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

We derive the probability for chiral oscillation of Majorana neutrinos based on quantum field theory. Since the Hamiltonian under the Majorana mass term does not conserve lepton number, the eigenstates of lepton number change continuously over time. Therefore, the transition amplitude is described by the inner product of the eigenstates of lepton number at the time of the neutrino production and the detection. With the Bogoliubov transformation, we successfully relates the lepton number eigenstates at different times. This method enables us to understand the time variation of lepton number induced by chiral oscillations in terms of transition probabilities. We also present the physical picture that emerges through the Bogoliubov transformation.

💡 Research Summary

The paper presents a quantum‑field‑theoretic (QFT) derivation of the probability for chiral oscillations of a Majorana neutrino. The authors start from the observation that the Majorana mass term violates lepton‑number conservation, so the lepton‑number operator (L(t)) is explicitly time‑dependent and its eigenstates evolve continuously. They define the transition amplitude as the inner product between the lepton‑number eigenstate at the production time (t_i) and the eigenstate at the detection time (t_f).

To make this construction concrete they first quantize a single‑flavour Majorana field while deliberately excluding the zero‑momentum mode. The exclusion is implemented via delta‑function constraints in the path‑integral, which removes the problematic zero‑mode contributions to the anticommutation relations. The resulting Hamiltonian contains only non‑zero momentum modes and can be expressed in terms of two‑component chiral fields (\eta) and (\eta^\dagger) with the canonical equal‑time anticommutator ({\eta(x),\eta^\dagger(y)}=\delta^3(x-y)-1/V).

The field is expanded using massless plane‑wave spinors (\varphi_\pm(\pm\mathbf{n}p)) and creation/annihilation operators (a(\mathbf{p},t), b(\mathbf{p},t)). By rescaling these operators they introduce dimensionless fermionic operators (\alpha(\mathbf{p},t)) and (\beta(\mathbf{p},t)) that satisfy standard anticommutation relations. Bilinear operators (N{\alpha,\beta}(\pm\mathbf{p},t)) (number operators) and Cooper‑pair‑like operators (B_{\alpha,\beta}(\mathbf{p},t)=\alpha(\mathbf{p},t)\alpha(-\mathbf{p},t)) (and similarly for (\beta)) are defined. These obey a closed algebra, and the Hamiltonian separates into independent (p)-sectors: (H=\sum_{\mathbf{p}\in A} h(\mathbf{p})), with each (h(\mathbf{p})) commuting with all others.

The time evolution of the fundamental operators is derived from the Heisenberg equation. The result is a Bogoliubov transformation:

\