Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity techniques

Innes Bigaran; Pedro Pasquini; Stephen J. Parke

Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity techniques

T0 review · 0 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For the decay of a light scalar into two charged leptons and two neutrinos, the spin-summed amplitude squared for Majorana neutrinos equals the Dirac result plus a term of order $m_\nu^2(k\cdot\bar{k})$ that vanishes as the neutrino mass…

desk verdict A solid, explicit spinor-helicity calculation that confirms the Dirac-Majorana confusion theorem for scalar decay into two charged leptons and two neutrinos, with a sound phase-invariance argument that pins down the correct antisymmetrization. read the letter →

arxiv 2507.07180 v1 pith:ZFALJRES submitted 2025-07-09 hep-ph hep-exhep-th

Innes Bigaran , Stephen J. Parke , Pedro Pasquini This is my paper

classification hep-phhep-exhep-th

keywords Dirac-MajoranaconfusiontheoremMajorananeutrinosDiracspinor-helicityamplitudesquantumstatisticsneutrinomasssuppressionantisymmetrizationscalardecaytoleptons

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether the quantum statistics of neutrinos can be used to tell Dirac from Majorana particles in a decay with two neutrinos in the final state. For the decay of a light scalar into two charged leptons and two neutrinos, the authors compute all helicity amplitudes with massive spinor-helicity methods and show that the spin-summed amplitude squared for Majorana neutrinos differs from the Dirac result only by the term $m_\nu^2(k\cdot\bar{k})$. Because that term vanishes as $m_\nu \to 0$, the Dirac-Majorana confusion theorem holds for this process: correlations of the charged leptons cannot distinguish the two neutrino natures in the massless limit. The paper also identifies exactly where earlier claims of a violation went wrong: either the incorrect antisymmetrization of the Dirac amplitudes was used, or the helicity sum was performed coherently before squaring, which produces unphysical interference terms.

What carries the argument

The machinery is the massive spinor-helicity formalism, in which a massive momentum $p$ and spin vector $s$ are decomposed into two massless momenta, $p_p \equiv \tfrac{1}{2}(p+ms)$ and $q_p \equiv \tfrac{1}{2}(p-ms)$, so that helicity amplitudes can be evaluated with massless spinor techniques, with a momentum-dependent phase $\psi$ carried by each massive spinor. The load-bearing step is the choice of antisymmetrization that converts Dirac amplitudes into Majorana amplitudes: Option A pairs $D[L_1,R_2]$ with $D[R_2,L_1]$ (and similarly for the other helicities), which is the unique choice that preserves phase invariance and enforces the Pauli exclusion principle; Option B, which pairs $D[L_1,R_2]$ with $D[L_2,R_1]$, is rejected because it produces two independent opposite-helicity amplitudes, leaves a same-helicity amplitude that fails to vanish for identical neutrino momenta, and yields interference terms that depend on the arbitrary spinor phases. With Option A, the summed squared amplitudes of Dirac and Majorana neutrinos differ only by the identity in eq. (22), $m_\nu^2(k\cdot\bar{k})$ versus zero.

What would settle it

Compute the fully spin-summed amplitude squared for $\Phi\to\ell^+\ell^-\nu\nu$ using the Option B antisymmetrization ($D[L_1,R_2]-D[L_2,R_1]$) with two different choices of the spinor phases. The paper's claim implies the interference term changes with the phase choice, so the result is basis-dependent and cannot be a physical observable; if instead the result is phase-independent and differs from the Dirac squared amplitude by a term that does not vanish as $m_\nu\to0$, the central claim is wrong.

Watch

Extended reading notes

Core claim

The central claim is that the decay $\Phi(Q)\to l^+(\bar{k})\,\nu_1(p_1)\,l^-(k)\,\nu_2(p_2)$, with Standard Model left-chiral weak interactions, satisfies the Dirac-Majorana confusion theorem. With massless charged leptons, the spin-summed Dirac amplitude squared is proportional to $(k\cdot p_1)(\bar{k}\cdot p_2)+(k\cdot p_2)(\bar{k}\cdot p_1)$, while the Majorana amplitude squared is the same expression minus $m_\nu^2(k\cdot\bar{k})$; the extra term is the interference produced by antisymmetrization and vanishes in the zero-mass limit. The paper finds that there is exactly one physical way to antisymmetrize the Dirac amplitudes into Majorana amplitudes, which pairs each Dirac amplitude with the one obtained by interchanging which neutrino is the antiparticle without flipping helicities; the other mathematically possible pairing leads to amplitudes that violate the Pauli exclusion principle and whose interference depends on arbitrary spinor phases. The same conclusion holds when charged-lepton masses are included, and the spinorial structure is cross-checked against the related process $e^+e^- \to NN$.

Load-bearing premise

The conclusion rests on requiring physical amplitudes to be invariant under the arbitrary phase redefinitions of the massive spinors described in the appendix; if that phase-invariance requirement is dropped, the rejected antisymmetrization yields a non-vanishing Dirac-Majorana difference even as the neutrino mass goes to zero.

Editorial extensions

If this is right

In this decay, any helicity-summed observable that distinguishes Dirac from Majorana neutrinos is suppressed by a factor of order $m_\nu^2/M_\Phi^2$; the only phase-space region where the relative difference becomes of order one is where both neutrinos are non-relativistic in the parent rest frame, a region whose volume is of order $4m_\nu^2/M_\Phi^2$.
The discrepancy with earlier claims of a Dirac-Majorana distinction in $B^0\to\ell^+\ell^-\nu\nu$ is traced to three unphysical choices: omitting the alternative momentum assignment for the anti-neutrino, coherently summing the neutrino helicities before squaring, and using the Option B antisymmetrization.
Including charged-lepton masses does not change the conclusion: the Dirac-Majorana difference remains of the same form, with the Majorana rate no longer exactly zero at the both-neutrinos-at-rest threshold but still helicity-suppressed there.
The zero-mass coincidence holds amplitude by amplitude, not merely after spin summation, because the left-chiral interactions leave only one Dirac amplitude unsuppressed and the corresponding Majorana amplitudes tend to it.
Beyond the Standard Model, adding a right-handed $W$ or a $Z'$ can lift the confusion theorem; in the $W_R$ case the charged-lepton helicities suppress the new interference for light leptons but less so for $\tau^+\tau^-$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The phase-invariance criterion used to select Option A can be turned into a practical test for any proposed Dirac-vs-Majorana observable in multi-neutrino processes: if the observable changes when the massive spinors are rephased as in the appendix, it is not physical. (This sharpening is ours; the paper states the criterion but does not phrase it as a universal test.)
Applied to the related decay $B\to K\nu\nu$, the paper's logic implies that within the Standard Model the neutrino-pair correlations obey the confusion theorem, so a reported Dirac-Majorana distinction there would be evidence of new physics rather than of quantum statistics.
The phase-space analysis suggests an experimental strategy: because the only Standard Model region with an unsuppressed relative difference is the threshold where both neutrinos are nearly at rest, searches for new-physics contributions to these decays should be optimized for that phase-space corner even though its Standard Model yield is tiny.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper is worth reading: it settles the recent dispute over refs [8-10] for this process class, and it does so with an explicit calculation rather than a general argument. The new technical content is real. The authors construct massive spinor-helicity amplitudes for Phi -> l+l- nu nu, show that the correct antisymmetrization of Dirac amplitudes to Majorana amplitudes is Option A, and give a phase-invariance criterion in Appendix A.2 that excludes Option B because the Option B interference term changes under the spinor rephasings of eq. (A10). That argument is the hinge of the paper, and it holds up: physical amplitudes cannot depend on the arbitrary phases of the intermediate one-particle spinors used to build them, so any construction that does is not an S-matrix element. The independent check against Ma-Pantaleone in Appendix B strengthens the result.

The paper also does the right things with the limiting behavior. The m_nu -> 0 limit is smooth, the charged lepton mass case is treated consistently, and the final difference between Majorana and Dirac spin-summed squares is just m_nu^2 (k . kbar), confirming the DMCT for this decay. The diagnosis of what went wrong in refs [8-10] is precise: either they used the wrong antisymmetrization or a coherent helicity sum that is not basis independent.

The soft spots are modest. The phase-invariance criterion is sound, but it carries the entire weight of excluding Option B; a reader who demanded a derivation from first principles might want more, though I think the criterion is basic enough. The paper does not prove a general DMCT, only this process class, and the authors say so themselves. Some spinor manipulations are asserted rather than shown, but they are checkable and the cross-check catches any gross error. There is no code or data, but none is needed for a fully analytic calculation. The figures are illustrative rather than quantitative, which is fine.

Who gets value: anyone working on Dirac-Majorana distinction, especially the authors of refs [8-10] and people designing B0 decay searches. This deserves a serious referee and, in my view, acceptance after a careful check of the spinor phase conventions. I would cite it.

Referee Report

0 major / 5 minor

Summary. This paper uses massive spinor-helicity methods to compute tree-level amplitudes for the decay Φ → l+ l- νν with left-chiral W exchange. It presents explicit Dirac helicity amplitudes, discusses two possible antisymmetrizations to Majorana amplitudes, and selects Option A using the requirement that physical amplitudes be independent of spinor phase conventions. With that choice, the spin-summed Majorana amplitude squared equals the Dirac result plus a term proportional to −mν²(k·k̄), which vanishes as mν → 0, confirming the Dirac-Majorana confusion theorem for this process. The paper also treats massive charged leptons, applies the result to B0 → μ−ν μ+ν, and explains why previous claims of a violation relied on either an unphysical coherent spin sum or the wrong antisymmetrization.

Significance. If correct, the result resolves a recent controversy in a concrete Standard Model process and provides a transparent method for treating Majorana fermions in spinor-helicity calculations. The calculation is explicit, contains no fitted parameters, and is cross-checked against the independent trace result of Ma and Pantaleone for the crossing-related process e+e− → NN. The phase-invariance criterion that selects Option A is a standard and well-motivated requirement for an S-matrix element, and Appendix B gives strong independent support for the Option A amplitudes. The paper also offers a useful diagnostic of where earlier claims of Dirac-Majorana distinguishability go wrong, and its discussion of BSM extensions in Section V is clearly scoped and sensible.

minor comments (5)

[Section II] There is a typo in the opening sentence ('spinor-helicity techinique'), and in Section III A 'abreviation' should be 'abbreviation'.
[Section III C] The transition from the individual squared amplitudes in Eq. (19) to the spin-summed result in Eq. (21) is stated rather than shown; a short derivation of the cancellation of the trace terms, or a pointer to an appendix, would make the key result easier to verify.
[Figures 3-5] The axis labels in Figures 3-5 appear with placeholder or garbled symbols (for example '□i' instead of a differential decay-rate symbol), and the legends should explicitly define what 'i = M' and 'i = D' denote; as printed the figures are hard to read.
[Section III D, Eq. (23)] The quantities γp1 and γp2 should be explicitly defined as the neutrino Lorentz factors in the Φ rest frame immediately before Eq. (23), since the expression is frame-dependent.
[Section IV, Eq. (39)] There is a mismatched parenthesis in D[λ2, λ1) in Eqs. (39)-(43); the notation should be made uniform with the rest of the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained, uses no fitted parameters, and is cross-checked against an external trace calculation.

full rationale

The paper's central claim, that the spin-summed Majorana amplitude squared equals the Dirac one plus m_nu^2 (k·kbar), is derived from explicit helicity amplitudes. The Dirac amplitudes in eq. (11) follow from the massive spinor definitions in eq. (6) and the Fierz identity in eq. (10); the Majorana amplitudes in eqs. (17)-(18) are antisymmetrizations of those Dirac amplitudes. The summed results in eqs. (16) and (21) are obtained by direct summation over the listed helicity configurations, and the mass-dependent difference is an explicit output, not an input. The selection of Option A over Option B is argued from Pauli exclusion and from the phase-rephasing invariance of physical amplitudes in eqs. (A10)-(A14); Option B is rejected because it produces phase-dependent interference terms, not because the conclusion was assumed. The only self-citations, refs. [12], [13] and [19], concern the standard spinor-helicity technique and do not carry the DMCT claim. The result is independently checked against the external trace calculation of ref. [16] in Appendix B, and the paper explicitly states in the Introduction that no general model-independent proof of the DMCT presently exists. No parameter is fitted, and no prediction is equivalent by construction to an input. Circularity score is therefore 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The model scalar Φ is introduced purely as a demonstrative example of the B0 decay structure (Sections III and IV), not as a new physical entity. No free parameters are fitted; all masses are inputs. The central argument relies on standard QFT assumptions.

assumptions (4)

standard math Standard QFT spinor completeness and Dirac equation for massive fermions
Used to construct massive spinors in terms of massless momenta pp and qp (Section II, eqs. (4)-(6)).
domain assumption Physical amplitudes are invariant under arbitrary phase redefinitions of the constituent spinors
Invoked in Appendix A.2 (eqs. A10-A14) to distinguish physical Option A from unphysical Option B antisymmetrization; this is the load-bearing premise that rules out a potential DMCT violation.
domain assumption The weak interaction is left-chiral for neutrinos in the SM
The entire helicity suppression pattern and the specific form of the amplitudes (Section III.A) rely on γL couplings; DMCT validity is tied to this.
domain assumption Majorana neutrinos are identical particles requiring antisymmetrized amplitudes
Used in Section III.B to construct M amplitudes from Dirac amplitudes; the physical choice is Option A.

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Cite this review

Pith. "Pith review of Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity techniques." pith.science (2026). https://pith.science/paper/ZFALJRES

@misc{pith2026250707180,
  author       = {Pith},
  title        = {Pith review of: Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity techniques},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZFALJRES}},
  note         = {Machine review of arXiv:2507.07180}
}

read the original abstract

Recently, there has been interest in the applicability of quantum statistics to distinguish Dirac from Majorana neutrinos in multi-neutrino final states. In particular, debate has arisen over the validity of the Dirac-Majorana confusion theorem in these processes, i.e. that any distinction between the Dirac and Majorana processes goes to zero as the neutrino mass goes to zero. Here we approach this problem equipped with spinor-helicity methods generalized for massive Dirac and Majorana fermions. We explicitly calculate all helicity amplitudes for the decay of a light scalar particle to two neutrinos and two oppositely charged leptons. This allows us to pinpoint the crucial steps which could lead to claims of a violation of the confusion theorem. We show that if the correct anti-symmetrization of Dirac to Majorana amplitudes is used, identification of which is clear in this framework, and all relevant contributions are appropriately summed, a scalar decay into two charged leptons and two neutrinos satisfies the Dirac-Majorana confusion theorem.

Figures

Figures reproduced from arXiv: 2507.07180 by the authors.

**Figure 4.** FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** FIG. 5. The differential decay rates of Φ versus the square root of the product of the energy of the two neutrinos, [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

For a Dirac amplitude which is not helicity suppressed , such as L1R2, the anti-symmetrized is with a Dirac amplitude that is doubly helicity suppressed , L1R2, and similarly vice versa

work page

[2] [2]

This anti-symmetrisation corresponds to the simple interchange between fermion and antifermion for the neutrinos, without any additional spin flip

For a Dirac amplitude that is singly helicity suppressed , such as L1L2, the anti-symmetrized is with another singly suppressed Dirac amplitude, L1L2. This anti-symmetrisation corresponds to the simple interchange between fermion and antifermion for the neutrinos, without any additional spin flip. The helicity suppression is generated by the left-chiral n...

work page

[3] [3]

All of these spinors satisfy the appropriate Dirac equation, and are eigenstates of the spin matrix, γ5/s

A F ull Set of Spinors The full set of spinors for a massive fermion used in this work is given in Table A 1 and relationships for the phase ψ is given in Table II. All of these spinors satisfy the appropriate Dirac equation, and are eigenstates of the spin matrix, γ5/s. They are normalized and orthogonal as follows: U λ(p, s) Uλ′(p, s) = 2m δλλ′, V λ(p, ...

work page

[4] [4]

In fact, we can show the interference term for the anti-symmetrization of type B has effectively no physical meaning

A T echnical Argument Against Anti-symmetrization Option B An extra argument can be made against the anti-symmetrization choice of Option B. In fact, we can show the interference term for the anti-symmetrization of type B has effectively no physical meaning. To see this, consider the most general phase change in the spinors given in Table A 1 that is also...

work page

[5] [5]

For example, choose pp1 ∥ ¯k and pp2 ∥ k then all but one of the Majorana amplitudes in eq

Different Bases Charge-Lepton basis: In the neutrino rest frame, chose the neutrino spin to be aligned with the direction of one of the charged leptons in that frame and since they are both massless momenta they remain parallel under any boost. For example, choose pp1 ∥ ¯k and pp2 ∥ k then all but one of the Majorana amplitudes in eq. (17) and (18) are re...

work page 2020

[6] [6]

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B. Kayser, “Majorana Neutrinos and their Electromagnetic Properties,” Phys. Rev. D 26 (1982) 1662

work page 1982

[7] [7]

Electromagnetic Properties and Decays of Dirac and Majorana Neutrinos in a General Class of Gauge Theories,

R. E. Shrock, “Electromagnetic Properties and Decays of Dirac and Majorana Neutrinos in a General Class of Gauge Theories,” Nucl. Phys. B 206 (1982) 359–379

work page 1982

[8] [8]

Neutrinoless Double beta Decay in SU(2) x U(1) Theories,

J. Schechter and J. W. F. Valle, “Neutrinoless Double beta Decay in SU(2) x U(1) Theories,” Phys. Rev. D 25 (1982) 2951

work page 1982