10 January 2002

Physics Letters B 524 (2002) 319-331

www. elsevier. com/locate/npe

Searching for the CP-violation associated with Majorana neutrinos

S. Pascolia b, S.T. Petcovab1, L. Wolfensteinc

a Scuola Internazionale Superiore di Studi Avanzati, I-34014 Trieste, Italy b Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34014 Trieste, Italy c Department of Physics, Carnegie-Mellon University, Pittsburgh, PA 34014, USA

Received 29 October 2001; accepted 26 November 2001 Editor: G.F. Giudice

Abstract

The effective Majorana mass which determines the rate of the neutrinoless double beta ((fifi)0v-) decay, |(m)|, is considered in the case of three-neutrino mixing and massive Majorana neutrinos. Assuming a rather precise determination of the parameters characterizing the neutrino oscillation solutions of the solar and atmospheric neutrino problems has been made, we discuss the information a measurement of |(m)| > (0.005-0.010) eV can provide on the value of the lightest neutrino mass and on the CP-violation in the lepton sector. The implications of combining a measurement of |(m)| with future measurement of the neutrino mass mVe in 3H fi-decay experiments for the possible determination of leptonic CP-violation are emphasized. © 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Experiments on atmospheric and solar neutrinos have produced convincing evidence of neutrino oscillations [1-7]. Ongoing and planned experiments, including long baseline ones, aim to determine the parameters for these oscillations. Assuming mixing of only three neutrinos, these are the magnitudes of the elements of the 3 x 3 unitary lepton mixing matrix—the Maki-Nakagawa-Sakata-Pontecorvo (MNSP) mixing matrix [8,9], a CP-violating phase, and the mass-squared difference parameters, say, Am^ and Am21. In principle, long baseline experiments at neutrino factories can distinguish the alternatives of a

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.

(i) hierarchical neutrino mass spectrum and of (ii) neutrino mass spectrum with inverted hierarchy [10,11]. If we number (without loss of generality) the neutrinos with definite mass in such a way that mi < m2 < m3, case (i) corresponds to Am3)1 = Am^tm » Am21 = Am2ol, while in case (ii) we have Am31 =

Am2tm » Am32 = ^o^ where Am2tm and Am2ol are the values of the neutrino mass-squared differences

inferred from the atmospheric and solar neutrino data. These experiments cannot determine, however, the actual neutrino masses, that is, the value of the lightest neutrino mass m1. Furthermore, assuming the massive neutrinos are Majorana particles, as we will in this paper, there are two more parameters, two Majorana CP-violating phases, associated with the MNSP mixing matrix [12] (see also [13]). The neutrino oscillation experiments cannot provide information on the Majorana CP-violating phases [12,14] as well. This Letter

0370-2693/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S0370-2693(01)01403-4

is concerned with the prospects and problems in determining or constraining these three parameters, assuming the others have been well determined. The mass mi is of interest, e.g., in cosmology since massive neutrinos at present are the only non-baryonic dark matter constituents known. Knowing the neutrino mass spectrum is fundamental for understanding the origin of the neutrino masses and mixing. The Majorana CP-violating phases indicate the relation between CP violation and lepton number violation; a major goal is to identify any possibility of detecting this CP-violation.

[21-24], aim at reaching a sensitivity to values of |(m)| ~ 0.01 eV, which are considerably smaller than the presently existing most stringent upper bounds (2) and (3).

The results of the 3H j-decay experiments studying the electron spectrum, which measure the electron (anti-)neutrino mass mVe, are of fundamental importance, in particular, for getting information about the neutrino mass spectrum. The Troitzk [25] and Mainz [26] experiments have provided stringent upper bounds on mv :

2. Neutrinoless double f-decay and 3H f-decay experiments

The process most sensitive to the existence of massive Majorana neutrinos (coupled to the electron in the weak charged lepton current) is the neutrinoless double beta ((№)0v-) decay (see, e.g., [15,16]). If the (Pfi)ov-decay is generated only by the left-handed (LH) charged current weak interaction through the exchange of virtual massive Majorana neutrinos, the probability amplitude of this process is proportional in the case of Majorana neutrinos having masses not exceeding a few MeV to the so-called "effective Majorana mass parameter", |(m)| (see, e.g., [17]). A large number of experiments are searching for (j3j3)0v -decay of different nuclei at present (a rather complete list is given in [16]). No indications that (PP)0v-decay takes place have been found so far. A stringent constraint on the value of the effective Majorana mass |(m)| was obtained in the 76Ge Heidelberg-Moscow experiment [18]:

< 0.35 eV, 90% C.L.

Taking into account a factor of 3 uncertainty associated with the calculation of the relevant nuclear matrix element (see, e.g., [15,16]) we get

\(m) \ < (0.35-1.05) eV, 90% C.L. (2)

The IGEX Collaboration has obtained [19]:

\ (m) \ < (0.33-1.35) eV, 90% C.L. (3)

A sensitivity to |(m)| ~ 0.10 eV is foreseen to be reached in the currently operating NEMO3 experiment [20], while the next generation of (PP)0v-decay experiments CUORE, EXO, GENIUS, MOON

mVe < 2.5 eV [25],

mve < 2.9 eV [26] (95% C.L.).

There are prospects to increase substantially the sensitivity of the 3H j -decay experiments and probe the region of values of mve down to mve ^ (0.3-0.4) eV [27]2 (the KATRIN project).

It is difficult to overestimate the importance of the indicated future (jj)0v-decay and3H j-decay experiments for the studies of the neutrino mixing: these are the only feasible experiments which can provide information on the neutrino mass spectrum and on the nature of massive neutrinos. Such information cannot be obtained [12,14], as we have indicated, in the experiments studying neutrino oscillations. The measurement of |(m)| > 0.02 eV and/or of mVe > 0.4 eV can give information, in particular, on the type of neutrino mass spectrum [30-32]. As we will discuss, it is only by combining a value of |(m)| and a value of, or a sufficiently stringent upper limit on, mve one might hope to detect Majorana CP-violation.

3. A brief summary of the formalism

As it is well known, the explanation of the atmospheric and solar neutrino data in terms of neutrino oscillations requires the existence of 3-neutrino mix-

2 Cosmological and astrophysical data provide information on the sum of the neutrino masses. The current upper bound reads (see, e.g., [28] and the references quoted therein): Jj mj < 5.5 eV. The future experiments MAP and PLANCK may be sensitive to [29] J2j mj = 0.4 eV.

ing in the weak charged lepton current:

vlL = Yh UliviL,

where vlL, l = e,^,r, are the three left-handed flavour neutrino fields, vjL is the left-handed field of the neutrino vj having a mass mj and U is the MNSP neutrino mixing matrix [8,9]. If vj are Majorana neutrinos with masses not exceeding few MeV, as will be assumed in what follows, the effective Majorana mass |(m)| of interest can be expressed in the form

\(m)\ = \m1|Ue1|2 + m2|Ue2|2 eia21 + ms|Ues|2 eia31\,

where a21 and a31 are the two Majorana CP-violating phases3 [12] (see also [13]). If CP-invariance holds, one has [33-35] a21 = kn, a31 = k'n, k, k' = 0, 1,2, ____In this case

V21 = e

^31 = e!"31 =±1,

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

The quantities relevant for Eq. (6) to be determined in neutrino oscillation experiments in the case of three-neutrino mixing are Am^lm, Am0, the mixing angle, 00, constrained by the solar neutrino data, and the mixing angle, 0, determined from the probability that the atmospheric neutrino oscillations involve ve. At present 0 is limited by the data from the CHOOZ [36] and Palo Verde [37] experiments, but in the future it should be determined, e.g., in long baseline neutrino oscillation experiments [11,38,39].

We can number (without loss of generality) the neutrino masses in such a way that m1 <m2 < m3. The neutrino masses can be expressed in terms

of the lightest neutrino mass m\ and, e.g., J An and Amj2 (see, e.g., [40-42]):

m2 = + Afflji,

m3 = ^m2 + AfBj! + Am32 .

(8) (9)

3 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.

For Am^tm inferred from the neutrino oscillation interpretation of the atmospheric neutrino data we have:

Am 2tm = Am31 = Am21 + Am22. (10)

In the case of normal neutrino mass hierarchy,

Am| = Am21, (11)

AmQ = Am^2

|C/ell=COS00yi-|C/e3|2,

|C7e2|=sin00yi-|C7e3|2, |C/e3|2 = sin20. (12) For the inverted neutrino mass hierarchy one has [43]:

(13) and

|C/e2l=COS00yi-|C/el|2,

|C/e3| =sin0OA/l - |C/el|2, |f/el|2 = sin20. (14)

In our analysis we will consider values of m1 varying from 0 to 2.9 eV—the upper limit from the 3H j3-decay data, Eq. (4). As m1 increases from 0, the three neutrino masses get closer in magnitude.4 For m1 > 0.2 eV, the neutrino masses are quasi-degenerate and the differences between the cases of hierarchical spectrum and the spectrum with inverted hierarchy essentially disappear.

Given the values of Am0, 00, Arn;;tm and of 0, the effective Majorana mass |(m)| depends, in general, on three parameters: the lightest neutrino mass m1 and on the two CP-violating phases a21 and a31. It depends also on the "discrete ambiguity" expressed in Eqs. (11)-(14) and related to the two possible types of neutrino mass spectrum—the hierarchical and that with inverted hierarchy. As is obvious from Eqs. (8)-(11) and (13), the knowledge of m1 would allow to determine the neutrino mass spectrum.

In the discussion which follows we use the best fit value for Am^lm, obtained in the analysis of the

4 For the values of Am~^tm obtained in [44], one has neutrino mass spectrum with hierarchy (with partial hierarchy) or with inverted hierarchy (partial inverted hierarchy) for [31] m1 ^ 0.02 eV (0.02 eV < m1 < 0.2 eV).

atmospheric neutrino data in [44],

^Dbfv = 2-5 X 10-3 eV2.

In what regards the parameters Am0 and 00, in most of the discussion we assume they lie in the region of the large mixing angle (LMA) MSW solution of the solar neutrino problem, although we comment briefly on how our conclusions would change in the cases of the LOW—quasi-vacuum oscillation (LOW-QVO) solution and of the small mixing angle (SMA) MSW solution. The most recent analyses [45-48] show that the current solar neutrino data, including the SNO results, favor the LMA MSW and the LOW-QVO solutions. To illustrate our discussion and conclusions we use the best fit value of Amg found in [46],

m^L^r = 4.5 X 10"

5 eV2,

(AmQ) BFV

three values of cos200 from the LMA solution region,5 and two values of the mixing angle 0, constrained by the CHOOZ and Palo Verde data.

4. Constraining or determining the lightest neutrino mass m1 and/or the Majorana CP-violating phases

If the (fifi)0v-decay of a given nucleus will be observed, it would be possible to determine the value of |(m)| from the measurement of the associated lifetime of the decay. This would require the knowledge of the nuclear matrix element of the process. At present there exist large uncertainties in the calculation of the (fifi)0v-decay nuclear matrix elements (see, e.g., [15,16]). This is reflected, in particular, in the factor of ~ (2-3) uncertainty in the upper limit on |(m)|, which is extracted from the experimental lower limits on the (fifi)0v-decay half life-time of 76Ge. The observation of a (fifi)0v-decay of one nucleus is likely to lead to the searches and eventually to observation of the decay of other nuclei. One can expect that such a progress, in particular, will help to solve completely the problem of the sufficiently precise calculation of

5 In our further discussion we assume cos 200 ^ 0, which is favored by the analyses of the solar neutrino data [45-48]. The modification of the relevant formulae and of the results in the case cos 200 < 0 is rather straightforward.

the nuclear matrix elements for the (fifi)0v-decay. Taking the optimistic point of view that the indicated problem will be resolved in one way or another, we will not discuss in what follows the possible effects of the currently existing uncertainties in the evaluation of the (fifi)0v-decay nuclear matrix elements on the results of our analysis.

In this section we consider the information that future (fifi)0v-decay and/or 3H fi-decay experiments can provide on the lightest neutrino mass m1 and on the CP-violation generated by the two Majorana CP-violating phases a21 and a31. The results are summarized in Fig. 1 (normal neutrino mass hierarchy) and in Fig. 2 (inverted hierarchy).

We shall discuss first the case of Am0 = Am21 (Eqs. (11), (12)).

4.1. Normal mass hierarchy: Amg = Am21

If Am0 = Am2i,

221, for any given solution of the solar neutrino problem LMA MSW, LOW-QVO, SMA MSW, as can be shown, |(m)| can lie anywhere between 0 and the present upper limits, given by Eqs. (2) and (3). This conclusion does not change even under the most favorable conditions for the determination of |(m)|, namely, even when Am2tm, Afflg, 0q and 0 are known with negligible uncertainty, as Fig. 1 indicates. The further conclusions that are illustrated in Fig. 1 are now summarized. We consider the case of the LMA MSW solution of the solar neutrino problem.

Case A. An experimental upper limit on |(m)|, |(m)| < |(m)|exp, will determine a maximal value of m 1, m 1 < (m 1 )max. The latter is fixed by the equality:

(m 1)max:

(mi cos2 0Q - Jm\ + Am% sin2 0Q) (l - \Ue312)

y»î2 + Am2tr

I^e3|2

Given m 1 = 0 and Am q , the sign of the last term in the left-hand side of the inequality depends on the value of cos 200: the positive sign corresponds to cos 20q < AmQ sin2 00/m2 (i.e., to cos200 = 0), while the negative sign is valid forcos200 > Am0 sin2 00/m\.

For the quasi-degenerate neutrino mass spectrum one has m1 » Am0, Am^tm, m1 = m2 = m3 = mVe,

Fig. 1. The dependence of |(m)| on m in the case of Am0 = Am^ (normal hierarchy of neutrino masses) for the LMA MSW solution of the solar neutrino problem. The three vertical left (right) panels correspond to |Ue312 = 0.01 (0.05), while the two upper, the two middle and the two lower panels are obtained respectively for cos200 = 0.10; 0.30; 0.54. The figures are obtained for the best fit values of Am^tm and Am0, given in Eqs. (15) and (16). In the case of CP-conservation the allowed values of |(m)| are constrained to lie on (i) the solid line if ^21 = n31 = 1, (ii) on the dashed line if n21 = — n31 = 1, (iii) on the dotted lines if n21 = n31 =—1, and (iv) on the dash-dotted lines if n21 = —n31 = —1. The region colored in grey (not including these lines) requires CP-violation ("just CP-violation" region).

Fig. 2. The same as Fig. 1 for the inverted hierarchy, Amgg = Am^. The three vertical left (right) panels correspond to |Ue312 = 0.005 (0.05), while the two upper, the two middle and the two lower panels are obtained respectively for cos 29q = 0.10; 0.30; 0.54. The figures are obtained for the best fit values of Am^j^ and Amg, given in Eqs. (15) and (16). If CP-invariance holds, the allowed values of |(m)| are constrained to lie on (i) the solid line if n21 = n31 = 1, (ii) on the dashed line if n21 = n31 =— 1, (iii) on the dotted line if n21 = —n31 =— 1, and (iv) on the dash-dotted lines if n21 = —^31 = 1 for |Ue312 = 0.05 and on (v) the solid line if n21 = n31 =±1, (vi) on the dotted line if n21 = —131 = ±1 for | Ue312 = 0.005. The region colored in grey (not including the indicated lines) requires CP-violation.

and up to corrections ~ Am0 sin2 00/(2m2) and Am2m|Ue3|2/(2rn2) one finds:

|(m)|exp

(m 1 )max =

| cos200(1 -|Ue3|2) |Ue3 |21 *

If | cos 200(1 — | Ue312) — | Ue3121 is sufficiently small, the upper limit on mve obtained in 3H j -decay experiments could yield a more stringent upper bound on m1 than the bound following from the limit on

|(m)|.

Case B. A measurement of |(m)| = (|(m)|)exp > 0.02 eV would imply that m 1 > 0.02 eV and thus a neutrino mass spectrum with partial hierarchy or of quasi-degenerate type [31]. The lightest neutrino mass will be constrained to lie in the interval, (m 1)m;n < m1 < (m 1)max, where (m1)max and (m1)m;n are determined, respectively, by Eq. (17) and by the equation:

(m 1)min:

(mi cos2 0© + ym\ + A m% sin2 00) (l - \Ue3\2)

+ ym2 + Am2m|C7e3|2=|<m>|exp. (19)

The limiting values of m1 correspond to the case of CP-conservation. For Am0 ^ m1, (i.e., for Am0 < 10~4 eV2), (mi )min to a good approximation is independent of 00, and for y Am2tm |£/e3 2 <<C ni i. which takes place in the case we consider as |Ue3|2 < 0.05, we have (m^min = (|(m)|)exp. For |Ue3|2 « cos20;, which is realized in the illustrative cases in Fig. 1 for |Ue3|2 < 0.01, practically all of the region between (m 1 )min and (m 1)max, (m 1)min < m1 < (m 1)max, corresponds to violation of the CP-symmetry. If |Ue3|2 is non-negligible with respect to cos200, e.g., if | Ue312 = (0.02-0.05) for the values of cos 20; used to derive the right panels in Fig. 1, one can have (m 1 )min < m1 < (m 1)max if CP-symmetry is violated, as well as in two specific cases of CP-conservation. One of these two CP-conserving values of m1, corresponding to n21 = —n31 = —1, can differ considerably from the two limiting values (see Fig. 1).

A measured value of |(m)| satisfying (|(m)|)exp < (|(m)|)max, where (|(m)|)max = «1 = mve in the case of a quasi-degenerate neutrino mass spectrum, and

(I (m)|)max = C/A^ sin2 0©)(1 -\Ue3+ x

|Ue3|2 if the spectrum is hierarchical (i.e., if m1 ^

m2 ^ m3), would imply that at least one of the two CP-violating phases is different from zero: a21 = 0 or/and a31 = 0; in the case of a hierarchical spectrum that would also imply a21 = a31.

In general, the knowledge of the value of |(m)| alone will not allow to distinguish the case of CP-conservation from that of CP-violation.

Case C. It might be possible to determine whether CP-violation due to the Majorana phases takes place in the lepton sector if both |(m)| and mve are measured. Since prospective measurements are limited to (mve)exp > 0.35 eV, the relevant neutrino mass spectrum is of quasi-degenerate type (see, e.g., [31]). In this case one has m1 > 0.35 eV, m1 = m2 = m3 = mve and

\ (m) \ — mve \ cos2 00(1 — |Ue3|2)

+ sin200(1 - |Ue3|2)eia21

+ |Ueз|2eiaзl\. (20)

If we can neglect | Ue312 in Eq. (20) (i.e., if cos 200 > |Ue3|2), a value of mve = m1, satisfying (m 1)min < mve < (m1 )max, where (m1)m{n and (m1)max are determined by Eqs. (19) and (17), would imply that the CP-symmetry does not hold in the lepton sector. In this case one would obtain correlated constraints on the CP-violating phases a21 and a31 [31,50]. This appears to be the only possibility for demonstrating CP-violation due to Majorana CP-violating phases in the case of Am0 = Am21 under discussion. In order to reach a definite conclusion concerning CP-violation due to the Majorana CP-violating phases, considerable accuracy in the measured values of |(m)| and mve is required. For example, if the oscillation experiments give the result cos200 < 0.3 and |(m)| = 0.3 eV, a value of mve between0.3 eVand 1.0 eV would demonstrate CP-violation. However, this requires better than 30% accuracy on both measurements. The accuracy requirements become less stringent if the upper limit on cos 200 is smaller.

If cos200 > |Ue3|2 but |Ue3|2 cannot be neglected in (20), there exist two CP-conserving values of mve in the interval (m 1)m;n < mve < (m 1)max. The one that can significantly differ from the extreme values of the interval corresponds to a specific case of CP-conservation—to n21 = —n31 = —1 (Fig. 1).

Case D. A measured value of mVe, (mVe)exp > 0.35 eV, satisfying (mve) exp > (m 1)max, where (m1)max is determined from the upper limit on |(m)|, Eq. (17), in the case the (PP)0v -decay is not observed, might imply that the massive neutrinos are Dirac particles. If (PP)0v-decay has been observed and |(m)| measured, the inequality (mvjexp > (m1)max, with (m 1)max determined from the upper limit or the value of Eq. (17), would lead to the conclusion that there exist contribu-tion(s) to the (PP)0v -decay rate other than due to the light Majorana neutrino exchange (see, e.g., [51] and the references quoted therein) that partially cancels the contribution from the Majorana neutrino exchange.

A measured value of |(m)|, (|(m)|)exp > 0.01 eV, and a measured value of mve or an upper bound on mve such that mve < (m 1)min, where (m 1)min is determined by Eq. (19), would imply that there are contributions to the (PP)0v-decay rate in addition to the ones due to the light Majorana neutrino exchange (see, e.g., [52]), which enhance the (PP)0v-decay rate and 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.

Case E. An actual measurement of |(m)| < 10-2 eV is unlikely, but it is illustrated in Fig. 1 to show the interpretation of such a result. There always remains an upper limit on m1. As |(m)| decreases, there appears a finite lower limit on m1 as well. Both the upper and the lower limits on m 1 approach asymptotic values which depend on the values of Am0, Am^lm, cos200 and |Ue3|2, but are independent of |(m)| (Fig. 1). For cos200 > 21Ue312, the maximum and minimum asymptotic values of m1 are determined by the expressions:

(-i]3i7Am2tm|C/e312 cos2 <90 ± [Am2m|Ue3|4cos400

— (Am2m |Ue314 — Am0 sin4 0q)

x cos200]1/^ cos-1 200. (21)

For the maximum asymptotic value we have (m 1)max

= m1+) with n31 = —1. If further

Am2tm|Ue3|4cos4 00 » | (Am2tm | Ue314 — Am0 sin4 00) cos 2001

(which requires | Ue312 = (0.02-0.05)), the expression for the asymptotic value of interest is given approximately by

9 COS2 0q

Amitm\Ue3\ -—

cos 200

and is typically in the range (m1)max = (0.7-3.0) x 10—2 eV (Fig. 1, right panels). If, however,

Am20 sin4 00

» max^ Am2m|Ue3|4,Am2m|Ue3|

4 cos4 9q cos200

one typically finds: (m1)max = (0.3-1.0) x 10—2 eV (Fig. 1, left panels).

For the minimum asymptotic value of m1 we have (m 1)m;n = m1+ with n31 = 1 if Am0 sin4 00 > Am^lm|Ueз|4, and (m1)min = m^] with n31 = —1 if Am0 sin4 00 < Am2tm|Ue3|4.

Over certain interval of values of |(m)|, which depends on |Ue3|2, on the values of the difference of the Majorana CP-violating phases, (a31 — a21), and on cos200, the lower limit on m1 goes to zero, as is shown in Fig. 1. This interval, |(m)|— < |(m)| < |(m)|+, is given by

./Â^ sin2 0q( 1 - |C7e31:2) ± ^Arn2m |C7e31

and has a width of 2y Am2tm | Ue312.

For a given |(m)| from the indicated interval we have 0 < m1 < (m 1)max, with (m 1)max determined by Eq. (17). Further, the limiting value of m1 = (m 1)max, as well as at least one and up to three internal values of m1 from the interval 0 < m1 < (m 1)max in the simplified case we are analyzing are CP-conserving (Fig. 1). The remaining values of m1 from the interval 0 < m1 < (m 1)max are CP-violating.

It should be noted also that one can have |(m)| = 0 for m1 = 0 in the case of CP-invariance if n21 = — n31 and the relation

sin2 0Q(1 - IC7e312) = ■JAffljta IC7e312

holds. Finally, there would seem to be no practical

possibility to determine the Majorana CP-violating phases.

The analysis of the Cases A-E for the LOW-QVO solution of the solar neutrino problem leads to the same qualitative conclusions as those obtained above for the LMA MSW solution. The conclusions differ, however, in the case of the SMA MSW solution and we will discuss them next briefly. An experimental upper limit on |(m)| (Case A) in the range |(m)|exp > 10—2 eV, would imply in the case of the SMA MSW solution,

m1 < I (m)

(1 — 2|Ue3|

2\— 1

For values of |(m)| > 10—2 eV, the maximum and minimum values of mi are extremely close: (mi)min = |(m)|exp. As a result, a measurement of |(m)| (Case B) practically determinesm1, m1 = |(m)|.However, there is no possibility to determine or constrain the Majorana CP-violating phases. Thus, no information about CP-violation generated by the Majorana phases can be obtained by the measurement of |(m)| (or of |(m)| and mve) [31]. If both |(m)| > 0.02 eV and mve > 0.35 eV would be measured (Case C), the relation m1 = (|(m)|)exp = (mve)exp should hold. The conclusions in the Cases D and E are qualitatively the same as for the LMA MSW solution.

4.2. Inverted mass hierarchy: Am© = Am^

Consider next the possibility of a neutrino mass spectrum with inverted hierarchy, which is illustrated in Fig. 2. A comparison of Figs. 1 and 2 reveals two major differences in the predictions for |(m)|: if Amg = Am^2, (i) even in the case of m1 ^ m2 = m3 (i.e., even if m1 ^ 0.02 eV), |(m)| can exceed ~ 10—2 eV and can reach the value of ~ 0.08 eV [30, 31], and (ii) a more precise determination of Am2tm, Amg, 0© and sin2 0 = | Ue112, can lead to a lower limit on the possible values of |(m)| [31]. For the LMA and the LOW-QVO solutions, min(|(m)|) will depend, in particular, on whether CP-invariance holds or not in the lepton sector, and if it holds—on the relative CP-parities of the massive Majorana neutrinos. All these possibilities are parametrized by the values of the two CP-violating phases, a21 and a31, entering into the expression for |(m)|. The existence of a significant lower limit on the possible values of |(m)| depends crucially in the cases of the LMA and LOW-QVO solutions on the minimal value of | cos 20© |, allowed

by the data: up to corrections ~ 5 x 10 3 eV we have for these two solutions (see, e.g., [30,31,43]):

LMA, LOW-QVO:

in(|<m>|)LMA= yAfflji,,, | cos20©|(l - \Uel\2)

± 0(- 5 x 10—3 eV)

The min(|(m)|) in Eq. (22) is reached in the case of CP-invariance and n21 = — n31 = ±1. If cos200 = 0 is allowed, values of |(m)| smallerthan = 5 x 10—3 eV and even |(m)| = 0 would be possible. If, however, it will be experimentally established that, e.g., | cos2001 > 0.20, we will have min(|(m)|) = 0.01 eV if Am2tm and |Ue112 lie within their 90% C.L. allowed regions found in [49]6 (i.e., |Ue112 < 0.055, Am2tm = (1.4-6.1) x 10—3 eV2). According to the latest analysis of the solar neutrino data (including the SNO results) performed in [45], for the LMA MSW solution one has cos200 > 0.30 (0.50) at 99% (95%) C.L.

For the SMA MSW solution one has in the case of Am20 = Am322 under discussion:

SMA MSW:

min(|<m)|)SMA = |<m>|

yAm2m(l-|C/ei|2)±0(~5x 10"3 eV)

where |Ue1|2 is limited by the CHOOZ data. Using the current 99% (90%) C.L. allowed values of Am2tm and |Ue1|2, derived in [49], one finds min(|(m)|) = 0.030 (0.050) eV

We shall discuss next briefly the implications of the results of future (jj)0v -decay and 3H j -decay experiments. We follow the same line of analysis we have used for neutrino mass spectrum with normal hierarchy. Consider the case of the LMA MSW solution of the solar neutrino problem.

Case A. An experimental upper limit on |(m)|, |(m)| < |(m)|exp, which is larger than the minimal

value of |(m)|, Km)^, predicted by taking into account all uncertainties in the values of the relevant input parameters (Am^, Am0, 00, etc.), |(m)|exp >

6 If, for instance, | cos 2001 > 0.30; 0.50, then under the same conditions one will have min(|(m)|) = 0.015; 0.025 eV.

|(m)|3min, will imply an upper limit on m1, m1 < (m 1)max. The latter is determined by the equality:

(m 1)max:

(Jm\ + Affljta - Am2Q cos2 6>0

- y»i2 + Amatm sin2 0O) (1 - |Ue112)

± m1|Ue1|2

The term m1 |Ue112 enters with a plus (minus) sign if the difference between the two terms in the big round brackets in the left-hand side of the equation is negative (positive). For the quasi-degenerate neutrino mass spectrum (m1 » Am0, Am^lm, m1 = m2 = m3 = mve), (m1)max is given by Eq. (18) in which |Ue312 is replaced by | Ue112. Correspondingly, the conclusion that if | cos200(1 — |Ue112) — |Ue112| is sufficiently small, the upper limit on m1 = mVe, obtained in 3H fi -decay, can be more stringent than the upper bound on m 1, implied by the limit on |(m)|, remains valid.

An experimental upper limit on |(m)|, which is smaller than the minimal possible value of |(m)|, |(m)|exp < |(m)|mhin, would imply that either (i) the neutrino mass spectrum is not of the inverted hierarchy type, or (ii) that there exist contributions to the (fifi)0v-decay rate other than due to the light Majorana neutrino exchange (see, e.g., [51]) that partially cancel the contribution from the Majorana neutrino exchange. The indicated result might also suggest that the massive neutrinos are Dirac particles.

Case B. A measurement of

(m)| = |(m)

> ^Arn2m (1 - |UQ\ I2) = (0.04-0.08) eV, where we have used the 90% C.L. allowed regions

and |Ue1|2 from [49], would imply the

existence of a finite interval of possible values of m1,

(m 1 )min < m1 < (m 1)max, with (m 1)max and (m 1)min

given respectively by Eq. (24) and by (m 1)min:

mi\Uei\2 + (Jm\ + Affljta - Acos26>0

+ y»î2 +Am2tm sin2 <90)

X (1 — |Ue1|^ =

In this case m1 > 0.04 eV and the neutrino mass spectrum is with partial inverted hierarchy or of quasi-degenerate type [31]. The limiting values of m1 correspond to CP-conservation. For Am0 ^ m2, i.e., for Am0 < 10—4 eV2, (m1)min is to a good approximation independent of 00 and we have:

y((mi)min)2 + Am2m(l - |C/ei|2) = (l<WI>l)exp-

For negligible |Ue112 (i.e., |Ue112 < 0.01 for the values of cos 200 in Fig. 2), essentially all of the interval between (m 1)min and (m 1W, (m 1)min < m1 < (m 1)max, corresponds to violation of the CP-symmetry. If the terms ~ |Ue112 cannot be neglected in Eqs. (24) and (25) (i.e., |Ue1|2 = (0.02-0.05) for the values of cos 200 in Fig. 2), there exists for a fixed |(m)|exp two CP-conserving values of m1 in the indicated interval, one of which differs noticeably from the limiting values (m 1)min and (m 1)max and corresponds to n21 = —n31 = 1 (Fig. 2).

In general, measuring the value of |(m)| alone will not allow to distinguish the case of CP-conservation from that of CP-violation. In principle, a measurement of mVe, or even an upper limit on mVe, smaller than (m 1)max, could be a signal of CP-violation. However, unless cos200 is very small, the required values of mVe are less than prospective measurements. For example, as is seen in Fig. 2, upper left panel, for cos 20q = 0.1 and |(m)| = 0.03 eV, one needs to find mve < 0.35 eV to demonstrate CP-violation.

If the measured value of |(m)| lies in the interval |(m)|— < |(m)| < |(m)|+, where

7Am2tm - Am% cos2 00

Am a2tm

00K1 — |Ue1|^,

we would have (m 1)m;n = 0. The values of m1 satisfying 0 < m1 < (m1)max, where (m1)max is determined by Eq. (24), correspond to violation of the CP-symmetry (Fig. 2).

Case C. As Fig. 2 indicates, the discussions and conclusions are identical to the discussions and conclusions in the same cases for the neutrino mass spectrum with normal hierarchy, except that instead of Eq. (20)

we have

||Uel|2 + cos2 0© (1 -|Uei|2)eia21 + sin20© (1 -^el^e^l,

(mi)max and (m i)min are determined by Eqs. (24) and (25), and |^e3|2 must be substituted by |^e1|2 in the relevant parts of the analysis.

Case D. If mVe is measured and (mVe )exp > 0.35 eV but the (jj)0v -decay is not observed or is observed and (mve)exp > (mi)max, where (mi)max is determined by Eq. (17), the same considerations and conclusions as in Case D for the normal hierarchy mass spectrum apply.

A measured value of |(m}|, (|(m)|)exp > 0.1 eV, in the case when the measured value of mVe or the upper bound on mve are such that mve < (mi)min, where (m i)m;n is determined by Eq. (25), would lead to the same conclusions as in Case D for the normal hierarchy mass spectrum.

Case E. It is possible to have a measured value of |(m}| < i0-2 eV in the case of the LMA MSW solution and neutrino mass spectrum with inverted hierarchy under discussion only if cos200 is rather small, cos 20q < 0.2. A measured value of |(m}| < |(m}|min would imply that either the neutrino mass spectrum is not of the inverted hierarchy type, or that there exist contributions to the (jj)0v -decay rate other than due to the light Majorana neutrino exchange that partially cancel the contribution from the Majorana neutrino exchange.

The above conclusions hold with minor modifications (essentially of the numerical values involved) for the LOW-QVO solution as well. In the case of the SMA MSW solution we have, as is well-known, sin2 00 « i and Am0 < i0-5 eV2 (see, e.g., [46]). Consequently, the analog of Eq. (i8) in Case A reads

miw = |m>|exp(i - 2|Uei|2)

The conclusions in the Cases B-D are qualitatively the same as in the case of neutrino mass spectrum with normal hierarchy. In particular, a measured value of

would essentially determine mi, mi = (|(m}|)exp. No information about CP-violation generated by the Majorana phases can be obtained by the measurement of |(m}|, or of |(m}| and mVe. If both |(m}| and mVe > 0.35 eV are measured, the relation mi = (|(m}|)exp = (mVe)exp should hold. If it is found that |(m}| =

yAm2tm(l - |£/ei|2), one would have 0 < m\ < (mi)max, where (mi)max is determined by Eq. (24) in which effectively sin2 00 = 0, cos2 00 = i, and Am2 = 0. Finally, a measured value of

= |<m>|+ = V Amatrn

yArn2m(l-|C7el|2)

= A.m2tm (l — \US\ |2),

would either indicate that there exist new additional contributions to the (PP)0v -decay rate, or that the SMA MSW solution is not the correct solution of the solar neutrino problem.

5. Conclusions

Neutrino oscillation experiments can never tell the actual neutrino masses (that is, the lowest mass mi), whether neutrinos are Majorana, and, if so, whether there are Majorana CP-violating phases associated with the AL = 2 neutrino mass. Neutrinoless doublebeta decay experiments can, in principle, answer the first two questions, but cannot by themselves provide information about CP-violation. Here we have analyzed how, given optimum information from neutrino oscillation and (PP)0v-decay experiments, a measurement of neutrino mass from3 H j3 -decay could, in principle, give evidence for Majorana CP-violating phases, even though no CP-violation would be directly observed. The indicated possibility requires quite accurate measurements and holds only for a limited range of parameters.

Note added

After the completion of the present Letter we became aware of the very recent work [52], where some of the topics we discuss are also considered but within a somewhat different approach.

Acknowledgements

The work of L.W. was carried out in part at the Aspen Center for Physics and was supported in part by the US Department of Energy under grant DE-FG02-91ER407682. S.T.P. would like to acknowledge with gratefulness the hospitality and support of the SLAC Theoretical Physics Group, where part of the work on the present study was done. The work of S.T.P. was supported in part by the EEC grant ERBFMRXCT960090. S.P. would like to thank the Theoretical Physics Group of the University of Sussex where part of this work was done.

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