Double beta decay to the first $2^+$ state within a boson expansion formalism with a projected spherical single particle basis

The Gamow-Teller transition operator is written as a polynomial in the dipole proton-neutron and quadrupole charge conserving QRPA boson operators, using the prescription of the boson expansion technique of Belyaev-Zelevinski type. Then, the $2\nu\beta\beta$ process ending on the first $2^+$ state in the daughter nucleus is allowed via one, two and three boson states describing the odd-odd intermediate nucleus. The approach uses a single particle basis which is obtained by projecting out the good angular momentum from an orthogonal set of deformed functions. The basis for mother and daughter nuclei have different deformations. The GT transition amplitude as well as the half lives were calculated for ten transitions. Results are compared with the available data as well as with some predictions obtained with other methods.

One of the most exciting subject of nuclear physics is that of double beta decay. The interest is generated by the fact that in order to describe quantitatively the decay rate one has to treat consistently the neutrino properties as well as the nuclear structure features.
The process may take place in two distinct ways: a) by a 2νββ where decay the initial nuclear system, the mother nucleus, is transformed in the final stable nuclear system, usually called the daughter nucleus, two electrons and two anti-neutrinos; b) by the 0νββ process where the final state does not involve any neutrino. The latter decay mode is especially interesting since one hopes that its discovery might provide a definite answer to the question whether the neutrino is a Majorana or a Dirac particle. The contributions over several decades have been reviewed by many authors. [1,2,3,4,5,6].
Although none of the double beta emitters is a spherical nucleus most formalisms use a single particle spherical basis.
In the middle of 90's we treated the 2νββ process in a pnQRPA formalism using a projected spherical single particle basis which resulted in having a unified description of the process for spherical and deformed nuclei [7,8]. Recently the single particle basis [9,10] has been improved by accounting for the volume conservation while the mean field is deformed [11,12]. The improved basis has been used for describing quantitatively the double beta decay rates as well as the corresponding half lives [13,14]. The results were compared with the available data as well as with the predictions of other formalisms. The manners in which the physical observable is influenced by the nuclear deformations of mother and daughter nuclei are in detail commented. Two features of the deformed basis are essential: a) the single particle energy levels do not exhibit any gap; b) the pairing properties of the deformed system are different from those of spherical system. These two aspects of the deformed nuclei affect the overlap matrix of the pnQRPA states of mother and daughter nuclei. Moreover, considering the Gamow-Teller (GT) transition operator in the single particle-space generated by the deformed mean-field, one obtains an inherent renormalization with respect to the one acting in a spherical basis.
In Ref. ([15]) we studied the higher pnQRPA effects on the GT transition amplitude, by means of the boson expansion technique for a spherical single particle basis. Considering higher order boson expansion terms in the transition operator, significant corrections to the GT transition amplitude are obtained especially when the strength of the two body particle-particle (pp) interaction approaches its critical value where the lowest dipole energy is vanishing. As we showed in the quoted reference, there are transitions which are forbidden at the pnQRPA level but allowed once the higher pnQRPA corrections are included. An example of this type is the 2νββ decay leaving the daughter nucleus in a collective excited state 2 + . The electrons resulting in this process can be distinguished from the ones associated to the ground to ground transition by measuring, in coincidence, the gamma rays due to the transition 2 + → 0 + in the daughter nucleus [16].
The aim of this letter is to study the double beta decay 0 + → 2 + where 0 + is the ground state of the emitter while 2 + is a single quadrupole phonon state describing the daughter nucleus. The procedure is the boson expansion method as formulated in our previous paper [15] but using a projected spherical single particle basis.
In order to fix the necessary notations and to be self-contained, in the present work we describe briefly the main ideas underlying the construction of the projected single particle basis.
The single particle mean field is determined by a particle-core Hamiltonian: where H sm denotes the spherical shell model Hamiltonian while H core is a harmonic quadrupole boson (b + µ ) Hamiltonian associated to a phenomenological core. The interaction of the two subsystems is accounted for by the third term of the above equation, written in terms of the shape coordinates α 00 , α 2µ . The quadrupole shape coordinates are related to the quadrupole boson operators by the canonical transformation: where k is an arbitrary C number. The monopole shape coordinate is determined from the volume conservation condition. In the quantized form, the result is: AveragingH on the eigenstates of H sm , hereafter denoted by |nljm , one obtains a deformed boson Hamiltonian whose ground state is, in the harmonic limit, described by a coherent with |0 b standing for the vacuum state of the boson operators and d a real parameter which simulates the nuclear deformation. On the other hand, the average ofH on Ψ g is similar to the Nilsson Hamiltonian [17]. Due to these properties, it is expected that the best trial functions to generate a spherical basis are: The projected states are obtained by acting on these deformed states with the projection operator The subset of projected states : are orthogonal with the normalization factor denoted by N I nlj . Although the projected states are associated to the particle-core system, they can be used as a single particle basis. Indeed, when a matrix element of a particle like operator is calculated, the integration on the core collective coordinates is performed first, which results in obtaining a final factorized expression: one factor carries the dependence on deformation and one is a spherical shell model matrix element.
The single particle energies are approximated by the average of the particle-core Hamiltonian H ′ =H − H core on the projected spherical states defined by Eq.(0.7): The off-diagonal matrix elements of H ′ is ignored at this level. Their contribution is however considered when the residual interaction is studied.
As shown in Ref. [9], the dependence of the new single particle energies on deformation is similar to that shown by the Nilsson model [17]. The quantum numbers in the two schemes are however different. Indeed, here we generate from each j a multiplet of (2j + 1) states distinguished by the quantum number I, which plays the role of the Nilsson quantum number Ω and runs from 1/2 to j and moreover the energies corresponding to the quantum numbers K and -K are equal to each other. On the other hand, for a given I there are 2I +1 degenerate sub-states while the Nilsson states are only double degenerate. As explained in Ref. [9], the redundancy problem can be solved by changing the normalization of the model functions: Due to this weighting factor the particle density function is providing the consistency result that the number of particles which can be distributed on the (2I+1) sub-states is at most 2, which agrees with the Nilsson model. Here α stands for the set of shell model quantum numbers nlj. Due to this normalization, the states Φ IM α used to calculate the matrix elements of a given operator should be multiplied with the weighting factor 2/(2I + 1).
Finally, we recall a fundamental result, obtained in Ref. [12], concerning the product of two projected states, which comprises a product of two core components. Therein we have proved that the matrix elements of a two body interaction corresponding to the present scheme are very close to the matrix elements corresponding to spherical states projected from a deformed state consisting of two spherical single particle states times a single collective core wave function. The small discrepancies of the two types of matrix elements could be washed out by using slightly different strengths for the two body interaction in the two methods.
As we already stated, in the present work we are interested to describe the Gamow-Teller two neutrino double beta decay of an even-even deformed nucleus. In our treatment the Fermi transitions, contributing about 20% to the total rate, and the "forbidden" transitions are ignored, which is a reasonable approximation for the two neutrino double beta decay in medium and heavy nuclei. The 2νββ process is conceived as two successive single β − transitions. The first transition connects the ground state of the mother nucleus to a magnetic dipole state 1 + of the intermediate odd-odd nucleus which subsequently decays to the first state 2 + of the daughter nucleus. The second leg of the transition is forbidden within the pnQRPA approach but non-vanishing within a higher pnQRPA approach [15]. The states, involved in the 2νββ process are described by the following many body Hamiltonian: The operator c † τ αIM (c τ αIM ) creates (annihilates) a particle of type τ (=p,n) in the state Φ IM α , when acting on the vacuum state |0 . In order to simplify the notations, hereafter the set of quantum numbers α(= nlj) will be omitted. The two body interaction consists of three terms, the pairing, the dipole-dipole particle hole (ph) and the particle-particle (pp) interactions. The corresponding strengths are denoted by G τ , χ, χ 1 , respectively. All of them are separable interactions, with the factors defined by the following expressions: The remaining operators from Eq.(0.10) can be obtained from the above defined operators by hermitian conjugation.
The one body term and the pairing interaction terms are treated first through the standard BCS formalism and consequently replaced by the quasiparticle one body term τ IM E τ a † τ IM a τ IM . In terms of quasiparticle creation (a † τ IM ) and annihilation (a τ IM ) operators, related to the particle operators by means of the Bogoliubov-Valatin transformation, the two body interaction terms, involved in the model Hamiltonian, can be expressed just by replacing the operators (3.2) by their quasiparticle images. Thus, the Hamiltonian terms describing the quasiparticle correlations become a quadratic expression in the dipole and quadrupole two quasiparticles and quasiparticle density operators: Since the pnQRP A treatment of the dipole-dipole interaction in the particle-hole (ph) and pp channels run in an identical way as in our previous publications [13,14], here we don't give any detail about building the dipole proton-neutron phonon operator : We just mention that the amplitude are determined by the pnQRPA equations and the normalization condition.
The charge conserving QRP A bosons are determined by the QRP A equations associated to the matrices: (0. 16) In order to distinguish between the phonon operators acting in the RPA space associated to the mother and daughter nuclei respectively, one needs an additional index. Also, an index labeling the solutions of the RPA equations is necessary. Thus, the two kinds of bosons will be denoted by: Acting with Γ † 1µ (ik) and Γ † 1µ (f k) on the vacuum states |0 i and |0 f respectively, one obtains two sets of non-orthogonal states describing the intermediate odd-odd nucleus. By contrast, the states Γ † 2 (ik)|0 i and Γ † 2 (f k)|0 f describe different nuclei, namely the initial and final ones participating in the process of 2νββ decay. The mentioned indices are however omitted whenever their presence is not necessary.
Within the boson expansion formalism the transition GT operators are written as polynomial expansion in terms of the QRPA boson operators with the expansion coefficients determined such that the mutual commutation relations of the constituent operators A † 1µ (pn), A 1µ (pn), B † 1µ (pn), B 1µ (pn) be preserved in each order of approximation [18]. One arrives at the expressions: where the expansion coefficients are those given in Ref. [15] . the reciprocal value of the 2νββ half life can be factorized as: where F 2 is the Fermi integral which characterizes the phase space of the process while the second factor is the GT transition amplitude which, in the second order of perturbation theory, has the expression: Here ∆E 2 = ∆E + E 1 + , with E 1 + standing for the experimental energy for the first state 1 + . The The term E k,m from the denominator of Eq. (0.21) is the average of the energies of the mother and daughter states |k, m normalized to the average energy of the first pnQRP A states 1 + in the initial and final nuclei. Calculations were performed for the following 10  [13,14] for ground to ground transition, we don't present them again. The strength of the QQ interaction was fixed by requiring that the first root of the QRPA equation for the quadrupole charge conserving boson is close to the experimental energy of the first 2 + state. The results of the fitting procedure are given in Table I.
Having the RPA states defined, the GT amplitude has been calculated by means of The final results are collected in Table II. Therein one may find also the available experimental data as well as some theoretical results obtained with other approaches. One notices that the half life is influenced by both the phase space integral (through the Q-value) and the single particle properties which determine the transition amplitude. Indeed, for 128 Te and 134 Xe the small Q-value causes a very large half life, while in 48 Ca the opposite situation is met. By contrary the Q value of 110 Pd is about the same as for 76 Ge but, due to the specific single particle and pairing properties of the orbits participating coherently to the process, the half life for the former case is more than three orders of magnitude less than in the later situation. Since in Ref. [19] a spherical single particle basis and a renormalized pnQRPA approach are used, the transition matrix elements are larger and the half lives shorter than in our calculations, although the higher RPA formalism in the quoted reference is similar [21] to the one proposed by us in Ref. [15] and used in the present work. It is worth mentioning the good agreement between our prediction for 100 Mo and that of Ref. [20] obtained with a deformed SU(3) single particle basis.
It is worth mentioning that the double beta transitions to excited states have been considered by several authors in the past, but the calculations emphasized the role of the transition operator and some specific selection rules. Many of calculations regarded the neutrinoless process. Thus, in Ref. [22] it was shown that the neutrinoless transition to the excited 0 + for medium heavy nuclei might be characterized by matrix element which are larger than that of ground to ground transition and that happens since in the first transition, the change of the K quantum number is less. In Ref. [23] it has been stated that the 0 + → 2 + matrix element depends on the left-right current coupling and not on the neutrino mass. However according    with the approximate interaction [σ(1) × σ(2)] λ=2 t + (1)t + (2) in order to test some selection rules. Thus, this interaction was used in Ref. [24] for the transition 0 + → 2 + of 48 Ca, using a single j calculation. It has been proved that the matrix element for this transition is suppressed due to the signature selection rules. The transition to 0 + 1 was examined for A = 76, 82, 100, 136 nuclei by assuming light and heavy Majorana neutrino exchange mechanism and triliniar R-parity contribution. Higher RPA as well as renormalization effects for the nuclear matrix elements were included [25].
Here we show that the transition 0 + → 2 + in a 2νββ process is allowed by renormalizing the GT transition operator with some higher RPA corrections which results in making the