Search for long-lived particles using displaced vertices and missing transverse momentum in proton-proton collisions at $\sqrt{s}=13~\mathrm{TeV}$

A search for long-lived particles produced in proton-proton collisions at a center-of-mass energy of $13~\mathrm{TeV}$ at the CERN LHC is presented. The search is based on data collected by the CMS experiment in 2016--2018 and corresponding to a total integrated luminosity of $137~\mathrm{fb}^{-1}$. This search is designed to be sensitive to long-lived particles with mean proper decay lengths between $0.1$ and $1000~\mathrm{mm}$ whose decay products produce a final state with at least one displaced vertex and missing transverse momentum. The observation is consistent with the standard model background prediction, and the results are used to constrain split supersymmetry models with different mean proper decay lengths and gluino masses. Gluinos with mean proper decay lengths between $1$ and $100~\mathrm{mm}$ are excluded at $95\%$ confidence level in models for which the gluino mass is below $1800$ and $2000~\mathrm{GeV}$ when the difference in the gluino and neutralino masses is, respectively, $100$ and $200~\mathrm{GeV}$. These limits are the most stringent to date for signal models with proper decay lengths up to $100~\mathrm{mm}$. $$\textit{This document has been revised with respect to the version dated August 21, 2023.}$$