Neutrino Oscillation Explained

• Neutrino oscillation is a quantum phenomenon where neutrinos change flavor during propagation due to interference among mass eigenstates, as confirmed by diverse experiments.
• The three-flavor paradigm uses the PMNS matrix with six parameters, including mixing angles and mass-squared differences, to accurately describe oscillation probabilities.
• Matter effects modify neutrino oscillations in various environments, highlighting experimental strategies and theoretical challenges in determining mass ordering and CP violation.

Neutrino oscillation is the quantum phenomenon wherein a neutrino produced in a specific flavor (electron, muon, or tau) can later be detected as a different flavor. This conversion is a direct result of the fact that the flavor eigenstates—the interaction states defined by the weak charged current—are nontrivial superpositions of the mass eigenstates. As these mass eigenstates propagate with slightly different phases, their interference leads to flavor change probabilities that oscillate with distance and energy. The oscillation process, which has been confirmed by solar, atmospheric, reactor, and accelerator neutrino experiments, provides direct evidence that neutrinos have mass and that lepton flavor is not conserved in propagation, extending the Standard Model of particle physics. The three-flavor paradigm describes these phenomena in terms of six fundamental parameters: three mixing angles, two independent mass-squared differences, and one Dirac CP-violating phase (Denton, 14 Jan 2025).

1. Theoretical Formalism and Flavor–Mass Structure

Neutrino oscillation arises because the weak-interaction flavor eigenstates $|\nu_\alpha\rangle$ ($\alpha = e, \mu, \tau$) are nontrivial linear combinations of the mass eigenstates $|\nu_i\rangle$ ($i = 1,2,3$). This relationship is encoded by the unitary Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix:

$$|\nu_\alpha\rangle = \sum_{i=1}^3 U^*_{\alpha i} |\nu_i\rangle.$$

Each mass eigenstate evolves with a phase factor $\exp(-i m_i^2 L / 2E)$ over propagation distance $L$ (assuming $E \gg m_i$). The transition amplitude for a flavor change $\nu_\alpha \to \nu_\beta$ is then:

$$A(\nu_\alpha \to \nu_\beta;L,E) = \sum_{i=1}^3 U^*_{\alpha i} U_{\beta i} \exp\left(-i\frac{m_i^2 L}{2E}\right).$$

The oscillation probability is:

$$P(\nu_\alpha \to \nu_\beta;L,E) = |A(\nu_\alpha \to \nu_\beta;L,E)|^2.$$

The PMNS matrix is conveniently parameterized as

$U = R_{23} \cdot R_{13}(\delta) \cdot R_{12},$

where $R_{ij}$ is a rotation in the $ij$-plane and $\delta$ is the Dirac CP-violating phase.

2. Oscillation Parameters and the Three-Flavor Paradigm

Within the three-flavor framework, oscillations are governed by six parameters (Denton, 14 Jan 2025):

• Two mass-squared differences: $\Delta m^2_{21} \equiv m_2^2 - m_1^2$ (“solar splitting”) and $\Delta m^2_{31}$ (“atmospheric splitting”; with $\Delta m^2_{32}$ related).
• Three mixing angles: $\theta_{12}$ (solar), $\theta_{23}$ (atmospheric), and $\theta_{13}$ (reactor).
• One Dirac CP phase: $\delta$.

The explicit three-flavor probability in vacuum is:

$$\begin{aligned} P(\nu_\alpha \rightarrow \nu_\beta; L, E) = &\ \delta_{\alpha\beta} - 4 \sum_{i>j} \mathrm{Re}(U_{\alpha i}U^*_{\beta i} U^*_{\alpha j}U_{\beta j}) \sin^2\left( \frac{\Delta m_{ij}^2 L}{4E} \right) \ &\pm 8 J \sin\left(\frac{\Delta m_{21}^2 L}{4E}\right) \sin\left(\frac{\Delta m_{31}^2 L}{4E}\right) \sin\left(\frac{\Delta m_{32}^2 L}{4E}\right), \end{aligned}$$

where $J$ is the Jarlskog invariant, quantifying CP violation:

$$J = \frac{1}{8} \cos\theta_{13} \sin2\theta_{13} \sin2\theta_{12} \sin2\theta_{23} \sin\delta.$$

The oscillation length for a given mass-squared difference is $L^{\mathrm{osc}}_{ij} = 4\pi E/\Delta m_{ij}^2$.

3. Role of the Matter Effect

As neutrinos propagate through matter, coherent forward scattering with electrons modifies their effective potential, altering the oscillation pattern (Denton, 14 Jan 2025). The effective Hamiltonian in the flavor basis acquires a term proportional to the electron density $n_e$:

$$H_{\mathrm{matter}} = \frac{1}{2E} U \mathrm{diag}(0, \Delta m_{21}^{2}, \Delta m_{31}^{2}) U^\dagger + V,$$

where $V = \mathrm{diag}(V_{CC}, 0, 0)$, $V_{CC} = \sqrt{2} G_F n_e$, and $G_F$ is the Fermi constant. This “matter effect” (Wolfenstein effect) can resonantly enhance oscillations (MSW effect), which is critical for solar and some long-baseline terrestrial experiments.

At high energy or high density, matter effects cause the effective mixing angles and mass splittings to differ significantly from vacuum values, modifying both survival and appearance probabilities. For example, the high-energy solar neutrino survival probability is

$$P^{(\text{solar, HE})}(\nu_e \to \nu_e) \approx c_{13}^4 s_{12}^2 + s_{13}^4,$$

with $c_{13} = \cos\theta_{13}$, $s_{12} = \sin\theta_{12}$, etc.

4. Experimental Determination of Oscillation Parameters

Oscillation parameters are extracted via disappearance and appearance experiments:

Probe Type	Channel(s)	Primary Parameters	Experimental Realization
Disappearance	$P(\nu_e \to \nu_e)$, $P(\nu_\mu \to \nu_\mu)$	$\theta_{12}$, $\theta_{13}$, $\Delta m^2$	Solar (SNO, Borexino); KamLAND; Daya Bay
Appearance	$P(\nu_\mu \to \nu_e)$, $P(\nu_\mu \to \nu_\tau)$	$\theta_{13}$, $\delta$, mass ordering	T2K, NOvA, Super-Kamiokande

The measurement of $\theta_{13}$, previously thought to be “small,” was established to be relatively large ($\sim$9°) by Daya Bay and RENO in 2012, enabling enhanced sensitivity to CP violation and the mass ordering (Weiler, 2013).

Global fits combine data from solar, atmospheric, reactor, and accelerator neutrino experiments, yielding precise measurements of $\sin^2\theta_{12} \approx 0.308$, $$\Delta m^2_{21} \approx 7.41 \times 10^{-5} \text{ eV}^2$$, and $\Delta m_{31}^2 \sim 2.4$–$2.7 \times 10^{-3} \text{ eV}^2$ (Balantekin et al., 2013).

5. Phenomenological Consequences and Current Status

Neutrino oscillation provides unambiguous proof for nonzero neutrino masses and large flavor mixing—a major breakthrough in particle physics (Bellini et al., 2013). The framework predicts:

• Two distinct mass-squared splittings (solar and atmospheric scales).
• Large mixing angles: $\theta_{12}$, $\theta_{23}$; moderately large $\theta_{13}$.
• Oscillation probabilities sensitive to experimental $L/E$ ratios, CP phase, and matter effects.
• Survival and appearance channel measurements across a variety of energies and baselines show sinusoidal (or, with matter effects, distorted) dependence on $L/E$.

Solar, reactor, atmospheric, and long-baseline accelerator experiments robustly confirm the three-flavor paradigm, with remaining ambiguities concerning the sign of $\Delta m^2_{31}$ (normal vs. inverted hierarchy), the exact value of $\delta$, and the possible octant of $\theta_{23}$ (Denton, 14 Jan 2025).

6. Theoretical and Experimental Challenges

Open issues include (Bellini et al., 2013, Denton, 14 Jan 2025):

• Determining the absolute neutrino mass scale (oscillation experiments are sensitive only to differences).
• Elucidating the mass ordering (normal or inverted hierarchy).
• Measuring the CP-violating phase $\delta$ in the lepton sector, which may relate to the matter–antimatter asymmetry of the Universe.
• Establishing whether $\theta_{23}$ is exactly maximal ($45°$) or deviates (octant problem).
• Clarifying whether neutrinos are Dirac or Majorana particles (addressed by neutrinoless double beta decay searches).
• Testing for additional “sterile” states not coupling to standard weak interactions (motivated by anomalies in LSND, MiniBooNE, reactor, and gallium data) (Bellini et al., 2013).

Future experiments—such as DUNE, Hyper-Kamiokande, and JUNO—are designed to address these outstanding questions.

7. Model-Building and Beyond the Standard Oscillation Paradigm

Many theoretical models aim to explain the observed pattern of mixing angles and mass splittings (Weiler, 2013). Early flavor symmetry models invoked schemes such as tri-bimaximal mixing, which is now excluded by the measured size of $\theta_{13}$; current efforts use discrete flavor groups and “flavon” fields to generate the correct mixings. These models seek to embed oscillation parameters in deeper structures, potentially connecting neutrino properties to grand unification, lepton-number violation, or other new physics beyond the Standard Model.

The quantitative and experimental success of the three-flavor oscillation model has firmly established neutrino oscillation as a probe of flavor structure, CP violation, and physics at high scales, but crucial aspects of the underlying theory remain to be elucidated through ongoing and future measurements.