Practical Dirac-Majorana Confusion Theorem

• The pDMCT is a theoretical result demonstrating that kinematic differences between Dirac and Majorana neutrinos are suppressed by mν² when full phase-space and spin summation are applied.
• It employs standard model (V–A) interactions where integrating over undetected neutrino variables renders the observable differences practically unmeasurable under typical laboratory conditions.
• Special experimental setups or non-standard interactions, such as exclusive kinematic reconstructions or CP-violating mediators, can circumvent the theorem’s suppression to yield observable effects.

The Practical Dirac-Majorana Confusion Theorem (commonly abbreviated as pDMCT or DMCT) encapsulates the empirical indistinguishability, in conventional kinematic observables, between Dirac and Majorana neutrinos in the limit of vanishing neutrino masses for a broad class of processes governed by Standard Model (SM) interactions. This theorem plays a pivotal role in the phenomenology of neutrino physics, delineating the fundamental and practical limitations in probing the nature of neutrinos through laboratory experiments, and establishing the need for targeted search strategies that go beyond naive kinematic integration. The theorem's status, exceptions, and generalizations are matters of ongoing precise theoretical analysis and experimental planning.

1. Formal Statement and Mathematical Structure

The pDMCT states: in any SM process where the final state contains an undetected neutrino-antineutrino ($\nu$, $\bar\nu$) pair and the observable integrates over their full phase space and over spins, the difference in any kinematic observable between the Dirac and Majorana hypotheses is suppressed by the square of the neutrino mass, becoming negligible as $m_\nu \to 0$ (Kim, 2023, Kim et al., 2021, Bigaran et al., 9 Jul 2025).

Mathematically, if $\mathscr M^D = \mathscr M(p_1,p_2)$ is the amplitude for Dirac neutrinos, and for Majorana neutrinos one must antisymmetrize,

$$\mathscr M^M = \frac{1}{\sqrt{2}} \left[\mathscr M(p_1, p_2) - \mathscr M(p_2, p_1)\right],$$

then after spin summation,

$$|\mathscr M^D|^2 - |\mathscr M^M|^2 = \frac{1}{2}\left[|\mathscr M(p_1,p_2)|^2 - |\mathscr M(p_2,p_1)|^2\right] + \operatorname{Re}\left[\mathscr M(p_1,p_2) \mathscr M^*(p_2,p_1)\right].$$

Integrating over all neutrino phase space, the symmetric integration ensures that the first term cancels, and the second is suppressed by $m_\nu^2$ due to helicity flips required for the interference term. For typical processes,

$O_D - O_M \propto m_\nu^2,$

which is totally negligible for $m_\nu \lesssim \textrm{eV}$-scale (Kim, 2023, Bigaran et al., 9 Jul 2025).

2. Physical Interpretation, Assumptions, and Domain of Applicability

The confusion theorem is not a consequence of any deep quantum field-theoretic identity, but rather a phenomenological result arising from the SM flavor structure (V–A weak currents), the practical impossibility of resolving individual neutrino quantum numbers (helicity, flavor, lepton number), and the necessity to integrate over unobservable neutrino momenta (Kim, 2023). The theorem presumes:

• Only SM (V–A) couplings and neutral currents, or charged-current processes where the neutrino pair is untagged
• Analyticity in $m_\nu$ through zero in the amplitudes
• Summation over all spins; neutrino polarization is undetectable at sub-eV
• No access to kinematic configurations that permit unique reconstruction of the individual neutrino momenta (with rare exceptions; see below)

Historically, the theorem arose from analysis of SM processes such as $Z \to \nu \bar{\nu}$, $e^+ e^- \to \nu \bar{\nu}$, $B \to K \nu \bar{\nu}$, where integrating out the neutrino variables ensures the practical impossibility of distinguishing Dirac from Majorana neutrinos (Kim, 2023).

3. Explicit Examples and Calculational Illustrations

The theorem applies to a broad class of SM-mediated observables, as seen in the following processes (Kim, 2023, Kim et al., 2021):

Process	Coupling	Outcome of Dirac–Majorana difference
$Z \to \nu \bar{\nu}$	NC	$\propto m_\nu^2$
$e^+ e^- \to \nu \bar{\nu}$	NC	$\propto m_\nu^2$
$B \to K \nu \bar{\nu}$	FCNC (loop)	$\propto m_\nu^2$
$\ell \to \ell' \nu \bar{\nu} \gamma$	CC	$\propto m_\nu^2$ after integration
CE$\nu$NS on spin-zero targets	NC	$\propto m_\nu^2$ (SM), see below

Amplitude-level calculations using spinor-helicity methods show no misalignment with this theorem: All observable differences after correct antisymmetrization and helicity summation vanish as $m_\nu^2 / E^2$ (with $E$ the typical energy scale) (Bigaran et al., 9 Jul 2025). Gravitational scattering experiments (Schwarzschild backgrounds) likewise yield identical results for Dirac and Majorana fermions at leading order (Lai et al., 2021).

4. Known Loopholes, Limitations, and Methods to Evade the Theorem

Despite its generality, the pDMCT is not fundamental and can be evaded under precisely delineated circumstances (Kim, 2023, Kim et al., 2021, Kim et al., 2024):

• Special Kinematic Reconstruction: In processes such as $B^0 \to \mu^+ \mu^- \nu_\mu \bar{\nu}_\mu$, imposing a back-to-back kinematic configuration in the rest frame uniquely determines the neutrino and antineutrino 4-momenta from measured charged-lepton momenta. In this scenario, antisymmetrized Majorana amplitudes yield a non-zero difference from Dirac, manifesting as $O(1)$ distortions in angular/differential distributions, independent of $m_\nu$ (Kim et al., 2021, Kim et al., 2024). This results from not integrating over all neutrino variables, thus preserving the quantum-statistical signature of Majorana statistics.
• Non-Standard Interactions: Introducing new operators (e.g. scalar, tensor, or right-handed neutral currents) can disrupt the complete cancellation and result in unsuppressed Dirac–Majorana differences, even for vanishing $m_\nu$ (Kim, 2023, Delepine et al., 24 Jan 2026). For instance, in neutral vector boson ($Z'$) extensions with CP-violating phases, Majorana neutrinos couple only to the imaginary part, and the distinction scales as $\epsilon^2 \sin^2\phi$ (with $\epsilon$ a small new-physics coupling and $\phi$ the CP phase), rather than $m_\nu^2/E^2$ (Delepine et al., 24 Jan 2026).
• Vector Torsion Effects: In weak field gravity contexts, while pure gravitational and axial-torsion couplings cannot distinguish the two types, vector torsion in the background field contributes for Dirac but cancels for Majorana neutrinos. This provides, at least theoretically, a geometric interaction sensitive to the neutrino's nature (Lai et al., 2021).
• Incomplete Phase-Space Integration and Quantum Measurement: If a measurement projects the neutrinos into distinguishable final states (e.g., via lepton number or helicity), the necessary antisymmetrization for Majorana pairs no longer applies, and the Dirac–Majorana distinction in quantum statistics becomes unobservable (Kim et al., 2024).

5. Practical and Experimental Implications

Within the SM, and under typical experimental conditions where neutrinos are not detected and all kinematic variables are integrated out, observable Dirac–Majorana rate differences are unmeasurably small: for $m_\nu/E \lesssim 10^{-8}$, the effect is orders of magnitude below experimental sensitivity (Kim, 2023, Bigaran et al., 9 Jul 2025).

Experimental searches circumvent this practical indistinguishability by:

• Focusing on exclusive kinematic regions (e.g., fully reconstructible back-to-back settings in multi-body decays) (Kim et al., 2021).
• Searching for lepton-number violating (LNV) processes (e.g., neutrinoless double beta decay $0\nu\beta\beta$), which are strictly forbidden for Dirac neutrinos.
• Probing for new physics signatures in neutral-current scattering, such as CE$\nu$NS with vector bosons and observable CP-violating effects (Delepine et al., 24 Jan 2026).
• High-precision measurements in rare meson decays, using techniques that can infer missing momenta to differentiate distributions under the Dirac and Majorana hypotheses (Kim et al., 2024).

6. Theoretical Generalizations and Ongoing Controversies

Recent work generalizes the pDMCT in the context of physics beyond the SM. If new mediators (e.g., $Z'$ bosons) possess complex, CP-violating couplings, the character of Dirac–Majorana confusion is altered: for spin-zero targets, the Majorana contribution depends solely on the imaginary part of the coupling, lifting the $m_\nu^2/E^2$ suppression of the difference in cross-sections (Delepine et al., 24 Jan 2026). This introduces new observables in coherent neutrino-nucleus scattering, offering direct sensitivity to both the neutrino's nature and CP structure.

There has been technical debate regarding the extent to which particular kinematic selections, such as the back-to-back configuration in four-body decays, genuinely evade the confusion theorem. Some analyses assert that after a complete and consistent phase space integration, the distinction again vanishes (Márquez et al., 2023). Others emphasize the critical role of not projecting onto distinguishable neutrino states, and maintaining a quantum-statistical interference term in the observable, as clarified in (Kim et al., 2024, Kim et al., 2021).

7. Summary Table: Conditions Establishing or Evading pDMCT

Condition	Dirac–Majorana Difference?	Reference
Full integration, SM V–A, undetected ν	$\propto m_\nu^2/E^2$ (negligible)	(Kim, 2023, Bigaran et al., 9 Jul 2025)
Exclusive kinematic region (reconstructed)	$O(1)$ in distribution	(Kim et al., 2021, Kim et al., 2024)
Presence of CP-violating $Z'$	$\propto \epsilon^2 \sin^2\phi$	(Delepine et al., 24 Jan 2026)
Gravitational/axial torsion only	None	(Lai et al., 2021)
Vector torsion	Yes	(Lai et al., 2021)

In summary, the practical Dirac-Majorana confusion theorem, while robust for SM processes involving unresolved neutrino pairs, is a consequence of phase-space integration, helicity summation, and SM symmetry structure—not a fundamental prohibition. Its domain of applicability can be circumvented via carefully designed observables, kinematic selections, or the introduction of SM extensions with new interactions, thereby allowing experimental distinction between Dirac and Majorana neutrinos under special circumstances (Kim, 2023, Kim et al., 2021, Delepine et al., 24 Jan 2026).