Limits on Effective Masses of Light and Heavy Majorana Neutrinos for Positron Emitting Modes of Double Beta Decay

Double beta decay is a rare weak interaction process in which two identical nucleons inside the nucleus decay with or without the emission of neutrinos. If the neutrinoless double beta decay is observed, the (e + DBD) processes will play a crucial role in discriminating the finer issues like the dominance of Majorana neutrino mass or the right-handed current. In the present work, we have obtained the limits on the effective mass of light and heavy Majorana neutrinos for the electron-positron conversion and double positron-emitting modes of 96 Ru, 106 Cd, 124 Xe, and 130 Ba isotopes, using the nuclear transition matrix elements NTMEs M (0ν) and M (0N) for light and heavy Majorana neutrinos obtained in projected Hartree-Fock Bogoliubov (PHFB) model. The predicted half-lives and corresponding extracted limits on heavy neutrino mass <M N > is discussed. We have also calculated nuclear sensitivities ξ (0ν) and ξ (0N) due to the light and heavy neutrino exchange, respectively. Finally, the mass limits are obtained using various phase space factors (PSF), and the effect of this PSF on mass limits is discussed.


Introduction
The nuclear double beta ( ) decay is characterized by two modes. They are the twoneutrino double-beta ( ) decay and the neutrinoless double beta ( ) decay. These modes can further be classified into double electron ( ) emission, double positron ( ) emission, electron-positron conversion ( ) and double electron capture (ECEC). The latter three processes are energetically competing, and we refer to them as positron double beta decay (e + DBD) modes as given in Eqs. (1) and (2).
The experimental and theoretical study of the (e+DBD) decay has been excellently reviewed [1][2][3][4][5][6][7][8][9][10][11][12][13]. Also, the observation of flavor oscillation of neutrinos at solar neutrino [14,15], atmospheric neutrino [16][17][18], and reactor neutrino experiments [19,20] confirmed the conclusion that neutrinos are massive particles. However, it is generally agreed that the observation of 0νββ decay can clarify a number of issues regarding the nature of neutrinos, namely the origin of neutrino mass (Dirac vs. Majorana), the absolute scale on neutrino mass, the type of a hierarchy, and CP violation in the leptonic sector, etc. [21]. On the other hand, observing (e + DBD) 0ν decay modes will help decide issues such as the dominance of the mass mechanism or right-handed currents [22]. The experimental and theoretical study of (ECEC) 0ν decay mode has not been attempted in a great deal so far, even though the kinetic energy released in this decay mode is the largest. The emission of one real photon is forbidden for the 0 + → 0 + transition if atomic electrons are absorbed from the K-shell. Therefore, one must consider various processes such as internal pair production, internal conversion, etc. [4]. The decay rates of the processes mentioned above have to be calculated at least by the third-order perturbation theory. Hence, the experimental, as well as theoretical study of (e+DBD)0ν decay, has been restricted to only (β + β + ) 0ν and (εβ + ) 0ν modes. The complex structure of nuclei in general, and of mass region 96 < A < 156 in particular, is due to the subtle interplay of pairing and multipolar correlations present in the effective two-body interaction. The mass region A  100 offers a nice example of shape transitions at N = 60. The nuclei are soft vibrators for neutron numbers N <60 quasirotors for N >60. Nuclei with neutron numbers N = 60 are transitional nuclei. Hence, it is clear that deformation plays a crucial role in reproducing the properties of these nuclei. The effects of pairing and quadrupolar correlations on the NTMEs of the (β − β − ) 0ν mode have been studied in the interacting shell model (ISM) [23].
In the projected Hartree-Fock Bogoliubov (PHFB) model, the role of deformation effects due to quadrupolar [24] and multipolar correlations [25] has also been studied. In the PHFB model, the interplay of pairing and deformation degrees of freedom is treated simultaneously and on equal footing. However, the structure of the intermediate odd Z-odd N nuclei, which provide information on the single-β decay rates and the distribution of GT strengths, cannot be studied in the present version of the PHFB model. In spite, the PHFB model along with the Pairing plus quadrupole (PQQ) interaction [26] in conjunction with the summation method has been successfully applied to study the (e + DBD) 2ν decay of 96 Ru, 106 Cd, 124 Xe and 130 Ba [27] isotopes for the 0 + → 0 + transition, not in isolation but together with other observed nuclear spectroscopic properties, namely yeast energy spectra, reduced B(E2:0 + → 2 + ) transition probabilities, quadrupole moments Q(2 + ), and gyromagnetic factors g (2 + ). This success of the PHFB model has prompted us to apply the same to study the 0 + → 0 + transition of ( ) and ( ) modes for the nuclei mentioned above. The article is organized as follows. A brief description of theoretical formalism is presented in section 2. In section 3, we have calculated limits on effective masses of light and heavy Majorana neutrinos and analyzed them along with nuclear sensitivities. The different phase space factors and their effect on light and heavy Majorana neutrino mass limits are discussed. Finally, some concluding remarks are presented in section 4.

Theoretical Formalism
In the neutrino mass mechanism, the half-lives for the transition of ( ) and ( ) modes in the 2n mechanism are given by [4,28] [ Here, β denotes the (εβ + ) 0ν /(β + β + ) 0ν modes. is the integrated kinematical factor and m p (m e ) being the proton (electron) mass. The effective light and heavy neutrino masses are given by where U ei is the unitary mixing matrix, and m i is the Majorana neutrino mass.
Here, K = 0v, 0N denotes the exchange of light and heavy Majorana neutrino mechanism.
In the closure approximation, Fermi and Gamow-Teller, nuclear transition matrix elements (NTMEs), ( ) and ( ) are written as Here I,F denote the initial and final states. The neutrino potential H(r) arising due to the exchange of light neutrino [29] is defined as E N , E I , E F being the energies of intermediate, initial, and final states. The effect due to the finite size of nucleons (FNS) is taken into account by a dipole type of form factor, making the replacement with Λ = 850 MeV and g V (q 2 ), g A (q 2 ) are real functions of a Lorenz scalar q 2 . The values of these form factors in the zero momentum transfer limit are the vector and axial-vector coupling constants, respectively. The values taken for these coupling constants are g v =1.0 and g A =1.254. In the PHFB model, configuration mixing takes care of long-range correlations. The short-range correlation (SRC) effect, arising mainly from the repulsive nucleon-nucleon potential due to exchange of ρ and ω mesons, is incorporated through phenomenological Jastrow type of correlation using the Miller and Spencer parameterization by the prescription [30] with a = 1.1 fm -2 , b = 0.68 fm -2 and c = 1.0 In the PHFB model, the calculation of the NTMEs M (K) of the ( ) and ( ) modes are carried out as follows. The two basic ingredients of the PHFB model are an independent quasiparticle mean-field solution and the projection technique. To start with, amplitudes (u im , v im ) and expansion coefficients C ij,m required to specify the axially symmetric HFB intrinsic state | ⟩ with K = 0, is obtained by carrying out the HFB calculation by minimizing the expected value of the effective Hamiltonian. Subsequently, states with good angular momentum J are obtained from | ⟩ using the standard projection technique [31] given by where ( ) and ( ) are the rotation operator and the rotation matrix, respectively. Further with the creation operators and ̄ defined as Finally, the NTMEs M (K) for the ( ) and ( ) modes are calculated by using the closure approximation in the PHFB model as where The NTMEs M (K) for the ( ) and ( ) modes are calculated by evaluating the matrices [ ] and ( ) using expressions (21) and (22) respectively and then using these values in equation (18), the required NTMEs are obtained with 20 Gaussian quadrature points in the range (0,π).

Results of (εβ + ) 0ν and (β + β + ) 0ν modes
In The matrix elements ( ) and ( ) for light and heavy Majorana neutrinos are calculated in the PHFB model using short-range correlation and finite-size effect and are given in columns 5 and 6, respectively. From the Table, it is seen that the extracted limits on m ν and M N are not so much stringent as in the case of (ββ -) 0ν decay. It is observed that better limits are obtained for (εβ + ) 0ν mode even if the limits on half-lives of (εβ + ) 0ν and (β + β + ) 0ν modes are same. The best-obtained limits are ⟨ ν ⟩ < 1.9x10 2 eV for 106 Cd isotope and ⟨ ⟩>> 2.12x10 4 GeV for 130 Ba isotope in case of (εβ + ) 0ν mode while ⟨ ν ⟩ < 1.07x10 3 eV and ⟨ ⟩>> 1.37x10 4 GeV for 106 Cd isotope in case of (β + β + ) 0ν mode. As the extracted limits on the effective neutrino masses ⟨ ⟩ and ⟨ ⟩ are not stringent enough, it is more meaningful to calculate half-lives of (εβ + ) 0ν and (β + β + ) 0ν modes, which will be helpful in the design of future experimental setups. Hence, the half-lives of (εβ + ) 0ν and (β + β + ) 0ν modes for ⟨ ν ⟩ = 50 meV are calculated, and we have extracted corresponding limits on heavy neutrino mass ⟨ ⟩, as shown in Table 2. In the absence of stringent limits on the effective neutrino masses ⟨ ν ⟩ and ⟨ ⟩, we can calculate the nuclear sensitivity, defined as [37]     K K M G 01 8 10   (24) where K stands for 0ν or 0N mode, and an arbitrary normalization factor 10 8 is introduced to make the nuclear sensitivity of the order of unity. It is observed that, in general, nuclear sensitivities for (εβ + ) 0ν mode are larger than those for (β + β + ) 0ν mode. Further, in case of light neutrinos, the nuclear sensitivities for 106 Cd, 96 Ru, 124 Xe, and 130 Ba isotopes are in decreasing order of magnitude for (εβ + ) 0ν and (β + β + ) 0ν modes. In case of heavy neutrinos, the nuclear sensitivities show decreasing behavior for 106 Cd, 124 Xe, 130 Ba, and 96 Ru isotopes respectively for (εβ + ) 0ν mode while for (β + β + ) 0ν mode, similar behavior is observed except 96 Ru isotope having nuclear sensitivity greater than 130 Ba isotope.

Effect of phase space factors on effective light and heavy Majorana neutrino mass limits
In order to observe the effect of phase space factors on effective light ⟨ ν ⟩ and heavy ⟨ ⟩ Majorana neutrino mass limits, we have presented the phase factors (PSF) for (εβ + ) 0ν and (β + β + ) 0ν modes in units of yr -1 calculated by Kotila and Iachello [32] and Doi and Kotani [38] in Table 3. Kotila and Iachello have calculated PSF using exact Dirac wave functions with finite nuclear size and electron screening, while Doi and Kotani obtained the results by using approximate electron wave functions. For comparison, we have also included the PSF calculated by Boehm and Vogel [39], but their results are not available for (εβ + ) 0ν mode. Using these PSF, we have obtained the limits on effective masses of light ⟨ ν ⟩ and heavy ⟨ ⟩ Majorana neutrinos for the (β + β + ) 0ν and (εβ + ) 0ν modes of 96 Ru, 106 Cd, 124 Xe, and 130 Ba isotopes and are shown in Table 4 for comparison.
It is observed that the mass limits ⟨ ν ⟩ on light neutrinos obtained by PSF of Kotila and Iachello (KI) and Doi and Kotani (DK) are nearly same but increase by a factor of 2.6 approximately in case of (β + β + ) 0ν mode for 96 Ru, 106 Cd, 124 Xe isotopes and by a factor of 3.5 nearly for 130 Ba isotope for PSF of Boehm and Vogel (BV). In the case of (εβ + ) 0ν mode, the mass limits do not change much for PSF of KI and DK, while results of BV are not present for this mode. In the case of heavy Majorana neutrinos, once again, the mass limits ⟨ ⟩ are approximately the same for PSF of KI and DK for the two modes, while for (β + β + ) 0ν mode, the limits decrease by the same factors as in the case of light neutrinos for 96 Ru, 106 Cd, 124 Xe, and 130 Ba isotopes. Thus it is concluded that the limits obtained on effective masses of light ⟨ ν ⟩ and heavy ⟨ ⟩ Majorana neutrinos for the (β + β + ) 0ν and (εβ + ) 0ν modes of 96 Ru, 106 Cd, 124 Xe and 130 Ba isotopes using the PSF of KI and DK are nearly the same while results differ by a factor of 2.6 and 3.5 approximately for PSF of BV in case of 96 Ru, 106 Cd, 124 Xe, and 130 Ba isotopes respectively.

Conclusion
We have calculated the limits on the effective mass of light ⟨ ν ⟩ and heavy ⟨ ⟩ Majorana neutrinos for the (εβ + ) 0ν and (β + β + ) 0ν modes of 96 Ru, 106 Cd, 124 Xe and 130 Ba isotopes using matrix elements calculated in PHFB model. The best limits are obtained for 106 Cd isotope. Also, the half-lives of the isotopes mentioned above for ⟨ ν ⟩ = 50meV is calculated in case of (εβ + ) 0ν and (β + β + ) 0ν modes and corresponding limits on heavy neutrino mass, ⟨ ⟩ are extracted. The highest limit is obtained for 106 Cd isotope corresponding to (εβ + ) 0ν and (β + β + ) 0ν modes. Further, the nuclear sensitivities are calculated, and in general, the sensitivities for (εβ + ) 0ν mode are found larger than those of (β + β + ) 0ν mode. The three different phase space factors are used to see the effect on mass limits ⟨ ν ⟩ and ⟨ ⟩ of light and heavy Majorana neutrinos and it is observed that the mass limits due to two PSF using exact and approximate electrons wave functions by KI and DK respectively do not change much but differ by a large factor due to PSF of BV.