A search for lepton-flavor-violating decays of the Z boson into a $\tau$-lepton and a light lepton with the ATLAS detector in proton-proton collisions at $\sqrt{s}=$ 13 TeV at the LHC

In the original Standard Model with the massless neutrinos, the lepton flavour number is strictly conserved per each generation. But, in experimentally discovered neutrino oscillations neutrinos can change the flavour, leading to the violation of the lepton flavour number. As a consequence, lepton flavour number violation is possible in other processes via loop diagrams, corresponding to tiny branching ratios at the order of $10^{-54}$. Lepton flavour number violation is predicted in several Beyond the Standard Model theories such are certain Supersymmetric Models, multi Higgs doublet models and others. As those theories predict lepton flavour violation at much larger rates which are accessible by experiments, an observation of lepton-flavour violating process would be a sign of new Beyond the Standard Model physics. The analysis presented in this thesis describes a search for lepton flavour violation in the $Z\rightarrow\tau e$ and $Z\rightarrow\tau \mu$ processes with hadronic $\tau$-lepton decays. The search is performed using the 2015+2016 proton-proton collision corresponding to 36.1 fb$^{-1}$ recorded by the ATLAS detector at the centre-of-mass energy $\sqrt{s} =$ 13 TeV. The backgrounds are estimated using a data-driven fake-factor method for the processes where the $\tau$-lepton is faked by a jet and using the Monte Carlo simulation for the backgrounds contributing with a real $\tau$-lepton or a $\tau$-lepton faked by another lepton. Dedicated selections are used to suppress the backgrounds and to define a signal-enriched region where the final fit is performed. The final discriminating variable is constructed using the neural networks approach, where a neural network is trained to differentiate between the signal and the major background processes by "learning" from their kinematical differences. Finally, the neural network score is fitted in the signal region and the upper limit on the branching ratio is set at $5.84\times10^{-5}$ for the $Z\rightarrow\tau e$ and $2.45\cdot10^{-5}$ for the $Z\rightarrow\tau \mu$ channel.