This thesis presents the search for a charged-lepton flavour violation via a model-independent Higgs-like boson decaying to a muon and a tau lepton, $H\rightarrow\mu\tau$. The validation of high-$p_T$ tau lepton identification is performed with the measurement of $Z\rightarrow\tau^+\tau^-$ production cross-section, using 2\;$fb^{-1}$ of integrated luminosity from $pp$ collisions at $\sqrt{s}=8\;$ TeV collected by the LHCb experiment at the LHC in 2012. The tau leptons are reconstructed in both leptonic and hadronic decay channels. The cross-section, restricted to events with both tau leptons having a transverse momentum greater than 20 GeV/c and a pseudorapidity between 2 and 4.5, and with a tau pair invariant mass between 60 and 120 GeV/$c^2$, is measured to be $$ \sigma_{Z\rightarrow\tau^+\tau^-}} = 95.20 \pm 2.13 \pm 4.79 \pm 0.17 \pm 1.10 \pb. $$ The uncertainties are statistical, systematic, from the LHC beam energy, and from the integrated luminosity. The results are compatible with the lepton universality hypothesis in $Z$ decays, and are in agreement with NNLO Standard Model predictions. Using the validated tau-lepton identification and detection efficiencies, the search for $H\rightarrow\mu\tau$ covering 99\% of tau decay modes found no statistically significant excess. The upper limit on the cross-section times branching fraction, $\sigma_{gg\rightarrow H\rightarrow\mu\tau$, at 95\% confidence level is set, ranging from about 22 pb for $m_H = 45$ GeV/$c^2$ to 4 pb at 195 GeV/$c^2$. Assuming the Standard Model Higgs, the limit on the braching fraction is $\mathcal{B}(H\rightarrow\mu\tau) < 25.7\%$, corresponding to a Yukawa coupling of $\sqrt{|Y_{\mu\tau}|^2 + |Y_{\tau\mu}|^2} < 1.69\times10^{-2}$.