  We explore the scenarios where the only accessible new states at the
electroweak scale consist of a pair of color-singlet electroweak particles,
whose masses are degenerate at the tree level and split only by electroweak
symmetry breaking at the loop level. For the sake of illustration, we consider
a supersymmetric model and study the following three representative cases with
the lower-lying states as (a) two spin-1/2 Higgsino SU(2)$_L$ doublets, (b) a
spin-1/2 wino SU(2)$_L$ triplet and (c) a spin-0 left-handed slepton SU(2)$_L$
doublet. Due to the mass-degeneracy, those lower-lying electroweak states are
difficult to observe at the LHC and rather challenging to detect at the $e^+
e^-$ collider as well. We exploit the pair production in association with a
hard photon radiation in high energy $e^+ e^-$ collisions. If kinematically
accessible, such single-photon processes at $e^+e^-$ colliders with polarized
beams enable us to characterize each scenario by measuring the energy and
scattering angle of the associated hard photon, and to determine the spin of
the nearly invisible particles unambiguously through the threshold behavior in
the photon energy distribution.
