⚛️ phenomenology

Neutrino mass variables in 3 active and 2 sterile neutrino scenario

This paper investigates the phenomenology of a $3+2$ neutrino scenario incorporating one eV-scale and one sub-eV sterile neutrino, demonstrating how this framework alters constraints on absolute mass observables and assessing the complementary potential of upcoming experiments like KATRIN, Project 8, and LEGEND-1000 to probe physics beyond the standard three-flavor paradigm.

Original authors: Srubabati Goswami, Hemanth M., Debashis Pachhar, N Rajeev

Published 2026-03-19

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Srubabati Goswami, Hemanth M., Debashis Pachhar, N Rajeev

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is a giant, complex orchestra. For decades, physicists believed this orchestra had exactly three main instruments playing the melody of the universe: the electron neutrino, the muon neutrino, and the tau neutrino. These three "active" neutrinos are like the violin, cello, and flute—they interact with everything around them and are well understood.

However, in recent years, some musicians in the audience (experimental data) started hearing strange, faint notes that didn't fit the standard three-instrument score. These anomalies suggested that there might be two more instruments hiding in the shadows, playing quietly. These are called sterile neutrinos. They are "sterile" because they don't play with the other instruments (they don't interact with the Standard Model forces); they only exist by mixing their sound with the active ones.

This paper is a deep dive into a specific theory: What if there are exactly two of these hidden instruments? One is heavy (like a bass drum, around 1 eV), and one is very light (like a whisper, under 1 eV). The authors, a team of physicists from India and Russia, asked: If these two hidden instruments exist, how does it change the music of the universe, and can we hear them?

Here is the breakdown of their findings using simple analogies:

1. The Three-Flavor vs. The 3+2 Scenario

The Old View (3-Flavor): We thought there were only three neutrinos. Their masses were like three steps on a ladder.
The New View (3+2): The authors propose a ladder with five steps. The bottom three are the known active neutrinos. The top two are the new "sterile" ones.
- The Heavy Sterile (eV-scale): This one is motivated by "short-baseline" experiments (like LSND and MiniBooNE) that saw weird spikes in data. It's like a heavy bass note that shakes the floor.
- The Light Sterile (Sub-eV): This one is motivated by tensions in long-distance experiments (like T2K and NOvA) and solar neutrino data. It's a subtle, high-pitched note that might explain why the solar music sounds slightly off.

2. The Four Possible "Tuning" Schemes

Just as you can arrange the steps of a ladder in different orders, the authors looked at four different ways these five neutrinos could be arranged by mass. They gave them catchy names:

SSN & SSI: The two heavy sterile notes are at the very top of the ladder, with the active ones below.
SNS & SIS: One sterile note is at the very bottom (lighter than the active ones), and the other is at the top.

3. The Three Ways to "Listen" for Them

The paper checks if these hidden instruments can be detected using three different "microphones" (observables):

A. The Cosmological Microphone (The Big Picture)

The Concept: Cosmologists measure the total weight of all neutrinos in the universe (the "Sum of Masses"). If you add two extra heavy instruments to the orchestra, the total weight goes up.
The Finding: The universe has a strict "weight limit" set by the Big Bang's afterglow (Cosmic Microwave Background).
The Result: The authors found that for most of their four arrangements, adding these two extra neutrinos makes the universe too heavy. It's like trying to fit a grand piano and a tuba into a small elevator that can only hold a suitcase.
- Verdict: The "SSI" and "SIS" arrangements are completely ruled out by current data. The "SSN" and "SNS" arrangements are only allowed if the neutrinos are incredibly light and the universe is very specific about how they were born.

B. The Beta Decay Microphone (The KATRIN Experiment)

The Concept: This experiment (KATRIN) looks at the energy of electrons flying off during radioactive decay. If a heavy sterile neutrino is hiding, it creates a tiny "kink" or distortion in the energy curve, like a bump in a smooth road.
The Finding: Adding the heavy sterile neutrino pushes the predicted energy bump higher.
The Result: Current experiments (KATRIN) can't rule these scenarios out yet, but the future Project 8 experiment (which will be much more sensitive) could act like a super-powered stethoscope. It might be able to hear the "SSI" and "SIS" arrangements entirely, effectively silencing those theories.

C. The Double Beta Decay Microphone (The LEGEND Experiment)

The Concept: This looks for a rare event where two neutrons turn into two protons and two electrons without emitting neutrinos. This only happens if neutrinos are their own antiparticles (Majorana particles). The strength of this signal depends on the "effective mass."
The Finding: This is where it gets tricky. The new sterile neutrinos can interfere with the active ones.
- Constructive Interference: Like two sound waves adding up to make a louder noise, the sterile neutrinos could make the signal huge and easy to find.
- Destructive Interference: Like noise-canceling headphones, the sterile neutrinos could cancel out the active ones, making the signal vanish completely.
The Result: In some scenarios, the signal could be so small that even the most sensitive future detectors (like LEGEND-1000) might hear nothing, even if the neutrinos exist. This makes the search harder but also more exciting because finding a "null result" could actually prove the existence of these hidden particles through their cancellation effect.

The Big Picture Conclusion

The authors conclude that the "3+2" idea (3 active + 2 sterile) is a very crowded room.

Cosmology is the bouncer: It's already kicking out most of the possible arrangements because they make the universe too heavy.
Beta Decay is the next check: Upcoming experiments will likely rule out even more arrangements.
Double Beta Decay is the final test: It offers a unique way to see these particles, but only if they don't cancel each other out perfectly.

In simple terms: The paper says, "We think there are two hidden neutrinos. We've mapped out all the ways they could be arranged. Unfortunately, the universe's weight limit already bans most of those arrangements. But if they do exist in the few remaining spots, the next generation of experiments will either hear them clearly or prove they are hiding so well they cancel themselves out."

It's a story of a cosmic mystery where the clues are getting tighter, and the next few years of experiments will tell us if the "ghost" neutrinos are real or just a trick of the light.

1. Problem Statement

While the standard three-flavor neutrino oscillation framework successfully explains most experimental data, persistent anomalies remain:

Short-baseline anomalies: Results from LSND, MiniBooNE, and Gallium source experiments (GALLEX, SAGE, BEST) suggest the existence of light sterile neutrinos with masses at the eV scale ( $\Delta m^2 \sim 1 \text{ eV}^2$ ).
Long-baseline/Solar tensions: Tensions between T2K and NO $\nu$ A data, and the absence of the expected upturn in the solar neutrino energy spectrum, motivate the existence of sub-eV sterile states ( $\Delta m^2 \sim 10^{-2}$ to $10^{-5} \text{ eV}^2$ ).

The standard 3+1 model (one sterile neutrino) struggles to reconcile appearance and disappearance datasets simultaneously. This paper investigates the 3+2 scenario (three active + two sterile neutrinos), incorporating both an eV-scale sterile state and a sub-eV sterile state. The primary goal is to analyze how this extended framework impacts absolute neutrino mass observables: the cosmological sum of masses ( $\Sigma$ ), the effective electron neutrino mass ( $m_\beta$ ) from beta decay, and the effective Majorana mass ( $m_{\beta\beta}$ ) from neutrinoless double beta decay ( $0\nu\beta\beta$ ).

2. Methodology

A. Theoretical Framework

The authors extend the standard PMNS mixing matrix to a $5 \times 5$ unitary matrix ( $U_{5\times5}$ ) involving two sterile states ( $s_1, s_2$ ).

Mixing Parameters: The framework introduces 7 new mixing angles, 5 new Dirac CP phases, and 2 new Majorana phases.
Mass Ordering Schemes: Four distinct mass hierarchy schemes are defined based on the relative positioning of active and sterile mass eigenstates ( $m_1, m_2, m_3$ $m_{1}, m_{2}, m_{3}$ active; $m_4, m_5$ $m_{4}, m_{5}$ sterile):
1. SSN (Sterile-Sterile-Normal): $m_5 > m_3 > m_4 > m_2 > m_1$ (or variations).
2. SSI (Sterile-Sterile-Inverted): $m_5 > m_2 > m_1 > m_4 > m_3$ (or variations).
3. SNS (Sterile-Normal-Sterile): $m_5 > m_3 > m_2 > m_1 > m_4$ .
4. SIS (Sterile-Inverted-Sterile): $m_5 > m_2 > m_1 > m_3 > m_4$ .

B. Input Parameters and Constraints

Oscillation Data: Global fit values for active neutrino parameters ( $\theta_{12}, \theta_{13}, \theta_{23}, \Delta m^2_{21}, \Delta m^2_{31}$ ) are taken from recent fits (NO and IO).
Sterile Benchmarks:
- Heavy sterile mass splitting: $\Delta m^2_{s1} \equiv m_5^2 - m_{lightest}^2 = 1.3 \text{ eV}^2$ (motivated by LSND/MiniBooNE).
- Light sterile mass splitting: $\Delta m^2_{s2} \equiv m_4^2 - m_{lightest}^2 = 10^{-2} \text{ eV}^2$ and $10^{-4} \text{ eV}^2$ .
- Mixing angles ( $|U_{e4}|^2, |U_{e5}|^2$ ) are constrained by MINOS, MINOS+, Daya Bay, and Bugey-3 data.
Cosmological Constraints: The analysis uses a 10-parameter cosmological model (10-PCM) combining Planck 2018, BAO, Hubble, and Supernova Ia data. This yields limits on the effective number of relativistic species ( $N_{eff} \approx 3.11$ ) and the sum of neutrino masses ( $\Sigma < 0.16 \text{ eV}$ ). The authors account for partial thermalization of sterile neutrinos, where the contribution to $\Sigma$ is scaled by $\Delta N_{eff}$ .

C. Observables Calculated

Cosmological Sum ( $\Sigma$ ): $\Sigma = \sum m_i$ (modified for partial thermalization).
Beta Decay Mass ( $m_\beta$ ): $m_\beta = \sqrt{\sum |U_{ei}|^2 m_i^2}$ .
Neutrinoless Double Beta Decay Mass ( $m_{\beta\beta}$ ): $m_{\beta\beta} = |\sum U_{ei}^2 m_i|$ , including Majorana phases.

3. Key Results

A. Cosmological Constraints ( $\Sigma$ )

Tension: The presence of eV-scale sterile neutrinos generally drives $\Sigma$ well above the cosmological upper limit of $0.16 \text{ eV}$ .
Allowed Regions:
- SSN: Allowed only for very small lightest neutrino masses ( $m_{lightest} \lesssim 0.013 - 0.016 \text{ eV}$ ).
- SSI & SIS: Completely disfavored/excluded by the 10-PCM bound because the mass of the lightest active state in these hierarchies is already $\sim 0.05 \text{ eV}$ , causing the total sum to exceed limits even with minimal sterile contributions.
- SNS: Allowed only for $\Delta m^2_{s2} = 10^{-4} \text{ eV}^2$ and very small $m_{lightest}$ ; excluded for $\Delta m^2_{s2} = 10^{-2} \text{ eV}^2$ .

B. Beta Decay ( $m_\beta$ )

Shift in Sensitivity: The eV-scale sterile contribution dominates $m_\beta$ , shifting predictions upward compared to the 3-flavor case.
Projected Sensitivity:
- Current KATRIN limits ( $m_\beta < 0.45 \text{ eV}$ ) are compatible with most scenarios.
- Project 8 (sensitivity $\sim 40 \text{ meV}$ ) has the power to exclude the SSI and SIS scenarios entirely.
- Project 8 can also rule out the SNS scenario for $\Delta m^2_{s2} = 10^{-2} \text{ eV}^2$ and probe a significant portion of the SSN and SNS parameter space for $\Delta m^2_{s2} = 10^{-4} \text{ eV}^2$ .

C. Neutrinoless Double Beta Decay ( $m_{\beta\beta}$ )

Interference Effects: The interplay between active and sterile contributions, mediated by Majorana phases, leads to constructive or destructive interference.
SSN & SNS: Destructive interference can suppress $m_{\beta\beta}$ down to $\mathcal{O}(10^{-4}) \text{ eV}$ , potentially hiding the signal even in the Inverted Ordering (IO) regime. For SSN with $\Delta m^2_{s2} = 10^{-4} \text{ eV}^2$ , the "cancellation region" is extended.
SSI: Partial cancellations can relax the lower bound on $m_{\beta\beta}$ compared to standard IO, allowing for lighter neutrino masses ( $m_{lightest} < 0.6 \text{ eV}$ ) that would otherwise be excluded.
SIS: Suppression is possible only in a narrow parameter region; the quasi-degenerate regime remains dominated by active neutrinos.
Future Probes: Next-generation experiments like LEGEND-1000 (sensitivity $\sim 10^{-3} \text{ eV}$ ) will be crucial in testing these cancellation regions and distinguishing between the minimal 3-flavor paradigm and the 3+2 scenario.

4. Significance and Conclusion

This work provides a comprehensive phenomenological analysis of the 3+2 sterile neutrino model, bridging short-baseline anomalies and long-baseline tensions.

Complementarity: The study highlights the critical role of complementary probes. Cosmology provides the strongest constraints, effectively ruling out many mass ordering schemes (SSI, SIS) and large portions of the parameter space. Beta decay experiments (Project 8) offer the next line of defense, capable of excluding specific hierarchies. Neutrinoless double beta decay (LEGEND-1000) is essential for probing the Majorana nature and testing sterile-induced cancellations that could otherwise hide new physics.
Exclusion Power: The authors demonstrate that current cosmological data already exclude significant regions of the 3+2 parameter space. Future data from KATRIN, Project 8, and LEGEND-1000 will provide decisive tests for the remaining viable scenarios.
Theoretical Implication: The 3+2 framework offers a rich phenomenology where sterile states can significantly alter absolute mass observables. However, the tight constraints from cosmology and upcoming experiments suggest that if such sterile states exist, they must reside in very specific, finely tuned regions of the parameter space (e.g., specific mass orderings and partial thermalization scenarios).

In summary, while the 3+2 scenario offers a potential solution to experimental anomalies, it faces severe tension with cosmological bounds, and its remaining viable parameter space is poised to be rigorously tested by the next generation of neutrino experiments.