by $\rm QFit$, are mostly found close to the vertical direction, as illustrated in Figure \[costhcutfig\] (top).

After the pre-selection, the angular separation, $\rm \Psi$, between the reconstructed track and the GC’s direction is used in an optimisation of the model rejection factor [@mrf] as a function of the WIMP mass, $\rm M_{WIMP}$. For each WIMP mass and annihilation channels listed in the Tables \[tab:qresultsone\]+\[tab:qresultstwo\]+\[tab:qresultsthree\]+\[tab:qresultsfour\], the selected value $\rm \Psi$ is the one that minimises the average 90% confidence level (C.L.) upper limit on the $\rm \nu_{\mu}+\bar{\nu}_{\mu}$ flux, $\rm \overline{\Phi}_{\nu_{\mu}+\bar{\nu}_\mu}$, defined as:

$$\rm{\overline{\Phi}_{\nu_{\mu}+\bar{\nu}_\mu} = \frac{\bar{\mu}^{90\%}}{\sum\limits_{i} \bar{A}_{eff}^{i}(M_{WIMP}) \times T_{eff}^{i}}} \, ,
\label{mrfeq}$$

where the index $\rm i$ denotes the periods with different detector configurations, $\rm \bar{\mu}^{90\%}$ is the average upper limit of the background from scrambled data at $90$% C.L. (computed using a Poisson distribution in the Feldman-Cousins approach [@feldmancousins]), and $\rm {T_{eff}^{i}}$ is the effective livetime for each detector configuration. The effective area averaged over the neutrino energy, $\rm{\bar{A}_{eff}^{i}(M_{\rm WIMP})}$, is defined as:

$$\begin{aligned}
\rm{\bar{A}_{eff}^{i}(M_{\rm WIMP})} & = & 
\rm{\frac{\int_{E_{\nu_{\mu}}^{th}}^{M_{WIMP}} \left ( A_{eff}^{\nu_{\mu},i}+A_{eff}^{\bar{\nu}_{\mu},i} \right ) \, \frac{dN_{\nu_{\mu}}}{dE_{\nu_{\mu}}}\Big|_{\oplus} dE_{\nu_{\mu}}}
{2 \, \int_{0}^{M_{\rm WIMP}}\frac{dN_{\nu_{\mu}}}{dE_{\nu_{\mu}}} dE_{\nu_{\mu}}}} \, ,
\label{aeffeq}\end{aligned}$$

[./Accept\_All\_BBFitS\_AAFitM\_LogScale\_Other.eps]{} (70,15)[![Example of the acceptance $\rm \bar{A}_{eff}(M_{\rm WIMP}) \times T_{eff}$ ($\rm m^{2}\cdot yr$) to 