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

This paper critiques a recent study (arXiv:2507.07180) by arguing that its proposed ad-hoc symmetrization of Dirac neutrino amplitudes lacks physical justification and incorrectly violates lepton number conservation within the Standard Model.

The Gist

The Big Picture: A Scientific "He Said, She Said"

Imagine a group of physicists (the authors of this paper: Kim, Murthy, and Sahoo) who proposed a new way to tell the difference between two types of invisible particles: Dirac neutrinos and Majorana neutrinos.

Think of these neutrinos like two types of twins:

• Dirac Neutrinos: Like distinct twins, one is a "brother" (neutrino) and one is a "sister" (antineutrino). They look different and have different "identities" (like carrying different amounts of "lepton number," a type of charge).
• Majorana Neutrinos: Like identical twins who are their own mirror images. A neutrino and an antineutrino are actually the same person.

The authors of this paper argued that by looking at how these particles behave in a specific experiment (a particle decay), we could use the rules of Quantum Statistics (the rulebook for how identical particles behave) to figure out which type of twin we are dealing with.

However, another group of physicists (Bigaran, Parke, and Pasquini) published a paper (Ref [1]) saying, "No, you can't do that. Your math is wrong."

This paper is the response. It is a "rebuttal" where the original authors say, "Actually, your math is wrong, and here is why your critique doesn't hold up."

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The Core Conflict: The "Symmetrization" Mistake

The main disagreement is about a mathematical step called symmetrization.

The Analogy: The Two-Child Party

Imagine a party where two children, Tim (a boy) and Tom (a girl), are running around.

• The Rule: In physics, if you have two identical particles (like two identical twins), you have to treat them as a single unit. If you swap their positions, the math has to account for that swap because you can't tell them apart. This is called symmetrization (or antisymmetrization for fermions).
• The Situation: In the experiment, Tim and Tom are running around, but the detectors are too far away to see them clearly. We only see the other guests at the party.

The Critics' Argument (Ref [1]):
The critics say: "Since we can't see Tim and Tom, and we don't know who is who, we should just assume they could be swapped. So, in our math, we should add a second version of the calculation where we pretend Tim is Tom and Tom is Tim, then add them together."

The Authors' Rebuttal (This Paper):
The authors say: "That is completely wrong! Tim and Tom are not identical. One is a boy, one is a girl. Even if we can't see them, they are still distinct people with different identities.

• The Mistake: By forcing the math to treat them as identical (swapping them), the critics are breaking the fundamental rules of the universe.
• The Consequence: In the Standard Model of physics, swapping a neutrino and an antineutrino is like swapping a positive charge with a negative charge. If you do that in the math, you are accidentally saying that electric charge (or lepton number) is not conserved. It's like calculating a bank transaction where money magically appears out of nowhere just because you forgot to look at the receipt.

Key Points Explained Simply

1. The "Invisible" Particle Problem

The critics argued: "Since the neutrinos are invisible, we must treat them as identical."
The authors reply: "That's like saying, 'Because I can't see the driver and the passenger in a car, I must assume they are the same person.'

• Reality: In physics, the math (the amplitude) is calculated based on what could happen, assuming we know everything. Whether a detector actually sees the particle or not doesn't change the laws of physics.
• The Fix: If a particle is invisible, we don't change the rules of the game; we just ignore that part of the data in the final calculation (we integrate over the phase space). We don't need to "fake" the math by swapping identities.

2. The "Lepton Number" Violation

The authors point out a fatal flaw in the critics' math.

• The Analogy: Imagine a law that says "You must always have a red ball and a blue ball."
• The Critics' Math: By swapping the neutrino and antineutrino, they accidentally create a scenario where you have two red balls or two blue balls.
• The Result: This violates the "Law of Conservation of Lepton Number." In the Standard Model, this is a hard rule. The authors say the critics' method breaks this rule, which means their method is fundamentally broken.

3. The "Spin" Confusion

The critics claimed the authors' math was vague about "spin" (a quantum property like a tiny internal compass).
The authors clarify: "We didn't ignore spin; we just didn't write it out in every single line because it's handled automatically by the standard rules of the Weak Force (the force that causes these decays). It's like saying, 'We didn't list the color of every car in the parking lot, but we know they are all cars.' The math works perfectly fine without listing every single detail explicitly."

The Conclusion: Who is Right?

The authors conclude that:

1. The Critics are Wrong: Their method of "symmetrizing" (swapping) the math for Dirac neutrinos is an "ad hoc" (made-up) fix that has no basis in real physics. It breaks the rules of the Standard Model.
2. The Original Idea Stands: Their original work is correct. Quantum statistics can help distinguish between Dirac and Majorana neutrinos, but only if you do the math correctly without forcing the particles to be identical when they aren't.
3. The "Confusion Theorem": There is a famous idea called the "Dirac-Majorana Confusion Theorem" which says, "If you can't see the neutrinos, you can't tell them apart." The authors agree this is true if you do the math right (by integrating over all possibilities). However, they argue that the critics tried to prove this by using wrong math, which invalidates their critique.

Summary in One Sentence

The authors are telling the critics: "You tried to fix a math problem by pretending two different people are the same person, which breaks the laws of physics; our original math was correct, and your critique is based on a misunderstanding of how invisible particles are handled in quantum mechanics."

Technical Summary

Based on the provided text, here is a detailed technical summary of the paper "Comments on 'Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity techniques'" by C. S. Kim, M. V. N. Murthy, and Dibyakrupa Sahoo.

1. Problem Statement

The paper addresses a theoretical dispute regarding the distinction between Dirac and Majorana neutrinos in high-energy physics processes, specifically focusing on the Practical Dirac-Majorana Confusion Theorem (pDMCT).

• Context: The authors are responding to a critique (Ref. [1] by Bigaran, Parke, and Pasquini) of their earlier works [2–4].
• The Conflict: Ref. [1] argues that for processes involving unobserved neutrino-antineutrino pairs (e.g., $B^0 \to \mu^- \bar{\nu} \mu^+ \nu$), the amplitude squared for Dirac neutrinos must be symmetrized with respect to the exchange of the neutrino and antineutrino 4-momenta ($p_1 \leftrightarrow p_2$) to allow for a fair comparison with the Majorana case.
• The Authors' Stance: Kim et al. argue that this symmetrization is physically incorrect for Dirac neutrinos. They contend that Dirac neutrinos and antineutrinos are distinguishable particles; therefore, their amplitude squared should not be symmetrized under momentum exchange, even if the particles are not detected. They assert that Ref. [1]'s approach violates fundamental principles of the Standard Model (SM), specifically lepton number conservation.

2. Methodology and Theoretical Framework

The authors employ Standard Quantum Field Theory (QFT) and the Spinor-Helicity formalism (implicitly via their previous work) to analyze the decay of the $B^0$ meson into a muon pair and a neutrino-antineutrino pair ($B^0 \to \mu^- \bar{\nu} \mu^+ \nu$).

• Amplitude Construction:
  • They define the Dirac amplitude as $M_D(p_1, p_2)$, where $p_1$ and $p_2$ are the momenta of the antineutrino and neutrino, respectively.
  • They emphasize that the amplitude includes explicit dependence on 4-momenta and spins, though they often use simplified notation $M(p_1, p_2)$ for brevity.
  • The calculation follows standard QFT procedures: taking the modulus square of the amplitude, summing over final spins, and averaging over initial spins (trivial for spin-0 $B^0$).
• Critique of Ref. [1] Methodology:
  • The authors analyze the mathematical steps taken in Ref. [1], specifically Eqs. (16) and (35), where an "ad hoc" second term is added to the Dirac amplitude squared.
  • They argue that Ref. [1] conflates experimental detection limitations (missing particles) with theoretical amplitude construction.

3. Key Contributions and Arguments

A. Rejection of Ad Hoc Symmetrization for Dirac Neutrinos

The central technical contribution is the refutation of the claim that the Dirac amplitude squared must be symmetrized under $p_1 \leftrightarrow p_2$ when neutrinos are unobserved.

• Distinguishability: Dirac neutrinos ($\nu$) and antineutrinos ($\bar{\nu}$) are distinct particles with different lepton numbers. Unlike identical Majorana fermions, there is no Pauli exclusion principle or quantum statistics requirement to symmetrize their wavefunctions or amplitudes.
• Lepton Number Violation (LNV): The authors demonstrate that the symmetrization proposed in Ref. [1] effectively forces a summation over terms where lepton number is violated.
  • In the SM, $W^+ \to \ell^+ \nu$ and $W^- \to \ell^- \bar{\nu}$ conserve lepton number.
  • The symmetrized term in Ref. [1] implies a process where $W^+ \to \ell^+ \bar{\nu}$ (or similar LNV configurations), which is forbidden in the pure SM Dirac framework.
• Phase Space vs. Amplitude: The authors clarify that non-detection of particles is handled by integrating over the unobservable phase space, not by symmetrizing the amplitude itself. The amplitude square is a theoretical quantity calculated assuming all momenta are known; experimental constraints are applied after the calculation during the integration step.

B. Clarification of Spin and Helicity Notation

• The authors correct a misinterpretation in Ref. [1] regarding their notation. Ref. [1] claimed the amplitude vanishes when $p_1 = p_2$.
• Kim et al. clarify that their notation $M(p_1, p2)$ implicitly includes spin indices. The correct statement is that the amplitude vanishes only when both momenta and spins are identical ($p_1 = p_2$ and $s_1 = s_2$).
• They reiterate that their calculation sums over all final spins and does not rely on specific helicity states for the neutrino pair, contrary to the assumptions made in Ref. [1].

C. Domain of Validity for pDMCT

• The authors reaffirm that the Practical Dirac-Majorana Confusion Theorem (pDMCT) holds true when integrating over the momenta of the neutrino pair.
• However, they argue that Ref. [1] achieves the same mathematical result (confusion between Dirac and Majorana) through an incorrect physical mechanism (ad hoc symmetrization).
• The authors' work focuses on identifying exceptions to pDMCT where the nature of the neutrino can be distinguished:
  1. When neutrino momenta are fixed (directly or indirectly via kinematics).
  2. When New Physics effects are considered.

4. Results

• Refutation of Ref. [1]: The authors conclude that the critique in Ref. [1] is fundamentally flawed because it relies on a procedure (symmetrizing Dirac amplitudes) that has no basis in the Standard Model and violates lepton number conservation.
• Validation of Original Works: The authors confirm that their original calculations in Refs. [2–4] are correct, utilizing standard QFT without ad hoc modifications.
• Theoretical Consistency: The paper establishes that the distinction between Dirac and Majorana neutrinos relies on the identical nature of the particles (Majorana) versus their distinguishability (Dirac), and this distinction must be preserved in the amplitude construction regardless of detector capabilities.

5. Significance

• Preservation of Standard Model Principles: The paper defends the strict application of lepton number conservation in the SM, preventing the introduction of unphysical terms (LNV) into Dirac neutrino calculations under the guise of "fair comparison."
• Clarification of Theoretical vs. Experimental Limits: It provides a crucial pedagogical distinction between how theoretical amplitudes are constructed (assuming full knowledge of momenta) and how experimental observables are derived (via phase space integration over unobserved variables).
• Impact on Neutrino Physics: By invalidating the specific critique in Ref. [1], the authors reinforce the validity of their previous findings that quantum statistics can distinguish Dirac and Majorana neutrinos under specific kinematic conditions or New Physics scenarios, keeping the door open for experimental searches beyond the pDMCT limit.

In summary, this paper serves as a rigorous technical rebuttal, asserting that the proposed method to distinguish neutrino types in Ref. [1] is mathematically coincidental but physically incorrect, and that the standard QFT approach used by Kim et al. remains the valid framework for analyzing these processes.