Constraints on Neutrino Mixing Parameters By Observation of Neutrinoless Double Beta Decay

Assuming positive observation of neutrinoless double bata decay together with the CHOOZ reactor bound, we derive constraints imposed on neutrino mixing parameters, the solar mixing angle \theta_{12} and the observable mass parameter_{\beta} in single beta decay experiments. We show that 0.05 eV<<m>_{\beta}<2 eV at the best fit parameters of the LMA MSW solar neutrino solution by requiring the range of the parameter_{\beta \beta} deduced from recently announced double beta decay events at 95 % CL with +/- 50 % uncertainty of nuclear matrix elements.


I. INTRODUCTION
While the good amount of evidences for the neutrino mass and the lepton flavor mixing have been accumulated [1][2][3], we still lack observational indications of how large is the absolute mass of the neutrinos. To our understanding to date it may show up in only a few places, the single [4] or the double beta decay [5] experiments as well as future cosmological observations [6]. Other potential possibilities for hints of absolute mass of neutrinos include Z-burst interpretation of highest energy cosmic rays [7].
Among these various experimental possibilities the neutrinoless double beta decay experiments seems to have relatively higher sensitivities. The most stringent bound on effective mass parameter m ββ (see eq. (2) for definition) is now m ββ < 0.35 eV, which comes from Heidelberg-Moscow group [8]. Furthermore, a wide variety of proposals for future facilities as well as ongoing attempt with greater sensitivies are actively discussed. They include NEMO [9], GENIUS [10], CUORE [11], MOON [12], XMASS [13], and EXO [14] projects.
These high-sensitivety experiments open the enlighting possibility of discovering neutrinoless double beta decay events, not just placing an upper bound on m ββ by its nonobservation.
Therefore, it is of great importance to completely understand what kind of informations can be extracted if such discovery is made.
We discuss in this paper in a generic three flavor mixing framework the constraints on neutrino masses and mixing by positive observation (as well as nonobservation) of neutrinoless double beta decay. The constraints imposed on neutrino mixing parameters by neutrinoless double beta decay have been discussed by many authors. They include the ones in early epoch [15], those in "modern era" in which real constraints on solar mixing parameters are started to be extracted [16,17], and the ones in "post-modern era" where the analyses are performed in a complehensive manner in the framework of generic three flavor neutrino mixing [18].
In a previous paper, we have made a final step in the series of analyses by proposing a way of expressing the constraints solely in terms of observables in single and double beta decay [19]. By using the framework, we discussed the possibility of placing lower bound on |U e3 | 2 assuming positive observation in direct mass measurement in single beta decay and an upper limit on m ββ in double beta decay experiments. It is a natural and logical step for us to examine next the alternative case of positive observation of neutrinoless double beta decay events.
Timely enough, an evidence of the neutrinoless double beta decay has just been reported by Klapdor-Kleingrothaus and collaborators [20]. Since the confidence levels of the claimed evidence are about 2 and 3 σ in Bayesian and Partcle Data Group methods, respectively, we must wait for confirmation by further data taking, or by other groups to conclude that neutrinos are Majorana particles. Nevertheless, we feel that the peak in the relevant kinematic region in their experiments is too prominent to be simply ignored.
As will become clear as we proceed it is essential to combine the constraint on |U e3 | 2 = s 2 13 imposed by the reactor experiments [21]. One of the key points in our subsequent discussion is that the double beta and the reactor bounds cooperate to produce a stringent constraint on absolute mass scale of neutrinos and the mixing angle θ 12 which is responsible for the solar neutrino problem.

II. CONSTRAINTS FROM NEUTRINOLESS DOUBLE BETA DECAY
Let us start by defining our notations. We use throughout this paper the standard notation of the MNS matrix [22]: (1) Using the notation, the observable in neutrinoless double beta decay experiments can be expressed as where m i (i=1, 2, 3) denote neutrino mass eigenvalues, U ei are the elements in the first low of the MNS matrix, and β and γ are the extra CP-violating phases characteristic to Majorana neutrinos [23], for which we use the convention of Ref. [16].
We define the neutrino mass-squared difference as ∆m 2 ij ≡ m 2 j − m 2 i in this paper. In the following analysis, we must distinguish the two different neutrino mass patterns, the normal (∆m 2 23 > 0) vs. inverted (∆m 2 23 < 0) mass hierarchies. We use the convention that m 3 is the largest (smallest) mass in the normal (inverted) mass hierarchy so that the angles θ 12 and θ 23 are always responsible for the solar and the atmospheric neutrino oscillations, respectively. We therefore often use the notations |∆m 2 23 | ≡ ∆m 2 atm and ∆m 2 12 ≡ ∆m 2 ⊙ to emphasize that they are experimentally measurable quantities. Because of the hierarchy of mass scales, ∆m 2 ⊙ /∆m 2 atm ≪ 1, ∆m 2 12 can be made always positive as far as θ 12 is taken in its full range [0, π/2] [24].
In order to derive constraint on mixing parameters we need the classification.
Case I: Case II: However, examination of the case II reveals that it does not lead to useful bounds. Therefore, we only discuss the case I in the rest of this paper.

A. Joint constraint by upper bounds on m ββ and reactor experiments
Since we try to utilize the experimental upper bound on m ββ , m ββ ≤ m max ββ , we derive the lower bound on m ββ . It can be obtained in the following way; Noticing that the right-hand-side (RHS) of (5) has a minimum at cos 2β = −1, we obtain the inequality We note that the RHS of (6) is a decreasing function of s 2 13 , and hence takes a minimum value for the maximum value of s 2 13 which is allowed by the limit placed by the reactor experiments [21]. We denote the maximum value as s 2 13 (CH) throughout this paper. Numerically, (While the precise value of the CHOOZ constraint actually depends upon the value of ∆m 2 atm [21], we do not elaborate this point in this paper.) Using the constraint we obtain It can be rewritten as the bound on cos 2θ 12 = cos 2θ ⊙ as where c 2 13 (CH) ≡ 1 − s 2 13 (CH).

B. Joint constraint by lower bounds on m ββ and reactor experiments
A positive observation of neutrinoless double beta decay will lead to the experimental lower bound on m ββ , m ββ ≥ m min ββ , which we use to place new bound on neutrino mixing parameters. Toward the goal we note, similarly as (5), that m ββ ≤ c 2 13 m 2 1 c 4 12 + m 2 2 s 4 12 + 2m 1 m 2 c 2 12 s 2 12 cos 2β + m 3 s 2 13 , whose RHS is maximized by taking cos 2β = +1 and s 2 13 = s 2 13 (CH) in the last term and c 2 13 = 1 in front of the square root. (A more refined treatment entails the same excluded region.) One can then derive an inequality similar to (7); By rewriting (10) we obtain the other upper bound on cos 2θ 12 ; To summarize, we have derived in this section the two kinds of upper bound on cos 2θ 12 (lower bound for cos 2θ 12 < 0) by using the assumed experimental constraint m min ββ ≤ m ββ ≤ m max ββ imposed by neutrinoless double beta decay experiments.

III. CONSTRAINTS EXPRESSED BY EXPERIMENTAL OBSERVABLES
We rewrite the bounds on solar mixing angle in terms of measurable quantities. Toward the goal we note that three neutrino masses m i (i=1,2,3) can be expressed by the two ∆m 2 and a remaining over-all scale m H . We assign m H to the mass of the highest-mass state, m 3 in the normal mass hierarchy (∆m 2 23 > 0), and m 2 in the inverted mass hierarchy (∆m 2 23 < 0), respectively. We have argued in our previous paper [19] that in a reasonable approximation one can regard m H as the observable m β in direct mass measurements in single beta decay experiments. * We have noticed that the identification is exact in two extreme cases of degenerate and hierarchical mass spectra. Then, the three mass eigenvalues of neutrinos can be represented solely by observables; ∆m 2 atm , ∆m 2 ⊙ , and m β in a good approximation. In each neutrino mass pattern, we have the expressions of three mass eigenvalues: Normal mass hierarchy (∆m 2 23 > 0); Inverted mass hierarchy (∆m 2 23 < 0); It is instructive to work out the form of constraint in the degenerate mass approximation, It is easy to show that in the degenerate mass limit the bound (8) becomes On the other hand, the bound (9) gives the inequality m β ≥ m min ββ in the degenerate mass limit. (To show this one may go back to (9), rather than using (11).) * While we used the linear formula derived by Farzan, Peres and Smirnov [25] m β = n j=1 m j |U ej | 2 n j=1 |U ej | 2 (12) with n being the dimension of the subspace of (approximately) degenerate mass neutrinos, this point remains valid even if we use an alternative quadratic expression [26].

IV. ANALYSIS OF THE DOUBLE BETA-REACTOR JOINT CONSTRAINTS
We analyze in this section the joint constraints derived in the foregoing sections and try to extract the implications. Let us start by examining the case of recent obsevation announced in [20] which gives rise to 0.11 eV ≤ m ββ ≤ 0.56 eV and 0.05 eV ≤ m ββ ≤ 0.84 eV if ±50 % uncertainty of the nuclear matrix elements are considered, each at 95 % CL. In Fig. 1 we present on m β -cos 2θ 12 plane the constraint (8)  uncertainty of the nuclear matrix elements, respectively. The regions surrounded by these lines are allowed. The slope of m β -dependence of (11) is so large that the dashed line looks like a vertical line, which implies the inequality m β ≥ m min ββ . We have derived it earlier in the degenerate mass limit, but it is generically true if ∆m 2 ⊙ is smaller than other relevant mass squared scales. Only the case of normal mass hierarchy (∆m 2 23 > 0) is shown in Fig. 1; the case of inverted hierarchy (∆m 2 23 < 0) gives an almost identical result except for a slight shift of the dashed line toward smaller m β by ≃ 10 %. Fig. 1 are the 95 % CL allowed regions of cos 2θ 12 for the large mixing angle (LMA) MSW solution (indicated by the shaded region between thin solid lines) and the low (LOW) MSW solution (indicated by the shaded region between thin dashed lines) of the solar neutrino problem [27]. There are several up to date global analyses of the solar neutrino data with similar results of allowed region of mixing parameters [28]. Therefore, we just quote the result obtaind by Krastev and Smirnov in the last reference in [28].  (11) and (8), respectively. Thus, we have a clear prediction for direct mass measurements using a single beta decay with observation of double beta decay events. With use of the numbers given in [20], for example, the observable m β must fall into the region 0.05 eV ≤ m β ≤ 2 eV (0.11 eV ≤ m β ≤ 1.3 eV) with (without) uncertainty of nuclear matrix elements at the best fit parameters of the LMA MSW solution. (The best fit value is tan 2 θ 12 = 0.35, or cos 2θ 12 = 0.48 in the last reference in [28].) Within the allowed region the cancellation between three mass eigenstates can take place for appropriate values of Majorana phases that allow (typically) a factor of 2-3 larger values of m β compared with the measured value of m ββ . At around maximal mixing (cos 2θ 12 ≃ 0), which is allowed by 95 % CL in the LOW solution, the cancellation is so efficient that much larger values of m β is allowed.

Superimposed in
Therefore, there are still ample room for hot dark matter mass neutrinos both in the LMA and the LOW solutions.
It should be emphasized that a finite value of m ββ does imply a lower bound on m β , as indicated in Fig. 1; a vanishingly small m β cannot be consistent with finite m ββ in double beta decay experiments. The sensitivity of the proposed KATRIN experiment is expected to extend to m β ≤ 0.3 eV [29]. On the other hand, the present 68 % CL limit quoted in [20] without nuclear element uncertainty is 0.28 eV ≤ m ββ ≤ 0.49 eV. Therefore, if the limit is further tightened by additional data taking in the future, both experiments can become inconsistent, giving an another opportunity of cross checking.
In Fig. 2, we demonstrate the approximate scaling relation obeyed by the constraint (8) by taking m β / m max ββ as the abscissa in a wide range of the m max ββ in degerarate mass region, 0.1 eV < ∼ m max ββ < ∼ 1 eV. The scaling is exact in the degenerate mass limit as shown in (15). The relation is useful to estimate the allowed region of m β for a given value of m max ββ which is not explicitly examined in this paper. The most stringent bound to date on m β is from the Mainz collaboration [30], m β ≤ 2.2 eV (95 % CL). (A similar bound m β ≤ 2.5 eV (95 % CL) is derived by the Troitsk group [31].) As we can see in Fig. 2 that the double beta bound with the CUORE sensitivity region m ββ < ∼ 0.3 eV [11] becomes stronger than the Mainz bound for the LMA MSW solution but not for the LOW MSW solution in their 95 % CL regions.
In Fig. 3, we present the similar allowed regions for hypothetical discovery of neutrinoless double beta decay events which would produce the experimental bounds 0.01 eV ≤ m ββ ≤ 0.03 eV. It is to examine how the constraint changes in some other situation of discovery with different mass parameter ranges. We note that even such deep region of sensitivity will be explored by several experiments [10,[12][13][14].
For this case, the bounds for the normal and the inverted mass hierarchies start to split as shown in Fig. 3. In the case of inverted mass hierarchy the lower bound on m β is replaced by the trivial bound m β ≥ ∆m 2 atm which is more restrictive. The latter is indicated by the dash-dotted line in Fig. 3b. It is also evident that the constraint from double beta decay is so stringent that the limit on m β is tightened to be m β < ∼ 0.2 eV for the LMA MSW solution.
In conclusion, we have demonstrated in this and the previous papers the mutual intimate relationship between observation and/or nonobservation in single beta decay and neutrinoless double beta decay experiments. We hope that it stimulates even richer future prospects not only in double beta decay experiments but also in direct mass measurements using single beta decay.
Finally, some remarks are in order: (1) If the LMA MSW solution is the case and if the KamLAND experiment [32] that just started data taking can measure cos 2θ 12 within 10 % level accuracy, the upper limit of m β can be accurately determined with ∼ 20 % accuracy.
(2) In this paper we have derived constraints imposed on neutrino mixing parameters by observation of neutrinoless double beta decay events and the CHOOZ reactor bound on |U e3 | 2 in the generic three flavor mixing framework of neutrinos. Suppose that neutrinoless double beta decay events are confirmed to exist and the single beta decay experiments detect neutrino mass outside the region of the bound derived in this paper. What does it mean?
It means either that double beta decay would be mediated by some mechanisms different from the usual one with Majorana neutrinos, or the three flavor mixing framework used in this paper is too tight.
Nore added: After submitting the first version of our paper to the electronic archive, we became aware of the works which address relatively model-independent implication of the results reported in [20], or critically comment on the interpretation of the events. References [33] and [34] are the incomplete lists of them.  (8) and (11), respectively; the allowed region is inside these three lines. The bold and the normal lines are for the ranges of mass parameter 0.05 eV ≤ m ββ ≤ 0.84 eV and 0.11 eV ≤ m ββ ≤ 0.56 eV corresponding, respectively, with and without ±50 % uncertainty of nuclear matrix elements. The mixing parameters are fixed as ∆m 2 atm = 3 × 10 −3 eV 2 and ∆m 2 ⊙ = 4.8 × 10 −5 eV 2 . Also shown as shaded region are the allowed regions of cos 2θ 12 at 95 % CL for the LMA (the region between thin solid lines) and LOW (the region between thin dashed lines) MSW solutions.