Fermionic dark matter via UV and IR freeze-in and its possible X-ray signature

Non-observation of any dark matter signature at various direct detection experiments over the last decade keeps indicating that immensely popular WIMP paradigm may not be the actual theory of particle dark matter. Non-thermal dark matter produced through freeze-in is an attractive proposal, naturally explaining null results by virtue of its feeble couplings with the Standard Model (SM) particles. We consider a minimal extension of the SM by two gauge singlet fields namely, a $\mathbb{Z}_2$-odd fermion $\chi$ and a pseudo scalar $\tilde{\phi}$, where the former has interactions with the SM particles only at dimension five level and beyond. This introduces natural suppression in the interactions of $\chi$ by a heavy new physics scale $\Lambda$ and forces $\chi$ to be a non-thermal dark matter candidate. We have studied production of $\chi$ in detail taking into account both ultra-violate (UV), infra-red (IR) as well as mixed UV-IR freeze-in and found that for $10^{10}{\rm GeV}\leq\Lambda\leq 10^{15}{\rm GeV}$, $\chi$ is dominantly produced via UV and mixed UV-IR freeze-in when reheat temperature $T_{RH}\gtrsim 10^4$ GeV and below which the production is dominated by IR and mixed freeze-in. Furthermore, we have considered the cascade annihilation $\chi\bar{\chi} \rightarrow \tilde{\phi}\tilde{\phi}\rightarrow 4\gamma$ to address the longstanding $\sim3.5$ keV X-ray line observed from various galaxies and galaxy clusters. We have found that the long-lived intermediate state $\tilde{\phi}$ modifies dark matter density around the galactic centre to an effective density $\rho_{eff}$ which strongly depends on the decay length of $\tilde{\phi}$. Finally, the allowed parameter space in $\Lambda-g$ plane ($g$ is the coupling between $\chi\bar{\chi}$ and $\tilde{\phi}$) is obtained by comparing our result with the XMM Newton observed X-ray flux from the centre of Milky Way galaxy in $2\sigma$ range.