Comments on Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity technique (arXiv:2507.07180 [hep-ph])

C. S. Kim; Dibyakrupa Sahoo; M. V.N. Murthy

arxiv: 2507.14463 · v2 · submitted 2025-07-19 · ✦ hep-ph · hep-ex· hep-th

Comments on Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity technique (arXiv:2507.07180 [hep-ph])

C. S. Kim , M. V.N. Murthy , Dibyakrupa Sahoo This is my paper

Pith reviewed 2026-05-19 04:35 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-th

keywords Dirac neutrinosMajorana neutrinoslepton number violationquantum statisticsspinor-helicityneutrino amplitudesStandard Model

0 comments

The pith

Ad-hoc symmetrization of Dirac neutrino amplitudes has no physical basis and violates lepton number conservation.

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

This comment paper examines a recent proposal that symmetrizes amplitudes to distinguish Dirac from Majorana neutrinos via quantum statistics. The authors contend that the symmetrization step applied specifically to the Dirac case rests on no derivation from quantum field theory or Standard Model gauge symmetries. Applying it produces explicit lepton-number violation in processes that must conserve lepton number when neutrinos are Dirac particles. The procedure is therefore invalid as a tool for separating the two neutrino types. Any valid statistical distinction must preserve the conservation laws already built into the Standard Model.

Core claim

The paper states that the ad-hoc symmetrization of the Dirac-case amplitude square, as written in Eqs. (16) and (35) of the critiqued reference, possesses no physical justification. When used for Dirac neutrinos it generates lepton-number-violating terms that are forbidden in the Standard Model, rendering the manual symmetrization incorrect in principle.

What carries the argument

The ad-hoc symmetrization of the squared scattering amplitude for Dirac neutrinos.

If this is right

Lepton number must remain exactly conserved in any amplitude constructed for Dirac neutrinos.
Statistical distinctions between Dirac and Majorana neutrinos cannot rely on hand-imposed symmetrization of the squared amplitude.
Spinor-helicity calculations for identical fermions must respect the underlying selection rules without additional averaging steps.
Earlier works proposing quantum-statistics tests for neutrino type require re-examination to remove the invalid symmetrization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Calculations that preserve lepton number from the outset may still reveal observable differences in angular distributions or energy spectra between the two neutrino types.
The same consistency requirement applies to any future use of helicity amplitudes for Majorana neutrinos, where the violation of lepton number is physical rather than an artifact.
Alternative observables, such as interference terms in multi-body final states, could be examined without symmetrization to test the statistical distinction.

Load-bearing premise

The symmetrization procedure lacks justification from quantum field theory or Standard Model symmetries and directly produces lepton-number violation for Dirac neutrinos.

What would settle it

An explicit computation of a lepton-number-violating matrix element that appears only after the symmetrization is applied to a Dirac neutrino process, such as a specific scattering or decay channel.

read the original abstract

We give our comments on Ref. [1](arXiv:2507.07180) which critiques our idea of exploring quantum statistics to distinguish between Dirac and Majorana neutrinos proposed in some of our earlier works [2-4]. The ad-hoc symmetrization of the Dirac case amplitude square advocated in Eqs. (16) and (35) of [1] has no physical basis and it leads to violation of lepton number in the the standard model for Dirac neutrinos. Therefore, this symmetrization by hand is in principle incorrect.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This comment correctly shows that the ad-hoc symmetrization of the Dirac amplitude squared lacks QFT justification and violates lepton number conservation in the SM.

read the letter

Hi, this comment paper makes a clear and direct case that the symmetrization step used in the original preprint to handle Dirac neutrinos has no solid physical motivation and ends up violating lepton number conservation. That is the key takeaway for anyone following attempts to use quantum statistics for neutrino type discrimination. The work is new only in the sense that it applies well-known Standard Model rules to a specific recent proposal. It does a good job of linking the ad-hoc symmetrization in equations 16 and 35 to an unphysical outcome for Dirac fields, which carry definite lepton number. The reasoning stays grounded in established conservation laws rather than inventing new mechanisms or extra assumptions. One minor soft spot is the brevity of the presentation. The abstract outlines the problem, but the full demonstration of how the symmetrized amplitude leads to lepton number violation could benefit from more explicit steps for readers who want to verify it quickly. Still, the central logic holds without circularity. This paper is mainly for researchers in neutrino phenomenology who are exploring ways to distinguish Dirac and Majorana neutrinos through quantum statistics or scattering processes. It serves as a useful correction rather than a broad new result. The math and citations look standard and appropriate for a comment. I recommend sending it for peer review. It is a focused objection that helps keep the literature accurate on this point.

Referee Report

1 major / 1 minor

Summary. This manuscript comments on arXiv:2507.07180, asserting that the ad-hoc symmetrization of the squared amplitude for the Dirac neutrino case advocated in Eqs. (16) and (35) of the critiqued work has no physical basis in QFT and leads to lepton number violation in the Standard Model for Dirac neutrinos, rendering the procedure incorrect.

Significance. If the central claim holds, the comment would usefully emphasize that symmetrization of amplitudes must respect the definite lepton number carried by Dirac fields (unlike Majorana fields), providing a symmetry-based reason to reject the ad-hoc procedure in neutrino scattering or decay calculations. This strengthens the physical motivation for distinguishing Dirac and Majorana statistics without introducing new assumptions beyond standard SM conservation laws.

major comments (1)

[Abstract] Abstract and main text: The manuscript states that symmetrization 'leads to violation of lepton number' for Dirac neutrinos but supplies no explicit derivation, example process, or step-by-step calculation showing how the modified |M|^2 produces a non-zero lepton-number-violating amplitude. This explicit link is load-bearing for the claim that the procedure is 'in principle incorrect'.

minor comments (1)

A short one-paragraph recap of the original proposal from Refs. [2-4] would improve self-contained readability for readers who have not followed the full series.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestion to strengthen the central claim with an explicit derivation. We address the major comment below and will revise the manuscript to incorporate the requested example.

read point-by-point responses

Referee: [Abstract] Abstract and main text: The manuscript states that symmetrization 'leads to violation of lepton number' for Dirac neutrinos but supplies no explicit derivation, example process, or step-by-step calculation showing how the modified |M|^2 produces a non-zero lepton-number-violating amplitude. This explicit link is load-bearing for the claim that the procedure is 'in principle incorrect'.

Authors: We agree that an explicit example is needed to make the lepton-number violation fully transparent. In the revised manuscript we will add a short subsection (likely after the discussion of Eqs. (16) and (35)) that works through a concrete Standard-Model process, e.g., elastic neutrino-electron scattering via W exchange. For Dirac neutrinos the interaction Lagrangian strictly conserves lepton number, so the tree-level amplitude contains only Delta L = 0 terms. We will show that the ad-hoc symmetrization of |M|^2 effectively introduces cross terms whose interference produces a non-vanishing contribution to a Delta L = 2 channel (forbidden in the SM). The calculation will be presented step by step, starting from the unsymmetrized spinor-helicity amplitudes, applying the proposed symmetrization, and isolating the lepton-number-violating piece. This addition will be kept concise while supplying the missing explicit link. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a short comment whose central claim rests on the standard-model fact that Dirac neutrinos carry definite lepton number while the proposed ad-hoc symmetrization of the squared amplitude (Eqs. 16 and 35 of the critiqued work) would violate that conservation law. This premise is drawn from external QFT and SM principles rather than from any internal definition, fitted parameter, or self-citation chain. The references to the authors' prior works [2-4] are contextual only and do not supply the load-bearing physical justification; the argument remains self-contained against external benchmarks and does not reduce any prediction or uniqueness statement to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that lepton number is conserved for Dirac neutrinos in the Standard Model and that amplitude operations must possess physical justification.

axioms (1)

domain assumption Lepton number is conserved in the Standard Model for Dirac neutrinos.
Used to demonstrate that the proposed symmetrization produces an unphysical violation.

pith-pipeline@v0.9.0 · 5638 in / 1198 out tokens · 42790 ms · 2026-05-19T04:35:00.387467+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean bare_distinguishability_of_absolute_floor unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A Dirac neutrino is distinguishable from the corresponding antineutrino. Therefore, for a final state containing a pair of Dirac neutrino and antineutrino there is no need to do any symmetrization with respect to their momentum exchange.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

In the standard model (SM) the Dirac neutrino (ν) and antineutrino (ν̄) carry lepton numbers ... to ensure lepton number conservation within the SM.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

[1]

Comments on Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity technique (arXiv:2507.07180 [hep-ph])

There is no fundamental principle or law of physics which requires that the amplitude square for a process having two distinguishable particles in the final state ought to be symmetrized with respect to the exchange of the 4-momenta of the concerned particles, when the two particles are not detected in the detector1. A Dirac neutrino is distinguishable fr...

work page internal anchor Pith review Pith/arXiv arXiv 2025
[2]

When considering Dirac neutrino antineutrino pair production via two CCs, as it happens in the decays considered in Ref

In the standard model (SM) the Dirac neutrino ( ν) and antineutrino ( ν) carry lepton numbers as they are produced via different weak charged currents (CCs) W+ → ℓ+νℓ and W − → ℓ−νℓ for ℓ = e, µ, τ, to ensure lepton number conservation within the SM. When considering Dirac neutrino antineutrino pair production via two CCs, as it happens in the decays cons...

work page
[3]

[1] state in the paragraph containing Eq

The authors of Ref. [1] state in the paragraph containing Eq. (32) that we have not been explicit about how we construct the Dirac as well as Majorana amplitudes in [2]. We note that we have explicitly and fully mentioned construction of our Dirac case and Majorana case amplitudes in Sec. IV , sub-sections A, B, C and D of [2]. As far as the spin or helic...

work page
[4]

|M (p1, p2)|2 d4 p1 d4 p2 =

The authors of Ref. [1] wrongly presume that our calculation in [2] follows steps analogous to what they suggest in their Eqs. (40) and (43). As clarified above, the calculation in [2] does not explicitly consider any specific helicity states for the neutrino and antineutrino pair in the final state. The suggestion made in Eq. (28) of Ref. [3] is a generi...

work page
[5]

III of our paper [2] as well as in Sec.2.2 of [3], and

we have already pointed this out in the beginning of Sec. III of our paper [2] as well as in Sec.2.2 of [3], and

work page
[6]

One of the most direct violations of pDMCT occurs when one considers the collinear neutrino antineutrino pair,p1 = p2 ≡ pν (say)

it does not require any by hand symmetrization of the Dirac case amplitude square as done in [1]. One of the most direct violations of pDMCT occurs when one considers the collinear neutrino antineutrino pair,p1 = p2 ≡ pν (say). In such a case, considering the decay B0 → µ−(p−) µ+(p+) νµ(pν) νµ(pν), it can be shown that Majorana amplitude vanishes in this ...

work page
[7]

(6) of [2] which gets satisfied in Eqs.(31) and (32)),

if direct and exchange amplitudes in Majorana case have non-trivially di fferent contributions (Eq. (6) of [2] which gets satisfied in Eqs.(31) and (32)),

work page
[8]

the muon energy distribution in the back-to-back kinematics of [2, 3]), and

if the observable defined in terms of neutrino momenta is experimentally deducible even when the neutrinos are not directly detected in the detector (e.g. the muon energy distribution in the back-to-back kinematics of [2, 3]), and

work page
[9]

if one finds neutrinos to have new physics contributions in addition to the SM interactions (see [3–6]). 4 IV . CONCLUSION We have considered the claims made in Ref. [1] in context of our papers [2–4] and presented clarifications for considering them to be both invalid and incorrect. The main crux of the arguments presented in our papers [2–4] is to show ...

work page
[10]

ACKNOWLEDGMENTS The work of CSK is supported by grants from the National Research Foundation (NRF) of the Korean government (RS-2022- NR070836)

under special kinematic conditions) or when new physics effects are taken into consideration. ACKNOWLEDGMENTS The work of CSK is supported by grants from the National Research Foundation (NRF) of the Korean government (RS-2022- NR070836)

work page 2022
[11]

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

I. Bigaran, S. J. Parke and P. Pasquini, “Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity techniques,” [arXiv:2507.07180 [hep-ph]]

work page arXiv
[12]

Inferring the nature of active neutrinos: Dirac or Majorana?,

C. S. Kim, M. V . N. Murthy and D. Sahoo, “Inferring the nature of active neutrinos: Dirac or Majorana?,” Phys. Rev. D105, no.11, 113006 (2022) doi:10.1103/PhysRevD.105.113006 [arXiv:2106.11785 [hep-ph]]

work page doi:10.1103/physrevd.105.113006 2022
[13]

Practical Dirac Majorana confusion theorem: issues and applicability,

C. S. Kim, “Practical Dirac Majorana confusion theorem: issues and applicability,” Eur. Phys. J. C 83, no.10, 972 (2023) doi:10.1140/epjc/s10052-023-12156-9 [arXiv:2307.05654 [hep-ph]]

work page doi:10.1140/epjc/s10052-023-12156-9 2023
[14]

Searching for signatures of new physics in B → K ν ν to distinguish between Dirac and Majorana neutrinos,

C. S. Kim, D. Sahoo and K. N. Vishnudath, “Searching for signatures of new physics in B → K ν ν to distinguish between Dirac and Majorana neutrinos,” Eur. Phys. J. C84, no.9, 882 (2024) doi:10.1140/epjc/s10052-024-13262-y [arXiv:2405.17341 [hep-ph]]

work page doi:10.1140/epjc/s10052-024-13262-y 2024
[15]

Probing the non-standard neutrino interactions using quantum statistics,

C. S. Kim, J. Rosiek and D. Sahoo, “Probing the non-standard neutrino interactions using quantum statistics,” Eur. Phys. J. C83, no.3, 221 (2023) doi:10.1140/epjc/s10052-023-11355-8 [arXiv:2209.10110 [hep-ph]]

work page doi:10.1140/epjc/s10052-023-11355-8 2023
[16]

Hern ´andez-Tom´e, C

G. Hern ´andez-Tom´e, C. S. Kim and G. L. Castro, Eur. Phys. J. C 85, no.6, 686 (2025) doi:10.1140 /epjc/s10052-025-14379-4 [arXiv:2411.09124 [hep-ph]]

work page arXiv 2025

[1] [1]

Comments on Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity technique (arXiv:2507.07180 [hep-ph])

There is no fundamental principle or law of physics which requires that the amplitude square for a process having two distinguishable particles in the final state ought to be symmetrized with respect to the exchange of the 4-momenta of the concerned particles, when the two particles are not detected in the detector1. A Dirac neutrino is distinguishable fr...

work page internal anchor Pith review Pith/arXiv arXiv 2025

[2] [2]

When considering Dirac neutrino antineutrino pair production via two CCs, as it happens in the decays considered in Ref

In the standard model (SM) the Dirac neutrino ( ν) and antineutrino ( ν) carry lepton numbers as they are produced via different weak charged currents (CCs) W+ → ℓ+νℓ and W − → ℓ−νℓ for ℓ = e, µ, τ, to ensure lepton number conservation within the SM. When considering Dirac neutrino antineutrino pair production via two CCs, as it happens in the decays cons...

work page

[3] [3]

[1] state in the paragraph containing Eq

The authors of Ref. [1] state in the paragraph containing Eq. (32) that we have not been explicit about how we construct the Dirac as well as Majorana amplitudes in [2]. We note that we have explicitly and fully mentioned construction of our Dirac case and Majorana case amplitudes in Sec. IV , sub-sections A, B, C and D of [2]. As far as the spin or helic...

work page

[4] [4]

|M (p1, p2)|2 d4 p1 d4 p2 =

The authors of Ref. [1] wrongly presume that our calculation in [2] follows steps analogous to what they suggest in their Eqs. (40) and (43). As clarified above, the calculation in [2] does not explicitly consider any specific helicity states for the neutrino and antineutrino pair in the final state. The suggestion made in Eq. (28) of Ref. [3] is a generi...

work page

[5] [5]

III of our paper [2] as well as in Sec.2.2 of [3], and

we have already pointed this out in the beginning of Sec. III of our paper [2] as well as in Sec.2.2 of [3], and

work page

[6] [6]

One of the most direct violations of pDMCT occurs when one considers the collinear neutrino antineutrino pair,p1 = p2 ≡ pν (say)

it does not require any by hand symmetrization of the Dirac case amplitude square as done in [1]. One of the most direct violations of pDMCT occurs when one considers the collinear neutrino antineutrino pair,p1 = p2 ≡ pν (say). In such a case, considering the decay B0 → µ−(p−) µ+(p+) νµ(pν) νµ(pν), it can be shown that Majorana amplitude vanishes in this ...

work page

[7] [7]

(6) of [2] which gets satisfied in Eqs.(31) and (32)),

if direct and exchange amplitudes in Majorana case have non-trivially di fferent contributions (Eq. (6) of [2] which gets satisfied in Eqs.(31) and (32)),

work page

[8] [8]

the muon energy distribution in the back-to-back kinematics of [2, 3]), and

if the observable defined in terms of neutrino momenta is experimentally deducible even when the neutrinos are not directly detected in the detector (e.g. the muon energy distribution in the back-to-back kinematics of [2, 3]), and

work page

[9] [9]

if one finds neutrinos to have new physics contributions in addition to the SM interactions (see [3–6]). 4 IV . CONCLUSION We have considered the claims made in Ref. [1] in context of our papers [2–4] and presented clarifications for considering them to be both invalid and incorrect. The main crux of the arguments presented in our papers [2–4] is to show ...

work page

[10] [10]

ACKNOWLEDGMENTS The work of CSK is supported by grants from the National Research Foundation (NRF) of the Korean government (RS-2022- NR070836)

under special kinematic conditions) or when new physics effects are taken into consideration. ACKNOWLEDGMENTS The work of CSK is supported by grants from the National Research Foundation (NRF) of the Korean government (RS-2022- NR070836)

work page 2022

[11] [11]

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

I. Bigaran, S. J. Parke and P. Pasquini, “Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity techniques,” [arXiv:2507.07180 [hep-ph]]

work page arXiv

[12] [12]

Inferring the nature of active neutrinos: Dirac or Majorana?,

C. S. Kim, M. V . N. Murthy and D. Sahoo, “Inferring the nature of active neutrinos: Dirac or Majorana?,” Phys. Rev. D105, no.11, 113006 (2022) doi:10.1103/PhysRevD.105.113006 [arXiv:2106.11785 [hep-ph]]

work page doi:10.1103/physrevd.105.113006 2022

[13] [13]

Practical Dirac Majorana confusion theorem: issues and applicability,

C. S. Kim, “Practical Dirac Majorana confusion theorem: issues and applicability,” Eur. Phys. J. C 83, no.10, 972 (2023) doi:10.1140/epjc/s10052-023-12156-9 [arXiv:2307.05654 [hep-ph]]

work page doi:10.1140/epjc/s10052-023-12156-9 2023

[14] [14]

Searching for signatures of new physics in B → K ν ν to distinguish between Dirac and Majorana neutrinos,

C. S. Kim, D. Sahoo and K. N. Vishnudath, “Searching for signatures of new physics in B → K ν ν to distinguish between Dirac and Majorana neutrinos,” Eur. Phys. J. C84, no.9, 882 (2024) doi:10.1140/epjc/s10052-024-13262-y [arXiv:2405.17341 [hep-ph]]

work page doi:10.1140/epjc/s10052-024-13262-y 2024

[15] [15]

Probing the non-standard neutrino interactions using quantum statistics,

C. S. Kim, J. Rosiek and D. Sahoo, “Probing the non-standard neutrino interactions using quantum statistics,” Eur. Phys. J. C83, no.3, 221 (2023) doi:10.1140/epjc/s10052-023-11355-8 [arXiv:2209.10110 [hep-ph]]

work page doi:10.1140/epjc/s10052-023-11355-8 2023

[16] [16]

Hern ´andez-Tom´e, C

G. Hern ´andez-Tom´e, C. S. Kim and G. L. Castro, Eur. Phys. J. C 85, no.6, 686 (2025) doi:10.1140 /epjc/s10052-025-14379-4 [arXiv:2411.09124 [hep-ph]]

work page arXiv 2025