$\widetilde{\tau}$ searches at future e$^+$e$^-$ colliders

The direct pair-production of the superpartner of the $\tau$-lepton, the $\widetilde{\tau}$, is one of the most interesting channels to search for SUSY in: the $\widetilde{\tau}$ is likely to be the lightest of the scalar leptons, and is one of the most experimentally chalanging ones. The current model-independent $\widetilde{\tau}$ limits come from LEP, while limits obtained at the LHC do extend to higher masses, but are model-dependent. The future Higgs factories will be powerful facilities for SUSY searches, offering advantages with respect to previous electron-positron colliders as well as to hadron machines. In order to quantify the capabilities of these future $e^+e^-$ colliders, the "worst-case" scenario for $\widetilde{\tau}$ exclusion/discovery has been studied, taking into account the effect of the $\widetilde{\tau}$ mixing on $\widetilde{\tau}$ production cross-section and detection efficiency. To evaluate the latter, the ILD concept, originally developed for the International Linear Collider (ILC), and the ILC beam conditions at a centre-of-mass energy of 500 GeV have been used for detailed simulations. The obtained exclusion and discovery reaches extend to only a few GeV below the kinematic limit even in the worst-case scenario. The results of the detailed simulation study are then discussed in view of the experimental environment of other proposed Higgs factory projects.