  Terrestrial planets currently in the habitable zones around M dwarfs likely
experienced a long-term runaway greenhouse condition because of a slow decline
in host-stellar luminosity in its pre-main sequence phase. Accordingly, they
might have lost significant portions of their atmospheres including water vapor
at high concentration by hydrodynamic escape induced by the strong stellar XUV
irradiation. However, the atmospheric escape rates remain highly uncertain due
partly to a lack of understanding of the effect of radiative cooling in the
escape outflows. Here we carry out 1-D hydrodynamic escape simulations for an
H$_{2}$-H$_{2}$O atmosphere on a planet with mass of $1M_{\oplus}$ considering
radiative and chemical processes to estimate the atmospheric escape rate and
follow the atmospheric evolution during the early runaway greenhouse phase. We
find that the atmospheric escape rate decreases with the basal H$_{2}$O/H$_{2}$
ratio due to the energy loss by the radiative cooling of H$_{2}$O and chemical
products such as OH and H$_{3}^{+}$: the escape rate of H$_{2}$ becomes one
order of magnitude smaller when the basal H$_{2}$O/H$_{2}=0.1$ than that of the
pure hydrogen atmosphere. The timescale for H$_{2}$ escape exceeds the duration
of the early runaway greenhouse phase, depending on the initial atmospheric
amount and composition, indicating that H$_{2}$ and H$_{2}$O could be left
behind after the end of the runaway greenhouse phase. Our results suggest that
temperate and reducing environments with oceans could be formed on some
terrestrial planets around M dwarfs.
