Searching for displaced vertices with a gaseous tracker for a future e$^+$e$^-$ Higgs factory

This paper presents results of the first full simulation study addressing prospects for observation of long-lived particles (LLPs) with the International Large Detector (ILD), operating at the International Linear Collider (ILC) at $\sqrt{s}=250$ GeV. Neutral LLP production, resulting in a displaced vertex signature inside the ILD's time projection chamber (TPC), is considered. We focus on scenarios interesting from the experimental perspective and perform a search based on displaced vertex finding inside the TPC volume. Two experimentally challenging scenarios are explored: the first involving very soft final states due to a small mass splitting between a heavy LLP and the dark matter particle to which it decays, and the second with production of a light and therefore highly boosted LLP resulting in almost colinear vertex tracks. The expected limits on the signal production cross section are presented for a wide range of the LLP proper lifetimes corresponding to $c\tau$ from 0.1 mm to 10 km.