The SNO Solar Neutrino Data, Neutrinoless Double-Beta Decay and Neutrino Mass Spectrum

Assuming 3- ν mixing and massive Majorana neutrinos, we analyze the implications of the results of the solar neutrino experiments, including the latest SNO data, which favor the LMA MSW solution of the solar neutrino problem with tan 2 θ (cid:12) < 1, for the predictions of the eﬀective Majorana mass j <m> j in neutrinoless double beta (( ββ ) 0 ν -) decay. Neutrino mass spectra with normal mass hierarchy, with inverted hierarchy and of quasi-degenerate type are considered. For cos 2 θ (cid:12) (cid:24) > 0 . 26, which follows (at 99.73% C.L.) from the SNO analysis of the solar neutrino data, we ﬁnd signiﬁcant lower limits on j <m> j in the cases of quasi-degenerate and inverted hierarchy neutrino mass spectrum, j <m> j (cid:24) > 0 . 03 eV and j <m> j (cid:24) > 8 . 5 (cid:2) 10 − 3 eV, respectively. If the spectrum is hierarchical the upper limit holds j <m> j (cid:24) < 8 . 2 (cid:2) 10 − 3 eV. Correspondingly, not only a measured value of j <m> j 6 = 0, but even an experimental upper limit on j <m> j of the order of few (cid:2) 10 − 2 eV can provide information on the type of the neutrino mass spectrum; it can provide also a signiﬁcant upper limit on the mass of the lightest neutrino m 1 . A measured value of j <m> j (cid:24) > 0 . 2 eV, combined with data on neutrino masses from the 3 H β − decay experiment KATRIN


Introduction
With the publication of the new results of the SNO solar neutrino experiment [1,2] (see also [3]) on i) the measured rates of the charged current (CC) and neutral current (NC) reactions, ν e + D → e − + p + p and ν l (ν l ) + D → ν l (ν l ) + n + p, ii) on the day-night (D-N) asymmetries in the CC and NC reaction rates, and iii) on the spectrum of the final state e − in the CC reaction, further strong evidences for oscillations or transitions of the solar ν e into active neutrinos ν µ,τ (and/or antineutrinosν µ,τ ), taking place when the solar ν e travel from the central region of the Sun to the Earth, have been obtained. The evidences for oscillations (or transitions) of the solar ν e become even stronger when the SNO data are combined with the data obtained in the other solar neutrino experiments, Homestake, Kamiokande, SAGE, GALLEX/GNO and Super-Kamiokande [4,5].
Global analysis of the solar neutrino data [1,2,3,4,5], including the latest SNO results, in terms of the hypothesis of oscillations of the solar ν e into active neutrinos, ν e → ν µ(τ ) , show [1] that the data favor the large mixing angle (LMA) MSW solution with tan 2 θ < 1, where θ is the angle which controls the solar neutrino transitions. The LOW solution of the solar neutrino problem with transitions into active neutrinos is only allowed at approximately 99.73% C.L. [1]; there do not exist other solutions at the indicated confidence level. In the case of the LMA solution, the range of values of the neutrino mass-squared difference ∆m 2 > 0, characterizing the solar neutrino transitions, found in [1] at 99.73% C.L. reads: LMA MSW : 2.2 × 10 −5 eV 2 ∼ < ∆m 2 ∼ < 2.0 × 10 −4 eV 2 (99.73% C.L.).
The best fit value of cos 2θ in the LMA solution region is given by (cos 2θ ) BF V = 0.50. Strong evidences for oscillations of atmospheric neutrinos have been obtained in the Super-Kamiokande experiment [6]. As is well known, the atmospheric neutrino data is best described in terms of dominant ν µ → ν τ (ν µ →ν τ ) oscillations. The explanation of the solar and atmospheric neutrino data in terms of neutrino oscillations requires the existence of 3-neutrino mixing in the weak charged lepton current (see, e.g., [7,8]).
Rather stringent upper bounds on |< m >| have been obtained in the 76 Ge experiments by the Heidelberg-Moscow collaboration [29], |< m >| < 0.35 eV (90%C.L.), and by the IGEX collaboration [30], |< m >| < (0.33 ÷ 1.35) eV (90%C.L.). Taking into account a factor of 3 uncertainty in the calculated value of the corresponding nuclear matrix element, we get for the upper limit found in [29]: |< m >| < 1.05 eV. Considerably higher sensitivity to the value of |< m >| is planned to be reached in several (ββ) 0ν −decay experiments of a new generation. The NEMO3 experiment [31], which began taking data in 2001, and the cryogenic detector CUORE [32], are expected to reach a sensitivity to values of |< m >| ∼ = 0.1 eV. An order of magnitude better sensitivity, i.e., to |< m >| ∼ = 10 −2 eV, is planned to be achieved in the GENIUS experiment [33] utilizing one ton of enriched 76 Ge, and in the EXO experiment [34], which will search for (ββ) 0ν −decay of 136 Xe. Two more detectors, Majorana [35] and MOON [36], are planned to have sensitivity to |< m >| in the range of f ew × 10 −2 eV.
The fact that the solar neutrino data implies a relatively large lower limit on the value of cos 2θ , eq. (2), has important implications for the predictions of the effective Majorana mass parameter in (ββ) 0ν -decay [10,11] and in the present article we investigate these implications.

The SNO Data and the Predictions for the Effective Majorana
Mass |<m>| According to the analysis performed in [1], the solar neutrino data, including the latest SNO results, strongly favor the LMA solution of the solar neutrino problem with tan 2 θ < 1. We take into account these new development to update the predictions for the effective Majorana mass |< m >| , derived in [10], and the analysis of the implications of the measurement of, or obtaining a more stringent upper limit on, |< m >| performed in [10,11]. The predicted value of |< m >| depends in the case of 3-neutrino mixing of interest on (see e.g. [10,11,25]): i) the value of the lightest neutrino mass m 1 , ii) ∆m 2 and θ , iii) the neutrino mass-squared difference which characterizes the atmospheric ν µ (ν µ ) oscillations, ∆m 2 atm , and iv) the lepton mixing angle θ which is limited by the CHOOZ and Palo Verde experiments [39,40]. The ranges of allowed values of ∆m 2 and θ are determined in [1], while those of ∆m 2 atm and θ are taken from [41]. Given the indicated parameters, the value of |< m >| depends strongly [10,11] on the type of the neutrino mass spectrum, as well as on the values of the two Majorana CP-violating phases, present in the lepton mixing matrix.
We number the massive neutrinos (without loss of generality) in such a way that m 1 < m 2 < m 3 . In the analysis which follows we consider neutrino mass spectra with normal mass hierarchy, with inverted hierarchy and of quasi-degenerate type [10,11,21,22,23,24,26]. In the case of neutrino mass spectrum with normal mass hierarchy (m 1 (<) m 2 m 3 ) we have ∆m 2 ≡ ∆m 2 21 and sin 2 θ ≡ |U e3 | 2 , while in the case of spectrum with inverted hierarchy (m 1 m 2 ∼ = m 3 ) one finds ∆m 2 ≡ ∆m 2 32 and sin 2 θ ≡ |U e1 | 2 . In both cases one can choose ∆m 2 atm ≡ ∆m 2 31 . It should be noted that for m 1 > 0.2 eV ∆m 2 atm , the neutrino mass spectrum is of the quasi-degenerate type, m 1 ∼ = m 2 ∼ = m 3 , and the two cases, ∆m 2 ≡ ∆m 2 21 and ∆m 2 ≡ ∆m 2 32 , lead to the same predictions for |< m >| .

Normal Mass Hierarchy: ∆m 2 ≡ ∆m 2 21
If ∆m 2 = ∆m 2 21 , the effective Majorana mass parameter |< m >| is given in terms of the oscillating parameters ∆m 2 , ∆m 2 atm , θ and |U e3 | 2 , which is constrained by the CHOOZ data, as follows [10]: The effective Majorana mass |< m >| can lie anywhere between 0 and the present upper limits, as Fig. 1 (left panels) indicates. This conclusion does not change even under the most favorable conditions for the determination of |< m >| , namely, even when ∆m 2 atm , ∆m 2 , θ and θ are known with negligible uncertainty [11]. Our further conclusions for the case of the LMA solution of the solar neutrino problem [1] are illustrated in Fig. 1 (left panels) and are summarized below.
Taking into account the new constraints on the solar neutrino oscillating parameters following from the SNO data [1] does not change qualitatively the conclusions reached in ref. [10,11]. The upper limit on |< m >| for given m 1 reads: where (cos 2 θ ) MIN and (sin 2 θ ) MAX are the values corresponding to (tan 2 θ ) MAX , and (∆m 2 atm ) MAX is the maximal value of ∆m 2 atm allowed for the |U e3 | 2 MAX [41]. For the allowed values of ∆m 2 and tan 2 θ from the LMA solution region [1], eqs. (1) and (2), we get for m 1 < 0.02 eV: |< m >| ≤ |< m >| MAX 8.2 × 10 −3 eV. The maximal value of |< m >| corresponds to the case of CP-conservation and ν 1 , ν 2 and ν 3 having identical CP-parities, There is no significant lower bound on |< m >| because of the possibility of mutual compensations between the terms contributing to |< m >| and corresponding to the exchange of different virtual massive Majorana neutrinos. Furthermore, the uncertainties in the oscillation parameters do not allow to identify a "just-CP violation" region of values of |< m >| [10] (a value of |< m >| in this region would unambiguously signal the existence of CP-violation in the lepton sector, caused by Majorana CP-violating phases). However, if the neutrinoless double beta-decay will be observed, the measured value of |< m >| , combined with information on m 1 and a better determination of the relevant neutrino oscillation parameters, would allow to determine whether the CP-symmetry is violated due to Majorana CP-violating phases, or to identify which are the allowed patterns of the massive neutrino CP-parities in the case of CP-conservation (for a detailed discussion see ref. [11]). The new element in the predictions for the effective Majorana mass |< m >| is the existence of a lower bound on the possible values of |< m >| (Fig. 1, left panels). For m 1 ∼ > 0.07 eV this lower bound is significant, |< m >| ∼ > 10 −2 eV. In the case of quasi-degenerate spectrum, m 1 > 0.2 eV, the lower bound reads: |< m >| ∼ > 0.035 eV.
For a given m 1 ≥ 0.02 eV, the minimal value of |< m >| , |< m >| MIN , is given by where again (∆m 2 atm ) MAX is the maximal allowed value of ∆m 2 atm for the |U e3 | 2 MAX [41]. The upper bound on |< m >| , which corresponds to CP-conservation and η 21 = η 31 = +1 (ν 1 , ν 2 and ν 3 possessing identical CP-parities), can be found for given m 1 by using eq. (6). For the allowed values of m 1 ≥ 0.02 eV (which is limited from above by the 3 H β−decay data [37,38]), |< m >| MAX is limited by the upper bounds obtained in the (ββ) 0ν -decay experiments [29,30]: For values of |< m >| , which are in the range of sensitivity of the future (ββ) 0ν -decay experiments, there exists a "just-CP-violation" region. This is illustrated in Fig. 2, where we show |< m >| /m 1 for the case of quasi-degenerate neutrino mass spectrum (m 1 > 0.2 eV, m 1 m 2 m 3 ), as a function of cos 2θ . The "just-CP-violation" interval of values of |< m >| /m 1 is determined by Taking into account eq. (2) and the existing limits on |U e3 | 2 , this gives 0.67 < |< m >| /m 1 < 0.85. The mass m 1 in the case of interest is m 1 mν e and can be measured in the KATRIN experiment, provided m 1 ∼ > 0.35 eV.

Inverted Neutrino Mass
The new predictions for |< m >| again differ substantially from those obtained before the appearance of the latest SNO data due to the existence of a significant lower bound on |< m >| for every value of m 1 : even in the case of m 1 m 2 ∼ = m 3 (i.e., even if m 1 0.02 eV), we get |< m >| ∼ > 8.5 × 10 −3 eV (see Fig. 1, right panels). Actually, the minimal value of |< m >| , min(|< m >| ), depends on whether CP-invariance holds or not in the lepton sector, and if it holds -it depends on the relative CP-parities of the massive Majorana neutrinos. The effective Majorana mass |< m >| can be considerably larger than in the case of a hierarchical neutrino mass spectrum [10,23]. The maximal value of |< m >| corresponds to CP-conservation and η 21 = η 31 = +1 and for given m 1 reads: where (cos 2 θ ) MIN and (sin 2 θ ) MAX are the values corresponding to (tan 2 θ ) MAX , and |U e1 | 2 MIN is the minimal allowed value of |U e1 | 2 for the (∆m 2 atm ) MAX . For the allowed ranges of the neutrino oscillation parameters found in refs. [1,41], the maximal allowed value of |< m >| is | < m > | MAX 0.08 eV.
The existence of a relevant lower bound on |< m >| in the case of the LMA solution, |< m >| ∼ > 8.5× 10 −3 eV, is a consequence of the fact that cos 2θ is found to be significantly different from zero. The minimal value of |< m >| , |< m >| MIN , is reached in the case of CP-invariance and η 21 = −η 31 = −1, and is determined by: where (cos 2 θ ) MIN and (sin 2 θ ) MAX are the values corresponding to (tan 2 θ ) MAX , and |U e1 | 2 MAX is the maximal allowed value of |U e1 | 2 for the (∆m The discussion and conclusions in the case of the spectrum of quasi-degenerate type and with partial inverted hierarchy are identical to those in the same case for the neutrino mass spectrum with normal hierarchy given in sub-section 2.1, Case B, except for the maximal and minimal values of |< m >| , |< m >| MAX and |< m >| MIN , which for a fixed m 1 are determined by: where |U e1 | 2 MIN (|U e1 | 2 MAX ) in eq. (12) (in eq. (13)) is the minimal (maximal) allowed value of |U e1 | 2 given the (∆m 2 atm ) MAX ((∆m 2 atm ) MIN ). For any value of m 1 ≥ 0.02 eV, the lower bound on |< m >| reads |< m >| ∼ > 0.01 eV.

The Effective Majorana Mass and the Determination of the Neutrino Mass Spectrum
The existence of a lower bound on |< m >| , |< m >| MIN , in the quasi-degenerate and inverted mass hierarchy (∆m 2 = ∆m 2 32 ) cases implies that the future (ββ) 0ν -decay experiments might allow to determine the type of the neutrino mass spectrum (under the general assumptions of 3-neutrino mixing and massive Majorana neutrinos, (ββ) 0ν -decay generated only by the (V-A) charged current weak interaction via the exchange of the three Majorana neutrinos, neutrino oscillation explanation of the solar and atmospheric neutrino data). This conclusion is valid not only under the assumption that the (ββ) 0ν -decay will be observed in these experiments and |< m >| will be measured, but also in the case only a sufficiently stringent upper limit on |< m >| will be derived.
More specifically, as is illustrated in Fig. 3, the following statements can be made: 1. a measurement of |< m >| = |< m >| exp > 0.20 eV, would imply that the neutrino mass spectrum is of the quasi-degenerate type (m 1 > 0.20 eV) and that there are both a lower and an upper limit on m 1 , (m 1 ) min ≤ m 1 ≤ (m 1 ) max . The values of (m 1 ) max and (m 1 ) min are fixed respectively by the equalities |< m >| MIN = |< m >| exp and |< m >| MAX = |< m >| exp , where |< m >| MIN and |< m >| MAX are given by eqs. (7) and (6); 2. if |< m >| is measured and is found to lie in the interval 8.5 × 10 −2 eV ∼ < |< m >| exp ∼ < 2.0 × 10 −1 eV, one could conclude that either i) ∆m 2 ≡ ∆m 2 21 and the spectrum is of the quasi-degenerate type (m 1 > 0.20 eV) or with partial hierarchy (0.02 eV ≤ m 1 ≤ 0.2 eV), with 8. 4. a measurement or an upper limit on |< m >| , |< m >| ∼ < 8.0 × 10 −3 eV, would lead to the conclusion that the neutrino mass spectrum is of the normal mass hierarchy type, ∆m 2 ≡ ∆m 2 21 , and that m 1 is limited from above by m 1 ≤ (m 1 ) max 5.8×10 −2 eV, where (m 1 ) max is determined by the condition |< m >| MIN = |< m >| exp , with |< m >| MIN given by eq. (7). A measured value of (or an upper limit on) the effective Majorana mass |< m >| ∼ < 0.03 eV would disfavor (if not rule out) the quasi degenerate mass spectrum, while a value of |< m >| ∼ < 8×10 −3 eV would rule out the quasi degenerate mass spectrum, disfavor the spectrum with inverted mass hierarchy and favor the hierarchical neutrino mass spectrum.
If the minimal value of cos 2θ inferred from the solar neutrino data, is somewhat smaller than that in eq. (2), the upper bound on |< m >| in the case of neutrino mass spectrum with normal hierarchy (∆m 2 ≡ ∆m 2 21 , m 1 0.02 eV) might turn out to be larger than the lower bound on |< m >| in the case of spectrum with inverted mass hierarchy (∆m 2 ≡ ∆m 2 32 , m 1 0.02 eV). Thus, there will be an overlap between the regions of allowed values of |< m >| in the two cases of neutrino mass spectrum at m 1 0.02 eV. The minimal value of cos 2θ for which the two regions do not overlap is determined by the condition: where we have neglected terms of order (sin 2 θ) 2 MAX . For the values of the neutrino oscillation parameters used in the present analysis this "border" value turns out to be cos 2θ ∼ = 0.25.
Let us note that [11] if the (ββ) 0ν -decay is not observed, a measured value of mν e in 3 H β-decay experiments, (mν e ) exp ∼ > 0.35 eV, which is larger than (m 1 ) max , (m νe ) exp > (m 1 ) max , where (m 1 ) max is determined as in the Case 1 (i.e., from the upper limit on |< m >| , |< m >| MIN = |< m >| exp , with |< m >| MIN given in eq. (7)), might imply that the massive neutrinos are Dirac particles. If the (ββ) 0ν -decay has been observed and |< m >| measured, the inequality (mν e ) exp > (m 1 ) max , would lead to the conclusion that there exist contribution(s) to the (ββ) 0ν -decay rate other than due to the light Majorana neutrino exchange which partially cancel the contribution due to the Majorana neutrino exchange.
A measured value of |< m >| , (|< m >| ) exp > 0.08 eV, and a measured value of mν e or an upper bound on mν e , such that mν e < (m 1 ) min , where (m 1 ) min is determined by the condition |< m >| MAX = |< m >| exp , with |< m >| MAX given by eq. (12), would imply that [11] there are contributions to the (ββ) 0ν -decay rate in addition to the ones due to the light Majorana neutrino exchange (see, e.g., [42]), which enhance the (ββ) 0ν -decay rate. This would signal the existence of new ∆L = 2 processes beyond those induced by the light Majorana neutrino exchange in the case of left-handed charged current weak interaction.

Conclusions
Assuming 3-ν mixing and massive Majorana neutrinos, we have analyzed the implications of the results of the solar neutrino experiments, including the latest SNO data, which favor the LMA MSW solution of the solar neutrino problem with tan 2 θ < 1, for the predictions of the effective Majorana mass |< m >| in (ββ) 0ν -decay. Neutrino mass spectra with normal mass hierarchy, with inverted hierarchy and of quasi-degenerate type are considered. For cos 2θ ≥ 0.26, which follows (at 99.73% C.L.) from the analysis of the solar neutrino data performed in [1], we find significant lower limits on |< m >| in the cases of quasi-degenerate and inverted hierarchy neutrino mass spectrum, |< m >| ∼ > 0.03 eV and |< m >| ∼ > 8.5 × 10 −3 eV, respectively. If the neutrino mass spectrum is hierarchical (with inverted hierarchy) the upper limit holds |< m >| ∼ < 8.2 × 10 −3 (8.0 × 10 −2 ) eV. Correspondingly, not only a measured value of |< m >| = 0, but even an experimental upper limit on |< m >| of the order of f ew × 10 −2 eV can provide information on the type of the neutrino mass spectrum; it can provide also a significant upper limit on the mass of the lightest neutrino m 1 . A measured value of |< m >| ∼ > 0.2 eV, which would imply a quasi-degenerate neutrino mass spectrum, combined with data on neutrino masses from the 3 H β−decay experiment KATRIN (an upper limit or a measured value), might allow to establish whether the CP-symmetry is violated in the lepton sector. Further reduction of the LMA solution region due to data, e.g., from the experiments SNO, KamLAND and BOREXINO, leading to an increase (a decreasing) of the current lower (upper) bound of cos 2θ can strengthen further the above conclusions.

Note Added.
After the work on the present study was essentially completed, few new global analyses of the solar neutrino data have appeared [43,44]. The results obtained in [43] do not differ substantially from those derived in [1]; in particular, the (99.73% C.L.) minimal allowed values of cos 2θ in the LMA solution region found in [1] and in [43] practically coincide. Smaller minimal allowed values of cos 2θ of the order of ∼ 0.1 (99.73% C.L.) and larger maximal allowed values of ∆m 2 of the order of 4 × 10 −4 eV 2 (99.73% C.L.) have been obtained in the analyses performed in [44]. At present it is not clear to us what is the source of these differences. Using the LMA values of cos 2θ and ∆m 2 found in [44], the maximal value of |< m >| in the case of a hierarchical neutrino mass spectrum (m 1 0.02 eV) increases somewhat to 10 −2 eV 2 , while the minimal one in the case of the spectrum with inverted hierarchy is f ew × 10 −3 eV.