Neutrino Oscillations in Vacuum

• Neutrino oscillations in vacuum are a quantum phenomenon where neutrinos change flavors due to interference among distinct mass eigenstates.
• The oscillation probability is governed by mixing angles, mass-squared differences, and the ratio of propagation distance to energy, as demonstrated in various experiments.
• Field-theoretic and wave-packet models provide robust frameworks to account for production, propagation, detection, and potential decoherence effects.

Neutrino oscillations in vacuum refer to the quantum phenomenon whereby neutrinos produced in a definite flavor state (such as electron, muon, or tau neutrinos) are detected after propagation as a different flavor. This behavior fundamentally stems from the nontrivial mixing of flavor and mass eigenstates, combined with the existence of nonzero neutrino mass splittings. The oscillation arises through quantum interference between propagating neutrino mass eigenstates, with a probability that depends sensitively on the ratio of propagation distance to energy, mass-squared differences, and mixing angles. The theoretical framework encompasses both quantum-mechanical and quantum-field-theoretic treatments, with robust experimental ramifications in accelerator, reactor, atmospheric, and solar neutrino observations. The following presents a comprehensive treatment of the structure, derivation, and phenomenological implications of neutrino oscillations in vacuum.

1. Flavor–Mass Mixing Framework

Neutrino flavor states (|ν_α⟩, α = e, μ, τ), defined as the eigenstates created and detected by charged-current weak interactions, are not eigenstates of the free Hamiltonian. The propagation eigenstates are the neutrino mass eigenstates (|ν_i⟩, i = 1,...,n), with definite masses m_i. The relationship between flavor and mass bases is given by a unitary mixing matrix U: $$|ν_\alpha⟩ = \sum_{i=1}^{n} U^*_{\alpha i} |ν_i⟩\,,$$ where for n=3, U is the PMNS matrix. In the two-flavor approximation, the mixing can be represented as a 2×2 rotation matrix,

$$U(\Theta) = \begin{pmatrix} \cos\Theta & \sin\Theta \ -\,\sin\Theta & \cos\Theta \end{pmatrix} \,.$$

Consequently, flavor eigenstates prepared at production or detection are quantum superpositions over the mass eigenbasis (Bilenky, 2016, Weinheimer, 2010).

2. Quantum-Mechanical Evolution and Vacuum Oscillation Probability

In the Schrödinger picture, assuming a free Hamiltonian diagonal in the mass basis, each mass eigenstate evolves as a plane wave: $$|ν_i(t)⟩ = e^{ -i E_i t} |ν_i⟩\,, \qquad E_i = \sqrt{p^2 + m_i^2} \,.$$ Expressing the flavor state at time t,

$$|ν_\alpha(t)⟩ = \sum_{i=1}^{n} U^*_{\alpha i}\, e^{- i E_i t} |ν_i⟩\,,$$

the amplitude to be detected as flavor β after time t (or, equivalently, after distance L ≈ t in natural units) is

$$A_{\alpha \to \beta}(L) = \sum_{i} U_{\beta i} U^*_{\alpha i} e^{-i E_i L}\,,$$

with the vacuum oscillation probability given by

$$P_{α\to\beta}(L) = |A_{α\to\beta}|^2 = \sum_{i,j} U_{\beta i} U^*_{\alpha i} U_{\beta j}^* U_{\alpha j} e^{-i(E_i - E_j) L}\,.$$

In the ultrarelativistic limit (E ≫ m_i), expanding E_i = E + m_i^2/(2E) and setting L ≈ t,

$$P_{α\to\beta}(L,E) = \sum_{i,j} U_{\beta i} U^*_{\alpha i} U_{\beta j}^* U_{\alpha j} \exp\left[ -i \frac{Δm_{ij}^2}{2E} L\right]\,,$$

with Δm_{ij}^2 = m_i^2 - m_j^2 (Bilenky, 2016, Upadhyay et al., 2011, Bilenky, 2012, Fujikawa, 7 May 2024).

The two-flavor form reduces to

$$P_{α\to\beta}(L,E) = \sin^2(2\Theta) \sin^2\left( \frac{Δm^2 L}{4E} \right),$$

and the oscillation length is λ_{osc} = 4πE/Δm^2 (Weinheimer, 2010).

3. Field-Theoretic, Wave-Packet, and Scattering Theory Approaches

The quantum field theory (QFT) formalism treats the production, propagation, and detection as a single Feynman diagram, with external source and detector particles described as localized wave packets (Naumov et al., 2010, Kovalenko et al., 2022, Grimus, 2023, Dobrev et al., 14 Apr 2025). The flavor transition probability emerges from the interference of amplitudes with different neutrino masses exchanged internally. The statistically averaged transition probability, after integrating over all unmeasured degrees of freedom, exhibits the same phase structure as the quantum-mechanical approach: $$P_{α\to\beta}(L) = \sum_{i, j} U_{\alpha i} U^*_{\beta i} U^*_{\alpha j} U_{\beta j} e^{-i Δm_{ij}^2 L/(2E)} \quad$$ (Kovalenko et al., 2022, Grimus, 2019, Dobrev et al., 14 Apr 2025).

Wave-packet treatments incorporate the effect of finite spatial and temporal localization at the source and detector. In this case, the propagation amplitude includes additional suppression terms reflecting the decoherence arising from wave-packet separation: $$P_{α\to\beta}(L) = [\text{coherent terms}] \times \exp\left[ - (L/L_{\rm coh})^2 \right],$$ where the coherence length is typically

$$L_{\rm coh} \sim \frac{4\sqrt{2} E^2}{|\Delta m^2| \sigma_p}\,,$$

with σp the momentum width of the packet. In laboratory and astrophysical settings, L{\rm coh} is much greater than relevant baselines, so complete coherence is effectively maintained (Perez et al., 2013, Naumov et al., 2010, Ciuffoli et al., 2020).

Explicit QFT calculations with generic wave-packet source and detector states confirm that, under conditions where all involved states cannot kinematically distinguish the different neutrino masses, the standard oscillation probability emerges robustly (Grimus, 2019, Dobrev et al., 14 Apr 2025).

4. Key Physical Parameters and Dependencies

The neutrino oscillation probability in vacuum arises from interference between different mass eigenstates and is a function of the following parameters:

• Mixing Matrix U: Elements U_{αi} specify the admixture of mass eigenstates in each flavor state. For n=3, U is parametrized by three mixing angles (θ{12}, θ{13}, θ_{23}) and a Dirac CP phase δ (Bilenky, 2016, Bilenky, 2012).
• Mass-Squared Differences Δm²_{ij}: The squared mass gaps between eigenstates set the oscillation frequencies in L/E. Measured splittings include |Δm^2_{21}| ≈ 7.5 × 10^−5 eV² (solar) and |Δm^2_{31}| ≈ 2.4 × 10^−3 eV² (atmospheric) (Bilenky, 2016).
• Oscillation Length:

$L_{osc}^{(ij)} = \frac{4\pi E}{|Δm_{ij}^2|}\,,$

which dictates the spatial period over which the probability cycles (Bilenky, 2012).

• CP-Violating Phase δ: Enters as imaginary components in appearance probabilities and leads to differences between neutrino and antineutrino oscillation probabilities. Observable only in the presence of three-flavor mixing (Bilenky, 2016, Bilenky, 2012).
• Coherence: If the spatial separation between mass eigenstate packets at the detector exceeds their spatial width, the interference terms average out and oscillations are damped. For typical neutrino parameters, L_{\rm coh} greatly exceeds experimental baselines, and such decoherence is negligible unless there is environmental sensitivity to emission or detection times (Kovalenko et al., 2022, Grimus, 2023, Ciuffoli et al., 2022, Ciuffoli et al., 2020, Oksanen et al., 21 Nov 2024).

5. Approximations, Regimes, and Assumptions

Standard analyses employ several central approximations:

• Plane-Wave Approximation: The mass eigenstates are treated as plane waves with definite momentum and energy. The oscillation phase arises from their relative propagation (Bilenky, 2016, Weinheimer, 2010).
• Ultrarelativistic Limit: The neutrino energy is much larger than its mass, E ≫ m_i, so E_i ≈ E + m_i^2/(2E) and L ≈ t (Bilenky, 2016, Fujikawa, 7 May 2024). This matches all known experimental conditions for active neutrino flavors.
• Neglect of Matter Effects: Pure vacuum oscillations assume no forward scattering with background media (Earth, Sun, etc.). Matter effects must be included for neutrino propagation through dense environments, modifying the effective mixing and oscillation length (Bilenky, 2016).
• Wave-Packet Localization: The finite spatial and momentum width of source and detector states are included to model production and detection localization, leading to possible decoherence at extreme baselines (Naumov et al., 2010, Perez et al., 2013).
• No Baseline or Time-Tagging Decoherence: When the partner particles at production or detection are not measured, and the experimental time window substantially exceeds the intrinsic time delays due to mass splittings, no additional decoherence appears (Ciuffoli et al., 2020, Grimus, 2019).
• Two-Flavor and Three-Flavor Reductions: If only one Δm^2 and one mixing angle are relevant (or if one mixing is small), the full n-flavor formulas reduce to effective two-flavor expressions (Upadhyay et al., 2011, Bilenky, 2012).

6. Extensions and Physical Interpretation

Neutrino oscillations are inherently a manifestation of quantum coherence and superposition. They probe mass differences rather than absolute masses, and are sensitive to the full structure of the lepton mixing matrix. Coherent forward scattering off the Brout-Englert-Higgs vacuum can be formulated as a refractive index problem; the "refractive vacuum" approach yields the same oscillation formulas from a manifestly covariant wave perspective (Oksanen et al., 21 Nov 2024, Tureanu, 2023).

The field-theoretic derivations confirm that oscillation probabilities correspond to interference between amplitudes for different mass eigenstates propagating within the same energy window, provided that neither the source nor the detector measures the absolute neutrino mass (Kovalenko et al., 2022, Grimus, 2023, Grimus, 2019). The role of the source and detector time windows, as well as the finite lifetime of the source particle, can be transparently included in the QFT and scattering theory frameworks (Dobrev et al., 14 Apr 2025).

Majorana neutrinos—where each mass eigenstate is its own antiparticle—yield identical vacuum oscillation probabilities compared to Dirac neutrinos, since any additional Majorana CP phases cancel in the transition probability (Fujikawa, 7 May 2024, Perez et al., 2011).

Laboratory, reactor, atmospheric, and astrophysical neutrino experiments have now observed vacuum oscillation signatures over many orders of magnitude in energy and baseline length, providing direct evidence for nonzero neutrino mass splittings and nontrivial mixing (Bilenky, 2016).

7. Analogs, Simulations, and Extensions

The underlying physics of neutrino oscillations is formally analogous to wave interference in classical physics. Optical experiments with birefringent crystals, where polarization states correspond to flavor and refractive indices correspond to mass, precisely reproduce the mathematics of two-flavor neutrino oscillations in vacuum (Weinheimer, 2010). Similarly, quantum walks can simulate Dirac neutrino oscillations, providing an alternate discretized view that reproduces the continuum Dirac equation and oscillation formulae in the appropriate limit (Molfetta et al., 2016).

Investigations into more general settings, such as curved spacetime or in the presence of conformal scalar couplings, indicate that the basic oscillation structure is robust, with path-integral-weighted corrections to the oscillation phase or length, but unitarity and coherence remain preserved provided suitable conditions on localization and propagation are satisfied (Hammad et al., 2022, Blasone et al., 2018).

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References: (Weinheimer, 2010, Bilenky, 2016, Fujikawa, 7 May 2024, Naumov et al., 2010, Perez et al., 2013, Upadhyay et al., 2011, Bilenky, 2012, Grimus, 2023, Kovalenko et al., 2022, Dobrev et al., 14 Apr 2025, Dvornikov, 2010, Blasone et al., 2018, Grimus, 2019, Tureanu, 2023, Kovalenko et al., 2022, Ciuffoli et al., 2022, Ciuffoli et al., 2020, Oksanen et al., 21 Nov 2024, Molfetta et al., 2016, Hammad et al., 2022, Perez et al., 2011)