\begin{document}$ pp\to hjj $\end{document} with \begin{document}$ h\to ss $\end{document}, in which the long-lived scalar decays as \begin{document}$ s\to b\bar b $\end{document}, and use the VBF tag jets to identify the hard-scatter vertex. Its prompt central tracks, timed by a CMS-MTD-like barrel layer that also times the central tracks of the displaced vertex, define a per-event start time, turning each displaced vertex into a four-dimensional object. Because a fast simulation cannot determine the absolute normalization of the heavy-flavour and instrumental displaced-vertex backgrounds, we take as the primary result a normalization-free figure of merit: the ratio of the timing-enhanced sensitivity to the spatial-only sensitivity. Adding the timing layer improves this ratio by a factor of \begin{document}$ \simeq2.8 $\end{document} at a nominal working point and by up to an order of magnitude for a tighter delayed-vertex requirement, because the heavy-flavour background that survives the spatial selection is prompt in collision time. The improvement increases with decay length, extending sensitivity into the long-lifetime regime in which purely spatial searches lose tracker acceptance. The gain is essentially mass-independent at a decay length of \begin{document}$ c\tau\simeq100 \,{\rm{mm}} $\end{document}, where the trend with scalar mass reverses: lighter scalars benefit more at short lifetimes because of their larger boost, whereas heavier scalars benefit more at long lifetimes as the lighter states leak out of the tracker."> Vector-boson-fusion timing references for four-dimensional displaced-vertex searches -
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