Softly broken lepton number L_e - L_μ - L_τ with non-maximal solar neutrino mixing

We consider the most general neutrino mass matrix which leads to $\theta_{13} = 0$, and present the formulae needed for obtaining the neutrino masses and mixing parameters in that case. We apply this formalism to a model based on the lepton number $\bar L = L_e - L_\mu - L_\tau$ and on the seesaw mechanism. This model needs only one Higgs doublet and has only two right-handed neutrino singlets. Soft $\bar L$ breaking is accomplished by the Majorana mass terms of the right-handed neutrinos; if the $\bar L$-conserving and $\bar L$-breaking mass terms are of the same order of magnitude, then it is possible to obtain a consistent $\bar L$ model with a solar mixing angle significantly smaller than $45^\circ$. We show that the predictions of this model, $m_3 = 0$ and $\theta_{13} = 0$, are invariant under the renormalization-group running of the neutrino mass matrix.