  At the high luminosity electron-positron circular colliders like FCC-ee in
CERN and CEPC in China it will possible to measure very precisely $e^+e^-\to
Z\gamma$ process with subsequent Z decay into particles invisible in the
detector, that is into three neutrina of the Standard Model and possibly into
other weakly coupled neutral particles. Apart from the measurement of the total
invisible width (which is not the main subject of this work) this process may
be used as a source of $Z$ coupling to electron neutrino -- known very poorly.
This is possible due to the presence of the $t$-channel $W$ exchange in the
$e^+e^-\to \nu_e \bar\nu_e \gamma$ channel which deforms slightly spectrum of
the photon. We are going to show experimental investigation of this effect, for
$\sim 10$ inverse atobarn accumulated luminosity, which can provide measurement
of the $Z-\nu_e$ coupling with statistical error of order 1\%. The estimation
of the systematic experimental error will require more work, but most likely it
will be of similar size.
