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We report on a search for a new $Z'$ ($L_\mu-L_\tau$) vector boson performedat the NA64 experiment employing a high energy muon beam and a missingenergy-momentum technique. Muons from the M2 beamline at the CERN Super ProtonSynchrotron with a momentum of 160 GeV/c are directed to an active target. Asignal event is a single scattered muon with momentum $<$ 80 GeV/c in the finalstate, accompanied by missing energy, i.e. no detectable activity in thedownstream calorimeters. For a total statistic of $(1.98\pm0.02)\times10^{10}$muons on target, no event is observed in the expected signal region. Thisallows us to set new limits on part of the remaining $(m_{Z'},\ g_{Z'})$parameter space which could provide an explanation for the muon $(g-2)_\mu$anomaly. Additionally, our study excludes part of the parameter space suggestedby the thermal Dark Matter relic abundance. Our results pave the way to exploreDark Sectors and light Dark Matter with muon beams in a unique andcomplementary way to other experiments.

Exploration of the Muon $g-2$ and Light Dark Matter explanations in NA64 with the CERN SPS high energy muon beam