Search for new physics in rare four-body charm decays at LHCb

This dissertation presents a search for physics beyond the Standard Model in $D^0\to\pi^+\pi^-\mu^+\mu^-$ and $D^0\rightarrow K^+K^-\mu^+\mu^-$ decays at the LHCb experiment. These rare four-body decays of neutral charm mesons receive contributions from electroweak flavour-changing neutral current $c\to u\ell^+\ell^-$ transitions and are unique probes for potential new heavy degrees of freedom and for additional sources of $CP$ violation in the up-type quark sector. Using a sample of proton-proton collisions corresponding to an integrated luminosity of 2fb$^{-1}$ recorded at a center-of-mass energy of 8TeV in 2012, the first observation of $D^0\to\pi^+\pi^-\mu^+\mu^-$ and $D^0\rightarrow K^+K^-\mu^+\mu^-$ decays is reported. Furthermore, their branching fractions are measured. With an increased data set, corresponding to an integrated luminosity of 5fb$^{-1}$ and recorded at center-of-mass energies of $7,8$ and $13$TeV in the years 2011-2016, the first measurement of the $CP$ asymmetry $A_{CP}$, the forward-backward asymmetry in the lepton system $A_{FB}$ and the triple-product asymmetry $A_{2\phi}$ in these decays is presented. All observables are also investigated as functions of the dimuon mass. The results are consistent with Standard Model predictions.