Single production of the gauge boson $W$ via polarized $e^{-}\gamma$ collisions in the littlest Higgs model

In the framework of the littlest Higgs($LH$) model, we study single production of the standard model($SM$) gauge boson $W_s^{-}$ and the heavy gauge boson $W_H^{-}$ via polarized $e^{-}\gamma$ collisions. We find that the corrections of the $LH$ model to the cross section $\sigma (W_s^{-})$ might be observed only for the scale parameter $f\le 1.5TeV$ and the mixing parameter $c^{\mathrm{\prime }}\ge 0.4$ in future high energy linear $e^{+}{e}^{-}$ collider($LC$) experiment with the center-of-mass($CM$) energy $\sqrt{S}=500GeV$ and a yearly integrated luminosity $\mathrm{\pounds }=100f{b}^{-1}$. However, with a suitably chosen polarization of the initial electron and position beams, the possible signals of the heavy gauge boson $W_H^{-}$ can be easily detected via $e^{-}\gamma$ collisions in future $LC$ experiment with $\sqrt{S}=3TeV$ and $\mathrm{\pounds }=500f{b}^{-1}$.