\section{Introduction}
In 1998 the paper "Unified approach to the classical statistical analysis
of small signals" [\citet{FaC1998}] appeared, in which the classical approach
to the construction of confidence intervals is discussed in detail.
The paper has received great attention
and has stimulated the discussion about the calculation
of confidence intervals. Two of the main results of the paper can be stated as
follows:
\begin{itemize}
  \item By using the likelihood ratio as ordering quantity one obtains
confidence intervals which automatically change over from two-sided intervals
to upper limits
as the signal becomes statistically less significant
("Unified approach").
  \item This eliminates undercoverage caused by basing this choice (of
quoting an upper limit or a confidence interval) on the data
("flip-flopping").
\end{itemize}

These attractive features have induced many experimenters to follow the
approach proposed by Feldman and Cousins.
The purpose of the present paper is to
point out possible misunderstandings of the above statements and to give
alternative solutions to the flip-flopping problem.

Another result of the paper [\citet{FaC1998}] is, that an approach which
uses of the likelihood ratio as ordering quantity yields finite and
physical intervals, in cases where central intervals are empty or unphysical.
This subject is not discussed here.
\section{Neyman construction of confidence intervals and coverage}
\label{Section:Neyman}
To be specific the same example is considered as in [\citet{FaC1998}], a Poisson
process with background, where the mean background is known:
\begin{eqnarray}
P(n|\mu )\;=\;\dfrac{(\mu +b)^n}{n!}\times \exp[-(\mu +b)]
\label{eq:Pnmu}
\end{eqnarray}
\begin{tabular}{ll}
$n$           & is the measured number of signal plus background events \\
$\mu$         & is the average number of signal events \\
$b$           & is the average number of background events,
                $b$ is assumed to be exactly known \\
$P(n|\mu )$   & is the probability of measuring $n$, given $\mu$ and $b$
\end{tabular} \\

\noindent For the discussion in this Section and for later comparisons
with the results from [\citet{FaC1998}], the likelihood ratio
\begin{eqnarray}
R(n,\mu)\;=\;\dfrac{P(n|\mu )}{P(n|\mu_{best})}
\label{eq:likratio}
\end{eqnarray}
\noindent is assumed as ordering principle, when constructing the
acceptance region in
$n$ for an individual $\mu$. $\mu_{best}$ is the value of $\mu$ which maximizes
$P(n|\mu )$, at fixed $n$, where only physically allowed values of $\mu$ are
considered. For a given $\mu$, values of $n$ are added to the {\bf acceptance
region} $[n_1(\mu),\;n_2(\mu)]$ in decreasing order of the ratio $R(n,\mu)$, until
the sum $p\;=\;\sum_n P(n|\mu)$
agrees with the desired confidence level $\alpha$. Such an ordering
principle is naturally implied by the theory of likelihood ratio tests.

For a given measurement $n$,
the {\bf confidence interval} $I\;=\; [\mu_1,\;\mu_2]$ for $\mu$ is found by
including in $I$ all those $\mu$ which contain $n$ in their acceptance region.
The limits $\mu_1$ and $\mu_2$ obviously depend on $n$. The set of
confidence intervals $[\mu_1(n),\;\mu_2(n)]$ is called {\bf confidence belt}.

Upper limits of $\mu$ are determined in the same way as confidence intervals,
except that the ordering quanity is now $R(n,\mu)\;=\;n$. The resulting
"acceptance regions" are one-sided in this case: $[n_{low}(\mu), \infty]$.
The upper limit
$\mu_{up}(n)$ of $\mu$ at fixed $n$ is given by that $\mu$, for which the
lower edge $n_{low}(\mu)$ of the
acceptance region coincides with $n$. In general, upper limits of $\mu$ at a
certain confidence level differ from the upper ends of confidence intervals
at the same confidence level (see below).

Considering an ensemble of experiments with arbitrary but fixed $\mu$,
this fixed value of
$\mu$ will be contained in the confidence belt in a fraction $\alpha$ of all
experiments. This follows from the way the confidence intervals are
constructed: Given $n$, the confidence interval is the set of all those $\mu$
which
contain $n$ in their acceptance region. Confidence intervals with this property
 are said to have exact
coverage. Coverage is an important feature of Neyman-constructed confidence
intervals [\citet{Neyman1937}].

For upper limits the corresponding statement reads: Given an ensemble of
experiments with arbitrary but fixed $\mu$, this fixed value of $\mu$
will be less than
$\mu_{up}(n)$ in a fraction $\alpha$ of all experiments. In the following,
this definition of upper limit will be called "standard definition", and the
corresponding upper limit as "{\bf standard upper limit}".

It is evident that for both the confidence belt and for the standard upper limit
the specification of the confidence level is essential. For the confidence belt,
the ordering principle is relevant in addition. The ordering principle for
the standard upper limit is fixed.

In the present example small under or overcoverage occurs due to the
discreteness of $n$. In order to avoid this problem, in the calculations
presented below the factorial in the discrete Poisson function
$P(n|\mu )$ is replaced by Euler's Gamma function, which is then
normalized to 1. Euler's Gamma function, which is defined for all real
values of $n$, agrees with the discrete Poisson function at all integer
arguments $n$. The normalization factor differs from 1 by less than
0.1\% for $\mu\geq 3$, the maximum difference occuring at $\mu = 0$,
where it is less than 2 \%.

It should be stressed that it is not proposed to do this replacement in
practical applications. It is applied here only for the purpose of this paper,
to allow a discussion which is not affected by under or overcoverage due to
the discreteness of $n$.

In the general Neyman construction the ordering principle is not specified. In
the present paper, besides the ordering principle based on the likelihood ratio
(\ref{eq:likratio}) also the ordering principle based on central intervals is
considered.
\section{Discussion of Neyman-constructed confidence intervals and standard upper
limits}
\label{section:neyman}
In the previous Section the construction of confidence intervals and of
standard upper
limits is fully defined. In order to understand that this procedure of
defining confidence intervals and standard upper limits
ensures coverage, it is instructive to consider how
coverage would be tested by Monte Carlo simulations: One would generate
an ensemble of experiments with fixed $\mu_0$, throwing $n$ according to
eq.(\ref{eq:Pnmu}). An acceptance region $[n_1(\mu_0),\;n_2(\mu_0)]$
would be defined as explained in the previous Section. Although the confidence
belt is not yet completely defined by knowing the acceptance region for a single
$\mu_0$, one can already say that the confidence belt would contain $\mu_0$ in a
fraction $\alpha$ of all cases, because for those $n$ which lie in the
acceptance region of $\mu_0$ (which happens in a fraction $\alpha$ of all
cases)
$\mu_0$ would be added to the confidence interval  $[\mu_1(n),\;\mu_2(n)]$.
These statements are valid for all physically allowed values of $\mu$. One
can conclude that coverage is confirmed.

Thus, the way these confidence intervals and standard upper limits
are defined ensures exact coverage (disregarding effects which occur when the
problem involves discrete values). Obviously, the construction is independent
of the decision, whether to quote a standard upper limit, a confidence interval
or both.
Therefore, also coverage is guaranteed independent of such a decision.
The experimenter may thus decide to quote a standard upper limit, a confidence
interval or both, at equal or different confidence levels, without violating coverage.
Here, "coverage" is understood as explained in Section \ref{Section:Neyman}.
Coverage in a very specific scenario is discussed in Section
\ref{section:flip-flopping}.

\begin{table}
\caption{\label{tab:prob}Integrated probabilities $p_1$, $p$
and $p_2$ below, in an above the acceptance regions respectively,
as functions of $\mu$.The confidence
level chosen is $\alpha$ = 90\%.}
\begin{tabular}{c|ccc}
 $ \;\;\;\;\mu\;\;\;\;$    &  $\;\;\;\;\;p_1\;\;\;\;$  &  $\;\;\;\;\;p\;\;\;\;$
&  $\;\;\;\;\;p_2\;\;\;\;$  \\
\hline
  0.0   &     0.000   &     0.9   &     0.100  \\
  0.5   &     0.000   &     0.9   &     0.100  \\
  1.0   &     0.016   &     0.9   &     0.084  \\
  1.5   &     0.035   &     0.9   &     0.065  \\
  2.0   &     0.046   &     0.9   &     0.054  \\
  2.5   &     0.053   &     0.9   &     0.047  \\
  3.0   &     0.056   &     0.9   &     0.044  \\
  3.5   &     0.057   &     0.9   &     0.043  \\
  4.0   &     0.057   &     0.9   &     0.042  \\
  4.5   &     0.057   &     0.9   &     0.043  \\
  5.0   &     0.056   &     0.9   &     0.044  \\
  5.5   &     0.056   &     0.9   &     0.044  \\
  6.0   &     0.056   &     0.9   &     0.044  \\
  6.5   &     0.056   &     0.9   &     0.044  \\
  7.0   &     0.056   &     0.9   &     0.044  \\
  7.5   &     0.056   &     0.9   &     0.044  \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{\label{tab:limits}Lower and upper edge ($n_1$ and $n_2$)
of the 90\% c.l. acceptance region of $n$, 90\% c.l. lower limit and
95\% c.l. lower limit of $n$, as functions of $\mu$.}
\begin{tabular}{c|cc|cc}
  $\;\;\;\mu\;\;\;$    &  $\;\;\;\;n_1\;\;\;$  &  $\;\;\;\;n_2\;\;\;$
&  $\;\;\;n_{90\%}(\mu)\;$ &  $\;\;n_{95\%}(\mu)\;\;$ \\
\hline
  0.0   &     0.00   &     5.34   &     1.00  & 0.62 \\
  0.5   &     0.00   &     6.01   &     1.28  & 0.84 \\
  1.0   &     0.53   &     6.91   &     1.59  & 1.09 \\
  1.5   &     1.13   &     7.92   &     1.92  & 1.36 \\
  2.0   &     1.59   &     8.85   &     2.26  & 1.65 \\
  2.5   &     2.01   &     9.70   &     2.61  & 1.96 \\
  3.0   &     2.39   &    10.48   &     2.98  & 2.28 \\
  3.5   &     2.75   &    11.20   &     3.35  & 2.61 \\
  4.0   &     3.09   &    11.87   &     3.72  & 2.95 \\
  4.5   &     3.43   &    12.51   &     4.10  & 3.29 \\
  5.0   &     3.78   &    13.15   &     4.49  & 3.65 \\
  5.5   &     4.14   &    13.79   &     4.87  & 4.00 \\
  6.0   &     4.50   &    14.43   &     5.26  & 4.36 \\
  6.5   &     4.87   &    15.06   &     5.66  & 4.73 \\
  7.0   &     5.23   &    15.69   &     6.05  & 5.09 \\
  7.5   &     5.61   &    16.32   &     6.45  & 5.47 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{\label{tab:limitsofmu}Lower and upper edge ($\mu_1$ and $\mu_2$)
of the 90\% c.l. confidence interval of $\mu$, 90\% c.l. standard upper limit and
95\% c.l. standard upper limit of $\mu$, as functions of $n$.}
\begin{tabular}{c|cc|cc}
  $\;\;\;n\;\;\;$    &  $\;\;\;\;\mu_1\;\;\;$  &  $\;\;\;\;\mu_2\;\;\;$
&  $\;\;\;\mu_{90\%}(n)\;$ &  $\;\;\mu_{95\%}(n)\;\;$ \\
\hline
  0.0   &     0.00   &     0.61   &     0.00   &     0.00 \\
  0.5   &     0.00   &     0.98   &     0.00   &     0.00 \\
  1.0   &     0.00   &     1.38   &     0.02   &     0.86 \\
  1.5   &     0.00   &     1.90   &     0.87   &     1.75 \\
  2.0   &     0.00   &     2.50   &     1.64   &     2.58 \\
  2.5   &     0.00   &     3.16   &     2.36   &     3.35 \\
  3.0   &     0.00   &     3.87   &     3.04   &     4.08 \\
  3.5   &     0.00   &     4.59   &     3.71   &     4.81 \\
  4.0   &     0.00   &     5.30   &     4.39   &     5.50 \\
  4.5   &     0.00   &     6.01   &     5.04   &     6.21 \\
  5.0   &     0.00   &     6.69   &     5.67   &     6.89 \\
  5.5   &     0.11   &     7.36   &     6.31   &     7.56 \\
  6.0   &     0.49   &     8.04   &     6.94   &     8.21 \\
  6.5   &     0.81   &     8.69   &     7.56   &     8.89 \\
  7.0   &     1.04   &     9.34   &     8.19   &     9.54 \\
  7.5   &     1.29   &    10.01   &     8.81   &    10.19 \\
  8.0   &     1.54   &    10.64   &     9.41   &    10.84 \\
  8.5   &     1.81   &    11.29   &    10.03   &    11.47 \\
  9.0   &     2.08   &    11.93   &    10.64   &    12.11 \\
  9.5   &     2.37   &    12.56   &    11.24   &    12.74 \\
\hline
\end{tabular}
\end{table}

It should also be noted that coverage has to be obeyed at fixed $\mu_0$, and
for coverage it is irrelevant how far $\mu_0$ is away from the edges
$\mu_1(n)$ and $\mu_2(n)$ or whether $\mu_1(n)$ is given by the lowest
physically allowed value of $\mu$. This is characteristic for a Frequentist
approach. The discussion in Section \ref{section:unifiedapproach} will refer
to this feature.

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{FaC2uplneu
\caption{\label{fig:FaC}90\% c.l. confidence belt (full circles), 90\%
c.l. standard upper limit (stars) and 95\% c.l. standard upper limit (open circles)
for the average number of signal events $\mu$
in the presence of a Poisson background with known mean $b=3.0$.}
\end{figure}

Assuming a confidence level of $\alpha$ = 90\%, setting $b$ equal to 3
and using the likelihood ratio as ordering quantity,
one obtains for the present example the confidence belt shown in
Fig.\ref{fig:FaC} (full circles).
Also shown in Fig.\ref{fig:FaC} are the 90\% c.l. standard
upper limit (stars) and the 95\% c.l. standard upper limit (open circles)
of $\mu$.

As can be seen, the 90\% c.l. standard upper limit and the 90\% c.l. confidence
belt in Fig.\ref{fig:FaC} are perfectly compatible with the corresponding limit
and confidence belt respectively in Figs. 5 and 7 of [\citet{FaC1998}].
Replacing the discrete Poisson distribution by the renormalized Euler's
function is therefore a valid procedure, for the purpose of this paper.

Table \ref{tab:prob} gives for each $\mu$ the integrated probabilities
below, in and above the acceptance region of $n$:
$p_1=\int_{n<n1}P(n|\mu ){\rm d}n$, $p=\int_{n1\leq n \leq n2}P(n|\mu ))
{\rm d}n$, $p_2=\int_{n>n2}P(n|\mu )){\rm d}n$. By construction, $p$ is
always equal to $\alpha$ = 90\%. As a result of the ordering principle chosen,
$p_1$ is generally different from $p_2$, $p_1$ being
less than $p_2$ for $\mu\;<\;2.4$ and greater than $p_2$ for
 $\mu\;>\;2.4$.

In Table \ref{tab:limits} the lower and upper edge ($n_1$ and $n_2$)
of the 90\% c.l. acceptance region of $n$, the 90\% c.l. lower limit
$n_{90\%}(\mu)$ of $n$ and the 95\% c.l. lower limit $n_{95\%}(\mu)$
of $n$ are given as functions of $\mu$. The inverse functions
of $n_1(\mu)$, $n_2(\mu)$, $n_{90\%}(\mu)$ and $n_{95\%}(\mu)$
are the functions  $\mu_2(n)$, $\mu_1(n)$, $\mu_{90\%}(n)$ and
 $\mu_{95\%}(n)$ respectively, which define the confidence intervals
and standard upper limits of $\mu$ as functions of $n$. They are listed in Table
\ref{tab:limitsofmu}.

By definition, for a fixed $\mu_0$, $n$ lies within the acceptance region
$[n_1(\mu_0),\;n_2(\mu_0)]$ if and only if $\mu_0$ lies in the confidence
belt $[\mu_1(n),\;\mu_2(n)]$. This means that the probability that
$[\mu_1(n),\;\mu_2(n)]$ covers the value $\mu_0$ is equal to $p$ = $\alpha$.

Also by construction, $n>n_1(\mu )$ if and only if $\mu < \mu_2(n)$, implying
that the probability for $\mu < \mu_2(n)$ is equal to $p+p_2$. Thus
$[\mu_1(n),\;\mu_2(n)]$ is a confidence interval at the $\alpha$-confidence
level.
If $\mu_2(n)$ is interpreted as a standard upper limit, its coverage  $(p+p_2)$
depends strongly
on $\mu$, making this interpretation problematic.
The situation is different for central intervals, for
which $p_1= (1-\alpha )/2$, $p$ = $\alpha$ and
$p_2=(1-\alpha )/2$. In the case of central intervals, $\mu_2$ is
a standard upper limit at the confidence level $p+p_2\;=\;(1+\alpha)/2$, at all $\mu$.

\section{The flip-flopping problem}
\label{section:flip-flopping}
In this Section a specific scenario is considered,
in which the experimenter decides to quote a standard upper limit at the c.l. $\beta$ if $n$
is below and a confidence interval at the c.l. $\alpha$ if $n$ is above a certain
value $n_0$. It should be noted that this very specific scenario is different from the
scenario which is assumed when constructing confidence intervals or standard
upper limits
(see Section \ref{Section:Neyman}). In the latter scenario one talks independently
about coverage for a Neyman-constructed confidence interval or coverage for a
Neyman-constructed standard upper limit, and there is no condition involving the
measurement $n$. Therefore, the coverages for the two scenarios
are not expected to be identical. In addition, the coverage $\gamma$ for the specific
scenario depends on the choice of $\alpha$ and $\beta$. For central intervals all
coverages of interest can be directly given, without the need of complicated calculations.

In [\citet{FaC1998}] the special case $\alpha = \beta = 90$ \% is discussed for
an approach with central intervals. One obtains a coverage of
$\gamma =  \beta -p_2=\alpha -p_2= (3\alpha -1)/2=85$ \%, at low $\mu$.
The fact that $\gamma\neq \alpha$ is called
in [\citet{FaC1998}] violation of coverage, or flip-flopping problem. Talking of
"violation of coverage" means that
a coverage of $\gamma = \alpha$ is expected. As pointed out above, there is no
justification for that.

The actual problem is that $\gamma$ is different for different regions of $\mu$.
If $\gamma$ were the same for all $\mu$, one could quote a unique confidence level
($=\gamma$) for the specific scenario and there would be no problem.

The behaviour $\gamma\neq \alpha$ at low $\mu$ is due to the fact that
with the chosen
alternative between a standard upper limit (at a c.l. $\beta$=90 \%)
and a confidence interval (at the c.l. $\alpha$=90\%) the upper edge
$\mu_2(n)$ of the confidence belt, in the region $n<n_0$, is shifted. Knowing this,
it is easy to find a way of avoiding the flip-flopping problem: One has to define
alternatives which preserve the upper edge $\mu_2(n)$, at all $\mu$. In the case of
central intervals the alternatives are a $(1+\alpha)/2=$
95\% c.l. standard upper limit and a $\alpha=$
90\% c.l.confidence interval. With this choice the upper edge $\mu_2(n)$ of the
confidence belt is not changed, because for central intervals $\mu_2(n)$, if
interpreted as standard upper limit, has a c.l. of 95\%. Since the upper edge of
the confidence belt is preserved, coverage is fulfilled. Thus the flip-flopping problem
can be easily avoided by choosing appropriate confidence levels for the standard
upper limit and the confidence intervals.
\section{The approach by Feldman \& Cousins}
\label{section:unifiedapproach}
In the approach described in [\citet{FaC1998}] a confidence belt is determined according
to a Neyman construction, where the likelihood ratio is used as ordering quantity.
Coverage is therefore guaranteed for the confidence level $\alpha$ chosen:
If an experiment with fixed $\mu_0$ were repeated many times one would obtain
a set of confidence intervals $[\mu_1(n), \mu_2(n)]$, and the relation
$\mu_1(n)<\mu_0 < \mu_2(n)$ would be correct in a fraction $\alpha$ of all cases.
Because for small $n$ the lower edge
$\mu_1(n)$ coincides with the physical limit of $\mu$, the set of
relations (which define the confidence belt) can also be written as
\begin{eqnarray}
\qquad\qquad \mu_0\;<\;\mu_2(n)\qquad\qquad {\rm for}\;\; n\leq n_0 \qquad\qquad
                                             {\rm (a)} \nonumber\\
                                             \label{eq:FCuplim} \\
\qquad\qquad \mu_1(n)\;<\;\mu_0\;<\;\mu_2(n)\qquad\qquad {\rm for}\;\; n>n_0 \qquad\qquad
                                             {\rm (b)} \nonumber
\end{eqnarray}
where $n_0$ is the largest $n$ for which $\mu_1(n)$ coincides with the physical
limit. For $n\leq n_0$, $\mu_2(n)$ is called "upper limit" in  [\citet{FaC1998}], and
it will be called F\&C limit in the following: the F\&C limit is defined as the
upper edge of a confidence interval, whose lower edge coincides with the physical
limit of $\mu$.
Again, out of all relations (\ref{eq:FCuplim}) a fraction $\alpha$ of them would
be staisfied, and coverage would be fulfilled. However, one should not call the F\&C limit
an "$\alpha$ c.l. upper limit", because the relation  $\mu_0\;<\;\mu_2(n)$ is not
satisfied in a fraction $\alpha$ of all cases, and, moreover, this fraction depends on $\mu_0$.

In [\citet{FaC1998}] the presence of two kinds of intervals ((a) and (b))
is expressed as "This choice (of using the likelihood ratio as ordering quantity)
yields intervals which automatically change over from upper limits to two-sided
intervals as the signal becomes more statistically significant (Unified approach)".

This is correct, when the term "upper limit" is understood as the F\&C limit.
One should note, however, that this is not a special feature of
an approach which uses the likelihood ratio as ordering quantity.
Approaches with other ordering
principles, like central intervals, symmetric intervals or highest-probability
intervals, exhibit the same behaviour. The term "Unified approach" is therefore
equally well justified for these approaches.

The approach in [\citet{FaC1998}] is also presented as a solution to the
flip-flopping problem (see Section \ref{section:flip-flopping}). This could be
understood such that the approach in [\citet{FaC1998}] provides an alternative between a
standard upper limit and a confidence interval. However, this approach rather provides
an alternative between the F\&C limit and a confidence interval. The F\&C limit
can therefore be easily misunderstood as a standard upper limit.

As noted in Section \ref{section:neyman}, in a Frequentist approach
the value of the physical limit of $\mu$ is irrelevant, except that the
acceptance regions for $n$ can only be determined for physically allowed
values of $\mu$. The definition of "upper limit" as the upper edge of a
confidence interval whose lower edge coincides with the physical limit, is therefore
not a Frequentist type of definition and may lead to confusion.
In order to exclude any misunderstanding it is recommended in the present paper
to call the F\&C limit "upper edge of the confidence belt", even if the lower
edge coincides with the physical limit of $\mu$.
Differences between the F\&C limit and the standard upper limit were also pointed
out by the authors of [\citet{FaC1998}].

From this discussion it follows, that the coverage properties of the F\&C limit,
if interpreted as a standard upper limit, are of great interest.
From Table \ref{tab:prob} one can see that (for $\alpha=90$\%) this coverage $(p+p_2)$
varies between 94,4 \% and 100 \%. Compared to
the confidence level $\alpha$ of the confidence belt, the F\&C limit
$\mu_2$, if interpreted as standard upper limit,
is very conservative, with a coverage around $(1+\alpha)/2$=95\% or an
overcoverage around $(1+\alpha )/2-\alpha =(1-\alpha )/2$=5\%.

In the case of central intervals the upper edge of the confidence interval has
the nice property that it is a standard upper limit with a well-defined confidence
level, namely $(1+\alpha)/2$, at all $\mu$.

Coming back to the flip-flopping problem: In the approach by F\&C this problem
is avoided, because introducing the alternative between an F\&C limit and a
confidence interval doesn't change the original confidence belt,
for which coverage is fulfilled. This is analogous
to the approach with central intervals, where the original confidence belt
is preserved by choosing the alternative between a $(1+\alpha)/2$ c.l. standard upper limit
and a $\alpha$ c.l. confidence interval (see Section \ref{section:flip-flopping}).
\section{Summary and Conclusions}
The paper discusses two of the main results of [\citet{FaC1998}].
One of them concerns the so-called flip-flopping problem, which according to
[\citet{FaC1998}] exists for central intervals in the specific scenario, where
the experimenter decides to quote a standard upper limit or a confidence interval,
depending on the observed data. In the present paper it is shown that this
problem is due to an expectation which is not justified. The problem
can be easily avoided by choosing appropriate confidence levels
for the standard upper limit and the confidence belt.

Another result of [\citet{FaC1998}] concerns the Unified approach.
In [\citet{FaC1998}] the term "upper limit" denotes the upper edge of a
confidence interval whose lower edge coincides with the physical limit of $\mu$.
With this definition of upper limit, upper ends of confidence intervals
automatically change over to "upper limits" as the signal becomes weaker.
This transition from confidence
intervals to upper limits is not a special
property of the approach using the likelihood ratio as ordering quantity, because
the same transition takes place in approaches with different ordering priciples,
like central intervals, symmetric intervals or highest-probability intervals.
Thus the term "Unified approach" is equally well justified for these approaches.
The approach with central intervals has the additional advantage that the upper end
of the confidence interval
is a standard upper limit at a well defined confidence level, at all $\mu$.

The Unified approach is presented in the F\&C paper as a solution to the
flip-flopping problem, where the experimenter quotes alternatively a
standard upper limit or a confidence interval, depending on the measurement.
The F\&C limit can therefore easily be misunderstood as a standard
upper limit. For this reason the coverage properties of the F\&C limit are
investigated. It is shown that the coverage of the F\&C limit,
if it is interpreted as standard upper limit,
depends strongly on the parameter $\mu$ to be determined.
Compared to the confidence level $\alpha$ of the confidence belt,
the upper limit is very conservative, with a coverage around $(1+\alpha )/2$ or
an overcoverage around $(1+\alpha )/2-\alpha =(1-\alpha )/2$. More precisely,
for $\alpha\;=\;90$ \%, the coverage of the upper limit varies between
94,4 \% and 100 \%, and a unique confidence level cannot be assigned to it.
Differences between the F\&C
limit and a standard upper limit were already pointed out in
[\citet{FaC1998}]. In order to exclude any misunderstanding, it is
proposed in the present paper to call the F\&C limit "upper edge of
the confidence interval", even if its lower edge coincides with the
physical limit.

In this paper only one example was discussed in detail. The conclusions and
recommendations, however, are also valid for any other application, in which
confidence and upper limits are determined using the likelihood ratio
as ordering quantity. Though, the applications are restricted to the cases
with only one unknown ($\mu$) and no nuisance paramter, except if the
nuisance parameter is exactly known ($b$).

Finally it should be emphasized that the criticism expressed in this paper
does not refer to the Neyman construction of confidence intervals itself
or to the use of the likelihood ratio as ordering quantity.


\section{Introduction}
\label{sec:1}

Studies of galaxy populations much benefit from the coordinated efforts of multiwavelength observations, which allow for a deeper insight into different galaxy properties and evolution. One of the best examples of these coordinated efforts is the {\em Great Observatories Origins Deep Survey (GOODS)} (P.I.: M. Giavalisco), with  320 arcmin$^2$ of the sky fully covered with deep observations from X-rays through radio wavelengths. Both North and South GOODS fields have been designed to be a unique case of {\em HST} coverage: they have deep and homogeneous maps in four broad-band {\em Advanced Camera for Surveys (ACS)} filters $B, V, I_{775}$ and $z_{850}$. The large amount of science produced over the last years using GOODS data demonstrates the importance of {\em HST} Treasury and other major-telescope Legacy programs.

Near and mid-IR selected galaxies constitute quite unbiased tracers of two different astrophysical properties: stellar mass and star-formation/AGN activity, respectively.   Although, in general, these galaxies populations are studied separately, a joint analysis of results allows to better understand the interplay between star formation, AGN activity and stellar-mass growth at different cosmic times. In the study of near- and mid-IR-selected galaxies, deep {\em HST} data have had a major role in the following aspects: 1) the determination of  optical spectral energy distributions (SEDs), 2) the computation of precise ($\sigma \approx 3-5\%$) photometric redshifts, as a complement to existing spectroscopic redshifts and 3) galaxy morphology. The two former aspects have been of key importance to conduct complete studies of galaxy luminosity evolution and stellar mass assembly.

\section{Near-IR-selected galaxies}
\label{sec:2}

An important landmark in the study of near-IR-selected galaxies has been the K20 survey \cite{cim02}, which included part of the GOODS-South among their fields. The first science results obtained by the K20 survey have soon been extended after the progressive public releases of the deeper {\em ISAAC-VLT} $J_s, H$ and $K_s$-band data in the GOODS-South (P.Is.: C. Cesarsky, E. Giallongo). Finally, after the launch of {\em Spitzer} in 2003 (P.I.: M. Werner), both GOODS fields have been observed with {\em IRAC} at 3.6, 4.5, 5.8 and 8.0 $\rm \mu m$ (P.I.: M. Dickinson).

At the typical depths of the GOODS data (21-22 Vega mag), $K_s$-selected galaxies mostly span the redshift range $z=0-1.5$, but  $\sim 20\%$ of these galaxies are found at higher $z=1.5-5.0$ redshifts \cite{cap05}\cite{cap06k}\cite{gra06}. The latter include Extremely Red Galaxies (ERGs),  which have been the subject of many studies looking for the progenitors of present-day massive ellipticals \cite{cap05}\cite{smail02}\cite{roch03}\cite{cap04}\cite{roch06}.

The evolution of the rest-frame $K_s$-band luminosity function (LF) has been studied up to $z \approx 1.5$ with the K20 survey \cite{pozz03}, and then extended up to redshift $z \approx 2.5$ using GOODS and other datasets \cite{cap05} \cite{cap06k} \cite{sar06}\cite{arn07}. All these works agree in a  mild but still significant increase of the bright end of this LF from $z\approx0$ to $z\approx2.0-2.5$. At the same time, the overall number density of galaxies decreases,  making the near-IR luminosity density to be nearly constant with redshift (cf. Figure \ref{fig1}).

Multiwavelength SED modelling has also allowed to obtain stellar-mass estimates for near-IR-selected galaxies. The redshift evolution of galaxy number densities is differential with stellar mass: while the density of moderately massive galaxies ($M\sim 1 \times10^{11} \, \rm M_\odot$) continuosly increases from redshift $z\sim4$ down to $z\sim0.5$, virtually all of the most massive ($M>2-3 \times 10^{11} \, \rm M_\odot$) galaxies appear to be in place by $z\sim1.5-2.0$ \cite{cap06k}\cite{sar05}\cite{tho05}\cite{fran06}\cite{ren06}\cite{pozz07}. The formation epoch of the most massive galaxies seems to be constrained to the redshift range $z\sim 2-4$, as no evidence of such galaxies has been found at $z>4$ in the GOODS-South field, indicating that these objects must be very rare (if any exists) at very high redshifts \cite{dun07}.

Moreover, during the last years, a general consensus has  been achieved on the redshift evolution of the total stellar mass density:  half of the stellar mass has been assembled in galaxies before the Universe was 30-40\% of its present age \cite{cap05}\cite{cap06k}\cite{fon04}.

\begin{figure}
\centering
\includegraphics[width=9cm,angle=270]{nearmidirldens.eps}
\caption{The evolution of the IR luminosity densities associated with the stars already locked in galaxies (rest-$K_s$) and with on-going star formation ($5-1000 \, \rm \mu m$). The cosmology adopted has $\rm H_0=70 \, km \,s^{-1}\, Mpc^{-1}$, $\rm \Omega_{\rm M}=0.3$ and $\rm \Omega_{\rm \Lambda}=0.7$.}
\label{fig1}
\end{figure}

\section{Mid-IR-selected galaxies and their link to near-IR galaxies}
\label{sec:3}

The study of mid-IR selected galaxies has been a subject of major interest in European astronomy since the launch of the {\em Infrared Space Observatory (ISO)} e.g. \cite{aus99}\cite{gen00}\cite{fra01}.   The amount of energy contained in the extragalactic IR background ($\lambda \approx 5-1000 \, \rm \mu m$) is comparable to that in the optical background \cite{dole06}, which emphasizes the importance of understanding the role of mid- and far-IR galaxy populations  in the general context of galaxy evolution.

After {\em ISO} set the first constraints on the mid-IR Universe up to redshift $z\sim1$,  {\em Spitzer} has allowed us  to prove that mid-IR galaxies have in fact a significant contribution up to higher $z\sim2-3$ redshifts, e.g.\cite{cap06b}. A sort of `downsizing' effect is observed in the typical stellar masses of the galaxies hosting the bulk of IR activity: at $z\sim2$, most of the IR activity is found in $\raisebox{-0.13cm}{~\shortstack{$>$ \\[-0.07cm] $\sim$}}~ 10^{11} \, \rm M_\odot$ galaxies, while at $z=0.5-1.0$, an important fraction of the IR activity is concentrated in intermediate-mass galaxies, with  $10^{10} < \rm M < 10^{11} \, M_\odot$ \cite{cap06b}\cite{ham05}.

A result that might have come as a surprise is the large fraction of massive galaxies that are experiencing an ultra-luminous IR phase at high redshifts \cite{daddi05}\cite{cap06c}. Very recent results suggest that the AGN contribution to this IR phase at $z\sim2$ might be much more important than previously thought \cite{daddi07}.

The mid-IR LF strongly evolves from $z=0$ to 1 \cite{lefl05}\cite{cap07}. But, after removing to our best current knowledge the AGN contribution at high redshifts, the mid-IR LF associated with star formation shows only a modest evolution from $z=1$ to $z=2$ \cite{cap07}. This is reflected in the observed evolution of the bolometric IR  ($5-1000 \, \rm \mu m$) luminosity density, as shown in Figure \ref{fig1}. From this figure, it is clear that the amount of energy associated with on-going star formation is significantly larger than the light produced by the stars already locked in galaxies at $z\raisebox{-0.13cm}{~\shortstack{$>$ \\[-0.07cm] $\sim$}}~ 0.6-0.7$, while the balance is reversed at lower redshifts.  The  $(5-1000) \, \rm \mu m$ luminosity related to star formation can be interpreted as the gradient of the evolution of the assembled stellar mass. By $z\sim0.7$, 80\% of the present-day stellar mass is already in place \cite{cap06k}, so there is only a minor amount of star formation still needed afterwards to finish assembling our local Universe.


\section{Introduction}\label{intro}

Redshifted absorption systems lying along the sight-lines to distant
quasars are important probes of the early-to-present day Universe.  In
particular, damped Lyman-$\alpha$ absorption systems (DLAs), where
$N_{\rm HI}\ge2\times10^{20}$ \scm, are useful as they account for at
least 80\% of the neutral gas mass density in the Universe
\citep{phw05}. Since the Lyman-$\alpha$ transition occurs in the
ultra-violet band, direct ground based observations of this
transition are restricted to redshifts of $z\gapp1.7$, although
observations of the H{\sc \,i} ~spin-flip transitions at $\lambda_{\rm
rest}=21$ cm can probe neutral hydrogen from $z=0$, thereby providing a useful
complement to the high redshift optical data. Furthermore, 21-cm
observations can be used to confirm and complement reported variations
in the fundamental constants over the history of the Universe
\citep{mwf03}. Unfortunately, redshifted radio absorption systems are
currently very rare with only 50 H{\sc \,i}\ 21-cm absorption systems known
(summarised in \citealt{cwm+06}), 17 of which are DLAs, with only nine
of these having sufficiently high quality UV/optical and 21-cm data
\citep{tmw+06}.  We are therefore currently undertaking a survey of,
as yet unsearched, DLAs occulting radio-loud quasars in order to
increase the number of 21-cm absorbing systems and in this letter we
report our first detection, in the intriguing DLA towards 1622+238
(3C\,336).

\section{Observations and Data Reduction}\label{obs}

The observations were performed on 9--10 July 2006 with the Westerbork
Synthesis Radio Telescope (WSRT) as part of our survey
(described in \citealt{ctp+07}). The front-end was the UHF-high
receiver tuned to 857.68 MHz (or $z_{\rm abs}=0.6561$,
\citealt{rt00}), backed with a band-width of 5 MHz over 2048 channels
(dual polarisation), giving a channel width of 0.85 \hbox{${\rm km\ s}^{-1}$}. The two
orthogonal polarisations (XX \& YY) were used in order to allow the
removal of any polarisation-dependent radio frequency interference
(RFI). After the flagging out of time-dependent RFI, 12.0 hours of
good data remained, although there was some RFI remaining on some
baselines, particularly in the XX polarisation. After further
flagging, 63 full and partial baseline pairs remained. Additionally,
frequencies below 856.6 MHz (channels above 1500) were removed due to
a steep 1 Jy dip in the flux.

The data were reduced using the {\sc miriad} interferometry reduction
package, with which we extracted a summed spectrum from the emission
region of the continuum map. Upon this, we
found absorption towards the peak of the radio continuum apparent in
both polarisations. After subtracting the continuum with a first order
polynomial from the line free channels of the combined polarisations,
a single Gaussian gives the fit shown in Fig.~\ref{1622}.
\begin{figure}
\vspace{11.5cm}
\special{psfile=velplot.eps hoffset=-65 voffset=-25 hscale=65 vscale=65 angle=0}
\caption{Top: The bandpass calibrated spectrum of 3C\,286. Middle: The 21-cm spectrum of
1622+238. The Gaussian fit gives a peak line
depth of $30.8$ mJy (giving $\tau=0.0088$, against a 3.55 Jy continuum
flux density), a centroid at $-47\pm5$ \hbox{${\rm km\ s}^{-1}$}\ and a FWHM of 235 \hbox{${\rm km\ s}^{-1}$}\
for the line. The centroid gives a frequency of 857.815 MHz, which
corresponds to a redshift of $z_{\rm abs} = 0.65584\pm0.00003$,
cf. the values determined for the Lyman-{$\alpha$} absorption: 0.6561
\citep{rt00} and 0.6564 \citep{clwb98}. The r.m.s. noise is 4.7 mJy
per each 10 \hbox{${\rm km\ s}^{-1}$}\ channel. Bottom: The absorption of Mg{\sc \,i}
(unsaturated) and Mg{\sc \,ii} (saturated) at $z=0.656$ towards
1622+238 (from ESO service-mode program 69.A-0371).}
\label{1622}
\end{figure}

The obvious caveat with the detection is the lack of flat baseline in
the spectrum above $\approx300$ \hbox{${\rm km\ s}^{-1}$}\ due to the lost channels.
However, the spectra obtained in each individual polarisation are
fitted by very similar Gaussians. Furthermore, if instead we use
3C\,286 for the bandpass calibration, which was observed at the start
of the run, rather than 3C\,48, which was observed at the end, the
data give very similar profiles (again with the 1 Jy dip in the flux
below 856.6 MHz), which are fitted by the same Gaussian parameters as
before. As would be expected from a real feature towards an unresolved
($<75''\times42''$) background source, the line is present in the
centre and surrounding pixels, but disappears with increasing
proximity from the centre pixel. We also produced a cube in which
1622+238 is offset from the phase centre: While the absorption
was still apparent towards 1622+238, no feature was seen at the
phase centre. As a further check, we used 3C\,48 to
bandpass calibrate 3C\,286, and, as seen from Fig. \ref{1622} (top
panel), there is no overall ripple in the spectrum.

\section{Results and Discussion}

\subsection{Observational results}
\label{res}

At $\tau=0.0088$, the peak optical depth is at the lower end of the
DLAs detected in 21-cm absorption (table 1 of \citealt{cmp+03}),
although the large line-width gives a velocity integrated optical
depth of $\int\tau dv = 2.2\pm0.2$ \hbox{${\rm km\ s}^{-1}$}, which takes it to joint 5th
place in the line strength stakes (Table \ref{res}).  In the optically
thin regime ($\tau\lapp0.3$), the total atomic hydrogen column
density, $N_{\rm HI}$ [\scm], of the absorbing gas in a homogeneous
cloud is related to the velocity integrated optical depth of the 21-cm
line via
\begin{equation}
N_{\rm HI}=1.823\times10^{18}\,\frac{T_{\rm s}}{f}\int\!\tau\,dv\,,
\end{equation}
where $T_{\rm s}$ [K] is the spin temperature of the gas, $f$ is the
 covering factor of the flux by the absorber and $\int\tau dv$ is the
 velocity integrated optical depth of the 21-cm absorption. If the
 21-cm and Lyman-$\alpha$ absorption arise in the same cloud
 complexes, the total neutral hydrogen column density of
 $2.3\times10^{20}$ \scm\ \citep{rt00},  gives $T_{\rm
 s}/f\approx60$~K, the lowest value yet detected in a
 redshifted source (Table \ref{res}).
\begin{table}
\centering
\begin{minipage}{90mm}
\caption{The DLAs detected in 21-cm absorption listed in order of
increasing spin temperature/covering factor ratio. $z_{\rm abs}$ is
the redshift of the DLA, $\int\tau dv$ is the velocity integrated
optical depth of 21-cm absorption [\hbox{${\rm km\ s}^{-1}$}], $N_{\rm HI}$ is the total
neutral hydrogen column density [\scm] of the absorber (see
\citealt{cwbc01}), which is unknown for the
DLAs towards 0248+430 and 2351+456. The optical identification of the
absorber (ID) is given: D--dwarf, L--LSB, S--spiral, U--unknown.
$T_{\rm s}/f$ is the inferred spin temperature/covering factor ratio
[K]. The final column gives the 21-cm absorption reference. See
\citet{cmp+03} for details. \label{res}}
\begin{tabular}{@{}l c c c c r c@{}}
\hline
QSO &$z_{\rm abs}$ & $\int\tau dv$& $\log N_{\rm HI}$ & ID & $T_{\rm s}/f$ & Ref. \\
\hline
1622+238$^a$ & 0.6561 & 2.2 & 20.4 & -- & 60 & C07\\
0809+483$^{b}$ & 0.4369 & 0.89 & 20.2 & S & 120 & BM83\\
0235+164 &  0.523385 & 14 & 21.7 & S &200 & R76\\
1629+120 & 0.5318 & 0.49 &  20.5 & S &310 & KC03\\
0458--020 & 2.03945 & 7.2 & 21.7 & U & 380 & W85\\
0827+243 & 0.5247 & 0.26 & 20.3 & S & 420 & KC01\\
1229--021 & 0.39498 & 0.66 & 20.8 & S & 520 & BS79\\
1127--145 & 0.3127 & 2.7 & 21.7 & L & 1000 & L98,CK00\\
1328+307$^c$ & 0.692154 & 0.93 & 21.3 & L & 1200 & BR73\\
0738+313 & 0.2212 & 0.34 & 20.8 & D & 1300 & L98\\
0438--436 & 2.347 & 0.22 & 20.8 & U & 1600 &K06\\
1157+014 & 1.94362 & 2.2 & 21.8 & L& 1600 & W81 \\
0201+113 & 3.386 & 0.71 & 21.3 & U & 2000 & K07\\
0952+179 & 0.2378  & 0.11 & 21.3 & L & 9900 &  KC01\\
\hline
0248+430 & 0.394 & 3.0 & -- & U & -- & LB01 \\
2351+456 & 0.779452 & 13 &-- & U & -- & D04\\
\hline
\end{tabular}
{Notes: $^a$3C\,336, $^{b}$3C\,196, $^{c}$3C\,286.\\ References: BR73:
\citet{br73}, R76: \citet{rbb+76}, BS79: \citet{bs79}, W81: \citet{wbj81}; BM83:
\citet{bm83}, W85: \citet{wbt+85}, L98: \citet{lsb+98}, CK00:
\citet{ck00}, KC01: \citet{kc01a}, LB01: \citet{lb01}, KC03:
\citet{kc02}, D04: \citet{dgh+04}, K06: \citet{kse+06}, K07:
\citet{kcl06}, C07: This paper.}
\end{minipage}
\end{table}
Since both the spin temperature and covering factor are generally unknown, the
 degeneracy in the ratio is best left intact \citep{cmp+03,cw06},
 although for $f\leq1$ (by definition) this means that $T_{\rm
 s}\leq60$ K.

The 21-cm absorption profile in this absorber is much wider than that
found in other DLAs ($\leq53$ \hbox{${\rm km\ s}^{-1}$}, see table 1 of \citealt{cmp+03})
and more like the typical H{\sc \,i}\ emission profile of a galaxy
(e.g. \citealt{ksk+04}). We note also, in the other DLAs the 21-cm
profile-width is always narrower than the metal-ion lines (typically
FWHM$_{\rm 21cm}\approx0.2\,\Delta V_{\rm MgII}$, \citealt{ctp+07}),
but in this case FWHM$_{\rm 21cm}\approx2\,\Delta V_{\rm MgII}$, which
suggests very different kinematics from those of the typical DLA.  In
fact, for other DLAs the spin temperature is often compared with the
kinetic temperature (e.g. \citealt{lb01}), which should be equivalent
for single cloud in thermodynamic equilibrium. Applying $T_{\rm
k}\approx22\times{\rm FWHM}^2$ to 1622+238 gives the unphysical
$T_{\rm k}\sim10^{6}$ K, confirming that kinematical motions are
chiefly responsible for the wide profile.

\subsection{The nature of the {\boldmath$z=0.656$} absorber towards 1622+238}

\label{nat}
\subsubsection{Possible absorber hosts}
\label{poss}

In Fig. \ref{overlay} we show the HST WFPC2 (F702W filter) optical
image of the field towards 3C\,336 ($z_{\rm em}=0.927$) overlaid with
the radio maps. Although there is compact 8.4 GHz emission from the
QSO itself, most of the radio emission is from the extended lobes,
which span an enormous $28''$ \citep{bhl+94,cwc05}.
\begin{figure*}
\vspace{15.6cm}
\special{psfile=overlay-point2.ps hoffset=-60 voffset=-175  hscale=100 vscale=100 angle=0}
\caption{The central part of the HST image of \citet{sdm+97} overlaid
with the 8.4 GHz [inner/red contours] and 1.4 GHz [middle/green] radio
emission \citep{cwc05}.  We also show the FIRST 1.4 GHz radio map
[outer lobe/blue]. We do not show the 858 MHz continuum emission from
our observations as the quasar is unresolved by the $75''\times42''$
WSRT beam. The contours scale from 10\% of the peak emission, and for
the sake of clarity, shown only up to 50\% in the 1.4 GHz maps. The
only $z=0.656$ galaxy found by \citet{sdm+97} is highlighted. The
circular region, centered at this galaxy, shows the size of a region
which contains 0.88\% of the total 858 MHz flux density (not
necessarily located here, see text). The linear distance indicated is
calculated for a redshift of $z=0.656$ using
$H_{0}=75$~km~s$^{-1}$~Mpc$^{-1}$, $\Omega_{\rm matter}=0.27$ and
$\Omega_{\Lambda}=0.73$ (used throughout the paper). North is up and
east to the left. The HST image and radio maps are courtesy of Teddy
Cheung and the Very Large Array's FIRST (Faint Images of the Radio Sky
at Twenty Centimetres) survey.}
\label{overlay}
\end{figure*}
The DLA towards this QSO is somewhat of a mystery, since the only
galaxy in the HST field with a redshift of $z_{\rm abs}=0.656$ must
have a projected disk impact parameter of $\approx65\,h^{-1}$ kpc
\citep{sdm+97}. Since no other galaxy within this impact parameter has
ever been found, \citet{sdm+97} suggest, but stipulate as unlikely,
that the absorption may be due to an under-luminous late-type spiral,
in which an invisible disk spans at least all of the way from the
identified $z=0.656$ galaxy to the QSO sight-line.

As an alternative, the authors suggest that there may be other
galaxies within this impact parameter, although as these are
undetected, they would have to be extremely faint ($L<0.05L_{\rm
K}^*$). Another possibility is offered by the galaxy of unknown
redshift close to the optical hotspot, or even the hotspot
itself, both of which are located close to a radio lobe
sight-line \citep{cwc05}. If these (and perhaps some other low
luminosity galaxies), were part of a group, which included the
$z=0.656$ galaxy, the redshift criterion would be satisfied, while
perhaps providing a large H{\sc \,i}\ reservoir located closer to the QSO
sight-line. Furthermore, the H{\sc \,i}\ could be largely dispersed over the
region through the mutual tidal interactions of these galaxies
\citep{sdm+97}.

\subsubsection{Implications of the 21-cm absorption}

From Fig. \ref{overlay}, we see that, although the identified
$z=0.656$ galaxy is located close to the sight-line of a radio lobe,
it remains sufficiently remote to be covering a negligible amount of
flux\footnote{Although we would expect the 1420 MHz emission in the
rest frame of the galaxy (858 MHz) to be further extended than the
21-cm emission measured in our rest frame.}. Although we cannot
determine the location of the absorbing region within the radio-bright
region, we may obtain an estimate of its extent: The circular region
centered on the $z=0.656$ galaxy represents the physical area which
encompasses 0.88\% of the total 858 MHz flux. That is, this is the
smallest region that can give the observed optical depth. So although
the visible feature identified as the $z=0.656$ galaxy does not
intercept much radio flux, a cold, invisible disk of $\gapp20$~kpc
(3.3\hbox{$^{\prime\prime}$} ) radius could cover significant radio emission (as shown
in the figure), although, as per the reservations of \citet{sdm+97},
it may not necessarily be centered upon this galaxy. In fact, the
21-cm and Lyman-\AL\ absorption may not be coincident with one another,
in light of the relative profile widths and offset in the profiles
(Fig. \ref{1622}).

For a zero intrinsic velocity offset, one can estimate the variability of the
constant $x\equiv \alpha^2 g_{\rm p} \mu $, where $\alpha$ is the fine
structure constant, $g_{\rm p}$ the proton $g$-factor and $\mu\equiv
m_{\rm e}/m_{\rm p}$ is the electron-to-proton mass ratio. In such a
case, the relative change in the value of $x$ between redshift $z$ and
a terrestrial ($z=0$) value is given by (e.g. \citealt{tw80})
\begin{equation}\label{equ:dxx}
\frac{\Delta x}{x} \equiv \frac{x_z-x_0}{x_0}
= \frac{z_{\rm UV}-z_{\rm 21} } {1+z_{\rm 21}},
\end{equation}
where $z_{\rm 21}$ and $z_{\rm UV}$ represent the observed absorption
redshifts for 21-cm and a UV heavy-element transition,
respectively. Using the Gaussian fit centroid, $z_{\rm
21}=0.65584\pm0.00003$, and the stongest Mg {\sc \,i} component,
$z_{\rm UV}=0.656062\pm0.000003$, we obtain $\Delta x/x =
(1.34\pm0.18)\times 10^{-4}$. This is an order of magnitude larger
than any of the eleven other values which have been determined (see
figure 1 of \citealt{tmw+06}), suggesting that the velocity offset
between the 21-cm and UV absorption is not primarily due to the values
of the fundamental constants at $z=0.656$.

Therefore the observed $\approx50$ km s$^ {-1}$ velocity difference,
in conjunction with the atypical 21-cm line-width, suggests a physical
offset between the cold, 21-cm absorbing, and warm, UV absorbing,
gas. However, although not completely coincident, hydrogen is observed
in both regimes, within the same field (possibly remote from the QSO
sight-line), at (nearly) the same redshift, and so we find it unlikely
that the radio and UV absorption are not related in some way. The
width of the 21-cm absorption in comparison to the optical lines, and
that of other DLAs, may be the result of an atypical situation
where we have an extended absorber occulting an extended radio
source. However, all of the known DLAs associated with spiral galaxies
also occult large radio sources (figure 4 of
\citealt{cmp+03}). Furthermore, although located much closer to the
spirals than the other DLAs in terms of $T_{\rm s}/f$, due to the wide
profile, the 21-cm absorption towards 1622+238 may be somewhat of an
anomaly (Fig.~\ref{Toverf})\footnote{Note that if FWHM$_{\rm
21cm}\approx0.2\,\Delta V_{\rm MgII}$, as per the other DLAs, $T_{\rm
s}/f$ would be $\approx600$ K, placing this bang in the middle of the
spirals in Fig.~\ref{Toverf}.}.
\begin{figure}
\vspace{6.5cm}
\special{psfile=Toverf.ps  hoffset=-20 voffset=230 hscale=75 vscale=75 angle=270}
\caption{The spin temperature/covering factor ratio versus the
absorption redshift for the DLAs searched for in 21-cm absorption. The
symbols represent the 21-cm detections and the shapes represent the
type of galaxy with which the DLA is associated: circle--unknown type,
star--spiral, square--dwarf, triangle--LSB. The arrows show the lower
limits and all of these bar one (0454+039 at $z_{\rm abs}=0.8596$)
have unknown host identifications. The unfilled symbols show the new
detections since \citet{cmp+03}, from which the figure is adapted:
\citet{kse+06,kcl06} and 1622+238 (unfilled star), on the assumption
that the 21-cm absorption arises in the same gas as the Ly-\AL\
absorption.}
\label{Toverf}
\end{figure}

If the system is bound and the wide 21-cm profile is purely the result
of rotational broadening (i.e. not an outflow), the total line-width
of $\pm280$ \hbox{${\rm km\ s}^{-1}$}, leads to a dynamical mass estimate of $M_{\rm
dyn}=1.8\times10^{10}\,r/\sin^2 i$ M$_\odot$, where $r$ [kpc] is the edge of
the cold H{\sc \,i}\ distribution [or the 858 MHz emission, whichever comes
first]. In the case where this gas is located close to the QSO sight-line, so
that the identified $z=0.656$ galaxy is actually near the edge of
the absorbing disk or group\footnote{Assuming $i$ is the inclination
of the most rapidly orbiting galaxies and neglecting their individual
rotations.}, i.e. $r\approx75$ kpc \citep{clwb98}, the dynamical mass
becomes $M_{\rm dyn}\sim1\times10^{12}/\sin^2 i$ M$_\odot$. If the gas is
due to a large spiral, centered on the $z=0.656$ galaxy\footnote{Which
may be indicated by the single Gaussian fit, although signal-to-noise
ratio of the spectrum (Fig. \ref{1622}) is far too low to resolve
possible contributions from smaller discrete galaxies.}, by assuming
circular motions and the same inclination as the visible ``nucleus''
(55$^{\circ}$, \citealt{clwb98}), the deprojected velocities and impact
parameter give $M_{\rm dyn}\sim2\times10^{12}$ M$_\odot$\ (for $r\approx75$
kpc). Although large, the derived masses are close to the values
derived for the two most local spirals, the Milky Way ($M_{\rm
dyn}\sim2\times10^{12}$~M$_\odot$, \citealt{we99}) and M31 ($M_{\rm
dyn}\sim1\times10^{12}$~M$_\odot$, \citealt{ewg+00}), and certainly
reasonable for a group of smaller galaxies.

Combining this dynamical mass with the luminosity of $L<0.25\,L^*$
\citep{sdm+97}\footnote{Note that, from local studies, the vast
majority (87\%) of DLAs are believed to be due to galaxies of
luminosities $<L^*$ \citep{zvb+05}.}, gives a mass-to-light ratio of
$M_{\rm dyn}/L\gapp400$ M$_\odot$/L$_\odot$. Such ratios ($\sim1000$ M$_\odot$/L$_\odot$),
define the hypothesised low density galaxies (LDGs), which can range
in mass from $10^{9}$ to $10^{12}$M$_\odot$\ \citep{jhhp97}. In an LDG a low
density disk resides in a dark halo, with the surface density below
that required by the Toomre criterion for instabilities to form
($\lapp10^{20}$ \scm, \citealt{jhhp97,voj02}), and so there is no
ongoing star formation: From deep HST H\AL\ imaging, \citet{blc+01}
determine a star formation rate of $<0.15$ M$_\odot$\ yr$^{-1}$ kpc$^{-2}$
in the 3C\,336 field. Another possible example of an LDG may already
have been observed as the ``dark hydrogen cloud'', VIRGOHI 21 in the
Virgo Cluster \citep{mdd+05}. This has a mass-to-light ratio of
$M_{\rm dyn}/L_B> 500$ M$_\odot$/L$_\odot$\ and spans at least 16 kpc in
extent. If it is a galaxy, this has a mass of $M_{\rm dyn}>10^{11}$
M$_\odot$, although, as may be the case towards 1622+238, the large amount
of gas could be due to tidal debris. Where ever the H{\sc \,i}\ (Ly-\AL\ \&
21-cm) absorption arises towards 1622+238, it is clear that this would
not be the first time a large amount of unidentified neutral hydrogen
has been detected.

\section{Summary}

We have detected 21-cm absorption in the $z_{\rm abs}=0.656$ damped
Lyman-$\alpha$ absorber towards 1622+238 (3C\,336). If the absorption
arises in the same gas as the Lyman-$\alpha$ absorption, the spin
temperature of the gas is $T_{\rm s}\leq60$ K, the lowest yet found in
a DLA. The low spin temperature is a consequence of the extremely wide
profile, which, most atypically, is wider than the Mg{\sc \,ii} line.
This could be due to a large absorber occulting what we know to be
large radio source. The 21-cm is also offset from the unsaturated
Mg{\sc \,i} absorption by $\approx-50$ \hbox{${\rm km\ s}^{-1}$}, which corresponds to
$\Delta x/x \approx1.3\times 10^{-4}$ ($x\equiv \alpha^2 g_{\rm p} \mu
$), which is an order of magnitude larger than that expected from
other DLAs, suggesting that the offset is not dominated by different
values of the fundamental constants at $z=0.656$. It may indicate,
instead, that the radio and UV absorption are unrelated, although a
lower Lyman-$\alpha$ column density would give gas significantly
colder than 60 K. Furthermore, it seems somewhat strange that the
hydrogen absorbers detected in two different regimes at the same
redshift would be unrelated, especially considering that none of these
seems to arise directly along the QSO sight-line.
Regarding this, our results cannot distinguish from where the hydrogen
absorption arises, whether this be the disk of a large underluminous
spiral, an extremely faint galaxy lying close to the QSO sight-line or
a possible tidal interaction, between other faint galaxies, which is
dragging significant amounts of gas across the sight-line.

\section*{Acknowledgments}

We would like to thank Raffaella Morganti for coordinating the WSRT
observations, Alan Bridle and Teddy Cheung for their radio images, and
Bob Carswell, Martin Zwaan, Steve Longmore, Alan Pedlar, Rob Beswick,
Matt Owers and Matthew Whiting for their advice. This research has
made use of the NASA/IPAC Extragalactic Database (NED) which is
operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with the National Aeronautics and Space
Administration.  This research has also made use of NASA's
Astrophysics Data System Bibliographic Services.