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Assuming a diagonal Majorana neutrino mass matrix, we investigate theneutrino Yukawa textures which lead to a non-zero reactor mixing angle$\theta_{13}$. The neutrino effective coupling matrix $\kappa^{eff}$ ispre-diagonalized by a constant mixing pattern $V_{\nu}$ with a vanishing$\theta^{\nu}_{13}$. The resulting pre-diagonal symmetrical matrix $\kappa$ isset to be four texture zeros with two types of off-diagonal elements nonzero,which is $\kappa_{13}$ and $\kappa_{23}$, respectively. With the expectation ofsimple textures we thoroughly classify the linear combinations, $\alpha_{i}$,$\beta_{i}$ and $\gamma_{i}$ of Yukawa elements $\lambda_{ij}$ in a same row,according to the values vanishing or not. Each set of the classifications canlead to a Yukawa texture which may have implications for the discrete flavormodel buildings. We also present a model based on $A_{4}$ according to one setof the constraints on the three combinations with a specific choice of acoefficient in Yukawa texture.

Obtaining nonvanishing $\theta_{13}$ with constrained neutrino Yukawa matrix and implications for flavor model buildings