$ZZ\gamma$ and $Z\gamma\gamma$ couplings at linear $e^{+} e^{-}$   collider energies with the effects of Z polarization and initial state   radiation
  The constraints on the neutral gauge boson couplings, $ZZ\gamma$ and $Z\gamma\gamma$, are investigated at linear $e^{+}e^{-}$ collider energies through the $Z\gamma$ production with longitudinal and transverse polarization states of the final Z boson. Because of high energy of linear electron- positron beams, initial state radiation (ISR) considerably changes the production cross section. We obtain an increase in the cross section by a factor of 2-3 due to ISR for transverse polarization and by a factor of 10-100 for longitudinal polarization states depending on the energy.   We find the 95% C.L. limits on the CP conserving form factors $h_{3}^{Z}$, $h_{4}^{Z}$, $h_{3}^{\gamma}$ and $h_{4}^{\gamma}$ with integrated luminosity 500$fb^{-1}$ and $\sqrt{s}=$0.5, 1, 1.5 TeV energies. It is shown that the longitudinal polarization of the Z boson, together with ISR, can improve sensitivities by factors 2-3. 
S. Atag and I. Sahin