$\tilde\tau$ searches at the ILC

The colour neutral sector is the one most relevant for the suggestions SUSY presentsto the problems of the the SM - the hierarchy problem, naturalness, dark matter, themuon g-2 enigma. This sector is both for theoretical reasons, and from the results ofglobal fits, expected to be light. But, unfortunately, it is also the sector for whichcurrent energy frontier colliders, viz. the LHC, is less well adapted.In contrast, any future high energy electron-positron collider offers an excellent facilityfor SUSY searches in general, and for searches in the colour neutral sector in particular.Among the particles in this part of the SUSY spectrum, the superpartner of the $\tau$-lepton,the $\widetilde{\tau}$, plays an important role. The $\widetilde{\tau}$ is likely to be the lightest of thesfermions, so to be the first SUSY particle that could be observed. As a benchmark for thepower of SUSY searches, the $\widetilde{\tau}$ is the prime candidate, as it can for good reasons beregarded as the worst and thus most general scenario for the searches: If one can find the$\widetilde{\tau}$, then any alternative next-to-lightest SUSY particle (NLSP) will also be findable.In this contribution, a detailed study of the direct $\widetilde{\tau}$ pair-production at the InternationalLinear Collider (ILC) has been performed, showing the capability of this collider for determiningstau exclusion/discovery limits in a model-independent way.The studies were done using the full detector simulation and reconstruction procedures of theInternational Large Detector concept (ILD) at the ILC. The simulation included all SM backgrounds,as well as the beam induced ones. A detailed study of the effect of beam induced backgrounds, asoverlay-on-physics and - for the first time - overlay-only events, in the stau limits was performed.The analysis of the worst mixing for stau searches at the ILC conditions is also included.We have thus carefully evaluated all possible complications, both theoreticallyand experimentally, and are confident that this analysis indeed confirms the statement that at alinear e$^+$e$^-$ collider, SUSY will be discovered if the NLSP mass is up to just a few GeV belowthe kinematic limit of the collider