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Constraining 0vββ

A class of discrete flavor–symmetry–based models predicts constrained neutrino mass matrix schemes that lead to specific neutrino mass sum–rules (MSR). We show how these theories may constrain the absolute scale of neutrino mass, leading in most of the cases to a lower bound on the neutrinoless double beta decay effective amplitude.


Introduction
The discovery of neutrino oscillations implies non-vanishing neutrino masses and mixing providing one of the most solid indications for physics beyond the Standard Model. The fact that neutrinos have very tiny masses, in contrast to charged leptons and quarks, and that two of the mixing angles are large, is among the deepest theoretical puzzles in particle physics. Since neutrinos carry no electric charge, they are expected on general grounds to be Majorana particles [1], leading to the existence of lepton number violating processes. This intriguing possibility will be hopefully confirmed by the observation of neutrinoless double beta decay (0νββ) processes [2,3]. Indeed, upcoming 0νββ experiments are expected to improve the sensitivity by up to about one order of magnitude.
The observed pattern of lepton mixing angles suggests the existence of an underlying flavor symmetry of some sort, either an Abelian [4] symmetry or a non-Abelian one [5]. In the former case one typically obtains texture zeros for the mass matrices but is unable to predict mixing angles. In contrast, non-Abelian flavor symmetries are potentially more powerful, allowing also in principle for mixing angle predictions. As an example, several realizations of non-Abelian discrete flavor symmetry schemes lead to an effective neutrino mass matrix which corresponds to a numerical (parameter-free) prediction for lepton mixing. A popular example of such neutrino mass matrix M ν is the tri-bimaximal (TBM) [6] type, characterized by which depends only on three complex parameters x, y and z. In the mass eigentstate basis the three complex parameters x, y and z would correspond to three neutrino mass parameters plus two Majorana CP phases [1,7], as the Dirac phase disappears since θ 13 = 0. Many such schemes are characterized by a specific (complex) relation among the parameters x, y and z (see [8] for a complete list of models that predicts such relationships), leaving only two free complex parameters, further reducing the number of independent model parameters describing the lepton sector. In the mass basis these correspond to only two independent neutrino mass eigenvalues (the other follows from the existence of a neutrino mass sum rule), plus two Majorana CP phases (as mentioned, the Dirac phase is unphysical).

Mass relations
In this section we focus on a general sub-class of mass matrices leading to a numerical (parameter-free) prediction for the lepton mixing matrix, where the following types of mass relations hold: Here m ν i = m 0 i denote neutrino mass eigenvalues, up to a Majorana phase factor, while χ and ξ are free parameters which specify the model, taken to be positive without loss of generality. For the sake of completeness, we also consider a fourth mass relation, .
These kind of mass sum-rule can be obtained in the scenario of inverse seesaw mechanism [9,10,11], which arises when introducing a fermion singlet S with opposite lepton number with respect to the right-handed neutrinos, so that the effective light neutrino mass matrix is Assuming m D and µ to be proportional to the identity matrix and M ∼ M T BM , it is straightforward to show that we can obtain the mass sum-rule of type (D).

Lower bound for neutrinoless double-β decay
Let us first consider the amplitude for neutrinoless double-β decay within a flavor-generic model. One can plot the effective neutrino mass parameter |m ee | determining the 0νββ decay amplitude, as a function of the lightest neutrino mass. As is well-known, by varying the neutrino oscillation parameters ∆m 2 atm , ∆m 2 sol , θ 12 , θ 13 , θ 23 in their allowed ranges one obtains two types of relatively broad bands in the (|m ee |, m ν light ) plane corresponding to normal and inverse hierarchy spectra.
In this "generic" case there is a lower bound on the neutrinoless double-β decay effective mass parameter |m ee | only in the case of inverse mass hierarchy: due to the possibility of destructive interference among the light neutrinos from the effect of having opposite CP signs or due to the effect of majorana phases, no lower bound can be established for the case of normal hierarchy [12,13,14].
Let us now turn to the case where MSR relations like (A),(B),(C) and (D) hold. As discussed above these can be obtained in flavour models where the neutrino mass matrix only depends on two independent free parameters, so that the resulting mixing angles are fixed, like for example for the tri-bimaximal or bimaximal mixing patterns.
For definiteness here we focus on the case where the rotation in the neutrino sector is of tri-bimaximal form. Corrections from higher dimensional operators and/or from the charged lepton sector can yield θ 13 = 0.
Hence we retain the TBM approximation as a useful starting point to obtain our MSR relations. However, when evaluating a lower bound on the effective neutrino mass parameter |m ee | determining the neutrinoless double-β decay amplitude, we include explicitly the effects of non-vanishing θ 13 . We do this by taking the values at 3 σ determined in Ref. [15]. Such a lower bound can be obtained from the following procedure. We first consider that the neutrino masses are complex parameters, where the two Majorana phases are encoded in m ν 2 and m ν 3 , i.e.
As shown in Fig. 1, the neutrino mass sum-rule can then be interpreted geometrically as a triangle in the complex plane, whose area provides a measure of Majorana CP violation 1 . Each Figure 1. Neutrino mass sum-rule triangle in the complex plane.
of the above equations (6), (7) and (8) can be split into two independent equations for the real and imaginary parts. For simplicity let us start from the idealized case where the neutrino oscillation parameters ∆m 2 atm , ∆m 2 sol , θ 12 , θ 13 , θ 23 are perfectly well-measured quantities. In this case one can extract the two Majorana phases α and β as functions of the base of the triangle, which is determined by m 0 1 in case of normal hierarchy (NH) or by m 0 3 in case of inverted hierarchy (IH), as well as the parameters χ and ξ labeling the particular model under consideration.
These relations obtained can then be inserted into the general expression of |m ee |: |m ee | = c 2 12 c 2 13 m 1 + s 2 12 c 2 13 e iα m 2 + s 2 13 e iβ m 3 .
For each (χ, ξ) model this effective mass parameter depends on a single parameter, namely the length of the triangle base, which gives a measure of the abolute scale of neutrino mass. For instance, for case (A) this procedure gives: ξ cos α m 0 2 + χ cos β m 0 3 = m 0 1 , ξ sin α m 0 2 + χ sin β m 0 3 = 0 , 1 As the area shrinks to zero one obtains the CP-conserving limits corresponding to the four independent choices of CP sign [12,13].
The lower bound for the lightest neutrino mass can be obtained from our MSR, using the triangle inequality in the complex plane as suggested in [16], see Fig.(1) for a schematic view. We must first select the biggest side of the triangle; calling them x 1 , x 2 and x 3 , then the triangle inequality |x i | ≤ |x j | + |x k | must be fulfilled, where |x i | ≡ M ax(|x 1 |, |x 2 |, |x 3 |) and i = j = k 2 . In case (A) and assuming NH for the neutrino mass spectrum, we always have (χm 0 2 , ξm 0 3 ) > m 0 1 and the largest side of the triangle can be either χm 0 2 or ξm 0 3 ; so we must consider separately these two cases. After rewriting two masses in terms of the two squared mass differences, we can obtain a lower limit for the lightest neutrino mass from the triangle inequality |x i | ≤ |x j | + |x k |. For the other cases we follow the same procedure. The lower bound on the lightest neutrino mass obtained in this way is then used to estimate the lower bound for |m ee | from the general expression in eq. (9). Notice that, although we focus here on TBM schemes, some of the MSR considered in our analysis can also be derived using bimaximal mixing as a starting point.  Table 1. Minimal values for the effective 0νββ decay mass parameter |m ee |, in eV, see text for details.
The results obtained from the procedure discussed above are summarized in Tab.1, where we report the lower limits of |m ee | corresponding to different integer choices of (χ, ξ) between 1 and 3 and for each of the four MSR considered in eqs. (2)-(5), for both normal and inverted hierarchies. The cases already discussed in the literature are indicated by a †, see [8] for a complete list of references. The entries denoted with the symbol ( ) represent situations that do not satisfy the inequality for any value of the lightest neutrino mass m light . Cases marked by a (-) correspond to models which, as far as we can tell, have not been considered. Some comments are in order. First let us consider the effect of a possible non-zero effect of θ 13 . In Fig 2 we show the prediction for |m ee | as function of m light obtained from the MSR 1 √ m 2 + 2 √ m 3 = 1 √ m 1 . The red band corresponds to the normal hirarchy case and the green band the inverse hierarchy case, varying the values of θ 13 , θ 23 and θ 12 within their 3σ C.L. interval. By looking at Fig. 2 Figure 2. |m ee | as a function of the lightest neutrino mass corresponding to the mass sum-rule [11]. The gray and yellow bands are the allowed range for |m ee | for the normal hierarchy (NH) and inverse hierarchy (IH) respectively. The red (NH) and green(IH) are the predicted values for the MSR varying the values of θ 13 , θ 23 and θ 12 to 3σ C.L. range. one sees that, indeed, the 0νββ lower bound is sensitive to the value of θ 13 .
One also finds that, as expected on general gounds, all inverse hierarchy schemes corresponding to various choices of (χ, ξ) within sum-rules A-D have a lower bound for the parameter |m ee |. However, the numerical value obtained depends on the MSR scheme, signalling that not all values within the corresponding band in Fig. 2 are covered.
On the other hand, even though normal hierarchy models do not lead to a lower bound on the 0νββ amplitude due to the possibility of destructive interference amongst the light neutrinos, one finds that the possibility of full cancellation is precluded for all schemes in the table, except for the (2,1) case considered in Ref. [17] and the (3,2) scheme, both of which correspond to MSR of type (C). All other NH MSR schemes considered here imply a minimum value for the 0νββ decay amplitude 3 .
In particular, the maximal value we have found for the lower bound on |m ee | is |m ee | = 0.061 eV, obtained in correspondence with the set of values (χ, ξ) = (3, 3) for the case (A) in IH. Such a value for |m ee | lies within the sensitivity of upcoming experiments; hence it would be interesting, from the model building point of view, to find from first principles a flavorsymmetry-based model predicting such a mass relation.
Mass sum rules are classified in four different categories, some have already been considered in the literature. For each case, we have first extracted the allowed numerical values of |m ee |, for both mass orderings of the neutrino mass eigenstates and we have then given the behaviour 3 Of course some of the bounds are phenomenologically less interesting since they fall outside realistic sensitivities of coming experiments. of |m ee | as a function of the lightest neutrino mass. Although our MSR schemes were obtained within the TBM anzatz, we have computed the possible values of |m ee | considering all the neutrino parameters (including a non-vanishing θ 13 ) within their 3σ allowed ranges. In most MSR schemes one finds a lower bound for the 0νββ amplitude, even for NH spectra. We find that the most favourable case (large lower bound) corresponds to a sum-rule of type (A) obtained in correspondence of the set of values (χ, ξ) = (3, 3), |m ee | = 0.061 eV.