  We study the effective interactions of the fermionic, scalar and vector dark
matter (DM) with leptons and neutral electroweak gauge Bosons induced by the
higher dimensional effective twist-2 tensor operators. We constrain these
lepto-philic, $\tau^\pm$-philic and $U(1)_Y$ gauge Boson B-philic effective
interactions of DM with the visible world from the WMAP and Planck data. The
thermally averaged indirect DM pair annihilation cross-section and the
DM-electron direct-detection cross-section for such DM candidates are observed
to be consistent with the respective experimental data. Constraining
coefficients of the effective operators from the low energy LEP data for the DM
$\le$ 80 GeV, we further study their sensitivities in the pair production of
such DM $\ge$ 50 GeV in association with mono-photon at the proposed ILC and
perform the $\chi^2$ analysis to obtain the 99.73\% C.L. acceptance contours in
the $m_{\rm DM}-\Lambda_{\rm eff}$ plane from the two dimensional differential
distributions of the kinematic observables. We observe that ILC has rich
potential to probe the contribution of such effective operators.
