Fast track segment finding in the Monitored Drift Tubes (MDT) of the ATLAS Muon Spectrometer using a Legendre transform algorithm

Many of the physics goals of ATLAS in the High Luminosity LHC era, including precision studies of the Higgs boson, require an unprescaled single muon trigger with a 20 GeV threshold. The selectivity of the current ATLAS first-level muon trigger is limited by the moderate spatial resolution of the muon trigger chambers. By incorporating the precise tracking of the MDT, the muon transverse momentum can be measured with an accuracy close to that of the offline reconstruction at the trigger level, sharpening the trigger turn-on curves and reducing the single muon trigger rate. A novel algorithm is proposed which reconstructs segments from MDT hits in an FPGA and find tracks within the tight latency constraints of the ATLAS first-level muon trigger. The algorithm represents MDT drift circles as curves in the Legendre space and returns one or more segment lines tangent to the maximum possible number of drift circles. This algorithm is implemented without the need of resource and time consuming hit position calculation and track fitting procedures. A low-latency pure-FPGA implementation of a Legendre transform segment finder will be presented. This logic evaluates in parallel a total of 128 possible track segment angles for each MDT drift circle, calculating in a fast FPGA pipeline the offset of each segment candidate from an arbitrary origin for each angle and circle. The (angle, offset) pairs, corresponding to the MDT drift circles in one station, are used to fill a 2D histogram and the segment finder returns the position and angle of the maximum peak, corresponding to the most likely tangent line, this defines the reconstructed segment. Segments are then combined to calculate the muon's transverse momentum with a parametric approach which accounts for varying magnetic field strength throughout the muon spectrometer.