⚛️ phenomenology

Neutrinoless double $\beta$ decay and leptogenesis in seesaw model

This paper argues that the right-handed neutrino field in the Type I seesaw model is not a Majorana fermion, thereby precluding neutrinoless double beta decay, and proposes that both leptogenesis and neutrinoless double beta decay can be realized by redefining the model with C-symmetric Majorana fermions via a Bogoliubov transformation.

Original authors: Kazuo Fujikawa

Published 2026-06-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kazuo Fujikawa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Case of Mistaken Identity

Imagine you are trying to solve a mystery about the universe. Scientists have a popular theory called the Type I Seesaw Model. This theory tries to explain two big things:

Why neutrinos (tiny ghost-like particles) have mass.
How the universe ended up with more matter than antimatter (a process called leptogenesis).

For a long time, physicists thought the key player in this story was a specific type of particle they called a "Majorana fermion." They believed this particle was its own antiparticle, like a coin that is the same on both sides. If this were true, it would explain how neutrinos get mass and how the universe created an imbalance between matter and antimatter.

However, this paper argues that the "coin" everyone is looking at is actually a trick.

The Problem: The "Ghost" That Can't Exist

The author, Kazuo Fujikawa, points out a fundamental rule of physics in our 4-dimensional world (3 dimensions of space + 1 of time). He says:

"You cannot have a 'Majorana-Weyl' fermion."

The Analogy:
Think of a chiral fermion (like a neutrino) as a left-handed glove.
Think of a Majorana fermion as a glove that is its own mirror image (a glove that can be worn on either hand).

In the standard Seesaw Model, physicists tried to combine a "left-handed glove" with its "mirror image" to make a "Majorana glove." But Fujikawa says that in our 4D world, the laws of geometry (the Lorentz group) make it impossible to stitch a left-handed glove and a right-handed glove together to make a single, self-identical object without breaking the rules of the game.

The particle the Seesaw model usually calls "Majorana" (which is a mix of a right-handed neutrino and its charge-conjugate) is not a true Majorana fermion. It's a "fake" Majorana. Because it's not a true Majorana, it cannot perform the magic trick required for neutrinoless double beta decay (a rare radioactive process where two neutrons turn into two protons without emitting neutrinos).

The Result: If you stick to the standard definitions, the Seesaw model fails to explain neutrinoless double beta decay.

The Solution: Changing the "Vacuum" (The Bogoliubov Trick)

So, if the standard particle isn't a real Majorana fermion, how do we fix the theory to make it work?

Fujikawa proposes a clever mathematical "magic trick" called a Bogoliubov transformation.

The Analogy:
Imagine you have a room full of people (particles). In the standard view, the "floor" (the vacuum) is empty.
Fujikawa says: "Let's change the definition of the floor."

By using a mathematical transformation (similar to how physicists describe superconductors or how a spinning top behaves differently depending on how you look at it), he redefines what the "empty space" (vacuum) looks like.

Before the trick: The particles are a messy mix of left and right hands that don't quite fit together. They aren't true Majorana fermions.
After the trick: By shifting the "floor," those same messy particles suddenly organize themselves into true Majorana fermions.

It's like taking a jumbled pile of puzzle pieces and realizing that if you rotate the whole box 90 degrees, the pieces suddenly snap together perfectly to form a complete picture.

Why This Matters: Two Problems Solved at Once

Once this "floor" is changed and the particles are redefined as true Majorana fermions, two major problems in physics get solved naturally:

Neutrinoless Double Beta Decay: Now that we have real Majorana fermions, the theory can explain this rare radioactive decay. The "magic" of the particle being its own antiparticle is restored.
Leptogenesis (Matter vs. Antimatter): The universe needs a reason why we have more matter than antimatter. The paper shows that with these newly defined Majorana fermions, the heavy particles in the early universe could decay in a way that creates an imbalance.
- Note: The paper explains that this imbalance starts as a difference in "handedness" (chirality) and is later converted into a difference in "lepton number" (matter vs. antimatter) through weak interactions.

The "Gauge" Issue: It Depends on How You Look

The paper also touches on a technical detail about how we measure these things.

Unitary Gauge: This is like looking at a building through a window where the curtains are drawn. You see the structure clearly, but you miss some details about the air outside. In this view, the "lepton number" (matter count) isn't obvious.
Landau Gauge: This is like opening the curtains. You see the "charged Higgs scalars" (extra particles) that were hidden before.

Fujikawa shows that while the math looks different depending on which "gauge" (window) you use, the physical result is the same: Leptogenesis happens. The asymmetry between matter and antimatter is generated, provided you account for the interactions correctly.

Summary

The Claim: The standard way of defining the "Majorana" particle in the Seesaw model is mathematically inconsistent in our 4D world. It's a "fake" Majorana.
The Fix: You must use a mathematical transformation (Bogoliubov) to redefine the vacuum state of the universe.
The Payoff: Once you do this, the particles become real Majorana fermions. This allows the theory to successfully explain:
1. Why neutrinos have mass.
2. How neutrinoless double beta decay can happen.
3. How the universe created more matter than antimatter (Leptogenesis).

In short, the paper says: "The Seesaw model is a great idea, but we've been looking at the particles wrong. Once we change our perspective (the vacuum), everything fits together perfectly."

Technical Summary: Neutrinoless Double $\beta$ Decay and Leptogenesis in Seesaw Model

Problem Statement
The paper addresses a fundamental inconsistency in the standard interpretation of the Type I seesaw model within four-dimensional ( $d=4$ ) field theory. Specifically, the fermion combination $\nu_R + C\nu_R^T$ , commonly identified as a Majorana fermion in this model, is shown not to satisfy the definition of a Majorana fermion under the standard charge conjugation ( $\hat{C}$ ) and parity ( $\hat{P}$ ) transformations. This discrepancy arises from the theorem regarding the absence of Majorana-Weyl fermions in the representation of the $d=4$ proper Lorentz group. Consequently, the standard formulation fails to naturally describe neutrinoless double $\beta$ decay (which requires genuine Majorana fermions) and presents conceptual difficulties in defining the lepton number asymmetry required for leptogenesis.

Methodology
The author employs a rigorous field-theoretic analysis based on the consistent definitions of discrete symmetries ( $C$ , $P$ , and $CP$) in $d=4$ .

Symmetry Definitions: The analysis utilizes a representation where $\gamma_5$ is diagonal. The charge conjugation $\hat{C}$ is defined to preserve chirality (mapping $\nu_L \to \nu_R^C$ and vice versa), while parity $\hat{P}$ includes an extra factor of $i$ characteristic of Majorana fermions.
Diagonalization: The Type I seesaw Lagrangian is analyzed using the Autonne-Takagi factorization to diagonalize the complex symmetric mass matrix, yielding mass eigenstates $\psi_+$ and $\psi_-$ .
Canonical Transformation: To resolve the issue that the resulting eigenstates $\psi_\pm$ are not conventional Majorana fermions under $\hat{C}$ , the author introduces a generalized Pauli-Gursey canonical transformation (analogous to the Bogoliubov transformation in BCS theory). This transformation mixes neutrinos and antineutrinos, effectively redefining the vacuum state.

Key Contributions and Results

Identification of Non-Majorana States: The paper demonstrates that the fermion $\psi_+ = \nu_R + C\nu_R^T$ (and similarly $\psi_-$ ) is not a Majorana fermion under the standard $\hat{C}$ transformation defined in $d=4$ . While these fields are invariant under a "pseudo-C" (chirality-changing charge conjugation), such an operator is ill-defined for chiral projections in $d=4$ due to the absence of Majorana-Weyl fermions. Therefore, identifying these fields as Majorana fermions in the conventional sense is theoretically inconsistent.
Reformulation via Bogoliubov Transformation: By applying the canonical transformation $O$ (Eq. 19), the original chiral fields are mapped to new fields $N$ and $N^C$ . In this new basis, the physical eigenstates $\Psi_\pm$ are defined as:
$\Psi_+ = \frac{1}{\sqrt{2}}(N + N^C), \quad \Psi_- = \frac{1}{\sqrt{2}}(N - N^C)$
These $\Psi_\pm$ satisfy the standard Majorana conditions ( $C\Psi_\pm^T = \pm \Psi_\pm$ ) under the conventional $\hat{C}$ and $\hat{P}$ transformations.
Neutrinoless Double $\beta$ Decay: The reformulation ensures that the light neutrino component ( $\Psi_-$ ) is a genuine Majorana fermion, thereby allowing for the description of neutrinoless double $\beta$ decay, which was previously problematic in the standard Type I seesaw formulation.
Leptogenesis Mechanism: The paper constructs a scenario where leptogenesis occurs naturally. The decay of heavy Majorana neutrinos ( $\Psi_+$ ) into light Majorana neutrinos ( $\Psi_-$ ) and Higgs particles generates a chirality asymmetry due to $CP$ violation in the Higgs couplings. This chirality asymmetry is subsequently converted into a lepton number asymmetry through weak interactions. The analysis shows that while lepton number is not defined in the Higgs coupling sector alone, the full process involving weak interactions yields an observable lepton number asymmetry.
Gauge Dependence: The paper notes that the generation of lepton number asymmetry in the Higgs coupling sector is not electroweak gauge independent. In the Landau gauge, charged Higgs scalars ( $H^\pm$ ) appear, and their fate is linked to the asymmetry of charged leptons via electromagnetic charge conservation.

Significance and Claims
The paper claims that the standard "Majorana" particles in the Type I seesaw model (Eq. 8) cannot be assigned an ordinary physical meaning within the framework of conventional $d=4$ field theory. The author asserts that to consistently describe both neutrinoless double $\beta$ decay and natural leptogenesis, it is essential to redefine the Majorana fermions by introducing a new vacuum state via an analogue of the Bogoliubov canonical transformation. This approach resolves the conflict between the chirality of the seesaw fields and the requirements for Majorana fermions in $d=4$ , providing a consistent theoretical basis for these phenomena without invoking unphysical pseudo-symmetries. The work emphasizes that the lepton number violation in leptogenesis arises from the interplay between the heavy neutrino decay (generating chirality asymmetry) and subsequent weak interactions, rather than from the Higgs sector alone.