Searches for new physics in violation of strong $CP$ symmetry and lepton universality with the LHCb experiment

In this thesis two analyses are presented, both searches for hints of new physics in data recorded by the LHCb detector at CERN. In the first one, a search for the \CP-violating strong decays $\eta \to \pi^+\pi^-$ and $\eta^\prime(958) \to \pi^+\pi^-$ is made, analysing the $\pi^+\pi^-$ mass spectra coming from the decays $D^+ \to \pi^+\pi^+\pi^-$ and $D_s^+ \to \pi^+\pi^+\pi^-$. The dataset corresponds to an integrated luminosity of 3.0\,fb$^{-1}$ of $pp$ collision data recorded during LHC Run~1 (2011 and 2012), and 0.3\,fb$^{-1}$ recorded in Run~2 (2015). No evidence is seen, and upper limits at 90\% confidence level are set on the branching fractions of both channels.\\ The second analysis is a sensitivity study for a test of lepton universality in semileptonic charm decays, aiming to measure the ratio of branching fractions $R^{\mu/e} = \frac{\mathcal{B}(\Dz\to\Km \mup \neum)}{\mathcal{B}(\Dz\to \Km \ep\neue)}$. The dataset corresponds to an integrated luminosity of 2.0\,fb$^{-1}$ of $pp$ collision data recorded during LHC Run~2 (2015 and 2016). The study is performed on a small fraction of the total dataset using toy simulation to describe the signal and background components, and leads to an expected reduction of the uncertainty on the current measurement by about one order of magnitude, neglecting systematic effects. A determination of electron identification efficiencies, using a new method, is also presented.