$B-L$ model with $A_4\times Z_3\times Z_4$ symmetry for $3+1$ active$-$sterile neutrino mixing

We construct a multiscalar and nonrenormalizable $B-L$ model with $A_4\times Z_3\times Z_4$flavorsymmetrywhichsuccessfullyexplainstherecent$3+1$ active-sterile neutrino data. The tiny neutrino mass the mass hierarchy are obtained by the type-I seesaw mechanism. The hierarchy of the lepton masses is satisfied by a factor of $v_H \left(\frac{v_l}{\Lambda}\right)^2 \sim 10^{-4}, $\mathrm{G}\mathrm{e}\mathrm{V}$$ of the electron mass compared to the muon and tau masses of the order of $\frac{v_H{v}_l}{\mathrm{\Lambda }}\sim {10}^{-1}\mathrm{G}\mathrm{e}\mathrm{V}$. The recent $3+1$ active-sterile neutrino mixings are predicted to be $0.015 \leq|U_{e 4}|^2\leq 0.045$,$0.004 \leq|U_{\mu 4}|^2\leq 0.012$,$0.004 \leq|U_{\tau 4}|^2\leq 0.014$fornormalhierarchyand$0.020\leq|U_{e 4}|^2\leq 0.045$,$0.008 \leq|U_{\mu 4}|^2\leq 0.018$,$0.008\leq|U_{\tau 4}|^2\leq 0.022$ for inverted hierarchy. Sterile neutrino masses are predicted to be $0.7 \lesssim m_s , ($\mathrm{e}\mathrm{V}$) \lesssim 3.16$fornormalhierarchyand$2.6 \lesssim m_s , ($\mathrm{e}\mathrm{V}$) \lesssim 7.1$ for inverted hierarchy. For three neutrino scheme the model predicts $0.3401 \leq \sin^2\theta_{12}\leq 0.3415, , 0.460 \leq \sin^2\theta_{23}\leq 0.540,, -0.60 \leq \sin\delta_{CP}\leq -0.20$ for normal hierarchy and $0.3402 \leq \sin^2\theta_{12}\leq 0.3416,, 0.434\leq\sin^2\theta_{23}\leq 0.610,, -0.95 \leq \sin\delta_{CP}\leq -0.60$forinvertedhierarchy\mathrm{.}Theeffectiveneutrinomassesarepredictedtobe$35.70 \leq \langle m_{ee}\rangle [\mbox{meV}] \leq 36.50$in3+1schemeand$3.65 \leq \langle m^{(3)}{ee}\rangle [\mbox{meV}] \leq 4.10$ in three neutrino scheme for NH while $160.0 \leq \langle m{ee}\rangle [\mbox{meV}] \leq 168.0$in3+1schemeand$47.80 \leq \langle m^{(3)}_{ee}\rangle [\mbox{meV}] \leq 48.70$ in three neutrino scheme for for IH which are all in agreement with the recent experimental data.