description of the SD NME and the resulting log f t of the transition 76 As(2 − ) → 76 Se(0 + gs ) over the previously obtained values (in a slightly adjusted basis) in the works [39,40]. Adjustments have been done also in the works [41][42][43] with varying success. A more comprehensive study of the effects on the magnitudes of the NMEs coming from the sizes of the single-particle model spaces, adjustments of the single-particle energies and orbital occupancies was discussed in [44].
The GT β ± sum strengths from the present QRPA calculations satisfy the GT Ikeda sum rule [45]. Then, one can safely say that when the standard Woods-Saxon mean field is used with the BCS occupancies the pnQRPA predicted SD NMEs are consistently much larger than the observed SD NMEs. This difference stems from both the nuclear medium (non-nucleonic and other) effects and the deficiencies of the pnQRPA and the mean field. Disentangling one from the other is not straightforward and easy, but it seems that in spite of all the fine tuning of the nuclear mean field and the resulting pnQRPA calculation there is always a good portion of the observed NME that cannot be reached by the calculations and this remaining portion could be mostly coming from the nuclear medium effect. Hence, the presently extracted reduction factors should be considered in the pnQRPA calculations of the SD NMEs related to double beta decays and astro-neutrino interactions The reduction factor, being defined as the ratio of the true NME to the model calculation, depends on how completely the nucleonic and nonnucleonic effects are included explicitly in the model, and thus may be different for other models, like the interacting shell model (ISM) and the interacting boson approximation (IBA2) [30].
Furthermore, it seems that missing couplings of the simple model configurations to more complicated many nucleon configurations and the use of small single-particle model spaces contribute to the reduction factor. The effect on the M 0ν has recently been pointed out in [46].
The reduction factors of k τ σ and k NM are of the same order of magnitude, being in accord with the universal ansatz of the coupling constants of g NN and g N [47,48]. Then one needs to include explicitly both the τ σ NN −1 and N −1 excitations in the calculation to evaluate the relative contributions to the reduction of the SD matrix element. Precise studies of the SD2 − GR strength distribution and the sum of the SD strengths are useful to get the relative contributions of k τ σ and k NM .
The reductions found in the SD and GT matrix elements show that the SD transition of the p-wave τ σ mode is much modified in nuclei by the τ σ p-wave polarizations of NN −1 , N −1 and other correlations, analogously to the case of the GT transitions where the reduction stems from the s-wave τ σ polarizations.
Actually the reduction due to the isobar can be related to the quenching of g A and the summed axial vector strengths. Summed GT strengths observed in (p, n)/(n, p) are claimed to be 0.88 ±0.06 of the sum rule value, indicating no appreciable quenching effect [51,52] but a shift of the GR strength [53] (see Fig. 1). Recent highprecision ( 3 He, t) experiment [54] shows a quenching of around 0.5. Summed SD strengths in medium heavy nuclei are not observed, and the SD sum rule is sensitive to the nuclear structure [55]. The summed GT strengths and the GT quenching problem will be discussed elsewhere.
The reduction of the axial vector (AV) 2 − single β matrix elements could have a strong impact on the 0νββ experiments [2,4,10]. The AV component of M 0ν depends on |M(AV)| 2 . Thus, if the renormalization factor of k NM ≈ 0.5 would also be applied to other multipole AV NMEs, the AV 0νββ NME would be reduced with respect to the standard QRPA one by a factor of up to around 4, and the total 0νββ NME by a factor around 2, depending on the relative AV component.
It should be noted that the 0νββ process is a two-body process with the neutrino potential of H ij , so that the 0νββ transition operator is not given by a separable form like the 2νββ operator. Since the neutrino potential is of the Coulomb type, 1/r ij , the NME M 0ν (2 − ) via 2 − intermediate states may be approximately given by a product of single β ± NMEs of M − (SD2 − ) and M + (SD2 − ) via the 2 − intermediate states like the 2νββ matrix element M 2ν is given by the GT type NMEs via 1 + intermediate states. Even one could see something like single-state dominance for the SD 2 − mode as seen for some of the 2νββ decaying nuclei for the GT 1 + mode [56][57][58]. Actually, low-lying intermediate states contribute to the 2νββ NME [4,27,28]. There are 2-5 low lying 2 − states in the present ββ nuclei. Accordingly, the reduction effects on the SD 2 − single β ± NME may appear also in the 0νββ NME as well.
Theoretical model calculations with adequate nucleonic and non-nucleonic correlations are needed to evaluate M 0ν accurately.
The present studies of the single β ± NMEs of M ± (SD2 − ) and the obtained reduction coefficients may help QRPA and other model calculations of M 0ν and evaluations for the AV weak astroneutrino-nuclear cross sections and astro-neutrino-nuclear synthesis rates.