Generating $theta_{13}$ from sterile neutrinos in $mu - tau$ symmetric models

The smallness of the $theta_{13}$ mixing angle as observed in neutrino oscillation experiments can be understood through an approximated $mu - tau$ exchange symmetry in the neutrino mass matrix. Using recent oscillation neutrino data, but assuming no textit{CP} violation, we study $mu-tau$ breaking parameter space to establish the conditions under which such a breaking could have a perturbative origin. According to the so-obtained conditions, we suggest that a sterile neutrino, matching LSND/MiniBooNE neutrino oscillation results, could provide the necessary ingredients to properly fix atmospheric and $theta_{13}$ mixing angles to observable values, without exceeding the sterile neutrino fraction bound in solar oscillations. In such a scenario, we analyze the general effect of a fourth neutrino on the prediction for the effective $m_{ee}$ majorana mass parameter.