\section{Slater's proof}

In classical hypergeometric theory, formulae concerning summation,
transformation and contiguous relations are provided in order to
manipulate expressions that represent special functions.
One of those is Kummer's non-terminating summation theorem:
\begin{equation} \label{e:kummerth}
 {}_2F_1\left[{a,b \atop 1+a-b};-1\right]
     = \frac{ \Gamma(1+\frac{a}{2}) \: \Gamma(1+a-b) }
            { \Gamma(1+a) \: \Gamma(1+\frac{a}{2}-b) } \ \mbox{,}
\end{equation}
in which $ Re(b) < 1 $ ensures the series to be convergent.

The first step in Slater's proof is to prove the following quadratic
transformation, due to Kummer:
\begin{equation} \label{e:kummerqua}
 {}_2F_1\left[{a,b \atop 1+a-b};z\right]
     = (1-z)^{-a}
       \: {}_2F_1\left[{\frac{a}{2},\frac{1}{2}+\frac{a}{2}-b
                       \atop 1+a-b};\frac{-4\:z}{(1-z)^2}\right] \ \mbox{.}
\end{equation}
The proof of this transformation is as follows.\\
The right-hand side of (\ref{e:kummerqua}) can be written:
\begin{eqnarray*} \displaystyle
& & (1-z)^{-a} \: \sum_{k=0}^\infty
               \frac{(\frac{a}{2})_k \: (\frac{1}{2}+\frac{a}{2}-b)_k}
                    {(1+a-b)_k \: k!}
               \: \frac{(-4)^k \: z^k}{(1-z)^{2\:k}}\\
&=& \sum_{k=0}^\infty
    \frac{(\frac{a}{2})_k \: (\frac{1}{2}+\frac{a}{2}-b)_k \: (-4)^k}
         {(1+a-b)_k \: k!}
    \: z^k \: (1-z)^{-a-2\:k} \ \mbox{,}\\
&=& \sum_{k=0}^\infty
    \frac{(\frac{a}{2})_k \: (\frac{1}{2}+\frac{a}{2}-b)_k \: (-4)^k}
         {(1+a-b)_k \: k!}
    \: z^k \: {}_1F_0\left[{a+2\:k \atop -};z\right]

                                   \quad \mbox{by the binomial theorem,}\\
&=& \sum_{k=0}^\infty \sum_{n=0}^\infty
    \frac{(\frac{a}{2})_k \: (\frac{1}{2}+\frac{a}{2}-b)_k \: (-4)^k}
         {(1+a-b)_k \: k!}
    \: \frac{(a+2\:k)_n}{n!} \: z^{k+n} \ \mbox{.}
\end{eqnarray*}
Extracting the coefficient of $z^N$ yields:
\begin{eqnarray}

 &   & \sum_{k=0}^{N}
    \frac{(\frac{a}{2})_k \: (\frac{1}{2}+\frac{a}{2}-b)_k
          \: (-4)^k \: (a+2\:k)_{N-k}}
         {(1+a-b)_k \: (N-k)! \: k!} \nonumber \\
\qquad & = & \frac{(a)_N}{N!}
    \: \sum_{k=0}^{N}
       \frac{(\frac{1}{2}+\frac{a}{2}-b)_k \: (a+N)_k \: (-N)_k}
            {(1+a-b)_k \: (\frac{1}{2}+\frac{a}{2})_k \: k!}
 \label{e:beforesaal}
\end{eqnarray}
since:
\[ \frac{1}{(N-k)!} = \frac{(-1)^k \: (-N)_k}{N!} \]
and
\[ (a+2\:k)_{N-k} = \frac{(a)_{N+k}}{(a)_{2\:k}}
                  = \frac{(a)_N \: (a+N)_k}{4^k \: (\frac{a}{2})_k
                    \: (\frac{1}{2}+\frac{a}{2})_k} \ \mbox{.} \]
The right-hand side of (\ref{e:beforesaal}) is summable by
Saalsch{\" u}tz theorem \cite[(III.2)]{b-slater}:
\[
   \frac{(a)_N}{N!}
   \: \frac{(\frac{1}{2}+\frac{a}{2})_N \: (1-b-N)_N}
           {(1+a-b)_N \: (\frac{1}{2}-\frac{a}{2}-N)_N} \ \mbox{.}
\]
As this is the coefficient of $z^N$ in the right-hand side
of (\ref{e:kummerqua}), we obtain:
\begin{eqnarray*}
   \sum_{N=0}^{\infty} \frac{(a)_N}{N!}
      \: \frac{(\frac{1}{2}+\frac{a}{2})_N \: (1-b-N)_N}
              {(1+a-b)_N \: (\frac{1}{2}-\frac{a}{2}-N)_N} \: z^N
  & = & \sum_{N=0}^{\infty} \frac{(a)_N \: (b)_N}{(1+a-b)_N \: N!} \: z^N \\
  & = & {}_2F_1\left[{a,b \atop 1+a-b};z\right] \ \mbox{,}
\end{eqnarray*}
which proves Kummer's quadratic transformation.\\

The second step is to set $z$ to $-1$ in (\ref{e:kummerqua}). Therefore
\begin{equation} \label{e:kummerquaspe}
 {}_2F_1\left[{a,b \atop 1+a-b};-1\right]
     = 2^{-a}
       \: {}_2F_1\left[{\frac{a}{2},\frac{1}{2}+\frac{a}{2}-b
                       \atop 1+a-b};1\right] \ \mbox{.}
\end{equation}
Note that these series are convergent if $ Re(b) < 1 $.\\
We can sum the series on the right-hand side
by Gauss's theorem \cite[(III.3)]{b-slater}. This yields
\begin{equation}
 {}_2F_1\left[{a,b \atop 1+a-b};-1\right]
 = 2^{-a}
   \: \frac{\Gamma(1+a-b) \: \Gamma(\frac{1}{2})}
           {\Gamma(1+\frac{a}{2}-b) \: \Gamma(\frac{1}{2}+\frac{a}{2})} \ \mbox{.}
\end{equation}
Since $ \Gamma(\frac{1}{2}+\frac{a}{2}) \: \Gamma(\frac{a}{2}+1)
             = 2^{a} \: \Gamma(\frac{1}{2}) \: \Gamma(a+1) $,
we finally obtain Kummer's theorem:
\begin{equation}
 {}_2F_1\left[{a,b \atop 1+a-b};-1\right]
     = \frac{ \Gamma(1+a-b) \: \Gamma(1+\frac{a}{2})}
            { \Gamma(1+\frac{a}{2}-b) \: \Gamma(1+a)} \ \mbox{.}
\end{equation}

\section{A new proof}

In formula (\ref{e:kummerth}), let us replace $a$ by $a+2\:n$
where $n$ is a new variable that stands for a nonnegative integer.
The resulting formula
\begin{equation} \label{e:kummervar}
 {}_2F_1\left[{a+2\:n,b \atop 1+a+2\:n-b};-1\right]
     = \frac{ \Gamma(1+\frac{a}{2}+n) \: \Gamma(1+a+2\:n-b) }
            { \Gamma(1+a+2\:n) \: \Gamma(1+\frac{a}{2}+n-b) }
\end{equation}
is clearly equivalent to (\ref{e:kummerth}).\\
The introduction of the free parameter $n$ is the key to this new
computer-assisted proof.

Formula (\ref{e:kummervar}) can be written as:
\begin{equation} \label{e:p1}
    \frac{\sum_k f(n,k)}{S(n)} = 1 \ \mbox{,}
\end{equation}
where \[ \displaystyle f(n,k) = \frac{(a+2\:n)_k \: (b)_k}{(1+a+2\:n-b)_k}
                  \: \frac{(-1)^k}{k!} \]
and   \[ \displaystyle S(n) = \frac{\Gamma(1+\frac{a}{2}+n) \: \Gamma(1+a+2\:n-b)}
                    {\Gamma(1+a+2\:n) \: \Gamma(1+\frac{a}{2}+n-b)} \ \mbox{.} \]
(The coefficient $2$ in the substitution $a \rightarrow a+2\:n$ makes $S(n)$
hypergeometric in $n$.)\\
Formula (\ref{e:p1}) can be rewritten as:
\begin{equation} \label{e:p2}
\sum_k F(n,k) = 1 \qquad \mbox{where }
F(n,k) = \frac{f(n,k)}{S(n)} \mbox{.}
\end{equation}
We shall actually prove this last formula. Using Gauthier's Maple
package \texttt{HYPERG} \cite{b-user}, we apply Zeilberger's algorithm
to obtain:
\begin{equation} \label{e:kummerWZ}
F(n,k) - F(n+1,k) = G(n,k+1) - G(n,k) \ \mbox{,}
\end{equation}
with $ G(n,k) = F(n,k) \: C(n,k) $ and $ \displaystyle C(n,k)
= -\frac{(b-1)\:k}{(1+a+2\:n-b+k) \: (a+2\:n)} $.\\
This $C(n,k)$ is the so-called certificate of the WZ-pair $(F,G)$.\\
After summing (\ref{e:kummerWZ}) for $ k $ from $ 0 $ to $K$,
the right-hand side telescopes:
\begin{eqnarray}
 \sum_{k=0}^{K} F(n,k) - \sum_{k=0}^{K} F(n+1,k)
 & = & \sum_{k=0}^{K} G(n,k+1) - \sum_{k=0}^{K} G(n,k) \\
 & = & G(n,K+1) - G(n,0) \\
 \label{e:WZsum}
 & = & F(n,K+1) \: C(n,K+1) \ \mbox{.}
\end{eqnarray}
We now let $K$ tend to infinity. Clearly,
\[ \lim_{K \rightarrow +\infty} C(n,K+1) = -\frac{b-1}{a+2\:n} \ \mbox{.} \]
To determine $ \displaystyle \lim_{K\rightarrow+\infty} F(n,K+1) $, we use
the well-known estimate (see e.g. \cite{b-tom}):
       \begin{equation} \label{e:lim}
       \frac{\Gamma(a+k)}{\Gamma(b+k)} \sim k^{a-b}
       \quad \mbox{as } k \rightarrow \infty \ \mbox{.}
       \end{equation}
First,
\begin{eqnarray*}
 F(n,k) & = & \frac{(a+2\:n)_k \: (b)_k}{(1+a+2\:n-b)_k}
              \: \frac{(-1)^k}{k!} \: \frac{1}{S(n)} \ \mbox{,} \\
        & = & \frac{\Gamma(a+2\:n+k) \: \Gamma(b+k) \: (-1)^k}
                   {\Gamma(1+a+2\:n-b+k) \: \Gamma(k+1)}
              \: T(n) \ \mbox{,}
\end{eqnarray*}
where
$\displaystyle T(n)=\frac{(a+2\:n)\:\Gamma(1+\frac{a}{2}+n-b)}
                         {\Gamma(b)\:\Gamma(1+\frac{a}{2}+n)}$
does not depend on $k$.\\
Then,
\begin{eqnarray*}
 |F(n,k)| & \sim & k^{a+2n-(1+a+2n-b)} \: k^{b-1} \: T(n) \ \mbox{,} \\
          & \sim & k^{2b-2} \: T(n) \ \mbox{.}
\end{eqnarray*}
We know that $ Re(b)<1 $ (necessary condition for the convergence
of the series in (\ref{e:kummerth})).\\
Therefore
\[ \lim_{K \rightarrow +\infty} F(n,K+1) = 0 \ \mbox{.} \]

Expressing as before $F(n,k)$ in terms of Gamma functions and using the same
aymptotics, it is easy to find the limit of $F(n,k)$ when $n$ tends to
infinity:
\begin{eqnarray*}
  F(n,k)  & = & \frac{(a+2\:n)_k \: (b)_k}{(1+a+2\:n-b)_k}
                \: \frac{(-1)^k}{k!}
                \: \frac{\Gamma(1+a+2\:n) \: \Gamma(1+\frac{a}{2}+n-b)}
                       {\Gamma(1+\frac{a}{2}+n) \: \Gamma(1+a+2\:n-b)} \\
          & = & \frac{\Gamma(a+2\:n+k) \: \Gamma(1+a+2\:n-b)}
                     {\Gamma(a+2\:n)\:\Gamma(1+a+2\:n-b+k)}
                \: \frac{(b)_k \: (-1)^k}{k!} \\
          &   & \times \frac{\Gamma(1+a+2\:n) \: \Gamma(1+\frac{a}{2}+n-b)}
                        {\Gamma(1+a+2\:n-b) \: \Gamma(1+\frac{a}{2}+n)} \\
 |F(n,k)| &\sim& \left| (2\:n)^{k} \: (2\:n)^{-k} \: \frac{(b)_k \: (-1)^k}{k!}
                \: (2\:n)^b \: n^{-b} \right| \ \mbox{.}
\end{eqnarray*}
So,
\begin{equation}
\lim_{n \rightarrow +\infty} F(n,k)
   = \frac{(b)_k \: (-1)^k}{k!} \: 2^b \ \mbox{.}
\end{equation}

By (\ref{e:lim}), we have $ 1/S(n) \sim 2^b $ when $n$ tends to infinity, so
for all $n \geq n_0$, $1/S(n) < A$ where $A$ is a positive constant.
Therefore,
\begin{equation} \label{e:convS}
 |F(n,k)| = \left|\frac{f(n,k)}{S(n)}\right| \leq A \: |f(n,k)| \ \mbox{.}
\end{equation}
Moreover,
\begin{equation} \label{e:majorF1}
 |f(n,k)| =    \frac{ |(a+2\:n)_k| \: |(b)_k| }{ |(1+a+2\:n-b)_k| \: k! }
          \leq \frac{ |(b)_k| }{ k! } \ \mbox{,}
\end{equation}
because
\[
  \forall i \geq 0 \ \mbox{,} \qquad
  \frac{|a+2\:n+i|}{|1+a+2\:n-b+i|} < 1
  \quad \mbox{when } n \mbox{ is big enough,}
\]
and
\begin{eqnarray*}
  &      & \frac{ |(a+2\:n)_k| }{ |(1+a+2\:n-b)_k| } \\[0.2cm]
  &   =  & \frac{ | (a+2\:n) \: (a+2\:n+1) \ldots (a+2\:n+k-1) |}
                { | (1+a+2\:n-b) \: (1+a+2\:n-b+1) \ldots (1+a+2\:n+b+k-1) |}
           \ \mbox{,} \\[0.2cm]
  & \leq & \frac{ |a+2\:n| \: |a+2\:n+1| \ldots |a+2\:n+k-1| }
                { |1+a+2\:n-b| \: |1+a+2\:n-b+1| \ldots |1+a+2\:n+b+k-1| }
           \ \mbox{,} \\[0.2cm]
  & \leq & 1 \ \mbox{.}
\end{eqnarray*}
We can finally establish (use (\ref{e:convS}), (\ref{e:majorF1})
and (\ref{e:lim})) that
\begin{equation} \label{e:majorF2}
 |F(n,k)| \leq A \: \left| \frac{(b)_k}{k!} \right|
                  \sim A \: \left| \frac{k^{b-1}}{\Gamma(b)} \right|  \mbox{.}
\end{equation}

For $Re(b)<0$, this last expression is the general term of a convergent series,
so $\sum_{k \geq 0} F(n,k) $ is dominated by a convergent series.

The limit of formula (\ref{e:WZsum}) when $K$ tends to infinity gives
\[
   \sum_{k=0}^{\infty} F(n,k) - \sum_{k=0}^{\infty} F(n+1,k) = 0 \ \mbox{.}
\]
Hence, the sum $\sum_{k=0}^{\infty} F(n,k) $ is independent of $n$. Then
\begin{eqnarray*}
\sum_{k=0}^{\infty} F(n,k)
   & = & \lim_{n \rightarrow +\infty} \sum_{k=0}^{\infty} F(n,k) \ \mbox{,} \\
   & = & \sum_{k=0}^{\infty} \lim_{n \rightarrow +\infty} F(n,k) \ \mbox{,}
\end{eqnarray*}
where the last equality is justified by dominated convergence.
Moreover,

\begin{eqnarray*}
\sum_{k=0}^{\infty} F(n,k)
   & = & \sum_{k=0}^{\infty}
         \lim_{n \rightarrow +\infty} (F(n,k) \ \mbox{,} \\
   & = & \sum_{k=0}^{\infty} 2^b \:  \frac{(b)_k \: (-1)^k}{(k)!} \\
   & = & 2^b \: \sum_{k=0}^{\infty} \: \frac{(b)_{k} \: (-1)^k}{k!}
         \ \mbox{,} \\
   & = & 2^b \: (1-(-1))^b = 1 \ \mbox{,} \qquad \mbox{by binomial theorem.}
\end{eqnarray*}
This is precisely formula (\ref{e:p2}). So, we proved ---~with the restriction $
Re(b)<0 $~--- the validity of Kummer's theorem
(specializing $n$ by $0$ in (\ref{e:kummervar})).

Kummer's theorem (\ref{e:kummerth}) can be rewritten as:
\[
\sum_k M(a,b,k) = \frac{ \Gamma(1+\frac{a}{2}) \: \Gamma(1+a-b) }
                       { \Gamma(1+a) \: \Gamma(1+\frac{a}{2}-b) } \ \mbox{,}
\]
with
$\displaystyle M(a,b,k) = \frac{(a)_k \: (b)_k}{(1+a-b)_k \: k!} \: (-1)^k $.\\
We derive by Zeilberger's algorithm, a recurrence for $M$ with respect
to the parameter $b$:
\[
    (a-2\:b) \: M(a,b,k) + (-2\:a+2\:b) \: M(a,b+1,k)
  = G'(a,b,k+1) - G'(a,b,k)
\]
with $ G'(a,b,k) = M(a,b,k) \: \frac{(a-b+k)\:k}{b} $ and $ Re(b)<0 $.
After summing both sides with respect to $k$ from $0$ to $K-1$, it gives
\[
    (a-2\:b) \: \sum_{k=0}^{K-1} M(a,b,k)
  + (-2\:a+2\:b) \: \sum_{k=0}^{K-1} M(a,b+1,k)
  = G'(a,b,K) - G'(a,b,0) \ \mbox{,}
\]
and then we let $K$ tend to infinity to obtain
\[
    (a-2\:b) \: \sum_{k=0}^{\infty} M(a,b,k)
  + (-2\:a+2\:b) \: \sum_{k=0}^{\infty} M(a,b+1,k)
  = 0 \ \mbox{,}
\]
because $ G'(a,b,0) = 0 $ and
$ \displaystyle \lim_{K \rightarrow +\infty} G'(a,b,K) = 0 $.

We have now for any $b$ with $Re(b)<0$
\begin{eqnarray*}
  \sum_{k=0}^{\infty} M(a,b+1,k)
  & = & \frac{\frac{a}{2}-b}{a-b} \: \sum_{k=0}^{\infty} M(a,b,k) \ \mbox{,} \\
  & = & \frac{\frac{a}{2}-b}{a-b}
        \: \frac{ \Gamma(1+\frac{a}{2}) \: \Gamma(1+a-b) }
                { \Gamma(1+a) \: \Gamma(1+\frac{a}{2}-b) } \ \mbox{,} \\
  & = & \frac{ \Gamma(1+\frac{a}{2}) \: \Gamma(a-b) }
             { \Gamma(1+a) \: \Gamma(\frac{a}{2}-b) } \ \mbox{.}
\end{eqnarray*}
Let us replace $b+1$ by $B$, we finally conclude that
\[
   {}_2F_1\left[{a,B \atop 1+a-B};-1\right]
   = \frac{ \Gamma(1+\frac{a}{2}) \: \Gamma(1-a-B) }
          { \Gamma(1+a) \: \Gamma(1+\frac{a}{2}-B) } \ \mbox{,}
\]
for all values of $B$ such that $Re(B)<1$.

\textbf{Remark:} We can also derive this result directly
from (\ref{e:majorF2}) by a convergence-acceleration argument.
For this, we define a new sequence $ H(n,k) := F(n,2\:k) + F(n,2\:k+1) $,
and we find that $\sum_{k \geq 0} H(n,k)$ is dominated by a convergent series.
Hence, we have
\begin{eqnarray*}
\sum_{k=0}^{\infty} F(n,k)
   & = & \lim_{n \rightarrow +\infty} \sum_{k=0}^{\infty} F(n,k) \ \mbox{,} \\
   & = & \lim_{n \rightarrow +\infty}
         \sum_{k=0}^{\infty} (F(n,2\:k)+F(n,2\:k+1)) \ \mbox{,} \\
   & = & \lim_{n \rightarrow +\infty} \sum_{k=0}^{\infty} H(n,k)
     =   \sum_{k=0}^{\infty} \lim_{n \rightarrow +\infty}  H(n,k) \ \mbox{,} \\
   & = & \sum_{k=0}^{\infty}
         \lim_{n \rightarrow +\infty} (F(n,2\:k)+F(n,2\:k+1)) \ \mbox{,} \\
   & = & \sum_{k=0}^{\infty} 2^b \:
           \left(  \frac{(b)_{2\:k} \: (-1)^{2\:k}}{(2\:k)!}
            + \frac{(b)_{2\:k+1} \: (-1)^{2\:k+1}}{(2\:k+1)!} \right) \\
   & = & 2^b \: \sum_{k=0}^{\infty} \: \frac{(b)_{k} \: (-1)^k}{k!}
         \ \mbox{,} \\
   & = & 1 \ \mbox{.}
\end{eqnarray*}

\section{Other summation theorems}

The combination of Zeilberger's algorithm and asymptotic estimates
suggests an approach to automatically prove identities involving
hypergeometric series. It is well suited to summation theorems.
As an example, we now give two apparently new proofs of classical
hypergeometric identities then a list of summation theorems
that the package \texttt{HYPERG} can prove automatically.

\subsection{Bailey's theorem}

\begin{equation} \label{e:baileyth}
{}_2F_1\left[{a,1-a \atop b};\frac{1}{2}\right]
       = \frac{ \Gamma(\frac{b}{2}) \: \Gamma(\frac{1+b}{2}) }
              { \Gamma(\frac{a+b}{2}) \: \Gamma(\frac{1-a+b}{2}) } \ \mbox{,}
\end{equation}
Following the same scheme, let $ b \rightarrow b+2\:n $.
Formula (\ref{e:baileyth}) can be written:
\begin{equation} \label{e:bai1}
\sum_k F(n,k) = 1 \ \mbox{,}
\end{equation}
where
$ \displaystyle F(n,k) = \frac{f(n,k)}{S(n)} $, \\
with
$ \displaystyle
f(n,k) = \sum_k \frac{(a)_k \: (1-a)_k}{(b+2\:n)_k}
        \: \left(\frac{1}{2}\right)^k \quad \mbox{and} \quad
S(n) = \frac{\Gamma(\frac{b}{2}+n) \: \Gamma(\frac{1}{2}+\frac{b}{2}+n)}
            {\Gamma(\frac{a+b}{2}+n) \: \Gamma(\frac{1-a+b}{2}+n)} \ \mbox{.} $\\
Zeilberger's algorithm applied to $F(n,k)$ succeeds to finding a WZ-pair:
\begin{equation} \label{e:baileyWZ}
F(n,k) - F(n+1,k) = G(n,k+1) - G(n,k) \ \mbox{,}
\end{equation}
where
$ \displaystyle G(n,k) = F(n,k) \: \left(-2\:\frac{k}{b+2\:n+k}\right) $.\\
As with Kummer's theorem, we have:
\begin{eqnarray}
G(n,0)                              & = & 0 \ \mbox{,} \\
\lim_{k \rightarrow +\infty} G(n,k) & = & 0 \ \mbox{,}
\end{eqnarray}
and so, $ \displaystyle \sum_k F(n,k) $ is independent of $n$.\\
In this case, $ \displaystyle \lim_{n \rightarrow +\infty} F(n,k)
= \delta_{k,0} $, so that:
\begin{equation}
\sum_{k=0}^{\infty} F(n,k)
  = \lim_{n \rightarrow +\infty} \sum_{k=0}^{\infty} F(n,k)
  = \sum_{k=0}^{\infty} \lim_{n \rightarrow +\infty} F(n,k)
  = \sum_{k=0}^{\infty} \delta_{k,0} = 1 \ \mbox{,}
\end{equation}
which proves Bailey's theorem.

\subsection{Dixon's theorem}

\begin{eqnarray} \label{e:dixonth}
&   & {}_3F_2\left[{a,b,c \atop 1+a-b,1+a-c};1\right] \nonumber \\
& = & \frac{ \Gamma(1+a-b) \: \Gamma(1+a-c) \: \Gamma(1+\frac{a}{2})
                \: \Gamma(1+\frac{a}{2}-b-c) }
              { \Gamma(1+a) \: \Gamma(1+\frac{a}{2}-b)
                \: \Gamma(1+\frac{a}{2}-c) \: \Gamma(1+a-b-c) } \ \mbox{,}
\end{eqnarray}
where $ Re(2+a-2\:b-2\:c) > 0 $.\\
Once again, we replace $ a $ by $a+2\:n$. With the same notation
(\ref{e:bai1}), we now have:
\begin{eqnarray*}
f(n,k) & = & \sum_k \frac{(a+2n)_k \: (b)_k \: (c)_k}
                     {(1+a+2n-b)_k \: (1+a+2n-c)_k \: k!} \ \mbox{,} \\
        \mbox{and} \quad
S(n) & = & \frac{\Gamma(1+a+2n-b) \Gamma(1+a+2n-c)
             \Gamma(1+\frac{a}{2}+n) \Gamma(1+\frac{a}{2}+n-b-c)}
            {\Gamma(1+a+2n) \Gamma(1+\frac{a}{2}+n-b)
             \Gamma(1+\frac{a}{2}+n-c) \Gamma(1+a+2n-b-c)} \ \mbox{.}
\end{eqnarray*}
The WZ-pair given by Zeilberger's algorithm is
\begin{equation} \label{e:dixonWZ}
F(n,k) - F(n+1,k) = G(n,k+1) - G(n,k) \ \mbox{,}
\end{equation}
with
$ \displaystyle G(n,k) = F(n,k) \: C(n,k) $, where the computed
certificat $C(n,k)$ is:
{ \footnotesize
\begin{verbatim}
-(-2-8*a*n*c-a+4*b+4*c-2*n-2*n*b*c-2*a*b*k-2*a*k*c-a*b*c-8*a*n*b
-4*n*k*c-4*n*b*k+a*b+2*n*b-2*a^2*c-8*n^2*c+a*c+2*n*c-2*b^2+3*a^2
+12*a*n+12*n^2+12*a^2*n-2*a^2*b+24*a*n^2+a*k^2-8*n^2*b+2*n*k^2
+2*a^3+16*n^3-6*c*b+3*k*a^2+3*k*a+6*k*n+12*k*n^2+12*k*a*n+2*b^2*c
+2*b*c^2-2*c^2)*k/(2+a+2*n-2*b-2*c)/(1+a+2*n-c+k)/(1+a+2*n-b+k)
/(a+2*n)
\end{verbatim} }
\noindent It is easy to establish the following limits:
\begin{eqnarray*}
G(n,0)                              & = & 0 \ \mbox{,} \\
\lim_{k \rightarrow +\infty} G(n,k) & = &
\lim_{k \rightarrow +\infty} k^{2\:b+2\:c-2\:n-4-a} \: T(n) \\
                                    & = & 0  \ \mbox{.}
\end{eqnarray*}
Again, $ \displaystyle \sum_k F(n,k) $ is independent of $n$.\\
Since $ \displaystyle \lim_{n \rightarrow +\infty} F(n,k)
= \delta_{k,0} $, we conclude again that:
\begin{equation}
\sum_{k=0}^{\infty} F(n,k)
 = \lim_{n \rightarrow +\infty} \sum_{k=0}^{\infty} F(n,k)
 = \sum_{k=0}^{\infty} \lim_{n \rightarrow +\infty} F(n,k)
 = \sum_{k=0}^{\infty} \delta_{k,0} = 1 \ \mbox{.}
\end{equation}

\subsection{A list of theorems}

These formulas can automatically be proved using the \texttt{HYPERG} package:
\begin{itemize}
\item Gauss's theorem \cite[(III.3)]{b-slater}:
      \[ {}_2F_1\left[{a,b \atop c};1\right]
         = \frac{ \Gamma(c) \: \Gamma(c-a-b) }
                { \Gamma(c-a) \: \Gamma(-b+c) } \ \mbox{.} \]

\item another Dixon's theorem \cite[(III.10)]{b-slater}:
      \[
      {}_4F_3\left[{a,1+\frac{a}{2},b,c \atop \frac{a}{2},1+a-b,1+a-c};
                                                                      -1\right]
      =
      \frac{\Gamma(1+a-b) \: \Gamma(1+a-c)}
           {\Gamma(1+a) \: \Gamma(1+a-b-c)} \ \mbox{.}
      \]

\item another Dixon's theorem \cite[(III.12)]{b-slater}:
      \begin{eqnarray*}
      & & {}_5F_4\left[{a,\frac{a}{2}+1,b,c,d \atop \frac{a}{2},1+a-b,1+a-c,1+a-d};
                                                                       1\right]
      \\
      & = &
      \frac{\Gamma(1+a-b) \: \Gamma(1+a-c) \: \Gamma(1+a-d) \: \Gamma(1+a-b-c-d)}
           {\Gamma(a+1) \: \Gamma(1+a-b-c) \: \Gamma(1+a-b-d) \: \Gamma(1+a-c-d)} \ \mbox{.}
      \end{eqnarray*}

\end{itemize}


\section{Introduction}
Naturalness is the only problem of the SM that requires new physics at energies $\Lambda$
accessible to accelerators, $\Lambda\circa{<} 4\pi M_Z$.
Supersymmetry (SUSY) is one possible solution to the problem:
in this case the scale of new physics is given by the mass of the supersymmetric particles
(`sparticles'), that are expected to be just around the $Z$-boson:
$m_{\rm SUSY}=(\frac{1}{3}\div 3)M_Z$.

However experiments now say that most of the sparticles must be heavier than the $Z$ boson.
It is thus interesting to studying if the existing `conventional' supersymmetric models
(supergravity and gauge mediation)
prefer a chargino mass $M_\chi$ around $\sim 30\,{\rm GeV}$ or $\sim 300\,{\rm GeV}$.
We would like to discuss the following questions:
\cQ{
\begin{quotation}

Experiments are saying that naturalness is a problem for supersymmetry?

If yes, can we found a solution and learn something about the sparticle spectrum?

\end{quotation}}
{$\bullet$ Experiments are saying that naturalness is a problem for supersymmetry?
$\bullet$ If yes, can we found a solution and learn something about the sparticle spectrum?}

\section{Naturalness and fine-tuning}
Usually questions of this kind are discussed using fine tuning (`FT')\cite{FT}
as a quantitative measure of naturalness.
In a supersymmetric model the $Z$ boson mass squared, $M_Z^2(\wp)$, can be computed
as function of the parameters $\wp$ ($\mu$-term, gaugino masses, etc) of the model,
by minimizing the potential.
The FT (for example with respect to the $\mu$ term) is defined as
\cQ{$${\rm FT}(\mu)\equiv \frac{\mu^2}{M_Z^2}\frac{\partial M_Z^2}{\partial \mu^2}$$}
{${\rm FT}(\mu)\equiv (\mu^2/M_Z^2)(\partial M_Z^2/\partial \mu^2)$}
and tells how sensitive is the $Z$-mass with respect to variation of the parameters.

The result of a FT analysis in `minimal' supergravity (`mSuGra', sometimes also called
`constrained' suGra) MSSM
is the following\cite{FTLEP}:
the most recent bounds require a FT greater than about 6.
Is this an answer to our questions?
We believe not, due to the following reasons
\cQ{\begin{enumerate}}{}
\cQ{\item}{(1.)} Replacing $M_Z^2\to M_Z$ in the definition of the FT would reduce the minimal FT from 6 to 3.
\cQ{\item}{(2.)} Replacing $\mu^2\to \mu$ would increase the minimal FT from 6 to 12.
\cQ{\item}{(3.)} Should we worry if FT $> 5$, or 10, or 20, or\ldots?
\cQ{\item}{(4.)} The FT can be large in some cases  where nothing of unnatural happens\cite{FTcritica}.
\cQ{\end{enumerate}}{}
So,  the normalization of the FT is arbitrary and
setting an upper bound on the FT is a subjective choice.
Summarizing, a FT analysis only gives the following not very interesting statement:
\cQ{$$\hbox{
If we change the parameters of the mSuGra model by $1\%$,
the $Z$ mass changes by more than 3\%.}$$}
{``If we change the parameters of the mSuGra model by 1\% the $Z$ mass changes by more than 3\%''}
that does not seem to indicate a problem for supersymmetry.

\section{Naturalness and the allowed $\%$ of parameter space}
Since the FT does not give a useful answer, we approach the naturalness problem in a more direct way\cite{nat}t.
We will illustrate the question with various plots, all generated by the following procedure:
we consider the model that we want to study and

\begin{table}[t]
\caption{How naturally 3 different models generate the EW scale,
using 2 different ways of quantifying naturalness.\label{tab:exp}}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{c|ccc}
&\special{color cmyk 1. 1. 0.3 0} ElectroWeak scale  & Fine-tuning & acceptable \% of \special{color cmyk 0 0 0 1.}\\
&\special{color cmyk 1. 1. 0.3 0} $M_Z^2$ generated as & parameter & parameter space \special{color cmyk 0 0 0 1.}\\ \hline
\special{color cmyk 0 1. 1. 0.5} Standard Model\special{color cmyk 0 0 0 1.} & $\sum \frac{\alpha}{4\pi}({\cal O}}  \def\SU{{\rm SU}(\Lambda^2)+M^2)$&$\sim 10^{30}$&$\sim 10^{-30}$\\
\special{color cmyk 0 1. 1. 0.5} Techni-colour\special{color cmyk 0 0 0 1.}   & $\Lambda^2 e^{-4\pi b/\alpha}$ & $\sim 50$~ & $\sim 1$\\
\special{color cmyk 0 1. 1. 0.5} MSSM (mSuGra)\special{color cmyk 0 0 0 1.}  & $9 M_{1/2}^2-2\mu^2$ & $>6 $& $\sim 5\%$\\
\end{tabular}
\end{center}
\end{table}

\begin{enumerate}
\item We choose random values of ${\cal O}}  \def\SU{{\rm SU}(1)$ for all the dimensionless ratios of soft parameters
(like $\mu/M_{1/2}$);
\item We do not impose how heavy are the sparticles; the overall mass scale is instead fixed
by requiring that the the minimum of the one-loop effective potential $V(h)$
be at its physical value $M_Z=g_2\langle h\rangle=91.18\,{\rm GeV}$;

\item Now everything is fixed and everything can be computed.
Since we want to study naturalness we check if the obtained spectrum of sparticles is sufficiently
heavy for satisfying all accelerator bounds;

\item We repeat the previous steps many times and we compute
$\%\equiv \frac{\hbox{number of acceptable spectra}}{\hbox{total number of spectra}}$.

\end{enumerate}
This Monte Carlo-like procedure computes
how frequently numerical accidents can make the $Z$ boson
sufficiently lighter than the unobserved supersymmetric particles.
The main result is shown in table~1, where we compare
how natural are different models according to the FT and to our $\%$.
Both metods agree that the SM is perfectly unnatural in presence of high mass scales $M,\Lambda\circa{>}10^{16}\,{\rm GeV}$.
Fine-tuning says that techni-colour models generate the $Z$ scale in an unnatural way\cite{FTcritica},
while this is not true and our $\%$ is of order one
(concrete models would have various problems).
{\em The FT does not give a clear answer about the naturalness of the mSuGra MSSM ---
our procedure says that roughly $95\%$ of its parameter space
(with properly broken EW symmetry) is now excluded.}

Before analizing more models in more detail,
we comment about the solidity of the procedure that we follow.
It can be viewed as an efficient way of making plots with density of points {\em proportional} to 1/FT:
thus our $\%$ does not depend on the largely subjective normalization of the FT.
The $\%$ is normalized to be 100\% in absence of experimental bounds.
So a small $\%$ has a clear meaning.
Only {\em one aspect of the definition of the $\%$ is subjective\/}: in step~1 we must specify what we mean
with `order-one'.
In more general terms, we sample the parameter space according to some arbitrary
distribution of probability $p(\wp)$.
The results (like the allowed percentage of parameter space) depend on the choice of $p(\wp)$.
Actually this is a standard situation in inferential statistics:
for example an experimentalist that wants to report a lower bound on a neutrino mass $m_\nu$
needs to assume an arbitrary distribution of probability $p(m_\nu)$
(this assumption is explicit in the Bayesian approach\cite{Bayes}, where $p$ is called `prior distribution';
the situation is not better in the frequentistc approach, where different `confidence
intervals' are obtained using different techniques).

In conclusion the procedure that we will use
\cQ{\begin{itemize}}{}
\cQ{\item}{~~$\bullet$~~} is more intuitive (and simpler to compute) than FT;
\cQ{\item}{~~$\bullet$~~} does not tell that natural situations are unnatural;
\cQ{\item}{~~$\bullet$~~} does not tell that unnatural situations are natural
(differently from our $\%$, the FT only looks at how the $Z$ mass is obtained:
but other unnatural situations can be present, as we will exemplificate);
\cQ{\item}{~~$\bullet$~~} is not more subjective than --- say --- a 90\% C.L. bound on a neutrino mass.
\cQ{\end{itemize}}{

}
In short, our procedure is a sort of Bayesian analysis with an unconventional use of the
prior distribution: the only subjective aspect of our analysis is the
usual one, that no one knows how to to avoid.
So the real question is if this procedure is safe enough
(i.e.\ is there a decent choice of $p(\wp)$ that transforms the 5\% in table~1 into 50\%?):
we will discuss this question in the next section when we analyze concrete models.

\section{The supersymmetric naturalness problem}\label{masses}
In this section we discuss how serious is the supersymmetric naturalness problem
in the various different motivated scenarios of SUSY breaking.
We will not consider models with extra fields at low energy beyond the minimal ones present in the MSSM
and we will assume that $R$ parity is conserved.
To be conservative {\em we will consider as excluded only those spectra that violate the experimental
bounds coming from direct searches\/} at accelerators.

\subsection{A simple example}\label{example}
We now try to illustrate the previous discussions with a simple and characteristic example.
We consider the `most minimal' gauge mediation scenario with very heavy messengers
(one $5\oplus\bar{5}$ multiplet of the unified gauge group SU(5) with mass $M_{\rm GM}=10^{15}\,{\rm GeV}$).
We also assume that the unknown mechanism that generates the $\mu$-term
does not give additional contributions to the other parameters of the higgs potential.
Since the $B$ term vanishes at the messenger scale,
we obtain moderately large values of $\tan\beta\sim20$.
This model is a good example because it is not completely irrelevant and
\begin{itemize}

\item its spectrum is not very different from
a typical supergravity spectrum and is the most natural possibility within
`conventional' models.
It has the lowest FT (we fix the unprecisely measured parameter
$\lambda_t(M_{\rm GUT})$ at a small value (0.5)
that alleviates the naturalness problem).

\item it is simple because it has only two free parameters:
the overall scale of gauge mediated soft terms and the $\mu$ term.
The condition of correct electroweak breaking fixes the overall mass scale,
and only one dimensionless parameter remains free.
We choose it to be $\wp\equiv \mu/M_2$ (renormalized at low energy),
where $M_2$ is the mass term of the $\SU(2)_L$ gaugino.

\end{itemize}
In figure~1a we plot, as a function of $\wp$,
the lightest chargino mass in GeV.
The dark gray regions where $\mu$ is too small or too large are always excluded from our analysis
because the electroweak gauge symmetry cannot be properly broken.
Values of $\wp$ shaded in \Orange light gray\special{color cmyk 0 0 0 1.}{}
are excluded because some supersymmetric particle is too light
(in this example the chargino gives the strongest bound).
The chargino is heavier than its LEP2 bound
only in small ranges of the parameter space at $\mu/M_2\approx  2.3$ (left unshaded in fig.~\ref{fig:example})
close to the region where EW symmetry breaking is not possible
because $\mu$ is too large.
{\em The fact that the allowed regions are very small and atypical
is a naturalness problem for supersymmetry}.
About $90\%$ of the acceptable values of $\wp$ are now excluded
by the chargino mass bound.
This percentage would not be significantly different if we had used $\mu^2$ in place of $\mu$ as a parameter;
it could be significantly higher only choosing a rather ad-hoc parametrization.
Indirect bounds (that we neglect) are not completely negligible:
about half of the allowed parameter space of this example is excluded
because the $b\to s\gamma$ decay is too fast.

\begin{figure}[t]
\caption[SP]{
Fig.~1a: the naturalness problem in a simple supersymmetric model:
values of $\mu/M_2$ marked in dark gray are unphysical,
while \Orange light gray\special{color cmyk 0 0 0 1.}{} regions have too light sparticles.
Only the small white vertical band is now experimentally acceptable.
Fig.~1b: Scatter plot of chargino mass vs higgs mass
done with sampling density proportional to the naturalness probability
in minimal supergravity.
Only the \Green light gray\special{color cmyk 0 0 0 1.}{} sampling points are allowed.
The shading covers excluded regions.
Fig.~1c: naturalness distribution of chargino mass:
allowed spectra give the $\sim 5\%$ tail in \GreenDark dark gray\special{color cmyk 0 0 0 1.}{}.
\label{fig:example}}
\begin{center}
\begin{picture}(17,5)
\putps(-0.5,-1)(-0.5,-1){figMoriond}{figMoriond.ps}\special{color cmyk 0 1. 1. 0.5}
\put(2.5,4.5){fig.~1a}
\put(8.3,4.5){fig.~1b}
\put(14.1,4.5){fig.~1c}\special{color cmyk 0 0 0 1.}
\end{picture}
\cQ{\vspace{5mm}}{}
\end{center}\end{figure}

\subsection{Minimal supergravity}
``Minimal supergravity'' assumes common values $m_0$, $M_{1/2}$ and $A_0$
at the unification scale $M_{\rm GUT}\approx2\cdot10^{16}\,{\rm GeV}$
for the sfermion masses,
the three gaugino masses, and the $A$-terms.
The parameters $\mu_0$ and $B_0$ are free.
\cQ{Since we adopt standard notations, we do not list the exact definitions of
the various supersymmetric parameters.}{}
As explained in the previous section we randomly fix the dimensionless ratios
of the soft terms
and compute the overall supersymmetric
mass scale ``$m_{\rm SUSY}$'' from the minimization condition of the MSSM potential.
More precisely we scan the parameters
within the following ranges
$$m_0,|\mu_0|,|M_{1/2}|,|B_0|,|A_0|=(0\div 1)m_{\rm SUSY},\qquad
(a\div b)\equiv \hbox{a random number between $a$ and $b$}.$$
The alternative scanning procedure
\cQ{$$m_0=(\frac{1}{9}\div 3) m_{\rm SUSY},\qquad
|\mu_0|,|M_{1/2}|=(\frac{1}{3}\div 3) m_{\rm SUSY},\qquad
A_0,B_0=(-3\div 3) m_0$$}
{$m_0=(\frac{1}{9}\div 3) m_{\rm SUSY}$,
$|\mu_0|,|M_{1/2}|=(\frac{1}{3}\div 3) m_{\rm SUSY}$,
$A_0,B_0=(-3\div 3) m_0$}
(with the samplings of $m_0, M_{1/2}$ and $\mu_0$ done with flat density in logarithmic scale)
would give the same final results: {\em
\cQ{in the MSSM with soft terms mediated by `minimal supergravity'}{}
the present experimental bounds exclude $95\%$ of our parameter space.\/}

As in the previous example, reasonable different definitions of `order one'
(i.e.~of the prior Bayes distribution)
do not alleviate the naturalness problem:
we could make the naturalness problem apparently more dramatic
by restricting the dimensionless ratios to a narrow region
that does not include some significant part of the experimentally allowed region,
or by extending the range to include larger values that produce a larger spread
in the spectrum so that it is more difficult to satisfy all the experimental bounds.
Allowing one or few soft terms to be much smaller or much larger than the others does not help.
Complex soft terms only give problems with electric dipoles.

\smallskip

\cQ{We now exhibit the results of this analysis in a series of figures.}{}
In fig.~1b we show a scatter plot
with sampling density proportional to the naturalness probability.
The sampling points corresponding to excluded (still allowed) spectra are drawn in black,
(in \Green light gray\special{color cmyk 0 0 0 1.}{}).
Fig.~1b shows the correlation between the masses of the lightest chargino
and of the lightest higgs, $m_\chi$ and $m_h$.
We see that the most natural value for the chargino mass is around $30\,{\rm GeV}$
(rather than $300\,{\rm GeV}$):
however numerical accidents can sometimes make the chargino so light that it can be produced
at $B$-factories, or make it so heavy that it cannot be produced at LEP2.
Fig.~1b also shows that the experimental bounds on $m_\chi$ and $m_h$
are the most important ones in minimal supergravity
(and that the bound on $m_\chi$ is more important than the one on $m_h$).
The LEP2 bounds together with the assumption of gaugino mass universality at the unification scale
give stronger constraints on the mass of coloured sparticles than the direct bounds from Tevatron experiments.

In fig.~1c we show the same kind of results using a different format:
we plot the density of the points in fig.~1b as function of $m_\chi$.
We obtain a bell centered around $30\,{\rm GeV}$.
Allowed spectra are confined in fig.~1c to
the small ($\sim 5\%$) upper tail where the chargino is anomaously heavy.

\subsection{`Unified' supergravity}
If we only impose unification relations between soft terms,
the naturalness problem remains unchanged:
an unnatural cancellation remains necessary even if there are more parameters.
From the point of view of phenomenology, maybe the most interesting
new possibility is that a mainly right handed stop
can `accidentally' become significantly lighter than the other squarks.
A very light stop can mediate large loop effects (in $b\to s\gamma$, $\Delta m_B$, $\varepsilon_K$, \ldots)
and allows baryogenesis at the weak scale\cite{EWbaryogenesis}.
However this possibility is severely limited by naturalness considerations\cite{lightStop,nat}:
it requires not only that one numerical accident makes the $Z$ boson
sufficiently lighter than the unobserved chargino,
but also that a second independent numerical accident gives a stop lighter than the other squarks.
This unnatural combination occurs very rarely ($p\sim10^{-3}$) in our analysis.

\subsection{Gauge mediation}
For our purposes `gauge mediation' models\cite{GaugeSoft} can be characterized
saying that they predict the
soft terms $M_i$, $m_R$, $A$ of sparticles
in terms of their gauge charges $c_R^i$ in the following way:
$$
M_i(M_{\rm GM})=\frac{\alpha_i(M_{\rm GM})}{4\pi} M_0,\qquad
m_R^2(M_{\rm GM}) =  \frac{c^i_R }{\sqrt{n}}M_i^2(M_{\rm GM}) ,\qquad
A(M_{\rm GM})=0
$$
($R$ runs over the sfermions and $i$ over the gauginos).
These predictions depend on three unknown parameters:
the overall scale of soft terms $M_0$,
the messenger mass $M_{\rm GM}$ and the `number of messengers' $n$.
A computation shows that:
\cQ{\begin{itemize}}{}
\cQ{\item}{~~$\bullet$~~} If the messengers are heavy, $M_{\rm GM}\circa{>}10^{12}\,{\rm GeV}$
the gauge mediated sparticle spectrum, and the naturalness problem,
are not much different than in supergravity.
\cQ{\item}{~~$\bullet$~~} If $M_{\rm GM}\sim 10^{(6\div 10)}\,{\rm GeV}$ gauge mediation models predict a right handed
selectron mass significantly smaller than the higgs mass term which sets
the scale of electroweak symmetry breaking.
This makes the naturalness problem more acute\cite{GMFT}.
\cQ{\item}{~~$\bullet$~~} If the messengers are very light ($M_{\rm GM}\circa{<}100\,{\rm TeV}$)
and the lightest neutralino $N$ decays in a detectable way ($N\to \gamma\,\hbox{gravitino}$)
within the detector,
experimentalists can put a {\em very\/} strong bound on its mass
that makes the model very unnatural\cite{nat}.
\cQ{\end{itemize}}{}
\cQ{Extremely light messengers ($M_{\rm GM}\approx 10\,{\rm TeV}$)
could give rise to a more natural sparticle spectrum, but unknown NLO corrections,
that depend on unknown couplings between messengers, become relevant in this limiting case.}{}

\section{The credibility of supersymmetric models after LEP2}
If we employ the standard notation
\cQ{$$}{$}
p(A|B)\equiv \hbox{(probability of $A$ assuming that $B$ is true)}
\cQ{$$}{$}
we can compactely rewrite our main result as
\cQ{\begin{equation}}{$\special{color cmyk 1. 1. 0.3 0}}
p(\hbox{``LEP''}|\hbox{``SUSY''})=5\%\label{eq:5}
\cQ{\end{equation}}{$\special{color cmyk 0 0 0 1.}}
where \hbox{``LEP''}{} and \hbox{``SUSY''}{} are abbreviations for
\begin{eqnarray*}
\hbox{``LEP''} &=&\hbox{``no supersymmetric signal found at LEP and Tevatron experiments''}\\
\hbox{``SUSY''} &=&\hbox{``MSSM with minimal supergravity, with comparable soft terms''}
\end{eqnarray*}
Our result means that if \hbox{``SUSY''}{} is true, it is strange that
sparticles have not been discovered at LEP\footnote{We repeat that {\bf \hbox{``SUSY''}{} does not mean supersymmetry\/} ---
it is just a particular representative model often used for discussing capabilities
of accelerators and for other purporposes (we are also assuming that soft terms are comparable,
and that there is no special correlation between them).
However a similar naturalness problem is present in all `conventional' supersymmetric models.}.
We do {\bf not} claim that $p(\hbox{``SUSY''}|\hbox{``LEP''})=5\%$, i.e.\ that \hbox{``LEP''}{} exclude \hbox{``SUSY''}{} at $95\%$ C.L.
Nothing says that the inverse probability $p(\hbox{``SUSY''}|\hbox{``LEP''})$
is equal to the direct one $p(\hbox{``LEP''}|\hbox{``SUSY''})$.
However we can say something on the more interesting inverse probability because
the Bayes theorem\cite{Bayes}
says how they are related:
$$\frac{p(\hbox{``SUSY''}|\hbox{``LEP''})}{p(\hbox{``SUSY''})}=\frac{5\%}{1-95\%p(\hbox{``SUSY''})}$$
In order to apply the theorem we have introduced an additional quantity
$p(\hbox{``SUSY''})=$ degree of belief that \hbox{``SUSY''}{} (rather than some other
unspecified mechanism) is the solution to the SM naturalness problem
before knowing the experimental results.

The theorem says how much one must reduce its confidence in \hbox{``SUSY''}{} after knowing the LEP2 results.
Of course it gives obvious results.
For example who very strongly believes in $\hbox{``SUSY''}$ would insert $p(\hbox{``SUSY''})=1$
and get $p(\hbox{``SUSY''}|\hbox{``LEP''})=1$: experiments are explained by saying that we live in the
small allowed corner of the parameter space of \hbox{``SUSY''}{} with heavy sparticles.
Who believes strongly in supersymmetry could instead suspect that the true model has not yet been found.

\section{Conclusions}\label{fine}
In conclusion, the negative results of the recent searches for supersymmetric particles
pose a naturalness problem to all `conventional' supersymmetric models.
What could be the reason of these negative experimental results?
\cQ{\begin{itemize}}{}
\cQ{\item}{~~$\bullet$~~}  Maybe SUSY has escaped detection beacuse a numerical accident
has made the sparticles too heavy for LEP2.
This happens with $\sim5\%$ probability.
Once one accepts the presence of a numerical accident,
it is no longer extremely unlikely that the coloured sparticles have mass of few TeV (beyond the LHC reach) due to an
accidental cancellation stronger than the `minimal one'~\cite{nat};
\cQ{\item}{~~$\bullet$~~} Maybe SUSY has not been discovered because it does not exists at the weak scale;
\cQ{\item}{~~$\bullet$~~} Maybe SUSY exists at the weak scale, there is no unluck accident, but
some ingredient is missing in the `conventional' models.
\cQ{\end{itemize}}{

}
Some recent papers have studied few different possible ways of alleviating the naturalness problem:
\cQ{\begin{enumerate}}{}
\cQ{\item}{(1.)} A correlation between the $\mu$ term and the gaugino mass\cite{corr}.
However this correlation reduces the FT with respect to $\mu$, but not the FT
with respect to $\lambda_{\rm top}$ or to $\alpha_{\rm strong}$.
The problem is that the required value of $\mu/M_{1/2}$ depends on the values of
$\lambda_{\rm top},\alpha_{\rm strong},\tan\beta\ldots$
(i.e.\ even if the high-energy theory predicts a $Z$ boson ligher than sparticles,
the prediction is destabilized by large RGE corrections).
\cQ{\item}{(2.)}
Models with SUSY breaking mediated at lower energy could predict a light $Z$ boson,
not destabilized by quantum corrections.
However concrete models of this kind (gauge mediation\cite{GaugeSoft}
and higher dimensional Scherk-Schwarz\cite{SS}\footnote{The predictions
of these models depend on the form of the ultraviolet cutoff;
even for the cutoff used in\cite{SS} the dependence of the higgs potential
on the arbitrary matching scale
(beteween the full theory and the low energy 4-dimensional theory)
is so strong to require a NLO computation, which
result would depend on the unknown details of the high energy theory.} models)
turn out to be not more natural than supergravity.
\cQ{\item}{(3.)}
A sparticle spectrum more degenerate than the `conventional' one.
Since the Tevatron direct bound on the gluino mass is weaker than the indirect bound
obtained from LEP2 assuming gaugino mass universality,
it is possible to reduce the mass of coloured particles
(and consequently their large RGE corrections to the $Z$ mass, that sharpen the naturalness problem)
if gaugino mass universality is strongly broken\cite{FTred,bau,nat}.
The phenomenology is more interesting than in `conventional' models:
the gluino is lighter (could be discovered at Tevatron) and
the minimal cross section for detecting a neutralino dark matter is $\sim 50$ times higher.
However even this case is severely constrained by experimental bounds:
often at least one sparticle happens to be lighter than the $Z$ boson
and contradics one of the many experimental bounds.
\cQ{\end{enumerate}}{

}
In conclusion  it is certainly possible to debate if there is a supersymmetric naturalness problem,
sometimes with considerations that also make the SM natural,
thus killing the motivation for supersymmetry.
Once that the problem is accepted it is however a solid problem:
as usual it cannot be eluded with mild assumptions about the high energy theory.

\special{color cmyk 0 0 0 1.}
\frenchspacing
\section*{References}


\section{Introduction}

The cosmic X--ray background (XRB) above $\sim 1$ keV is the result of the
integrated emission of discrete sources, since the contribution of any
intergalactic hot medium must be negligible (Wright et al. 1994).
In the soft X--ray band from 0.5 to 2 keV the largest fraction of the XRB has
already been resolved into sources (Hasinger et al. 1998), most of which turned
out to be broad line active nuclei (Schmidt et al. 1998), i.e.
Quasi Stellar Objects (QSOs) and Seyfert galaxies of type~1.
The spectra of these sources are however too steep to reproduce also the hard
XRB at several tens of keV, where the bulk of the energy resides, and a
population of objects with flatter spectra is therefore required.

The most popular synthesis models of the XRB are based on the so--called
unification schemes for Active Galactic Nuclei (AGNs), where the orientation
of a molecular torus
surrounding the nucleus determines the classification of the source.
At a zeroth--order approximation level, sources observed along lines of
sight free from the torus
obscuration should have unabsorbed X--ray spectra and optical
broad lines (type 1 AGNs), while sources seen through the torus should
have absorbed
X--ray spectra and appear as narrow line objects in
the optical (type 2 AGNs, e.g. Seyfert 2 galaxies).

In this framework type 2 AGNs provide a natural class of sources
with X--ray spectra flattened by absorption. The intrinsic
X--ray luminosity function (XLF) of type 2 objects is unknown and has been
usually assumed to be the same as the one derived for type 1s (e.g. Boyle et
al. 1993), apart from a normalization factor. The cosmological
evolution has also been taken identical for type 1s and type 2s.
Under these assumptions it has been shown that the broad band 3--100 keV
spectrum of the XRB can be reproduced by an appropriate mix of
unabsorbed and absorbed AGNs (Matt \& Fabian 1994; Madau, Ghisellini \&
Fabian 1994; Comastri et al. 1995, hereafter Co95). The number ratio $R$
of type 2 to type 1 objects, as well as the distribution of the
absorbing column densities $N_{\rm H}$, are key parameters of the models;
these have been assumed to be independent of redshift and of intrinsic
source
luminosity, and have been treated as free parameters in the fitting
procedure.
Since the overall parameter space of the models is quite large and a good
fit to the XRB can be obtained with different set of values, it is
important to compare the model predictions with the largest number of
observational constraints. Indeed, Co95
showed that the source counts in the 0.5--2 keV
and 2--10 keV energy bands, as well as the redshift distributions, could
successfully be reproduced by their model.

Very recently an additional set of observational constraints
has become available.
Deep surveys from ROSAT have extended our knowledge to
the low luminosity part of the AGN XLF (Miyaji, Hasinger \& Schmidt
1999a, hereafter Mi99a).
Contrary to previous results (Boyle et al. 1993; Page et al. 1996; Jones
et al. 1997) a pure luminosity evolution (PLE) of AGNs with redshift is no
longer consistent with the data, and a luminosity
dependent density evolution (LDDE) is required.
From the X--ray data of an optically selected sample of Seyfert galaxies
Risaliti, Maiolino \& Salvati (1999) have determined the $N_{\rm H}$
distribution for local
Seyfert 2 galaxies, pointing out that a significant fraction of sources
have columns exceeding $N_{\rm H}=10^{25}$ cm$^{-2}$ and are therefore
completely thick to Compton scattering. The $R$ ratio between type 2s and
type 1s has been determined in the local Universe for low luminosity AGNs,
i.e. Seyfert galaxies (Maiolino \& Rieke 1995), while
the existence of a relevant number of high luminosity absorbed
sources, the so--called QSO~2s, which is a basic assumption of
previous models, is still uncertain (Akiyama et al. 1998).
An observational constraint to
the QSO~2 number density can be obtained from the infrared source
counts. Indeed, QSO~2s are expected to
have strong infrared counterparts, since
the dust present in the torus should re--emit in the IR band the
nuclear radiation absorbed by the gas.
The ultraluminous infrared galaxies (ULIRGs) discovered by IRAS are the
only local objects with QSO--like bolometric luminosities (Soifer et al.
1986; Kim \& Sanders 1998).
Thus, even if all ULIRGs were powered by a hidden AGN,
the local QSO~2s could not be more numerous than ULIRGs.
Finally, source counts in the 5--10 keV band have been derived for the
first time by the BeppoSAX satellite with the HELLAS survey
(Giommi et al. 1998; Comastri et al. 1999).

In the present paper we test the standard synthesis
model to verify if it remains compatible with the new data. These data
leave still some latitude to important parameters of the model, and various
choices are possible to fit the XRB equally well. However, in all cases
we find moderate but consistent evidence that at least some of the standard
assumptions have to be relaxed: extra hard spectrum AGNs are needed at
intermediate or high redshifts, in addition to those expected in the usual
scenario. The additional sources could be analogous to
local Seyfert 2s, if they evolve faster than type 1s, or they could be
other astrophysical sources not yet enlisted among the contributors to the
XRB. We discuss the observations which could distinguish between
the alternatives.

Throughout this paper the deceleration parameter and the Hubble constant
are given the values $q_{0}=0.5$ and $H_{0}=50$ km s$^{-1}$ Mpc$^{-1}$.

\section{AGN X--ray properties}

\subsection{The spectra}

After the observations of X--ray satellites like GINGA, ASCA and
BeppoSAX, different components
have been recognized in the X--ray spectra of
AGNs. Starting with Sey 1 galaxies, the basic component is a power law
with energy
spectral index $\alpha\sim0.9$ (Nandra et al. 1997a) and an
exponential
cut off at high energies. A mean value for the $e$--folding energy can
probably be set at $\sim 300$ keV, although the observed dispersion is
very high (Matt 1998).
Some of the primary radiation is reprocessed by an accretion disc and/or
the torus around the nucleus, producing a flattening of the spectral slope
above $\sim 10$ keV, and a strong iron line at 6.4 keV (Nandra \& Pounds 1994).
Below 1--1.5 keV a radiation excess with respect to the power law
emission is detected in a large fraction of Sey 1s (sometimes resulting
from a misfit of the ``warm absorber'' component).

The spectrum of QSO~1s is similar to that of Sey 1s, but there is no evidence
for the iron line and the reflection hump to be as common (Lawson \& Turner
1997). Assuming that the accretion disc produces most of the line and hump,
Nandra et al. (1997b) ascribe these differences to a higher ionization state
of the disc in higher accretion rate sources, so that in QSO~1s the
spectral features due to photoelectric processes are quenched. Recently,
Vignali et al. (1999) have derived a mean spectral slope of $\langle
\alpha \rangle =0.67\pm 0.11$ from a sample of 5 QSO~1s at redshifts above
2. Although the statistics is poor, this result seems to suggest that the
spectra of high redshift QSOs are flatter than those of local ones.

In Sey 2 galaxies the power law is cut off by photoelectric absorption at
energies increasing with the column density of the intercepted torus.
For highly absorbed objects the X--ray luminosity may be dominated by that
fraction of the nuclear radiation which is reflected off the torus surface
towards the observer. When $N_{\rm H}>10^{25}$ cm$^{-2}$ the
obscuring medium is completely thick to Compton scattering
and the spectrum is a pure reflection continuum as described by Lightman
\& White (1988), with a 2--10 keV luminosity about two orders of
magnitude lower than that of Sey 1s (Maiolino et al. 1998). On the
contrary, when $N_{\rm H}<10^{24}$ cm$^{-2}$ the medium is Compton--thin
and the spectrum is dominated by the component transmitted through the
torus. In the range
$10^{24}<N_{\rm H}<10^{25}$ cm$^{-2}$ both a transmitted and a reflected
component contribute to the observed luminosity, the Circinus galaxy
(Matt et al. 1999) being a typical example.
Also Sey 2 galaxies often have soft emission in excess of
the absorbed power law (Turner et al. 1997).
These soft excesses are however two orders of magnitude weaker than those
of Sey 1s of the same intrinsic luminosity and their nature is still
unclear (probably scattered or starburst radiation).

\subsection{The XLF and cosmological evolution}

The most recent results about the AGN XLF and cosmological evolution
have been obtained by Mi99a by combining data from several
ROSAT surveys. Down to a limiting flux of $10^{-15}$ erg s$^{-1}$
cm$^{-2}$, reached by the deep survey in the Lockman Hole, they
collected a sample of about 670 sources,
which is the largest X--ray selected sample of AGNs presently available.
The local XLF is described with a smoothed double power law of the
following form:

\begin{displaymath}
\phi(L_{\rm x})=\frac{{\rm d}\,\Phi(L_{\rm x})}{{\rm
d\,log}L_{\rm x}}={A}\,\left[(L_{\rm x}/{L_*})
      ^{{\gamma_1}}
         +(L_{\rm x}/{L_*})^{{\gamma_2}}
        \right]^{-1},
\end{displaymath}

\noindent where $L_{\rm x}$ is the observed 0.5--2 keV X--ray luminosity,
ranging from $10^{41.7}$ to $10^{47}$ erg s$^{-1}$.
The best fit values for the cosmology adopted here are:
$A=(1.57 \pm 0.11)\times 10^{-6}$ Mpc$^{-3}$,
$L_*=0.57^{+0.33}_{-0.19} \times 10^{44}$ erg s$^{-1}$, $\gamma_1=0.68\pm
0.18$ and $\gamma_2=2.26\pm 0.95$.

The XLF has been found to evolve from redshift 0 up to $z_{cut}=1.51\pm
0.15$, with an evolution rate which drops at low
luminosities according to the factor:
\begin{eqnarray}
e(z,L_{\rm x})=        \left\{
        \begin{array}{ll}
        (1+z)^{\max(0,{p1}-{\alpha}({\rm log}\;
          {L_{\rm a}} - {\rm log}\;L_{\rm x}))}
             & L_{\rm x}<L_{\rm a} \\
        (1+z)^{{p1}}
             & L_{\rm x}\ge L_{\rm a}\;;\\
        \end{array}
	\right.
       \nonumber
\end{eqnarray}

\noindent here $p1=5.4\pm 0.4$, $\alpha = 2.3\pm 0.8$ and
${\rm log} L_{\rm a}=44.2$ (fixed).
The X--ray AGNs have been observed at redshifts up to $z=4.6$ and there is no
evidence for a decline in their space density beyond $z\sim 3$, unlike what
is found in optical (Schmidt, Schneider \& Gunn 1995) and radio surveys
(Shaver et al. 1997).

We note that the XLF parametrization of Mi99a
is a preliminary result and is not a unique
representation of the ROSAT data. The extrapolation of the high
redshift XLF into the low luminosity range, where few data are available,
is not well constrained. Indeed, the number of
low luminosity, high redshift AGNs could be higher than
the Mi99a representation (Hasinger et al. 1999). Another cause of
uncertainty is the possible presence of type 2 AGNs in the
Mi99a sample: unlike previous works, where only (optical) type 1 AGNs
were included, Mi99a do not discriminate between type 1s and type 2s;
some of the latter could then appear in the ROSAT bandpass because of their
soft excesses and, for sources at high redshifts, because of the
$K$--correction. Objects with type 2 optical spectra are relatively
rare in the ROSAT sample (Hasinger et al. 1999). As for the X--ray
spectral type, within any given model the raw counts can be corrected
for the contribution of the absorbed sources, and these can
be subtracted from the XLF: in the following we consider also this approach,
and investigate the robustness of our conclusions with respect to the
correction. In general, the correspondence between optical and X--ray
spectral classification is broadly verified in the local Universe, albeit
with some blurring (see Section 2.3); at high redshifts the question is
still unsettled.

Previous works about the XLF of AGNs used a PLE model both in the soft
X rays, by combining observations from ROSAT and Einstein (Boyle
et al. 1993, 1994; Jones et al. 1997), and in the hard X rays from ASCA
data (Boyle et al. 1998a). In the Mi99a data the fit with a PLE model is
rejected with a high significance. A pure density evolution model is
marginally rejected, and LDDE models are preferred, even if several variants
are still being discussed.

\subsection{The number and column densities of local type 2 AGNs}

In the local Universe 5--10\% of the galaxies show Seyfert activity
(Maiolino \& Rieke 1995; Ho, Filippenko \& Sargent 1997). From
a sample of $\sim 90$ nearby Seyfert galaxies limited in the B
magnitude of the host galaxy, Maiolino \& Rieke (1995) derived an estimate
for the local ratio $R$ of type 2 to type 1 Seyferts.
From our point of view Seyfert types 1.8, 1.9 and 2, which
have flat X--ray spectra due to absorption by cold gas, can be grouped
as type 2 objects, while types 1, 1.2 and 1.5, which have steep X--ray
spectra without significant cold absorption ($N_{\rm H}<10^{21}$
cm$^{-2}$), can be grouped as type 1s. Here it is noted that
the relation between Seyfert type and X--ray absorption is not univocal.
By observing with ROSAT the complete sample of Piccinotti et
al. (1982), Schartel et al. (1997) showed that
at least a fraction of type 1 AGNs suffer from X--ray
absorption by more than $N_{\rm H}=10^{21}$ cm$^{-2}$.
However this fraction (grouping Seyfert 1, 1.2 and 1.5) is only 20\%, and
on average their $N_{\rm H}$ does not exceed $10^{22}$ cm$^{-2}$. The
inclusion of some
moderate--absorption type 1s should not change significantly our results.

By considering Seyfert types 1.8, 1.9 and 2 as type 2s, and Seyfert types 1,
1.2 and 1.5 as type 1s, Maiolino \& Rieke found $R$=4.0$\pm$0.9, in agreement
with the results of
Osterbrock \& Martel (1993) and, more recently, Ho et al. (1997).

From the Maiolino \& Rieke sample Risaliti et al. (1999)
have derived a distribution of X--ray column densities for Sey 2s.
The selection of the sample by means of optical narrow emission lines,
rather than in the X--rays,
should avoid biases against X--ray absorbed sources. It
turned out that most of the sources are affected by strong absorption,
$\sim 75\%$ of the objects having $N_{\rm H}>10^{23}$ cm$^{-2}$.
Furthermore, a significant fraction of sources ($>25\%$) are
absorbed by $N_{\rm H}>10^{25}$ cm$^{-2}$.
Their results are shown in Fig.~1 and compared with the $N_{\rm H}$
distribution assumed by Co95.

\begin{figure}
\epsfig{file=8484_fig1.ps,  width=9.cm, height=9.cm,
angle=0}
\caption{The normalized $N_{\rm H}$ distribution in Sey 2 galaxies
derived by Risaliti et al. (1999) [Panel a)], compared with
the distribution assumed by Co95 [Panel b)]. The shaded area represents
the fraction of sources for which only a lower limit of log$N_{\rm H}>24$
is available.}
\end{figure}

In the limited luminosity range of Seyfert galaxies Risaliti et al. (1999)
did not find evidence of a correlation between absorption and luminosity.
In the wider luminosity domain which includes the QSOs the evidence
is contradictory. Recent results from the IRAS 1--Jy survey (Kim \& Sanders
1998) show that the space density of ULIRGs at $z\stackrel{<}{_{\sim}}
0.1$ is similar to that of optically selected QSOs of
comparable bolometric luminosity. By considering that the number
of obscured
QSOs cannot exceed that of ULIRGs, and assuming that every ULIRG is powered
only by nuclear activity, we can set a conservative upper limit of
2 to the $R$ ratio at high luminosities.
The actual value of $R$ might be significantly lower. Indeed,
there is evidence that the fraction of AGN--powered ULIRGs
decreases from 50\% for IR luminosities $L_{\rm
IR}\stackrel{>}{_{\sim}} 1.7\times 10^{46}$ erg s$^{-1}$ to 15\% below
this value (Lutz et al. 1998), the remaining ones being dominated by
starburst activity.

\section{The model}

\subsection{The XRB spectrum}

Our models are completely analogous to those of the canonical lineage,
the only differences arising from updated input parameters. A key set
of such parameters is the one referring to the XLF and its cosmological
evolution, and for model A1 we adopt the results of Mi99a.
Strictly speaking, the XLF of Mi99a refers to the observed 0.5--2 keV
luminosities and could be considered as the 0.5--2 keV XLF in the rest
frame only by assuming simple power law spectra with energy index
$\alpha=1$ (i.e. zero $K$--correction). Indeed, in the rest frame energy
range seen by ROSAT at different redshifts, the spectra of type
1 AGNs assumed in our models do not differ significantly from a
power law with $\alpha=1$. Since we refer the Mi99a XLF to unabsorbed
AGNs, without correcting for the contribution of absorbed AGNs to the
ROSAT counts, model A1 might be biased in favor of a soft XRB.
We discuss the strength of this bias in connection with models A2 and B
in the following.

The absorption distribution in type 2 AGNs is no longer derived from best
fitting, instead it is taken equal to the local one, as measured by Risaliti
et al. (1999). The objects for which only a lower
limit is available, $N_{\rm H}>10^{24}$ cm$^{-2}$, have been
assigned to the bin $10^{24}<N_{\rm H}<10^{25}$ cm$^{-2}$.

Because of the evidence that the $R$ ratio decreases with the intrinsic
luminosity of the AGNs, at least locally, we introduce a change with respect
to the canonical scenario: the XLF is divided in two luminosity regions
as follows:

\begin{displaymath}
\phi(L_{\rm x})=\phi(L_{\rm x})e^{-\frac{L_{\rm x}}{L_s}}+\phi(L_{\rm
x})(1-e^{-\frac{L_{\rm x}}{L_s}})\;,
\end{displaymath}

\noindent with the 0.5--2 keV $e$--folding luminosity set
equal to $L_s$=10$^{44.3}$ erg s$^{-1}$, following Miyaji, Hasinger \&
Schmidt (1999b; hereafter Mi99b).
The first and second term represent the XLF of Sey 1s and QSO 1s,
respectively, and, apart from the exponential factors, are equal to
the Mi99a functions as given in Section 2.2.
The XLF of Sey 2s and QSO 2s are $R_{\rm S}$ and $R_{\rm Q}$ times the XLF
of the corresponding type~1 objects. In this parametrization we can explore
various hypotheses, including for instance the effects of eliminating
altogether the QSO 2s.

Following Co95, and the experimental evidence referred to in Section 2.1,
we assume that the basic spectrum for
type 1 AGNs is a power law with energy index $\alpha=0.9$ and
exponential cut off with $e$--folding energy $E_c=320$ keV. Below 1.5 keV
the soft excess is modeled with a power law of index 1.3. A reflection
component from the accretion disc has been included for Sey 1s with
relative normalization $f_d=1.29$ (Co95).
Beside the disc, we have also included for Sey 1s a
torus reflection component which is
normalized in accordance with the prescriptions of
Ghisellini, Haardt \& Matt (1994). In type 1 AGNs the relative
contribution of the torus at 30 keV is 29\% and 55\% for $N_{\rm H}=10^{24}$
and $10^{25}$ cm$^{-2}$, respectively. If we assume that the column density
of the torus is approximately the same for all obscured lines of sight,
from the measured $N_{\rm H}$ distribution we find that the torus contributes
on the average 28\% at 30 keV. The same disc and torus reflection
components of Sey 1s have been included also in the QSO spectra:
this is against the evidence at low redshifts, but mimics the harder
power law seen at high redshifts (Vignali et al. 1999), where most of the
XRB is produced. If anything, this assumption tends to reduce the need
for additional hard spectrum sources, thus strengthening our results.

\begin{figure}
\epsfig{file=8484_fig2.ps,  width=9.cm, height=9.cm,
angle=0}
\caption{The fit to the XRB. The data of higher intensity below 6 keV are
from ASCA (Gendreau et al. 1995), while those above 3 keV are a compendium
of various experiments including HEAO--1 A2 (Gruber 1992). The solid
line is the overall fit while the other labeled curves
represent the contributions of the different classes of sources.
The radiation excess due to the iron line is also shown.}
\end{figure}

Sey 2 spectra have been computed for different amounts of intrinsic
absorption (log$N_{\rm H}$=21.5, 22.5, 23.5, 24.5, 25.5) to cover all the
observed column densities.
In the Compton thin regime the adopted spectrum is that of Sey 1s
with a photoelectric cut off and a lower amount of disc reflection
($f_d=0.88$, Co95). In this regime the component reflected by the torus
does not contribute significantly to the observed radiation (5\% at 30 keV
for log$N_{\rm H}$=23.5, inclusive of orientation effects).
For the sources with log$N_{\rm H}$=25.5 we have adopted a pure reflection
continuum. The normalization of the spectrum is determined so as to reproduce
the contribution of thick tori to the flux of Sey 1s (55\% at 30 keV)
after correcting for orientation effects (Ghisellini, Haardt \& Matt,
1994). This approach predicts that the
2--10 keV continuum luminosity of completely Compton thick sources
is about 2\% of the typical luminosity of Sey 1s, in agreement with the
results of Maiolino et al. (1998).
A composite reflected/transmitted spectrum has been considered for
Circinus--like sources with log$N_{\rm H}$=24.5, where the reflected
and transmitted components have been normalized in analogy with the
previous cases.
\footnote{For log $N_{\rm H}$=24.5 the effects of Compton scattering begin
to be important. We have checked the error introduced by our approximation
with respect to the MonteCarlo simulations of Matt, Pompilio \& La Franca
(1999). The
counts remain unaffected, while the model XRB at 30 keV should be
reduced by 10--15\%, and even more type 2s should be included in order
to maintain the agreement with the data.}

We have modeled the soft excess of Sey 2s with a power law of index 1.3
and a normalization at 1 keV which is 3\% of the primary
de--absorbed power law. In analogy with type 1s, the spectra of QSO 2s
--if at all present-- are assumed to be identical to those of Sey 2s.
Finally, we have added to the input spectra
an iron emission line at 6.4 keV. Following Gilli et al. (1999) we have
considered lines with different equivalent widths according to the spectral
absorption, and have not included the iron line in the spectra of QSOs.

\begin{figure}
\epsfig{file=8484_fig3.ps,  width=9.cm, height=9.cm, angle=0}
\caption{The predictions of model A1 compared with the
source counts observed in the 0.5--2 keV band by ROSAT. In this and in the
following Figures, source counts are plotted as $S_{14}^{1.5}\times
N(>S)$, where $S_{14}$ is the flux in units of $10^{-14}$ erg s$^{-1}$
cm$^{-2}$. The AGN data (squares) are adapted from Mi99a.
Cluster counts (stars) at $1\times 10^{-14}$ erg s$^{-1}$ cm$^{-2}$
and $6\times 10^{-14}$ erg s$^{-1}$ cm$^{-2}$ are from
Rosati et al. (1995) and Jones et al. (1998), respectively. The cluster
surface density at $1.2\times 10^{-11}$ erg s$^{-1}$ cm$^{-2}$ is adapted
from De Grandi et al. (1999).
The fluctuation limits to the total counts (dotted area) are
adapted from Hasinger (1998).}
\end{figure}

In our model A1 we assume $R_{\rm Q}=0$, i.e. we do not include QSO~2s.
The Mi99a
XLF are integrated in the range $10^{41}<L_{\rm x}<10^{49}$ erg s$^{-1}$
up to $z_{max}=4.6$.
The contribution of
clusters of galaxies has been included by considering thermal
bremsstrahlung spectra with
a distribution of temperatures. We have adopted the 2--10 keV luminosity vs
temperature relation of David et al. (1993), and the 2--10 keV
X--ray luminosity function of Ebeling et al. (1997). The
cluster XLF is assumed not to evolve, and is integrated in the range
$10^{42}<L_{\rm x}<10^{47}$ erg s$^{-1}$ up to $z_{max}=2$.

The overall XRB spectrum resulting from the model is shown in Fig.~2 as a
solid line, which is the sum of the contibutions of the other labeled curves.
In order to fit the observed XRB spectrum we need a ratio $R_{\rm S}=4.2$,
in good agreement with the local value.
Above $\sim 1$ keV, where the XRB is completely extragalactic,
the model provides a good fit to the data from
ASCA (Gendreau et al. 1995) and the compilation of Gruber (1992) based on
HEAO--1 A2 measurements.
The contribution of the AGN iron line to the model XRB is found to be less
than 7\% at $\sim 6.4/(1+z_{cut})$ in agreement with the results of Gilli
et al. (1999) obtained in a different framework (PLE).
Clusters of galaxies are found to contribute to
the model XRB by $\sim 12\%$ at 1 keV, in agreement with the results of
Oukbir, Bartlett \& Blanchard  (1997).

\subsection{The X--ray source counts}

We now compare the predictions of model A1 with the observed
source counts in different X--ray bands. The results in the soft 0.5--2
keV band are shown in Fig.~3.
The expected AGN counts, which are dominated by
unabsorbed sources, agree with the data of Mi99a; the expected cluster
counts agree with the data of Jones et al. (1998), Rosati et al.
(1995), and De Grandi et al. (1999). Since the XLF and its evolution are
derived from the ROSAT counts,
this is no more than a self--consistency check; the slight overprediction
at low fluxes ($\sim$30\% at $\sim 2\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$)
is due to the $K$--displaced type~2 objects, as anticipated previously.

\begin{figure}
\epsfig{file=8484_fig4.ps, width=9.cm,
height=9.cm, angle=0}
\caption{The predictions of model A1 (lines are as in Fig.~3) compared with the
hard counts between 2 and 10 keV. The open square at $\sim 3\times
10^{-11}$ erg s$^{-1}$ cm$^{-2}$ represents the AGN surface
density in the Piccinotti et al. (1982) sample corrected by 20\% after Co95.
ASCA data are represented by the filled squares (Cagnoni et al. 1998), the
open circle at $\sim 2\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$
(Ueda et al. 1998) and the open square at $\sim 4\times 10^{-14}$ erg
s$^{-1}$ cm$^{-2}$ (Ogasaka et al. 1998). The open triangle at
$5\times 10^{-14}$ erg s$^{-1}$ cm$^{-2}$ is from BeppoSAX (Giommi et al.
1998).}
\end{figure}

In the hard 2--10 keV band the predictions of the model have to be compared
with the results of HEAO--1 A2 (Piccinotti et al. 1982), ASCA (Cagnoni,
Della Ceca \& Maccacaro 1998; Ogasaka et al. 1998; Ueda et al. 1998)
and BeppoSAX (Giommi et al. 1998).
At the flux limit of $S\simeq 3\times 10^{-11}$ erg s$^{-1}$ cm$^{-2}$
Piccinotti et al. (1982) found that AGNs and clusters of galaxies have the
same surface density of $1.1\times 10^{-3}$ deg$^{-2}$. However, the AGN
density found by these authors is likely to be overestimated by $\sim
20\%$ due to the local supercluster (Co95). As shown in Fig.~4, after
the Piccinotti et al. point is
corrected by 20\%, the model is in agreement with the data within $1\sigma$.
On the contrary the disagreement cannot be solved at
fainter fluxes. At $S\sim 2\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$
the AGN surface density expected in our model
is about a factor of 2 lower than the measurement of Cagnoni et al. (1998).
This corresponds to a $\sim 2\sigma$ discrepancy. When comparing the
model with the
data of the ASCA Large Sky Survey (Ueda et al. 1998) the discrepancy is
even larger.

The situation is worse still in the 5--10 keV band. The only
available counts in this band are from the HELLAS survey
performed by BeppoSAX (Giommi et al. 1998; Comastri et al. 1999). At the
flux of $\sim 2\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$ the observed surface
density is $2.7\pm0.7$ deg$^{-2}$, which is a factor of 4 ($\sim 3\sigma$)
above the predictions (Fig.~5).

\begin{figure}
\epsfig{file=8484_fig5.ps,  width=9.cm,
height=9.cm, angle=0}
\caption{The expected counts in model A1 (lines are as in Fig.~3) compared
with the 5--10 keV BeppoSAX data (Giommi et al. 1998; Comastri et al. 1999).
The open square represents the 20\% corrected AGN surface density found by
Piccinotti et al. (1982), converted into the 5--10 keV band. The
conversion has been performed by assuming a mean of the input spectra of
our model, weighted for the $N_{\rm H}$ distribution of the Piccinotti et
al. sample (Schartel et al. 1997).}
\end{figure}

\subsection{Correction for absorbed sources}

One of the main assumptions of model A1 is that the XLF and
evolution derived by Mi99a refer to unabsorbed AGNs. However, as discussed
in Sect. 2.2, this is likely not the case. In order to evaluate the
effects of our assumption, we have computed a different variant, A2,
which adopts the XLF and LDDE of Mi99b. These authors allow
self--consistently within their model for the $K$--correction, and for
the absorbed sources which, especially at faint fluxes and high redshifts,
appear in the ROSAT counts; thus the parameters they provide refer
to unabsorbed sources only. Of course, our model is different from theirs,
and the self--consistency is lost; however, this is likely to be a higher
order effect, and for a first order estimate we can include their
parameters in our computation. The results are shown in Fig.~6. Note that
around 1 keV the unabsorbed sources produce 30\% of the XRB, exactly as in
Mi99b. Note also that in
order to fit the XRB spectrum a ratio $R_{\rm S}=13$ is now required, which is
much higher than the local value and implies additional hard spectrum sources.
Since the contribution of type 2s has increased with respect to A1, in order
to make up for the reduced contribution of type 1s, the mean spectrum of the
population producing the XRB is harder, and the discrepancies between the
model predictions and the hard counts are somewhat reduced (though not
completely eliminated).

\begin{figure}
\epsfig{file=8484_fig6a.ps,  width=9.cm, height=7.cm,
angle=0}
\epsfig{file=8484_fig6b.ps,  width=9.cm, height=7.cm,
angle=0}
\epsfig{file=8484_fig6c.ps,  width=9.cm, height=7.cm,
angle=0}
\caption{From top to bottom, the same as in Figs.~2, 4, 5 but for model
A2. Curves are as in the previous Figures.
Note the very hard spectrum of the model XRB due
to the high contribution of absorbed sources.}
\end{figure}

\section{Discussion}

The main difference between models A1 and A2 is the fractional contribution
of type 1 AGNs to the XRB; this contribution is dominated by objects close
to the XLF break at redshifts close to $z_{cut}$, and is not well constrained
by the data. In the former model 60\% of the 1--keV XRB is due to type 1s, so
the local value of the type ratio $R_{\rm S}$ is sufficient to account for the
entire XRB; the average spectrum, though, is too soft, and the softness shows
up in a marginal discrepancy with the XRB spectrum at $>40$~keV (Fig.~2),
and unacceptable discrepancies with the hard counts (Figs.~4 and
5). In the latter model the type 1s account for only 30\% of the 1--keV XRB,
and making up the entire XRB requires an $R_{\rm S}$ much larger than the local
value; now the average spectrum is harder, the shape discrepancy disappears
and the count discrepancies are reduced (Fig. 6). By extrapolating from
these two models we can make qualitative predictions on still different
parametrizations of the Mi99a sample: for instance, models adopting density
evolution with a dependence on luminosity weaker than Mi99a would predict a
type 1 soft X--ray contribution higher than 60\%, and would miss the hard
counts by factors larger than Figs.~4 and 5. Density evolution with a
dependence on luminosity stronger than Mi99b (if at all acceptable) would
require $R_{\rm S}>13$, which is already three times the local value.

The main result of our analysis is precisely this one: no matter which
variant is adopted for the XLF and the evolution, the models which
incorporate the most recent observations within the standard prescriptions
always produce some discrepancy. The discrepancy may appear as an
underprediction of the observed hard counts, or a type 2 to type 1 ratio
higher than the observed local value, but in all cases it points to
additional hard spectrum sources at intermediate or high redshifts.

For the sake of completeness, we present in the Appendix a PLE model
with $R_{\rm Q}=R_{\rm S}$ (model B): this region of the parameter space
is not favored by the most recent data, but was adopted in practically
all previous works on the XRB. Note that model B has the same type 1 soft
X--ray contribution as model A2 (30\%), but the type 2 contribution here
is due to higher luminosity sources, which show up in
higher flux bins: indeed, the XRB spectrum is well fitted, the counts in
the ASCA band are matched, and the discrepancy with the HELLAS counts is
reduced to $2\sigma$. The cost to be paid is a number density
$R_{\rm Q}=7.7$, which --again-- is definitely higher than the local upper
limit $R_{\rm Q}<2$. Irrespective of the plausibility of PLE and QSO~2s,
we stress that even model B results in a discrepancy, and the discrepancy
is concordant with the results of the other, less controversial variants.

A population of absorbed or hard spectrum AGNs evolving more rapidly than
the type~1s could accomodate all the problems discussed above. In this
context one should be reminded that the hard counts already resolve
$\sim$30\% of the XRB at fluxes $\sim~5\times 10^{-14}$ erg cm$^{-2}$
s$^{-1}$, so they must converge rapidly just below these values.
The optical identifications of the counts in the 2--10 keV and
5--10 keV bands are still largely incomplete.
Up to now 34 X--ray sources detected in the ASCA LSS survey (Ueda et al.
1998) have been identified (Akiyama et al. 1998), and 28 objects turned out
to be AGNs. They are 22 broad line AGNs (type 1--1.5) with
$0<z<1.7$, and 6 type 2 AGNs with $0<z<0.7$. The number of identified
sources of the BeppoSAX HELLAS survey is lower, but the distribution of
the AGNs seems similar to the ASCA one: 7 broad
line QSOs with $0.2<z<1.3$ and 5 Seyferts 1.8--1.9 with $0.04<z<0.34$
(Fiore et al. 1999).

If one accepts these low redshift type 2 identifications,
one has to find a physical reason for a
convergence so recent in comparison with all other AGNs (BL--Lacs excepted)
and star forming galaxies. Alternatively, one could rely on the poor
statistics
to maintain that the hard counts are mostly due to optically empty fields,
containing very absorbed, very powerful sources at redshifts $>1$.

There are prospective candidates in both scenarios. In the low--$z$
hypothesis one could assume that ``normal'' Seyfert 2 galaxies evolve
more rapidly than type 1s, so that $R_{\rm S}$ increases with redshift
up to the required value. Not only the number ratio, but also the $N_{
\rm H}$ distribution could change with cosmic time (Franceschini et al. 1993).
Local Sey 2s are
associated with a star formation activity higher than Sey 1 and normal
galaxies ( Maiolino et al. 1995; Rodr\'{\i}guez-Espinoza et al. 1986),
so this assumption would have interesting implications on the star
formation history. One could also
invoke Advection Dominated Accretion Flows (ADAFs,
Di Matteo et al. 1998) whose luminosity is proportional to $\dot M^2$,
where $\dot M$ is the mass accretion rate,
and which become normal QSOs at large $\dot M$: thus, they should
evolve more rapidly than normal AGNs at intermediate redshifts, and
should undergo a change of class at high redshifts. In the high--$z$
hypothesis one should resort to ULIRGs, which indeed are absorbed and
powerful, and appear to evolve as fast as required [$(1+z)^{7.6\pm3.2}$,
Kim \& Sanders 1998]. As mentioned before there is evidence that the IR
emission of ULIRGs is powered both by starburst and AGN processes;
Kim, Veilleux \& Sanders (1998) and Lutz et al. (1998) find that the
fraction
of AGN--powered infrared luminous galaxies increases with the bolometric
luminosity, and reaches 30--50\% in the ULIRG range. While normal starbursts
are inefficient emitters in the hard X--rays, obscured AGNs in ULIRGs
could easily explain the hard counts.

Finally, it should be noted that the optical identifications of the hard
X--ray counts, scanty as they are, suggest that at the given X--ray flux
the type 1s are more numerous than the type 2s. Concordant evidence is
provided by recent BeppoSAX observations of the Marano field and the
Lockman hole (Hasinger et al. 1999): most of the BeppoSAX sources have
ROSAT counterparts, which in most cases are optically identified with
type 1 AGNs. This type composition is in agreement with the predictions
of, for instance, model A1 (Figs.~4 and 5). But if the hard counts in
excess of the model are attributed entirely to obscured AGNs, then the
predicted type ratio is reversed, with the type 2s more numerous than the
type 1s. The numbers involved are too small to draw any conclusion,
however they seem to suggest that some of the hard counts are due to flat
X--ray spectrum sources with type 1 optical spectra; a few similar sources
might have been found already in the ASCA LSS (Akiyama et al. 1998).

Clearly, a decisive progress in this area will require more numerous
and more secure identifications of hard X--ray counts; given the various
hypotheses, counterparts should be looked for not only at optical
wavelengths, but also in the infrared and submillimeter domains, where
AGN--dominated ULIRGs should be conspicuous.

\section{Summary and conclusions}

In this paper we have shown that the standard prescriptions for
synthesizing the XRB from the integrated emission of AGNs are
not consistent with a number of recent observational constraints,
and some of them must be relaxed.

We have worked out models (A1 and A2) which take into account detailed
input spectra of AGNs, the $N_{\rm H}$ distribution observed in local
Seyfert 2s, and the XLF and evolution newly determined from the largest
ROSAT sample. The latter data do not define a unique parametrization,
and the two models explore different variants. As prescribed by the
standard model, the XLF and evolution of type 2 AGNs are taken from
type 1s, and the spectra of both types are taken independent of redshift;
the only fitting parameter is the number ratio $R$ of type 2s to type 1s.
We find that model A1 reproduces the XRB and the soft counts with a ratio
$R$ compatible with the local value, but underestimates the hard counts.
Model A2 is less discrepant as far as the counts are concerned, but requires
a ratio $R$ definitely larger than observed locally.
We have also computed a model adopting a canonical pure luminosity
evolution (model B).
In agreement with the results of Co95, model B can reproduce the
XRB, the soft X--ray counts and the ASCA hard counts in the
2--10 keV band. It is also consistent within 2$\sigma$ (or discrepant at
2$\sigma$) with the preliminary BeppoSAX counts in the 5--10 keV band.
Nevertheless, it requires a number of type 2 QSOs much higher than the local
upper limit, and perhaps already ruled out by the deep X--ray surveys.

The discrepancies found in all models are to some extent model dependent,
but all of them point in the same direction, and suggest that hard
spectrum
sources at intermediate or high redshifts are needed in addition to the
predictions of the standard scenario.
The X--ray spectrum of these additional sources could be flattened by
absorption,
or could be intrinsically hard. In the former hypothesis
reasonable candidate counterparts could be rapidly evolving, ``normal''
Seyfert 2s. One should also note that a fraction of ULIRGs
seem to be powered by AGNs, and their cosmological evolution
seems faster than that of unabsorbed QSOs.
The alternate hypothesis could instead require the presence of ADAFs.
Optical identifications of the hard X--ray sources are still largely
incomplete and do not allow yet to decide between the various possibilities.

\begin{acknowledgements}

We are grateful to A. Comastri and G. Zamorani for a careful reading of
the manuscript, and to T. Miyaji G. Hasinger and M. Schmidt for permission
to use their LDDE model in advance of publication. Our presentation was
greatly improved by the comments of the referee, Prof. G. Hasinger. This
work was partly supported by the Italian Space Agency
(ASI) under grant ARS--98--116/22 and by the Italian Ministry for
University and Research (MURST) under grant Cofin98--02--32.

\end{acknowledgements}


\section*{Acknowledgments}
This work was supported in part by the DOE (at Chicago,
Fermilab and Case Western Reserve) and by
the NASA (at Fermilab through grant NAG 5-7092).