Future searches for SUSY at the LHC post Fermilab $(g-2)_\mu$

We assess the future directions for the search for supersymmetry at the Large Hadron Collider in view of the new precision results on the muon anomaly by the Fermilab Collaboration. The existence of a deviation of size 4.1$\sigma$ from the Standard Model prediction points to light sleptons and light weakinos in the mass range of few hundred GeV while the observation of the Higgs boson mass at $\sim 125$ GeV points to squark masses lying in the few TeV range. Thus a split sparticle spectrum is indicated. We discuss the possibility of such a split sparticle spectrum in the supergravity unified model and show that a splitting of the sfermion spectrum into light sleptons and heavy squarks naturally arises within radiative breaking of the electroweak symmetry driven by heavy gluinos ($\tilde g$SUGRA). We discuss the possible avenues for the discovery of supersymmetry at the LHC within this framework under the further constraint of the recent muon anomaly result from the Fermilab Collaboration. We show that the most likely candidates for early discovery of a sparticle at the LHC are the chargino, the stau, the smuon and the selectron. We present a set of benchmarks and discuss future directions for further work. Specifically, we point to the most promising channels for SUSY discovery and estimate the integrated luminosity needed for the discovery of these benchmarks at the High Luminosity LHC and also at the High Energy LHC.
Comment: Contribution to Snowmass