  We discuss one loop generation of the term connecting gauge fields from a
local hidden $SU(2)_H$ and the standard model $U(1)_Y$ introducing an $SU(2)_H$
doublet fermion with non-zero hypercharge and a scalar field in adjoint
representation. Then we obtain a kinetic mixing term between $SU(2)_H$ and
$U(1)_Y$ gauge fields after the adjoint scalar field developing vacuum
expectation value. We illustrate such a concrete scenario introducing a dark
matter model in an ultraviolet (UV) completion with local $SU(2)_H$ symmetry
where the scalar doublet is our dark matter candidate and its stability is
guaranteed by remnant $Z_2$ symmetry from $SU(2)_H$. Relic density of dark
matter is calculated focusing on the case in which dark matter annihilate into
known particles via $SU(2)_H$ gauge interactions with radiatively induced
kinetic mixing.
