When Oscillations Stop: Wave-Packet Decoherence in Neutrino Propagation

The textbook derivation of neutrino oscillation has a clean simplicity: a neutrino is produced as a flavor eigenstate, propagates as a coherent superposition of mass eigenstates with slightly different phases, and is detected as a flavor eigenstate again. The probability that a ${\nu }_{\mu }$ produced at the source is detected as a ${\nu }_e$ after a distance $L$ oscillates as a sine of $\Delta m^{2}L/4E$. The formula sits at the heart of the entire field.

But strip the textbook treatment to its plane-wave bones and a question is hiding underneath. Mass eigenstates with different energies and momenta propagate at different group velocities. If the source produces a localized wave packet — and any real source must — then those mass-eigenstate wave packets gradually separate in space. Once they no longer overlap, they can no longer interfere, and the oscillation pattern should wash out. The plane-wave formula assumes infinite coherence; the real world has finite coherence; and somewhere in between, neutrino oscillations have to stop.

This post is about the wave-packet treatment of neutrino oscillations: when coherence breaks down, how to estimate the distance over which it does, and why in nearly every experimental setting this question is — surprisingly — irrelevant.

The plane-wave shortcut and its tension

The standard derivation starts by writing the flavor state $|{\nu }_{\alpha }⟩={\sum }_i{U}_{\alpha i}^{\ast }|{\nu }_i⟩$ as a sum over mass eigenstates, lets each mass eigenstate pick up a phase $e^{-i{E}_{i}t+i{p}_{i}x}$ as it propagates, and computes the flavor-transition amplitude at the detector. The result, after some sleight of hand about whether one uses equal energies or equal momenta for the mass eigenstates, is the textbook formula

$P({\nu }_{\alpha }\to {\nu }_{\beta })={\delta }_{\alpha \beta }-4{\sum }_{i<j}\mathrm{Re}\left[U_{\alpha i}{U}_{\beta i}^{\ast }{U}_{\alpha j}^{\ast }{U}_{\beta j}\right]{\sin }^2\left(\frac{\Delta m_{ij}^{2}L}{4E}\right)+\cdots$

The “sleight of hand” matters more than it looks. Equal-energy and equal-momentum prescriptions give different intermediate states and only agree on the final answer because the differences are buried in negligible corrections. More importantly, the plane-wave states extend over all space, so they never “separate” — coherence is built in trivially and never lost. A consistent treatment requires localizing the neutrino in a wave packet, and then the question of coherence becomes physical and meaningful.

Two kinds of decoherence

The wave-packet approach yields two distinct ways for oscillations to wash out.

The first is wave-packet separation, sometimes called kinematic decoherence. The mass eigenstates have slightly different group velocities, $v_i=p_i/E_i\approx 1-m_i^2/(2E^2)$. The difference in velocity between eigenstates $i$ and $j$ is

$\Delta v_{ij}=\frac{\Delta m_{ij}^2}{2E^2}$

After a distance $L$, the two wave packets are separated by $\Delta v_{ij}L$. If this separation exceeds the wave-packet width ${\sigma }_x$, the packets no longer overlap at the detector, the interference between mass eigenstates is suppressed, and the oscillation amplitude is multiplied by a damping factor $\exp \left[-(L/L_{\mathrm{coh}}{)}^2\right]$ with

$L_{\mathrm{coh}}\approx \frac{4\sqrt{2}{E}^2{\sigma }_x}{|\Delta m_{ij}^2|}$

This is the coherence length. Beyond it, the flavor transition probabilities settle to constant flavor-averaged values rather than continuing to oscillate.

The second mechanism is observational averaging. Even in the absence of physical decoherence, finite energy resolution or finite source-extension averages the oscillation pattern over many periods once $L$ is large enough. This is not strictly a coherence effect — the mass eigenstates may still overlap perfectly — but it produces the same observable consequence: the oscillation pattern is unresolved and the probability looks constant. Most experimental “decoherence” is actually averaging in disguise.

How large is the wave packet?

The wave-packet width ${\sigma }_x$ is the central physical input. It is not a property of the neutrino itself but of the production and detection processes. The smaller of two scales typically controls it: the localization of the source and the localization of the detector.

For a pion-decay beam, the source localization is roughly the pion’s mean free path divided by the time-dilation factor — a centimetre or so in typical accelerator setups. For nuclear beta decay in a fixed lattice, the source localization is set by the thermal velocity of the parent atom, giving wave-packet widths on the order of $10^{-7}$ cm. For solar neutrinos the production region is dominated by collisional broadening, giving widths around $10^{-10}$ cm. Detection processes add similar widths.

To translate these into coherence lengths for the atmospheric mass splitting $\Delta m^2\approx 2.5\times 10^{-3}$ eV² and a GeV-scale neutrino, one finds:

• Pion-decay beam, ${\sigma }_x\sim$ cm: $L_{\mathrm{coh}}\sim 10^{15}$ km.
• Nuclear beta decay, ${\sigma }_x\sim 10^{-7}$ cm: $L_{\mathrm{coh}}\sim 10^6$ km.
• Solar production, ${\sigma }_x\sim 10^{-10}$ cm: $L_{\mathrm{coh}}\sim 10^3$ km.

All of these vastly exceed the baselines in any terrestrial oscillation experiment, and most exceed the Earth-Sun distance. Decoherence is not the limiting factor.

When does it ever matter?

There are a few regimes where wave-packet decoherence is physically relevant.

The first is astrophysical neutrinos. For neutrinos travelling from a distant source — say a blazar at gigaparsecs, or the cosmic neutrino background that has been free-streaming since the universe was a second old — the propagation distance is vastly larger than any conceivable coherence length for terrestrial wave packets. Oscillations completely average out, and the flavor ratio at Earth is the mass-eigenstate-averaged value. This is the regime invoked in the discussion of astrophysical flavor ratios for IceCube and KM3NeT.

The second regime involves propagation through dense matter with stochastic potentials. Random scattering or fluctuations in the matter density introduce phase noise that can destroy coherence faster than the kinematic mechanism would. This kind of decoherence has been considered in the context of solar neutrinos and supernova environments, but again the standard analyses find that the effect is small.

The third, and most speculative, possibility is fundamental decoherence — couplings of neutrinos to a stochastic environment such as quantum gravity, leading to flavor oscillations being damped on top of the standard kinematic damping. Several experiments have looked for this kind of departure from textbook oscillation; none has been seen, and current limits push the allowed decoherence scale to extreme values. The mere fact that we observe consistent oscillation patterns in solar, atmospheric, reactor and accelerator experiments places strong bounds on any underlying decoherence physics.

Why it usually doesn’t matter

Despite the conceptual importance of wave-packet treatment for a fully consistent derivation, the punch line for experimental practice is short: in essentially every terrestrial oscillation experiment, the coherence length is so much larger than the baseline that decoherence makes no observable difference. The plane-wave formula works. What one does have to worry about in practice is averaging: finite energy resolution, finite source extension, and finite detector resolution combine to smear out the oscillation pattern in any real measurement. These are not coherence problems in the quantum-mechanical sense; they are ordinary smearing effects that any experimenter has to model carefully. The wave-packet formalism is essential for conceptual completeness and for the rare regimes where it really matters, but the textbook formula remains the workhorse for analyzing every real experiment.

Summary

Neutrino oscillations require coherence between the mass eigenstates that compose a flavor state. The wave-packet treatment makes this concrete: the eigenstates propagate at slightly different group velocities, separate after a coherence length set by their wave-packet width and the mass splitting, and lose interference once the separation exceeds the width. For practical experimental setups the coherence length is enormous — vastly larger than any terrestrial baseline — so the plane-wave formula is an excellent approximation. Decoherence becomes important only at astrophysical scales, where it explains why neutrinos from cosmic sources arrive in mass-eigenstate-averaged flavor ratios, and as a tool for hunting fundamental coupling of neutrinos to an unknown decohering environment. In day-to-day oscillation analysis, observational averaging matters more than fundamental decoherence — and the textbook formula remains the tool of choice.