Wave Packet Modified Neutrino Propagator

• The paper introduces a wave packet formalism that replaces idealized plane waves with localized superpositions, enhancing the treatment of neutrino oscillation probabilities.
• Key results demonstrate modified coherence lengths and damping factors that are crucial for interpreting short and long baseline experimental data.
• The approach integrates spatial dispersion, matter effects, and field-theoretical corrections, offering a more accurate framework for neutrino propagation.

A wave packet modified neutrino propagator describes neutrino propagation incorporating the finite spatial and momentum uncertainty intrinsic to production and detection processes. Unlike the idealized plane-wave formulation, the wave packet approach replaces sharp energy-momentum eigenstates by localized superpositions, fundamentally altering propagation amplitudes and flavor transition probabilities. This construct is essential for rigorous treatments of coherence, decoherence, finite detector/source localization, and quantum field theoretical (QFT) consistency in neutrino oscillation phenomena.

1. Foundations of the Wave Packet Approach

The wave packet framework is motivated by the impossibility of producing or detecting mono-energetic neutrino states in realistic physical scenarios. Instead, neutrinos are generated and detected as part of localized processes; thus, their kinematical states are superpositions characterized by finite momentum (and position) spreads. This formalism corrects the inadequacies of the plane-wave treatment for describing coherence loss, event rates at short or long baselines, and modifications due to spatial resolution.

In the non-relativistic quantum mechanical setting, the propagator for a single mass eigenstate is given by the Green's function for a Schrödinger or Dirac equation, but extended to superpositions with Gaussian or more general envelopes in momentum space. In QFT, the neutrino appears as an internal line in Feynman diagrams bridging source and detector vertices, while the external states are represented by covariant wave packets. The resulting amplitude involves the convolution of the standard propagator with wave-packet form factors, fundamentally altering its analytic structure (Naumov et al., 18 Dec 2025, Naumov et al., 2014, Naumov et al., 2010).

2. Mathematical Structure and Construction

Single-Particle Quantum Mechanics

A mass eigenstate $|\nu_j\rangle$ produced at $x=0$ is typically represented as

$$|\nu_j(t)\rangle = \int \frac{dp}{\sqrt{2\pi}}\, f_j(p) \exp(-i E_j(p) t) |\nu_j,p\rangle$$

where $f_j(p)$ is a normalized envelope, often Gaussian: $$f_j(p) = \frac{1}{(2\pi \sigma_p^2)^{1/4}} \exp\Big(-\frac{(p-p_j)^2}{4\sigma_p^2}\Big)$$ with $p_j$ the mean momentum, $\sigma_p$ the width. The configuration-space solution is a wave packet propagating with group velocity $v_j = p_j/E_j$ and experiencing dispersion controlled by $\tau_j \sim E_j^3/(m_j^2 \sigma_p^2)$. The corresponding propagator is the time-evolved wave packet (Cheng et al., 2020, Chan et al., 2015).

Field-Theoretical Propagator

The QFT amplitude for a process with localized source (S) and detector (D) is mediated by a neutrino line connecting external wave packets. This leads to the modified coordinate-space propagator: $$J(X) = \int \frac{d^4q}{(2\pi)^4}\, \widetilde\delta_S(q-q_S) \widetilde\delta_D(q+q_D) \, \frac{(\gamma \cdot q + m)}{q^2-m^2+i\epsilon} \, e^{-i q \cdot X}$$ where $X = x_D - x_S$, $q_{S,D}$ are momentum transfers, and $\widetilde\delta_{S,D}$ are packet-induced smeared delta functions. In the "relativistic Gaussian packet" (RGP) model, these form factors are sharply peaked Gaussians in momentum space, with explicit Lorentz-covariant generalizations available (including "asymmetric" wave packets with tensor structures controlling shape and anisotropy) (Naumov et al., 2014, Korenblit et al., 2017).

Under the saddle-point approximation for macroscopic baselines, this integral yields

$G_{wp,\,j}(L,T) \approx \frac{\slashed{\bar{q}}_j + m_j}{2\bar{E}_j} \exp \left[ -i \bar{E}_j T + i \bar{\mathbf{q}}_j \cdot \mathbf{L} - \Sigma_j(L,T) \right]$

where $\Sigma_j$ encodes spatial and temporal localization: $$\Sigma_j(L,T) = \frac{1}{2}(T_S^{\mu\nu} + T_D^{\mu\nu}) X_\mu X_\nu$$ with $T_{S,D}^{\mu\nu}$ constructed from the external packet widths and orientation tensors (Naumov et al., 2014, Naumov et al., 18 Dec 2025).

3. Phenomenology: Oscillation Probability and Decoherence

The wave packet approach modifies the flavor transition amplitude and probability, introducing coherence lengths, localization suppressions, and dispersion effects. For Gaussian packets, the averaged probability is found to be (Cheng et al., 2020, An et al., 2016, Chan et al., 2015): $$P_{\alpha \to \beta}(L) = \sum_{i,j} U^*_{\alpha i} U_{\beta i} U_{\alpha j} U^*_{\beta j} \, \sqrt{\frac{1}{1+y_{ij}^2}} \exp[-\lambda_{ij}] \exp[-\gamma_{ij}] \exp[i \phi_{ij}]$$ with \begin{align*} \lambda_{ij} &= (L/L_\text{coh}^{ij})^2, \ L_\text{coh}^{ij} &= \frac{4\sqrt{2} E^2}{|\Delta m^2_{ij}| \sigma_E}, \ y_{ij} &= L/L_\text{dis}^{ij}, \quad L_\text{dis}^{ij} = \frac{2\sqrt{2} E^2}{|\Delta m^2_{ij}| \sigma_E^2}, \ \gamma_{ij} &\propto \left( \Delta m_{ij}^2 \sigma_x / (2 E^2) \right)^2. \end{align*} The standard oscillation phase $\phi_{ij} \simeq \Delta m^2_{ij} L / (2E)$ is modulated by small packet-induced corrections. Decoherence arises when $L \gg L_\text{coh}$ due to group velocity differences, suppressing interference. Spreading effects are parameterized by $y_{ij}$, and excessive coordinate localization yields suppression via $\gamma_{ij}$ (Cheng et al., 2020, Chan et al., 2015).

In short- or long-baseline limits, leading corrections to the inverse-square law for event rates appear as $1/L^2 - \kappa L^2$ (short baseline) or $1/L^2 + c_1/L^4$ (long baseline), directly reflecting the interplay between localization and propagation (Naumov et al., 18 Dec 2025, Naumov et al., 2022).

4. Specialized Formulations and Extensions

Mixed Parity and Squeezed Coherent States

The 1D Dirac equation admits squeezed coherent state solutions, with physical propagating wave packets requiring equal-weight mixing of even and odd parity eigenfunctions. For each mass eigenstate, the propagating state is constructed as

$$|\Phi(a_0)\rangle = \int da\,e^{-\gamma(a-a_0)^2} \big[ |\psi_+(a)\rangle + |\psi_-(a)\rangle \big]$$

with $|\psi_\pm(a)\rangle$ being parity eigenstates. Pure-parity packets are non-propagating due to exact "zitterbewegung" cancellation, while mixed-parity packets propagate with maximal group velocity (Wang et al., 2012).

Covariant Asymmetric and Interpolating Wave Packets

Advancements have introduced asymmetric wave packets (AWP), where the momentum-space shape tensor $\rho_{\mu\nu}$ leads to direction-dependent coherence lengths and elliptic spatial localization—contrasting with the isotropic RGP model where $L_\text{coh}\propto 1/\sigma_x$. The interpolating wave packet formalism uniquely specifies packet width via analytic properties of the Wightman functions and ensures correct propagation and causality in QFT amplitudes (Naumov et al., 2014, Korenblit et al., 2017).

Magnetic Field and Matter Effects

In the presence of magnetic fields, propagation eigenstates split into spin branches with modified group velocities and dispersion relations, affecting both flavor and spin coherence lengths: $$E_i^s(p) = \sqrt{ m_i^2 + p^2 + \mu_i^2 B^2 + 2 s \mu_i \sqrt{ m_i^2 B^2 + p^2 B_\perp^2 }}$$ with resulting oscillation and coherence lengths depending on both $\Delta m^2$ and the neutrino magnetic moment (Popov et al., 16 Jan 2024).

Matter effects modify the effective Hamiltonian and eigenenergies, leading to resonance phenomena (MSW effect) and matter-enhanced wave packet coherence/decoherence. A new "matter coherence length" $L^{\text{mcoh}}_{ij}$ emerges in the wave packet formalism, parametrizing enhancement or suppression of the oscillation amplitude due to the interplay between spatial localization and matter-induced splittings (Qin et al., 2012).

5. Experimental Implications and Limits

The Daya Bay experiment directly constrained the intrinsic relative momentum dispersion $\sigma_{\rm rel}$ of the neutrino wave packet by fitting survival probabilities to data, yielding $2.38\times 10^{-17} < \sigma_{\rm rel} < 0.23$ (95% C.L.), or, with spatial width constraints, $10^{-14} \lesssim \sigma_{\rm rel} < 0.23$ and $\sigma_x$ bounded between $10^{-11}$ cm and 2 m. In all conventional reactor oscillation experiments, decoherence and localization corrections are negligible at typical distances, validating the use of plane-wave oscillation formulas in practical analyses. However, at ultra-short baselines or with exceptionally high energy or spatial resolution, wave packet modifications can lead to observable effects, including inverse-square law violation at very short distances, potentially contributing to the so-called reactor antineutrino anomaly (An et al., 2016, Naumov et al., 18 Dec 2025).

6. Comparative Summary of Approaches

Approach	Core Objects	Key Corrections
Single-particle (QM, squeezed)	1D Dirac packets, mixed parity	Only mixed parity propagates, damping in $a$ space; no field theory propagator
QFT, symmetric/AWP	RGP or AWP with packet overlap tensors	Damping via $T^{\mu\nu} X_\mu X_\nu$; direction-dependent $L_\text{coh}$
Covariant interpolating	Unique Lorentz-analytic packet, composite overlap functions	Causal, local, width from analytic structure
Direct experiment (Daya Bay)	Gaussian packet, fit to data	Limits on $\sigma_{\rm rel}$, negligible decoherence at baseline

While single-particle quantum mechanics yields important insights, the fully consistent and experimentally applicable framework for a wave packet modified neutrino propagator is field-theoretical, encoding all source and detector smearing and ensuring Lorentz covariance, causality, and analytic control over decoherence and localization effects (Naumov et al., 2014, Naumov et al., 18 Dec 2025, Naumov et al., 2010, Korenblit et al., 2017, Cheng et al., 2020).

7. Outlook and Contemporary Significance

Wave packet modifications to the neutrino propagator are crucial for rigorous quantitative predictions in oscillation phenomenology, especially under experimental conditions with extremely short or long baselines, high spatial or energy resolution, or for probing subtle effects such as the reactor antineutrino anomaly, inverse-square law violation, and experimental signatures of loss or (rare) enhancement of coherence due to matter or external fields. The mature field-theoretical approaches with explicitly constructed wave packet envelopes, including asymmetric or interpolating forms, provide the definitive tools for confronting such effects and will underpin future high-precision oscillation, astrophysical, and short-baseline reactor experiments (Naumov et al., 18 Dec 2025, Naumov et al., 2014, Popov et al., 16 Jan 2024).