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Searching for $ν_μ\to ν_τ$ Oscillations with Extragalactic Neutrinos

We propose a novel approach for studying $ν_μ\to ν_τ$ oscillations with extragalactic neutrinos. Active Galactic Nuclei and Gamma Ray Bursts are believed to be sources of ultrahigh energy muon neutrinos. With distances of 100 Mpc or more, they provide an unusually long baseline for possible detection of $ν_μ\to ν_τ$ with mixing parameters $Δm^2$ down to $10^{-17}$eV$^2$, many orders of magnitude below the current accelerator experiments. By solving the coupled transport equations, we show that high-energy $ν_τ$'s, as they propagate through the earth, cascade down in energy, producing the enhancement of the incoming $ν_τ$ flux in the low energy region, in contrast to the high-energy $ν_μ$'s, which get absorbed. For an AGN quasar model we find the $ν_τ$ flux to be a factor of 2 to 2.5 larger than the incoming flux in the energy range between $10^2$ GeV and $10^4$ GeV, while for a GRB fireball model, the enhancement is 10%-27% in the same energy range and for zero nadir angle. This enhancement decreases with larger nadir angle, thus providing a novel way to search for $ν_τ$ appearance by measuring the angular dependence of the muons. To illustrate how the cascade effect and the $ν_τ$ final flux depend on the steepness of the incoming $ν_τ$, we show the energy and angular distributions for several generic cases of the incoming tau neutrino flux, $F_ν^0 \sim E^{-n}$ for n=1,2 and 3.6. We show that for the incoming flux that is not too steep, the signal for the appearance of high-energy $ν_τ$ is the enhanced production of lower energy $μ$ and their distinctive angular dependence, due to the contribution from the $τ$ decay into $μ$ just below the detector.