Top quark physics and search for physics beyond the Standard Model at the Large Hadron Collider

In 2016 the LHC produced a large sample of $\sqrt{s}=13$ TeV proton-proton collisions for the first time. The energy and luminosity of the data allow exploration of the top quark to a greater detail than ever before, involving standard model (SM) and hypothetical beyond SM (BSM) interactions. This thesis documents a measurement of the top quark pair production cross section in the $\mathrm{t}\overline{\mathrm{t}}\to (\ell \nu_{\ell}) (\tau_\mathrm{h} \nu_{\tau}) \mathrm{b}\overline{\mathrm{b}}$ channel, where $\ell$ is either an electron or a muon. The analysis is performed using a data sample corresponding to an integrated luminosity of $35.9\mathrm{fb}^{-1}$ collected in proton-proton collisions at $\sqrt{s} = 13$ TeV by the CMS detector. Assuming a top quark mass of 172.5 GeV, the measured total $\mathrm{t}\overline{\mathrm{t}}$ cross section $\sigma_{\mathrm{t}\overline{\mathrm{t}}}(\ell\tau_\mathrm{h}) = 781 \pm 7\,\text{(stat.)} \pm 62\,\text{(syst.)} \pm 20\,\text{(lum.)}\,\mathrm{pb}$ is in agreement with the SM expectations. This is the first measurement of the $\mathrm{t}\overline{\mathrm{t}}$ production cross section in proton-proton collisions at $\sqrt{s}=13$TeV that explicitly includes $\tau$ leptons, and it improves the relative precision with respect to the 7 and 8 TeV results. The ratio of the cross section in the $\ell\tau_\mathrm{h}$ final state to the light dilepton cross section $R_{\ell\tau_\mathrm{h}/\ell\ell}=0.973 \pm 0.009\,\text{(stat.)} \pm 0.066\,\text{(syst.)}$ and the ratio of the partial to the total width of the top quark decay $\Gamma(\mathrm{t}\to\tau\nu_{\tau}\mathrm{b})/\Gamma_{\text{total}}=0.1050 \pm 0.0009\,\text{(stat.)} \pm 0.0071\,\text{(syst.)}$ are also measured, improving the precision over the previous results. The higher precision is achieved through a better estimation of the background with misidentified $\tau$ leptons. The SM $\mathrm{t}\overline{\mathrm{t}}$ process provides a source of W boson decays to leptons, where the ratio of the W boson branching fractions $\mathcal{B}(\mathrm{W}\to\tau\nu_{\tau})/\mathcal{B}(\mathrm{W}\to\ell\nu_{\ell})$ can be measured to a high precision as a test of the lepton universality in W bosons. Using the methods developed in the cross section analysis, a feasibility study of the test in $\mathrm{t}\overline{\mathrm{t}}$ dilepton final states is carried out for the full Run2 data sample. The study demonstrates a possibility to achieve a measurement with the overall relative uncertainty of about 3%. It would be a significant contribution to the current world-best measurements.