Possibilities of future double beta decay experiments to investigate inverted and normal ordering region of neutrino mass

An overview of modern experiments on the search for neutrinoless double decay is presented. The obtained limits on the effective mass of the Majorana neutrino $\langle m_{\nu} \rangle$ are discussed taking into account the uncertainties in the value of the nuclear matrix elements (NMEs) and the value of the axial-vector constant $g_A$. Predictions for the values of $\langle m_{\nu} \rangle$ from the results of oscillation experiments and modern cosmological data are presented. The possibilities of the next generation experiments with sensitivity to $\langle m_{\nu} \rangle$ at the level of $\sim$ 10-50 meV (studying mainly the inverted ordering (IO) region) are discussed. %Description of the most developed and promising projects is presented. The prospects for studying the normal ordering (NO) region are discussed too. It is shown that the possibilities of studying the NO depend on the mass of the lightest neutrino m$_0$. In the limiting case of small mass (m$_0$ $\le$ 0.1 meV), the values of $\langle m_{\nu} \rangle$ $\approx$ 1-4 meV are predicted, which makes the study of this region inaccessible by the next generation experiments. But there is an allowed region of m$_0$ (7-30 meV) in the framework of NO, where the predicted values for $\langle m_{\nu} \rangle$ could be $\sim$ 10-30 meV and that is quite achievable for the next generation experiments. The possibility to rich in the future sensitivity to $\langle m_{\nu} \rangle$ at the level of $\sim$ 1-10 meV is also discussed.


INTRODUCTION
The interest in neutrinoless double decay increased significantly after the discovery of neutrino oscillations in experiments with atmospheric, solar, reactor and accelerator neutrinos (see, for example, discussions in [1,2,3]). This is due to the fact that the very existence of neutrino oscillations indicates that the neutrino has a nonzero mass. However, oscillation experiments are not sensitive to the nature of the neutrino mass (Dirac or Majorana) and do not provide information on the absolute scale of neutrino masses. Registration of neutrinoless double beta decay will clarify many fundamental aspects of neutrino physics (see, for example, discussions in [4,5,6]): (i) lepton number non-conservation; (ii) neutrino nature: whether the neutrino is a Dirac or a Majorana particle; (iii) absolute neutrino mass scale; (iv) the type of neutrino mass ordering (normal or inverted); (v) CP violation in the lepton sector (measurement of the Majorana CP-violating phases).
This process assumes a simple form, namely The discovery of this process is of fundamental interest, since it is practically the only way to establish the Majorana nature of neutrino. The Majorana nature of the neutrino would have interesting implications in many extensions of the Standard Model. For example the seesaw mechanism requires the existence of a Majorana neutrino to explain the lightness of neutrino masses [7,8,9,10]. A Majorana neutrino would also provide a natural explanation for the lepton number violation, and for the leptogenesis process which may explain the observed matter-antimatter asymmetry of the Universe [11].
The standard underlying mechanism behind neutrinoless double-beta decay is the exchange of a light Majorana neutrino. In this case, the half-life time of the decay can be presented as where G 0ν is the phase space factor, which contains the kinematic information about the final state particles, and is exactly calculable to the precision of the input parameters [12,13], g A is the axialvector coupling constant 1 , | M 0ν | is the nuclear matrix element, m e is the mass of the electron, and m ν is the effective Majorana mass of the electron neutrino, which is defined as m ν = | i U 2 ei m i | where m i are the neutrino mass eigenstates and U ei are the elements of the neutrino mixing Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix.
In contrast to two-neutrino decay (this decay has been detected -see review [18], for example), neutrinoless double beta decay has not yet been observed. The best limits on m ν are obtained for 136 Xe, 76 Ge, 130 Te, 100 Mo and 82 Se (see Section 3). The assemblage of sensitive experiments for different nuclei permits one to increase the reliability of the limit on m ν . Present conservative limit can be set as 0.23 eV at 90% C.L. (using conservative value from the KamLAND-Zen experiment). But one has to take into account that, in fact, this value could be in ∼ 1.5-2 times greater because of the possible quenching of g A (see recent discussions in [17]).
The main goal of next generation experiments is to investigate the IO region of neutrino mass ( m ν ≈ (14-50) meV). If one will not see the decay in this region then it will be necessary to investigate region with m ν < 14 meV.

PREDICTIONS ON M ν FROM NEUTRINO OSCILLATION AND COSMOLOGICAL DATA
Using the data of oscillatory experiments, one can obtain predictions for possible values of m ν . Usually a so-called "lobster" ("crab") plot is constructed, which shows the possible values of m ν , depending on the type of ordering and the mass of the lightest neutrino m 0 , which is unknown (see, for example, recent papers [6,19,20]). The cosmological constraints on Σm ν are used to limit the possible values of m 0 . In Fig. 1, predictions on the effective Majorana neutrino mass are plotted as function of the lightest neutrino mass m 0 . The 2σ and 3σ values of neutrino oscillation parameters are taken into account [21]. The PLANCK collaboration in a recent publication gives a limit of Σm ν < 0.12 eV [23], using the new CMB data with different large scale structure observations. This leads to a limitation on m 0 < 30 and < 16 meV for normal and inverted ordering, respectively. Taking into account PLANCK's limit, different regions of possible values of m ν are obtained depending on the type of ordering: 1) m ν ≈ 14-50 meV for all values of m 0 in the IO case.
2) In the NO case the situation is more complicated. The m ν can take values from practically 0 to 30 meV. And it has to be stressed that there is an allowed region of m 0 = 7-30 meV, where the m ν could be ∼ 10-30 meV and that is quite achievable for the next generation experiments. At m 0 = 10-30 meV, the NO and IO regions partially overlap and it will be difficult to uniquely determine the type of ordering. And only at m 0 < 10 meV it will be possible to reliably distinguish between the NO and IO. At m 0 = 1-10 meV, a strong decrease in the values of m ν is possible for certain values of the Majorana phases (nevertheless, the probability of almost total nullification m ν is sufficiently small [20]). At values of m 0 ≤ 0.1 meV the m ν ≈ 1-4 meV (the so-called "limiting" case).
A global analysis of all available data was carried out in [20] and it was shown that the NO is more preferable (at 3.5σ level). It was also demonstrated that Σm ν ≥ 0.06 eV for the NO case, and Σm ν ≥ 0.1 eV for the IO. Nevertheless, the question of the order of the neutrino masses is not yet fully clarified and experiments on a double beta decay can contribute to its solution. A limit on m ν below 14 meV could be used to rule out the IO scheme, assuming that neutrinos are Majorana fermion. On the other hand a positive detection of 0νββ decay in the range that corresponds to m ν > 14 meV would not give sufficient information to determine the mass ordering without an independent determination of m 0 . Finally, in the context of three neutrino mixing, neutrinoless double beta decay experiments alone will be able to determine the neutrino mass ordering only ruling out the inverted scheme, that is to say if the ordering is normal and m 0 ≤ 10 meV.
It is hoped that in a few years the value of Σm ν could be determined from cosmology (see, for example, discussions in [20,22]). This will help make a reliable conclusion about the type of ordering (for example, if the measured value will be less than 0.1 eV, it will mean that the NO is realized) and obtain information on the value of m 0 . And this, in turn, will improve the predictions for a possible range of m ν . For examlpe, in Ref. [19] it was demonstrated that if the sum of neutrino masses is found to satisfy Σm ν > 0.10 eV, then for NO case m ν > 5 meV for any values of the Majorana phases. Table 1 shows the best results for today on search for 0νββ decay for the most interesting nucleuscandidates for this process. Limits on the values of T 1/2 and m ν are given. To calculate m ν the NMEs from recent works [14,24,25,26,27,28,29,30,31,32,33,34] and the value g A = 1.27 have been used. One can see that the best modern experiments have reached a sensitivity of ∼ 10 25 -10 26 years for the half-life and ∼ 0.1-0.3 eV for the m ν . The spread in the values of the neutrino mass in each case is related to the currently existing uncertainties in the calculations of NMEs. Uncertainty in the values of NMEs is a factor of ∼ 2-3. As already noted, quenching of g A in the nucleus is possible and, as a result, the limits on the neutrino mass could be ∼ 1.5-2 times weaker. Table 1 shows that the most stringent limits on the effective mass of Majorana neutrino are obtained in experiments with 136 Xe, 76 Ge, 130 Te, 100 Mo and 82 Se. For some nuclei, Table 1 lists two limit values for T 1/2 and m ν . This is due to the fact that in some cases ( 136 Xe, 130 Te and 76 Ge) a large background fluctuation leads to too "optimistic" limits, substantially exceeding the "sensitivity" of the experiments. Therefore the values of the "sensitivity" of the experiments are also given in the Table 1. I believe that these values are although more conservative, but the most reliable. With this in mind, the conservative limit on m ν from modern double beta decay experiments is 0.23 eV (90% C.L.). Table 2 shows the best current and planned to start in 2018-2019 modern experiments that will determine the situation in the neutrinoless double beta decay in the coming years. It is seen that in the best of these experiments sensitivity to the m ν ∼ 0.04-0.2 eV will be achieved, which, apparently, will not be enough for verification of the IO region (because to observe the effect, it is necessary to see the signal at least at 3σ level; therefore, even the most sensitive experiments with the most favorable values of NMEs will not be able to register the decay). Table 3 shows the most promising planned experiments, which will be realized in ∼ 5-15 years. To test the IO region of neutrino masses, it is necessary to achieve sensitivity to m ν at the level of ∼ 14-50 meV.

POSSIBILITIES OF FUTURE DOUBLE BETA DECAY EXPERIMENTS TO INVESTIGATE IO REGION OF NEUTRINO MASS
Practically all experiments listed in Table 3 have a chance to register a 0νββ decay, but only CUPID, nEXO and LEGEND-1000 overlap quite well the range of m ν associated with the IO. Thus, if the IO is actually realized in nature and the neutrino is Majorana particle, then it is likely that the neutrinoless double beta decay will be registered in the experiment in ∼ 5-15 years. And the CUPID, nEXO and LEGEND-1000 experiments have the greatest chances to see the effect. But even these, the most sensitive experiments, do not guarantee the observation of the effect. At unfavorable values of NMEs and g A , the sensitivity of these experiments will be insufficient to completely cover the entire range of possible values of m ν for the IO. And one has to remember that in order to observe the effect it is necessary to have at least 3σ confidence level (in Table 3, the sensitivity is indicated at 90% C.L. (1.6σ)).

POSSIBILITIES OF FUTURE DOUBLE BETA DECAY EXPERIMENTS TO INVESTIGATE NO REGION OF NEUTRINO MASS
In the NO case, the following possible ranges of m ν can be distinguished: 1) 10-30 meV. In this case, 0νββ decay could be detected in the next generation experiments (see Table  3). But, for this area of mass, it will be difficult to distinguish the NO from IO. In this case additional information about m 0 is required.
2) 3-10 meV. In this case, detectors containing ∼ 1-10 tons of ββ isotope are required. And it is possible (in principle) to investigate this region of m ν in the future (sensitivity to T 1/2 on the level of ∼ 10 28 −10 29 yr will be needed).
3) 1-3 meV. In this case, detectors containing ∼ 10-100 tons of ββ isotope are required. It will be very difficult (if possible) to investigate this region of m ν in the future (sensitivity to T 1/2 on the level of ∼ 10 29 − 10 30 yr will be needed). 4) < 1 meV. This area is not available for observation in foreseeable future.
The possibility of studying 0νββ decay with sensitivity to neutrino mass on the level of ∼ 1-5 meV has been analysed in [60]. It was shown that the 3-5 meV region can be studied by detectors containing ∼ 10 tons of ββ isotope. Moreover, the detectors should have a sufficiently high efficiency (∼ 100%), good energy resolution (FWHM <1-2%), and low level of background in the investigated region (∼ 10 −6 − 10 −7 c/kev×kg×yr). In addition, the cost of an isotope becomes important and can seriously limit the feasibility of such experiments [60]. It was noted in [60] that 136 Xe, 130 Te, 82 Se, 100 Mo and 76 Ge are most promising isotopes, and the most suitable experimental techniques are low-temperature scintillation bolometers, gas Xe TPC and HPGe semiconductor detectors.
Summarizing all of the above, one can conclude that if we are dealing with the NO and m ν = 10-30 meV, then 0νββ decay could be registered in next-generation experiments (∼ 5-15 years from now). To study the range of m ν <10 meV, new, more sensitive experiments with the mass of the investigated isotope ∼ 1-10 tons ( m ν = 3-10 meV) or ∼ 10-100 tons ( m ν = 1-3 meV) are required. In more detail, such possible experiments are discussed in [60].

CONCLUSION
Thus, we can conclude that the present conservative limit on m ν from double beta decay experiments is 0.23 eV (90% C.L.). Within the next 3-5 years, the sensitivity of modern experiments will be brought to ∼ 0.04-0.2 eV. To study the IO region (0.014-0.05 eV), new generation experiments will be realised, which will achieve the required sensitivity in ∼ 5-15 years. If we are dealing with NO, then everything depends on the value of m ν that is realized in nature. If m ν = 10-30 meV, then this lies in the sensitivity region of the next generation experiments and 0νββ decay could be registered. If m ν = 3-10 meV, new, more sensitive experiments with ∼ 1-10 tons of ββ isotope are required (and it seems possible). For m ν = 1-3 meV experiments with of ∼ 10-100 tons of the isotope are required and it will be very difficult (if possible) to reach needed sensitivity in this case. If, however, m ν ≤1 meV, then apparently 0νββ decay will not be registered in the foreseeable future 2 .

AUTHOR CONTRIBUTIONS
The author confirms being the sole contributor of this work and approved it for publication.   Figure 1. Predictions on m ν from neutrino oscillations versus the lightest neutrino mass m 0 in the two cases of normal (the blue region) and inverted (the red region) spectra. The 2σ and 3σ values of neutrino oscillation parameters are considered [21]. The excluded region by cosmological data (Σm ν < 0.12 eV) m 0 is presented in yellow (> 30 meV for the NO and > 16 meV for the IO). Table 1. Best present limits on 0νββ decay (at 90% C.L.). To calculate m ν the NME from [14,24,25,26,27,28,29,30,31], phase-space factors from [12,13] and g A = 1.27 have been used. In case of 150 Nd NME from [32,33] and in case of 48 Ca from [34] were used in addition. The bold type denotes the so-cold "sensitivity" values (see text