Neutrino masses, mixing, Majorana CP-violating phases and (ββ)0ν decay

Predictions of the effective Majorana mass |⟨m⟩| in (ββ)0ν decay for 3-ν mixing and massive Majorana neutrinos are reviewed in the present study. The physics potential of the experiments, searching for (ββ)0ν decay and having sensitivity to |⟨m⟩|⪆0.01 eV for providing information on the type of the neutrino mass spectrum, the absolute scale of neutrino masses and on the Majorana CP-violation phases in the PMNS neutrino mixing matrix, is also discussed.


Introduction
There has been remarkable progress in the study of neutrino oscillations over the last several years. Experiments with solar, atmospheric and reactor neutrinos [1]- [6] have provided compelling evidence for the existence of neutrino oscillations driven by nonzero neutrino masses and neutrino mixing. Evidence for oscillations of neutrinos was obtained also in the first long-baseline accelerator neutrino experiment K2K [7].
The hypothesis of solar neutrino oscillations which, in 1967, were first predicted to cause a solar neutrino deficit [8] and later, in one variety or another, were considered as the most natural explanation for the observed [1,2] solar neutrino deficit (see e.g. [9]- [11]), has received a convincing confirmation from the measurement of the solar neutrino flux through the neutral current reaction on deuterium by the SNO experiment [3,4]. Analysis of the solar neutrino data obtained by Homestake, SAGE, GALLEX/GNO, SK and SNO experiments showed that the data favour the large mixing angle (LMA) MSW solution for the solar neutrino problem. The first results of the KamLAND reactor experiment [6] have confirmed (under the very plausible assumption of CPT invariance) the LMA MSW solution, establishing it essentially as a unique solution for the solar neutrino problem. This remarkable result brought us, after more than 30 years of research, initiated by the pioneer works of Pontecorvo [8,12] and the experiment of Davis et al [13], very close to a complete understanding of the true cause of the solar neutrino problem.
A combined two-neutrino oscillation analysis of the solar neutrino and KamLAND data, performed before the latest (salt-phase) SNO results were announced, identified two distinct solution subregions within the LMA solution region, LMA-I and II (see e.g. [14,15]). The bestfit values of two-neutrino oscillation parameters, namely the solar neutrino mixing angle θ and the mass-squared difference m 2 , in the two subregions, LMA-I and -II, read (see e.g. [14]) m 2I = 7.3 × 10 −5 eV 2 , m 2II = 1.5 × 10 −4 eV 2 and tan 2 θ I = tan 2 θ II = 0.46. The LMA-I solution was preferred statistically by the data. At 90% CL, it was found in (see e.g. [14]) In September 2003, the SNO collaboration published data from the salt phase of the experiment [4]. In particular, for the ratio of the CC and NC event rates, the collaboration found R CC/NC = 0.306 ± 0.026 ± 0.024 and, correspondingly, R CC/NC 0.41 at high CL. As was shown in [16], an upper limit of R CC/NC < 0.5 implies a significant upper limit on m 2 smaller than 2 × 10 −4 eV 2 : m 2 1.7 × 10 −4 eV 2 . Thus, the latest SNO data on R CC/NC imply stringent constraints on the LMA-II solution. A combined statistical analysis of data from the solar neutrino and KamLAND experiments, including the latest SNO results, showed [18] (see also e.g. [19]) that the LMA-II solution is allowed only at 99.13% CL. Furthermore, the data have substantially reduced the maximum allowed value of sin 2 θ (see [16,17]), thereby excluding the possibility of maximal mixing 2 at 5.4 standard deviations (S.D.) [4,18]. The best-fit value of θ corresponds to cos 2θ = 0.40, whereas, at 90% (95%) CL, one has cos 2θ 0. 24 (0.22). This has very important implications, in particular for predictions of the effective Majorana mass in neutrinoless double-beta decay, | m | [21]- [24].
There is also strong evidence for oscillations of the atmospheric ν µ (ν µ ) from the Zenith angle dependence of the sub-GeV and multi-GeV µ-like events, which were observed in the Super-Kamiokande experiment [25]. The experimental results are best described in terms of dominant two-neutrino ν µ → ν τ (ν µ →ν τ ) oscillations with maximal mixing, sin 2 2θ atm ∼ = 1. A combined analysis [26] of atmospheric neutrino data and the data from the K2K long-baseline accelerator experiment [7] shows that, at 90% CL, the neutrino mass-squared difference responsible for the atmospheric neutrino oscillations lies in the interval 2.0 × 10 −3 eV 2 m 2 atm 3.2 × 10 −3 eV 2 .
The m 2 atm best-fit value found in [26] reads m 2 atm | BF = 2.6 × 10 −3 eV. Preliminary results from an improved analysis of the SK atmospheric neutrino data, performed recently by the SK collaboration, gave [25] with best-fit value | m 2 atm | = 2.0 × 10 −3 eV 2 . Adding the K2K data [7], Fogli et al [27] find the same best-fit value and Recently, the SK collaboration presented the first evidence of an 'oscillation dip' in the L/E dependence, where L and E are the distance travelled by neutrinos and the neutrino energy respectively, of a particular selected sample of µ-like events 3 [28]. Such a dip is predicted due to the oscillatory dependence of the ν µ → ν τ (ν µ →ν τ ) oscillation probability on L/E: the ν µ → ν τ (ν µ →ν τ ) transitions of atmospheric neutrinos are predominantly twoneutrino transitions governed by vacuum oscillation probability. The dip in the observed L/E distribution corresponds to the first oscillation minimum of the ν µ (ν µ ) survival probability, P(ν µ → ν µ )(P(ν µ →ν µ )), as L/E increases starting from values for which m 2 atm L/(2E) 1 and P(ν µ → ν µ ) ∼ = 1. This beautiful result represents the first ever observation of a direct effect of the oscillatory dependence on L and E of the probability of neutrino oscillations in vacuum.
Interpretation of the solar and atmospheric neutrinos, and of the KamLAND data in terms of neutrino oscillations, requires the existence of three-neutrino mixing in the weak charged lepton current: Here, ν lL (where l = e, µ, τ) are the three left-handed flavour neutrino fields, ν jL is the lefthanded field of the neutrino ν j having a mass m j and U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix [29]. Actually, all existing neutrino oscillation data, except the data of the LSND experiment [30], 4 can be described if we assume the existence of three-neutrino mixing in vacuum, (equation (5)), and we will consider this possibility in what follows. 5 3 These are µ-like events for which the relative uncertainty in the experimental determination of the L/E ratio does not exceed 70%. 4 In the LSND experiment, indications for oscillationsν µ →ν e with ( m 2 ) LSND 1 eV 2 were obtained. The LSND results are being tested in the MiniBooNE experiment at Fermilab [31]. 5 For a similar analysis taking into account the LSND evidences for neutrino oscillations, see [32].
The PMNS mixing matrix U can be parametrized by three angles θ atm , θ and θ and, depending on whether the massive neutrinos ν j are Dirac or Majorana particles, by one or three CP-violating phases [33]- [35]. In the standard parametrization of U (see e.g. [36]), the three mixing angles are denoted as θ 12 , θ 13 where we have used the usual notations s ij ≡ sin θ ij , c ij ≡ cos θ ij ; δ is the Dirac CP-violation phase and α 21 and α 31 are two Majorana CP-violation phases [33]- [35]. If we identify the two independent neutrino mass-squared differences in this case, m 2 21 and m 2 31 , with the neutrino mass squared differences that induce the solar and atmospheric neutrino oscillations, m 2 = m 2 21 > 0, m 2 A = m 2 31 , we have θ 12 = θ , θ 23 = θ atm and θ 13 = θ. The angle θ is limited by the data from the CHOOZ and Palo Verde experiments [37,38]. The oscillations between flavour neutrinos are insensitive to the Majorana CP-violating phases α 21 and α 31 [33,39]. Information about these phases can be obtained, in principle, from the (ββ) 0ν -decay experiments [22,24], [40]- [43] (see also [44]- [46]). Majorana CP-violating phases might be at the origin of the baryon asymmetry of the Universe [47].
Somewhat better limits on sin 2 θ compared with the existing one can be obtained in the MINOS, OPERA and ICARUS experiments [49,50]. Various options are currently being discussed (experiments with off-axis neutrino beams, more precise reactor antineutrino and long-baseline experiments, etc; see e.g. [51,52]) on how to improve the sensitivity to sin 2 θ by at least a factor of 5 or more, i.e. to values of ∼0.01 or smaller. The combined 3-ν oscillation analysis of the solar neutrino, CHOOZ and KamLAND data also showed [18] that the allowed ranges of the solar neutrino oscillation parameters do not differ substantially for sin 2 θ < 0.05 from those derived in the two-neutrino oscillation analyses. At 90% CL, for instance, one finds 0.23 sin 2 θ 0.38 for sin 2 θ = 0.0, 0.25 sin 2 θ 0.36 for sin 2 θ = 0.04.
The m 2 best-fit value is practically the same for the two values of sin 2 θ: m 2 ∼ = 7.2 × 10 −5 eV 2 . Note that the atmospheric neutrino and K2K data do not allow one to determine the sign of m 2 atm . This implies that if we identify m 2 atm with m 2 31 for three-neutrino mixing, one can have m 2 31 > 0 or m 2 31 < 0. The two possibilities correspond to two different types of neutrino mass spectra: with normal hierarchy, m 1 < m 2 < m 3 , and with inverted hierarchy, m 3 < m 1 < m 2 (see e.g. [40]). We will use the terms normal hierarchical (NH) and inverted hierarchical (IH) for the two types of spectra in the case of strong inequalities between the neutrino masses, i.e. if m 1 m 2 m 3 and 6 m 3 m 1 < m 2 , respectively. The spectrum can also be of quasi-degenerate (QD) type: m 1 ∼ = m 2 ∼ = m 3 and m 2 1,2,3 | m 2 atm |. The sign of m 2 atm can be determined in very long baseline neutrino oscillation experiments at neutrino factories (see e.g. [54]), e.g. using combined data from long baseline oscillation experiments at the JHF facility and with off-axis neutrino beams [55]. Under some rather special conditions, it might be determined also in experiments with reactorν e [53,56].
As is well known, neutrino oscillation experiments allow one to determine differences of squares of neutrino masses, but not the absolute values of the masses. Information on the absolute values of neutrino masses of interest can be derived in the 3 H β-decay experiments studying the electron spectrum [57]- [59] and from cosmological and astrophysical data (see e.g. [60]- [62]). The currently existing most stringent upper bounds on the electron (anti-)neutrino mass mν e were obtained in the Troitzk [58] and Mainz [59] 3 H β-decay experiments and read mν e < 2.2 eV (95% CL).
We have mν e ∼ = m 1,2,3 for the QD neutrino mass spectrum. The KATRIN 3 H β-decay experiment [59] is expected to reach a sensitivity of mν e ∼ (0.20-0.35) eV, i.e. to probe the region of QD neutrino mass spectrum. Data from the WMAP experiment on the cosmic microwave background radiation was used to obtain an upper limit on the sum of neutrino masses [61]: A conservative estimate of all the uncertainties related to the derivation of this result (see e.g. [63]) leads to a less stringent upper limit, at least by a factor of ∼1.5 and possibly by a factor of ∼3. The WMAP and future PLANCK experiments can be sensitive to [60] j m j ∼ = 0.4 eV. Data on weak lensing of galaxies by large-scale structure, combined with data from the WMAP and PLANCK experiments may allow one to determine (m 1 + m 2 + m 3 ) with an uncertainty of [62] δ ∼ 0.04 eV. After the spectacular experimental progress made during the last several years in the studies of neutrino oscillations, further understanding of the structure of the neutrino masses and mixing, of their origins and of the status of the CP symmetry in the lepton sector requires a large and challenging programme of research to be pursued in neutrino physics. The main goals of such a research programme should include the following: • High-precision determination of the neutrino mixing parameters which control the solar and the dominant atmospheric neutrino oscillations, m 2 , θ and m 2 A , θ atm . • Measurement of, or improving by at least a factor of 5-10 the existing upper limit on, the value of the only small mixing angle θ (=θ 13 ) in the PMNS matrix U. • Determination of the type of the neutrino mass spectrum (NH or IH or QD).
• Determining or obtaining significant constraints on the absolute scale of neutrino masses, or on the lightest neutrino mass. • Determining the nature of massive neutrinos, which can be Dirac or Majorana particles.
• To establish whether the CP symmetry is violated in the lepton sector (i) due to the Dirac phase δ and/or (ii) due to the Majorana phases α 21 and α 31 if the massive neutrinos are Majorana particles. • Searching with increased sensitivity for possible manifestations, other than flavour neutrino oscillations, of the nonconservation of individual lepton charges L l , l = e, µ, τ, such as µ → e + γ and τ → µ + γ decays. • Understanding at a fundamental level the mechanism giving rise to the neutrino masses and mixing and to the L l nonconservation, i.e. finding the theory of neutrino mixing. Progress in the theory of neutrino mixing might also lead, in particular, to a better understanding of the possible relation between CP violation in the lepton sector at low energies and generation of the baryon asymmetry of the Universe.
Obviously, successful realization of the experimental part of this programme of research would be a formidable task and would require many years.
In the present paper, we will review the potential contribution of studies of neutrinoless double-beta ((ββ) 0ν ) decay of certain even-even nuclei, (A, Z) → (A, Z + 2) + e − + e − , to the programme of research outlined above. The (ββ) 0ν decay is allowed if the neutrinos with definite mass are Majorana particles (see e.g. [10,64,65] for reviews). Let us recall that the nature (Dirac or Majorana) of the massive neutrinos ν j is related to the fundamental symmetries of particle interactions. The neutrinos ν j will be Dirac fermions if the particle interactions conserve some lepton charge, e.g. the total lepton charge L. The neutrinos with definite mass can be Majorana particles if there does not exist any conserved lepton charge. As is well known, massive neutrinos are predicted to be of Majorana nature by the see-saw mechanism of neutrino mass generation [66], which also provides a very attractive explanation for the smallness of the neutrino masses and, through the leptogenesis theory [47], for the observed baryon asymmetry of the Universe.
If the massive neutrinos ν j are Majorana fermions, processes in which the total lepton charge L is not conserved and changes by two units, such as K + → π − + µ + + µ + and µ + + (A, Z) → (A, Z + 2) + µ − , should exist. The process most sensitive to the possible Majorana nature of the massive neutrinos ν j is the (ββ) 0ν decay (see e.g. [10]). 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 a few MeV, which will be assumed to hold throughout this paper, the dependence of the (ββ) 0ν -decay amplitude A(ββ) 0ν on the neutrino mass and mixing parameters factorizes in the effective Majorana mass m (see e.g. [10,65]): where M is the corresponding nuclear matrix element (NME) and α 21 and α 31 being the two Majorana CP-violating phases of the PMNS matrix 7 [33,35]. Let us note that if CP invariance holds, one has [67] In this case, represent the relative CP parities of the Majorana neutrinos ν 1 and ν 2 and ν 1 and ν 3 , respectively. It follows from equation (15) that measurement of | m | will provide information, in particular, on the neutrino masses. As equation (14) indicates, observation of the (ββ) 0ν decay of a given nucleus and measurement of the corresponding half lifetime would allow one to determine | m | only if the value of the relevant NME M is known with a relatively low uncertainty. The experimental searches for (ββ) 0ν decay have a long history (see e.g. [64,65]). Rather stringent upper bounds on | m | have been obtained in the 76 Ge experiments by the Heidelberg-Moscow collaboration [68]: Taking into account a factor of 3 uncertainty associated with the calculation of the relevant NME [65], we get The IGEX collaboration has obtained [69]: Evidence for (ββ) 0ν decay of 76 Ge, taking place with a rate corresponding to 0.11 eV | m | 0.56 eV (95% CL), is claimed to have been obtained in [70]. The results presented in [70] have been criticized in [71]. Even stronger evidence has reported recently in [72], where the following value of | m | has been given: The results reported in [72] will be checked in the currently running and future (ββ) 0ν -decay experiments (see below). However, it may take a very long time before a comprehensive check could be completed.
As we will discuss in what follows, the studies of (ββ) 0ν decay and the measurement of a nonzero value of | m | a few 10 −2 eV: • Can establish the Majorana nature of massive neutrinos. The (ββ) 0ν decay experiments are presently the only feasible experiments capable of doing that (see [10]). • Can give information on the type of the neutrino mass spectrum [21]- [23], [82] (see also [40,79]- [81]. More specifically, a measured value of | m | a 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 if ν 1,2,3 are Majorana particles [21,23]. • Can provide also unique information on the absolute scale of neutrino masses, or on the lightest neutrino mass (see e.g. [21,22,79]). • With additional information from other sources ( 3 H β-decay experiments or cosmological and astrophysical data and considerations) on the absolute neutrino mass scale, the (ββ) 0ν -decay experiments can provide unique information on the Majorana CP-violation phases α 21 and α 31 [22,24,40,42,43].
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-violation phases of the PMNS matrix, α 21 and α 31 (see equation (15)). Note that for the QD spectrum, Correspondingly, the two possibilities, m 2 ≡ m 2 21 and m 2 ≡ m 2 32 , lead effectively to the same predictions for | m |. 9

NH neutrino mass spectrum
For the NH neutrino mass spectrum, one has m 2 ∼ = m 2 , m 3 ∼ = m 2 atm and |U e3 | 2 ≡ sin 2 θ and, correspondingly, where we have neglected the term ∼m 1 in equation (22). In this case, one of the three massive Majorana neutrinos effectively 'decouples' and does not give a contribution to | m |; however, the value of | m | still depends on the Majorana CP-violation phase α 32 = α 31 − α 21 . This is a consequence of the fact that, in contrast with the case of massive Dirac neutrinos (or quarks), CP violation can take place in the mixing of only two massive Majorana neutrinos [33].
From equations (3) and (8)- (11), it follows that m 2 10 −2 eV, sin 2 θ 0.40, m 2 A 5.5 × 10 −2 eV (at 90% CL) and the largest neutrino mass enters into the expression for | m | with the factor sin 2 θ < 0.05. For these reasons, the predicted value of | m | is below 10 −2 eV: for sin 2 θ = 0.04 (0.02), one finds | m | 0.006 (0.005) eV. Using the bestfit values of the indicated neutrino oscillation parameters, we get even smaller values for | m |, | m | 0.005 (0.004) eV (see tables 1 and 2).Actually, from equation (21) and the allowed ranges of values of m 2 , m 2 A , sin 2 θ , sin 2 θ as well as of the lightest neutrino mass m 1 and the CP-violation phases α 21 and α 31 , it follows that for the NH spectrum, there can be a complete cancellation between the contributions of the three terms in equation (21) and one can have [22] | m | = 0.
Even though one of the three massive Majorana neutrinos 'decouples', the value of | m | depends on the Majorana CP-violating phase α 32 ≡ (α 31 − α 21 ). Obviously, 9 This statement is valid, within the convention m 1 < m 2 < m 3 we are using, as long as there are no independent constraints on the CP-violating phases α 21 and α 31 , which enter into the expression for | m |. For the spectrum with normal hierarchy, | m | depends primarily on α 21 (|U e3 | 2 1), whereas if the spectrum is with inverted hierarchy, | m | will depend essentially on α 31 − α 21 (|U e1 | 2 1). The maximal values of | m | (in units of 10 −3 eV) for the NH and IH spectra, and the minimal values of | m | (in units of 10 −3 eV) for the IH and QD spectra, for the best-fit values of the oscillation parameters and sin 2 θ = 0.0, 0.02 and 0.04. The results for the NH and IH spectra are obtained for m 2 atm | BF = 2.6 × 10 −3 eV 2 (2.0 × 10 −3 eV 2 for values in parentheses) and m 1 = 10 −4 eV, whereas those for the QD spectrum correspond to m 0 = 0.2 eV (from [23]).  [18], and of m 2 atm given in equation (6) (equation (8) for results in parentheses) (from [23]). The upper and lower limits correspond respectively to the CP-conserving cases α 32 = 0, or α 21 = α 31 = 0, ±π, and α 32 = ±π, or α 21 = α 31 + π = 0, ±π. Most remarkably, since cos 2θ ∼ 0.40 according to the solar neutrino and KamLAND data, we get a significant lower limit on | m |, typically exceeding 10 −2 eV, in this case [21,22] (tables 1 and 2). Using, for example, the best-fit values of m 2 A and tan 2 θ , one finds: | m | 0.017 eV. The maximal value of | m | is determined by m 2 A and can reach, as it follows from equations (3) and (4), | m | ∼ 0.050-0.055 eV. The indicated values of | m | are within the range of sensitivity of the next generation of (ββ) 0ν -decay experiments.
The expression for | m |, equation (23), permits us to relate the value of sin 2 (α 31 − α 21 )/2 to the experimentally measured quantities [24,40] | m |, m 2 atm and sin 2 2θ : A more precise determination of m 2 A and θ and a sufficiently accurate measurement of | m | could allow one to get information about the value of (α 31 − α 21 ), provided the neutrino mass spectrum is of the IH type.

Three QD neutrinos
In this case, it is convenient to introduce m 0 ≡ m 1 ∼ = m 2 ∼ = m 3 Similar to the case of the IH spectrum, one has For cos 2θ ∼ 0.40, favoured by the solar neutrino and the KamLAND data, one finds a nontrivial lower limit on | m |, | m | 0.07 eV. For the 90% CL allowed ranges of values of the parameters, one has | m | 0.043 eV (tables 1 and 2). Using the conservative cosmological upper bound on the sum of neutrino masses, we get | m | 0.70 eV. Also, in this case, one can obtain, in principle, direct information about one CP-violation phase from the measurement of | m |, m 0 and sin 2 2θ : The specific features of the predictions of | m | for the three types of neutrino mass spectra discussed above are evident from figures 1 and 2, where the dependence of | m | on m 1 for the LMA solution is shown. For instance, if m 2 = m 2 21 , which corresponds to a spectrum with normal hierarchy, | m | can lie anywhere between 0 and the currently existing upper limits, given by equations (19) and (20). This conclusion does not change even under the most favourable conditions for the determination of | m |, namely when m 2 atm , m 2 , θ and θ are known with negligible uncertainty, as seen in figures 1 and 2.

Constraining the lightest neutrino mass
By observing the (ββ) 0ν decay of a given nucleus, it would be possible to determine the value of | m | from the measurement of the associated lifetime of the decay. This would require knowledge of the NME of the process. At present, there exists large uncertainty in the calculation of (ββ) 0νdecay NMEs (see e.g. [65,84]). This is reflected, in particular, in the factor of ∼3 uncertainty in the upper limit on | m |, which is extracted from the experimental lower limits on the (ββ) 0νdecay half lifetime of 76 Ge. Recently, encouraging results on the problem of calculating the NMEs have been obtained in [85]. The observation of a (ββ) 0ν decay of one nucleus would probably lead to searches and, eventually, to the observation of decay of other nuclei. One can expect that such a progress, in particular, will help to solve the problem of sufficiently precise calculation of the NMEs for the (ββ) 0ν decay.
In this section, we consider briefly the information that future (ββ) 0ν decay and/or 3 H β-decay experiments can provide on the lightest neutrino mass m 1 , without taking into account the possible effects of currently existing uncertainties in the evaluation of (ββ) 0ν -decay NMEs.   [23]).
We get similar results for inverted mass hierarchy, m 2 ≡ m 2 32 , provided the experimental upper limit | m | exp is larger than the minimal value of | m |, | m |  2 32 would imply that m 1 0.02 eV and 0.04 eV respectively and, thus, a neutrino mass spectrum with partial hierarchy or of the QD type [40] (figures 1 and 2). The lightest neutrino mass will be constrained to lie in a rather narrow interval, (m 1 ) min m 1 (m 1 ) max . 10 11 If the measured value of | m | lies between the minimal and maximal values of | m |, which are predicted for the IH spectrum, m 1 again would be limited from above, but we would have (m 1 ) min = 0 (figures 1 and 2). A measured value of mν e , (mν e ) exp 0.20 eV, satisfying (mν e ) exp > (m 1 ) max , where (m 1 ) max is determined from the upper limit on | m | if the (ββ) 0ν decay is not observed, might imply that the massive neutrinos are Dirac particles. If (ββ) 0ν decay has been observed and | m | measured, the inequality (mν e ) exp > (m 1 ) max , with (m 1 ) max determined from the measured value of | m |, would lead to the conclusion that there exist contribution(s) to the (ββ) 0ν -decay rate other than from the light Majorana neutrino exchange (see e.g. [87] and references therein) that partially cancels the contribution from the Majorana neutrino exchange. . The results for the NH and IH spectra are obtained for m 1 = 10 −4 eV, whereas those for the QD spectrum correspond to m 0 = 0.2 eV (from [23]).

Determining the type of the neutrino mass spectrum
The possibility of distinguishing between the three different types of neutrino mass spectrum, NH, IH and QD, depends on the allowed ranges of values of | m | for the three spectra. More specifically, it is determined by the maximal values of | m | for the NH and IH spectra, | m | NH max and | m | IH max , and by the minimal values of | m | for the IH and QD spectra, | m | IH min and | m | QD min . These can be derived from equations (21), (24) and (26)  In tables 1 and 2 (taken from [23]), we show the values of (i) | m | NH max , (ii) | m | IH min and (iii) | m | QD min (m 0 = 0.2 eV), calculated for the best-fit and the 90% CL allowed ranges of values of tan 2 θ and m 2 in the LMA solution region. In table 3 (from [23]), 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.) 'prospective' uncertainties 12 of 5% (15%) on tan 2 θ and m 2 and of 10% (30%) on m 2 A . As evident from tables 1-3, 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 crucially also on the values of θ and | m |. The precision itself in the measurement of | m | in the next generation of (ββ) 0νdecay experiments, given the latter sensitivity limits of ∼(1.5-5.0) × 10 −2 eV, depends on the value of | m |. Precision in the measurements of tan 2 θ and m 2 , used to derive the numbers in table 3 can be achieved, e.g. in the solar neutrino experiments and/or in the experiments with reactorν e [53,88]. 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 data [25,28], its value will be determined with a ∼10% error (1 S.D.) by the MINOS experiment [49].
High-precision measurements of m 2 A , tan 2 θ and m 2 are expected to take place in 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 in the indicated period. Under these conditions, the highest 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) 0ν -decay NMEs. We will also assume in what follows that, by the time one or more (ββ) 0νdecay 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 NMEs will be unambiguously determined.
Following [43,82], we will parametrize the uncertainty in | m | resulting from imprecise knowledge of the relevant NMEs-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 NME and is the experimental error. An experiment measuring a (ββ) 0ν -decay half lifetime will thus determine a range of | m | corresponding to The currently estimated range of ζ 2 for experimentally interesting nuclei varies from 3.5 for 48 Ca to 38.7 for 130 Te (see e.g. table 2 in [65,84]). For 76 Ge and 82 Se, it is ∼10 [65]. The actual uncertainties can be smaller [85].
To be possible to distinguish between the NH and IH spectra, between the NH and QD spectra and between IH and QD spectra, the following inequalities must hold, respectively: These conditions imply, as can be demonstrated [82], upper limits on tan 2 θ that are a function of neutrino oscillation parameters and of ζ.
In figure 3 (taken from [82]), the upper bounds on tan 2 θ , for which one can distinguish the NH spectrum from the IH spectrum and from that of the QD type, are shown as a function of m 2 for m 2 A = 3 × 10 −3 eV 2 , sin 2 θ = 0.05 and 0.0 and different values of ζ. For the NH versus IH spectrum, results for sin 2 θ = 0.01 are also shown. For the QD spectrum, values of m 0 = 0.2; 1.0 eV are used.
The dependence of the maximal value of tan 2 θ of interest on m 0 and sin 2 θ in the NH versus QD case is rather weak as demonstrated in figure 3. This is not so in what concerns the dependence on sin 2 θ in the NH versus IH case: the maximal value of tan 2 θ under discussion can increase noticeably (e.g. by a factor of ∼1.2-1.5) when sin 2 θ decreases from 0.05 to 0. As it follows from figure 3, it would be possible to distinguish between the NH and QD spectra for the values of tan 2 θ favoured by the data for values of ζ ∼ = 3 or even somewhat higher. In contrast, the possibility of distinguishing between the NH and IH spectra for ζ ∼ = 3 depends critically on the value of sin 2 θ: as figure 3 indicates, this would be possible for the current best-fit value of tan 2 θ and, for instance, m 2 = (5.0-15) × 10 −5 eV 2 , provided sin 2 θ 0.01.
In figure 4 (taken from [82]), we show the maximal value of tan 2 θ permitting one to distinguish between the IH and QD spectra as a function of m 2 A , for sin 2 θ = 0.05 and 0.0,   These quantitative analyses show that if | m | is found to be nonzero in 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 QD spectra. Discriminating between the latter would be less demanding if m 0 is sufficiently large.

Constraining the Majorana CP-violation phases
The problem of detection of CP violation in the lepton sector is one of the most formidable and challenging problems in the study of neutrino mixing. As was noticed in [22], the measurement of | m | alone could exclude the possibility that the two Majorana CP-violation phases α 21 and α 31 , present in the PMNS matrix, are equal to zero. However, such a measurement cannot rule out, without additional input, the case of the two phases taking different CP-conserving values. The additional input needed for establishing CP violation could be, for example, measurement of neutrino mass mν e in the 3 H β-decay experiment KATRIN [59] or cosmological determination of the sum of the three neutrino masses [62], = m 1 + m 2 + m 3 or derivation of a sufficiently stringent upper limit on . At present, no viable alternative to the measurement of | m | for obtaining information on the Majorana CP-violating phases α 21 and α 31 exists or can be foreseen to exist in the next ∼8 years.
The possibility of obtaining information on the CP violation due to the Majorana phases α 21 and α 31 by measuring | m | was studied by a number of authors [24,40,41,42,44], and more recently, it was studied in [43,45]. Barger et al [45] considered in their analysis, in particular, the effect of uncertainty in the knowledge of NMEs on the measured value of | m |. After making a certain number of assumptions about the experimental and theoretical developments in the field of interest that may occur by the year 2020, 13 they claim to have shown 'once and for all that it is impossible to detect CP violation from (ββ) 0ν decay in the foreseeable future'. A different approach to the problem was used in [43], where an attempt was made to determine the conditions under which CP violation might be detected from a measurement of | m | and mν e or , or of | m | and a sufficiently stringent upper limit . We will summarize below results of the latter study.
The analysis in [43] is based on prospective input data on | m |, mν e , , tan 2 θ , etc. The effect of the NME uncertainty was included in the analysis. For example, for the IH spectrum (m 1 m 2 m 3 , m 1 < 0.02 eV), a 'just-CP-violating' region [40]-a value of | m | in this region would signal unambiguously CP violation in the lepton sector due to Majorana CP-violating phases-would be present if where (| m | exp ) max(min) is the largest (smallest) experimentally allowed value of | m |, taking into account both the experimental error on the measured (ββ) 0ν decay half lifetime and the uncertainty due to the evaluation of the NMEs. Condition (38) depends crucially on the value of (cos 2θ ) max and it is less stringent for smaller values of (cos 2θ ) max [22]. Using the parametrization given in equation (32), the necessary condition permitting us to establish, in principle, that the CP symmetry is violated due to the Majorana CP-violating phases reads Obviously, the smaller the (cos 2θ ) max and values, the larger the 'theoretical uncertainty', which might allow one to make conclusions concerning the CP-violation of interest. A similar analysis was performed for QD neutrinos mass spectrum. The results can be summarized as follows. The possibility of establishing that the Majorana phases α 21 and α 31 have CP-nonconserving values requires quite accurate measurements of | m | and, e.g. of mν e or , and holds only for a limited range of values of the relevant parameters. More specifically, to prove that CP violation associated with Majorana neutrinos takes place requires, in particular, a relative experimental error on the measured value of | m | not higher than 15-20%, a 'theoretical uncertainty'in the value of | m | due to imprecise knowledge of the corresponding NMEs smaller than a factor of 2, a value of tan 2 θ 0.55, and values of the relevant Majorana CP-violating phases (α 21 , α 32 ) typically within the ranges of ∼(π/2-3π/4) and ∼(5π/4-3π/2).

Conclusions
Future (ββ) 0ν -decay experiments have a remarkable physics potential. They can establish the Majorana nature of the neutrinos with a definite mass ν j . If the latter are Majorana particles, the (ββ) 0ν -decay experiments can provide unique information on the type of the neutrino mass spectrum and on the absolute scale of neutrino masses. They can also provide unique information on the Majorana CP-violation phases present in the PMNS neutrino mixing matrix. Knowledge of values of the relevant (ββ) 0ν -decay NMEs with a sufficiently low uncertainty is crucial for obtaining quantitative information on the neutrino mass and mixing parameters from a measurement of (ββ) 0ν -decay half lifetime.

Acknowledgments
The present review is based on the author's joint publications with S M Bilenky, L Wolfenstein, S Pascoli and W Rodejohann. The author is grateful to them for very fruitful and enjoyable collaborations. The author would also like to thank Professors T Kugo and M Nojiri and the other members of the Yukawa Institute for Theoretical Physics (YITP), Kyoto, Japan, where part of this work was done, for the kind hospitality extended to him. This work was partially supported by the Italian INFN under the programme 'Fisica Astroparticellare'.
Notes added. After the submission of this article for publication, the KamLAND experiment reported new data [89] on the spectrum of e + produced by reactorν e :ν e + p → e + + n. The data correspond to a statistics of 766.3 Ty and clearly show a distortion of the e + spectrum, compatible with that expected forν e oscillations, driven by the neutrino mass-squared difference responsible for the solar neutrino transitions, m 2 and lying in the LMA-I subregion of the LMA solution region. A combined analysis of the global solar neutrino and KamLAND 766 Ty data in terms of neutrino oscillations reveals [91] that the LMA-II solution is excluded at 4σ. A three-flavour neutrinos oscillation analysis of the data shows [91] that sin 2 θ < 0.05 at 99.73% CL and that, for sin 2 θ = 0.0 (0.02), for instance, one has, at 90% CL, 0.23 sin 2 θ 0.34 and 7.5 × 10 −5 eV 2 m 2 9.2 (9.0) × 10 −5 eV 2 . The corresponding best-fit values read: sin 2 θ ∼ = 0.28 and m 2 ∼ = 8.4 × 10 −5 eV 2 . We see that the maximal allowed values of sin 2 θ are smaller than those in equations (8) and (9), whereas the best-fit and the minimal values of m 2 are larger than those used (equations (10) and (11)). The former implies that the minimal values of | m | for IH and QD neutrino mass spectra are somewhat larger than those shown in figures 1 and 2; the latter means that the predictions for | m | in the case of the NH neutrino mass spectrum, illustrated in figures 1 and 2, will be somewhat modified. However, taking into account implications of the new KamLAND data will lead, in general, to minor modifications of the results on the effective Majorana mass | m | and the physics potential of the (ββ) 0ν -decay experiments, presented in this paper.