⚛️ phenomenology

Generalizing the Dirac-Majorana Confusion Theorem: The Role of CP-Violating Phases in New Physics Vector Interactions

This paper generalizes the Dirac-Majorana Confusion Theorem by demonstrating that CP-violating flavor-changing neutral current interactions mediated by a new vector boson ( $Z'$ ) lift the kinematic mass suppression in Coherent Elastic Neutrino-Nucleus Scattering, thereby enabling the experimental distinction between Dirac and Majorana neutrinos through the imaginary component of the interaction.

Original authors: David Delepine, A. Yebra

Published 2026-02-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: David Delepine, A. Yebra

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

The Big Mystery: Are Neutrinos Their Own Twins?

Imagine you have a mysterious particle called a neutrino. For decades, physicists have been arguing over a fundamental question: Is a neutrino its own antiparticle?

The "Dirac" View: Think of a neutrino like a human. You have a distinct "self" (the neutrino) and a distinct "twin" (the antineutrino). They look similar but are fundamentally different people.
The "Majorana" View: Think of a neutrino like a perfectly symmetrical snowflake. If you flip it over, it looks exactly the same. In this world, the neutrino is its own antiparticle.

Finding out which one is true is one of the biggest quests in physics. Usually, the only way to tell them apart is to look for a very rare event called "Neutrinoless Double Beta Decay." But if that doesn't happen, can we tell the difference in other ways?

The Old Rule: The "Confusion Theorem"

For a long time, physicists believed in a rule called the Practical Dirac-Majorana Confusion Theorem.

The Analogy: Imagine you are trying to tell the difference between a human and a snowflake by watching them run a race.

The rule says: "Unless the snowflake is incredibly heavy (which it isn't), it will run at the exact same speed as the human. You can't tell them apart."
In Physics terms: At high energies, the difference between Dirac and Majorana neutrinos is so tiny (suppressed by the square of their tiny mass) that it's impossible to measure. They look identical in almost every experiment.

The New Idea: Breaking the Rules with a "Magic Door"

The authors of this paper say: "Wait a minute! That rule only works if we look at the neutrino running straight ahead. What if we build a Magic Door that only opens for specific types of runners?"

They propose a new scenario involving:

A New Force: A new particle called a $Z'$ boson (think of it as a new kind of messenger).
Flavor-Changing: This messenger doesn't just let neutrinos pass; it lets them change their identity (e.g., a "muon-neutrino" turns into an "electron-neutrino") as they pass through.
CP Violation: This is the "magic" ingredient. It means the laws of physics treat the forward direction differently than the backward direction (like a screw that only turns one way).

How the "Filter" Works

Here is where the magic happens. The authors discovered that the "Majorana" nature of the neutrino acts like a strict bouncer at a club.

The Scenario:
Imagine a club with two types of guests: Humans (Dirac) and Snowflakes (Majorana). They are trying to get through a special "Flavor-Changing" door guarded by the $Z'$ messenger.

The Human (Dirac Neutrino):
- The Human can walk through the door using any key. They can use the "Real" key (standard physics) or the "Imaginary" key (the new CP-violating physics).
- Result: They get in easily. The signal is strong.
The Snowflake (Majorana Neutrino):
- Because the snowflake is perfectly symmetrical (it's its own twin), the laws of physics (Fermi-Dirac statistics) act as a filter.
- If the Snowflake tries to use the "Real" key (the standard part of the interaction), the door slams shut. The symmetry cancels it out perfectly.
- However, if the Snowflake tries to use the "Imaginary" key (the part of the new physics that violates CP symmetry), the door opens.
- Result: The Snowflake only gets in if the new physics is "weird" (CP-violating). If the new physics is "normal" (CP-conserving), the Snowflake is completely blocked.

The Big Discovery

The paper proves that if this new "Magic Door" ( $Z'$ ) exists and has the right kind of "weirdness" (CP violation), we can finally tell the difference between Humans and Snowflakes without needing them to be heavy.

If Neutrinos are Dirac (Humans): They interact strongly with the new force, creating a big, noticeable signal.
If Neutrinos are Majorana (Snowflakes): They are blocked from the main interaction. They only interact weakly through the "weird" CP-violating path.

This means the difference isn't hidden by their tiny mass anymore; it's hidden by whether the new physics is "normal" or "weird."

Why This Matters for Real Experiments

The authors suggest looking at two specific types of experiments to catch this:

COHERENT (The "Silent" Detector):
- This experiment shoots neutrinos at a tank of liquid Argon (which has atoms with zero spin, acting like a perfect filter).
- The Prediction: If neutrinos are Dirac, the new force would make the detector see more hits than expected (a linear increase). If neutrinos are Majorana, the main interaction is blocked, and they would see fewer hits (only a tiny, quadratic increase).
- Current Status: The data so far looks like the Standard Model (no huge extra hits). This actually hints that neutrinos might be Majorana, because the "Human" signal is missing!
DUNE (The "Shape" Detector):
- This experiment looks at how neutrinos bounce off electrons.
- The Prediction: If they are Dirac, the energy distribution of the bounces will look "lopsided" (asymmetric). If they are Majorana, the distribution will look perfectly "balanced" (symmetric) because the main interaction is gone.

The Takeaway

The paper rewrites the rulebook. It says: "We don't need to wait for neutrinos to be heavy to tell them apart. If we find a new force that changes neutrino flavors and breaks symmetry, the neutrino's own nature (Dirac vs. Majorana) will act as a filter, either letting the signal through or blocking it completely."

It turns the search for the neutrino's identity from a game of "finding a needle in a haystack" into a game of "checking if the door is locked." If the door is locked (Majorana), the signal disappears. If it's open (Dirac), the signal is loud and clear.

1. Problem Statement

The fundamental nature of the neutrino—whether it is a Dirac fermion ( $\nu \neq \bar{\nu}$ ) or a Majorana fermion ( $\nu = \nu^c$ )—remains an open question in particle physics. While Neutrinoless Double Beta Decay ( $0\nu\beta\beta$ ) is the primary probe for Majorana nature, it has not yet been observed.

The Practical Dirac-Majorana Confusion Theorem (PDMCT), established by Kayser and Shrock, posits that in lepton-number-conserving processes (like elastic scattering), the difference between Dirac and Majorana cross-sections is suppressed by the factor $(m_\nu/E_\nu)^2$ . Given the sub-eV neutrino mass and typical MeV-GeV experimental energies, this suppression renders the two hypotheses experimentally indistinguishable within the Standard Model (SM).

The paper addresses the question: Can this kinematic suppression be lifted in the presence of New Physics (NP), specifically through Flavor-Changing Neutral Currents (FCNC) and CP violation?

2. Methodology

The authors propose a theoretical framework extending the Standard Model with a new neutral vector boson, $Z'$ , which mediates Flavor-Changing Neutral Currents (FCNC) between neutrino flavors ( $\alpha \neq \beta$ ).

Effective Lagrangian: They introduce a vector interaction with complex couplings $g_{\alpha\beta}^V$ that carry a CP-violating phase $\phi$ . The Lagrangian is constructed to be Hermitian:
$\mathcal{L}_{int} \supset -Z'_\mu (g_{\alpha\beta}^\nu \gamma^\mu \nu_\beta + g_{\alpha\beta}^{*\nu} \nu_\beta \gamma^\mu \nu_\alpha)$
Fermi-Dirac Statistics & Majorana Condition: The core of the methodology relies on applying the Majorana condition ( $\nu = \nu^c$ $ν = ν^{c}$ ) and Fermi-Dirac statistics to the bilinear covariants.
- For a Majorana field, the vector current $\bar{\nu}_\alpha \gamma^\mu \nu_\alpha$ vanishes identically for diagonal transitions ( $\alpha = \beta$ ).
- For non-diagonal transitions ( $\alpha \neq \beta$ ), the authors derive selection rules showing how the Hermitian structure of the Lagrangian interacts with the Majorana condition.
Scattering Processes: The formalism is applied to two inclusive scattering processes where the final neutrino flavor is not detected:
1. Elastic Neutrino-Electron Scattering (ES): $\nu_\alpha e^- \to \nu_x e^-$ .
2. Coherent Elastic Neutrino-Nucleus Scattering (CE $\nu$ NS): $\nu_\alpha + A \to \nu_x + A$ .
Target Selection: To isolate vector interactions, the authors focus on spin-zero nuclear targets (e.g., $^{40}\text{Ar}$ , $^{76}\text{Ge}$ ). This suppresses the axial-vector contribution, which would otherwise obscure the vector current effects.

3. Key Contributions

The paper's primary contribution is the Generalization of the PDMCT. The authors demonstrate that the "confusion" between Dirac and Majorana neutrinos is not solely a function of kinematic mass suppression but is fundamentally governed by the CP structure of the new interaction.

The "CP-Filter" Mechanism:
- Dirac Neutrinos: Interact via the full modulus of the coupling $|g_{\alpha\beta}|^2$ . They are sensitive to both the real (CP-conserving) and imaginary (CP-violating) parts of the vector interaction. In the diagonal channel, they experience interference between the SM and NP amplitudes.
- Majorana Neutrinos: Due to Fermi statistics, the diagonal vector current vanishes. For flavor-changing transitions, the Majorana condition acts as a strict filter:
  - The real (CP-conserving) component of the vector coupling cancels out exactly.
  - Only the imaginary (CP-violating) component survives.
  - The effective coupling becomes proportional to $2i \text{Im}(g_{\alpha\beta}^V)$ .
Lifting the Suppression: Consequently, the difference in cross-sections is no longer suppressed by $(m_\nu/E_\nu)^2$ . Instead, the distinguishability depends on the CP-violating phase $\phi$ . If $\phi \to \pi/2$ (maximal CP violation), the Majorana signal is maximized; if CP is conserved ( $\phi = 0$ ), the Majorana vector FCNC contribution vanishes entirely.

4. Results

The authors derive specific phenomenological signatures for Dirac vs. Majorana neutrinos in inclusive scattering experiments:

Cross-Section Behavior:
- Dirac Case: The cross-section is dominated by the interference between the SM and the NP vector current in the diagonal channel. This leads to a term linear in the NP coupling strength ( $\epsilon$ ) and an asymmetric spectral shape in the recoil energy distribution.
- Majorana Case: The diagonal SM-NP interference is strictly zero. The signal arises solely from flavor-changing transitions driven by the CP-violating imaginary part. The cross-section scales as $\epsilon^2 \sin^2(\phi)$ and produces a symmetric spectral distortion.
Application to CE $\nu$ NS (e.g., COHERENT):
- Using spin-zero targets (like Liquid Argon), the axial background is removed.
- If neutrinos are Dirac, a deviation from the SM prediction should appear linearly with the coupling strength.
- If neutrinos are Majorana, the deviation is quadratically suppressed ( $\epsilon^2$ ) and depends entirely on the CP phase.
- The current COHERENT data, which aligns well with the SM, is interpreted as a potential hint for Majorana neutrinos in the presence of a light $Z'$ , as it naturally suppresses the large linear deviations expected for Dirac neutrinos.
Application to DUNE:
- The DUNE Near Detector, measuring inclusive $\nu-e$ scattering, could distinguish the two hypotheses by analyzing the recoil electron energy distribution. A symmetric distribution would favor the Majorana hypothesis driven by CP-violating FCNC, while an asymmetric distribution would support the Dirac hypothesis.

5. Significance

This work fundamentally alters the strategy for determining the neutrino's nature:

Beyond $0\nu\beta\beta$ : It suggests that the neutrino nature can be probed via high-energy scattering experiments, not just low-energy lepton-number-violating decays.
Interplay of CP and Statistics: It reveals a deep connection between the quantum statistics of neutrinos (Majorana vs. Dirac) and CP violation. The "Majorana nature" acts as a filter that eliminates CP-conserving vector interactions, leaving only CP-violating ones.
Experimental Guidance: It provides a clear roadmap for current and future experiments (COHERENT, DUNE, future CE $\nu$ NS detectors). By using spin-zero targets and analyzing spectral shapes (symmetric vs. asymmetric) and the scaling of deviations (linear vs. quadratic), experiments can place constraints on the CP-violating phase of New Physics or potentially identify the Majorana nature of neutrinos without relying on mass suppression arguments.

In summary, the paper argues that if New Physics involves CP-violating FCNC vector interactions, the "Practical Dirac-Majorana Confusion Theorem" is generalized: the distinction between the two neutrino types becomes a macroscopic observable dependent on the CP phase, rather than a negligible kinematic effect.