\section{Introduction}
\label{sec:intro}

An ultimate goal of the study of quasar (QSO) absorption lines is to
develop a comprehensive understanding of the kinematic,
chemical, and ionization conditions of gaseous structures in
early--epoch galaxies and to chart their cosmic evolution.
For a comprehensive physical picture of any given absorption
system, both high resolution spectra of a wide range of chemical and
ionization species and the empirically measured properties of the
associated galaxies are required.
For $z\sim 1$, shortly following the epoch of peak star formation, the
association between {{\rm Mg}\kern 0.1em{\sc ii}~$\lambda\lambda 2796, 2803$} absorption and galaxies is well
established (\cite{bb91}; \cite{steidel95}), and their kinematics,
though complex and varied, are consistent with being coupled to the
galaxies themselves (\cite{csv96}; \cite{cc98}).
It is unfortunate, however, that for $z \sim 1$, the spectroscopic
data of are not of uniform, high quality due to the need for large
amounts of space--based telescope time to observe ultraviolet
wavelengths.
Presently, any comprehensive analyses of low redshift systems for
which the low ionization species (i.e.\ {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}, {\hbox{{\rm Fe}\kern 0.1em{\sc ii}}}, {\hbox{{\rm Mg}\kern 0.1em{\sc i}}}) have
been observed at high resolution with HIRES/Keck (see Churchill, Vogt,
\& Charlton 1999b\nocite{cvc99}) must incorporate low resolution
FOS/{\it HST\/} spectra of the intermediate and high ionization
species, especially the strong {{\rm C}\kern 0.1em{\sc iv}~$\lambda\lambda 1548, 1550$}, {\NVdblt}, and {\OVIdblt}
doublets, and of several other important low ionization species.

Presently, it is not clear if these low resolution data can be used to
place meaningful constraints on the chemical and ionization conditions
of the clouds in {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} selected absorbers.
In this paper, we investigated this issue in a pilot study, since an
affirmation would imply that a larger sample could be studied using
existing data from the {\it HST\/} Archive.
We would then be able to address the broader implications for galaxy
formation scenarios based upon the inferred metallicities, abundance
patterns, and inferred relative spatial distribution of the low and
high ionization absorbing
gas clouds.

\begin{figure*}[th]
\figurenum{1}
\plotfiddle{Churchill.fig1.eps}{2.8in}{0}{60}{60}{-236}{-74}
\protect\caption[Churchill.fig1.eps]
{\footnotesize
The HIRES/Keck {{\rm Mg}\kern 0.1em{\sc ii}~$\lambda\lambda 2796, 2803$} doublet profiles at $R=45,000$.  Three
systems, A, B, and C, have been identified at  $z_{A} = 0.92540$,
$z_{B} = 0.92760$, and $z_{C} = 0.93428$.  The labeled bar bar ticks
give the systemic redshifted wavelengths for the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} doublets in
these systems. The solid line through the data is a synthetic spectrum
from a Voigt profile decomposition.  The vertical ticks above the
continuum give the locations of the Voigt profile
components. \label{fig:order28}}
\end{figure*}

Under the assumptions of photoionization and/or collisional ionization
equilibrium, we developed a technique in which it was assumed that
the number of clouds and their kinematics are obtained by Voigt profile
decomposition of the high resolution {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} spectra.
We then used the lower resolution profiles from the FOS data to place
constraints on the range of chemical and ionization conditions in
these clouds.
We also explored the idea that the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} could arise in relatively
low ionization clouds embedded in a higher ionization and more
extended medium (see \cite{bergeron94}; \cite{iap}).
More specifically, we set out to answer three questions:
(1) Assuming the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds measured with HIRES are photoionized,
can we construct model clouds that are consistent with the many
low and intermediate ionization species captured in the FOS data?
(2) If so, are we required to infer an additional (presumedly low
density and diffuse) component to account for higher ionization
absorption from {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}}?
(3) If so, can this diffuse component be made consistent with
photoionized only, collisionally ionized only, or photo plus
collisionally ionized gas?

For this paper, we chose the three systems at $z _{\rm abs} =
0.9254$, $0.9276$, and $0.9343$ along the line of sight toward
PG~$1206+459$ ($z_{\rm em} = 1.16$) because they are exceptionally
rich in low, intermediate, and high ionization ultraviolet transitions
(\cite{bt96}; \cite{uvconf}; \cite{kp13}).
The HIRES/Keck {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} profiles are illustrated in
Figure~\ref{fig:order28}.
Two of the systems are kinematically ``complex'' and are separated by
$\sim 300$~{\hbox{km~s$^{-1}$}}.
The third system is more isolated, being $\sim +1000$~{\hbox{km~s$^{-1}$}} from the
other two.
This system is classified as a ``weak'' {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorber [defined by
$W_{\rm r}(2796) < 0.3$~{\AA} (\cite{weakmgII})].
The highest ionization transitions, {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}} and {\hbox{{\rm O}\kern 0.1em{\sc vi}}}, are
seen to have a total kinematic spread of $\sim 1000$~{\hbox{km~s$^{-1}$}} coincident
with the three system seen in {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorption (\cite{uvconf}).

In the QSO field, Kirhakos et~al.\  (1994\nocite{kirhakos94}) found three
bright galaxies with angular separations from the quasar of 5.6, 8.6,
and 9.0{\arcsec} and $g$ magnitudes 21.1, 21.5, and 22.3,
respectively.
The  8.6{\arcsec} galaxy has detected {\hbox{[{\rm O}\kern 0.1em{\sc ii}]}} $\lambda 3727$ with flux
$9 \times 10^{-17}$ ergs {\hbox{cm$^{-2}$}} s$^{-1}$ at $z=0.93$ (\cite{thimm95}).
At this redshift, the QSO--galaxy impact parameters are $29$, $45$,
and $47~h^{-1}$~kpc ($q_0 = 0.05$).
There are $\sim 10$ galaxies with $21 \leq g \leq 22$ within
100{\arcsec} of the QSO (\cite{kirhakos94}).
This is an overdensity by a factor of $\sim 3$ compared to field
galaxies (\cite{tyson88}).
Thus, it is of interest to entertain the possibility of a group
environment for these absorbers.

In \S~\ref{sec:data} we describe the data and its analysis.
In \S~\ref{sec:models} we outline our modeling technique and
simplifying assumptions.
A synopsis of the model results are given in \S~\ref{sec:results}.
Details on how the data were used to constrain the models and how
various ionizing spectral energy distributions modify these models
are given in Appendices~\ref{app:constraints} and \ref{app:models}.
In \S~\ref{sec:discussion}, we compare and contrast the system
properties, and in \S~\ref{sec:onthe}, we discuss what might be
inferred about the relative spatial distribution of the low and high
ionization gas.
We summarize in \S~\ref{sec:conclusion}.

\section{Data and Analysis}
\label{sec:data}

\subsection{HIRES/Keck}

The optical data were obtained with the HIRES spectrometer
(\cite{vogt94}) on the Keck~I telescope on 23 January 1995 UT under
clear and stable conditions with a seeing of $\sim 0.6${\arcsec}.
The spectral resolution is $\sim 6.6$~{\hbox{km~s$^{-1}$}} ($R=45,000$), with a
sampling of 3 pixels per resolution element.
The  signal--to--noise ratio is $\sim 50$ per resolution element.
The {\hbox{{\rm Fe}\kern 0.1em{\sc ii}}} $\lambda 2344$, $2374$, $2383$, $2587$, and $2600$
transitions were captured at similar signal--to--noise ratio.

\begin{figure*}[th]
\figurenum{2}
\plotfiddle{Churchill.fig2.eps}{3.9in}{0}{75}{75}{-236}{-274}
\protect\caption[Churchill.fig2.eps]
{\footnotesize
The kinematics of the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} systems are shown, with individual
clouds seen in {\hbox{{\rm Fe}\kern 0.1em{\sc ii}}} $\lambda 2383$, {\hbox{{\rm Fe}\kern 0.1em{\sc ii}}} $\lambda 2600$,
{\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} $\lambda 2796$, and {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} $\lambda 2803$ absorption aligned
in rest--frame velocity.  Systems A and B are shown together in the
left panel and system C is shown in the right panel.  The zero--point
is defined at $z=0.92760$. The solid lines through the data are
synthetic spectra from a Voigt profile decomposition.  The ticks above
the continuum give the locations of the Voigt profile components.
\label{fig:vpfits}}
\end{figure*}

The HIRES spectrum was reduced with the IRAF\footnote{IRAF is
distributed by the National Optical Astronomy Observatories, which are
operated by AURA, Inc., under contract to the NSF.} {\it Apextract\/}
package for echelle data.  The detailed steps for the reduction are
outlined in Churchill (1995\nocite{lotr}).
The spectrum was extracted using the optimized routines of Horne
(1986\nocite{horne86}) and Marsh (1989\nocite{marsh89}).
The wavelengths were calibrated to vacuum using the IRAF task {\it
ecidentify\/}, which models the full 2D echelle format.
The absolute wavelength scale was then corrected to heliocentric
velocity.
The continuum normalization was performed using the IRAF {\it sfit\/}
task.
Objective and unbiased identification of absorption features (without
regard to their association with the studied systems) was performed as
described in Churchill et~al.\  (1999b\nocite{cvc99}), using the
methodology of Schneider et~al.\  (1993\nocite{schneider93}).

Three systems, which we hereafter call A, B, and C, are observed at
redshifts $z_{A} = 0.92540$, $z_{B} = 0.92760$, and $z_{C} = 0.93428$.
In Figure~\ref{fig:order28}, we present the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} doublet profiles.
The doublets are marked by the labeled bar above the normalized
continuum.
The solid curve through the data is a model spectrum generated using
Voigt profile (VP) decomposition.
The free parameters are the number of VP components (``clouds'') and,
for each, its redshift, column density, and Doppler $b$ parameter.
We have used the program MINFIT (\cite{cwcthesis}), which performs a
$\chi^{2}$ minimization while minimizing the number of clouds using
a specified confidence level and the standard F--test.
The adopted VP decomposition had six clouds in system A, five clouds
in system B, and a single cloud in system C.

The HIRES data and the VP decompositions are shown in
Figure~\ref{fig:vpfits} with the profiles aligned in rest--frame
velocity.
In Table~\ref{tab:vpfits}, we present the cloud properties, including
individual redshifts, velocities with respect to $z = 0.92760$, column
densities, $b$ parameters, and the ratios $\log \{
N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}})/N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}) \} $.
Only system B was found to have measurable {\hbox{{\rm Fe}\kern 0.1em{\sc ii}}}.
The $1\sigma$ upper limits on {\hbox{{\rm Fe}\kern 0.1em{\sc ii}}} were obtained for each cloud
from the equivalent width limits of the $\lambda 2600$ transition.
For {\hbox{{\rm Fe}\kern 0.1em{\sc ii}}} in system A, we measured a mean $1\sigma$ column density
upper limit of $N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}}) \leq 10^{11.1}$~{\hbox{cm$^{-2}$}} for the six clouds
using the technique of stacking (\cite{norris83}).

\phantom{dummy space.}

\subsection{FOS/{\it HST\/}}

The ultraviolet data were obtained with the Faint Object Spectrograph
(FOS) on the {\it Hubble Space Telescope\/} as part of the QSO
Absorption Line Key Project.
The data acquisition, their reduction, and the objective absorption
line lists are presented in Jannuzi et~al.\  (1998\nocite{kp13}).
Their fully reduced G190H and G270H spectra have kindly been made
available for this study.
The spectra have a resolution of $R=1300$ ($\sim 230$~{\hbox{km~s$^{-1}$}}) and cover
the approximate wavelength intervals $1600$ to $2313$~{\AA} and $2225$
to $3280$~{\AA} for the G190H and G270H settings, respectively.

For the most part, we have adopted the Key Project continuum fits and
line identifications.
Some refinement was needed near the {\SiIVdblt} doublet ($2680 \leq
\lambda _{\rm obs}  \leq 2715$~{\AA} on G270H), which lies on the red
wing of the broad {\hbox{{\rm Ly}\kern 0.1em$\alpha$}} emission line, and near the {\SiIIdblt}
doublet ($2290 \leq \lambda _{\rm obs} \leq 2310$~{\AA} on G190H),
which lies at the spectrum edge.
We have re--fit the continuum across these regions.

A Lyman limit break is present at $\sim 1760$~{\AA} (however, see
Stengler--Larrea 1995\nocite{kp5}; Jannuzi\nocite{kp13} et~al.\ 1998).
We extrapolated the continuum fit of Jannuzi et~al.\  below the break
starting at the break shoulder ($\lambda = 1895$~{\AA}).
This technique preserved the measured optical depth, or the break
ratio, $F_{+}/F_{-}$ (\cite{schneider93}), while yielding a reasonable
approximation to the shape of the recovery (see
Figure~3$a$).
Due to the high density of lines, the Jannuzi et~al.\  continuum may have
been systematically low by 5--10\% in the region 1790 to 1820~{\AA}
and our extrapolation may have propagated this systematic offset.
Nonetheless, this error has a negligible effect on the measured break
ratio of $2.5\pm0.4$, which we obtained from the unnormalized as well
as the normalized spectrum.
This ratio implies $\tau_{\rm LL} \simeq 0.9$ and a total neutral
hydrogen column density of $N({\hbox{{\rm H}\kern 0.1em{\sc i}}}) \simeq 10^{17.2}$~{\hbox{cm$^{-2}$}}.

Out of the $\sim 70$ QSOs analyzed by the QSO Absorption Line Key Project,
PG~$1206+459$ is one of eight for which the line identifications
(IDs) were subject to greater uncertainty (the {\hbox{{\rm Ly}\kern 0.1em$\alpha$}}
forest is ubiquitous blueward of 2623~{\AA} and at least four metal
systems are present).
Using the HIRES data for cross checks with the Jannuzi\nocite{kp13}
et~al.\   line IDs in the FOS spectrum, we have made a table of
transitions, including those detected and those used for constraining
the models.
In Table~\ref{tab:fosIDs} (end of paper), we have listed the observed
wavelengths, line IDs, ionization potentials, and notes on any
blending.
For example, we identified the Lyman series down to {\hbox{{\rm Ly}\kern 0.1em$\epsilon$}}, beyond
which the three systems blend together.
The FOS spectrum provides a large number of chemical and ionization
species, including
{\hbox{{\rm Ly}\kern 0.1em$\alpha$}}, {\hbox{{\rm Ly}\kern 0.1em$\beta$}}, {\hbox{{\rm Ly}\kern 0.1em$\gamma$}}, {\hbox{{\rm Ly}\kern 0.1em$\delta$}}, {\hbox{{\rm Ly}\kern 0.1em$\epsilon$}}, {\hbox{{\rm C}\kern 0.1em{\sc ii}}} $\lambda 1036$ and
$\lambda 1334$, {\hbox{{\rm C}\kern 0.1em{\sc iii}}} $\lambda 977$, {{\rm C}\kern 0.1em{\sc iv}~$\lambda\lambda 1548, 1550$}, {\hbox{{\rm N}\kern 0.1em{\sc ii}}} $\lambda
915$, {\hbox{{\rm N}\kern 0.1em{\sc iii}}} $\lambda 989$, {\NVdblt}, {\OVIdblt}, {\SiIIdblt} and
$\lambda 1260$, {\hbox{{\rm Si}\kern 0.1em{\sc iii}}} $\lambda 1206$, {\SiIVdblt}, and {\SVIdblt}.
These species represent a wide range of ionization potentials from
13.6~eV for {\hbox{{\rm H}\kern 0.1em{\sc i}}} to 138.1~eV for {\hbox{{\rm O}\kern 0.1em{\sc vi}}}.

\section{The Models}
\label{sec:models}

Assuming the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds are photoionized, our goal was to
determine if we can obtain useful constraints on their range of chemical
and ionization conditions using the FOS/{\it HST\/} spectra.
Within this context, we also explored the possibility of a high
ionization ({\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}}), presumedly diffuse, component.
In principle, this high ionization phase could be photoionized,
collisionally ionized, or photo plus collisionally ionized gas (a
spatially segregated two--phase high ionization component).

For both the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds and the high ionization phase, the
extragalactic ultraviolet ionizing background spectrum of Haardt
\& Madau (1996\nocite{handm96}) for $z=1$ was assumed.
Using the {\hbox{[{\rm O}\kern 0.1em{\sc ii}]}}~$\lambda 3727$ detection and constraints measured by
Thimm (1995\nocite{thimm95}), we explored the range of allowed
contributions (modifications to the Haardt \& Madau background) from
various galactic spectral energy distributions.
We find that galactic contributions, within the allowed ranges
explored, do not modify {\it general\/} conclusions based upon the
assumption of a pure Haardt \& Madau background (see
Appendix~\ref{app:models}).

\subsection{The {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} Clouds}

We used the photoionization code CLOUDY (version 90.4;
\cite{ferland}).
The free parameters are the neutral hydrogen column density,
$N({\hbox{{\rm H}\kern 0.1em{\sc i}}})$, the metallicity\footnote{In this work we use the notation
${\rm [X/Y]} = \log ({\rm X/Y}) - \log ({\rm X/Y})_{\odot}$ and for
metallicity use $Z \equiv \log Z/Z_{\odot}$.}, $Z$, the abundance
pattern, and the ionization ``parameter'', $U$, which is defined as
the ratio of the number of hydrogen ionizing photons to the hydrogen
number density, $n_H$ (including ionized, neutral, and molecular
forms).
For the Haardt \& Madau spectrum and  normalization, a simple relation
between ionization parameter and hydrogen number density, $\log U =
-5.2 - \log n_{H}$, holds.

For a given abundance pattern, the ratio $N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}})/N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$
uniquely determines the ionization parameter, $U$.
Once $U$ is determined, the measured $N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$ fixes $\log
N({\hbox{{\rm H}\kern 0.1em{\sc i}}}) + Z \simeq C_{1}$, for a cloud in photoionization
equilibrium.
For a given $Z$, the constant $C_{1}$ is constrained by the Lyman
series transitions in the FOS data.
Throughout, we assume $Z \leq 0$.
To characterize the abundance pattern, we used the ratio of
$\alpha$--group species to Fe--group species, [$\alpha$/Fe].
Abundance ratios measured in Galactic stars show a clear range of
$0 \leq [\alpha/{\rm Fe}] \leq +0.5$ (\cite{jtl96}), which we adopt
as a reasonable range for the studied systems.
Since all clouds have measured $N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$, an $\alpha$--group
element, it follows that $[\alpha/{\rm Fe}] + Z \simeq C_{2}$.
Thus, for clouds with measured $N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}})/N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$, the only
arbitrarily chosen free parameter is the abundance pattern.
One selects a $Z$ and [$\alpha$/Fe] (fixes $C_{2}$), determines
$N({\hbox{{\rm H}\kern 0.1em{\sc i}}})$ by constraining the photoionization models with the Lyman
series transitions, and thus determines $C_{1}$ (an example of
this process is given in Appendix~\ref{app:constraints}).

By exploring a large range of $Z$ and [$\alpha$/Fe], we verified the
above relationships.
When only an upper limit is available on $N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}})/N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$ for a
given cloud, one has two arbitrarily chosen parameters, [$\alpha$/Fe]
and $N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}})/N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$, which together uniquely determine the
ionization parameter.
The constraints on the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} cloud ionization parameters, abundance
pattern, and metallicities are fairly tight and robust; even when
$N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}})/N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$ was an upper limit, the {\hbox{{\rm Si}\kern 0.1em{\sc ii}}}, {\hbox{{\rm Si}\kern 0.1em{\sc iii}}}, and
{\hbox{{\rm Si}\kern 0.1em{\sc iv}}} ratios were key for constraining the ionization parameter,
independent of the abundance pattern (see
Appendix~\ref{app:constraints}).
In a given system, for clouds in which $N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}})/N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$ was
only an upper limit, we assumed that they were identical
vis--\'{a}--vis their {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} column densities (had identical
metallicities and abundance patterns).
This yielded model clouds with identical $Z$, and [$\alpha$/Fe], but
unique $N({\hbox{{\rm H}\kern 0.1em{\sc i}}})$, due to their unique $N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$.
The allowed range of cloud to cloud variations within a system, if
desired, can be obtained from the two relations giving $C_{1}$ and
$C_{2}$ (as long as the total $N({\hbox{{\rm H}\kern 0.1em{\sc i}}})$ is held constant).

To narrow parameter space, we began with a grid of photoionization
models with CLOUDY, where the grid was defined for
(1) $\log \{ N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}}) / N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}) \} $ from $-1$ to $-4$ in
intervals of 0.5 dex (this provides the ionization parameter, $U$),
(2) $N({\hbox{{\rm H}\kern 0.1em{\sc i}}})$ from $10^{14}$ to $10^{18}$~{\hbox{cm$^{-2}$}} in 1 dex intervals,
(3) $Z$, from $-2.0$ to $+0.4$ in intervals of 0.2 dex, and
(4) solar and $[\alpha/{\rm Fe}] = +0.5$ abundance pattern.
Once the parameter space was narrowed, we ran CLOUDY in its optimized
mode tuned to the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} column densities, to obtain the adopted
models.

\begin{figure*}[ht]
\plotfiddle{Churchill.fig3a.eps}{5.5in}{0}{60}{60}{-236}{121}
\plotfiddle{Churchill.fig3b.eps}{0.in}{0}{60}{60}{-236}{-74}
\figurenum{3$a$--$b$}
\protect\caption
{\footnotesize
The normalized FOS/{\it HST\/} spectrum (histogram) and the tuned model
predictions (not fits).  The ticks mark locations of the constraint
transitions identified for systems A, B, and C.  Three models are
shown. The thick solid spectrum includes the photoionized {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}
clouds, the photoionized diffuse component for systems B and C, and the
photo plus collisionally ionized diffuse component for system
A.  The narrow solid spectrum includes the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds and the
photoionized diffuse components for each system.  The dotted--line model
includes the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds only and is shown to illustrate the
required high ionization phases.
---($a$) The Lyman break and Lyman series.  See text regarding the
continuum fit, which is probably low in the region of 1760~{\AA} to
1820~{\AA}.  Note the {\hbox{{\rm N}\kern 0.1em{\sc ii}}} $\lambda 916$ in system B and the
{\SVIdblt} in systems A and B.
---($b$) The {\hbox{{\rm Ly}\kern 0.1em$\gamma$}}, {\hbox{{\rm C}\kern 0.1em{\sc iii}}} $\lambda 977$ and {\hbox{{\rm N}\kern 0.1em{\sc iii}}} $\lambda 989$
predictions.  Note the sensitivity of {\hbox{{\rm C}\kern 0.1em{\sc iii}}} to the model components.
}
\end{figure*}

\begin{figure*}[ht]
\figurenum{3$c$--$d$}
\plotfiddle{Churchill.fig3c.eps}{5.5in}{0}{60}{60}{-236}{121}
\plotfiddle{Churchill.fig3d.eps}{0.0in}{0}{60}{60}{-236}{-74}
\protect\caption
{\footnotesize
Same as Figure 3$a$ and 3$b$.
---($c$) The {\hbox{{\rm Ly}\kern 0.1em$\beta$}}, {\OVIdblt}, and {\hbox{{\rm C}\kern 0.1em{\sc ii}}} $\lambda 1036$ predictions.
The wavelength calibration may be shifted blueward in the region
$\sim 1980$ to 2005~{\AA}.  Virtually no {\hbox{{\rm O}\kern 0.1em{\sc vi}}} resides in the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}
clouds, but must arise in a higher ionization phase.
---($d$) The {\hbox{{\rm Si}\kern 0.1em{\sc iii}}} $\lambda 1206$, {\hbox{{\rm Ly}\kern 0.1em$\alpha$}}, and {\NVdblt} predictions.
Note the sensitivity of {\hbox{{\rm Ly}\kern 0.1em$\alpha$}} and {\hbox{{\rm N}\kern 0.1em{\sc v}}} to the model components.  The
{\hbox{{\rm Si}\kern 0.1em{\sc iii}}} arises primarily in the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
}
\end{figure*}

\begin{figure*}[ht]
\figurenum{3$e$--$f$}
\plotfiddle{Churchill.fig3ef.eps}{5.5in}{0}{60}{60}{-236}{121}
\plotfiddle{Churchill.fig3gh.eps}{0.0in}{0}{60}{60}{-236}{-74}
\protect\caption
{\footnotesize
Same as Figure 3$a$ and 3$b$.
---($e$) The {\SiIIdblt} doublet.  The {\hbox{{\rm Si}\kern 0.1em{\sc ii}}} also arises primarily
in the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
---($f$) The {\hbox{{\rm Si}\kern 0.1em{\sc ii}}} $\lambda 1260$ prediction.  A blend ({\hbox{{\rm Ly}\kern 0.1em$\alpha$}}?) must
be present in the red wing.
---($g$) The {\hbox{{\rm C}\kern 0.1em{\sc ii}}} $\lambda 1334$ prediction.  As with {\hbox{{\rm Si}\kern 0.1em{\sc ii}}}
and {\hbox{{\rm Si}\kern 0.1em{\sc iii}}}, the {\hbox{{\rm C}\kern 0.1em{\sc ii}}} arises primarily in the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.  Note
the strong blend ({\hbox{{\rm Ly}\kern 0.1em$\alpha$}}?) in the red wing.
---($h$) The {\SiIVdblt} doublet predictions.  Since the {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} also
arises primarily in the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds, the {\hbox{{\rm Si}\kern 0.1em{\sc ii}}}, {\hbox{{\rm Si}\kern 0.1em{\sc iii}}} and
{\hbox{{\rm Si}\kern 0.1em{\sc iv}}} ratios placed tight constraints on the cloud ionization
conditions.
}
\end{figure*}

\begin{figure*}[ht]
\figurenum{3$i$}
\plotfiddle{Churchill.fig3i.eps}{2.82in}{0}{60}{60}{-236}{-70}
\protect\caption
{\footnotesize
Same as Figure 3$a$ and 3$b$.
The (self--blending) {{\rm C}\kern 0.1em{\sc iv}~$\lambda\lambda 1548, 1550$} doublet predictions.  For
systems A and B, note that a fair fraction of the {\hbox{{\rm C}\kern 0.1em{\sc iv}}} arises in the
{\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds, but the majority must arise in a higher ionization
phase. \label{fig:3i} \label{fig:figure3}}
\end{figure*}

Model clouds are drawn from the grid and a synthetic FOS spectrum is
generated from the model column densities for the transitions listed
in Table~\ref{tab:fosIDs} (end of paper).
The simulated FOS spectrum is generated by modeling the absorption
from each transition by Voigt profiles convolved with the FOS
instrumental spread function.
The Doppler parameters of the modeled transitions are determined from
the observed {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} $b$ parameters and the kinetic temperature, $T$,
output by the CLOUDY models, typically 5000 to 30,000~K.
The CLOUDY temperature is used to estimate the thermal component of
$b({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$, from which the turbulent $b$ parameter is computed
from the relation, $b_{tot}^{2} = b_{thermal}^{2} + b_{turb}^{2}$.
The total $b$ parameter of any transition can then be estimated from
its CLOUDY thermal $b$ and the turbulent $b$.
The synthetic spectrum is then superimposed on the observed FOS
spectrum and the $\chi^{2}$ is calculated pixel by pixel in regions of
interest as an indicator of the goodness of the model.

\subsection{High Ionization Gas}

As will be shown, the model {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds could not account for even
a small fraction of the absorption strengths of the higher ionization
species (i.e.\ {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {{\rm S}\kern 0.1em{\sc vi}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}}).
Thus, we postulated a high ionization diffuse component not seen
in {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorption.
A maximal diffuse scenario provides the low ionization limit of the
{\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds such that their contribution to the moderate and high
ionization species is minimized.
A minimal diffuse scenario provides the high ionization limit of the
{\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds such that their contribution to the moderate and high
ionization species is maximized.

For a postulated high ionization component, we used {\hbox{{\rm C}\kern 0.1em{\sc iv}}} and {\hbox{{\rm O}\kern 0.1em{\sc vi}}}
to constrain model cloud properties for both a solar abundance
pattern, $\hbox{[C/O]} = 0$, and an oxygen to carbon enhancement of
$\hbox{[C/O]} = -0.5$.
For constraints on the models from {{\rm S}\kern 0.1em{\sc vi}}, we use $[{\rm S}/{\rm H}] =
0$, but note that sulfur is enhanced by $\sim 0.5$ dex for $[{\rm
Fe}/{\rm H}] < 0$.
We avoid using {\hbox{{\rm N}\kern 0.1em{\sc v}}} as a primary constraint because the chemical
enrichment processes for nitrogen can lead to wide variations in its
abundance (\cite{wheeler89}).
We adopt $\hbox{[N/O]} = 0$ for this work, but occasionally discuss
possible variations in this ratio.

To examine the range of properties of a diffuse component, we did the
following systematic explorations.
For each system, we first assumed that any required high ionization
gas is in a {\it single}, broad component.
The adopted models of the high ionization components are obtained
using the same techniques illustrated in Appendix~\ref{app:constraints},
but the primary constraints are the residual strengths of the {\hbox{{\rm C}\kern 0.1em{\sc iv}}}
and {\hbox{{\rm O}\kern 0.1em{\sc vi}}} absorption unaccounted by the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
Also, consistency with any unaccounted {\hbox{{\rm C}\kern 0.1em{\sc iii}}}, {\hbox{{\rm Si}\kern 0.1em{\sc iii}}}, {\hbox{{\rm Si}\kern 0.1em{\sc iv}}},
{{\rm S}\kern 0.1em{\sc vi}} and {\hbox{{\rm N}\kern 0.1em{\sc v}}} is required.
We then explored the possibility that the {\hbox{{\rm C}\kern 0.1em{\sc iv}}} arises in a few
``{\hbox{{\rm C}\kern 0.1em{\sc iv}}}--only'' clouds while the {\hbox{{\rm O}\kern 0.1em{\sc vi}}} arises in a diffuse low
density high ionization phase that is separate from the {\hbox{{\rm C}\kern 0.1em{\sc iv}}} clouds,
which are separate from the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
The {\hbox{{\rm N}\kern 0.1em{\sc v}}} and {{\rm S}\kern 0.1em{\sc vi}} absorption profiles were critical for
constraining these ``{\hbox{{\rm C}\kern 0.1em{\sc iv}}} cloud'' models.

We systematically explored if contributions from both photoionization
and collisional ionization are consistent with the data.
For collisional ionization, we have drawn from the equilibrium models
of Sutherland \& Dopita (1993\nocite{sutherland93}), with solar
abundances.
In these models, the relative column densities of various species,
including that of neutral hydrogen, are a unique function of
temperature for a given metallicity.
Thus, under the model assumptions, the remaining free parameter is the
temperature.
To do so, we located the lowest ionization level of a photoionized
diffuse component consistent with the intermediate and moderately high
ionization species, after taking into account the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
A collisional ionization model  was then tuned to any unaccounted
absorption in {\hbox{{\rm N}\kern 0.1em{\sc v}}} and {\hbox{{\rm O}\kern 0.1em{\sc vi}}}.

\section{Model Results}
\label{sec:results}

Our intent is to present a general picture of both the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} cloud
and high ionization diffuse component properties within the context of
our modeling.
The adopted model properties are presented in Tables~\ref{tab:cloudpars},
\ref{tab:PICs}, and \ref{tab:DICs}.
In Table~\ref{tab:cloudpars}, we list the physical parameters of the
twelve {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds for the scenarios of a maximal or minimal diffuse
component.
Typical model parameters for each system are given in
Table~\ref{tab:PICs}.
The tabulated values serve as a guide; in the following discussion we
quote allowed ranges in the cloud properties based upon
\S~\ref{sec:models}.
The diffuse component properties are listed in Table~\ref{tab:DICs}.
The results described below are for the Haardt \& Madau
(1996\nocite{handm96}) extragalactic background.
In Appendix~\ref{app:models}, we describe our explorations with several
galactic/starburst radiation fields (representing both somewhat
extreme and ``normal'' cases) and argue that our general conclusions
are not altered.
In Figure~3, we present the synthetic spectrum of our
models superimposed upon the FOS spectrum.
The model transitions are labeled with three--point ticks, which
give the locations of systems A, B and C, from blue to red,
respectively.
Three synthetic spectra are shown.
The dotted--line spectrum is of the twelve photoionized {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}
clouds.
The thin solid--line spectrum includes both the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds and
the single--phase photoionized diffuse components.
The thick solid--line spectrum includes a two--phase photo plus
collisionally ionized component in system A.

\subsection{System A $(z=0.9254)$}

Since $N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}})/N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$ was not measured in any of the system A
clouds, the clouds were assumed identical vis--\'{a}--vis their
$N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$.
The {\hbox{{\rm Si}\kern 0.1em{\sc ii}}} to {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} ratio tightly constrains the ionization level
of these six clouds.
There is only a very small range of allowed ionization conditions,
with $\log U \simeq -2.3$.
To avoid super solar metallicity, the mean $[\alpha / {\rm Fe}]$
ranges from $+0.3$ to $+0.5$, which corresponds to $0.0 \leq Z \leq
-0.2$, respectively, for the best match to the Lyman series.
We obtained $10^{14.9} \leq N({\hbox{{\rm H}\kern 0.1em{\sc i}}}) \leq 10^{15.6}$~{\hbox{cm$^{-2}$}} in each
of the six clouds; system A makes a negligible contribution to the
Lyman limit break.

A highly ionized phase, not seen in {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}, is required
to account for the observed {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}} absorption.
For a single component, a Doppler width of $50 \leq b \leq 100$~{\hbox{km~s$^{-1}$}}
is consistent with the data.
We ran a series of simulations, focused on the blended {\hbox{{\rm C}\kern 0.1em{\sc iv}}} profile,
to explore the best combination of Doppler widths for {\hbox{{\rm C}\kern 0.1em{\sc iv}}} and {\hbox{{\rm O}\kern 0.1em{\sc vi}}}
in systems A and B, and to determine the allowed ranges for fits with
with $\chi ^{2}_{\nu} \leq 2$.
Based upon these simulations, we adopted $b=70$~{\hbox{km~s$^{-1}$}}.
This component is consistent with either a single--phase photoionized
diffuse medium with $\log U \simeq -1.2$, or a two--phase photo plus
collisionally ionized diffuse medium with $\log U \simeq -1.26$ and
$T\simeq 3\times 10^{5}$~K, respectively.
All phases must be metal rich, $Z\sim 0$, in order to not overproduce
the Lyman series.
It is not possible to establish which of the two provides a better
description of the high ionization transitions.
The latter yields a better match to the {\hbox{{\rm N}\kern 0.1em{\sc v}}} absorption.
However, $[{\rm N}/{\rm O}] \simeq +0.15$ in the photoionized only
phase could alternatively explain the {\hbox{{\rm N}\kern 0.1em{\sc v}}} absorption.
If the collisional component is present, further tests showed that
{\hbox{{\rm C}\kern 0.1em{\sc iv}}} arising in a few clouds with smaller $b$ parameters was not
ruled out.
The bottom line is that a highly ionized phase is required, whether
the {\hbox{{\rm C}\kern 0.1em{\sc iv}}} arises in a single or multiple clouds and the {\hbox{{\rm O}\kern 0.1em{\sc vi}}} and
{\hbox{{\rm N}\kern 0.1em{\sc v}}} in a collisionally ionized phase, or whether the {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}},
and {\hbox{{\rm O}\kern 0.1em{\sc vi}}} all arise in a single highly photoionized phase.

For the photoionization scenario, the diffuse cloud size increased as
$b$ was decreased from $70$~{\hbox{km~s$^{-1}$}}, primarily due to the curve of
growth behavior of {\hbox{{\rm C}\kern 0.1em{\sc iv}}}; its column density increases rapidly with
decreasing $b$ and is effectively constant for $b>70$~{\hbox{km~s$^{-1}$}} in the
relevant range of equivalent width.
Based upon exploration of the allowed range of $b$ parameters, an
upper limit on the size of the diffuse component is $\sim 30$~kpc for
$b = 50$~{\hbox{km~s$^{-1}$}}.
However, the best models, incorporating both a $\chi^{2}_{\nu}$ fit to
the {\hbox{{\rm C}\kern 0.1em{\sc iv}}} profile and matching to the {\hbox{{\rm N}\kern 0.1em{\sc v}}}, {{\rm S}\kern 0.1em{\sc vi}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}}
profiles, had $b \geq 65$~{\hbox{km~s$^{-1}$}}, which yielded sizes of
$\sim 10$--$20$~kpc.

\phantom{dummy space.}

\phantom{dummy space.}

\subsection{System B $(z=0.9276)$}

A range of ionization conditions ($-3.2 \leq \log U \leq -2.7$) for
the clouds are found for this system, primarily based upon the
observed {\hbox{{\rm Fe}\kern 0.1em{\sc ii}}} to {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} ratios in clouds 8, 9, and 11, and upon
{\hbox{{\rm Si}\kern 0.1em{\sc ii}}}, {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} and the other low ionization species for clouds 7 and
10.
However, the low ionization limits (maximal diffuse scenario) are
ruled out because the {\hbox{{\rm Si}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc iii}}}, and a large fraction of the
{\hbox{{\rm C}\kern 0.1em{\sc iii}}} must arise in the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
The Lyman limit break is primarily produced by clouds 8 and 11.
For these clouds, the metallicity must be $Z \geq -0.2$, even for an
$\alpha$--group enhanced abundance pattern, or else the Lyman limit
break would be too large.
In clouds 7 and 10,  the allowed abundance pattern range is
$ 0.0 \leq [\alpha/{\rm Fe}] \leq +0.5$, corresponding to $ -0.1 \geq
Z \geq -0.6$ (the lower limit is constrained by the Lyman series).
The {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} arises primarily in cloud 10, the most highly ionized,
lowest density, and extended ($\sim 5$~kpc) {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} cloud of the five.
Even for the minimal diffuse scenario (high ionization limit), the
predicted {\hbox{{\rm C}\kern 0.1em{\sc iv}}} absorption is still well below the observed strength.

Thus, an additional high ionization component, not seen in
{\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorption, is required to account for the {\hbox{{\rm C}\kern 0.1em{\sc iv}}},
{\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}} absorption.
This component is consistent with a single--phase photoionized diffuse
medium ($b \simeq 70$~{\hbox{km~s$^{-1}$}}, based upon the simulations of the {\hbox{{\rm C}\kern 0.1em{\sc iv}}}
profile) with a solar abundance pattern and near solar metallicity.
The bulk of the {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}} absorption
arises in this component, whereas the {\hbox{{\rm N}\kern 0.1em{\sc iii}}} and {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} absorption
arise primarily in cloud 10.
The {\hbox{{\rm C}\kern 0.1em{\sc iii}}} arises in both the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds and the diffuse
component.
In Figure~3$a$, we point out the {\hbox{{\rm N}\kern 0.1em{\sc ii}}} $\lambda 916$
absorption, which accounts for the reduced flux in the Lyman break.
We also point out the weak {\SVIdblt} absorption, which arises in the
diffuse component (the $\lambda 933$ transition is blended with
{\hbox{{\rm Ly}\kern 0.1em$6$}} from system C).
It is not possible for a collisionally ionized phase to
substantially contribute.

As with system A, we obtained a maximum line of sight size, $S$, of
$30$~kpc for the highly ionized diffuse component, but found a
preferred size range of $10$--$20$~kpc.
For $b \leq 50$~{\hbox{km~s$^{-1}$}}, the larger cloud size elevated absorption in
all high ionization species such that, in particular, {{\rm S}\kern 0.1em{\sc vi}} was
significantly overproduced.
A reduced $[{\rm S}/{\rm H}]$ would yield reduced sulfur absorption,
but is unlikely because sulfur is usually enhanced relative to solar
and is known to not suffer dust depletion (\cite{jtl96}).

\subsection{System C $(z=0.9343)$}

Though system C is a single weak {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} cloud, it is best described
by two photoionized phases.
This inference is based upon a self--consistent match to the Lyman
series, which is obtained when both a narrow and a broad component are
included.
The narrower low ionization {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} cloud accounts for the {\hbox{{\rm H}\kern 0.1em{\sc i}}} in
the  {\hbox{{\rm Ly}\kern 0.1em$\gamma$}}, {\hbox{{\rm Ly}\kern 0.1em$\delta$}}, and {\hbox{{\rm Ly}\kern 0.1em$\epsilon$}} absorption, whereas the broader high
ionization diffuse component contributes significantly to the {\hbox{{\rm Ly}\kern 0.1em$\alpha$}}
and {\hbox{{\rm Ly}\kern 0.1em$\beta$}} profiles.

The lower ionization phase produces the narrow {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}, {\hbox{{\rm Si}\kern 0.1em{\sc iii}}}, and
{\hbox{{\rm Si}\kern 0.1em{\sc iv}}} in a smaller cloud.
We obtained $\log U \simeq -2.6$ and the range $-1.0 \geq Z \geq -1.5$
for $0.0 \leq [\alpha / {\rm Fe}] \leq +0.5$.
The higher ionization phase, with  $\log U \simeq -1.3$, has $b\sim
40$~{\hbox{km~s$^{-1}$}}, which yields a size $S\sim 10$~kpc.
It is not possible for this high ionization component to be strictly
collisionally ionized nor is it possible for it to be a two--phase
photo plus collisionally ionized component; any contribution from a
collisionally ionized phase is insignificant.

\section{Discussion}
\label{sec:discussion}

As shown in Figure~\ref{fig:vpfits}, system A is comprised of six
distinct {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds with a total velocity spread of $\sim
200$~{\hbox{km~s$^{-1}$}} and is $\sim -300$~{\hbox{km~s$^{-1}$}} from system B, which has five
clouds spread over $\sim 100$~{\hbox{km~s$^{-1}$}}.
System C is $\sim +1000$~{\hbox{km~s$^{-1}$}} from system B and is comprised of a
single, resolved {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} cloud.
In Figure~3, we show the normalized FOS spectrum with
simulated spectra superimposed.
These low resolution data reveal that each system is rich in
multiple chemical species covering a wide range of ionization
potentials and that the chemical and ionization conditions differ from
system to system.

One motivation for our study was simply to ascertain if a multiphase
medium was required to explain the strong {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}} and {\hbox{{\rm O}\kern 0.1em{\sc vi}}}
absorption lines.
In all three systems, we were required to postulate a higher
ionization component that is not seen in {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorption.
We emphasize that (1) the overall {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} cloud properties are well
constrained by the data and modeling, and (2) the allowed cloud to
cloud variations are constrained tightly enough that no scenario even
remotely modifies the requirement for a {\hbox{{\rm C}\kern 0.1em{\sc iv}}}--{\hbox{{\rm N}\kern 0.1em{\sc v}}}--{\hbox{{\rm O}\kern 0.1em{\sc vi}}} high
ionization phase to explain the data.
To the accuracy afforded by the FOS spectrum, each of these high
ionization phases is well described by a single component with $b
\sim 70$~{\hbox{km~s$^{-1}$}} (based upon simulations).
Given the large $b$ parameters and the range of sizes derived from the
models (Table~\ref{tab:DICs}), this high ionization gas is likely to
have a line of sight extent of $10 \leq S \leq 20$~kpc, and thus may
be a surrounding medium in which the $\sim 0.1$~kpc {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds are
embedded.

\subsection{Comparison of System Properties}

Consider the cloud to cloud variations in system B, which is a Lyman
limit system.
The line of sight velocity spread of the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds is $\sim
100$~{\hbox{km~s$^{-1}$}}, which implies they are bound within a galactic potential.
If the clouds are equally illuminated by the extragalactic background,
then the presence of {\hbox{{\rm Fe}\kern 0.1em{\sc ii}}} in three of the clouds (8, 9, and 11)
implies that they are more dense, more shielded from the ionizing
flux, and/or iron--group enriched relative to the other two clouds (7
and 10).
This suggests that clouds 8, 9, and 11 may be spatially contiguous
(relatively speaking), in that they may share similar histories of
iron--group enrichment from Type Ia SNe.
In the Galaxy, the association of Type Ia explosions with the
kinematically old disk implies that some events take place at large
scale heights, so there is uncertainty in how much iron--rich gas
is driven into galactic halos (\cite{wheeler89}).

The unique cloud in system B is cloud 10, which gives rise to a broader
absorption profile ($b  \simeq 13$~{\hbox{km~s$^{-1}$}}) and has no detectable {\hbox{{\rm Fe}\kern 0.1em{\sc ii}}}.
It has a higher ionization condition and gives rise to the majority of
the {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} absorption (see Figure~\ref{fig:stis}).
It also has the largest $N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$.
Models yield that it is extended ($\sim 5$~kpc) and has lower
metallicity.
This leads us to conservatively speculate that cloud 10 is more akin
to a halo--like cloud or to a so--called Galactic high velocity cloud.
The ratio $N({\hbox{{\rm Si}\kern 0.1em{\sc iv}}})/N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$ may be a useful indicator
of the differing local environments of clouds in higher redshift
systems.

System C classifies as a ``weak'' {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorbers, defined by
$W_{\rm r}(2796) < 0.3$~{\AA} (\cite{weakmgII}).
From a sample of thirty such systems over the redshift range $0.4 < z
< 1.4$, Churchill et~al.\  found a wide range of
$W_{\rm r}({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}})/W_{\rm r}({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$ and
$W_{\rm r}({\hbox{{\rm C}\kern 0.1em{\sc iv}}})/W_{\rm r}({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$, presumedly due to variations in
abundance pattern and ionization conditions, including single phase
and multiphase.
These {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorbers are sub--Lyman limit systems with $Z \geq -1$
and [some with $Z > 0$ (\cite{cl98})].
Apart from its line--of--sight proximity ($< 500$~{\hbox{km~s$^{-1}$}}) to system B,
system A would classify as a weak {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorber.
The $\sim 200$~{\hbox{km~s$^{-1}$}} kinematic spread of the six
clouds in system A, and the large $N({\hbox{{\rm C}\kern 0.1em{\sc iv}}})/N({\hbox{{\rm Si}\kern 0.1em{\sc iv}}})$ ratio in the
diffuse component are suggestive of lower ionization clouds moving
within a high ionization galactic corona, or halo, where the ratio
$N({\hbox{{\rm C}\kern 0.1em{\sc iv}}})/N({\hbox{{\rm Si}\kern 0.1em{\sc iv}}})$ is expected to be large (\cite{savage97}).
Is it possible that the system C {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} cloud arises in a similar
environment as the six system A clouds, but that the line of sight
happens to sample only one cloud?
If so, the {\hbox{{\rm C}\kern 0.1em{\sc iv}}} absorption strength in system C would be comparable
to that of system A, and it is significantly weaker.
For weak systems, it may be that strong, broad {\hbox{{\rm C}\kern 0.1em{\sc iv}}} absorption implies
a larger number of {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds with a larger kinematic spread.

\subsection{Profile Anatomy: Model Predictions}
\label{sec:anatomy}

\begin{figure*}[th]
\figurenum{4}
\plotfiddle{Churchill.fig4.eps}{6.5in}{0}{70}{70}{-227}{-37}
\protect\caption[Churchill.fig4.eps]
{\footnotesize
A selection of predicted high resolution profiles for systems A and B
based upon our models.  This simulated STIS/{\it HST\/} spectrum has
$R = 30,000$ ($v \sim 10$~{\hbox{km~s$^{-1}$}}) and signal--to--noise ratio of 30.
Short ticks mark the velocities based upon the HIRES {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds
and long ticks mark the diffuse component centroids.  For system A,
the three--phase model (photoionized {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds and photo plus
collisionally ionized diffuse component) is shown and, for system B,
the two--phase model (photoionized clouds and diffuse component) is
shown.  The dotted--line curves are the profiles of the blended
individual {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.  The solid--line curves are the
photoionized diffuse component profiles and the dash--dot curves are
of the collisionally ionized diffuse component.
\label{fig:stis}}
\end{figure*}

In Figure~\ref{fig:stis}, we present simulated STIS/{\it HST\/}
spectra with $R=30,000$, two pixels per resolution element, and a
signal--to--noise ratio of $30$ for the first four transitions in the
Lyman series, and for the {\hbox{{\rm Si}\kern 0.1em{\sc iv}}}, {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}} profiles
of systems A and B.
These spectra were generated assuming Voigt profiles with the
properties listed in Tables~\ref{tab:PICs} and \ref{tab:DICs}.
These model profiles can be compared directly to observed data (from
STIS/{\it HST\/}) and thus provide a direct test of the models.
We show the contributing components as smooth curves.
The dotted--line curves are the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds, with their velocity
centroids marked with the short ticks above the continuum.
The solid curves are from the photoionized diffuse component and
the dash--dot curves in system A represents the collisionally ionized
diffuse component.

\subsubsection{The Lyman Series}

Here, we emphasize the importance of the Lyman series.
For all three systems, the {\hbox{{\rm Ly}\kern 0.1em$\alpha$}} and {\hbox{{\rm Ly}\kern 0.1em$\beta$}} profiles in the FOS
spectra were significantly broader than could be fully accounted
by the {\hbox{{\rm H}\kern 0.1em{\sc i}}} obtained soley from the model {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
However, the narrower {\hbox{{\rm Ly}\kern 0.1em$\gamma$}}, {\hbox{{\rm Ly}\kern 0.1em$\delta$}}, and {\hbox{{\rm Ly}\kern 0.1em$\epsilon$}} profiles were
fully accounted by the {\hbox{{\rm H}\kern 0.1em{\sc i}}} in these clouds.
Based upon the curve of growth behavior of the Lyman series, we found
that the addition of a very broad ($b>50$~{\hbox{km~s$^{-1}$}}), lower $N({\hbox{{\rm H}\kern 0.1em{\sc i}}})$
component naturally explained the deeper and broader {\hbox{{\rm Ly}\kern 0.1em$\alpha$}} and {\hbox{{\rm Ly}\kern 0.1em$\beta$}}
profiles, without overproducing the narrower {\hbox{{\rm Ly}\kern 0.1em$\gamma$}}, {\hbox{{\rm Ly}\kern 0.1em$\delta$}}, and
{\hbox{{\rm Ly}\kern 0.1em$\epsilon$}} profiles (or modifying the Lyman limit break).

Such behavior of the Lyman series in low resolution data is likely to
be a strong indication that a broad, perhaps highly ionized diffuse
component is present.
In Figure~\ref{fig:stis}, we illustrate this behavior as it would be
seen in higher resolution spectra.
Note that the widths of {\hbox{{\rm Ly}\kern 0.1em$\alpha$}} and {\hbox{{\rm Ly}\kern 0.1em$\beta$}} are dominated by a
broad, high ionization component, whereas the higher order transition
widths are dominated by the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
In high resolution spectra, a diffuse component gives rise to a
broadened, shallow wing when the profiles are saturated (see {\hbox{{\rm Ly}\kern 0.1em$\alpha$}} and
{\hbox{{\rm Ly}\kern 0.1em$\beta$}} in system B) and/or suppresses the recovery of the flux to the
continuum level between clouds in the profile centers (see {\hbox{{\rm Ly}\kern 0.1em$\beta$}} and
{\hbox{{\rm Ly}\kern 0.1em$\gamma$}} in system A).

\subsubsection{High Resolution Metal Lines}

In system A, the {\hbox{{\rm C}\kern 0.1em{\sc iv}}} profiles have structure due to the lower
ionization {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
The {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} absorption predominantly arises in the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds, and
thus closely traces the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} kinematics.
Together, these profiles are tantalizing similar to those observed by
Savage, Sembach, \& Cardelli (1994\nocite{savage94}) along the line of
sight to HD~167756 in the Galaxy.
We quote, ``The sight line contains at least two types of highly
ionized gas.  One type gives rise to a broad {\hbox{{\rm N}\kern 0.1em{\sc v}}} profile, and the
other results in a more structured {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} profile.  The {\hbox{{\rm C}\kern 0.1em{\sc iv}}}
profile contains contributions from both types of highly ionized
gas.''
We also find similarities between the model {\hbox{{\rm C}\kern 0.1em{\sc iv}}}
profiles and those in the damped {\hbox{{\rm Ly}\kern 0.1em$\alpha$}} systems at $z_{\rm
abs}=3.3901$ toward Q$0000-262$ and $z_{\rm abs}=2.2931$ toward
Q$0216+080$ (Lu et~al.\  1996\nocite{lu96}, Figures 2 and 3).

In system B, cloud 10 is clearly unique among the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
This cloud has the largest ionization parameter and accounts for the
majority of the {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} absorption, which was clearly constrained to
arise in the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
Since the ionization conditions of clouds 8, 9, and 11 were set by
their measured $N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}})/N({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$, the ionization condition in
cloud 10 was well constrained by the ratio $N({\hbox{{\rm Si}\kern 0.1em{\sc ii}}})/N({\hbox{{\rm Si}\kern 0.1em{\sc iv}}})$;
there was no alternative but for cloud 10 to dominate the {\hbox{{\rm Si}\kern 0.1em{\sc iv}}}
absorption.
The point is that this cloud may arise in a spatially distinct
environment and have a unique formation history from the other clouds
in system B (see \S~\ref{sec:discussion}).
The {\hbox{{\rm C}\kern 0.1em{\sc iv}}} profile is dominated by a diffuse higher
ionization phase, whereas the {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} is predominantly due to higher
density clouds.
Some lines of sight through the Galaxy also exhibit narrow {\hbox{{\rm Si}\kern 0.1em{\sc iv}}}
profiles and broad {\hbox{{\rm C}\kern 0.1em{\sc iv}}} profiles (for examples, see
\cite{sembach94}; \cite{savsem94}).
In contrast, the {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} and {\hbox{{\rm C}\kern 0.1em{\sc iv}}} profiles in the $z_{\rm
abs}=2.8268$ damped {\hbox{{\rm Ly}\kern 0.1em$\alpha$}} system towards Q$1425+603$ (\cite{lu96})
appear to arise in the same phase.

A synthetic STIS/{\it HST\/} spectrum for System C (not shown) reveals
a narrow {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} profile and broad {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}} profiles.
The {\hbox{{\rm C}\kern 0.1em{\sc iv}}} profile exhibits a slightly deep, narrow core due to the
low ionization phase.
However this contribution would be lost for the expected noise levels
in observed data, unless this narrow component was off--center
by $\sim 40$~{\hbox{km~s$^{-1}$}} relative to the broad component (within
the uncertainty of our modeling such an offset is not ruled out).

\section{On the Nature of the High Ionization Phase}
\label{sec:onthe}

Within 100{\arcsec} of the QSO, Kirhakos et~al.\
(1994\nocite{kirhakos94}) identified $10$ galaxies with $21 \leq g
\leq 22$.
Compared to field galaxies, this is a slight overdensity by a factor
of a few to several (\cite{tyson88}).
Identified within 10{\arcsec} of the QSO are three bright galaxies
with impact parameters $29$, $45$, and $47~h^{-1}$~kpc, respectively
($q_0 = 0.05$).
At $z = 0.93$, Thimm (1995\nocite{thimm95}) detected a {\hbox{[{\rm O}\kern 0.1em{\sc ii}]}} $\lambda
3727$ flux of $9 \times 10^{-17}$ ergs {\hbox{cm$^{-2}$}} s$^{-1}$ from the
galaxy at $45~h^{-1}$~kpc.

Given these facts, we entertain the possibility that the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}
absorption systems arise in individual galaxies either in a small
group or possibly in a cluster.
Such a possibility would have interesting implications in view of the
types of mechanisms that could give rise to highly ionized gas in such
environments.

\subsection{Intragroup Corona?}

Mulchaey et~al.\  (1996\nocite{mulchaey96}) predicted that poor groups
that are rich in spiral galaxies have hot, diffuse coronae that are
cooler than the X--ray coronae surrounding poor groups dominated by
E/S0 galaxies.
This high ionization intragroup material is predicted to have $T\sim 2
\times 10^{6}$~K and to primarily give rise to strong {\hbox{{\rm O}\kern 0.1em{\sc vi}}}, whereas
any {\hbox{{\rm C}\kern 0.1em{\sc iv}}} or {\hbox{{\rm N}\kern 0.1em{\sc v}}} absorption is predicted to arise in the proximity
of the galaxies themselves (\cite{mulchaey96}).
As such, the {\hbox{{\rm O}\kern 0.1em{\sc vi}}} profiles would be broader than the {\hbox{{\rm N}\kern 0.1em{\sc v}}} and
{\hbox{{\rm C}\kern 0.1em{\sc iv}}}.
The same would hold for the Lyman series (\cite{verner94}).

Our models can fully explain the {\hbox{{\rm O}\kern 0.1em{\sc vi}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm C}\kern 0.1em{\sc iv}}} in a single
component of $T \sim 3 \times 10^{4}$~K gas (the possible collisional
component in system A has $T \sim 3 \times 10^{5}$~K).
The observed {\hbox{{\rm O}\kern 0.1em{\sc vi}}} or {\hbox{{\rm H}\kern 0.1em{\sc i}}} widths are not broader than those of the
{\hbox{{\rm C}\kern 0.1em{\sc iv}}} and {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and their $\sim 70$~{\hbox{km~s$^{-1}$}} $b$ parameters are not
suggestive of gas that is kinematically akin to the $\sim 200$~{\hbox{km~s$^{-1}$}}
dispersion of a galaxy group.
Furthermore, the {\hbox{{\rm O}\kern 0.1em{\sc vi}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm C}\kern 0.1em{\sc iv}}} profiles are clearly
aligned with the three {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorption redshifts, suggesting that
the high ionization material is spatially coincident with the
individual {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} systems.
The upper limit on the size of the photoionized diffuse components
is $S\leq30$~kpc (assuming photoionization equilibrium), and the
inferred $Z\sim 0$ metallicities suggest that the high ionization
material surrounds the galaxies and has been enriched by them.
Significantly lower metallicities are expected if the gas was
intragroup material left over from the formation processes.
We do not favor an interpretation of the data in which the
high ionization material arises in a intragroup or common halo.

\subsection{Galactic Coronae?}

The highly ionized material in these three $z\sim 1$ galaxies may
be more akin to galactic coronae (see Spitzer
1956\nocite{spitzer56}, 1990\nocite{spitzer90}; Savage et~al.\
1997\nocite{savage97}), material stirred up by energetic mechanical
processes, such as galactic fountains.
In this scenario, the gas is concentrated around individual galaxies
which presumably provide a source of support, heating, and chemical
enrichment.

In the Galaxy, $b \simeq 60$~{\hbox{km~s$^{-1}$}} was measured for {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, and was
observed to slightly increase with the ionization level of the
transition (\cite{savage97}).
Simulations of the allowed range of $b$ parameters for systems A and B
yielded similar Doppler widths, with $b \simeq 70$~{\hbox{km~s$^{-1}$}} providing
a good match to the {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}} profiles for both
systems.
If the high ionization diffuse components in these systems are
arising in coronae analogous to that surrounding the Galaxy, they may
have similar turbulent processes supporting their scale heights.

The radial extent of the Galactic corona is unknown, but for local
galaxies, radio maps reveal {\hbox{{\rm H}\kern 0.1em{\sc i}}} extends to tens of kpc
(\cite{corbelli89}; \cite{vangorkom93}), beyond which the hydrogen
becomes optically thin and a highly ionized extension is expected
(\cite{maloney93}; \cite{corbelli93}; \cite{dove94}).
The impact parameters of the candidate galaxies are $\sim
30$--$50~h^{-1}$~kpc, with the orientation of the line of sight through
each galaxy unknown.
Absorption, even from non--spherical absorbers, is not unexpected at
an impact parameter of $\sim 50$ kpc (see Figure~1 of \cite{cc96}).
For the Galaxy, Savage et~al.\  (1997\nocite{savage97}) measured the
effective scale heights of $5.1$, $4.4$, and $3.9$~kpc for {\hbox{{\rm Si}\kern 0.1em{\sc iv}}},
{\hbox{{\rm C}\kern 0.1em{\sc iv}}}, and {\hbox{{\rm N}\kern 0.1em{\sc v}}}, respectively.
We find a preferred size of $10$--$20$~kpc for the size of the high
ionization gas at $z\sim 0.93$, with an upper limit of $\sim 30$~kpc.
Since these ``sizes'' (or the path length through an oriented
structure) are comparable to twice the Galactic scale height, the
distribution of high ionization is consistent with Galactic--like
coronae.

A scenario in which the high ionization gas arises in individual
galactic coronae is in contrast to that proposed by Lopez et~al.\
(1998\nocite{lopez98}) for a system at $z\sim 1.7$ in the spectra of
HE~$1104-181$~A, B.
In a double line of sight study, Lopez et~al.\  found {\hbox{{\rm O}\kern 0.1em{\sc vi}}} profiles
consistent with $110 \leq b \leq 180$~{\hbox{km~s$^{-1}$}} and that the extent of
the highly ionized gas was $\sim 100$~kpc.
These inferences also differ from those of Bergeron et~al.\
(1994\nocite{bergeron94}), who found that the {\hbox{{\rm O}\kern 0.1em{\sc vi}}} phase in the
$z_{\rm abs} \sim 0.8$ {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorber toward PKS~$2145+064$ was at
least $50$~kpc in extent.

\section{Conclusions}
\label{sec:conclusion}

We have studied the kinematic, chemical, and ionization conditions of
three metal--line absorption systems at $z\sim 1$ seen in the
PG~$1206+459$ spectrum.
The  systems were selected by the presence of {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorption in a
high resolution spectrum, and were chosen as a pilot study of a larger
program designed to chart the physical conditions and evolution of
absorbing gas in galaxies.
The {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} profiles are shown in Figure~\ref{fig:order28}, with each
system designated A, B, and C.
Rich absorption line data from FOS/{\it HST\/} (\cite{kp13}) revealed
strong {\hbox{{\rm C}\kern 0.1em{\sc iv}}}, {\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}} absorption, as well as {\hbox{{\rm H}\kern 0.1em{\sc i}}} and many
other species and transitions covering a wide range of ionization
potentials.
Ground--based imaging data (\cite{kirhakos94}) revealed three
candidate absorbing galaxies within 10{\arcsec} of the QSO and
possibly a group of galaxies within 100{\arcsec} of the QSO.

The main goals of the study were to see if multiphase gas was
required to explain the strong high ionization absorption line data,
and to infer some level of information on the gas metallicities and
spatial distributions.
Assuming the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds are photoionized, we found the range of
chemical and ionization conditions consistent with both the high
resolution and low resolution data.
We then postulated the presence of high ionization components not seen
in {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorption to explain the unaccounted {\hbox{{\rm C}\kern 0.1em{\sc iii}}}, {\hbox{{\rm C}\kern 0.1em{\sc iv}}},
{\hbox{{\rm N}\kern 0.1em{\sc v}}}, and {\hbox{{\rm O}\kern 0.1em{\sc vi}}} absorption.

We briefly summarize the main results of our study:

1.
For systems A and B, we were required to postulate a high ionization
phase in addition to the lower ionization {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds.
System C could be made marginally consistent with a single--phase
absorber, though a two--phase absorber is strongly preferred due to
the nature of the Lyman series absorption.
We infer that, in these systems, the lower ionization {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds
arise in high ionization diffuse gas; each of these absorption systems is
comprised of a multiphase gaseous medium.
We find that the high ionization phase of system A {\it could\/} arise
in multiple, narrower  ``{\hbox{{\rm C}\kern 0.1em{\sc iv}}} clouds''.
For system B, such {\hbox{{\rm C}\kern 0.1em{\sc iv}}} clouds are ruled out.

2.
The absorbing gas in both the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds and the high ionization
components are consistent with photoionized clouds.
In systems B and C, a collisionally ionized phase is ruled out.
Only in system A could the data be made consistent with a three--phase
absorber, incorporating the photoionized {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds, a highly
photoionized diffuse component, and a collisionally ionized component.
This three--phase model provided a more consistent match to the {\hbox{{\rm N}\kern 0.1em{\sc v}}}
absorption in this system.
In this three--phase scenario, the {\hbox{{\rm C}\kern 0.1em{\sc iv}}} could arise in a few
narrower components.
However, $[{\rm N}/{\rm O}] \simeq 0.15$ in the highly photoionized
component, instead of the assumed $[{\rm N}/{\rm O}] = 0$, would
remove the need for the collisionally ionized gas and rule out the
three--phase absorber.

3.
We find no evidence that the {\hbox{{\rm O}\kern 0.1em{\sc vi}}} gas is in a separate
and very highly ionized diffuse phase that encompasses the {\hbox{{\rm C}\kern 0.1em{\sc iv}}} and
{\hbox{{\rm N}\kern 0.1em{\sc v}}} absorption.
Based upon the $b\sim 70$~{\hbox{km~s$^{-1}$}} profile widths, inferred $Z\sim 0$
metallicities, $3\times 10^{4}$~K temperatures, inferred $S\leq
30$~kpc sizes, and the clear redshift alignment of the high ionization
transitions with the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} systems, we suggest that the high
ionization gas is analogous to the Galactic corona in that it traces
the galaxies themselves and does not appear to exhibit the
characteristics predicted for intragroup or intracluster material.
The {\hbox{{\rm O}\kern 0.1em{\sc vi}}} likely arises in the same phase as the {\hbox{{\rm C}\kern 0.1em{\sc iv}}} and {\hbox{{\rm N}\kern 0.1em{\sc v}}}.
The {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} is constrained to arise in the same phase as the {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}
clouds, whereas {\hbox{{\rm C}\kern 0.1em{\sc iii}}} arises in both the clouds and the high
ionization phase.

4.
We have found cloud to cloud variations in the chemical and ionization
conditions in the five {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} clouds of system B.
Three of the clouds are likely to be iron--group enriched and have
higher densities and low ionization conditions.
The majority of the neutral hydrogen giving rise to the Lyman break in
the FOS spectrum is from these clouds.
A lower metallicity {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} cloud giving rise to a broader absorption
profile is interspersed in velocity with these clouds, likely has an
$\alpha$--group enhanced abundance pattern, and gives rise to the
majority of the {\hbox{{\rm Si}\kern 0.1em{\sc iv}}} absorption.
We speculate that this cloud may be similar to a halo--like cloud,
and suggest that the ratio $N({\hbox{{\rm Si}\kern 0.1em{\sc iv}}})/({\hbox{{\rm Mg}\kern 0.1em{\sc ii}}})$ may be a useful
indicator for discriminating between clouds in different parts of
high redshift galaxies.

The most compelling reason why we favor the scenario in which the high
ionization diffuse material is coupled to the galaxies is the clear
kinematic separation of the {\hbox{{\rm O}\kern 0.1em{\sc vi}}} profiles corresponding to systems A
and B.
Each of the inferred diffuse components must be centered (at least
roughly) on the systems, and they must be distinct from one another
in velocity space.
There are additional, if less compelling, arguments.
The inferred $b$ parameters in the model diffuse components are
in the same regime as those found for the Galactic corona
(\cite{savage97}), further suggesting that the material is galaxy
associated.
The inferred metallicities are high, $Z \sim 0$, which is best
understood if the material had been enriched by its host galaxy and
further suggests that the origin and source of enrichment of the
diffuse gas is related to star forming parts of galaxies.
The enrichment could be due to {\it in situ\/} star formation as gas
clouds collide and cool in the galactic halos (\cite{ss92}),
or could be due to galactic fountain processes from the galactic disks
(\cite{spitzer90}).

As a speculative aside, we ask if Galactic--like coronae {\hbox{{\rm O}\kern 0.1em{\sc vi}}}
absorbers are likely to be a common form of {\hbox{{\rm O}\kern 0.1em{\sc vi}}} systems at
$z\leq 1$, as opposed to group--halos.
Burles \& Tytler (1996\nocite{bt96}) have shown that absorbers
selected by the presence of {\hbox{{\rm O}\kern 0.1em{\sc vi}}} have the same redshift path density
as Lyman limit--{\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorbers at $\left< z \right> = 0.9$.
As we have found here, {\hbox{{\rm O}\kern 0.1em{\sc vi}}} can be associated with a Lyman limit
system when the absorbing gas is segregated into multiple ionization
phases.
If multiphase absorption is common in {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} absorbers, some {\hbox{{\rm O}\kern 0.1em{\sc vi}}}
might arise in Galactic--like corona.

It would seem that this pilot study has shown that wholesale study of
the kinematic, chemical, and ionization conditions of {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}
absorbers, using high resolution {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}} profiles and the available
low resolution {\it HST} spectra, would yield a improved understanding
of galactic gas at early epochs.
In order to assess the robustness of our modeling, we have synthesized
high resolution STIS/{\it HST\/} spectra of the ultraviolet
transitions (presented in Figure~\ref{fig:stis}).
The modeling techniques applied in this paper can be directly tested
by comparing these predicted profiles with those observed with
STIS/{\it HST}.
If our approach proves to yield an accurate description of the gas,
then wholesale modeling can be embarked upon for roughly 50 {\hbox{{\rm Mg}\kern 0.1em{\sc ii}}}
systems without requiring large amounts of space based telescope time
to acquire high resolution spectra of high quality.

\acknowledgments

This work was supported by NSF AST--9529242 and AST--9617185 and by NASA
NAG5--6399 and AR--07983.01--96A from STScI.
Special thanks to Buell Jannuzi, Sofia Kirhakos, and Don Schneider for
generously supplying the FOS/{\it HST\/} spectra and for helpful
discussion regarding the reduction and analysis of the FOS data.
We are thankful for comments from the anonymous referee that
resulted in an improved manuscript.
We thank Karen Knierman and Jane Rigby for assistance in producing the
stacked spectrum of the system A clouds for the limit on $N({\hbox{{\rm Fe}\kern 0.1em{\sc ii}}})$,
and are grateful to Steven S. Vogt for the HIRES spectrograph.


\section{Introduction}

With the discovery of the doubly charmed baryon $\Xi_{cc}$ \cite{SELEX:2002wqn,LHCb:2017iph,LHCb:2018pcs,LHCb:2018zpl,LHCb:2021eaf,LHCb:2022rpd}, the triply heavy baryon has become the only one absent in the baryon family. The masses of the triply heavy baryon have been calculated in kinds of models theoretically \cite{Faustov:2021qqf,Tazimi:2021ywr,Wang:2020avt,Bhavsar:2018tad,Vijande:2015faa,Llanes-Estrada:2013rwa,Wang:2011ae,Llanes-Estrada:2011gwu,Alomayrah:2020qyw,Yang:2019lsg,Shah:2019jxp,Rai:2017hue,Shah:2017jkr,Martynenko:2007je,Azizi:2014jxa,Aliev:2012tt,Zheng:2010zzc,Zhang:2009re,Patel:2008mv,Jia:2006gw} and different results have been gotten. Exploring the triply baryon in experiment is important for filling the particle spectra and testing the theories. To produce the triply heavy baryon, three pairs of heavy quarks need to be produced. This makes it hard to calculate the cross sections of the triply heavy baryon production. The total cross sections and the differential cross sections of the production of these baryons at LHC have been calculated in \cite{Chen:2011mb,Wu:2012wj}. There, the authors simplified the calculation utilizing the characteristic of the spin, the color configurations of the triply heavy baryons and the small relative momenta among the constituent quarks. The production of the triply charmed baryon in $e^+e^-$ annihilation at $Z$-boson pole have been estimated in \cite{Baranov:2004er}, where the mass of the charm quark has been ignored at some places in the matrix. The production of $\Omega_{ccc}$ from quark gluon plasma in heavy ion collisions has been calculated in \cite{Becattini:2005hb}. And the production of the triply baryons from the fragmentation processes have been calculated in \cite{GomshiNobary:2005ur,MoosaviNejad:2017rvi,GomshiNobary:2006tzy,GomshiNobary:2004mq,Delpasand:2019xpk}.

In this paper, we report the study on the production of the triply heavy baryons $\Omega_{ccc}$ and $\Omega_{bbb}$ at $e^+e^-$ colliders. Here, we calculate the direct production of these triply heavy baryons exactly.
Because the constituent quarks of the triply heavy baryon are all heavy flavored, the NRQCD \cite{Bodwin:1994jh} can be used to describe the triply heavy baryon. And, the production of the $\Omega_{QQQ}$ can be factorized into two parts, the short-distance coefficient which describes the process of the production of the
triple heavy quark pairs, and the long-distance matrix element which
describes the formation of the triply heavy baryon from point-like
three heavy quarks.\footnote{$\Omega_{QQQ}$ means the triply heavy baryon $\Omega_{ccc}$ or $\Omega_{bbb}$. And $Q$ denotes the $c$ or $b$ quark.}
To achieve the calculation, we will reduce the number of the Feynman diagrams which we have to calculate, as we have done in \cite{Chen:2011mb,Wu:2012wj}.

In the next section we will show the details on the calculation of these two parts, the short-distance coefficient and the long-distance matrix element, explicitly. In Sec.\ref{sec:NANDC}, we will list the numerical results and conclusions. And in Sec.\ref{sumra}, we will give a brief summary.

\section{Production of THE $\Omega_{ccc}$ and $\Omega_{bbb}$ at $e^+e^-$ colliders}
\label{sec:Production}
As a baryon, the triply heavy baryon must be in color singlet. The color wave function of the $\Omega_{QQQ}$ must be $\frac{1}{\sqrt{6}}\varepsilon^{\xi_1\xi_2\xi_3}Q_{1\xi_1}Q_{2\xi_2}Q_{3\xi_3}$
 with $\xi_i$ ($i=$1,2,3) being the color index of the valence quark $Q_i$. As a ground state, the orbital angular momentum wave function is symmetrical. The exchange antisymmetry of the identical fermions implies that the $\Omega_{QQQ}$ must be the spin-symmetrical state and
the spin of it must be $\frac{3}{2}$.

Because of the large masses of the heavy quarks, the NRQCD can be used to describe the triply heavy baryons. In NRQCD, we describe the normalized wave function of the $\Omega_{QQQ}$ as,

\begin{eqnarray} \label{nrqcd}
|\Omega_{QQQ},\frac{3}{2},S_Z\rangle&&=\sqrt{2M}\int\frac{d^3\vec{q}_1}{(2\pi)^3}\frac{d^3\vec{q}_2}{(2\pi)^3}\frac{1}{\sqrt{3!}} \nonumber\\
&&\sum_{\xi_1,\xi_2,\xi_3}\sum_{\eta_1,\eta_2,\eta_3}\frac{\varepsilon^{\xi_1\xi_2\xi_3}}{\sqrt{6}}
\langle
\frac{3}{2},S_Z|\eta_1,\eta_2,\eta_3\rangle \nonumber\\
&&\frac{1}{\sqrt{2E_12E_22E_3}}\psi(\vec{q}_1,\vec{q}_2)
|Q_1,\xi_1,\eta_1,\vec{q}_1\rangle\nonumber\\
&&|Q_2,\xi_2,\eta_2,\vec{q}_2\rangle
|Q_3,\xi_3,\eta_3,\vec{q}_3\rangle, \ \ \ \
\end{eqnarray}
\text{where, $\vec{q}_3=-\vec{q}_1-\vec{q}_2$, and} \ \
\begin{eqnarray}
\langle Q_i,\xi_{i},\eta_{i},&&\vec{q}_i|Q_j,\xi_j,\eta_j,\vec{q}_j\rangle \nonumber \\ &&=\delta_{Q_iQ_j}\delta_{\eta_i\eta_j}\delta_{\xi_i\xi_j}(2\pi)^32E_f\delta^{(3)}(\vec{q}_i-\vec{q}_j),\nonumber
\end{eqnarray}
 with $\eta_i$ and ($E_i$, $\vec{q}_i$)
($i=1,2,3$) being the spin and the four-momentum
of the $Q_i$ heavy quark; $M$ being the mass of the baryon $\Omega_{QQQ}$; $\langle \frac{3}{2},S_Z|\eta_1,\eta_2,\eta_3\rangle$
being the Clebsch-Gordan (C-G) coefficient; $S_Z$ being the
third component of the spin of the baryon, and
$\psi(\vec{q}_1,\vec{q}_2)$
being the wave function of the baryon in the momentum space which is normalized as follows
\begin{eqnarray} \label{nor-wa}
\int\frac{d^3\vec{q}_1}{(2\pi)^3}\frac{d^3\vec{q}_2}{(2\pi)^3}\psi^*(\vec{q}_1,\vec{q}_2)\psi(\vec{q}_1,\vec{q}_2)=1\;.
\end{eqnarray}

The production of $\Omega_{QQQ}$ at $e^+e^-$ colliders can be factorized into two parts, the short-distance coefficient corresponding to the production of the three $Q\bar{Q}$ pairs and the long-distance matrix element describing the three heavy quark $Q$ coupling to the triply heavy baryon $\Omega_{QQQ}$. In the heavy quark limit, the dependence of the short distance on the momenta, $q_1$ and $q_2$, can be neglected in the leading order. Namely, the momenta of the produced three identical heavy quarks are the same. As a result, the long-distance matrix is proportional to the wave function of the baryon at the origin,
 \begin{eqnarray} \label{wave}
\Psi(0,0)=\int\frac{d^3q_1}{(2\pi)^3}\frac{d^3q_2}{(2\pi)^3}\psi(q_1,q_2)\;.
\end{eqnarray}

 Now, the amplitude of the process $e^+e^-\rightarrow \Omega_{QQQ}\bar{Q}\bar{Q}\bar{Q}$ can be written as,
 \begin{eqnarray} \label{cross}
&&A(e^+e^-\to\Omega_{QQQ}\bar{Q}\bar{Q}\bar{Q}) =  \nonumber \\
&&  \frac{\sqrt{2M}}{\sqrt{(2m)^3}}
\frac{\Psi(0,0)}{\sqrt{3!}} \mathcal{M}(e^+e^-\to
(QQQ)_1^{(\frac{3}{2},S_Z)}\bar{Q}_1\bar{Q}_2\bar{Q}_3)\;, \ \ \ \
\end{eqnarray}
in which, $m$ is the mass of the heavy quark $Q$, and $\mathcal{M}(e^+e^-\to
(QQQ)_1^{(\frac{3}{2},S_Z)}\bar{Q}\bar{Q}\bar{Q})$ is the short-distance coefficient, namely, the matrix element of the process $e^+e^-\to
(QQQ)_1^{(\frac{3}{2},S_Z)}\bar{Q}\bar{Q}\bar{Q}$.\footnote{ $(QQQ)_1^{(S,S_Z)}$ means the total spin and the
third component of the total spin of the three identical $Q$-quarks are $\frac{3}{2}$ and $S_Z$, respectively; the momenta of the three quarks are the same; and the three heavy quarks couple to a color singlet.}

Now, let us consider the short-distance coefficient. The contribution to the production of the three heavy quark pairs
 comes from the $e^+e^-$ annihilation process mainly,

 \begin{eqnarray}\label{anni}
&&e^-(k_1)+e^+(k_2)\rightarrow Z^*/\gamma^*\rightarrow
Q(p_1,\xi_1)+Q(p_2,\xi_2)\nonumber \\
&&+Q(p_3,\xi_3)+\bar{Q}(p_4,\chi_1)+\bar{Q}(p_5,\chi_2)+\bar{Q}(p_6,\chi_3), \ \ \ \ \ \
\end{eqnarray}
where, $k_1$ and $k_2$ are the 4-momenta of the electron and position;
$p_i$ ($i$=1,6) are the 4-momenta of the produced $Q$ and $\bar{Q}$ quarks, with $p_1=p_2=p_3$; $\xi_j$ and $\chi_j$($j=$1,2,3) are the color indices of the
$Q_{j}$-quarks and $\bar{Q}_{j}$-quarks, respectively. The produced three pairs of heavy quarks in process (\ref{anni}) have six permutations denoted as, $(Q_1\bar{Q}_iQ_2\bar{Q}_jQ_3\bar{Q}_k)$ with $i,j,k$=1,2,3 and $i\neq j\neq k$.\footnote{$(Q_1\bar{Q}_iQ_2\bar{Q}_jQ_3\bar{Q}_k)$ ($i,j,k$=1,2,3 and $i\neq j\neq k$) means the quark $Q_1$ in the same fermion line with the antiquark $\bar{Q}_i$, the quark $Q_2$ in the same fermion line with the antiquark $\bar{Q}_j$ and the quark $Q_3$ in the same fermion line with the antiquark $\bar{Q}_k$.} In the calculation, we will disregard the contribution of the electroweak interaction between the heavy quarks for the process (\ref{anni}). As a result, there are seven inequivalent topology structures for each of the six permutations as the same as the one in \cite{Chen:2011mb}. So, there are 42 topology structures totally for the produced heavy quarks. Inserting the $e^+e^-\gamma^*$ vertex and $e^+e^-Z^*$ vertex into the 42 topology structures in all the allowed positions in the tree level, we get 576 Feynman diagrams for the process (\ref{anni}).

 As pointed out above, the color configuration of the produced three heavy quarks is $\frac{1}{\sqrt{6}}\varepsilon^{\xi_1\xi_2\xi_3}Q_{1\xi_1}Q_{2\xi_2}Q_{3\xi_3}$. Setting $T^a=\frac{\lambda^a}{2}$ with $\lambda^a $ $(a=1,\ldots,8)$ being the Gell-Mann matrices and using the color-flow method in \cite{Chen:2011mb}, we can get the color factors of the 576 Feynman diagrams. The color factors of the last two diagrams in Fig.\ref{9dth} are both,
 \begin{eqnarray}\label{co-fa7}
\sum_{a,b,c}\sum_{\xi_1,\xi_2,\xi_3}\frac{1}{\sqrt{6}}\varepsilon^{\xi_1\xi_2\xi_3}f^{abc}(T^a)_{\xi_l\chi_i}(T^b)_{\xi_m\chi_j}(T^b)_{\xi_n\chi_k}=0\;.\nonumber
\end{eqnarray}
in which, $f^{abc}$ ($(a,b,c=1,\ldots,8)$) is the structure constant of the $SU(3)$ group, and the index $\xi_i$ ($i=1,2,3$) appears twice, in $\varepsilon^{\xi_1\xi_2\xi_3}$ directly and as one of the indices $\xi_l$, $\xi_m$ and $\xi_n$ indirectly. And we see the result is independent onto the indices $i,j,k,l,m,$ and $n$. As a result, we get the conclusion that the total contribution to the amplitude of the process (\ref{anni}) from the 74 Feynman diagrams involving a three-gluon vertex vanishes because of the color factors of these Feynman diagrams are all zero. The number of the Feynman diagrams which we need to consider reduces to 504. The color factors of the first seven Feynman diagrams in Fig.\ref{9dth} are all the same,
\begin{eqnarray}\label{co-fa1}
\sum_{a,b}\sum_{\xi_1,\xi_2,\xi_3}\frac{1}{\sqrt{6}}\varepsilon^{\xi_1\xi_2\xi_3}(T^a)_{\xi_l\chi_i}&&(T^aT^b)_{\xi_m\chi_j}(T^b)_{\xi_n\chi_k} \nonumber \\
&&=(-1)^N\frac{4}{9}\frac{1}{\sqrt{6}}\varepsilon^{\chi_i\chi_j\chi_k}\;, \nonumber
\end{eqnarray}
in which $N$ is the number of permutations of transforming the set $(\xi_l,\xi_m,\xi_n)$ to the set $(\xi_1,\xi_2,\xi_3)$. For each of the Feynman diagrams, there is a Feynman factor $(-1)^{N^*}$ where $N^*$ equals the total number of permutations of transforming the set $(l,m,n)$ to the set $(1,2,3)$ and transforming the set $(i,j,k)$ to the set $(1,2,3)$. Subsuming the Feynman factor into the corresponding color factor, we find that all the color factors of the remaining 504 Feynman diagrams are the same,
\begin{eqnarray}\label{co-fa}
C_{col}=(-1)^{N^*}(-1)^N\frac{4}{9}\frac{1}{\sqrt{6}}\varepsilon^{\chi_i\chi_j\chi_k}=\frac{4}{9}\frac{1}{\sqrt{6}}\varepsilon^{\chi_1\chi_2\chi_3}\;.\nonumber
\end{eqnarray}

\begin{figure*}
\centering
\includegraphics[width=0.42\textwidth]{1fd.eps}
\includegraphics[width=0.42\textwidth]{2fd.eps}\\
\includegraphics[width=0.42\textwidth]{3fd.eps}
\includegraphics[width=0.42\textwidth]{4fd.eps}\\
\includegraphics[width=0.42\textwidth]{5fd.eps}
\includegraphics[width=0.42\textwidth]{6fd.eps}\\
\includegraphics[width=0.42\textwidth]{7fd.eps}
\includegraphics[width=0.42\textwidth]{8fd.eps}\\
\includegraphics[width=0.42\textwidth]{9fd.eps}
\caption{Nine typical Feynman diagrams for the process (\ref{anni}). The indices $i,j,k,l,m,n=1,2,3$ (with $i\neq j\neq k$ and $l\neq m\neq n$). }\label{9dth}
\end{figure*}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{3cpt.eps}
\caption{Distributions of the differential production cross section versus the transverse momentum of the $\Omega_{ccc}$ produced at CEPC with $\sqrt{s}=91.2\; {\rm GeV}$. }\label{3cpt}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{3bpt.eps}
\caption{Distributions of the differential production cross section versus the transverse momentum of the $\Omega_{bbb}$ produced at CEPC with $\sqrt{s}=91.2\; {\rm GeV}$. }\label{3bpt}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{3cs.eps}
\caption{The production cross section of the $\Omega_{ccc}$ produced by $e^+e^-$ annihilation at different energies. }\label{3c}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{3bs.eps}
\caption{The production cross section of the $\Omega_{bbb}$ produced by $e^+e^-$ annihilation at different energies.}\label{3b}
\end{figure}

Now, let's consider the remained 504 Feynman diagrams. For each one of the six permutations given above, $(Q_1\bar{Q}_iQ_2\bar{Q}_jQ_3\bar{Q}_k)$ with $i,j,k$=1,2,3, there are 84 Feynman diagrams, in which 42 ones correspond to the process $e^-+e^+\to Z^* \to Q_1Q_2Q_3\bar{Q}_1\bar{Q}_2\bar{Q}_3$ and the other 42 ones correspond to the process $e^-+e^+\to \gamma^* \to Q_1Q_2Q_3\bar{Q}_1\bar{Q}_2\bar{Q}_3$. In our calculation, the momenta of the produced identical heavy quarks are considered to be equal. As a result, we find that the contribution of every 84 Feynam diagrams corresponding to each one of the six permutations are the same.  So, the total amplitude for the process $e^+e^-\to
(QQQ)_1^{(\frac{3}{2},S_Z)}\bar{Q}\bar{Q}\bar{Q}$
\begin{eqnarray} \label{m-c-ee}
  \mathcal{M}(e^+e^-\to
(QQQ)_1^{(\frac{3}{2},S_Z)}\bar{Q}\bar{Q}\bar{Q})&=&\sum_{k=1}^{504}C_{col} \,\Gamma_{k}  \nonumber \\
         &=&3!\sum_{k=1}^{84}C_{col} \,\Gamma^{'}_{k}\;,
\end{eqnarray}
where $\Gamma_{k}$ ($k=1,2....,504$) are the matrices without color factors of the Feynman diagrams, and $\Gamma^{'}_{k}$ ($k=1,2....,84$) are the matrices without color factors of the 84 Feynman diagrams corresponding to the permutation $(Q_1\bar{Q}_1Q_2\bar{Q}_2Q_3\bar{Q}_3)$.

\section{Numerical results and discussions}\label{sec:NANDC}

Now, the cross section of the progress (\ref{anni}) can be written as
\begin{eqnarray}
 \sigma=&&\frac{1}{3!}\int\sum_{S_Z,\varsigma_i}\frac{(2\pi)^4}{2\hat{s}}
  \delta^4(k_1+k_2-P-p_4-p_5-p_6)
\nonumber\\
 &&d\Pi_4\frac{1}{4}\sum_{s_{1},s_{2},\chi_i}|\mathcal{A}(e^+e^-\to
  \Omega_{QQQ}\bar{Q}\bar{Q}\bar{Q})|^2,
\end{eqnarray}
with
\begin{eqnarray}
d\Pi_4=\frac{d^3P}{(2\pi)^32E}\frac{d^3p_4}{(2\pi)^32E_{p_4}}
   \frac{d^3p_5}{(2\pi)^32E_{p_5}}
   \frac{d^3p_6}{(2\pi)^32E_{p_6}}, \ \ \ \ \nonumber
\end{eqnarray}
where $\varsigma_i$ ($i=1,2,3$), $s_1$ and $s_2$ are the spins of the antiquarks $\bar{Q}_i$, $e^+$ and $e^-$, respectively.

To do the numerical calculation, the parameters are taken as follows:
\begin{eqnarray} \label{parame}
 &&m_c=1.5\,{\rm GeV}, \; m_b=4.9\,{\rm GeV},\; m_Z=91.18\,{\rm GeV}, \nonumber \\
  && sin^2\theta_w=0.224,\; \Gamma_{_Z}=2.49\,{\rm GeV}, \;  \alpha(m_Z)=1/127.95,\;\nonumber \\
 && |\Psi_{\Omega_{ccc}}(0,0)|^2=0.36\cdot10^{-3}\,{\rm GeV^6} \;\nonumber \\
 &&  \text{and}\ \   |\Psi_{\Omega_{bbb}}(0,0)|^2=0.189\,{\rm GeV^6}.
\end{eqnarray}

For the electromagnetic coupling constant, we adopt
\begin{eqnarray}
  \alpha(q)&=&\frac{\alpha(m_Z)}{1-\frac{\alpha(m_Z)}{3\pi}{\rm log}(\frac{q^2}{m_Z^2})},
\end{eqnarray}
 where, $q$ denotes the energy scale of the electromagnetic coupling constant, and we take $q=\frac{\sqrt{s}}{2}$ in this paper with $\sqrt{s}$ being the colliding energy in the center-of-mass frame. And the strong coupling constant, we adopt
\begin{eqnarray}
  \alpha_s(\mu)&=&\frac{\alpha_s(m_Z)}{1+\frac{b_0}{2\pi}\alpha_s(m_Z){\rm log}(\frac{\mu}{m_Z})},\\
  \text{with} \ \    b_0&=&11-\frac{2}{3}n_f,\nonumber
\end{eqnarray}
where $\mu$ is the energy scale and
$\alpha_s(m_Z)=0.118$. For the production of the $\Omega_{ccc}$, $n_f$ is taken to be 4 when $\mu\leq 2m_b$
and 5 when $\mu$ is larger than 2$m_b$. And for the production of the $\Omega_{bbb}$, $n_f$ is taken to be 5. For comparison, we take two
different values for $\mu$, i.e., $\mu$=$\sqrt{s}/2$ and $\mu$=$\mu_R/2$, where $\mu_R$ is the transverse mass of the produced baryon, namely
$\mu_R^2$=$p_T^2$+$M^2$.\footnote{Here, $p_T$ means the transverse momentum of the produced triply heavy baryon.}

SuperKEKB, the asymmetric beam energy $e^-e^+$ collider, is an upgrade of the KEKB accelerator facility. The target integrated luminosity of 50 $ab^{-1}$ to be collected by the Belle II experiment. And a high luminosity $e^+e^-$ collider named the Circular Electron-Positron Collider (CEPC) has been proposed by the Chinese particle physics community. The CEPC is designed to operate at three different modes, as a Higgs factory at $\sqrt{s}=240\,{\rm GeV}$, as a $Z$ factory at $\sqrt{s}=91.2\,{\rm GeV}$ and performing $WW$ threshold scans around $\sqrt{s}=160\,{\rm GeV}$ \cite{An:2018dwb}. We calculate the production cross sections of the baryons $\Omega_{ccc}$ and $\Omega_{bbb}$ at CEPC with the energy $\sqrt{s}=91.2\,{\rm GeV}$ and $\sqrt{s}=160\,{\rm GeV}$, and at Belle II with the center-of-mass energy $\sqrt{s}=10.58\,{\rm GeV}$. The results are shown in TABLE. \ref{cross1t}.

\begin{table}[htb]
\begin{tabular}{|c|c|c|c|c|}
\hline
-&$\sqrt{s}$&$\mu$&$\Omega_{ccc}$&$\Omega_{bbb}$ \\
\hline
CEPC&91.2 {\rm GeV}& $\sqrt{s}/2$&0.00204(5)&$0.332(4)\times10^{-3}$\\
-&-&$\mu_R/2$&0.0124(3)&$0.952(4)\times10^{-3}$\\
-&160 {\rm GeV}& $\sqrt{s}/2$&$0.214(9)\times10^{-5}$&$0.61(2)\times10^{-6}$\\
-&-&$\mu_R/2$&$0.108(9)\times10^{-4}$&$0.197(5)\times10^{-5}$\\
\hline
Belle II&10.58 {\rm GeV}& $\sqrt{s}/2$&$0.249(1)\times10^{-5}$&-\\
-&-&$\mu_R/2$&$0.707(3)\times10^{-5}$&-\\
\hline
\end{tabular}
\caption{Production cross sections (in unit $fb$) of the $\Omega_{ccc}$ and $\Omega_{bbb}$ at the CEPC and Belle II. }
\label{cross1t}
\end{table}

Here, we also calculate the differential cross sections $d \sigma/dp_T$ of the production of the $\Omega_{ccc}$ and $\Omega_{bbb}$ at $\sqrt{s}=91.2 \ \ {\rm GeV}$, which are shown in Figs. \ref{3cpt} and \ref{3bpt}. The line for the $p_T$ distribution are not smooth due to the Vegas simulation error which can be improved by increasing the event numbers. We calculate the production of the $\Omega_{ccc}$ and $\Omega_{bbb}$ at the $e^+e^-$ collider with different colliding energies, which are shown in Fig.\ref{3c} and Fig.\ref{3b}.

As we see that both the integrated cross sections and differential cross
 sections are proportional to $|\Psi(0,0)|^2$, $\alpha^2(q)$and $\alpha_s^4(\mu)$.
 So the numerical results can be changed by one even two orders using
different the wave function at the origin of the triply heavy
baryons, different running coupling constants and different energy
scale choices. We can get the conclusion from the numerical results shown in Figs.\ref{3c} and \ref{3b}, that the production cross sections of the $\Omega_{ccc}$ and $\Omega_{bbb}$ take their maximum values at $\sqrt{s}=m_Z$.
From the numerical results in Table \ref{cross1t}, we know that it is impossible to observe $\Omega_{ccc}$ at SuperKEKB, if there is no other models to produce them at the $e^+e^-$ colliders. Also, we see there are around 32-198 events for the production of $\Omega_{ccc}$ and 5-15 events for the production of $\Omega_{bbb}$, at CEPC with $16 \,ab^{-1}$ integrated luminosity running at $91.2 \, {\rm GeV}$.

\section{Summary}\label{sumra}
As a summary, we have studied the production of the $\Omega_{ccc}$ and $\Omega_{bbb}$ at the SuperKEKB and the CEPC. From the numerical results, we conclude that it is impossible to find the triply heavy baryons at SuperKEKB, and it is hard to find $\Omega_{ccc}$ and $\Omega_{bbb}$ at the CEPC because of the fewer events.

\hspace{2cm}

\noindent {\bf Acknowledgments:} The authors thank Doctor Jian-Wei Xu for helpful discussions and important suggestions on the manuscript. The
work of S. Z. Wu was supported by the Natural Science Foundation of China under Grants No. 11347200.