Constraining four neutrino mass patterns from neutrinoless double beta decay

All existing data on neutrino oscillations (including those from the LSND experiment) imply a four neutrino scheme with six different allowed mass patterns. Some of the latter are shown to be disfavored by using a conservative upper bound on the ββ0ν nuclear decay rate, if neutrinos are assumed to be Majorana particles. Comparisons are also made with restrictions from tritium β-decay and cosmology. 1) Permanent address: Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India Any observation of neutrinoless nuclear double beta decay would imply lepton nonconservation and a nonzero neutrino Majorana mass Mee. The latter is defined as Mee = ∑ i miU 2 ei , (1) wheremi is the nonnegative ith physical Majorana mass and Uei the matrix element which mixes the electrons neutrino νe with the mass eigenstate νi. There is now a considerable amount of flavor oscillation data from solar [1] and atmospheric [2] neutrinos, all of which can be accommodated within the standard picture of three neutrinos νe, νμ, ντ with tiny masses. The best fits yield two independent squared mass differences among the neutrinos: ∆S ∼ 4 × 10 eV 2 for the solar case and ∆A ∼ 3×10−3 eV 2 for the atmospheric one, the favored values of the corresponding mixing angles being sin 2θS ∼ 0.66 and sin 2θA ∼ 1. The following question then emerges: how is the overall mass scale of the neutrinos constrained? Specifically, how does one pin down the sum of the physical neutrino masses Σν which controls the neutrino component of dark matter and hence neutrino effects on structure formation? If we assume the neutrinos to be Majorana particles, there is a link between Σν and Mee. This link has been the subject of several recent investigations [3-8]. In particular, Barger et al [3] have given upper and lower bounds on Σν in terms of Mee,∆A and θS, neglecting ∆S in comparison with ∆A. When the small mixing angle relevant to unobserved neutrino oscillations at the CHOOZ reactor [9] is ignored, their inequalities become particularly simple, namely 2Mee + √ M2 ee ±∆A < Σν < 2Mee | cos 2θS| + √ M2 ee cos2 2θS ±∆A . (2) In eq.(2) the + (−) sign refers to the normal (inverted) three neutrino mass hierarchy m1 ≤ m2 < m3 (m1 < m2 ≤ m3). The inequality Mee > √ ∆A is then automatically implied for the inverted hierarchy case. However, such considerations completely ignore another item of neutrino flavor oscillation information, namely the data [10] from the LSND experiment. These data can be explained by ν̄μ → ν̄e (and νμ ↔ νe) oscillations with a mass squared difference ∆L= O(1) eV 2 and a small mixing angle θL= O(10−2). This note is addressed to a generalization of eq.(2) to include the LSND results. A fourth light neutrino νs, which is not electroweak active and is hence called sterile, is needed along with νe, νμ and ντ to simultaneously explain the solar, atmospheric and LSND anomalies. Of course, it follows from the recent SNO [11] and Super-K [2] results that the final state to which the solar νe or the atmospheric νμ oscillates cannot be a purely sterile species. On the other hand, orthogonal linear combinations of ντ and νs are still allowable final states in these oscillations. 2) The mass ordering m1 < m2 < m3 with nonnegative m’s has been chosen by definition. 2 FIGURE 1. Six allowed of patterns of masses, grouped into two schemes, for the four neutrino scenario. Vertical separations symbolize mass squared differences pertinent to solar, atmospheric and LSND oscillations. Comprehensive analyses [12-14] have recently been made of all current data on solar, atmospheric and LSND oscillations, together with constraints from other accelerator and reactor data, by considering the four neutrinos νe, νμ, ντ and νs. The conclusion is that the four neutrino picture is not excluded, though the required fits are not of particularly high quality. Once one considers a four neutrino scenario, with the experimental input ∆S << ∆A << ∆L, the mass spectrum of the neutrinos becomes an issue of central importance. There are six possible four neutrino mass patterns, as shown in Fig. 1, that are a priori compatible [12–14] with the data These group into two schemes called [15] (3+1) and (2+2). The (3+1) scheme, consisting of four possibilities a, b, c, d (c.f. Fig.1) is characterized by three close-by neutrino masses separated from the fourth by a gap O ( √ ∆L). Here the sterile neutrino is only slightly mixed with the active ones. It is therefore a weak component in solar and atmospheric neutrino oscillations, but mainly provides a description of the LSND effect. In the (2+2) scheme, comprising two possibilities A and B (cf. Fig.1), there are two pairs of nearly degenerate states, separated by a gap O ( √ ∆L). In this pattern two orthogonal linear combinations of νs and ντ with comparable coefficients make up the final states to which the solar νe and the atmospheric νμ oscillate. Oscillation phenomenology alone cannot distinguish between different patterns within any of these schemes. However, a distinction does become possible when nuclear ββ0ν decay is taken into account [16], assuming that the neutrinos are Majorana particles. Turning towards mixing aspects, let us define the unitary transformation 3) Our ordering for the physical masses is always m1 < m2 < m3 < m4. 3