Selection rules in the ββ decay of deformed nuclei

The 2 ν ββ decay half-lives of six nuclei, whose decays were previously reported as theoretically forbidden, are calculated by including the pairing interaction, which mixes diﬀerent occupations and opens up the possibility of the decay. All allowed channels for the 0 ν ββ decay are also computed. The estimated 2 ν ββ half-lives suggest that measurements in 244 Pu may ﬁnd positive signals, and that planned experiments would succeed in detecting the ββ 2 ν decay in 160 Gd. Limits for the zero neutrino mode, in the analyzed deformed emitters, are predicted. Neutrinoless double beta decay ( ββ 0 ν ), if detected, would oﬀer deﬁnitive evidence that the neutrino is a Majorana particle, i.e. that it is its own

antiparticle [1,2]. It would also provide the information needed to determine neutrino masses, complementary to the one obtained from solar and atmospheric neutrino experiments [3,4].
Theoretical nuclear matrix elements are needed to convert experimental half-life limits, which are available for many ββ-unstable isotopes [2,5], into constrains for the effective Majorana mass of the neutrino and the contribution of right-handed currents to the weak interactions. Thus, these matrix elements are essential to understand the underlying physics [6,7,8]. The two neutrino mode of the double beta decay (ββ 2ν ) is allowed as a second order process in the standard model. It has been detected in ten nuclei [5] and it has served to test a variety of nuclear models [2].
The pseudo SU(3) approach has been used to describe many low-lying rotational bands, as well as B(E2) and B(M1) intensities, in rare earth and actinide nuclei, both with even-even and odd-mass numbers. The theoretical results show, in general, a very good agreement with the data [9]. The ββ half-lives of some heavy deformed nuclei, which may decay to the ground and excited states of the daughters, were evaluated for the two and zero neutrino emitting modes [10,11,12,13,14] using the pseudo SU(3) scheme. The predictions were in good agreement with the available experimental data for 150 Nd and 238 U. The double electron capture decay channel was studied for the decay of other three nuclei [15].
The simplest pseudo SU(3) model predicts the complete suppression of the ββ 2ν decay for the following five nuclei: 154 Sm, 160 Gd, 176 Yb, 232 Th and 244 Pu [10,14]. Recently, it was argued that the cancellation of the ββ 2ν decay in 160 Gd would suppress the background for the detection of the 0ν mode [16].
In the present contribution we extend the previous research [10,11,12,13,14,17] and evaluate the ββ half-lives of 154 Sm, 160 Gd, 170 Er, 176 Yb, 232 Th and 244 Pu using the pseudo SU(3) model. In these nuclei the 2νββ mode is forbidden when the most probable occupations are considered. To be able to evaluate finite half-lives, we were forced to include in the calculations states with different occupation numbers which can be mixed through the pairing interaction. The amount of this mixing is evaluated, and the possibility of observing the ββ decay is discussed for both the 2ν and 0ν modes.
Pseudo-spin symmetry [18] describes the quasi-degeneracy of the singleparticle orbitals with j = l − 1/2 and j = (l − 2) + 1/2 in the η shell. It allows these orbitals to be classified as pseudo-spin partners with quantum numbersj = j,η = η − 1 andl = l − 1. The first step in the pseudo SU ( description of any nuclei is to find the occupation numbers for protons (p) and neutrons (n) in the normal and abnormal parity states n N p , n N n , n A p , n A n . These numbers are determined by filling the Nilsson levels from below, as discussed in [10]. The deformations [19] and occupancies for the 12 isotopes studied in the present work are shown in Table 1.
In the ββ 2ν decay each Gamow-Teller operator annihilates a proton and creates a neutron in the same oscillator shell and with the same orbital angular momentum. As a consequence, the ββ 2ν decay is allowed only if the occupation numbers fulfill the following relationships This selection rule [10] forbids the ββ 2ν decay between the nuclei marked with (1) or without comments in Table 1. However, the pairing interaction allows the mixing between states in the same nuclei with pairs of nucleons transferred between different configurations. These excited configurations are indicated by (2) in Table 1. An energy difference ∆E is required to promote a pair of nucleons from the last occupied normal parity orbital to the next intruder orbital (or viceversa), in the deformed single particle Nilsson scheme. It is listed in the seventh column of Table 1.
The "leading SU(3) irreps", those which are the most bounded under the quadrupole -quadrupole interaction, are the dominant component of the ground state wave function in these heavy deformed nuclei, representing in most cases more than 60% of the total wave function [9]. For the eight rare earth isotopes listed in Table 1, this dominant wave function component can be written as For the actinide isotopes, the intruder sector is The proton, neutron, and total SU(3) irreps associated to each set of occupation numbers are listed in Table 2.
As a first approximation, we will describe the ground state of each nucleus as a linear combination of these two states: Many multipole-multipole pairing type interactions can remove a pair of nucleons from an unique parity orbital and create another pair in a normal parity one. In the present approach we are restricting the pairs of nucleons in intruder orbits to be coupled to J=0, i.e. to have seniority zero. Under this approximation the only term in the Hamiltonian which can connect states with different occupation numbers in the normal and unique parity sectors is pairing. In the present case, the Hamiltonian matrix has the simple form with h pair = Nucleus (2), 0 + |H pair |Nucleus(1), 0 + , whose explicit expression is given in [17]. The values of h pair , x 1 and x 2 are presented in the last two (1) (10, 4) (18, 0) where G 2ν is a kinematical factor which depends on Q ββ , the total kinetic energy available in the decay. The nuclear matrix element is being Γ the Gamow-Teller operator. The energy denominator contains the intermediate E N , initial E i and final E f energies. The kets |1 + N denote intermediate states.
The mathematical expressions needed to evaluate the nuclear matrix elements of the allowed g.s. → g.s. ββ decay in the pseudo SU(3) model were developed in [10]. Using the summation method described in [10], exploiting the fact that the two body terms of theSU(3) Hamiltonian commutes with the Gamow-Teller operator [11], resuming the infinite series and recoupling the Gamow-Teller operators, the following expression was found: where σ(p, n) are the 1-body Gamow-Teller matrix elements and the energy denominator E is determined by demanding that the excitation energy of the Isobaric Analog State equals the difference in Coulomb energies [10,17]. The index p, n refer to the orbitals i p 11/2 , i n 13/2 for 154 Sm, 160 Gd, 170 Er, and 176 Yb, and to the orbitals j p 13/2 , j n 15/2 for 232 Th and 244 Pu. As it was discussed in [10] Eq. (6) has no free parameters, being the denominator E a well defined quantity. The reduction to only one term comes as a consequence of the restricted proton and neutron spaces of the model. The initial and final ground states are strongly correlated with a very rich structure in terms of their shell model components. The evaluation of the matrix elements in the normal space of Eq. (6) is performed by using SU(3) Racah calculus to decouple the proton and neutron normal irreps, and expanding the annihilation operators in their SU(3) tensorial components.
For the six potential ββ emitters listed in Tables I and II, the ββ 2ν decay can only proceed through the second component of the ground state wave function, and for this reason it is proportional to the amplitude x 2 . Its explicit expression is given in [17].
For massive Majorana neutrinos one can perform the integration over the four-momentum of the exchanged particle and obtain a neutrino potential, which for a light neutrino (m ν < 10 MeV) has the form where E is the average excitation energy of the intermediate odd-odd nucleus and the nuclear radius R has been added to make the neutrino potential dimensionless. In the zero neutrino case this closure approximation is well justified [21]. The final formula, restricted to the term proportional to the neutrino mass, is [22,20] (τ where G 0ν is the phase space integral associated with the emission of the two electrons. The nuclear matrix elements M 0ν are [20] M 0ν ≡ |M GT 0ν − where the kets |0 + i and |0 + f denote the corresponding initial and final nuclear states, the quantities g V and g A are the dimensionless coupling constants of the vector and axial vector nuclear currents, and being σ the Pauli matrices related with the spin operator and t − the isospin lowering operator, which satisfies t − |n = |p . The superscript GT denotes  the Gamow-Teller spin-isospin transfer channel, while F indicates the Fermi isospin one. In the present work we use the effective value ( g A g V ) 2 = 1.0 [6]. Transforming this operator to the pseudo SU(3) space, we arrive to the expression where the subscript index NN, NA, . . . are indicating the normal or abnormal spaces of the fermion creation and annihilation operators, respectively. In previous works [10,11] we restricted our analysis to six potential double beta emitters which, within the approximations of the simplest pseudo SU(3) scheme, were also decaying via the 2ν mode. They included the observed 150 Nd → 150 Sm and 238 U → 238 Pu decays. In these cases two neutrons belonging to a normal parity orbital decay in two protons belonging to an abnormal parity one. The transition is mediated by the operator O α ApNn . In Table 3 the ββ 2ν decays of the six nuclei, previously reported as forbidden, are presented. Those with the larger Q ββ values have the larger phase-space integrals G 2ν . The ββ 2ν -decay matrix elements M GT 2ν are suppressed by a factor x 2 ≈ 1/3 compared with the "allowed" ones (see [10]), which reflects in ββ 2ν half-lives a factor 10 larger than in other nuclei with similar Q-values. The exception is 244 Pu, which for the deformation used has a large mixing, and the shorter ββ 2ν half-life, which is not far from the limits reported in the Livermore experiment [23]. The decay of 160 Gd is suppressed but it is still not far from the present limits [24], and large enough to be seen in the proposed experiments [16].
Those configurations in which the ββ 2ν transitions are forbidden can still be connected through the zero neutrino mode, due to presence of the neutrino  potential. In this way there are two terms in the ββ 0ν decay: one connecting to the basis state which has allowed ββ 2ν decay, and one to the state with forbidden ββ 2ν decay. The equations needed in the first case are the same employed in the study of allowed decays [10,11]. The second one involves the annihilation of two neutrons in normal parity orbitals, and the creation of two protons in normal parity orbitals (intruder-intruder in 154 Sm). The transition is mediated by the operator O α NpNn (O α ApAn ). A detailed description of the calculations involved is presented in [17]. ββ 0ν phase-space integrals, nuclear matrix elements and half-lives are shown in Table 4.
As a consequence of the explicit inclusion of deformation in the present model, the ββ 0ν half-lives are larger than those reported in [25]. In 160 Gd the ββ 0ν -decay half-life is at least three orders of magnitude larger than the ββ 2ν -decay half-life. It implies that the background suppression due to a large ββ 2ν half-life would be effective, although not as noticeably as was optimistically envisioned in [16]. In any case, the results presented strongly suggest that the planned experiments using GSO crystals [16] would be able to detect the ββ 2ν decay of 160 Gd, and to establish competitive limits to the ββ 0ν decay.
The present results consider only the dominant pseudo SU(3) irrep for each configuration. We have learned from realistic calculations, where the single particle term and pairing interactions induce the mixing of different irreps, that the leading irreps represent in most even-even heavy deformed nuclei at least 60 % of the total wave function [9]. The inclusion of spin dependent terms in the Hamiltonian, relevant to the description of the Gamow-Teller resonance, is not expected to strongly modify the ground state wave function of the even-even initial and final nuclei. This dominance lead us to expect that future calculations, which will take into account contributions from various irreps, would slightly affect the present predictions. Given the leading role play by the quadrupole-quadrupole interaction in heavy deformed nucleus, we are confident that the order of magnitude of the predicted ββ half-lives, when various irreps are included in the calculations, will remain unchanged, as compared to the results reported above.
Work supported in part by CONACyT, and by a CONACyT-CONICET agreement under the project Algebraic methods in nuclear and subnuclear physics. O.Civitarese is a fellow of the CONICET, Argentina.