Neutrino mass variables in 3 active and 2 sterile neutrino scenario

The three-flavor framework of neutrino oscillations successfully explains most experimental results, but persistent anomalies at short- and long-baseline experiments hint at the existence of additional light sterile states. In particular, eV-scale sterile neutrinos are motivated by LSND and MiniBooNE results, while sub-eV sterile states with mass-squared differences at the $10^{-2}$ and $10^{-5}$eV$^2$ scales have been proposed to address the T2K–NO$ν$A tension and the absence of the expected upturn in the solar neutrino energy spectrum, respectively. Such sterile states are singlets under the Standard Model gauge group and mix only through their admixture with active neutrinos. In this work, we investigate the phenomenology of the $3+2$ scenario, incorporating one eV-scale sterile neutrino together with a sub-eV state, and analyze their impact on absolute-mass related observables: the sum of neutrino masses $Σ$ constrained by cosmology, the effective electron neutrino mass $m_β$ from beta decay, and the effective Majorana mass $m_{ββ}$ probed in neutrinoless double beta decay. We demonstrate that the presence of two sterile states can significantly modify the allowed parameter space compared to the three-flavor and $3+1$ frameworks, with some mass-ordering schemes already disfavored by current cosmological and laboratory limits. Finally, we assess the implications of upcoming sensitivities from KATRIN, Project8, and LEGEND-1000, highlighting the complementary role of sub-eV sterile neutrinos in probing physics beyond the minimal three-flavor paradigm.

💡 Research Summary

The paper investigates the phenomenology of a “3 + 2” neutrino framework, in which the three active neutrinos of the Standard Model are supplemented by two sterile states: one at the eV scale (motivated by the LSND and MiniBooNE anomalies) and a lighter one with mass‑squared splittings of order 10⁻⁴–10⁻² eV² (proposed to alleviate the T2K–NOvA tension and the missing upturn in the solar neutrino spectrum). The authors first construct the full 5 × 5 unitary mixing matrix, parameterising it as a product of complex rotations that extend the usual PMNS matrix. This introduces seven new mixing angles, five new Dirac CP phases and two additional Majorana phases, greatly enlarging the parameter space relative to the minimal 3 + 1 scenario.

Four possible mass‑ordering schemes are defined: SSN (Sterile‑Sterile‑Normal), SSI (Sterile‑Sterile‑Inverted), SNS (Sterile‑Normal‑Sterile) and SIS (Sterile‑Inverted‑Sterile). In SSN and SSI the two sterile masses are the heaviest, while in SNS and SIS one sterile state lies above the active spectrum and the other below the lightest active mass. The authors write analytic expressions for the three absolute‑mass observables—(i) the cosmological sum Σ = ∑ m_i, (ii) the effective electron‑neutrino mass m_β measured in β‑decay, and (iii) the effective Majorana mass m_ββ probed in neutrinoless double‑beta decay—in terms of the lightest mass, the measured oscillation parameters (Δm²_sol, Δm²_atm, θ_12, θ_13, θ_23) and the new active‑sterile mixing elements (U_e4, U_e5, …).

Using the latest global fits for the three‑flavor parameters and benchmark values for the sterile mixing angles (θ_14≈7°, θ_15≈3°, θ_24≈5°, θ_25≈2°, etc.) together with Δm²_41≈1.3 eV² and Δm²_51 in the range 10⁻⁴–10⁻² eV², the authors perform a numerical scan over the remaining unknowns, including the two new Majorana phases. The main findings are:

1. Cosmological constraints – In the SSN and SSI schemes the total mass Σ is at least ~1 eV, far above the Planck limit Σ ≲ 0.12 eV (for fully thermalised neutrinos) and even above the more relaxed bound Σ ≲ 0.5 eV (partial thermalisation). Consequently these two orderings are essentially excluded unless one invokes exotic cosmologies that dramatically suppress sterile thermalisation.

2. β‑decay limits – For the SNS and SIS orderings the lighter sterile state reduces Σ to 0.2–0.4 eV, compatible with current cosmology. The effective β‑decay mass m_β lies in the range 0.3–0.6 eV, depending on the size of the active‑sterile mixings. The upcoming KATRIN sensitivity (≈0.2 eV) and the projected Project 8 reach (≈0.04 eV) will therefore be able to probe or rule out the remaining viable orderings.

3. Neutrinoless double‑beta decay – The effective Majorana mass m_ββ is highly sensitive to the two new Majorana phases (α_41, α_51). Constructive interference can push |m_ββ| up to ~0.1 eV, within reach of the next‑generation LEGEND‑1000 experiment (goal ≈0.01 eV). Destructive interference, however, can suppress |m_ββ| below 10⁻² eV, making detection challenging. Thus 0νββ experiments provide a complementary probe of the CP‑phase structure of the sterile sector.

4. Partial thermalisation scenarios – The authors introduce a phenomenological factor f (0 ≤ f ≤ 1) to scale the sterile contribution to Σ, mimicking models with new sterile‑sector interactions, low reheating temperatures, or altered expansion histories. For f ≈ 0.3 the SSN and SSI schemes become marginally compatible with cosmology, but then the β‑decay and 0νββ limits become extremely restrictive, essentially forcing the sterile mixings to be tiny.

Overall, the analysis shows that the inclusion of two sterile neutrinos dramatically reshapes the allowed regions of absolute‑mass observables. Current data already disfavour the two schemes where both sterile states are heavy (SSN, SSI). The remaining schemes (SNS, SIS) survive but occupy a relatively narrow slice of parameter space that will be decisively tested by the next generation of laboratory experiments (KATRIN, Project 8, LEGEND‑1000) and by improved cosmological measurements (CMB‑S4, galaxy surveys). The paper thus highlights the synergistic role of cosmology, β‑decay spectroscopy, and neutrinoless double‑beta decay in exploring physics beyond the minimal three‑flavor paradigm.