neutrinos resulting from the self-annihilation channels in SM particles are propagated to the Earth for 17 different WIMP masses in the range from 25 GeV to 10 TeV. To compute the initial energy spectra at the GC, five benchmark self-annihilation channels of WIMP-like dark matter particles into SM particles are used:

$$\rm{WIMP\,WIMP \rightarrow b\bar{b},W^{+}W^{-},\tau^{+}\tau^{-},\mu^{+}\mu^{-},\nu_{\alpha}\bar{\nu}_{\alpha}} \, .
\label{benchch}$$

Also, neutrinos flavours, $\rm \alpha = e,\mu,\tau$, can be produced directly or as subsequent decay products of the SM particles listed above. The propagation of the neutrinos includes the three-flavour neutrino oscillations in vacuum along the line of sight from the GC to the surface of the Earth using the values from Table 8 in Ref. [@oscs]. The muon neutrino energy spectrum can be written as:

![Example of muon neutrino energy spectra at the surface of the Earth $\rm dN_{\nu_{\mu}}/dE_{\nu_{\mu}}\Big|_{\oplus}$ as a function of their energy $\rm E_{\nu_{\mu}}$ for a WIMP mass $\rm M_{WIMP} = 360$ GeV. The primary self-annihilation channels: $\rm WIMP\,WIMP$ $\rm \rightarrow$ $\rm b\bar{b}\,(green)$, $\rm W^{+}W^{-}\,(blue)$, $\rm \tau^{+}\tau^{-}\,(red)$, $\rm \mu^{+}\mu^{-}\,(black)$, $\rm \nu_{\mu}\bar{\nu}_{\mu}\,(orange)$ are shown.[]{data-label="benchchfig"}](./Cirelli_ESpectra_M360_Linear_Smooth_New.eps){width="0.8\linewidth"}

$$\rm{\frac{dN_{\nu_{\mu}}}{dE_{\nu_{\mu}}}\Big|_{\oplus} = \sum_{\alpha} P(\nu_{\alpha}\rightarrow\nu_{\mu}) \frac{dN_{\nu_{\alpha}}}{dE_{\nu_{\alpha}}}\Big|_{\ominus}} \, ,
\label{nuoscs}$$

where $\rm dN_{\nu_{\mu}}/dE_{\nu_{\mu}}\Big|_{\oplus}$ is the muon neutrino energy spectrum at the Earth, $\rm P(\nu_{\alpha}\rightarrow\nu_{\mu})$ is the probability to 