Nonthermal Two Component Dark Matter Model for Fermi-LAT $\gamma$-ray excess and 3.55 keV X-ray Line

A two component model of nonthermal dark matter is formulated to simultaneously explain the Fermi-LAT results indicating a $\gamma$-ray excess observed from our Galactic Centre in the 1-3 GeV energy range and the detection of an X-ray line at 3.55 keV from extragalactic sources. Two additional Standard Model singlet scalar fields $S_2$ and $S_3$ are introduced. These fields couple among themselves and with the Standard Model Higgs doublet $H$. The interaction terms among the scalar fields, namely $H$, $S_2$ and $S_3$, are constrained by the application of a discrete $\mathbb{Z}_2\times \mathbb{Z}^\prime_2$ symmetry which breaks softly to a remnant $\mathbb{Z}^{\prime \prime}_2$ symmetry. This residual discrete symmetry is then spontaneously broken through an MeV order vacuum expectation value $u$ of the singlet scalar field $S_3$. The resultant physical scalar spectrum has the Standard Model like Higgs as $\chi_{{}_{{}_1}}$ with $M_{\chi_{{}_{{}_1}}}\sim 125$ GeV, a moderately heavy scalar $\chi_{{}_{{}_2}}$ with $50 \,\,{\rm GeV} \leq M_{\chi_{{}_{{}_2}}}\leq 80\,\,{\rm GeV}$ and a light $\chi_{{}_{{}_3}}$ with $M_{\chi_{{}_{{}_3}}} \sim 7$ keV. There is only tiny mixing between $\chi_{{}_{{}_1}}$ and $\chi_{{}_{{}_2}}$ as well as between $\chi_{{}_{{}_1}}$ and $\chi_{{}_{{}_3}}$. The lack of importance of domain wall formation in the present scenario from the spontaneous breaking of the discrete symmetry ${\mathbb{Z}_2^{\prime\prime}}$, provided $u\leq 10$ MeV, is pointed out. We find that our proposed two component dark matter model is able to explain successfully both the above mentioned phenomena $-$ the Fermi-LAT observed $\gamma$-ray excess (from the $\chi_{{}_{{}_2}} \rightarrow {\rm b} \bar{\rm b}$ decay mode) and the observation of the X-ray line (from the decay channel $\chi_{{}_{{}_3}}\rightarrow\gamma \gamma$) by the XMM-Newton observatory.