We propose a new experiment to measure the running of the electromagnetic coupling constant in the space-like region by scattering high-energy muons on atomic electrons of a low-Z target through the elastic process μe→μe. The differential cross section of this process, measured as a function of the squared momentum transfer t= q2< 0 , provides direct sensitivity to the leading-order hadronic contribution to the muon anomaly aμHLO. By using a muon beam of 150 GeV, with an average rate of ∼ 1.3 × 10 7 muon/s, currently available at the CERN North Area, a statistical uncertainty of ∼ 0.3% can be achieved on aμHLO after two years of data taking. The direct measurement of aμHLO via μe scattering will provide an independent determination, competitive with the time-like dispersive approach, and consolidate the theoretical prediction for the muon g-2 in the Standard Model. It will allow therefore a firmer interpretation of the measurements of the future muon g-2 experiments at Fermilab and J-PARC.
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