The probability for chiral oscillation of Majorana neutrino in Quantum Field Theory

This paper derives the probability for chiral oscillation of Majorana neutrinos within quantum field theory by utilizing the Bogoliubov transformation to relate lepton number eigenstates at different times, thereby characterizing the time variation of lepton number as a transition amplitude.

The Gist

The Big Picture: The Shape-Shifting Ghost

Imagine you have a ghost named Majorana. Unlike normal ghosts (Dirac neutrinos) that are clearly either "male" or "female," this ghost is a Majorana particle, meaning it is its own antiparticle. It's like a chameleon that can be both male and female at the same time, or rather, it can flip its identity.

For a long time, physicists have studied how these ghosts change flavors (like a ghost changing from a "lemon" ghost to a "lime" ghost). But this paper asks a different, stranger question: Can a ghost flip its "gender" (lepton number) just by existing and moving through time?

The authors, Takuya Morozumi and Tomoharu Tahara, say yes, but with a very specific catch: this only happens significantly when the ghost is moving slowly (non-relativistic). If the ghost is zooming near the speed of light, it barely changes.

The Problem: The "Lepton Number" Wallet

In physics, particles carry a "wallet" called Lepton Number.

• A normal neutrino has a wallet with +$1.
• An anti-neutrino has a wallet with -$1.

Usually, physics rules say you can't just lose or gain money from your wallet; it must be conserved. However, because Majorana neutrinos are their own antiparticles, the "bank" (the Hamiltonian) has a loophole. The rules of the bank allow the wallet to change contents over time.

The authors wanted to calculate exactly how likely it is for a neutrino starting with a +$1 wallet to end up with a -$1 wallet (or a mix) after traveling for a while.

The Method: The "Time-Traveling Vacuum"

To solve this, the authors had to build a very specific mathematical machine. Here is the analogy:

1. The Empty Room (The Vacuum): Imagine a room that is completely empty. In quantum physics, "empty" isn't truly empty; it's a bubbling soup of potential particles.
2. The Zero-Mode Glitch: When they tried to do the math, they found a glitch with particles that have zero momentum (standing still completely). It was like trying to count the air in a room that has no volume. To fix this, they had to mathematically "kick out" the zero-momentum particles from their calculation. It's like saying, "We are only counting the people who are actually moving."
3. The Time-Dependent Vacuum: Because the Majorana mass term (the rule that allows the identity flip) is always active, the "Empty Room" itself changes as time passes. The vacuum at the start of the experiment is different from the vacuum at the end.

The Magic Trick: The Bogoliubov Transformation

This is the core of their discovery. They used a mathematical tool called the Bogoliubov Transformation.

The Analogy:
Imagine you have a bag of marbles.

• Start: You put one red marble (a neutrino) into the bag.
• The Process: As time passes, the bag shakes (due to the Majorana mass).
• The Result: When you look inside later, you don't just see the red marble. You might see:
  1. The original red marble (Survival).
  2. The red marble plus a pair of blue marbles (an anti-neutrino pair) that popped out of the "empty" space.

The authors realized that the "flipping" of the neutrino isn't just the neutrino turning into an anti-neutrino. It's more like the neutrino creating a pair of anti-neutrinos from the vacuum and keeping one for itself, effectively changing the total "lepton number" of the system.

They used the Bogoliubov transformation to act like a translator. It translates the language of "what the particle looked like at the start" into "what the particle looks like at the end," accounting for all the extra particles that popped into existence from the vacuum.

The Results: Speed Matters!

The paper calculates the probability of this flip. Here is the punchline:

• The Fast Ghost (Relativistic): If the neutrino is zooming near the speed of light (like a bullet), it is very stubborn. It keeps its identity. The probability of it flipping is tiny. It's like a fast-moving car that doesn't have time to change lanes.
• The Slow Ghost (Non-Relativistic): If the neutrino is moving slowly (like a turtle), the "flip" happens easily. The probability oscillates back and forth between being a neutrino and being a "neutrino-plus-pair."
  • The Oscillation: It's like a pendulum. At time $T$, it's 100% a neutrino. At time $T+1$, it's 50% neutrino/50% flipped. At time $T+2$, it's 100% flipped. Then it flips back.

The "Three-Particle" Surprise

One of the most important findings is correcting a misconception.

• Old Idea: A neutrino turns into a single anti-neutrino.
• New Finding: A neutrino turns into a three-particle state: The original neutrino + a pair of anti-neutrinos created from the vacuum.

Think of it this way: You don't just turn into your enemy; you turn into yourself plus a clone of your enemy that you created out of thin air. The total "charge" of the system changes, but the math balances perfectly.

Why Does This Matter?

1. It's a New Kind of Oscillation: This isn't the standard "flavor" oscillation (like a lemon turning into a lime). This is a "chiral" oscillation (a change in the fundamental handedness/identity of the particle).
2. It Only Happens Slowly: This effect is negligible for the high-speed neutrinos we usually detect from the sun or supernovae. However, if we ever find a way to trap and study slow-moving Majorana neutrinos (perhaps in the early universe or in specific lab conditions), this "identity flip" could be a major signature.
3. Mathematical Consistency: The authors proved that their method gives the exact same answer for the "average lepton number" as previous studies, but their method is more robust because it doesn't rely on messy approximations. It treats the vacuum and the particles as a unified, time-changing system.

Summary in One Sentence

The authors used advanced quantum math to show that slow-moving Majorana neutrinos don't just change flavors; they periodically flip their identity by spawning pairs of anti-particles from the vacuum, a phenomenon that is invisible to fast-moving particles but fundamental to the nature of matter.

Technical Summary

1. Problem Statement

The paper addresses the phenomenon of chiral oscillation in Majorana neutrinos, a process where a neutrino transforms into a state with a different lepton number due to the Majorana mass term.

• Context: While standard flavor oscillations are well-understood, neutrino-antineutrino oscillations (specifically chiral oscillations driven by Majorana mass) present theoretical challenges. Previous frameworks often rely on perturbative approaches or assume relativistic limits where the effect is suppressed.
• The Gap: In the Majorana framework, the Hamiltonian does not conserve lepton number. Consequently, the eigenstates of the lepton number operator change continuously over time. Standard quantum mechanical oscillation formulas (like Pontecorvo's) are insufficient because they often assume time-independent eigenstates or fail to account for the non-perturbative creation of particle pairs from the vacuum in the non-relativistic regime.
• Goal: To derive the exact time-dependent transition probability for chiral oscillation using a rigorous Quantum Field Theory (QFT) framework that applies to both relativistic and non-relativistic neutrinos, explicitly handling the time-dependence of the vacuum and lepton number eigenstates.

2. Methodology

The authors employ a non-perturbative QFT approach based on the following key steps:

• Exclusion of Zero Momentum Mode:

  • The authors start with the path integral for a single-flavor Majorana neutrino.
  • They explicitly exclude the zero momentum mode ($p=0$) from the field expansion. This is crucial because the zero mode cannot be expressed using massless spinors, and its inclusion would prevent the lepton number operator from being simply defined as the difference between neutrino and antineutrino number operators.
  • This is achieved by introducing auxiliary fields ($\xi_0$) and applying Dirac bracket quantization to handle the resulting second-class constraints.

• Field Quantization with Time-Dependent Vacuum:

  • The field operator is expanded using massless plane wave spinors and creation/annihilation operators ($a, b$) associated with definite lepton numbers.
  • Because the mass term couples these operators, the vacuum state $|0, t\rangle$ becomes time-dependent. The vacuum at detection time $t_f$ is a superposition of the vacuum and multi-particle states at production time $t_i$.

• Bogoliubov Transformation:

  • The time evolution of the creation and annihilation operators is derived using the Heisenberg equation of motion with the Majorana Hamiltonian.
  • The relationship between operators at $t_i$ and $t_f$ is established via a Bogoliubov transformation. This transformation relates the lepton number eigenstates at different times.
  • Specifically, the vacuum at $t_f$ is expressed as a superposition of the vacuum, two-particle (Cooper pair), and four-particle states at $t_i$.

• Transition Amplitude Calculation:

  • The transition amplitude is defined as the inner product between the state at production (eigenstate of lepton number at $t_i$) and the state at detection (eigenstate of lepton number at $t_f$).
  • The authors calculate the S-matrix elements for transitions in the specific momentum sector ($p$) containing the neutrino and the complementary sectors ($q \neq p$) containing the vacuum.

3. Key Contributions

• Rigorous QFT Derivation: The paper provides a first-principles derivation of chiral oscillation probabilities without relying on perturbative approximations of the mass term.
• Physical Picture of Oscillation: It clarifies that chiral oscillation is not a simple transition from a single neutrino to a single antineutrino. Instead, it is a transition from a single neutrino state ($L=+1$) to a three-particle state consisting of the original neutrino plus an antineutrino Cooper pair ($\bar{\nu}\bar{\nu}$) created from the vacuum ($L=-1$).
• Resolution of Divergences: By summing over all possible final states in the vacuum sectors (q-sectors), the authors demonstrate that the total probability is unitary and finite, avoiding the divergences often encountered in perturbative treatments that require subtraction of vacuum contributions.
• Mass Basis Consistency: The paper proves in Appendix C that the probability for a neutrino to oscillate directly into a single antineutrino ($P_{\nu \to \bar{\nu}}$) is exactly zero in the mass basis, reinforcing that the observed effect is a multi-particle process.

4. Results

The authors derive explicit formulas for the survival probability ($P_{\nu \to \nu}$) and the chiral oscillation probability ($P_{\nu \to \nu\bar{\nu}\bar{\nu}}$) for a neutrino with momentum $p$ and velocity $v = |p|/E_p$:

• Survival Probability:
  $$P_{\nu_p \to \nu_p}(p, \tau) = |f(p, \tau)|^2 = 1 - (1 - v^2)\sin^2(E_p \tau)$$
• Chiral Oscillation Probability:
  $$P_{\nu_p \to \nu_p \bar{\nu}_{-p} \bar{\nu}_p}(p, \tau) = |g(p, \tau)|^2 = (1 - v^2)\sin^2(E_p \tau)$$
  Where $E_p = \sqrt{p^2 + m^2}$ and $\tau = t_f - t_i$.

Key Observations from the Results:

1. Velocity Dependence:
   • Relativistic Limit ($v \approx 1$): The factor $(1-v^2)$ approaches zero. Chiral oscillation is heavily suppressed, and the neutrino essentially remains a neutrino.
   • Non-Relativistic Limit ($v \ll 1$): The factor $(1-v^2) \approx 1$. The probabilities oscillate fully between 0 and 1 with a period $\Delta \tau \approx \pi/m$. The effect is maximal.
2. Unitarity: The sum of the survival and oscillation probabilities is exactly 1, satisfying unitarity without ad-hoc subtractions.
3. Lepton Number Expectation: The expectation value of the lepton number derived from these probabilities matches previous results obtained via direct operator expectation values, validating the probabilistic interpretation.

5. Significance

• Theoretical Clarity: The work resolves ambiguities regarding the nature of Majorana neutrino oscillations. It demonstrates that lepton number violation in this context is a dynamical process involving the creation of particle-antiparticle pairs from the vacuum, rather than a simple flavor flip.
• Applicability to Low-Energy Neutrinos: The framework is particularly significant for non-relativistic neutrinos (e.g., relic neutrinos or neutrinos in dense astrophysical environments), where chiral oscillation effects are predicted to be large, unlike in high-energy collider experiments where they are negligible.
• Foundation for Future Work: The authors establish a formalism that can be extended to multi-flavor cases and matter effects (MSW effect), providing a robust tool for analyzing neutrino physics in regimes where standard perturbative methods fail.

In summary, this paper successfully constructs a consistent QFT framework for Majorana chiral oscillations, revealing that the phenomenon is a non-perturbative, vacuum-driven transition to a three-particle state, with magnitude strictly dependent on the neutrino's non-relativistic nature.