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We propose a simple model in the type-III seesaw framework to explain theneutrino mass, asymmetric dark matter, baryon asymmetry of the Universe, andthe recently reported W-mass anomaly by CDF-II collaboration. We extend thestandard model with a vector-like singlet lepton ($\chi$) and a hyperchargezero scalar triplet ($\Delta$) in addition to three hypercharge zero tripletfermions($\Sigma_i~,i=1,2,3$). A $Z_2$ symmetry is imposed under which $\chi$and $\Delta$ are odd, while all other particles are even. As a result, thelightest $Z_2$ odd particle $\chi$ behaves as a candidate of dark matter. Inthe early Universe, the CP-violating out-of-equilibrium decay of heavy tripletfermions to the Standard Model lepton ($L$) and Higgs ($H$) generate a netlepton asymmetry, while that of triplet fermions to $\chi$ and $\Delta$generate a net asymmetric dark matter. The lepton asymmetry is converted to therequired baryon asymmetry of the Universe via the electroweak sphalerons, whilethe asymmetry in $\chi$ remains as a dark matter relic that we observe today.We introduce a singlet scalar $\phi$, with mass $m_\phi < m_\chi$, which notonly assists to deplete the symmetric component of $\chi$ through theannihilation process: $\bar{\chi} \chi \to \phi \phi$ but also paves a path todetect dark matter $\chi$ at direct search experiments through $\phi-H$ mixing.The $Z_2$ symmetry is broken softly resulting in an unstable asymmetric darkmatter with mass ranging from a few MeV to a few tens of GeV. The softly broken$Z_2$ symmetry also induces a vacuum expectation value (vev) of $\Delta$ due towhich the asymmetry in $\Delta$ disappears. Moreover, the vev of $\Delta$enhances the W-boson mass as reported by CDF-II collaboration with $7\sigma$statistical significance, while keeping the $Z$-boson mass intact.

Cogenesis of matter and dark matter from triplet fermion seesaw