Search for a dark matter mediator ($X$) in the $K^+ \to \mu^+ \nu_\mu X, X \to \gamma \gamma$ decay from the high-intensity $K^+$ beam of the CERN NA62 experiment

This PhD thesis work focuses on the first experimental search for the $K^+ \to \mu^+ \nu_\mu X, X \to \gamma \gamma$ process, with $X$ produced radiatively from the muon in the $K^+$ decays from the high-intensity beam of the NA62 experiment at CERN. $X$ is here assumed to be a new scalar particle, neutral with respect to the Standard Model (SM) interactions and coupling preferentially to muons. The existence of such a particle provides a possible and one of the last valuable solution to solve the long-standing discrepancy between the experimental value and the SM prediction of the muon anomalous magnetic moment $(g-2)_\mu$ with new physics below the electroweak scale. Moreover, $X$ could represent the mediator of a new interaction responsible for the dark matter (DM) thermal freeze out, assuming the DM has a mass in MeV-GeV scale and exhibits feeble couplings to the SM. The whole data sample collected by NA62 through the minimum bias trigger line during Run1 (2016-2017-2018) has been analysed, leading to preliminary results for the first experimental upper limits at 90% C.L. for the branching ratio of the $K^+ \to \mu^+ \nu_\mu X,  X \to \gamma \gamma$ decay in the $X$ mass ranges 20-100 MeV and 170-320 MeV.