currents: since these scatterings have different cross sections depending on the neutrino flavor, they contribute to decoherence [@Stol2].

A natural phenomenological estimate for the order of the decoherence rate is: $\Lambda_{\text{\tiny DEC}}\sim n\,v\,\sigma$ with $v$ the relative velocity of the incoming neutrinos, $n$ the density of the environmental leptons, and $\sigma$ the relevant scattering cross section whose values are known in the literature [@sigmanue; @sigmanunu]: $$\begin{aligned}
\label{eq:s1}\sigma_{\nu_{e},e}&\simeq&7\times10^{-42}(\mathrm{E}_{\nu}/\mathrm{GeV})\mathrm{cm}^2\,,\\
\sigma_{\nu_{\mu},e}&\simeq&10^{-42}(\mathrm{E}_{\nu}/\mathrm{GeV})\mathrm{cm}^2\,,\\
\sigma_{\nu_e,\nu_e}&\simeq&2,8\times10^{-47}(\mathrm{E}_{\nu}/\mathrm{GeV})\mathrm{cm}^2\,,\\
\label{eq:s2}\sigma_{\nu_e,\nu_{\mu}}&\simeq&4\times10^{-48}(\mathrm{E}_{\nu}/\mathrm{GeV})\mathrm{cm}^2\,.\end{aligned}$$ The average density of electrons in the outer space and in the atmosphere are respectively $n_e^{\text{\tiny OUT}}\sim1/\mathrm{m}^3$ and $n_e^{\text{\tiny ATM}}\sim2\times10^{26}/\mathrm{m}^3$, while the average density of neutrinos is about $n_{\nu}\sim10^8/\mathrm{m}^3$ everywhere. Assuming $v$ to be the velocity of light in vacuum, then one finds: $$\begin{aligned}
\Lambda_{\text{\tiny DEC}}^{\text{\tiny OUT}} \sim 10^{-43}
(E/\text{eV})\,\text{Hz}\,,\qquad
\Lambda_{\text{\tiny DEC}}^{\text{\tiny ATM}}\sim10^{-20}(E/\text{eV})
\,\text{Hz}.\end{aligned}$$ with $\Lambda_{\text{\tiny DEC}}^{\text{\tiny OUT[ATM]}}$ the decoherence rate in the out-space \[atmosphere\]. Neutrinos travel through the atmosphere within $\sim10^{-4}\,$s. Using this data with the time-of-flights and energies in Table I, for the decoherence damping factor of the cosmogenic neutrinos, one finds: $\sim10^{-5}$. For solar neutrinos instead, one gets: $\sim10^{-18}$, thus the damping of solar neutrino oscillations is hardly detectable, in agreement with experimental results [@decoexp; @vogel].

This estimate shows that, since environmental decoherence on neutrino oscillations is 