  We investigate a class of models where the supergravity model with the
standard model gauge group is extended by a hidden sector $U(1)_X$ gauge group
and where the lightest supersymmetric particle is the neutralino in the hidden
sector. We investigate this possibility in a class of models where the stau is
the lightest supersymmetric particle in the MSSM sector and the
next-to-lightest supersymmetric particle of the $U(1)_X$-extended SUGRA model.
In this case the stau will decay into the neutralino of the hidden sector. For
the case when the mass gap between the stau and the hidden sector neutralino is
small and the mixing between the $U(1)_Y$ and $U(1)_X$ is also small, the stau
can decay into the hidden sector neutralino and a tau which may be
reconstructed as a displaced track coming from a high $p_T$ track of the
charged stau. Simulations for this possibility are carried out for HL-LHC and
HE-LHC. The discovery of such a displaced track from a stau will indicate the
presence of hidden sector dark matter.
