Firmware implementation of a displaced muon reconstruction algorithm for the Phase-2 Upgrade of the CMS muon system

Displaced muons serve as a crucial indicator of novel physics extending beyond the standard model. Several theoretical frameworks posit the existence of muons with extended lifetimes, which can materialize through collisions at the LHC. This study presents the firmware implementation of an algorithm to reconstruct displaced muons for the Overlap Muon Track Finder (OMTF) of the CMS Level-1 Trigger System (L1T) targeting the High Luminosity Large Hadron Collider (HL-LHC). The firmware response is evaluated at the simulation level, comparing the results with those of the emulator, and then the algorithm is implemented on an FPGA.