  We study pure weak eigenstate Dirac fermionic dark matters (DM). We consider
WIMP with renormalizable interaction. According to results of direct searches
and the nature of DM (electrical neutral and being a pure weak eigenstate), the
quantum number of DM is determined to be $I_3=Y=0$. There are only two possible
cases: either DM has non-vanishing weak isospin ($I\neq 0$) or it is an
isosinglet (I=0). In the first case, the Sommerfeld enhancement is sizable for
large $I$, producing large $\chi^0\bar{\chi^0}\to VV$ rates. In particular, we
obtain large $\chi\bar\chi\to W^+ W^-$ cross section, which is comparable to
the latest bounds from indirect searches and $m_\chi$ is constrained to be
larger than few hundred GeV to few TeV. It is possible to give correct relic
density with $m_\chi$ higher than these lower bounds. In the second case, to
couple DM to standard model (SM) particles, a SM-singlet vector mediator $X$ is
required from renormalizability and the SM gauge quantum numbers. To satisfy
the latest bounds of direct searches and to reproduce the DM relic density at
the same time, resonant enhancement in DM annihilation diagram is needed. Thus,
the masses of DM and the mediator are related. Furthermore, this model is not
sufficient to explain the deviation in muon $g-2$.
