couplings, the $A_4$ scale should be close to GUT scale. As far as fine-tuning is concerned, we set a tolerance same as electron Yukawa in the standard model. Different $A_4$ breaking scale will also have different implications for the super-WIMP dark matter sector, requiring some amount of fine tuning in relevant Yukawa couplings related to mother particle and dark matter, as we discuss below.

Super-WIMP Dark Matter {#sec3}
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The dark matter in our setup is similar to the super-WIMP scenario [@Feng:2003uy] where a metastable WIMP decays into super-weakly interacting dark matter at late epochs. As can be seen from the dimension five Lagrangian , there are several terms which can give rise to decay of either $\psi$ or $\eta$, the particles of the $Z_4$ sector. At renormalisable level however, both of them are stable due to the unbroken $Z_4$ symmetry. While $\eta$ can be produced thermally in the early universe due to its electroweak gauge interactions, the singlet fermion $\psi$ has negligible thermal abundance. In a general super-WIMP formalism where $\eta$ is the metastable WIMP and $\psi$ is the DM candidate, one can write down the the decay width of $\eta$ into two dark matter particles $(\psi)$ as $$\Gamma_{\eta 