Physics Letters B 544 (2002) 239-250

www. elsevier. com/locate/npe

The SNO solar neutrino data, neutrinoless double beta-decay and

neutrino mass spectrum

S. Pascolia b, S.T. Petcovab1

a Scuola Internazionale Superiore di Studi Avanzati, I-34014 Trieste, Italy b Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34014 Trieste, Italy

Received 27 May 2002; received in revised form 30 July 2002; accepted 14 August 2002

Editor: G.F. Giudice

Abstract

Assuming 3-v 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 tan2 0q < 1, for the predictions of the effective Majorana mass |(m)| in neutrinoless double beta-decay. Neutrino mass spectra with normal mass hierarchy, with inverted hierarchy and of quasi-degenerate type are considered. For cos 2% > 0.26, which follows (at 99.73% C.L.) from the SNO analysis of the solar neutrino data, we find significant lower limits on |(m)| in the cases of quasi-degenerate and inverted hierarchy neutrino mass spectrum, |(m)| > 0.035 eV and |(m)| > 8.5 x 10-3 eV, respectively. If the spectrum is hierarchical the upper limit holds |(m)| < 8.2 x 10-3 eV. Correspondingly, not only a measured value of |(m)| = 0, but even an experimental upper limit on |(m)| of the order of few x 10-2eV 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 m1. © 2002 Elsevier Science B.V. All rights reserved.

1. 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, ve + D ^ e- + p + p and vl (yi) + D ^ vl (yi) + n + p, (ii) on the day-night (DN) asymmetries in the CC and NC reaction rates, and (iii) on the day and night event energy spectra, further strong evidences for oscillations or

E-mail address: petcov@he.sissa.it (S.T. Petcov).

1 Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria.

transitions of the solar v^ into active neutrinos (and/or antineutrinos v), taking place when the solar v^ travel from the central region of the Sun to the Earth, have been obtained. The evidences for oscillations (or transitions) of the solar v^ 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 [1-5], including the latest SNO results, in terms of the hypothesis of oscillations of the solar v^ into active neutrinos, v^ ^ , show [1] that the data favor the large mixing angle (LMA) MSW solution with tan2 0q < 1, where 0q is the angle which controls

0370-2693/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S0370-2693(02)02510-8

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 Am© > 0, characterizing the solar neutrino transitions, found in [1] at 99.73% C.L. reads:

LMA MSW: 2.2 x 10-5 eV2

< Am2 < 2.0 x 10-4eV2 (99.73% C.L.). (1)

The best fit value of Am© obtained in [1] is (Am|)BF = 5.0 x 10-5 eV2. The mixing angle 0© was found in the case of the LMA solution to lie in an interval which at 99.73% C.L. is determined by [1]

LMA MSW: 0.26 < cos 20© < 0.64 (99.73% C.L.).

The best fit value of cos 20© in the LMA solution region is given by (cos20©)BF = 0.50.

Strong evidences for oscillations of atmospheric neutrinos have been obtained in the Super-Kamio-kande experiment [6]. As is well known, the atmospheric neutrino data is best described in terms of dominant vM ^ vT (VM ^ VT) 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]).

Assuming 3- v mixing and massive Majorana neutrinos, we analyze the implications of the latest results of the SNO experiment for the predictions of the effective Majorana mass |(m)| in neutrinoless double beta (PP)0v-decay (see, e.g., [9-11]):

|(m)| = |m1|^e!|2 + m2|Ue2|Va21 + m3|Ue3|2e

Here m12 3 are the masses of 3 Majorana neutrinos with definite mass v1,2,3, Uej are elements of the lepton mixing matrix U —the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [12,13], and a21 and a31 are two Majorana CP-violating phases2 [14,15]. If CP-invariance holds, one has [16,17]

|2 J«31 I

2 We assume that the fields of the Majorana neutrinos vj satisfy the Majorana condition: C(vj)T = vj, j = 1, 2, 3, where C is the

charge conjugation matrix.

«21 = kn, «31 = k'n, k,k = 0, 1,2,..., and

n21 = eia21 =±1, n31 = eia31 =±1,

represent the relative CP-parities of the neutrinos 1 and v2, and v1 and v3, respectively.

The experiments searching for (PP)ov-decay test the underlying symmetries of particle interactions (see, e.g., [9]). They can answer the fundamental question about the nature of massive neutrinos, which can be Dirac or Majorana fermions. If the massive neutrinos are Majorana particles, the observation of (PP)ov-decay3 can provide unique information on the type of the neutrino mass spectrum and on the lightest neutrino mass [10,11,21-26]. Combined with data from the 3H p -decay neutrino mass experiment KATRIN [27], it can give also unique information on the CP-violation in the lepton sector induced by the Majorana CP-violating phases, and if CP-invariance holds—on the relative CP-parities of the massive Majorana neutrinos [10,11,22,28].

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 (PP )0 v -decay experiments of a new generation. The NEMO3 experiment [31], which will begin to take data in July of 2002, 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 (PP)0v-decay of 136Xe. Two more detectors, Majorana [35] and MOON [36], are planned to have sensitivity to |(m)| in the range of few x 10-2 eV.

In what regards the 3H p -decay experiments, the currently existing most stringent upper bounds on

3 Evidences for

-decay taking place with a rate corre-

sponding to 0.11 eV < |(m)| < 0.56 eV (95% C.L.) are claimed to have been obtained in [18]. The results announced in [18] have been criticized in [19,20].

the electron (anti-)neutrino mass m ye were obtained in the Troitzk [37] and Mainz [38] experiments and read mte < 2.2 eV. The KATRIN 3H fi-decay experiment [27] is planned to reach a sensitivity to m ye ~ 0.35 eV.

The fact that the solar neutrino data implies a relatively large lower limit on the value of cos 20©, Eq. (2), has important implications for the predictions of the effective Majorana mass parameter in v-decay [10,11] and in the present article we investigate these implications.

2. 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 tan2 0© < 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 m1, (ii) Am© and 0©, (iii) the neutrino mass-squared difference which characterizes the atmospheric vM ( vM) oscillations, Am^, and (iv) the lepton mixing angle 0 which is limited by the CHOOZ and Palo Verde experiments [39,40]. The ranges of allowed values of Am© and 0© are determined in [1], while those of Am^m and 0 are taken from [41] (we use the best fit values and the 99% C.L. results 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, a21 and a31 (see Eq. (3)), present in the lepton mixing matrix.

Let us note that if Am2tm lies in the interval Am2tm = (2.0 - 5.0) x 10-3 eV2, as is suggested by the current atmospheric neutrino data [6], its value will be determined with a high precision (~ 10% uncertainty) by the MINOS experiment [42]. Similarly, if Am© = (2.5 - 10.0) x 10-5 eV2, which is favored by the solar neutrino data, the KamLAND experi-

ment [43] will be able to measure Am© with an uncertainty of - (10 - 15)% (99.73% C.L.). Combining the data from the solar neutrino experiments and from KamLAND would permit to determine cos 20© with a high precision as well (~ 15% uncertainty at 99.73% C.L., see4, e.g., [44]). Somewhat better limits on sin2 0 than the existing one can be obtained in the MINOS experiment.

We number the massive neutrinos (without loss of generality) in such a way that m1 < m2 < m3. In the analysis which follows we consider neutrino mass spectra with normal mass hierarchy, with inverted hierarchy and of quasi-degenerate type [10,11,2124,26]. In the case of neutrino mass spectrum with normal mass hierarchy (m1 ^ (<)m2 ^ m3) we have Am© = Am21 and sin2 0 = |Ue3|2, while in the case of spectrum with inverted hierarchy (m1 ^ m2 = m3) one finds Am© = Am22 and sin2 0 = |Ue112. In both cases one can choose Am^m = Am^1. It should be

noted that for m\ > 0.2 eV ;>> ^ Am2tm, the neutrino mass spectrum is of the quasi-degenerate type, m1 = m2 = m3, and the two possibilities, Am© = Am21 and Am© = Am^2, lead to the same predictions for |(m)|.

2.1. Normal mass hierarchy: Am© = Am21

If Am© = Ami,

i21, the effective Majorana mass parameter |(m)| is given in terms of the oscillation parameters Amg, Am^lm, 0© and |^e3|2 which is constrained by the CHOOZ data, as follows [10]:

|(m>| = | (mi cos2 0© + Y m2 + Am© sin2 0©ela21 )

x (1 - \Ue3\2)+Jm2 + Arn2m \Ue3\2eia^ |.

The effective Majorana mass |(m>| can lie anywhere between 0 and the present upper limits, as Fig. 1 (left panels) shows.5 This conclusion does not change even under the most favorable conditions for the determination of |(m>|, namely, even when Ami;tm, Am©, 0© and 0 are known with negligible uncertainty [11]. Our

4 We thank C. Pena-Garay for clarifications on this point.

5 This statement is valid as long as m1 and the CP violating phases which enter the effective Majorana mass |(m>| are not constrained.

5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0

Login?! [eV]) Log(m,[eV])

Fig. 1. The dependence of |(m)| on m\ in the case of the LMA solution of the solar neutrino problem [1] (99.73% C.L.), for (i) Am^ = Am2j (left panels) and (ii) Amg = Am^2 (right panels) and for sin2 9 = 0.05 (upper panels), sin2 9 = 0.01 (middle panels), sin2 9 = 0.005 (lower panels). For Amg = Am2j, (sin2 9 = |C/e3l2), the allowed values of |(m)| are constrained to lie in the case of CP-conservation in the medium-grey regions (a) between the two thick solid lines if n21 = V31 = 1, (b) between the two long-dashed lines and the axes if n21 = — V31 = 1, (c) between the dash-dotted lines and the axes if n21 = — V31 = —1, (d) between the short-dashed lines if n21 = V31 =—1. For Amg = Am^2, (sin2 9 = |C/e112), the allowed regions for |(m)| correspond: for |Ue112 = 0.005 and |Ue112 = 0.01—to the medium-grey regions (a) between the solid lines if n21 = n31 = ±1, (b) between the dashed lines if n21 = — n31 = ±1, and for |Ue112 = 0.05—to the medium-grey regions (c) between the solid lines if n21 = n31 = 1, (d) between the long-dashed lines if n21 = n31 =—1, (e) between the dashed-dotted lines if n21 = — n31 = 1, (f) between the short-dashed lines if n21 = — ??31 =—1. In the case of CP-violation, the allowed region for |(m)| covers all the grey regions. Values of |(m)| in the dark grey regions signal CP-violation.

further conclusions for the case ofthe LMA solution of CaseA. mi < 0.02 eV, mi <m2 ^ m3. 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 oscillation parameters following from

the SNO data [1] does not change qualitatively the conclusions reached in Ref. [10,11]. The maximal value of |(m}|, |(m)|MAX, for given m1 reads:

|(m}|max

= (mi (cos2 00)MIN + 7»i?+(Am|)MAX x (sin2 6q)max) i1 - 1 ^e3 |max)

+(a m2t^max

where (cos2 0©)MIN and (sin2 0©)MAX are the values corresponding to (tan2 0©)max, and (Am^MAX is the maximal value of Am2tm allowed for the

i^3|max[41]-

For the values of Am© and tan2 0© from the LMA solution region [1], Eqs. (1) and (2), we get for mi ^ 0.02 eV: |(m}|MAX ^ 8.2 x 10-3 eV. Using the best fit values of the oscillation parameters found in Refs. [1,41], one obtains: |(m}|MAX ^ 2.0 x 10-3 eV. The maximal value of |(m}| corresponds to the case of CP-conservation and v1i2,3 having identical CP-parities, V21 = n31 = 1.

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). If neutrino-less double beta-decay will be observed, the measured value of |(m}| and a better determination of the relevant neutrino oscillation parameters, might allow to get information on the Majorana CP-violating phases; in particular, it might allow to identify the allowed patterns of the massive neutrino relative CP-parities in the case of CP-conservation (for a detailed discussion see Ref. [11]).

Case B. Neutrino mass spectrum with partial hierarchy (0.02 eV < m1 < 0.2 eV).

For m1 > 0.02 eV there exists a lower bound on the possible values of |(m}| (Fig. 1, left panels). Using the 99.73% C.L. allowed values of Am© and cos 20©

from [1], we find that this lower bound is significant, i.e., |(m}| > 10-2 eV, for m1 > 0.07 eV For the best fit values of the oscillation parameters obtained in [1, 41], one has |(m)| > 10-2 eVfor m1 > 0.02 eV.

For a fixed m1 > 0.02 eV, the minimal value of |(m}|, |(m)|MiN, is given by

|(m)|MiN = m1 (cos20©)min(1 - |^e3|MAX)

■ym2 + (Am2m)MAX|C/e3|

/ (Am|)max\ \ 4m i /'

where again (Am^MAX

value of Am2tm for the

is the maximal allowed atm u r the |^e3lMAX [41]. The upper bound on |(m}|, which corresponds to CP-conservation and n21 = n31 = +1 ( ^,2,3 possessing identical CP-parities), can be found for given m1 by using Eq. (6). For the allowed values of m1, 0.02 eV < m1 < 0.2 eV, we have |(m}| < 0.2 eV.

2.2. Inverted neutrino mass hierarchy: Am© = Am^2

If Am© = Am32,

the effective Majorana mass |(m}| is given in terms of the oscillation parameters Am©, Am^lm, 0© and |^e112 which is constrained by the CHOOZ data [10]:

|(m}| = \mi\Uei |2 + m2 + Am2tm - Am©

x cos20©(1 - lUe1l2)eia21

+ ym2 + Am2msin20©(l -\Uel\2)eia^ |.

The new predictions for |(m}| 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 m1: even in the case of m1 ^ m2 = m3 (i.e., even if m1 ^ 0.02 eV), we get

|(m}| > 8.5 x 10-3 eV

(see Fig. 1, right panels). Given the neutrino oscillation parameters, the minimal allowed value of |(m}| depends on the values of the CP violating phases a21 and a31.

Case A. «1 < 0.02 eV, «1 ^ m2 — m3.

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 n21 = n31 = +1, and for given m1 reads:

|(m>|MAX = m1|^e1|MlN

+ Qml + (AwIatm)MAX " (Aml)

(cos2 0©MIN

Jmj + (Am2atm)MAX (sin2 0Q]

X (1 -|^e1|Ml^) >

where (cos2 0©)MIN and (sin2 0©)MAX are the values corresponding to (tan2 0©)MAX, and | Ue1 |MIN is the minimal allowed value of |Ue1|2 for the (Am;;tm)MAX. For the allowed ranges of Am©, tan2 0©—Eqs. (1) and (2), and Am2tm and |Ue1|2 [41] (99% C.L.), and for the best fit values of the neutrino oscillation parameters found in [1,41], we get |(m>|MAX — 0.080 eV and |(m>|MAX — 0.056 eV, respectively.

There exists a non-trivial lower bound on |(m>| in the case of the LMA solution for which cos 20© is found to be significantly different from zero. For the 99.73% C.L. allowed values of Am© and cos20© [1], this lower bound reads: |(m>| > 8.5 x 10-3 eV. Using the best fit values of the oscillation parameters [1,41], we find:

|(m>| > 2.8 x 10-2 eV.

The lower bound is present even for cos 20© > 0.1: in this case |(m)| > 4 x 10-3 eV. The minimal value of |(m)|, |(m)|MIN, is reached in the case of CP-invariance and n21 = — n31 = -1, and is determined by:

|(m)|MIN = |m1|Ue1|MAX

" (Jml + (AwIatm)MIN " (Aml)

X (cos2 0©)

■ y»?? + (Am2tm)MIN (sin2 00)MAX)

X (1 -|Ue1|MA^) |>

where (cos2 0©)MIN and (sin2 0©)MAX are the values corresponding to (tan2 0©)max, and |Ue1 |MAX is the maximal allowed value of | Ue112 for the (Am;;tm)MIN.

In the two other CP conserving cases of n21 = n31 = ±1, the lower bound on |(m>| depends weakly on the allowed values of 0© and reads |(m>| > 0.03 eV.

If the neutrino mass spectrum is of the inverted hierarchy type, a sufficiently precise determination of Am^lm, 0© and |Ue1|2 (or a better upper limit on |Ue1|2), combined with a measurement of |(m>| in the current or future (Pfi)ov-decay experiments, could allow one to get information on the difference of the Majorana CP-violating phases (a31 - a21) [22]. The value of sin2(a31 - a21)/2 is related to the experimentally measurable quantities as follows [10, 22]:

2 «31 - «21

— 1 -

|(m>|2

(mj + Am2tm)(1 - |Ue1|2)V sin220©

|(m>|2

Am2tm(1 -|Ue1|2)Vsin220©

(m1 < 0.02 eV). The constraints on sin2(a31 - a21)/2 one could derive on the basis of Eq. (13) are illus-trated6 in Fig. 11 of Ref. [10]. Obtaining an experimental upper limit on |(m>| of the order of 0.03 eV would permit, in particular, to get a lower bound on the value of sin2(a31 - a21)/2 and possibly exclude the CP conserving case corresponding to a31 - a21 = 0 (i.e., n21 = n31 = ±1).

Case B. Spectrum with partial inverted hierarchy (0.02 eV < m1 < 0.2 eV).

The discussion and conclusions in the case of the spectrum with partial inverted hierarchy are identical to those in the same case for the neutrino mass spectrum with normal hierarchy given in Subsection 2.1, Case B, except for the maximal and minimal values of |(m>|, |(m>|MAX and |(m>|MIN, which for a fixed m1 are determined by:

|(m>|MAX — m1 |Ue1 |MiN

ym2 + (Am2m)MAX(l-|C7ei|

6 Note that the CP-violating phase «21 is not constrained in the case under discussion. Even if it is found that «31 - «21 = 0, , «21 can be a source of CP-violation in AL = 2 processes other than (PP)0 v-decay.

| {m ) 1min ^ | m 11 t/el ImAX - 7 '"l + (A'"atm)min

x (cos20©)min(1 — |^e1 |Ma^)i' (15)

|^e1 |MiN (Ue1 |Max) inEq. (14) (inEq. (15))beingthe minimal (maximal) allowed value of |Ue112 given the maximal (minimal) value (Am^MAX ((Am^MiN).

For any m1 > 0.02 eV, the lower bound on |(m}| reads: |(m}| > 0.01 eV. Using the best fit values of the neutrino oscillation parameters, obtained in [1,41], one finds: |(m}| > 0.03 eV.

2.3. Quasi-degenerate mass spectrum (m1 > 0.2 eV, m1 — m2 — m3 — m ye )

The new element in the predictions for |(m}| in the case of quasi-degenerate neutrino mass spectrum, m1 > 0.2 eV, is the existence of a lower bound on the possible values of |(m}| (Fig. 1). The lower limit on |(m}| is reached in the case of CP-conservation and n21 = n31 = —1. One finds a significant lower limit, |(m}| > 0.01 eV, if

(cos20©)min

> max(0.05' 1.5|^e3|MAx/(1 — Ue3|MAx)). (16)

More specifically, using the best fit value, and the 90% C.L. and the 99.73% C.L. allowed values, of cos20© from [1], we obtain, respectively: |(m}| > 0.10 eV, |(m}| > 0.06 eVand

|(m}| > 0.035 eV

(Figs. 1 and 3). These values of |(m}| are in the range of sensitivity of the current and future (PP)0v-decay experiments.

The upper bound on |(m}|, which corresponds to CP-conservation and n21 = n31 = +1 ( v1,2,3 possessing the same CP-parities), can be found for a given m1 by using Eq. (6). For the allowed values of m1 > 0.2 eV (which is limited from above by the 3H P-decay data [37,38], m1,2,3 — mye), |(«}|max is limited by the upper bounds obtained in the (PP)0v-decay experiments: |(m}| < 0.35 eV [29] and |(m}| < (0.33 — 1.35) eV [30].

in the case of CP conservation and n21 = ±n31 = + 1, |(m}| is constrained to lie in the interval [10] mte(1 — 2|^e3|MAX) < |(m}| < mye. An upper limit on |(m}| would lead to an upper limit on mye which is more stringent than the one obtained in the present 3H

j-decay experiments: for |(m}| < 0.35 (1.05) eV we have mVe < 0.41 (1.23) eV. Furthermore, the upper limit |(m}| < 0.2 eV would permit to exclude the CP-parity pattern n21 =±n31 =+1 for the quasi-degenerate neutrino mass spectrum.

If the CP-symmetry holds and n21 = ±n31 = -1, there are both an upper and a lower limits on |(m}|, mye ((cos20©)MIN(1 - lUe3 |Min) + Ue3lMlN < |(m}| < mye((cos20©)MAx(1 - |Ue3lMAX) + |Ue3lMAX). Using Eq. (2) and the results on |Ue3|2 fromRef. [41], one finds 0.26 mte < |(m}| < 0.67 mte. Given the allowed values of cos20©, Eq. (2), the observation of the (jj)0 v -decay in the present and/or future (jj)0v-decay experiments, combined with a sufficiently stringent upper bound on m ye ~ m12 3 from the tritium beta-decay experiments, m ve < |(m}|exp/ ((cos 20©)max( 1 -l Ue3 lMAX) +1 Ue3 lMAX), would allow one, in particular, to exclude the case of CP-conservation with n21 = ±n31 = -1 (Fig. 2).

For values of |(m}|, which are in the range of sensitivity of the future (jj)0v-decay experiments, there exists a "just-CP-violation" region. This is il-

Fig. 2. The dependence of |(m}|/m] on cos20© for the quasi-degenerate neutrino mass spectrum (m\ > 0.2 eV, m\ — m2 — m3 — mye). if CP-invariance holds, the values of |(m}|/m] lie: (i) for n21 = V3\ = 1—on the line |(m}|/m] = 1, (ii) for n21 = — n31 = 1—in the region between the thick horizontal solid and dash-dotted lines (in light grey and medium grey colors), (iii) for n21 = — V31 = — 1—in the light grey polygon with long-dashed and long-dashed-double-dotted line contours and (iv) for n21 = V31 = — 1—in the medium grey polygon with the short-dashed and long-dashed-double-dotted line contours. The "just-CP-violation" region is denoted by dark-grey color. The values of cos 20© between the doubly thick solid lines correspond to the lower and upper limits of the LMA solution regions found in Ref.[1] at 99.73% C.L.

lustrated in Fig. 2, where we show |(m)|/m1 for the case of quasi-degenerate neutrino mass spectrum, m1 > 0.2 eV, m1 ~ m2 ~ m3 ~ mye, as a function of cos 20©. The "just-CP-violation" interval of values of |(m)|/m1 is determined by

(cos20©)max(1 - | ue3 |Max |(m>|

< 1 - 2|Ue3|

Taking into account Eq. (2) and the existing limits on |Ue3|2, this gives 0.67 < |(m>|/m„e < 0.85. Information about the masses m12 3 = m ye can be obtained in the KATRIN experiment [27].

A rather precise determination of |(m>|, m1 = m ve, 0© and |Ue3|2 would imply an interdependent constraint on the two CP-violating phases a21 and a31 [10] (see Fig. 16 in [10]). For m1 = mye > 0.2 eV, the CP-violating phase a21 could be tightly constrained if | Ue312 is sufficiently small and the term in |(m>| containing it can be neglected, as is suggested by the current limits on | Ue312 :

2 aJ±_ 2

— 1 -

I {m)Y ?

sin2 20©

The term which depends on the CP-violating phase a31 in the expression for |(m)|, is suppressed by the factor |Ue3|2. Therefore, the constraint one could possibly obtain on cos a31 is trivial (Fig. 16 in [10]), unless |Ue312 - O(sin20©).

3. The effective majorana mass and the determination of the neutrino mass spectrum

The existence of a lower bound on |(m>| in the cases of inverted mass hierarchy (Am© = Am22) and quasi-degenerate neutrino mass spectrum, Eqs. (9) and (17), implies that the future v-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, v-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 v-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 (m1 > 0.20 eV) and that there are both a lower and an upper limit on m1, (m 1 )min m1 < (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 x 10-2 eV < |(m)|exp < 0.20 eV, one could conclude that either (i) Am© = Am21 and the spectrum is of the quasi-degenerate type (m1 > 0.20 eV) or with partial hierarchy (0.02 eV < m1 < 0.2 eV), with 8.4 x 10-2 eV < m1 < 1.2 eV, where the maximal and minimal values of m1 are determined as in the Case 1; or that (ii) Am© = Am^2 and the spectrum is quasi-degenerate (m1 > 0.20 eV) or with partial inverted hierarchy (0.02 eV < m1 < 0.2 eV), with (m 1)min = 2.0 x 10-2 eV and (m1)max = 1.2 eV, where (m 1)max and (m 1)min are given by the equalities |(m)|MIN = | (m)|exp and |(m)|MAX = |(m)|exp, and |(m)|MIN and |(m)|MAX are determined by Eqs. (15) and (14);

3. A measured value of |(m)| satisfying 8.5 x 10-3 eV < |(m)|exp < 8.0 x 10-2 eV, would imply that (see Fig. 3) either (i) Am© = Am21 and the spectrum is of quasi-degenerate type (m1 > 0.20 eV), with (m 1)max < 0.48 eV, or with partial hierarchy (0.02 eV < m1 < 0.2 eV), or that (ii) Am© = Am\2 and the spectrum is quasi-degenerate (m1 > 0.20 eV), or with partial inverted hierarchy (0.02 eV < m1 < 0.2 eV), or with inverted hierarchy (m1 < 0.02 eV), with only a significant upper bound on m 1, (m 1 )min = 0, (m 1 )max < 0.48 eV, where (m1)max is determined by the equation |(m)|MN = |(m)|exp, with |(m)|MIN given by Eq. (15);

4. A measurement or an upper limit on |(m)|, |(m)| < 8.0 x 10-3 eV, would lead to the conclusion that the neutrino mass spectrum is of the normal mass hierarchy type, Am© = Am21, and that m1 is limited from above by m1 < (m 1)max ~ 5.8 x 10-2 eV, where (m 1)max is determined by the con-

Fig. 3. The dependence of |(m}| on m\ in the case of the LMA solution, for Am© = A«2j and Am© = Amand for the best fit values (upper panel) and the 90% C.L. allowed values (middle panel) of the neutrino oscillation parameters found in Refs. [1,41]. The lower panel is obtained by using the 99.73% C.L. allowed values of Am© and cos 20© from [1] and the 99% C.L. allowed values of Ama2tm and sin2 0 from [41] (the latter article does not include results at 99.73% C.L.). in the case of CP-conservation, the allowed values of |(m}| are constrained to lie: for (i) Am© = A«2j and the middle and lower panels (upper panel)—in the medium-grey and light-grey regions (a) between the two lower thick solid lines (on the lower thick solid line) if n21 = n31 = 1, (b) between the two long-dashed lines and the axes (on the long-dashed line) if n21 = — n31 = 1, (c) between the two thick dash-dotted lines and the axes (on the dash-dotted lines) if n21 = —n31 =— 1, (d) between the three thick short-dashed lines and the axes (on the short-dashed lines) if n21 = n31 = —1; and for (ii) Am© = Am^ and the middle and lower panels (upper panel)—in the light-grey regions (a) between the two upper thick solid lines (on the upper thick solid line) if n21 = V31 = ±1, (b) between the dotted and the doubly-thick short-dashed lines (on the dotted line) if n21 = — n31 = — 1, (c) between the dotted and the doubly-thick dash-dotted lines (on the dotted line) if n21 = —V31 = +1. in the case of CP-violation, the allowed regions for |(m}| cover all the grey regions. Values of |(m}| in the dark grey regions signal CP-violation.

dition |(m}|MiN = | (m}|exp, with |(«}|min given by Eq. (7). For the allowed values of the oscillation parameters (at a given confidence level, Fig. 3), an upper bound on |(m}|, |(m}| < 8 x 10—4 eV, would imply an upper limit on m1, m1 < 0.01 eV—Fig. 3, middle panel, and m1 < 0.025 eV—Fig. 3, lower panel. For the best fit values of Am2tm, Am© , 0© and 0, the bound |(m}| < 8 x 10—4 eV would lead to a rather narrow interval of possible values of m1, 1 x 10—3 eV < m1 < 4 x 10—3 eV (Fig. 3, upper panel).

Thus, 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 x 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.

Using the best fit values of Am©, cos 20© from [1] and of Am2tm and sin2 0 from [41], we have found that (Fig. 3, upper panel): (i) |(m}| < 2.0 x 10—3 eV in the case of neutrino mass spectrum with normal hierarchy, (ii) 2.8 x 10—2 eV < |(m}| < 5.6 x 10—2 eV if the spectrum is with inverted hierarchy, and (iii) |(m}| > 0.10 eV for the quasi-degenerate mass spectrum. Therefore, if Am^, Am© and cos 20© will be determined with a high precision (~ (10 — 15)% uncertainty) using the data from the MiNOS, KamLAND and the solar neutrino experiments and their best fit values will not change substantially with respect to those used in the present analysis,7 a measurement of |(m}| > 0.03 eV would rule out a hierarchical neutrino mass spectrum (m1 ^ m2 ^ m3) even if there exists a factor of ~ 6 (or smaller) uncertainty in the value of |(m}| due to a poor knowledge of the corresponding nuclear matrix element(s). An experimental upper limit of |(m}| < 0.01 eV suffering from the same factor of ~ 6 (or smaller) uncertainty would rule out the quasi-degenerate mass spectrum (m1 = m2 = m3, m1 > 0.2 eV), while if the uncertainty under discussion is only by a factor which is not bigger than ~ 3.0, the spectrum with inverted

7 The conclusions that follow practically do not depend on sin2 0 < 0.05.

hierarchy (m1 <m2 — m3, m1 < 0.02 eV) would be strongly disfavored (if not ruled out).

If the minimal value of cos 20© 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 (Am© = Am21, m1 < 0.02 eV) might turn out to be larger than the lower bound on |(m)| in the case 4. Conclusions of spectrum with inverted mass hierarchy (Am© = Am^2, m1 ^ 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 m1 ^ 0.02 eV. The minimal value of cos 20© for which the two regions do not overlap is determined by the condition:

(cos20©)MIN

y(Am|)MAX + 2 y (Afflatm)MAX (sin2 0)max

2 y (Am2tm)MiN + y (A»J©)MAX 0i (Am|)MAX

\4(Am2tm)MiN

where we have neglected terms of order (sin2 0)mMAX. For the values of the neutrino oscillation parameters used in the present analysis this "border" value turns out to be cos 20© - 0.25.

Let us note that [11] if the (PP)0v-decay is not observed, a measured value of m Ve in H P -decay experiments, (mye)exp > 0.35 eV, which is larger than (m 1)max, (mVe)exp > (mOmax, 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 (PP)0v-decay has been observed and |(m)| measured, the inequality (mve )exp > (m1)max, would lead to the conclusion that there exist contribution(s) to the (PP)0v-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 m ye or an upper bound on m te, such that m te < (m 1)min, where (m 1)min is determined by the condition |(m)|MAX = |(m)|exp, with |(m)|MAX given by Eq. (14), would imply that [11] there are contributions to the (PP)0 -decay rate in addition to the ones due to the light Majorana neutrino

exchange (see, e.g., [45]), which enhance the (PP)0v-decay rate. This would signal the existence of new AL = 2 processes beyond those induced by the light Majorana neutrino exchange in the case of left-handed charged current weak interaction.

Assuming 3- v 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 tan2 0© < 1, for the predictions of the effective Majorana mass |(m)| in (PP)0v-decay. Neutrino mass spectra with normal mass hierarchy, with inverted hierarchy and of quasi-degenerate type are considered. For cos20© > 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 x 10-3 eV, respectively. If the neutrino mass spectrum is hierarchical (with inverted hierarchy), the upper limit holds |(m)| < 8.2 x 10-3 (8.0 x 10-2) eV Correspondingly, not only a measured value of |(m)| = 0, but even an experimental upper limit on |(m)| of the order of few x 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 m1. Further reduction of the LMA solution region due to data, e.g., from the experiments SNO, KamLAND and BOREXINO, leading, in particular, to an increase (a decreasing) of the current lower (upper) bound of cos 20© can strengthen further the above conclusions.

Using the best fit values of Am©, cos 20© from [1] and of Am2tm and sin2 0 from [41], we have found that (Fig. 3, upper panel): (i) |(m)| < 2.0 x 10-3 eV in the case of neutrino mass spectrum with normal hierarchy, (ii) 2.8 x 10-2 eV < |(m)| < 5.6 x 10-2 eVifthe spectrum is with inverted hierarchy, and (iii) |(m)| > 0.10 eV for the quasi-degenerate neutrino mass spectrum. Therefore, if Am^, Am© and cos20© will be determined with a high precision (— (10 - 15)% uncertainty) using the data from the MINOS, KamLAND and the solar neutrino experiments and their best fit values will not change substantially with respect to

those used in the present analysis, a measurement of |(m}| > 0.03 eV would rule out a hierarchical neutrino mass spectrum (m1 ^ m2 ^ m3) even if there exists a factor of ~ 6 uncertainty in the value of |(m}| due to a poor knowledge of the corresponding nuclear matrix element(s). An experimental upper limit of |(m}| < 0.01 eV suffering from the same factor of ~ 6 (or smaller) uncertainty would rule out the quasi-degenerate neutrino mass spectrum (m1 = m2 = m3, m1 > 0.2 eV), while if the uncertainty under discussion is by a factor not bigger than ~ 3.0, the spectrum with inverted hierarchy (m1 < m2 = m3, m1 < 0.02 eV) would be strongly disfavored (if not ruled out).

Finally, 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 3H j -decay experiment KATRIN,8 might allow to establish whether the CP-symmetry is violated in the lepton sector [48].

Note added After the work on the present study was essentially completed, few new global analyses of the solar neutrino data have appeared [49-52]. The results obtained in [49] do not differ substantially from those derived in [1]; in particular, the (99.73% C.L.) minimal allowed values of cos 20© in the LMA solution region found in [1] and in [49] practically coincide. The best fit values of Am© and cos 20© found in [1,49,51, 52] also practically coincide, with cos20©|BF lying in the interval (0.41-0.50) and Am©|BF — 5 x 10—5 eV2. The authors of [50] find a similar cos20©|BF, but a somewhat larger Am©|BF — 7.9 x 10—5 eV2. According to [50-52], the lower limit cos20© > 0.25 holds approximately at 94% C.L., 90% C.L. and 81% C.L., respectively. Larger maximal allowed values of Am© than that given in Eq. (1)—of the order of (4-5) x 10—4 eV2 (99.73% C.L.), have been obtained in the analyses performed in [50-52]. The authors of [1,49] used the full SNO data on the day and night event spectra [1] in their analyses, while the authors of [5052] did not use at all or used only part of these data.

8 information on the absolute values of neutrino masses in the range of interest might be obtained also from cosmological and astrophysical data, see, e.g., Refs. [46,47].

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