Phenomenology of Vanishing Effective Majorana Mass with a Sterile Neutrino under Cosmological and JUNO Constraints

This paper investigates the parameter space for a vanishing effective Majorana mass in a $3+1$ sterile neutrino framework, demonstrating that while cosmological bounds on the sum of neutrino masses constrain the sterile mixing angle and lightest active neutrino mass, the resulting cancellation mechanism renders the sterile mixing angle insensitive to the high-precision solar oscillation measurements expected from the JUNO experiment.

The Gist

The Big Picture: The "Ghost" Neutrino and the Vanishing Act

Imagine the universe is a grand orchestra. For a long time, we knew about three main instruments (the active neutrinos: electron, muon, and tau) that play a specific tune. We know they have mass and they "mix" (change identity) as they travel through space.

But, there have been some strange notes in the music (experimental anomalies) suggesting there might be a fourth, invisible instrument playing along. We call this the Sterile Neutrino. It's called "sterile" because it doesn't interact with anything else in the orchestra; it's a ghost that only talks to the other neutrinos.

This paper asks a very specific question: If this ghostly fourth neutrino exists, can it help the three known neutrinos perform a "magic trick" where they completely cancel each other out?

Specifically, the authors are looking at a process called Neutrinoless Double Beta Decay. Think of this as a rare event where two neutrons in an atom turn into two protons and spit out two electrons, but no neutrinos are seen. For this to happen, the "effective mass" of the neutrinos must be exactly zero.

The authors are investigating: Can the addition of a sterile neutrino create a perfect "destructive interference" (like noise-canceling headphones) that makes the neutrino mass disappear completely?

The Setup: The Tightrope Walk

To make this magic trick work, the universe has to be incredibly precise. The authors are juggling three heavy constraints:

1. The Cosmological Limit (The Weight Limit):
   Imagine the universe has a strict weight limit for the total mass of all neutrinos combined. Recent data from the Planck satellite and the DESI telescope (which maps the universe's structure) says, "The total weight of all neutrinos cannot exceed 0.072 eV." That is incredibly light—lighter than a single grain of sand in a massive desert.

   • The Analogy: It's like trying to fit a family of four people into a tiny elevator that only holds 150 lbs. If the family is too heavy, the elevator breaks.

2. The Mixing Angle (The Volume Knob):
   The sterile neutrino has to mix with the active ones just enough to be noticed, but not too much. This is controlled by a parameter called $\theta_{14}$ (theta-14). Think of this as a volume knob. If it's too low, the ghost is silent; if it's too high, the ghost drowns out the others.

3. The JUNO Precision (The Tuning Fork):
   A new experiment called JUNO is coming online to measure the "solar mixing angle" ($\theta_{12}$) with extreme precision. It's like a master tuner checking if the orchestra is perfectly in tune.

The Findings: What the Math Says

The authors ran millions of simulations (Monte Carlo sampling) to see if they could find a combination of settings where the neutrino mass vanishes ($|M_{ee}| = 0$) while respecting all the rules above.

Here is what they found, broken down by the two possible "mass orderings" (how the neutrinos are arranged):

1. The Inverted Hierarchy (The "Heavy" Arrangement)

Imagine the neutrinos are arranged like a pyramid where the bottom two are heavy.

• The Result: It's a bust.
• Why? The math shows that for this arrangement to work, the total mass of the neutrinos would have to be at least 0.1 eV. But the universe's weight limit (from DESI/Planck) is 0.072 eV.
• The Verdict: If the universe's weight limit is correct, this "Inverted" arrangement with a sterile neutrino is ruled out. The elevator is too small for this family.

2. The Normal Hierarchy (The "Light" Arrangement)

Imagine the neutrinos are arranged like a staircase, starting very light and getting heavier.

• The Result: It's still possible, but barely.
• The Catch: To make the mass vanish, the sterile neutrino's "volume knob" ($\theta_{14}$) has to be set to a very specific, narrow range.
• The Prediction: The mixing angle must be between 0.10 and 0.13.
  • If the mixing is too weak, the cancellation doesn't happen.
  • If the mixing is too strong, the total mass gets too heavy and violates the cosmological limit.
• The Verdict: This scenario is still alive, but it's walking a very tightrope.

The JUNO Factor: Does the New Tuner Matter?

The paper also checked if the new, super-precise JUNO experiment would change anything.

• The Surprise: Surprisingly, JUNO's high precision on the solar angle doesn't change the outcome much.
• Why? The "cancellation magic" is so sensitive to other factors (like the mysterious CP-violating phases, which are like hidden timing delays in the music) that JUNO's precise tuning gets "washed out." The ghost neutrino can adjust its other settings to hide the fact that JUNO is listening closely.

The "So What?" Conclusion

This paper is a reality check for physicists hoping to find a sterile neutrino that explains why neutrino mass might vanish.

1. The "Inverted" option is dead (unless our understanding of the universe's weight limit is wrong).
2. The "Normal" option is alive but fragile. It predicts a very specific range for how the sterile neutrino mixes with the others.
3. Future experiments are the judge.
   • If the KATRIN experiment (which measures neutrino mass directly) finds the mixing is different from the predicted 0.10–0.13 range, the whole theory collapses.
   • If future cosmological data tightens the weight limit even further (e.g., down to 0.06 eV), even the "Normal" option might be squeezed out of existence.

In simple terms: The universe is playing a game of "Hide and Seek" with a ghost neutrino. The rules of the game (cosmology) have gotten stricter. The ghost can still hide, but it has to stand in a very specific, tiny spot in the room. If it steps even a little to the left or right, it gets caught. The next generation of experiments will be the ones to see if the ghost is actually there or if it was just a trick of the light.

Technical Summary

1. Problem Statement

The paper addresses the tension between the theoretical possibility of a vanishing effective Majorana mass ($|M_{ee}| = 0$) in neutrinoless double beta decay ($0\nu\beta\beta$) and recent stringent experimental and cosmological constraints.

• Context: While neutrino oscillation experiments have established that neutrinos have mass and mix, fundamental questions remain regarding the absolute mass scale, mass ordering (Normal vs. Inverted Hierarchy), and the existence of sterile neutrinos (eV-scale).
• The Conflict: The existence of a sterile neutrino introduces new mass eigenstates and CP-violating phases. These can lead to destructive interference among active and sterile states, potentially causing $|M_{ee}|$ to vanish. However, recent cosmological data (Planck and DESI+CMB) place extremely tight upper bounds on the sum of neutrino masses ($\sum m_i$), which severely restricts the parameter space for sterile neutrino models.
• Objective: The authors investigate the viability of a 3+1 neutrino framework (3 active + 1 sterile) where $|M_{ee}|$ vanishes, specifically testing this scenario against the latest cosmological bounds ($\sum m_i < 0.072$ eV) and high-precision solar oscillation data from the JUNO experiment.

2. Methodology

The authors employ a Monte-Carlo sampling technique to explore the parameter space of the 3+1 model.

• Formalism:

  • They utilize a $4 \times 4$ unitary mixing matrix $U$ parameterized by three active-active rotations, three active-sterile rotations, and three Majorana CP phases ($\alpha, \beta, \gamma$).
  • The effective Majorana mass is defined as:
    $$|M_{ee}| = |m_1 c_{12}^2 c_{13}^2 c_{14}^2 + m_2 s_{12}^2 c_{13}^2 c_{14}^2 e^{2i\alpha} + m_3 s_{13}^2 c_{14}^2 e^{2i\beta} + m_4 s_{14}^2 e^{2i\gamma}|$$
  • The condition for vanishing $|M_{ee}|$ requires both the real and imaginary parts of the expression to be zero simultaneously. This allows the authors to solve for the sterile mixing angle $\sin^2\theta_{14}$ in terms of masses and phases.

• Constraints & Inputs:

  • Oscillation Data: NuFIT 6.0 global fit values for active neutrino parameters ($\theta_{12}, \theta_{13}, \theta_{23}, \Delta m^2_{21}, \Delta m^2_{31}$).
  • Sterile Parameters: Scanned ranges for $\sin^2\theta_{14}$ (0.0098–0.031) and $\Delta m^2_{41}$ (0.35–2 eV$^2$) consistent with short-baseline anomalies.
  • Cosmological Bounds:
    • Planck 2018: $\sum m_i < 0.12$ eV (95% CL).
    • DESI+CMB: $\sum m_i < 0.072$ eV (95% CL) – The most stringent constraint used.
  • JUNO Data: High-precision measurements of solar parameters ($\theta_{12}, \Delta m^2_{21}$) are incorporated to test sensitivity.

• Analysis Strategy:

  • Random uniform sampling of the lightest neutrino mass ($m_1$ or $m_3$) and Majorana phases.
  • Gaussian sampling of oscillation parameters.
  • Iterative consistency checks to find physical solutions where $|M_{ee}| \approx 0$ and $\sum m_i$ satisfies cosmological limits.

3. Key Contributions and Results

A. Impact of Cosmological Bounds on Mass Ordering

• Inverted Hierarchy (IH): The model predicts a lower bound on the sum of neutrino masses ($\sum m_i > 0.1$ eV) due to the requirement of vanishing $|M_{ee}|$ in the 3+1 framework. This directly conflicts with the DESI+CMB bound ($\sum m_i < 0.072$ eV).
  • Result: Inverted Hierarchy is ruled out in this specific 3+1 scenario under the DESI+CMB constraints.
• Normal Hierarchy (NH): A small region of parameter space remains viable.
  • Result: NH is allowed, but only for a very narrow range of the sterile mixing angle.

B. Constraints on Sterile Mixing ($\sin\theta_{14}$)

• NH Prediction: Under the DESI+CMB bound, the active-sterile mixing angle is tightly constrained to:
  $$0.10 < \sin\theta_{14} < 0.13$$
  This provides a specific, testable upper bound on the mixing strength.
• Generic Lower Bound: The model predicts a lower bound of $\sin\theta_{14} > 0.115$ regardless of the hierarchy, driven by the cancellation mechanism required for $|M_{ee}|=0$.

C. Role of Majorana Phases

• The analysis reveals that the Majorana phase $\gamma$ is the critical parameter for achieving cancellation.
• In the IH case, $\gamma$ is constrained to a narrow range ($75^\circ - 105^\circ$), with $\gamma \approx 90^\circ$ being disallowed.
• In the NH case, the constraints on $\gamma$ are less severe but still restrict the parameter space significantly when combined with cosmological limits.

D. Sensitivity to JUNO Precision

• The authors investigated whether JUNO's high-precision measurement of $\theta_{12}$ would further constrain the model.
• Finding: The sterile neutrino parameters (specifically $\sin\theta_{14}$) are not sensitive to the precision of $\theta_{12}$. The constraints imposed by precise $\theta_{12}$ are effectively "washed out" by the new cancellations driven by the additional CP-violating phases ($\gamma$) required to make $|M_{ee}|$ vanish.

4. Significance and Implications

• Theoretical Viability: The 3+1 framework with vanishing $|M_{ee}|$ is highly predictive but fragile. While it remains viable in the Normal Hierarchy, it is completely excluded in the Inverted Hierarchy by current cosmological data.
• Predictive Power: The model makes a sharp prediction for the sterile mixing angle in the Normal Hierarchy ($0.10 < \sin\theta_{14} < 0.13$). This range is within the reach of future precision experiments.
• Future Tests:
  • Cosmology: If future surveys (e.g., CMB-S4) push the $\sum m_i$ bound even lower (e.g., $\approx 0.06$ eV), the remaining Normal Hierarchy region may also be excluded.
  • Oscillation Experiments: Experiments like KATRIN (direct mass measurement) and next-generation oscillation facilities (DUNE, Hyper-K) will be crucial. KATRIN's current percent-level sensitivity to active-sterile mixing could independently rule out the Inverted Hierarchy scenario.
• Conclusion: The study highlights that the interplay between cosmological mass limits and the requirement for $0\nu\beta\beta$ cancellation creates a "tightly confined" window for sterile neutrinos. The survival of the 3+1 model depends critically on future experimental sensitivities to both the sum of neutrino masses and the active-sterile mixing angle.