Predictions for Neutrinoless Double-Beta Decay in the 3+1 Sterile Neutrino Scenario

We present accurate predictions of the effective Majorana mass $|m_{\beta\beta}|$ in neutrinoless double-$\beta$ decay in the standard case of $3\nu$ mixing and in the case of 3+1 neutrino mixing indicated by the reactor, Gallium and LSND anomalies. We have taken into account the uncertainties of the neutrino mixing parameters determined by oscillation experiments. It is shown that the predictions for $|m_{\beta\beta}|$ in the cases of $3\nu$ and 3+1 mixing are quite different, in agreement with previous discussions in the literature, and that future measurements of neutrinoless double-$\beta$ decay and of the effective light neutrino mass in $\beta$ decay or the total mass of the three lightest neutrinos in cosmological experiments may distinguish the $3\nu$ and 3+1 cases if the mass ordering is determined by oscillation experiments. We also present a relatively simple method to determine the minimum value of $|m_{\beta\beta}|$ in the general case of $N$-neutrino mixing.


I. INTRODUCTION
Neutrino flavor oscillations have been observed in many solar, reactor and accelerator experiments (see the recent reviews in Refs. [1,2]), in agreement with the currently standard paradigm of three-neutrino (3ν) mixing. The global fits of neutrino oscillation data in the framework of 3ν mixing [3][4][5] give us rather precise information on the values of the elements of the three mixing angles which parameterize the neutrino mixing matrix and on the values of the two independent neutrino squared-mass differences, the smaller "solar" squared-mass difference ∆m 2 SOL ≈ 7.5 × 10 −5 eV 2 and the larger "atmospheric" squared-mass difference ∆m 2 ATM ≈ 2.4 × 10 −3 eV 2 . However, the standard 3ν mixing paradigm has been challenged by indications in favor of short-baseline oscillations generated by a new larger squared-mass difference ∆m 2 SBL ∼ 1 eV 2 : the reactor antineutrino anomaly [6], which is a deficit of the rate ofν e observed in several short-baseline reactor neutrino experiments in comparison with that expected from the latest calculation of the reactor neutrino fluxes [7,8]; the Gallium neutrino anomaly [9][10][11][12][13], consisting in a short-baseline disappearance of ν e measured in the Gallium radioactive source experiments GALLEX [14] and SAGE [15]; the signal of short-baselineν µ →ν e oscillations observed in the LSND experiment [16,17]. The simplest extension of 3ν mixing which can describe these short-baseline oscillations taking into account other constraints is the 3+1 mixing scheme [18,19], in which there is an additional massive neutrino at the eV scale and the masses of the three standard neutrinos are much smaller. Since from the LEP measurement of the invisible width of the Z boson we know that there are only three active neutrinos (see Ref. [20]), in the flavor basis the additional massive neutrino corresponds to a sterile neutrino [21], which does not have standard weak interactions.
A fundamental questions that remains open is: are neutrinos Dirac or Majorana particles? This question cannot be investigated in neutrino oscillation experiments, where the total lepton number is conserved and there is no difference between Dirac neutrinos with a conserved total lepton number and truly neutral Majorana neutrinos, which do not have a conserved total lepton number. The most promising process which can reveal the Majorana nature of neutrinos is neutrinoless doublebeta decay, in which the total lepton number changes by two units (see the recent review in Ref. [22]).
From the present knowledge of the neutrino squaredmass differences and mixing angles it is possible to predict the possible range of values for the effective Majorana mass |m ββ | in neutrinoless double-beta decay as a function of the absolute scale of neutrino masses (see Ref. [22]), which is still unknown, up to the upper bound of about 2 eV at 95% C.L. established by the Mainz [23] and Troitsk [24] Tritium β-decay experiments.
The introduction of a sterile neutrino at the eV mass scale can change dramatically the prediction for the possible range of values for the effective Majorana mass in neutrinoless double-beta decay [13,[25][26][27][28][29][30][31][32][33]. In this paper we present accurate predictions for |m ββ | taking into account the results of the global fit of solar, atmospheric and long-baseline reactor and accelerator neutrino oscillation data presented in Ref. [3] and the results of an update [34,35] of the global fit of short-baseline neutrino oscillation data presented in Ref. [19]. We are particularly interested to determine accurately the conditions for which |m ββ | 0.01 eV, which may be probed experimentally in the near future (see Refs. [36][37][38][39][40][41]), and the conditions for which there can be a cancellation of the different mass contributions to |m ββ |, which leads to an unfortunate uncertainty for the possibility of ever observing neutrinoless double-beta decay (unless it is induced by new interactions and/or the exchange of new particles; see Refs. [42][43][44][45][46][47][48][49]).
The plan of this paper is as follows. In Section II we discuss the predictions for |m ββ | in the standard 3ν framework, taking into account the two possible normal and inverted mass orderings. In Section III we discuss how these predictions are modified in the 3+1 mixing framework. In Section IV we draw our conclusions.

II. THREE-NEUTRINO MIXING
In the standard three-neutrino (3ν) mixing framework, the effective Majorana mass in neutrinoless double-beta decay is given by where is the partial contribution of the massive Majorana neutrino ν k with mass m k . The elements U ek of the mixing matrix, which quantify the mixing of the electron neutrino with the three massive neutrinos, can have unknown complex phases, which generate the two complex phases α 2 and α 3 in Eq. (1). Since the values of these phases is completely unknown, all predictions of the value of |m ββ | must take into account all the possible range of these phases, from 0 to 2π. We use the results of the global fit of solar, atmospheric and long-baseline reactor and accelerator neutrino oscillation data presented in Ref. [3], which are given in terms of the mixing angles ϑ 12 , ϑ 13 that determine the absolute values of the first row of the mixing matrix U in the standard parameterization: The results for the neutrino squared-mass differences are expressed in terms of the solar and atmospheric squared mass differences, which are defined by where ∆m 2 jk = m 2 j − m 2 k . Given this assignment of the squared mass differences, it is currently unknown if the ordering of the neutrino masses is normal (NO), such that m 1 < m 2 < m 3 or inverted (IO), such that m 3 < m 1 < m 2 . We discuss these two cases separately in the following subsections.

A. Normal Ordering
In order to study the case of Normal Ordering (NO), we express the neutrino masses in terms of the lightest    Figure 1 shows the best-fit values and the 1σ, 2σ and 3σ allowed intervals of the three partial mass contributions to |m ββ | in Eq. (1) as functions of the lightest mass m 1 . We calculated the confidence intervals using the χ 2 function   with the partial χ 2 's extracted from Fig. 3 of Ref. [3], neglecting possible small correlations of the four mixing parameters 1 . For each value of m 1 we calculated the confidence intervals for one degree of freedom. Although this method is in principle better than the method based on the propagation of errors of the parameters used by many authors, in practice it leads to similar results, because the χ 2 's of ∆m 2 SOL , ∆m 2 ATM , sin 2 ϑ 12 and sin 2 ϑ 13 in Fig. 3 of Ref. [3] are very well approximated by quadratic functions, which correspond to Gaussian uncertainties for which the method of propagation of errors is valid.
From Fig. 1 one can see that for m 1 2 × 10 −3 eV the contribution of µ 2 is dominant and cannot be canceled by the smaller contributions of µ 1 and µ 3 for any values of the relative phase differences. In the interval m 1 ≈ (2 − 7) × 10 −3 eV cancellations are possible, mainly between µ 1 and µ 2 which have similar values, with the smaller contribution of µ 3 , which is about 2.3 times smaller than µ 2 . For m 1 7 × 10 −3 eV again there cannot be a complete cancellation, because the contribution of µ 1 is dominant.
The result for |m ββ | is shown in Fig. 3, where we have plotted separately the allowed bands for the four possible cases in which CP is conserved (α 2 , α 3 = 0, π) and the coefficients of the contributions are real. These are the extreme cases which determine the minimum and maximum values of |m ββ |. The areas between the CP-conserving curves correspond to values of |m ββ | which are allowed only in the case of CP violation [31,[50][51][52][53][54] (although there is no manifest CP violation [55]). Figure 3 shows also the 90% C.L. upper limit band for |m ββ | estimated in Ref. [22] from the results of the KamLAND-Zen experiment [56] taking into account the uncertainties of the nuclear matrix element calculations. The reliability of this upper limit is supported by the upper limits with the same order of magnitude following from the results of the Heidelberg-Moscow [57], IGEX [58], GERDA [59], NEMO-3 [60], CUORICINO [61], and EXO [62] experiments.
From Fig. 3 one can see that, in agreement with the discussion above, there can be a complete cancellation of the three partial mass contributions to |m ββ | for m 1 in the intervals given in Tab. I at different confidence levels.
The exact determination of the region in which there can be a complete cancellation of the partial mass contributions to |m ββ | in the general case of N -neutrino mixing can be done in the following relatively simple way 2 . For each value of the lightest mass m 1 let us denote by a the index of the largest mass contribution, i.e.
1 The only significant correlations discussed in Ref. [3] are those which involve the mixing angle ϑ 23 , which is irrelevant for neutrinoless double-β decay, and the Dirac phase, whose effect in neutrinoless double-β decay is masked by the two unknown Majorana phases. 2 Other ways are discussed in Refs. [30,63,64].
Then, we can consider the quantities The quantity m (+) ββ is always positive and represents the most favorable case, in which all the mass contribution add with the same phase, giving the maximum value of |m ββ | for any value of the unknown phases: The quantity m there is an intermediate value of the phases which gives |m ββ | = 0. This can be seen clearly by writing |m ββ | as with an unknown phase α and N −2 unknown phases ξ k , of which one can be fixed to zero. The only possibility to have |m ββ | = 0 can be realized for α = π if µ = µ a . This equality can occur only if where µ min and µ max are the minimum and maximum values of µ for any value of the unknown phases ξ k . Since we have always µ min ≤ µ a because of Eq. (12) and the inequality in Eq. (15) is the necessary and sufficient condition for having |m ββ | = 0 for some value of the unknown phases. Hence, the minimum value of |m ββ | is given by which can also be written as [63] 3 ββ as a function of m 1 for the best-fit values of the partial mass contributions, which is sufficient for the determination of the interval of m 1 for which there can be a complete cancellation of the partial mass contributions. One can see that in the case of 3ν mixing m (−) ββ is negative and it is possible that |m ββ | = 0 only in the interval of m 1 given in Tab. I.
Let us now consider the opposite possibility that |m ββ | is larger than about 0.01 eV, which is a value that may be explored experimentally in the near future. From Fig. 3 one can see that |m ββ | 0.01 eV can be realized only for m 1 0.008 eV. This range of m 1 corresponds to almost degenerate m 1 and m 2 , because ∆m 2 SOL ≈ 8.7× 10 −3 eV. Hence, it will be very difficult to measure |m ββ | if there is a normal hierarchy of neutrino masses (m 1 m 2 m 3 ) for any value of the unknown phases α 2 and α 3 in Eq. (1).
One can also see from Fig. 3 that |m ββ | 0.01 eV is realized independently of the values of the unknown phases α 2 and α 3 for m 1 0.04 eV, which is close to the region m 1 ∆m 2 ATM ≈ 0.05 eV in which all the three neutrino masses are quasidegenerate. Figure 3 gives a clear view of the possible values of |m ββ | depending on the scale of the lightest mass m 1 , but it is of little practical usefulness, because it will be very difficult to measure directly the value of m 1 . In practice, the investigation of the absolute values of neutrino masses is performed through the measurements of the effective electron neutrino mass in β-decay experiments [23,24] and through the measurement of the sum of the neutrino masses in cosmological experiments (see, for example, Ref. [65]). Hence, it is useful to calculate the allowed regions in the m β -|m ββ | and Σ-|m ββ | planes [66][67][68][69], which are shown in Figs. 4 and 5. In this case, the confidence intervals are calculated using the χ 2 function in Eq. (11) with two degrees of freedom. We have plotted separately the allowed bands for the four possible cases in which CP is conserved (α 2 , α 3 = 0, π), in order to show the regions in which CP is violated. Potentially the possibility of measuring values of |m ββ | and m β or Σ in these regions is very exciting for the discovery of CP violation generated by the Majorana phases 4 , but in practice such measurement is very difficult because it would require a precision which seems 4 The phases α 2 and α 3 in Eq. (1) depend on the values of one Dirac phase and two Majorana phases in the neutrino mixing matrix (see Ref. [70]). The Dirac phase can be measured in neutrino oscillation experiments and there is some indication on its value [3][4][5]. to be beyond what can be currently envisioned [71][72][73][74], especially taking into account the current uncertainty of the calculation of the nuclear matrix element in neutrinoless double-β decay (see Refs. [22,[75][76][77]). 5 show the same 90% C.L. upper limit band for |m ββ | as in Fig. 3. In addition, Fig. 4 shows 5 the 90% C.L. sensitivity on m β of the KATRIN experiment [78], which is scheduled to start data taking in 2016, and can be measured only in lepton-number violating processes such as neutrinoless double-β decay. In the future, if the Dirac phase will be measured, CP violation in neutrinoless double-β decay may provide information on the Majorana phases. 5 The most stringent current upper limits on m β obtained in the Mainz (m β < 2.3 eV at 95% C.L.) [23] and Troitsk (m β < 2.1 eV 95% C.L.) [24] experiments are out of the scale in Fig. 4.

B. Inverted Ordering
In the case of Inverted Ordering (IO), the expressions of the neutrino masses in terms of the lightest neutrino mass m min are: Figure 6 shows the best-fit values and the 1σ, 2σ and 3σ allowed intervals of the three partial mass contributions to |m ββ | in Eq. (1) as functions of the lightest mass m 3 . One can see that µ 1 is always dominant, because ϑ 12 is smaller than π/4 and |U e1 | > |U e2 | > |U e3 |. Therefore, in the case of Inverted Ordering there cannot be a complete cancellation of the three mass contributions to |m ββ | (Fig. 8 shows that m In the case of an Inverted Hierarchy (m 3 m 1 < m 2 ) we also have the upper bounds |m ββ | < 4.8 (1σ), 4.9 (2σ), 4.9 (3σ) × 10 −2 eV.
The next generations of neutrinoless double-beta decay experiments (see Refs. [36][37][38][39][40][41]) will try to explore the range of |m ββ | between the limits in Eqs. (27) and (28), testing the Majorana nature of neutrinos in the case of an Inverted Hierarchy. Figures 9 and 10 show the correlation between |m ββ | and the measurable quantities m β and Σ in the Inverted Ordering. Since in this case both m β and Σ have relatively large lower bounds (4.6×10 −2 eV and 9.4×10 −2 eV, respectively, at 3σ) there is a concrete possibility that near-future experiments will determine an allowed region in these plots if in nature there are only three neutrinos with Inverted Ordering.

III. 3+1 MIXING
In this section we consider the case of 3+1 mixing in which there is a new massive neutrino ν 4 at the eV scale which is mainly sterile. As explained in Section I, 3+1 mixing is motivated [18,19] by the explanation of the reactor, Gallium and LSND anomalies, which requires the existence of a new squared-mass difference ∆m 2 SBL ∼ 1 eV 2 . In this case, the effective Majorana mass in neutrinoless double-beta decay is given by with the partial mass contributions given by Eq. (2). The contribution of ν 4 enters with a totally unknown new phase α 4 that must be varied from 0 to 2π as α 2 and α 3 in order to calculate the predictions of the value of |m ββ |.
The absolute values of the relevant first row of the 4×4 mixing matrix U is given by the simple extension of the standard parameterization: |U e1 | = cos ϑ 14 cos ϑ 13 cos ϑ 12 , (30) |U e2 | = cos ϑ 14 cos ϑ 13 sin ϑ 12 , Since in the case of 3+1 neutrino mixing, as well as in any extension of the standard 3ν mixing, the ordering of the three standard massive neutrinos is not known, in the following two subsections we consider separately the two cases of Normal and Inverted Ordering of ν 1 , ν 2 , ν 3 . The values of their masses as functions of the lightest mass m min are given by Eqs.
neglecting the contributions of ∆m 2 SOL and ∆m 2 ATM , which are much smaller than ∆m 2 SBL . We calculated the confidence intervals using the χ 2 function with χ 2 3ν defined in Eq. (11) and χ 2 (∆m 2 SBL , sin 2 ϑ 14 ) obtained from an update [34,35] of the global fit of shortbaseline neutrino oscillation data presented in Ref. [19].
After Eq. (11) we noted that in the case of 3ν mixing our statistical method for the calculation of the uncertainty of |m ββ | and the usual method based on the propagation of errors lead to similar results, because the χ 2 's of the relevant 3ν mixing parameters are very well approximated by quadratic functions. On the other hand, the usual propagation of errors is inaccurate in the case of 3+1 mixing, because the marginal χ 2 's of ∆m 2 SBL and sin 2 ϑ 14 are not quadratic. Moreover, there are significant correlations between ∆m 2 SBL and sin 2 ϑ 14 (see Fig. 3 of Ref. [19]) which are taken into account in χ 2 (∆m 2 SBL , sin 2 ϑ 14 ).  A. Normal Ordering Figure 11 shows a comparison of the best-fit value and the 1σ, 2σ and 3σ allowed intervals of the partial contribution µ 4 as a function of the lightest mass m 1 with those of µ 1 , µ 2 , µ 3 , which are slightly different from those in Fig. 1 because of the contribution of ϑ 14 in Eqs. (30)- (32). One can see that in the 3+1 case it is not possible to get a total cancellation of |m ββ | in the interval m 1 ≈ (2 − 7) × 10 −3 eV as in the case of 3ν mixing (see Tab. I), because in this interval of m 1 the contribution of µ 4 is dominant. However, there is a range of higher values of m 1 between about 0.02 and 0.2 eV in which µ 4 and µ 1 have similar values, leading to a possible total cancellation. For m 1 0.2 eV a total cancellation is again not possible because the contribution of µ 1 is dominant. This behavior is confirmed by Fig. 2, where one can see that the value of m (−) ββ (see Eq. (13)) corresponding to the best-fit values of the partial mass contributions is negative for 0.035 m 1 0.1 eV. Figure 12 shows the allowed values of |m ββ | as a function of m 1 at different confidence levels. The corresponding intervals of m 1 for which there can be a total cancellation of |m ββ | are given in Tab. II.
In Fig. 12 we have plotted separately the allowed bands for the eight possible cases in which CP is conserved (α 2 , α 3 , α 4 = 0, π), that are the extreme cases which determine the minimum and maximum values of |m ββ |. The areas between the CP-conserving allowed bands correspond to values of |m ββ | which are allowed only in the case of CP violation. Unfortunately, these areas are visibly smaller than those in Fig. 3 in the case of 3ν mixing. This is due to the relatively large uncertainty of µ 4 , which can be seen clearly in Fig. 11. This uncertainty broadens the allowed bands corresponding to the CP-conserving cases, leaving little intermediate space. In any case, even if the uncertainty of µ 4 will be reduced in the future, there cannot be a region which is allowed only in the case of CP-violation for m 1 10 −2 eV, where µ 4 is dominant and the CP-violating phases are irrelevant. In fact, all the best-fit CP-conserving curves have approximately the same value for m 1 10 −2 eV.
As in the case of 3ν mixing, the plot in Fig. 12 of |m ββ | as a function of the lightest mass m 1 is useful because it gives a clear view of the different possibilities for the value of |m ββ |, but in practice it will be very difficult to determine experimentally an allowed region in this plot because of the difficulty of measuring the value of the lightest mass. Therefore, we calculated also the allowed regions in the m β -|m ββ | and Σ-|m ββ | planes shown in Figs. 13 and 14, with the quantities m β and Σ defined in Eqs. (22) and (23) as in the case of 3ν mixing in terms of the three standard neutrino masses only. The reason of this choice is that m β and Σ are measurable quantities also in the 3+1 scheme. Indeed, considering β decay, m β quantifies approximately the deviation of the endpoint of the electron spectrum due to neutrino masses smaller than the experimental energy resolution [79][80][81][82][83], whereas the effect of the larger mass m 4 is a kink of the Kurie function (see Ref. [70]). In cosmology, the effects of the larger mass m 4 can be disentangled from those of the smaller masses, because ν 4 becomes non-relativistic shortly after matter-radiation equality, much earlier than ν 1 , ν 2 , ν 3 . Moreover, it is possible that the contribution of m 4 to the energy density of the Universe is suppressed, for example by a large lepton asymmetry [84][85][86][87][88], or an enhanced background potential due to new interactions in the sterile sector [89][90][91][92][93][94][95], or a larger cosmic expansion rate at the time of sterile neutrino production [96], or MeV dark matter annihilation [97].
From Figs. 13 and 14 one can see that the intervals of m β and Σ for which there can be a complete cancellation of the three partial mass contributions to |m ββ | (given in Tab. II) are much larger than those in Figs. 4 and 5 for the standard 3ν mixing case, and |m ββ | 0.01 eV for any value of the unknown phases α 2 , α 3 , α 4 only for the relatively large values m β 0.25 eV and Σ 0.8 eV.
It is useful to compare the allowed regions m β -|m ββ | and Σ-|m ββ | planes obtained in the cases of 3ν and 3+1 mixing with Normal Ordering of the three lightest neutrinos. Figures 15 and 16 show this comparison for the 3σ allowed regions. One can see that, if the Normal Ordering will be established by oscillation experiments (see Refs. [1,2]), with measurements of m β and |m ββ | and/or Σ and |m ββ | it may be possible to distinguish 3ν mixing and 3+1 mixing if the measured values select a region which is allowed only in one of the two cases. It is interesting that there are two regions allowed only to 3+1 mixing: one with |m ββ | smaller than that in the case of 3ν mixing and one with |m ββ | larger than that in the case of 3ν mixing. At least a part of the second region is accessible to the next generation of neutrinoless double-beta decay experiments (see Refs. [36][37][38][39][40][41]). The only β-decay experiment under preparation with the aim of exploring the sub-eV region of m β is KATRIN [98], which will have a sensitivity of about 0.2 eV that is not sufficient to explore the upper part of the region in Fig. 15 allowed only in the case of 3+1 mixing. On the other hand, cosmological observation may be able to measure the sum of the three light neutrino masses down to the lower limit of about 5.6 × 10 −2 eV [99].

B. Inverted Ordering
The best-fit value and the 1σ, 2σ and 3σ allowed intervals of the partial mass contributions to |m ββ | in the case of 3+1 mixing with Inverted Ordering of the three lightest neutrinos are shown in Fig. 17. One can see that there  can be a total cancellation between the partial contribution µ 4 and the dominant µ 1 in the case of 3ν mixing (see Fig. 6) for m 3 0.1 eV. Indeed, Fig. 8 shows that the values of the partial mass contributions is negative for m 3 0.09 eV. Figure 18 depicts the allowed regions in in the m 3 -|m ββ | plane. Comparing Fig. 18 with Fig. 7 one can see that the predictions for |m ββ | are completely different in the 3ν and 3+1 cases if there is an Inverted Ordering of the three lightest neutrinos, in agreement with the discussions in Refs. [13,[25][26][27][28][29][30][31][32][33]. The ranges of values of m 3 for which there can be a complete cancellation of |m ββ | are given in Tab. III. Figures 19 and 20 show the allowed regions in the m β -|m ββ | and Σ-|m ββ | planes. Figures 21 and 22 show the comparison of the 3σ allowed regions in the same planes in the cases of 3ν and 3+1 mixing with Inverted Ordering of the three lightest neutrinos. If the Inverted Ordering will be established by oscillation experiments (see Refs. [1,2]), it will be possible to exclude 3ν mixing in favor of 3+1 by restricting m β and |m ββ | or Σ and |m ββ | in the corresponding large region at small |m ββ | allowed only in the 3+1 case.

IV. CONCLUSIONS
We have presented accurate calculations of the effective Majorana mass |m ββ | in neutrinoless double-β decay in the standard case of 3ν mixing and in the case of 3+1 neutrino mixing indicated by the reactor, Gallium and LSND anomalies (see Refs. [18,19]). We have taken into account the uncertainties of the standard 3ν mixing parameters obtained in the global fit of solar, atmospheric and long-baseline reactor and accelerator neutrino oscillation data presented in Ref. [3] and the uncertainties on the additional mixing parameters in the 3+1 case obtained from an update [34,35] of the global fit of short-baseline neutrino oscillation data presented in Ref. [19].
We have shown that the predictions for |m ββ | in the cases of 3ν and 3+1 mixing are quite different, in agreement with the previous discussions in Refs. [13,[25][26][27][28][29][30][31][32][33]. Our paper improves these discussions by taking into account the uncertainties of all the mixing parameters and presenting all the results at 1σ, 2σ and 3σ.
We have presented accurate comparisons of the allowed regions in the planes m β -|m ββ | and Σ-|m ββ | of measurable quantities, taking into account the two possibilities of Normal and Inverted Ordering of the three light lightest neutrinos. We have shown that future measurements of these quantities may distinguish the 3ν and 3+1 cases if the mass ordering is determined by oscillation experiments (see Refs. [1,2]).
We have also introduced in Section II A a relatively simple method to determine the minimum value of |m ββ | in the general case of N -neutrino mixing.