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$B-L$ 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 $A_4\times Z_3\times Z_4$ flavor symmetry which successfully explains the recent $3+1$ 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 $v_H \left(\frac{v_l}Λ\right)^2 \sim 10^{-4}\, \mathrm{GeV}$ of the electron mass compared to the muon and tau masses of the order of $\frac{v_H v_l}Λ \sim 10^{-1}\, \mathrm{GeV}$. The recent $3+1$ active-sterile neutrino mixings are predicted to be $0.015 \leq|U_{e 4}|^2\leq 0.045$, $0.004 \leq|U_{μ4}|^2\leq 0.012$, $0.004 \leq|U_{τ4}|^2\leq 0.014$ for normal hierarchy and $0.020\leq|U_{e 4}|^2\leq 0.045$, $0.008 \leq|U_{μ4}|^2\leq 0.018$, $0.008\leq|U_{τ4}|^2\leq 0.022$ for inverted hierarchy. Sterile neutrino masses are predicted to be $0.7 \lesssim m_s \, (\mathrm{eV}) \lesssim 3.16$ for normal hierarchy and $2.6 \lesssim m_s \, (\mathrm{eV}) \lesssim 7.1$ for inverted hierarchy. For three neutrino scheme the model predicts $0.3401 \leq \sin^2θ_{12}\leq 0.3415, \, 0.460 \leq \sin^2θ_{23}\leq 0.540,\, -0.60 \leq \sinδ_{CP}\leq -0.20$ for normal hierarchy and $0.3402 \leq \sin^2θ_{12}\leq 0.3416,\, 0.434\leq\sin^2θ_{23}\leq 0.610,\, -0.95 \leq \sinδ_{CP}\leq -0.60$ for inverted hierarchy. The effective neutrino masses are predicted to be $35.70 \leq \langle m_{ee}\rangle [\mbox{meV}] \leq 36.50$ in 3+1 scheme and $3.65 \leq \langle m^{(3)}_{ee}\rangle [\mbox{meV}] \leq 4.10$ in three neutrino scheme for NH while $160.0 \leq \langle m_{ee}\rangle [\mbox{meV}] \leq 168.0$ in 3+1 scheme and $47.80 \leq \langle m^{(3)}_{ee}\rangle [\mbox{meV}] \leq 48.70$ in three neutrino scheme for for IH which are all in agreement with the recent experimental data.