On the Neutrino Mass Spectrum and Neutrinoless Double-Beta Decay

Assuming 3-nu mixing, neutrino oscillation explanation of the solar and atmospheric neutrino data and of the first KamLAND results, massive Majorana neutrinos and neutrinoless double-beta decay generated only by the (V-A) charged current weak interaction via the exchange of the three Majorana neutrinos, we analyze in detail the possibility of determining the type of the neutrino mass spectrum by measuring of the effective Majorana mass || in neutrinoless double-beta decay. The three possible types of neutrino mass spectrum are considered: i) normal hierarchical (NH), m1<<m2<<m3, ii) inverted hierarchical (IH), m1<<m2 \simeq m3, and iii) quasi-degenerate (QD), m1 \simeq m2 \simeq m3 \geq 0.20 eV. The uncertainty in the measured value of || due to the imprecise knowledge of the relevant nuclear matrix elements is taken into account in the analysis. We derive the ranges of values of tan^2 theta_\odot, theta_\odot being the mixing angle which controls the solar neutrino oscillations, and of the nuclear matrix element uncertainty factor, for which the measurement of || would allow one to discriminate between the NH and IH, NH and QD and IH and QD spectra.


Introduction
The solar neutrino experiments Homestake, Kamiokande, SAGE, GALLEX/GNO, Super-Kamiokande (SK) and SNO [1,2,3,4], the data on atmospheric neutrinos obtained by the Super-Kamiokande (SK) experiment [5] and the results from the KamLAND reactor antineutrino experiment [6], provide very strong evidences for oscillations of flavour neutrinos. The evidences for solar ν e oscillations into active neutrinos ν µ,τ , in particular, were spectacularly reinforced by the first data from the SNO experiment [3] when combined with the data from the SK experiment [2], by the more recent SNO data [4], and by the just published first results of the KamLAND [6] experiment. Under the rather plausible assumption of CPT-invariance, the KamLAND data practically establishes [6] the large mixing angle (LMA) MSW solution as unique solution of the solar neutrino problem. This remarkable result brings us, after more than 30 years of research, initiated by the pioneer works of B. Pontecorvo [7] and the experiment of R. Davis et al. [8], very close to a complete understanding of the true cause of the solar neutrino problem.
The interpretation of the solar and atmospheric neutrino, and of the KamLAND data in terms of neutrino oscillations requires the existence of 3-neutrino mixing in the weak charged lepton current (see, e.g., [9,10]): Here ν lL , l = e, µ, τ , are the three left-handed flavor neutrino fields, ν jL is the left-handed field of the neutrino ν j having a mass m j and U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix [11]. If the neutrinos with definite mass ν j are Majorana particles, the process of neutrinoless double-beta ((ββ) 0ν -) decay will be allowed (for reviews see, e.g., [12,13]). If the (ββ) 0ν -decay is generated only by the (V-A) charged current weak interaction via the exchange of the three Majorana neutrinos ν j and the latter have masses not exceeding few MeV, the dependence of the (ββ) 0ν -decay amplitude on the neutrino mass and mixing parameters factorizes in the effective Majorana mass |< m >| (see, e.g., [12]): where α 21 and α 31 are two Majorana CP-violating phases 2 [14,15]. If CP-invariance holds, one has [16] α 21 = kπ, α 31 = k ′ π, where k, k ′ = 0, 1, 2, .... In this case represent the relative CP-parities of the neutrinos ν 1 and ν 2 , and ν 1 and ν 3 , respectively. The oscillations between flavour neutrino are insensitive to the Majorana CP-violating phases α 21 , α 31 [14,17] -information about these phases can be obtained in the (ββ) 0ν -decay experiments [18,19,20,21,22,23] (see also [24]). Majorana CP-violating phases, and in particular, the phases α 21 and/or α 31 , might be at the origin of the baryon asymmetry of the Universe [25]. One can express [26] (see also, e.g., [10,27,19]) the masses m 2,3 and the elements of the lepton mixing matrix entering into eq. (2) for |< m >|, in terms of the neutrino oscillation parameters measured in the solar and atmospheric neutrino and KamLAND experiments: m 2,3 -in terms of the neutrino mass squared differences ∆m 2 ⊙ and ∆m 2 A , driving the solar and atmospheric neutrino oscillations, and the mass m 1 , and |U ej | 2 , j = 1, 2, 3, -in terms of the mixing angle which controls the solar ν e transitions θ ⊙ , and of the lepton mixing parameter sin 2 θ limited by the data from the CHOOZ and Palo Verde experiments [28,29].
The observation of (ββ) 0ν -decay will have fundamental implications for our understanding of the symmetries of the elementary particle interactions 3 (see, e.g., [12]). Under the general and plausible assumptions of 3-ν mixing, neutrino oscillation explanation of the solar and atmospheric neutrino data, massive Majorana neutrinos and (ββ) 0ν -decay generated only by the (V-A) charged current weak interaction via the exchange of the three Majorana neutrinos, which will be assumed to hold throughout this study, the observation of (ββ) 0ν -decay can provide unique information on [27,19,21,22,32,33] i) the type of neutrino mass spectrum which can be normal hierarchical (NH), inverted hierarchical (IH), or quasi-degenerate (QD), ii) on the absolute scale of neutrino masses, and [18,19,20,21,22,23] iii) on the Majorana CP-violating phases α 21 and α 31 .
A measured value of |< m >| ∼ few × 10 −2 eV can provide, in particular, unique constraints on, or even can allow one to determine, the type of the neutrino mass spectrum in the case ν 1,2,3 are Majorana particles [33]. The solar neutrino data and the first KamLAND results 4 , favor relatively large value of cos 2θ ⊙ , cos 2θ ⊙ ∼ 0.40 [34,35,36,37,38]. A value of cos 2θ ⊙ > ∼ 0.25 would imply [33] the existence of significant lower bounds on |< m >| (exceeding 0.01 eV) in the cases of IH and QD neutrino mass spectrum, and of a stringent upper bound (smaller than 0.01 eV) if the spectrum is of the NH type. The indicated lower bounds are in the range of the sensitivity of currently operating and planned (ββ) 0ν -decay experiments.
Information on the absolute values of neutrino masses in the range of interest can also be derived in the 3 H β-decay neutrino mass experiment KATRIN [39], and from cosmological and astrophysical data (see, e.g., ref. [40]).
In the present article we study in detail the possibility of determining the type, or excluding one or more types, of neutrino mass spectrum by measuring of |< m >| in the next generation of (ββ) 0ν -decay experiments. The three possible types of spectra are considered 6 In our analysis we take into account, in particular, the uncertainty in the determination of |< m >| due to the imprecise knowledge of the relevant nuclear matrix elements. This permits us to determine the requirements which the possibility of distinguishing between i) the NH and IH, ii) the NH and QD, and iii) the IH and QD spectra, imposes on the uncertainty in the values of the (ββ) 0ν -decay nuclear matrix elements. We derive 3 Evidences for (ββ)0ν -decay taking place with a rate corresponding to 0.11 eV ≤ |< m >| ≤ 0.56 eV (95% C.L.) are claimed to have been obtained in [30]. The results announced in [30] have been criticized in [31]. 4 We assume throughout this study that CPT-invariance holds in the lepton sector. 5 The quoted sensitivities correspond to values of the relevant nuclear matrix elements from ref. [45]. 6 We work with the convention m1 < m2 < m3 and use the term "spectrum with normal (inverted) hierarchy" for the spectra with ∆m 2 ⊙ ≡ ∆m 2 21 (∆m 2 ⊙ ≡ ∆m 2 32 ), while we call "normal hierarchical (NH)" ("inverted hierarchical (IH)") the neutrino mass spectrum with normal (inverted) hierarchy and m1 ≪ m2, m3.
also the maximal values of tan 2 θ ⊙ for which the measurement of |< m >| would allow one to discriminate between the NH and IH, NH and QD and IH and QD spectra, for different given values of the nuclear matrix element uncertainty factor. An upper limit |< m >| < few × 10 −2 eV would imply a significant constraint on the type of neutrino mass spectrum in the case the massive neutrinos are Majorana particles, e.g., on the theories in which the neutrino masses are generated via the see-saw mechanism.
It should be noted that the determination of the type of neutrino mass spectrum, based on the measured value of |< m >|, would provide simultaneously unique information on the absolute neutrino mass scale [21,22,32,33]. Similar information cannot be obtained in the neutrino oscillation experiments, in which the sign of ∆m 2 A can be determined 7 (see, e.g., [48,49]) since neutrino oscillations depend on neutrino mass squared differences and are insensitive to the absolute neutrino mass scale. The sign of ∆m 2 A can be determined in very long base-line neutrino oscillation experiments at neutrino factories (see, e.g., [48]), and, e.g, using combined data from long base-line oscillation experiments at the JHF facility and with off-axis neutrino beams [49]. Under certain rather special conditions it might be determined also in experiments with reactorν e [50].

Neutrino Oscillation Data and the Effective Majorana Mass
The predicted value of |< m >| depends in the case of 3 − ν mixing on: i) ∆m 2 A , ii) θ ⊙ and ∆m 2 ⊙ , iii) the lightest neutrino mass, and on iv) the mixing angle θ. Using the convention m 1 < m 2 < m 3 , one has ∆m 2 A ≡ ∆m 2 31 , where ∆m 2 jk ≡ m 2 j − m 2 k , and m 3 = m 2 1 + ∆m 2 A , while either ∆m 2 ⊙ ≡ ∆m 2 21 or ∆m 2 ⊙ ≡ ∆m 2 32 . The two possibilities for ∆m 2 ⊙ correspond respectively to the two different types of neutrino mass spectrum -with normal and with inverted hierarchy. In the first case we have m 2 = m 2 Given ∆m 2 ⊙ , ∆m 2 A , θ ⊙ and sin 2 θ, the value of |< m >| depends strongly on the type of the neutrino mass spectrum, as well as on the values of the two Majorana CP-violating phases, α 21 and α 31 (see eq. (2)), present in the lepton mixing matrix. Let us note that in the case of QD spectrum, m 1 ∼ = m 2 ∼ = m 3 , m 2 1,2,3 ≫ ∆m 2 A , ∆m 2 ⊙ , |< m >| is essentially independent on ∆m 2 A and ∆m 2 ⊙ , and the two possibilities, ∆m 2 ⊙ ≡ ∆m 2 21 and ∆m 2 ⊙ ≡ ∆m 2 32 , lead effectively to the same predictions for |< m >| 8 .
The possibility of determining the type of the neutrino mass spectrum if |< m >| is found to be nonzero in the (ββ) 0ν -decay experiments of the next generation, depends crucially on the precision with which ∆m 2 A , θ ⊙ , ∆m 2 ⊙ , sin 2 θ and |< m >| will be measured. It depends also crucially on the values of θ ⊙ and of |< m >|. The precision itself of the measurement of |< m >| in the next generation of (ββ) 0ν -decay experiments, given the latter sensitivity limits of ∼ (1.5−5.0)×10 −2 eV, depends on the value of |< m >|.
The KATRIN experiment [39] can test the hypothesis of a QD spectrum 9 , provided m 1,2,3 ∼ = mν e > ∼ (0.35 − 0.40) eV. The KATRIN detector is designed to have a 1 s.d. error of 0.08 eV 2 on a measured value of m 2 νe . This experiment is expected to start in 2007. Assuming CPT-invariance, combined ν e → ν µ(τ ) andν e →ν µ(τ ) oscillation analyzes of the solar neutrino data and of the just published first KamLAND results [6], have already been performed in 7 In the convention in which the sign of ∆m 2 A = ∆m 2 31 is not fixed, the latter determines the ordering of the neutrino masses: ∆m 2 A > 0 corresponds to m1 < m2 < m3, while ∆m 2 A < 0 implies m3 < m1 < m2. 8 This statement is valid, within the convention m1 < m2 < m3 we are using, as long as there are no independent constraints on the CP-violating phases α21 and α31 which enter into the expression for |< m >|. In the case of NH spectrum, |< m >| depends primarily on α21 (|Ue3| 2 ≪ 1), while if the spectrum is with IH, |< m >| will depend essentially on α31 − α21 (|Ue1| 2 ≪ 1). 9 Given the allowed regions of values of ∆m 2 ⊙ and ∆m 2 A [5], one has a QD spectrum for m1,2,3 ∼ = mν e > 0.20 eV. [34,35,36,37,38]. All analyzes show that the data favor the LMA MSW solution with ∆m 2 ⊙ > 0 and tan 2 θ ⊙ < 1, all the other solutions (LOW, VO, etc.) being essentially ruled out. In Tables  1 and 2 we give the best-fit values and the 90% C.L. allowed ranges of ∆m 2 ⊙ and tan 2 θ ⊙ in the LMA solution region obtained in [34,35,36,37]. The best fit values are confined to the narrow intervals (∆m 2 In the two-neutrino ν µ → ν τ (ν µ →ν τ ) oscillation analysis of the SK atmospheric neutrino data performed in [5] the following best-fit value of ∆m 2 A was obtained: (∆m 2 A ) BF ∼ = 2.5 × 10 −3 eV 2 . At 99.73% C.L., ∆m 2 A was found to lie in the interval: (1.5 − 5.0) × 10 −3 eV 2 . According to the more recent combined analysis of the data from the SK and K2K experiments [51], one has ∆m 2 A ∼ = (2.7 ± 0.4) × 10 −3 eV 2 . In certain cases of our analysis we will use as illustrative "best-fit" For the indicated allowed ranges of values of ∆m 2 ⊙ and ∆m 2 A , the NH (IH) spectrum corresponds to m 1 < ∼ 10 −3 (2 × 10 −2 ) eV.
The existing solar neutrino and KamLAND data favor values of ∆m 2 35,36,37,38]. If ∆m 2 ⊙ lies in this interval, a combined analysis of the future more precise KamLAND results and of the solar neutrino data would permit to determine the values of ∆m 2 ⊙ and tan 2 θ ⊙ with high precision: the estimated (1 s.d.) errors on ∆m 2 ⊙ and on tan 2 θ ⊙ can be as small as ∼ (3 − 5)% and ∼ 5% (see, e.g., [54,51]).
Similarly, if ∆m 2 A lies in the interval ∆m 2 A ∼ = (2.0 − 5.0) × 10 −3 eV 2 , as is suggested by the current atmospheric neutrino data [5,51], its value will be determined with a ∼ 10% error (1 s.d.) by the MINOS experiment [55]. Somewhat better limits on sin 2 θ than the existing one can be obtained in the MINOS experiment [55] as well. Various options are being currently discussed (experiments with off-axis neutrino beams, more precise reactor antineutrino and long base-line experiments, etc., see, e.g., [56]) of how to improve by at least an order of magnitude, i.e., to values of ∼ 0.005 or smaller, the sensitivity to sin 2 θ. The high precision measurements of ∆m 2 A , tan 2 θ ⊙ and ∆m 2 ⊙ are expected to take place within the next ∼ (6 − 7) years. We will assume in what follows that the problem of measuring or tightly constraining sin 2 θ will also be resolved within the indicated period. Under these conditions, the largest uncertainty in the comparison of the theoretically predicted value of |< m >| with that determined in the (ββ) 0ν -decay experiments would be associated with the corresponding (ββ) 0νdecay nuclear matrix elements. We will also assume in what follows that by the time one or more (ββ) 0ν -decay experiments of the next generation will be operative (2009 − 2010) at least the physical range of variation of the values of the relevant (ββ) 0ν -decay nuclear matrix elements will be unambiguously determined.

Determining the Type of Neutrino Mass Spectrum
The possibility to distinguish between the three different types of neutrino mass spectrum in the 3-neutrino mixing case under discussion depends on the allowed ranges of values of |< m >| for the three spectra. More specifically, it is determined by the maximal values of |< m >| in the cases of NH and IH spectra and by the minimal values of |< m >| for the IH and QD spectra. For the NH neutrino mass spectrum (m 1 ≪ m 2 ≪ m 3 ), the maximal value of |< m >| is obtained in the case of CP-conservation and equal CP-parities of ν 1,2,3 : where s 2 ≡ sin 2 θ and we have neglected m 2 1 with respect to ∆m 2 ⊙ and ∆m 2 A . In the case of IH neutrino mass spectrum (m 1 ≪ m 2 ∼ = m 3 ), the effective Majorana mass lies in the interval [18,19] |< with where we have neglected m 1 . The minimal (maximal) value of |< m >|, |< m >| IH min (|< m >| IH max ), corresponds to CP-conservation and opposite (equal) CP-parities of the neutrinos ν 2 and ν 3 .
The minimal value of |< m >| for the quasi-degenerate (QD) neutrino mass spectrum ( where m 0 > ∼ 0.20 eV and we have neglected ∆m 2 ⊙ and ∆m 2 A with respect to m 2 0 . As eq. (7) shows, |< m >| QD min scales to a good approximation with m 0 . Correspondingly, the minimal allowed value of |< m >| for the QD mass spectrum is obtained for m 0 = 0.2 eV.
In Tables 1 and 2 we show the calculated i) maximal predicted value of |< m >| in the case of NH neutrino mass spectrum, ii) the minimal value of |< m >| for the IH spectrum, and iii) the minimal value of |< m >| for the QD spectrum (m 0 = 0.2 eV), for the best-fit and the 90% C.L. allowed ranges of values of tan 2 θ ⊙ and ∆m 2 ⊙ in the LMA solution region. In Table 3 we give the same quantities, |< m >| NH max , |< m >| IH min and |< m >| QD min , calculated using the best-fit values of the neutrino oscillation parameters, including 1 s.d. (3 s.d.) uncertainties of 5% (15%) on tan 2 θ ⊙ and ∆m 2 ⊙ and of 10% (30%) on ∆m 2 A . The maximal predicted value of |< m >| for the IH spectrum is given by |< m >| IH max ∼ = (∆m 2 A ) max . For the best-fit value [5,51] and the 99.73% C.L. allowed range [5] of ∆m 2 A we have, respectively, |< m >| IH max ∼ = 0.05 and 0.07 eV.
On the basis of the results shown in Tables 1 − 3, we can conclude, in particular, that the NH spectrum could be ruled out if the measured value of |< m >| exceeds approximately 0.9 × 10 −2 eV, where we have been rather conservative in choosing the maximal value.

Theoretical and Experimental Uncertainties in |< m >|
Following the notation in ref. [23], we will parametrize the uncertainty in |< m >| due to the imprecise knowledge of the relevant nuclear matrix elements -we will use the term "theoretical uncertainty" for the latter -through a parameter ζ, ζ ≥ 1, defined as: where (|< m >| exp ) MIN is the value of |< m >| obtained from the measured (ββ) 0ν -decay half lifetime of a given nucleus using the largest nuclear matrix element and ∆ is the experimental error. An experiment measuring a (ββ) 0ν -decay half-life time will thus determine a range of |< m >| corresponding to The currently estimated range of ζ for experimentally interesting nuclei varies from 3.5 for 48 Ca to 38.7 for 130 Te, see, e.g., Table 2 in ref. [13] and ref. [57]. We estimate, following again [23], the 1 s.d. error on the experimentally measured value of |< m >| by using the standard expression where E

Requirements on the Solar Neutrino Mixing Angle
We shall derive next the constraints tan 2 θ ⊙ must satisfy in order to be possible to distinguish between the three types of neutrino mass spectrum NH, IH and QD.
The smaller m 1 and/or ∆m 2 ⊙ , the closer the upper bound on tan 2 θ ⊙ of interest becomes to 1. The above analysis shows also that the upper bound on tan 2 θ ⊙ under discussion exhibits relatively strong dependence on the value of s 2 < ∼ 0.05: it increases by a factor of ∼ (1.2 − 1.5) when s 2 decreases from 0.05 to 0.
These simple quantitative analyses show that if |< m >| is found to be non-zero in the future (ββ) 0ν -decay experiments, it would be easier, in general, to distinguish between the spectrum with NH and those with IH or of QD type using the data on |< m >| = 0, than to distinguish between the IH and the QD spectra. Discriminating between the latter would be less demanding if m 0 is sufficiently large. The requirement of distinguishing between the NH and the QD spectra leads to the least stringent conditions. The above analyses also show that the possibility to distinguish between the IH and QD, and NH and QD, spectra depends rather weakly on the value s 2 , satisfying the existing upper limits [28,29,51]: the relevant upper bounds on tan 2 θ ⊙ decrease somewhat with decreasing of s 2 . This is not so in the case of NH versus IH spectra: the upper bound of interest can increase noticeably (e.g., by a factor of ∼ (1.2 − 1.5)) when s 2 decreases from 0.05 to 0.
It is worth noting that in contrast to the conditions which would allow one to establish on the basis of a measurement of |< m >| = 0 the presence of CP violation due to the Majorana CPviolating phases [23], the conditions permitting to distinguish between the three types of neutrino mass spectrum imply an upper limit on tan 2 θ ⊙ .
In Fig. 1 we show the upper bounds on tan 2 θ ⊙ , for which one can distinguish the NH spectrum from the IH spectrum and from that of QD type, as a function of ∆m 2 ⊙ for different values of ζ. As is seen from the figure, the dependence of the maximal value of tan 2 θ ⊙ of interest on m 0 in both cases is modest. Obviously, with the increasing of ∆m 2 ⊙ and/or s 2 , |< m >| NH max also increases. As a consequence, the maximal tan 2 θ ⊙ under discussion decreases, which means that the corresponding spectra become harder to distinguish.
As we have seen, in order to be possible to distinguish between the IH and the QD spectra eq. (16) should be fulfilled. Fig. 2 shows the upper bound on tan 2 θ ⊙ as implied by eq.

Requirements on ∆ and ζ
We will investigate now the requirements the experimental and theoretical uncertainties ∆ and ζ should satisfy in order to allow one to discriminate between the three different neutrino mass spectra if |< m >| is measured, or a significantly improved bound on |< m >| is obtained.

3.3.2
Testing the Normal Hierarchical Spectrum The NH neutrino mass spectrum would be ruled out if How restrictive this condition is depends on the value of |< m >| NH max . Assuming that the more precise measurements of tan 2 θ ⊙ , ∆m 2 A and sin 2 θ will not produce results very different from their current best-fit values, we can use the predictions for |< m >| NH max given in Table 3

3.3.3
Probing the Inverted Hierarchical Spectrum For the IH neutrino mass spectrum, |< m >| IH is constrained to lie in the interval given by eqs. (5) and (6). The IH spectrum can be ruled out if the experimentally measured value of |< m >|, with both the experimental error ∆ and the nuclear matrix element uncertainty factor ζ taken into account, lies outside the range given in eq. (5). There are two possibilities.

Case i).
where |< m >| IH max depends on the allowed values of ∆m 2 A and s 2 and is given in the captions of Tables 1 − 3. Using the parametrization (|< m >| exp ) MIN = y IH |< m >| IH max , y IH > 1, we are lead to For y IH = 5, 4, 3, 2, 1.5 this condition is fulfilled if ∆ < (0.28, 0.21, 0.14, 0.07, 0.03) (1 − s 2 ) eV. The larger the measured value of |< m >|, the larger is the maximal experimental error which still permits to rule out the IH spectrum. Alternatively, for a value of the experimental error ∆ = (0.2, 0.1, 0.05, 0.03) eV it would be possible to rule out the IH spectrum provided y IH > 3.9, 2.4, 1.7, 1.4, respectively.
Case ii). The spectra with inverted hierarchy can be ruled out also if: Since |< m >| IH min is of the order of 0.01 eV, the experimental uncertainty will be required to be even below this value, making it not within reach of the currently planed experiments, except possibly for the 10t version of GENIUS. For instance, for (|< m >| exp ) MIN = 0.01 eV and ∆ = 0.01 eV one finds, e.g., ζ < 1.1 if |< m >| IH min = 0.022 eV. Probing the IH neutrino mass spectrum requires that the following conditions be fulfilled: Using the fact that |< m >| IH min = cos 2θ ⊙ |< m >| IH max and the parametrization (|< m >| exp ) MIN ≡ y IH |< m >| IH max , the necessary conditions on ζ and y IH read In the most favorable situation in which σ(|< m >|)/|< m >| ≪ 1 and y IH = cos 2θ ⊙ , ζ is required to be ζ < 1/ cos 2θ ⊙ . For the present best-fit values of tan 2 θ ⊙ reported in Table 1, we obtain ζ < 2.7, 2.7, 2.4, 2.5. Let us note, however, that from experimental point of view this possibility is rather demanding: as a first approximation, ∆ has to be of the order of, or smaller than, the difference between the maximal and minimal values of |< m >| in the IH case. This difference is typically of the order of ∼ (0.02 − 0.04) eV, and does not exceed ∼ 0.06 eV. If conditions (26) are satisfied, in order to establish the IH spectrum both eqs. (11) and (15) with ζ = 1 should also be valid. Taking as illustrative values ∆m 2 ⊙ = 7.0 × 10 −5 eV 2 and ∆m 2 A = 3.0 × 10 −3 eV 2 , both conditions are satisfied for tan 2 θ ⊙ < ∼ 0.5 (0.6) if s 2 = 0.05 (0).

3.3.4
The Inverted Hierarchical versus the Quasi-Degenerate Spectrum Let us assume that a value of (|< m >| exp ) MIN of a few 10 meV has been found, thus ruling out the NH spectrum. The remaining question to ask in this situation would be whether the neutrino mass spectrum is of the IH or QD type. Distinguishing between the two types of spectra might be possible provided |< m >| IH max < |< m >| QD min .
Obviously, one can reach a definite conclusion concerning the type of the spectrum only if the value of |< m >| exp is larger than |< m >| QD min , or is smaller than |< m >| IH max .
i) |< m >| exp > |< m >| QD min : this is equivalent to ruling out the IH spectrum and thus to the case i) analyzed in Subsection 3.3.3, see eq. (23) and the discussion thereafter. ii) |< m >| exp < |< m >| IH max : using eqs. (6) and (8), we find that |< m >| exp < |< m >| IH max if This inequality practically coincides with the second condition in (25). It is more restrictive for smaller values of (∆m 2 A ) max and larger values of ∆. Equation (28) can hold only for a rather limited range of parameters, since the sum of (|< m >| exp ) MIN and ∆ has to be smaller than (∆m 2 A ) max ≃ 0.07 eV. Let us note that the various conditions discussed in this Section do not require any additional input from 3 H β-decay experiments or from cosmological and astrophysical measurements.

Distinguishing Between Different Neutrino CP-Parity Configurations
In this Section we will discuss whether a measurement of |< m >| = 0 might allow one to distinguish between some of the possible neutrino CP-parity configurations when the Majorana phases take CP-conserving values, α 21 , α 31 = 0, ±π. We will denote these configurations by The possibility of determining the values of the Majorana CP-violating phases in the general case of CP-non conservation has been discussed in detail in ref. [23].
Inspecting Tables 1 − 3 leads to the conclusion that it might be relatively easy to distinguish between the (+ − −) and (− + −) configurations in the case of the IH spectrum (i.e., |< m >| IH min ) and the (+ − −) and (− + −) configurations for the QD spectrum (i.e., |< m >| QD min ). The more interesting question is whether it might be possible to distinguish between the different CP-parity configurations for a given type of neutrino mass spectrum. We will study it briefly in what follows in the cases of IH and QD spectra 10 .

Quasi-Degenerate Spectrum
In the case of QD spectrum, the (+ − −) and (− + −), and the (+ + +) and (− − +), configurations are difficult to distinguish due to the smallness of the mixing parameter s 2 limited by the reactor antineutrino experiments [28,29]: the corresponding differences in the predicted values of |< m >| do not exceed ∼ 10%. Therefore we shall analyze again the possibility to discriminate between these two pairs. Taking into account that |< m >| QD , the indicated two pairs of CP-parity configurations can be distinguished if the following inequality holds: The above inequality leads to the condition

Conclusions
Assuming 3-neutrino mixing and massive Majorana neutrinos, (ββ) 0ν -decay induced only by the (V-A) charged current weak interactions, LMA MSW solution of the solar neutrino problem and neutrino oscillation explanation of the atmospheric neutrino data, we have studied the requirements on the "solar" mixing angle θ ⊙ , the nuclear matrix element uncertainty factor ζ and the experimental error on the effective Majorana mass |< m >|, ∆, which allow one to distinguish between, and/or test, the normal hierarchical (NH), inverted hierarchical (IH) and quasi-degenerate (QD) neutrino mass spectra if |< m >| = 0 is measured, or a stringent upper bound on |< m >| is obtained. The possibility to discriminate between the three types of spectra depends on the allowed ranges of values of |< m >| for the three spectra: it is determined by the maximal values of |< m >| in the cases of NH and IH spectra, |< m >| NH,IH max , and by the minimal values of |< m >| for the IH and QD spectra, |< m >| IH,QD min . These are reported in Tables 1 − 3. In deriving them we have used the values of the solar and atmospheric neutrino oscillation parameters, θ ⊙ , ∆m 2 ⊙ , ∆m 2 A and sin 2 θ, favored by the existing data [1,2,4,5,6,28,29] (Tables 1 and 2) and assumed prospected precisions of future measurements (Table 3).
For the currently favored values of the neutrino oscillation parameters and sin 2 θ = 0, the upper bound on tan 2 θ ⊙ permitting to distinguish the NH from the IH spectrum is satisfied even for ζ = 3. If sin 2 θ lies close to its present 99.73% C.L. upper limit of 0.05 [34] (see also [28,29]), the upper bound of interest decreases by up to 50% and values of ζ slightly lower than 3 might be required (Fig. 1). The possibility to discriminate between the NH and the QD spectra depends weakly on sin 2 θ and on the neutrino mass m 1,2,3 ∼ = m 0 , and the respective conditions are satisfied even for values of ζ exceeding 3 (Fig. 1). Without any additional input from 3 H beta-decay experiments and/or cosmological and astrophysical measurements, and given the values of tan 2 θ ⊙ and ∆m 2 A favored by the data, the IH and QD spectra can be distinguished only if ζ < ∼ 1.5 (Fig. 2).
Let us emphasize that the conditions which would allow one to establish the presence of CP violation due to the Majorana CP-violating phases using a measurement of |< m >| = 0 lead to a lower bound on tan 2 θ ⊙ and, in general, require ζ < 2. In contrast, the conditions permitting to distinguish between the three types of neutrino mass spectrum imply an upper limit on tan 2 θ ⊙ and in most of the cases can be satisfied even for ζ ≃ 3.
We have studied also the conditions on ζ and ∆ which would permit to rule out, or establish, the NH, IH and the QD mass spectra. Typically, the next generation of (ββ) 0ν -decay experiments will be able to rule out the QD mass spectrum if ζ < ∼ 3, and establish it if, e.g., the measured (|< m >| exp ) MIN ∼ 0.2 (0.1) eV (see eq. (8)) and the experimental error is ∆ ∼ 0.15 (0.05) eV. The NH spectrum can be excluded provided the measured value of |< m >| is, e.g, ∼ 10 (7) times larger than |< m >| NH max and the experimental error is ∆ < ∼ 0.12 (0.06) eV. The IH spectrum can be ruled out for ∆ < ∼ 0.07 (0.10) eV provided (|< m >| exp ) MIN is at least by a factor of ∼ 2.0 (2.5) larger than |< m >| IH max . Establishing the IH mass spectrum is quite demanding and requires a measurement of |< m >| with an error ∆ < ∼ 0.02 − 0.04 eV.
Finally, we have studied the possibility to distinguish between certain neutrino CP-parity configurations in the case of CP-conservation. Due to the smallness of sin 2 θ, there are two pairs of CP-parities in the cases of QD and IH spectra, the two different CP-parity patterns within each pair being indistinguishable. Given the best-fit values of tan 2 θ ⊙ , one can discriminate between these two pairs for the IH mass spectrum if ζ < ∼ 2. For the QD mass spectrum and if m 0 is measured in tritium β-decay experiments, relatively large m 0 > ∼ 1.5 eV and tan 2 θ ⊙ > ∼ 0.5 are necessary. If astrophysical and cosmological measurements provide m 0 , values of ζ < ∼ 2 are required.  Table 1: The best-fit values of tan 2 θ ⊙ and ∆m 2 ⊙ (in units of 10 −5 eV 2 ) in the LMA solution region, as reported by different authors. Given are also the calculated maximal values of |< m >| (in units of 10 −3 eV) for the NH spectrum and the minimal values of |< m >| (in units of 10 −3 eV) for the IH and QD spectra. The results for |< m >| in the cases of NH and IH spectra are obtained for m 1 = 10 −3 eV and the best-fit value of ∆m 2 A , ∆m 2 A = 2.7 × 10 −3 eV 2 [51], while those for the QD spectrum are derived for m 0 = 0.2 eV. In all cases sin 2 θ = 0.05 has been used. The chosen value of ∆m 2 A corresponds to |< m >| IH max = 52.0 × 10 −3 eV.  [36] 0.31 − 0.56 6.0 − 8.7 6.6 13.0 43.2 [37] 0.31 − 0.66 5.9 − 8.9 7.0 9.5 28.6 Table 2: The ranges of allowed values of tan 2 θ ⊙ and ∆m 2 ⊙ (in units of 10 −5 eV 2 ) in the LMA solution region, obtained at 90% C.L. by different authors. Given are also the corresponding maximal values of |< m >| (in units of 10 −3 eV) for the NH spectrum, and the minimal values of |< m >| (in units of 10 −3 eV) for the IH and QD spectra. The results for the NH and IH spectra are obtained for m 1 = 10 −3 eV, while those for the QD spectrum correspond to m 0 = 0.2 eV. ∆m 2 A was assumed to lie in the interval [51] (2.3 − 3.1) × 10 −3 eV 2 . This implies |< m >| IH max = 55.7 × 10 −3 eV. As in Table 1   The upper bound on tan 2 θ ⊙ , for which one can distinguish the NH spectrum from the IH spectrum and from that of QD type, as a function of ∆m 2 ⊙ for ∆m 2 A = 3 × 10 −3 eV 2 and different values of ζ (see eqs. (12) and (14)). The lower (upper) line corresponds to s 2 = 0.05 (0).