Final results of the search for $ u_{mu} to u_{e}$ oscillations with the OPERA detector in the CNGS beam

The OPERA experiment has discovered the tau neutrino appearance in the CNGS muon neutrino beam, in agreement with the 3 neutrino flavour oscillation hypothesis. The OPERA neutrino interaction target, made of Emulsion Cloud Chamber, was particularly efficient in the reconstruction of electromagnetic showers. Moreover, thanks to the very high granularity of the emulsion films, showers induced by electrons can be distinguished from those induced by $pi^0$s, thus allowing the detection of charged current interactions of electron neutrinos. In this paper the results of the search for electron neutrino events using the full dataset are reported. An improved method for the electron neutrino energy estimation is exploited. Data are compatible with the 3 neutrino flavour mixing model expectations and are used to set limits on the oscillation parameters of the 3+1 neutrino mixing model, in which an additional mass eigenstate $m_{4}$ is introduced. At high $Delta m^{2}_{41}$ $( gtrsim 0.1~textrm{eV}^{2})$, an upper limit on $sin^2 2theta_{mu e}$ is set to 0.021 at 90% C.L. and $Delta m^2_{41} gtrsim 4 times 10^{-3}~textrm{eV}^{2}$ is excluded for maximal mixing in appearance mode.