Impact of antiparticle degrees of freedom on neutrino flavor oscillations in frames of quantum field theory

Technical Summary: Impact of Antiparticle Degrees of Freedom on Neutrino Flavor Oscillations in Frames of Quantum Field Theory

Problem Statement
Neutrino flavor oscillations are traditionally described using quantum mechanical approaches, though these formalisms possess known shortcomings regarding covariance and the treatment of production and detection processes. Quantum Field Theory (QFT) offers a consistent framework where neutrinos are treated as virtual particles propagating between a source and a detector. In previous works extending this QFT approach to neutrino oscillations in external fields (such as background matter and magnetic fields), the author decomposed the neutrino propagators into particle and antiparticle contributions. In those calculations, only the particle contribution was retained, while the antiparticle contribution was assumed to be negligible for ultrarelativistic neutrinos. However, this assumption, while expected, had not been rigorously justified within the formalism. The central problem addressed in this work is to provide a careful justification for neglecting the antiparticle part of the propagator when calculating the matrix element for neutrino flavor oscillations.

Methodology
The study employs a QFT approach where the S-matrix element for neutrino flavor oscillations is calculated. The process involves the production of a neutrino at a source (via interaction with a heavy nucleus and an incoming charged lepton) and its detection at a distant detector (via interaction with another heavy nucleus and the production of an outgoing charged lepton).

• Formalism: The neutrinos are treated as virtual particles represented by the propagators of mass eigenstates ($\psi_a$). The flavor eigenstates are expressed as superpositions of mass eigenstates via a mixing matrix $U$.
• Propagator Decomposition: The vacuum propagator $S_a$ for a neutrino mass eigenstate is decomposed into a particle part, $S^{(\psi)}_a$, and an antiparticle part, $S^{(\bar{\psi})}_a$, distinguished by the sign of the imaginary term ($\pm i0$) in their denominators.
• Calculation Strategy: The author calculates the matrix element $\mathcal{M}_{\beta \to \alpha}$ by integrating the propagators over the virtual neutrino momentum. To simplify the derivation, they adopt the forward scattering approximation for leptons (assuming left-polarized plane waves) and consider oscillations in a vacuum.
• Analytic Evaluation: The integrals over the virtual neutrino momentum are evaluated in the complex plane. The author performs contour integration, closing the contour in the upper half-plane to ensure convergence for a positive propagation distance $L$. They explicitly compute the contributions from both the particle pole and the antiparticle pole.

Key Contributions and Results

1. Explicit Calculation of Contributions: The paper provides a detailed analytic derivation of the matrix element contributions from both the particle ($I_\psi$) and antiparticle ($I_{\bar{\psi}}$) parts of the propagator.
   • The particle contribution yields a term proportional to $e^{iEL}$ with a phase factor dependent on the mass squared ($e^{-i m_a^2 L / 2E}$).
   • The antiparticle contribution yields a term proportional to $e^{-iEL}$ with a similar phase structure but different amplitude characteristics.
2. Suppression of Antiparticle Terms: By comparing the magnitudes of the two contributions, the author demonstrates that for ultrarelativistic neutrinos ($m_a \ll E$), the ratio of the antiparticle contribution to the particle contribution scales as $m_a^2 / E^2$.
   $$\frac{|\mathcal{M}^{(\bar{\psi})}_{e \to \mu}|}{|\mathcal{M}^{(\psi)}_{e \to \mu}|} \sim \frac{m_a^2}{E^2} \ll 1$$
   This confirms that the antiparticle part of the propagator gives a negligible contribution to the matrix element in the ultrarelativistic limit.
3. Recovery of Standard Oscillation Probability: Using only the particle contribution, the author derives the transition probability $P_{\nu_e \to \nu_\mu}(L)$. The result reproduces the standard vacuum oscillation formula:
   $$P_{\nu_e \to \nu_\mu}(L) = \sin^2 2\theta \sin^2 \left( \frac{\Delta m^2}{4E} L \right)$$
   This validates that the simplified formalism (retaining only the particle propagator) is sufficient to recover the known physical results.

Significance and Claims
The primary significance of this work is the rigorous justification of a simplifying assumption used in previous QFT studies of neutrino oscillations in external fields. The author explicitly states that while the calculations were performed for neutrino oscillations in vacuum (chosen because they allow for complete analytic treatment), the results can be extrapolated to more complex scenarios involving external fields.

The paper claims that the decomposition of propagators and the subsequent exclusion of the antiparticle contribution is a valid procedure for ultrarelativistic neutrinos. This justification supports the continued use of the simplified QFT formalism in Refs. [9, 10] for studying neutrino interactions with background matter and magnetic fields, ensuring that the neglect of antiparticle degrees of freedom does not compromise the accuracy of the matrix element calculation in those contexts. The author remains modest, noting that the technical complexity of external field scenarios is higher, but the fundamental suppression mechanism demonstrated here remains applicable.