Neutrino wavepackets and coherence lengths

For a one-dimensional illustration, take equal nonspreading Gaussian mass-component envelopes $\psi_i(x,t)\propto\exp[-(x-v_it)^2/(4\sigma_x^2)]$ with their oscillatory phases factored out. The corresponding spatial probability density has rms width sigma x. This definition matters: calling an amplitude width and a density width by the same symbol changes factors of square root two. The effective width represents assumptions about preparation and detection; it is not a universal neutrino radius measured independently of the source and experiment.

In natural units the ultrarelativistic group velocity is approximately one minus mass squared over twice energy squared. Two components therefore separate by $d\simeq|\Delta m^2|L/(2E^2)$ after traveling baseline L. Their normalized envelope overlap is the integral of the product of the two shifted Gaussians. Completing the square gives an exponential of minus d squared over eight sigma x squared. This is the interference damping factor in this simplified common-time spatial-overlap description, with equal widths, negligible spreading, and initially coherent production.

Coherent propagation cannot restore interference that was absent at production. If energy-momentum information in the source, recoil, or detector distinguishes the emitted mass eigenstate, one must sum corresponding probabilities after tracing over those records. In typical ultrarelativistic preparations, the scale $|\Delta m^2|/(2E)$ estimates the tiny kinematic distinction to compare with relevant uncertainties. A complete criterion involves both energy and momentum uncertainties and external-state localization, rather than one blanket equal-energy or equal-momentum rule. Entanglement alone is not sufficient to destroy interference when the accompanying alternatives remain indistinguishable.

Likewise, localization and experimental resolution affect whether a phase pattern is observable. Source and detector position spreads much smaller than the oscillation period avoid washing out its spatial variation. In common Gaussian treatments an effective packet width small compared with that period also avoids a localization suppression, but this is not a universal production-coherence theorem for every nearly stationary source. Detailed source-detector calculations can require separate spatial and temporal uncertainties. The simple separation estimate assumes those additional conditions are already favorable instead of claiming to replace them.

Even when every event remains coherent, averaging a phase over a broad energy bin or source-distance distribution can suppress its cosine. This is statistical averaging over distinct events, not separation of mass components within one wavepacket. Finer reconstruction can recover information lost only to coarse binning if the underlying data support it. It does not generally undo genuine loss of accessible overlap or a physical record identifying the mass alternative. Both mechanisms may produce the same simple averaged probability, so a rate alone cannot identify which mechanism operated.

The coherence length is a smooth scale: at one such length our Gaussian interference factor is e to the minus one, not zero. Its energy-squared scaling differs from the oscillation length's linear energy scaling, which can help organize estimates but is conditional on holding the effective width fixed. Source environments can change that width with energy or process. A claim of decoherence should therefore specify the packet model, external uncertainties, energy distribution, detector response, and the observable sensitive to their differences. A free pure-state phase rotation is not decoherence.