Leptonic scalar portal: Origin of muon $g -2$ anomaly and dark matter?

We present a model explaining both the <math display="inline"><mrow><mn>4.2</mn><mi>σ</mi></mrow></math> muon <math display="inline"><mi>g</mi><mo>-</mo><mn>2</mn></math> anomaly and the relic density of dark matter (DM) in which DM interacts with the Standard Model (SM) via a scalar portal boson <math display="inline"><mi>φ</mi></math> carrying both dark and SM leptonic numbers, and mediating a nondiagonal interaction between the electron and muon that allows <math display="inline"><mi>e</mi><mo stretchy="false">↔</mo><mi>μ</mi></math> transitions. The <math display="inline"><mi>φ</mi></math> could be produced in high-energy electron scattering off a target nuclei in the reaction <math display="inline"><mi>e</mi><mi>Z</mi><mo stretchy="false">→</mo><mi>μ</mi><mi>Z</mi><mi>φ</mi></math> followed by the prompt invisible decay <math display="inline"><mi>φ</mi><mo stretchy="false">→</mo><mi>DM</mi></math> particles and searched for in events with large missing energy accompanied by a single outgoing muon in the final state. Interestingly, several events with a similar signature have been observed in a data sample of <math display="inline"><mo>≃</mo><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>11</mn></msup></math> electrons on target collected during 2016-2018 for the search for light dark matter in the NA64 experiment at the CERN SPS [D. Banerjee (NA64 Collaboration), Phys. Rev. Lett. 123, 121801 (2019).]. Attributing so far these events to background allows us to set first constraints on the <math display="inline"><mi>φ</mi></math> mass and couplings while leaving at the same time decisively probing the origin of these events and a large fraction of the remaining parameter space to a near exiting future with the upgraded NA64 detector or other planned experiments.