Absence of Majorana-Weyl fermions in d=4 and the theory of Majorana fermions

Technical Summary: Absence of Majorana-Weyl Fermions in d=4 and the Theory of Majorana Fermions

Problem Statement
The paper addresses a fundamental inconsistency in the field-theoretic definition of Majorana fermions within four-dimensional spacetime ($d=4$), specifically in the context of chiral theories like the Type I seesaw model. A common construction identifies a Majorana fermion as $\psi_+ = \nu_R + C\nu_R^T$, where $\nu_R$ is a right-handed chiral field and $C$ is the charge conjugation operator. The author argues that this identification relies on a "chirality-changing" charge conjugation ($\tilde{C}$) which is incompatible with the standard representation theory of Lorentz symmetry in $d=4$. Specifically, the theorem on the absence of Majorana-Weyl fermions in $d=4$ implies that a charge conjugation operator cannot simultaneously preserve chirality and satisfy the Majorana condition for a single chiral field. Consequently, the formal construction $\psi_+ = \nu_R + C\nu_R^T$ leads to indeterminate physical predictions, such as vanishing actions or contradictory transformation properties, when subjected to standard substitution rules in field theory.

Methodology
The analysis proceeds through the following steps:

1. Representation Analysis: The author utilizes the $\gamma_5$-diagonal representation to explicitly define charge conjugation ($C$) and parity ($P$) transformations for chiral fermions. Two distinct definitions of charge conjugation are contrasted:
   • Chirality-Preserving ($\hat{C}$): Maps $\nu_R \to C\nu_L^T$ and $\nu_L \to C\nu_R^T$. This is consistent with the doublet representation of chiral fermions (Dirac-type fields) and standard field theory.
   • Chirality-Changing ($\tilde{C}$): Maps $\nu_R \to C\nu_R^T$ and $\nu_L \to C\nu_L^T$. This is the operation often used in literature to justify $\psi_+ = \nu_R + C\nu_R^T$ as a Majorana fermion.
2. Application of the No-Go Theorem: The paper demonstrates that the chirality-changing operator $\tilde{C}$ satisfies $\tilde{C}\gamma_5\tilde{C}^{-1} = -\gamma_5$. In $d=4$, this property prevents the simultaneous existence of a Majorana condition and a definite chirality for a single field.
3. Diagonalization of the Seesaw Model: The Type I seesaw Lagrangian is diagonalized using a unitary transformation ($U$) to obtain mass eigenstates $\psi_+$ and $\psi_-$. The author shows that under the chirality-preserving $\hat{C}$, these states do not satisfy the Majorana condition $\psi = C\psi^T$. Under the chirality-changing $\tilde{C}$, the action and the fields themselves vanish due to the no-go theorem, rendering the physical interpretation indeterminate.
4. Canonical Transformation (Bogoliubov Analogue): To resolve the inconsistency, the author applies a generalized Pauli-Gursey canonical transformation (analogous to the Bogoliubov transformation in BCS theory). This transformation mixes the chiral fields and their charge conjugates to construct new fields $\psi_{M1}$ and $\psi_{M2}$ from a Dirac-type field $N$.

Key Contributions and Results

• Refutation of the Standard Construction: The paper proves that the formal object $\psi_+ = \nu_R + C\nu_R^T$ is not a valid Majorana fermion in $d=4$ field theory. Attempts to treat it as such using chirality-changing symmetry lead to the vanishing of the action and the failure of substitution rules (e.g., the relation $[(1+\gamma_5)/2]\psi_+ = \nu_R$ becomes inconsistent under the assumed symmetries).
• Resolution via Dirac-Type Fields: The author demonstrates that consistent Majorana fermions can only be constructed in $d=4$ by first defining a Dirac-type field $\psi$ (containing both chiralities) and then applying the original Majorana definition: $\psi_M = (\psi \pm C\psi^T)/\sqrt{2}$.
• Reformulation of the Seesaw Model: By applying a specific $6 \times 6$ orthogonal canonical transformation to the diagonalized seesaw Lagrangian, the author derives two consistent Majorana fermions, $\psi_{M1}$ and $\psi_{M2}$. These fields satisfy the standard Majorana conditions ($\psi_M = \pm C\psi_M^T$) under the chirality-preserving charge conjugation $\hat{C}$.
• Chiral Projection Properties: A crucial result is that the chiral projection of a consistent Majorana fermion, e.g., $[(1+\gamma_5)/2]\psi_{M1}$, is not a chiral fermion in the sense of a single Weyl spinor. Instead, it is a superposition of components that ensures the non-vanishing of the neutrino-less double beta decay amplitude.
• Neutrino-less Double Beta Decay: The paper clarifies that if one incorrectly identifies the seesaw solution $\psi_-$ with a Majorana fermion using chirality-changing symmetry, the amplitude for neutrino-less double beta decay vanishes. However, using the consistent formulation derived from the Dirac-type field, the amplitude is non-zero, as required for the process to occur.

Significance
The paper claims that the identification of $\psi_+ = \nu_R + C\nu_R^T$ as a Majorana fermion is a formal artifact that fails in rigorous field theory due to the absence of Majorana-Weyl fermions in four dimensions. The significance lies in establishing that a consistent field-theoretic description of Majorana fermions in chiral theories requires a two-step process: first, forming a Dirac-type field, and second, applying the Majorana projection. This formulation resolves logical inconsistencies regarding substitution rules and parity transformations and provides a robust theoretical basis for analyzing weak interaction processes, such as neutrino-less double beta decay, without relying on ill-defined chirality-changing symmetries. The work suggests that previous analyses relying on the direct identification of chiral combinations as Majorana fermions may have overlooked these fundamental constraints.