observe the oscillation $\rm \nu_{\alpha}\rightarrow\nu_{\mu}$, where $\rm \alpha = e,\mu,\tau$, and $\rm dN_{\nu_{\alpha}}/dE_{\nu_{\alpha}}\Big|_{\ominus}$ is the corresponding energy spectrum at the GC. Examples of muon neutrino energy spectra at the surface of the Earth, $\rm dN_{\nu_{\mu}}/dE_{\nu_{\mu}}\Big|_{\oplus}$, are shown in Figure \[benchchfig\] as a function of their energy $\rm E_{\nu_{\mu}}$ (GeV) (the spectra in muon anti-neutrinos are identical), for an indicative mass $\rm M_{WIMP} = 360$ GeV. In this figure, the three-flavour oscillations process in vacuum over the GC-Earth line of sight is used as expressed in Equation \[nuoscs\]. As with the dark matter search in the direction of the Sun [@dmsun], $\rm WIMP\,WIMP$$\rm \rightarrow$$\rm b\bar{b}$ is the softest channel of muon neutrino production. The hardest channel of muon neutrino production from the GC is the primary $\rm WIMP\,WIMP \rightarrow \nu_{\alpha}\bar{\nu}_{\alpha}$. It is calculated taking into account the contribution of electroweak corrections discussed in Ref. [@cirelli]. The three other channels, $\rm WIMP\,WIMP \rightarrow W^{+}W^{-},\,\tau^{+}\tau^{-},\,\mu^{+}\mu^{-}$, present a hard contribution to the full spectrum of muon neutrinos at Earth. It can be noted that the channels to $\rm \tau^{+}\tau^{-}$ and $\rm \mu^{+}\mu^{-}$ differ only for $\rm E_{\nu_{\mu}} < 100$ GeV in this example.

In order to be as model-independent as possible, a self-annihilation branching 