The $10^{-3}$ eV Frontier in Neutrinoless Double Beta Decay

The observation of neutrinoless double beta decay would allow to establish lepton number violation and the Majorana nature of neutrinos. The rate of this process in the case of 3-neutrino mixing is controlled by the neutrinoless double beta decay effective Majorana mass $|\langle m \rangle|$. For a neutrino mass spectrum with normal ordering, which is favoured over the spectrum with inverted ordering by recent global fits, $|\langle m \rangle|$ can be significantly suppressed. Taking into account updated data on the neutrino oscillation parameters, we investigate the conditions under which $|\langle m \rangle|$ in the case of spectrum with normal ordering exceeds $10^{-3}~(5\times 10^{-3})$ eV: $|\langle m \rangle|_\text{NO}>10^{-3}~(5\times 10^{-3})$ eV. We analyse first the generic case with unconstrained leptonic CP violation Majorana phases. We show, in particular, that if the sum of neutrino masses is found to satisfy $\Sigma>0.10$ eV, then $|\langle m \rangle|_\text{NO}>5\times 10^{-3}$ eV for any values of the Majorana phases. We consider also cases where the values for these phases are either CP conserving or are in line with predictive schemes combining flavour and generalised CP symmetries.

The observation of neutrinoless double beta decay would allow to establish lepton number violation and the Majorana nature of neutrinos. The rate of this process in the case of 3-neutrino mixing is controlled by the neutrinoless double beta decay effective Majorana mass | m |. For a neutrino mass spectrum with normal ordering, which is favoured over the spectrum with inverted ordering by recent global fits, | m | can be significantly suppressed. Taking into account updated data on the neutrino oscillation parameters, we investigate the conditions under which | m | in the case of spectrum with normal ordering exceeds 10 −3 (5 × 10 −3 ) eV: | m |NO > 10 −3 (5 × 10 −3 ) eV. We analyse first the generic case with unconstrained leptonic CP violation Majorana phases. We show, in particular, that if the sum of neutrino masses is found to satisfy Σ > 0.10 eV, then | m |NO > 5 × 10 −3 eV for any values of the Majorana phases. We consider also cases where the values for these phases are either CP conserving or are in line with predictive schemes combining flavour and generalised CP symmetries. Despite their elusiveness, neutrinos have granted us unique evidence for physics beyond the Standard Theory. Observations of flavour oscillations in experiments with solar, atmospheric, reactor, and accelerator neutrinos (see, e.g., [1]) imply both non-trivial mixing in the leptonic sector and above-meV masses for at least two of the light neutrinos. Neutrino oscillations, however, are blind to the absolute scale of neutrino masses and to the nature -Dirac or Majorana -of massive neutrinos [2,3].
In order to uncover the possible Majorana nature of these neutral fermions, searches for the leptonnumber violating process of neutrinoless double beta ((ββ) 0ν -)decay are underway (for recent reviews, see e.g. [4,5]). This decay corresponds to a transition between the isobars (A, Z) and (A, Z + 2), accompanied by the emission of two electrons but -unlike usual double beta decay -without the emission of two (anti)neutrinos. A potential observation of (ββ) 0ν -decay is feasible, in principle, whenever single beta decay is energetically forbidden, as is the case for certain eveneven nuclei. The searches for (ββ) 0ν -decay have a long history (see, e.g., [6]). The best lower limits on the halflives T 0ν 1/2 of this decay have been obtained for the isotopes of germanium-76, tellurium-130, and xenon-136: T 0ν 1/2 ( 76 Ge) > 8.0 × 10 25 yr reported by the GERDA-II collaboration [7], T 0ν 1/2 ( 130 Te) > 1.5 × 10 25 yr obtained from the combined results of the Cuoricino, CUORE-0, and CUORE experiments [8], and T 0ν 1/2 ( 136 Xe) > 1.07 × 10 26 yr reached by the KamLAND-Zen collaboration [9], with all limits given at the 90% CL.
In the standard scenario where the exchange of three Majorana neutrinos ν i (i = 1, 2, 3) with masses m i < 10 MeV provides the dominant contribution to the decay rate, the (ββ) 0ν -decay rate is proportional to the socalled effective Majorana mass | m |. Given the present knowledge of neutrino oscillation data, the effective Majorana mass is bounded from below in the case of a neutrino mass spectrum with inverted ordering (IO) [10], | m | IO > 1.4 × 10 −2 eV. Instead, in the case of a spectrum with normal ordering (NO), | m | can be exceptionally small: depending on the values of the lightest neutrino mass and of the CP violation (CPV) Majorana phases we can have | m | NO 10 −3 eV (see, e.g., [1]). Recent global analyses show a preference of the data for NO spectrum over IO spectrum at the 2σ CL [11,12]. In the latest analysis performed in [13] this preference is at 3.1σ CL.
In the present article we determine the conditions under which the effective Majorana mass in the case of 3neutrino mixing and NO neutrino mass spectrum exceeds the millielectronvolt value. We consider both the generic case, where the Majorana and Dirac CPV phases are unconstrained, as well as a set of cases in which the CPV phases take particular values, motivated by predictive schemes combining generalised CP and flavour symme-tries. Our study is a natural continuation and extension of the study performed in [14].

II. THE EFFECTIVE MAJORANA MASS
Taking the dominant contribution to the (ββ) 0ν -decay rate, Γ 0ν , to be due to the exchange of three Majorana neutrinos ν i (m i < 10 MeV; i = 1, 2, 3), one can write the inverse of the decay half-life, (T 0ν 1/2 ) −1 = Γ 0ν / ln 2, as T 0ν where G 0ν denotes the phase-space factor, which depends on the Q-value of the nuclear transition, and M 0ν is the nuclear matrix element (NME) of the decay. The former can be computed with relatively good accuracy whereas the latter remains the predominant source of uncertainty in the extraction of | m | from the data (see, e.g., [4,15]). The effective Majorana mass | m | is given by (see, e.g., [16]): with U being the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) leptonic mixing matrix. The first row of U is the one relevant for (ββ) 0ν -decay and reads, in the standard parametrization [1], Here, c ij ≡ cos θ ij and s ij ≡ sin θ ij , where θ ij ∈ [0, π/2] are the mixing angles, and δ and the α ij are the Dirac and Majorana CPV phases [2], respectively (δ, α ij ∈ [0, 2π]). The most stringent upper limit on the effective Majorana mass was reported by the KamLAND-Zen collaboration. Using the lower limit on the half-life of 136 Xe obtained by the collaboration and quoted in the Introduction, and taking into account the estimated uncertainties in the NMEs of the relevant process, the limit reads [9]: Neutrino oscillation data provides information on mass-squared differences, but not on individual neutrino masses. The mass-squared difference ∆m 2 responsible for solar ν e and very-long baseline reactorν e oscillations is much smaller than the mass-squared difference ∆m 2 A responsible for atmospheric and accelerator ν µ andν µ and long baseline reactorν e oscillations, ∆m 2 /|∆m 2 A | ∼ 1/30. At present the sign of ∆m 2 A cannot be determined from the existing data. The two possible signs of ∆m 2 A correspond to two types of neutrino mass spectrum: ∆m 2 A > 0 -spectrum with normal ordering (NO), ∆m 2 A < 0 -spectrum with inverted ordering (IO). In a widely used convention we are also going to employ, the first corresponds to the lightest neutrino being ν 1 , while the second corresponds to the lightest neutrino being ν 3 . Combined with the fact that in this convention ∆m 2 ≡ ∆m 2 21 > 0 we have: We also define m min ≡ m 1 (m 3 ) in the NO (IO) case. A NO or IO mass spectrum is additionally said to be normal hierarchical (NH) or inverted hierarchical (IH) if respectively m 1 m 2,3 or m 3 m 1,2 . In the converse limit of relatively large m min , m min ∼ > 0.1 eV, the spectrum is said to be quasidegenerate (QD) and m 1 m 2 m 3 . In this last case, the distinction between NO and IO spectra is blurred and ∆m 2 and |∆m 2 A | can usually be neglected with respect to m 2 min . In terms of the lightest neutrino mass, CPV phases, neutrino mixing angles, and neutrino mass-squared differences, the effective Majorana mass reads: where we have defined α 31 ≡ α 31 − 2δ. It proves useful to recast | m | NO and | m | IO given above in the form with m i > 0 (i = 1, 2, 3). It is then clear that the effective Majorana mass is the length of the vector sum of three vectors in the complex plane, whose relative orientations are given by the angles α 21 and α 31 . For the IO case, taking into account the 3σ ranges of ∆m 2 32 , ∆m 2 21 , sin 2 θ 12 , and sin 2 θ 13 summarised in Table I, one finds that there is a hierarchy between the lengths of the three vectors, m 3 < 0.1 m 2 and m 2 < 0.6 m 1 , which holds for all values of m min . In particular, m 3 = m min s 2 13 can be neglected with respect to the other terms since s 2 13 cos 2θ 12 . 1 The above implies that extremal values of | m | IO are obtained when the three vectors are aligned (α 21 = α 31 = 0, | m | IO is maximal) TABLE I. Ranges for the relevant oscillation parameters in the case of an IO neutrino spectrum, at the 3σ CL, taken from the global analysis of Ref. [13]. As in Table II, ∆m 2 A is obtained from the quantities defined in Ref. [13] using the best-fit value of ∆m 2  or when m 1 is anti-aligned with m 2,3 (α 21 = α 31 = π, | m | IO is minimal). It then follows that there is a lower bound on | m | IO for every value of m min [10]. This bound reads: , taking into account the 3σ ranges of cos 2θ 12 and sin 2 θ 13 . Measurements of the end-point electron spectrum in tritium beta decay experiments constrain the combination m β ≡ i |U ei | 2 m i . The most stringent upper bounds on m β , m β < 2.1 eV and m β < 2.3 eV, both at the 95% CL, are given by the Troitzk [17] and Mainz [18] collaborations, respectively. The KATRIN experiment [19] is planned to either improve this bound by an order of magnitude, or discover m β > 0.35 eV. Taking into account the 3σ ranges for the relevant mixing angles and mass-squared differences, the Troitzk bound constrains the lightest neutrino mass to be m min < 2.1 eV. Cosmological and astrophysical data constrain instead the sum Σ ≡ i m i . Depending on the likelihood function and data set used, the upper limit on Σ reported by the Planck collaboration [20] varies in the interval Σ < [0.34, 0.72] eV, 95% CL. Including data on baryon acoustic oscillations lowers this bound to Σ < 0.17 eV, 95% CL. Taking into account the 3σ ranges for the mass-squared differences, this last bound implies m min < 0.05 (0.04) eV in the NO (IO) case. One should note that the Planck collaboration analysis is based on the ΛCDM cosmological model. The quoted bounds may not apply in nonstandard cosmological scenarios (see, e.g., [21]).

III. THE CASE OF NORMAL ORDERING
We henceforth restrict our discussion to the effective Majorana mass | m | NO , for which there is no lower bound. In fact, unlike in the IO case, here the ordering of the lengths of the m i depends on the value of m min and cancellations in | m | NO are possible: one risks "falling" inside the "well of unobservability". We summarise in Table II the nσ (n = 1, 2, 3) ranges for the oscillation parameters relevant to (ββ) 0ν -decay in the NO case, obtained in the recent global analysis of Ref. [13]. Considering variations of oscillation parameters in the corresponding 3σ ranges, for m min ≤ 5 × 10 −2 eV there is an upper bound | m | NO ≤ 5.1 × 10 −2 eV (obtained for α 21 = α 31 = 0). In the limit of negligible m min , m 2 min |∆m 2 A |, one has | m | NO ∈ [0.9, 4.2]×10 −3 eV.
From inspection of Eqs. (5) and (7), the vector lengths explicitly read m 1 = m min c 2 12 c 2 13 , m 2 = ∆m 2 + m 2 min s 2 12 c 2 13 , and m 3 = ∆m 2 A + m 2 min s 2 13 . In Figure 1, these lengths are plotted as functions of m min for 3σ variations of oscillation parameters.
The requirement of having the effective Majorana mass above a reference value | m | 0 is geometrically equivalent to not being able to form a quadrilateral with sides m 1 , m 2 , m 3 , and | m | 0 . This happens whenever one of the lengths exceeds the sum of the other three. If however Thus, for values of m min and oscillation parameters for which m 2 > m 1 + m 3 + | m | 0 or m 1 > m 2 + m 3 + | m | 0 (see Figure 1) one is guaranteed to have | m | NO > | m | 0 independently of the choice of CPV phases α 21 and α 31 . There are instead values of m min for which the conditions m 2 < m 1 + m 3 + | m | 0 and m 1 < m 2 + m 3 + | m | 0 hold independently of the values of oscillation parameters within a given range. In such a case, values of α 21 and α 31 such that | m | NO < | m | 0 are sure to exist.
We summarise in Figure 2 the ranges of m min for which these different conditions apply (see caption). We vary oscillation parameters in their respective nσ (n = 1, 2, 3) intervals and focus on the millielectronvolt "threshold", | m | 0 = 10 −3 eV. We find that, for 3σ variations of the sin 2 θ ij and ∆m 2 ij , one is guaranteed to have | m | NO > 10 −3 eV if m min > 1.10 × 10 −2 eV. This corresponds to the lower bound Σ > 0.07 eV on the sum of neutrino masses. For 2σ variations, having m min < 2 × 10 −4 eV or m min > 9.9×10 −3 eV is enough to ensure | m | NO > 10 −3 eV.
If one takes instead the higher value | m | 0 = 5 × 10 −3 eV and allows the relevant oscillation parameters to vary  Table II). See text for details.  Table  II)  in their respective 3σ ranges, | m | NO > | m | 0 is guaranteed provided m min > 2.3 × 10 −2 eV, which corresponds to the lower bound Σ > 0.10 eV on the sum of neutrino masses. This lower bound on Σ practically coincides with min(Σ) in the case of IO spectrum. Thus, if Σ is found to satisfy Σ > 0.10 eV, that would imply that | m | exceeds 5 × 10 −3 eV, unless there exist additional contributions to the (ββ) 0ν -decay amplitude which cancel at least par- tially the contribution due to the 3 light neutrinos. If instead m min < 1.4 × 10 −2 eV, for all (3σ allowed) values of oscillation parameters there is a choice of α 21 and α 31 such that | m | NO < | m | 0 = 5 × 10 −3 eV. These results are shown graphically in Figure 3.
Let us briefly remark on the dependence of | m | NO on the CPV phases. For the present discussion, 3σ variations of oscillation parameters are considered. For all values of α 31 and > 0 there exist values of α 21 and m min such that | m | NO < , i.e. such that | m | NO is arbitrarily small. This is a consequence of the fact that, for any fixed oscillation parameters and α 31 , there is always a point m * min at which | m 1 (m * min ) + m 3 (m * min ) e iα 31 | = m 2 (m * min ). Instead, there are values of α 21 and > 0 for which, independently of α 31 and m min , one has | m | NO > , i.e. for which | m | NO cannot be arbitrarily small. This conclusion may be anticipated from the graphical results of Ref. [22], where the structure of the | m | NO "well" has been studied as a function of m min and α 21 . In fact, we find that for α 21 ∼ < 0.81π or α 21 ∼ > 1.19π, | m | NO cannot be zero at tree-level since | m 1 + m 2 e iα21 | > m 3 , strictly.
In Figure 4 we highlight the region of the (m min , α 21 ) plane in which | m | NO is guaranteed to satisfy | m | NO > 5 × 10 −3 eV, independently of α 31 and of variations of oscillation parameters inside their 3σ ranges.

IV. CP AND GENERALISED CP
Given the strong dependence of | m | on α 21 and α 31 , some principle which determines these phases is welcome. The requirement of CP invariance constrains the values of the CPV phases α 21 , α 31 , and δ to integer multiples of π [23][24][25], meaning the relevant CP-conserving val- Regions in the (mmin, α21) plane where different conditions on | m |NO apply. In the green (dark grey) region, | m |NO satisfies | m |NO > 5 × 10 −3 eV (| m |NO < 5 × 10 −3 eV) for all values of θij, ∆m 2 ij , and α 13 from the corresponding 3σ or defining intervals. In the red and grey regions, conditions analogous to those described in the caption of Figure 2 apply and are indicated. This figure is to be contrasted with Figure 3, where the dependence on α21 is not explicit.
ues are α 21 , α 31 = 0, π. Non-trivial predictions for the leptonic CPV phases may instead arise from the breaking of a discrete symmetry combined with a generalised CP (gCP) symmetry. We focus on schemes with large enough residual symmetry such that the PMNS matrix depends at most on one real parameter θ [26] and realisations thereof where the predictions for the CPV phases are unambiguous, i.e. independent of θ. For symmetry groups with less than 100 elements, aside from the aforementioned CP-conserving values, the non-trivial values α 21 , α 31 = π/2, 3π/2 are possible predictions [27][28][29][30][31][32][33].
From Figures 5 -8, one sees that for each value of m min there exist values of the effective Majorana mass which are incompatible with CP conservation. Some of these points may nonetheless be compatible with gCP-based predictive models. For IO, one sees there is substantial overlap between the bands with (α 21 , α 31 ) = (0, 0) and (0, π), between those of (π, 0) and (π, π), and between the four bands (π/2, k π/2), with k = 0, 1, 2, 3. In the case of NO, it is interesting to note that, for a fixed, gCP-compatible but not CP-conserving pair (α 21 , α 31 ), | m | NO is bounded from below at the 2σ CL, with the lower bound at or above the meV value, | m | NO ∼ > 10 −3 eV. We collect in Table III information on the lower bound on | m | NO for each pair of phases.

V. CONCLUSIONS
The observation of (ββ) 0ν -decay would allow to establish lepton number violation and the Majorana nature of neutrinos. In the standard scenario of three light neutrino exchange dominance, the rate of this process is controlled by the effective Majorana mass | m |. In the case of neutrino mass spectrum with inverted ordering (IO) the effective Majorana mass is bounded from below, | m | IO > 1.4 × 10 −2 eV, where this lower bound is ob- Blue and green bands correspond to (the indicated, with k = 0, 1) CP-conserving values of the phases (α21, α 31 ), for IO and NO neutrino mass spectra, respectively, while in red regions at least one of the phases takes a CP-violating value. Blue hatching is used to locate CP-conserving bands in the case of IO spectrum whenever IO and NO spectra regions overlap. The KamLAND-Zen bound of Eq. (4) is indicated. See also [1,14].
tained using the current 3σ allowed ranges of the relevant neutrino oscillation parameters -the solar and reactor neutrino mixing angles θ 12 and θ 13 , and the two neutrino mass-squared differences ∆m 2 21 and ∆m 2 23 . In the NO case, the effective Majorana mass | m | NO , under certain conditions, can be exceedingly small, | m | NO 10 −2 eV, suppressing the (ββ) 0ν -decay rate.
Currently taking data and next-generation (ββ) 0νdecay experiments seek to probe and possibly cover the IO region of parameter space, working towards the | m | ∼ 10 −2 eV frontier. In case these searches produce a negative result, the next frontier in the quest for (ββ) 0ν -decay will correspond to | m | ∼ 10 −3 eV.
Taking into account updated global-fit data on the 3neutrino mixing angles and the neutrino mass-squared differences, we have determined the conditions under which the effective Majorana mass in the NO case | m | NO exceeds the 10 −3 eV (5 × 10 −3 eV) value. The effective Majorana mass | m | NO of interest, as is well known, depends on the solar and reactor neutrino mixing angles θ 12 and θ 13 , on the two neutrino mass-squared differences ∆m 2  We have shown, in particular, that if the sum of the three neutrino masses is found to satisfy the lower bound Σ > 0.10 eV, that would imply in the case of NO neutrino mass spectrum | m | NO > 5 × 10 −3 eV for any values of the CPV phases α 21 and α 31 , unless there exist additional contributions to the (ββ) 0ν -decay amplitude which cancel at least partially the contribution due to the 3 light neutrinos. Yellow bands correspond to (the indicated, k = 0, 1, 2, 3) gCP-compatible but not CP-conserving values of the phases (α21, α 31 ). Blue bands correspond to CP-conserving phases (see Figure 5) and hatching indicates overlap with such regions, while red regions are not gCP-compatible (for the models under consideration, see text). The KamLAND-Zen bound of Eq. (4) is also indicated.
We have additionally studied the predictions for | m | IO and | m | NO in cases where the leptonic CPV phases are fixed to particular values, α 21 , α 31 − 2δ ∈ {0, π/2, π, 3π/2}, which are either CP conserving (see Figure 5) or may arise in predictive schemes combining generalised CP and flavour symmetries (see , lower bounds on the effective mass | m | NO for such choices of phases are given in Table III). We find that | m | NO ∼ > 10 −3 eV for all gCP-compatible but not CPconserving pairs of the relevant phases.
The searches for lepton number non-conservation performed by the neutrinoless double beta decay experiments are part of the searches for new physics beyond that predicted by the Standard Theory. They are of fundamental importance -as important as the searches for baryon number non-conservation in the form of, e.g., proton decay. Therefore if current and next-generation (ββ) 0ν -decay experiments seeking to probe the IO region of parameter space produce a negative result, the quest for (ββ) 0ν -decay should continue towards the | m | ∼ 5 × 10 −3 eV and possibly the | m | ∼ 10 −3 eV frontier.  Figure 7, with yellow bands corresponding to pairs (α21, α 31 ) with both phases being gCP-compatible but not CP-conserving.