Three-Flavor Oscillation: Patterns in L/E

In the three-flavor framework, neutrino oscillation is described by the PMNS mixing matrix and three independent mass-squared differences (only two are independent — the third is the sum). The probability that a neutrino produced as flavor $\alpha$ is detected as flavor $\beta$ after travelling distance $L$ at energy $E$ is: $P({\nu }_{\alpha }\to {\nu }_{\beta })={\delta }_{\alpha \beta }-4{\sum }_{i>j}\text{Re}(U_{\alpha i}^{\ast }{U}_{\beta i}{U}_{\alpha j}{U}_{\beta j}^{\ast }){\sin }^2\left(\frac{\Delta m_{ij}^{2}L}{4E}\right)+2{\sum }_{i>j}\text{Im}(U_{\alpha i}^{\ast }{U}_{\beta i}{U}_{\alpha j}{U}_{\beta j}^{\ast })\sin \left(\frac{\Delta m_{ij}^{2}L}{2E}\right)$

This expression is more complicated than the two-flavor formula but has rich structure. For different ranges of $L/E$, different terms dominate, revealing different combinations of mixing parameters.

This post is about the three-flavor oscillation patterns: how they look in $L/E$, what each regime tells us, and how experiments are designed to probe them.

The three-flavor parameters

The three mixing angles and one CP phase that characterize the PMNS matrix are:

• ${\theta }_{12}$ — solar mixing, approximately 33°
• ${\theta }_{23}$ — atmospheric mixing, approximately 45° (near-maximal)
• ${\theta }_{13}$ — reactor/short-baseline mixing, approximately 8.6°
• ${\delta }_{CP}$ — CP-violating phase, currently measured to be approximately $-90°$ but with significant uncertainty

The two independent mass-squared splittings are:

• $\Delta m_{21}^2\approx 7.4\times 10^{-5}$ eV² (solar splitting)
• $\Delta m_{31}^2\approx 2.5\times 10^{-3}$ eV² (atmospheric splitting; sign yet to be determined)

These six parameters (three angles, one phase, two splittings) fully describe three-flavor oscillation in vacuum. The mass ordering is the seventh discrete unknown.

Survival probability patterns

For the ν_e survival probability — the most common observable for solar and reactor experiments — the leading-order behavior is: $P({\nu }_e\to {\nu }_e)\approx 1-{\sin }^2(2{\theta }_{12}){\cos }^4({\theta }_{13}){\sin }^2\left(\frac{\Delta m_{21}^{2}L}{4E}\right)-{\sin }^2(2{\theta }_{13}){\sin }^2\left(\frac{\Delta m_{31}^{2}L}{4E}\right)$

The two sinusoidal terms have very different frequencies. The solar term oscillates slowly (period of order $10^4$ km/MeV); the atmospheric term oscillates rapidly (period of order $10$ km/MeV).

Probing different regimes

Different experiments are designed for different regimes:

Reactor short-baseline (Daya Bay, RENO, Double Chooz): $L/E\sim 0.5$ km/MeV. Sensitive primarily to ${\theta }_{13}$ via the rapid oscillation term. Daya Bay’s 2012 measurement of ${\theta }_{13}$ used this regime.

Reactor medium-baseline (JUNO): $L/E\sim 12$ km/MeV. Sensitive to both ${\theta }_{12}$, $\Delta m_{21}^2$ and the relative phase that distinguishes mass ordering. The unique sensitivity comes from the spectral shape of the oscillation pattern.

Reactor long-baseline (KamLAND): $L/E\sim 80$ km/MeV. Primarily sensitive to ${\theta }_{12}$ and $\Delta m_{21}^2$ via the slow oscillation. KamLAND’s 2003 measurement was the first to constrain solar oscillation parameters terrestrially.

Long-baseline accelerator (T2K, NOvA, DUNE): $L/E\sim 500$-1500 km/GeV at $E\sim 0.6-3$ GeV. Sensitive to atmospheric splitting, ${\theta }_{23}$, ${\theta }_{13}$, and CP phase ${\delta }_{CP}$. Measures both survival (${\nu }_{\mu }\to {\nu }_{\mu }$) and appearance (${\nu }_{\mu }\to {\nu }_e$).

Atmospheric (Super-K, IceCube): $L/E$ varies from $\sim 1$ km/GeV (downward) to $\sim 10^4$ km/GeV (upward through Earth). The full L/E range is sampled in a single measurement, providing one of the cleanest tests of the oscillation framework.

Appearance probability and CP violation

The ${\nu }_{\mu }\to {\nu }_e$ appearance probability — the principal observable for long-baseline accelerator experiments — has the structure: $P({\nu }_{\mu }\to {\nu }_e)\approx {\sin }^2({\theta }_{23}){\sin }^2(2{\theta }_{13}){\sin }^2\left(\frac{\Delta m_{31}^{2}L}{4E}\right)+\text{(CP-violating + matter terms)}$

The leading term is approximately:

• ${\sin }^2({\theta }_{23})\approx 0.5$
• ${\sin }^2(2{\theta }_{13})\approx 0.085$
• The oscillation factor at the first $\Delta m_{31}^2$ maximum is at $L/E\approx 500$ km/GeV

Combined, the leading appearance probability is approximately $4\%$ at the first oscillation maximum. This is small enough that long-baseline experiments need:

• Massive detectors (DUNE: 40 kt; Hyper-K: 260 kt fiducial)
• Intense beams ($10^{20}-10^{21}$ POT per year)
• Long running times

The CP-violating term appears at order: $\Delta P_{CP}\approx 8J{\sin }^2\left(\frac{\Delta m_{31}^{2}L}{4E}\right)\cdot \cos {\delta }_{CP}$ where $J$ is the Jarlskog invariant ($J\approx 0.03$). The CP-violating modulation of the appearance probability is at the 10-30% level — large enough to be detectable in a multi-year run.

Matter effects

In matter, the oscillation parameters are modified by the coherent forward scattering on electrons. The effect is most important for ${\nu }_{\mu }\to {\nu }_e$ appearance at long-baseline accelerator experiments. The ordering-dependent modification distinguishes normal from inverted ordering.

For neutrinos in normal ordering at DUNE-like baselines (1300 km), the matter effect enhances the appearance probability by approximately 30%. For inverted ordering, it suppresses it by a similar amount. For antineutrinos, the effects reverse.

This is the basis for DUNE’s mass-ordering measurement and a key reason DUNE was sited at a longer baseline than NOvA. Longer baseline = more matter = bigger ordering-dependent effect = better discrimination.

Vacuum vs. matter regime

For very short baselines ($L<10$ km at GeV energies, or $L<1000$ km at MeV energies), vacuum oscillation dominates and matter effects are negligible. For long baselines through Earth, matter effects become significant.

The crossover happens when: $E_{\nu }\sim \frac{\Delta m^2}{2V_e}$

For Earth-mantle density and atmospheric splitting, this is approximately $E_{\nu }\sim 6$ GeV. Below this energy, matter effects are subdominant; above it, they dominate.

This crossover is why JUNO (which uses sub-MeV reactor antineutrinos) is essentially a vacuum oscillation experiment, while DUNE (using GeV-scale neutrinos through 1300 km of Earth) operates in the matter-dominated regime.

What the patterns reveal

The three-flavor oscillation framework predicts specific patterns at every L/E scale. Modern measurements have confirmed most of these patterns:

• The atmospheric-scale oscillation at $L/E\sim 30$ km/GeV (Super-K, MINOS, IceCube)
• The reactor-baseline ${\theta }_{13}$-driven oscillation at $L/E\sim 1$ km/MeV (Daya Bay, RENO, Double Chooz)
• The solar/KamLAND oscillation at $L/E\sim 10^4$ km/GeV
• The intermediate-baseline pattern at $L/E\sim 12$ km/MeV (JUNO, coming)
• Appearance modes at long baselines (T2K, NOvA, soon DUNE)

Each measurement tests the framework in a different regime. Their consistent agreement establishes the three-flavor picture as the correct description of neutrino oscillation, with the remaining open questions being the discrete unknowns (mass ordering, octant of ${\theta }_{23}$, value of ${\delta }_{CP}$).

A unified pattern

What makes the three-flavor picture compelling is its predictive power. Once the six parameters are fixed by some experiments, the same parameters predict the patterns at all other L/E scales. This consistency is one of the field’s greatest successes.

The various experiments — solar, atmospheric, reactor, beam — are not separate phenomena but different views of the same underlying physics. The PMNS matrix, with its three angles and one phase, completely determines the oscillation pattern at any L/E. Each new measurement is a test of the framework’s consistency.

By 2030, with JUNO operational, DUNE running, and Hyper-K commissioning, all six parameters will be measured to high precision and the framework will have been tested in essentially all accessible regimes. Any deviation from the predicted patterns at that point would be a clean signal of new physics — sterile neutrinos, non-standard interactions, or something more exotic.

For now, the framework holds. The L/E patterns are confirmed across more than four orders of magnitude in scale. The three-flavor picture is, by every indication, the correct description.

That this works at all — that simple unitary matrix structure with six parameters can describe oscillation across nine decades of L/E — is itself a non-trivial fact about how neutrinos behave. The pattern is there. We measure it. The Standard Model, suitably extended, accounts for it.