Neutrinoless Double-Beta Decay and Realistic Shell Model

We report on the calculation of the neutrinoless double-beta decay nuclear matrix element for 76Ge within the framework of the realistic shell model. The effective shell-model Hamiltonian and the two-body transition operator describing the decay are derived by way of many-body perturbation theory. Particular attention is focused on the role played by the so-called Pauli blocking effect in the derivation of the effective operator.


Introduction
Neutrinoless double-β (0νββ) decay is an exotic nuclear process predicted by extensions of the Standard Model of particle physics. Observation of such a process would prove the non-conservation of lepton number, and show that neutrinos have a Majorana mass component (see Refs. [1,2] and references therein).
In the framework of light Majorana neutrino exchange, the half life of the 0νββ decay is inversely proportional to the square of the effective Majorana neutrino mass m ν ≡ | i U 2 ei m ν i |. Explicitly, it can be written as where G 0ν is the so-called phase-space factor (or kinematic factor), and M 0ν is the nuclear matrix element (NME) directly related to the wave functions of the parent and grand-daughter nuclei. At present, the phase-space factors involved in the double β decays of nuclei of experimental interest are calculated with great accuracy [3,4], therefore it is crucial to have precise values of the NME, both to improve the reliability of the 0νββ lifetime predictions, and to extract neutrino properties from the experimental results. Several nuclear structure models have been employed to provide NME values as precise as possible, the most largely employed being, at present, the Interacting Boson Model (IBM) [5][6][7], the Quasiparticle Random-Phase Approximation (QRPA) [8][9][10][11], Energy Density Functional methods [12], and the Shell Model (SM) [13][14][15][16][17][18][19].
Each model can be more suitable than another for a certain class of nuclei, and, when comparing the calculated NMEs, it can be seen that, at present, the results obtained with different approaches can differ by a factor of two or three (see for instance the review [20]).
It is worth noting that, all the above models are based on the use of a truncated Hilbert space to overcome the * e-mail: nunzio.itaco@unicampania.it computational complexity, but only within the realistic shell model an effective 0νββ operator, which takes into account the degrees of freedom which do not appear explicitly in the calculated wavefunctions, may be introduced. This approach has been pioneered by Kuo and coworkers [21,22], and more recently pursued by Holt and Engel [23]. Actually, in the SM approach it is possible to consider the correlations involving single-particle orbitals outside the model space deriving an effective operator. This is the so-called realistic shell-model approach where both the effective Hamiltonian and the transition operators are derived from a realistic free nucleon-nucleon (NN) potential V NN by way of the many-body theory [24,25].
In two recent papers [26,27], we have adopted the above approach to calculate properties related to the GT and 2νββ decays of nuclei 0νββ-candidates, with mass ranging from A = 48 to A = 136. Our results and their comparison with the experimental data have shown that many-body perturbation theory (MBPT) allows to derive effective SM Hamiltonians and transition operators that reproduce quantitatively the observed spectroscopic and decay properties, without resorting to an empirical quenching of the axial coupling constant g A .
On these grounds, we report in this paper on a preliminary calculation of the NME involved in the 0νββ decay of 76 Ge, that is currently investigated by GERDA experiment [28] at the Laboratori Nazionali del Gran Sasso (LNGS) of INFN, and by CDEX-1 experiment [29] at China Jinping Underground Laboratory.
Our theoretical framework is the many-body perturbation theory [25,26,30,31]; we start from a realistic nuclear potential, and derive an effective SM Hamiltonian and a 0νββ decay operator that are employed to calculate the wavefunctions of the ground states of 76 Ge and 76 Se and the 0νββ NME. We focus our attention on the role of the so-called "blocking effect", that takes into account the Pauli exclusion principle in systems with more than two valence nucleons [32,33] in the derivation of the two-body effective 0νββ operator.
The paper is organized as follows. In the next section we give a brief description of how we derive the shellmodel Hamiltonian and the two-body effective 0νββ decay operator. In Section 3, we report the results that we obtain for the 76 Ge NME, analyzing in detail the contribution arising from the core-polarization blocking diagrams. A brief summary is reported in the last section.

Outline of calculations
The starting point of our calculation is provided by the high-precision CD-Bonn NN potential [34], that is smoothed integrating out its repulsive high-momentum components by way of the so-called V low−k approach [35,36]. In this way we get a softer NN potential defined up to a cutoff Λ, that preserves the physics of the original CD-Bonn interaction. The value of Λ is chosen, as in many of our recent papers [26,27,[37][38][39] equal to 2.6 fm −1 , this value being a trade off between the need of minimizing the role of the missing three-nucleon force (3NF) [38] and that of ensuring the perturbative behavior of the potential. The Coulomb potential is explicitly taken into account in the proton-proton channel.
The harmonic oscillator (HO) potential U is introduced as an auxiliary one-body potential in order to break up the Hamiltonian for a system of A nucleons as the sum of a one-body term H 0 , which describes the independent motion of the nucleons, and a residual interaction H 1 : It is now possible to define a truncated model space in terms of the eigenvectors of H 0 . To study the 0νββ decay of 76 Ge we employ a model space spanned by the four 1p 3/2 , 1p 1/2 , 0 f 5/2 , 0g 9/2 proton and neutron orbitals outside the doubly-closed 56 Ni core, and we derive an effective shell-model Hamiltonian H eff , that takes into account the degrees of freedom that are not explicitly included in the shell-model framework.
We derive H eff by resorting to the many-body perturbation theory, an approach that has been developed by Kuo and coworkers through the 1970s [24,40]. More precisely, we use the well-knownQ box-plus-folded-diagram method [41], where theQ box is defined as a function of the unperturbed energy of the valence particles: where the operator P projects onto the model space and Q = 1 − P. In the present calculations theQ box is expanded as a collection of one-and two-body irreducible valence-linked Goldstone diagrams up to third order in the perturbative expansion [31,42].
Within this framework, it can be shown that the effective Hamiltonian H eff can be written in terms of theQ box derivatives [43] where ... The H eff for one-valence-nucleon systems provides the single-particle (SP) energies for our SM calculations, while the two-body matrix elements (TBMEs) are obtained from H eff derived for the nuclei with two valence nucleons, by subtracting the theoretical SP energies.
The so obtained SP energies and TBMEs can be found in [27], where it is also reported a detailed discussion of the perturbative properties of H eff .
The NME involved in 0νββ decay, M 0ν , may be expressed as where the matrix elements are defined as follows: with α = (GT, F), having neglected the tensor term, whose effect has been estimated to be of the order of few percent [15]. The operators O α mn have the following expression: and the neutrino potentials H α are defined using the closure approximation where f F,GT (qr) = j 0 (qr), E is the average energy of the virtual intermediate states used in the closure approximation, while the explicit expression of the form factors h α (q 2 ) can be found for instance in Ref. [44].
As mentioned before, we want to derive the effective 0νββ decay operator tailored for the chosen model space. To this end, we resort to the formalism presented by Suzuki and Okamoto in Ref. [25]. In this approach, a non-Hermitian effective operator Θ eff can be expressed as where the operators χ n are defined as: · · · whereΘ m ,Θ mn have the following expressions: withΘ It is worth noting that using Eqs. 5, 6, Eq. 10 may be then rewritten as enlightening the connection existing between the effective Hamiltonian and the effective operators. In present calculation for the effective two-body 0νββ decay operator, we arrest the χ n series to the χ 2 term. Since χ 3 depends on the first, second, and third derivatives ofΘ 0 andΘ 00 , and on the first and second derivatives of theQ box (see Eq. 13), our estimation of these quantities leads to evaluate χ 3 being at least one order of magnitude smaller than χ 2 . The calculation is performed starting from a perturbative expansion ofΘ 0 andΘ 00 , and in Fig. 1 we report all the two-body Θ 0 diagrams up to the first order in V low−k , the bare operator Θ being represented by a dashed line. The circle with a cross inside represents a first-order (V low−k − U)-insertion, arising from the −U term in the interaction Hamiltonian H 1 (see for example Ref. [31] for details).
The diagrams in Fig. 1 refer to the derivation of the effective operator for a system with two valence-nucleons. When dealing with nuclei with a larger number of valence nucleons, many-body diagrams come into play, accounting for the interaction via the two-body force of the manyvalence nucleons with core excitations as well as with virtual intermediate nucleons scattered above the model space. The two topologies of second-order connected three-valence-nucleon diagrams for a two-body operatorΘ are reported in Fig. 2 (diagrams (a) and (b)). These diagrams correct the Pauli-principle violation introduced by diagram (a 1 ) and (b 1 ) when one of the intermediate particle states is equal to m.  Figure 2. Three-body second-order diagrams that should appear in the perturbative expansion ofΘ. As in Fig. 1, the dashed line indicates the bare operator Θ, the wavy line the two-body potential V low−k . For the sake of simplicity, for each topology only one of the diagrams which correspond to the permutation of the external lines is drawn.
Since there are no shell-model codes, at present, able to handle three-body transition operators, we derive a density-dependent two-body operator from the three-body one by summing and averaging over one incoming and outcoming particle of the connected diagrams (a) and (b) of Fig. 2. In this way we take into account the filling of the model-space orbitals when dealing with more than two valence nucleons.
More precisely, first we calculate exactly the three-body connected diagrams (k) [( j a j b ) J , j m ] J |(k)|[( j c j d ) J , j n ] J , with k equal to a and b, and whose expressions can be found in Ref. [45], then we evaluate the corresponding density-dependent two-body diagram [( j a j b ) J |(k)|[( j c j d ) J , whose explicit expression is where the summation over m-index runs in the model space, and ρ m is the unperturbed occupation density of the orbital j m accordingly to the number of valence nucleons. The so obtained two-body contribution is finally added to the collection of the diagrams of the perturbative expansion ofΘ.

Results
In this Section we present the results of our SM calculations for the NME of 76 Ge. As previously mentioned, all the calculations have been performed employing theoretical SP energies, TBMEs, and effective transition operators, describing 76 Ge as a core of 56 Ni plus 4 protons and 16 neutrons interacting in a reduced model space spanned by the four 1p 3/2 , 1p 1/2 , 0 f 5/2 , 0g 9/2 orbitals.
In Ref. [27] we have compared the calculated lowenergy spectrum, the electromagnetic properties, the GT − strength distributions and the calculated NMEs of the 2νββ decay with the available experimental data, showing a quantitative agreement.
In order to see how much the choice of the effective Hamiltonian can affect the value of the NME, in Table 1 we compare our calculated NME using the bare 0νββ operator without any renormalization, with the values obtained in two other SM calculations performed using the same model space but with different empirical SM hamiltonians [15,46]. Table 1. Theoretical 0νββ NME calculated in the framework of the SM using different effective Hamiltonians. I and II are the results obtained by Menendez and coworkers [15] and by Sen'kov and coworkers [46], respectively. The 0νββ operator employed is not renormalized.
Our I II NME 3.40 2.96 3.25 As it can be seen, the theoretical results are in good agreement, the largest discrepancy non exceeding 15%.
In Table 2 we report the calculated 0νββ NME obtained using an effective operator derived by second order MBPT and taking into account all the intermediate states up to an excitation energy of 14 ω, which are enough to provide convergent NME values. To clarify the role of Pauli blocking diagrams we have derived two different operators with and without taking into account diagrams (a) and (b) of Fig. 2. The inspection of Table 2 shows that including Pauli blocking diagrams provides a further reduction of M 0ν by an amount around 10%, and that this effect is mainly due to the reduction of the Gamow-Teller contribution.
To better understand the action of Pauli blocking diagrams, we report in Table 3 the values of the diagrams reported in Fig. 2 for GT-operator corresponding to the most relevant configurations involved in the 76 Ge 0νββ decay.

Summary
In this paper we have reported on some preliminary results for the SM calculation of the nuclear matrix element involved in the 76 Ge 0νββ decay. The effective shell-model Hamiltonian and the two-body transition operator describing the decay are derived by way of many-body perturbation theory at third and second order, respectively. The role of the Pauli blocking three body diagrams that appear in the perturbative expansion of the effective operator when dealing with more than two valence nucleons, is taken into account in an approximate way by introducing two-body density-dependent diagrams.