%Title:  Exotic Bound State Production at Hadron Colliders
%Authors:  J. H. Kuehn and E. Mirkes
%Published: Phys. Lett. <B> B311 </B> (1993) 301--306.
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\begin{document}
%--------------------------------------
%\def\ref#1{{\bf[#1]}}
\def\ttb{t\bar t}
\def\qqb{Q\bar Q}
\def\gg{\gamma\gamma}
\def\bt{\beta_{t}}
\def\to{\rightarrow}
\def\RM{{R(0)^2\over 2\pi M}}
\def\MeV{{\rm MeV}}
\def\Lms#1{\Lambda_{\overline{MS}}^{(#1)}}
\def\asp{{\alpha_s\over\pi}}
\def\ams{\alpha_{\overline {MS}} }
\def\amsp{{\ams \over\pi}}
\def\eps{\epsilon}
\newcommand{\hoch}[1]{\mbox{\rule[0cm]{0cm}{#1}}}
%-----------------------------
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\vspace*{-2mm}
%\begin{center}
\thispagestyle{empty}
\noindent
\hfill TTP93--10\\
\mbox{}
\hfill  March  1993  \\   %                 \hfill \today \\
\vspace{0.5cm}
\begin{center}
  \begin{Large}
  \begin{bf}
EXOTIC BOUND STATE PRODUCTION \\
AT  HADRON COLLIDERS\footnote{\normalsize Supported by
 BMFT Contract 055KA94P} 
   \\
  \end{bf}
  \end{Large}
  \vspace{0.8cm}
  \begin{large}
   J.H. K\"uhn and E.\ Mirkes\\[5mm]
    Institut f\"ur Theoretische Teilchenphysik\\
    Universit\"at Karlsruhe\\
    Kaiserstr. 12,    Postfach 6980\\[2mm]
    7500 Karlsruhe 1, Germany\\
  \end{large}
  \vspace{4.5cm}
  {\bf Abstract}
\end{center}
\begin{quotation}
\noindent
Hadronic production of nonrelativistic boundstates of $b^{\prime}$ or
isosinglet quarks D with suppressed weak decays is investigated
for LHC and SSC energies. 
QCD corrections to production and decay rates are incorporated.
Large rates for final states from $\eta_{b^{\prime}}\rightarrow HZ$
are predicted.
\end{quotation}
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%\section{Introduction}
The theoretical and experimental aspects of hadronic 
quarkonium production have attracted a great deal of 
interest after the original discovery of $J/ \Psi$ and
$\Upsilon$.  Gluon 
fusion seemingly dominates the cross section with a small 
admixture of quark-antiquark annihilation in $p\bar{p}$ and 
$\pi^{-}p$
collisions at lower energies.  Given sufficiently high 
center of mass energy and luminosity, the same parton 
processes could also lead to toponium production at a future 
hadron collider.  The production rate of $1^{--}$ bound states 
through the reaction 
$g+g\rightarrow 1^{--}+g$
 is tiny.  The cross section for 
gluon-gluon fusion into the ${}^{1}S_{0}$  state $\eta_{t}$,
 however, would be 
sufficiently large to produce a large number of resonances.  
The experimental signal
that might conceivably be accessible
under realistic assumptions
is the $\gamma\gamma$ decay
mode of $\eta_{t}$ \cite{Rubbia}. However, this mode
 is severely reduced by the tiny branching ratio of 
order $10^{-4}$ down to $10^{-5}$
--- a consequence of the large single 
quark decay rate of $\eta_{t}$.
Detailed investigations which include 
higher order corrections for the production rate and a 
realistic QCD potential show that the window for the 
discovery of $\eta_{t}$ is extremely narrow.
Fairly optimistic 
assumptions have to be employed to access quark masses even 
in the range of 100 to 110 GeV \cite{letter,rev}.

However, the situation changes drastically if the single 
quark decay mode 
of the ${}^{1}S_{0}$ bound state
is absent or at least strongly suppressed,
 as is the case for a $b^{\prime}$ with 
suppressed couplings to the lighter u, c, and t quarks, or
 for 
isosinglet quarks denoted in the following by $D$.  
A fourth generation is still consistent with our
present experimental information if the associated neutrino
is sufficiently heavy, that is $m_{\nu}>m_{Z}/2$.
A  isosinglet quark is naturally predicted in some
grand unified theories. 
 The branching ratio for decay modes of 
these bound states with truly characteristic
final states such as
$\gamma\gamma,\, \gamma Z,\,ZZ,\, WW,\,HZ$
are no longer drastically suppressed and may 
even become dominant, as is the case
for the $HZ$ mode in the high mass region.
  These would be easily 
accessible in future experiments at LHC or SSC, for quark 
masses even up to 1000 GeV.  Predictions for these reactions 
in Born approximation have been derived in references
\cite{acta,barger,kurep}.

In a recent paper \cite{letter,rev},
QCD corrections were evaluated for the 
production of nonrelativistic ${}^{1}S_{0}$
bound states in hadronic 
collisions.  These corrections
 are positive and fairly 
large, justifying the renewed investigation of exotic bound 
state production.
The corrections 
 can be also applied
to the production of bound states
composed of a $b^{\prime}$ with $I_{3}=-{1}/{2}$
and $e_{Q}=-{1}/{3}$ or of a $D$ quark with $I_{3}=0$ and 
$e_{Q}=-{1}/{3}$.

In the following we specify
the general structure of the cross section for
$\eta_{b',D}$ production in hadronic collisions
\begin{equation}
h_{1}(P_{1})+h_{2}(P_{2})\rightarrow\eta_{b',D}+X
\label{hadwq}
\end{equation}
in the framework of perturbative QCD.
In (\ref{hadwq})
 $h_{1}$ and $h_{2}$ denotes unpolarized hadrons with 
momenta $P_{i}$.
The hadronic cross section in NLO is thus given by
\begin{equation}
\sigma^{H}(S)
=\int dx_{1} dx_{2}
f_{a}^{h_{1}}(x_{1},Q^{2}_{F})f_{b}^{h2}(x_{2},Q^{2}_{F}) 
 \hat{\sigma}^{ab}(s=x_{1}x_{2}S,
 \alpha_{s}(\mu^{2}),\mu^{2},Q^{2}_{F})
\end{equation}
where one sums over $a,b=q,\bar{q},g$.
$f_{a}^{h}(x,Q^{2}_{F})$
is the probability density to find parton $a$ with
fraction $x$ 
in hadron $h$ if it is probed at scale $Q_{F}^{2}$
and $\hat{\sigma}^{ab}$
 denotes the parton cross section for the process
\begin{equation}
a(x_{1}P_{1})+b(x_{2}P_{2})\rightarrow \eta_{b',D}+X
\end{equation}
from which collinear
initial state singularities have been factorized out at a
scale $Q_{F}^{2}$ 
and implicitly included in the scale-dependent parton
densities $f_{a}^{h}(x,Q_{F}^{2})$.
The partonic cross sections  $\hat{\sigma}^{ab}$ in NLO
 for the $gg, qg$ and $q\bar{q}$ initiated 
 reactions are given in \cite{rev}.  They 
are proportional to the square of the bound state
 wave function at the 
origin $R(0)$ 
and hence dependent on the QCD potential.  For the 
evaluation we employ the two-loop QCD potential $V_{J}$
\cite{Igi}.
 The 
numerical results depend critically on the value of the QCD 
parameter $\Lms{4}$
but are fairly insensitive to the actual implementation 
of the potential \cite{bemerkung}.
The results for the dimensionless quantity
 $|R(0)|^{2}/M^{3}_{\eta}$ ($M_{\eta}=2m_{b^{\prime},D}$) are
displayed in
 fig. 1 
 for different values of $\Lms{4}$
(200, 300 and 500 MeV)  
 to indicate the characteristic 
uncertainty in the prediction.
These $\Lambda$ values
correspond to 
$\alpha_{s\,\overline{MS}}^{(5)}
=0.109,\, 0.1165$ and 0.127 respectively
 (and to
 $\Lms{5}= 0.132,\, 0.207$ and 0.366 MeV).
  For the parton distribution functions 
we employ MT set B1 with $\Lms{4}=194\, \mbox{MeV}$ \cite{MT} and
work in the DIS factorization 
scheme.  The
renormalization and factorization scales are fixed 
at $M=M_{\eta}$.
We use the $\overline{{MS}}$ definition
 of $\alpha_{s}$ at two-loop
accuracy with six flavours
\begin{equation}
\alpha_{s\,\overline{MS}}(Q)=
\frac{12\pi}{(33-2n_{f})\ln(Q^{2}/\Lambda^{2})}
\left[1-\frac{6(153-19n_{f})}{(33-2n_{f})^{2}}
\frac{\ln\ln(Q^{2}/\Lambda^{2})}{\ln(Q^{2}/\Lambda^{2})}\right]
\end{equation}
 and 
 $\Lms{6}=58$ MeV to be consistent with the
$\Lms{4}$ value from the parton distributions.

  The $K-$factor and the
production cross section
 are shown in 
figs. 2 and 3a respectively 
 as functions of the bound state mass for three 
different $\Lambda$  values in the potential.
  The 
uncertainty from the choice of the parton distribution 
functions is shown in fig. 3b.
  The additional 
uncertainty from the choice of
the renormalization and factorization scales is estimated to 
be less than 10\% \cite{rev}.  

To obtain predictions for the interesting reactions
these production cross sections have to be multiplied by the 
various possible branching ratios.
It is convenient to normalize the decay rates relative 
to the two photon decay rate $\Gamma_{\gamma\gamma}$
with
\begin{equation}
\Gamma_{\gamma\gamma}=12e_{Q}^{4}\alpha^{2}\frac{|R(0)|^{2}}{M^{2}}
\label{gamborn}
\end{equation}
With $R_{ab}:=\Gamma_{ab}/ \Gamma_{\gamma\gamma}$ 
($ab=\gamma Z,\,HZ,\,ZZ,\,WW,\,gg,\,t\bar{t})$
one finds 
\newpage
\begin{eqnarray}
R_{\gamma Z}&=&
2\,\frac{v_{Q}^{2}}{e_{Q}^{2}y^{2}}
\left(\frac{2p}{M}\right)
\label{dgamz}
\\[2mm]
R_{HZ} &=&
\,\,\,\frac{a_{Q}^{2}}{e_{Q}^{4}}\frac{1}{\rho_{Z}^{2}y^{4}}
\left(\frac{2p}{M}\right)^{3} 
\label{dhz}
\\[2mm]
R_{ZZ} &=&
\,\,\,\frac{(v_{Q}^{2}+a_{Q}^{2})^{2}}{e_{Q}^{4}y^{4}}
\frac{1}{(1-2\rho_{Z})^{2}}
\left(\frac{2p}{M}\right)^{3} 
\label{dzz}
\\[2mm]
R_{WW} &=&
\,\,\,\frac{1}{2e_{Q}^{4}(4\sin^{2}\theta_{W})^{2}}
\frac{1}{(1-\rho)^{2}}
\left(\frac{2p}{M}\right)^{3} 
\label{dww}
\\[2mm]
R_{gg}&=&
\frac{2}{9}\,\frac{\alpha_{s}^{2}(M)}{
e_{Q}^{4}\alpha^{2}}
\label{dgg}
\\[2mm]
R_{t\bar{t}}&=&
6\,\frac{a_{Q}^{2} a_{t}^{2}}{e_{Q}^{4}y^{4}}
\frac{m_{t}^{2}M^{2}}{M_{Z}^{4}}
\left(\frac{2p}{M}\right)
\label{dtt}
    \end{eqnarray}
where
\begin{equation}
  \begin{array}{ll}
\rho_{i}\equiv M_{i}^{2}/M^{2}\,\, ,\hspace{1cm} &
\rho\equiv 2\left(1/4-\rho_{Q^{\prime}}+\rho_{W}\right)\,\, ,\\[2mm]
v_{Q}\equiv 2I_{3\,Q}-4e_{Q}\sin^{2}\theta_{W}\,\,,&
a_{Q}=2 I_{3\,Q} \hspace{1cm}
y\equiv 2\sin 2\theta_{W}
%p=
%[(M^{2}-(m_{a}+m_{b})^{2})(M^{2}-(m_{a}-m_{b})^{2})]^{1/2}/(2M)
\label{abb}
  \end{array}
\end{equation}
QCD corrections to the decay rates  will be discussed below.
In eq. (\ref{abb}) $I_{3\,Q}$ denotes the third component
 of the weak isospin
of 
the $Q-$type quark and $\theta_{W}$ is the Weinberg angle.
$p$ denotes the three-momentum of the final state particles.
The single quark decay mode which dominates for $\eta_{t}$ proceeds 
through the decay of the heavy quark into  $b+W$.  For a heavy 
sequential quark $b^{\prime}$ of isospin $I_{3}=-{1}/{2}$ the analog 
decay $b^{\prime}\rightarrow t^{\prime}+W$ is presumably strongly 
suppressed by a small 
mixing angle $\theta^{\prime}$
 if the mass of the isospin partner $t^{\prime}$ is 
larger than $m_{b^{\prime}}$.
Once ${\theta^{\prime}}^{2} < 10^{-3}$
 the annihilation decays dominate 
and single quark decays can be ignored for most practical purposes. 
 Similar 
considerations apply to decays of isosinglet quarks $D$ into 
$b+Z, t+W$ or $c+W$
which are also inhibited by small mixing angles.  For 
the subsequent discussion these single quark modes will be 
ignored.  The dominant decays are thus all 
 proportional to the 
wave function squared, the branching ratios
$\Gamma_{ab}/\sum\Gamma_{ab}$
 are independent 
of the wave function, and as a result can be predicted 
unambiguously.  

The WW mode needs further specification:  
The decay of $\eta_{b^{\prime}}$  into $WW$
 proceeds through the virtual isospin 
partner $t^{\prime}$ and hence depends on the mass of $t^{\prime}$.  
Present limits on the deviation of the $\rho-$parameter from one 
as deduced from electroweak precision measurements limit the 
mass splitting $\bigtriangleup m $
 between $m_{t^{\prime}}$ and $m_{b^{\prime}}$ in a drastic way and
 $\bigtriangleup m < 200 \,\,\mbox{GeV}$
can be considered a generous upper bound.
For simplicity we shall assume equal $b^{\prime}$ and $t^{\prime}$
masses 
($\rho_{Q^{\prime}}=1/4$ in eqs. (\ref{dww},\ref{abb}))
in the subsequent 
numerical evaluation.  

For bound states $\eta_{D}$ from isosinglet 
quarks  the WW decay may in principle 
proceed through virtual quarks of charge 2/3.  However, 
the $D$-$t$-$W$-coupling is suppressed by small mixing angles and 
the WW mode can therefore safely be neglected.
The $HZ$ mode is absent as a consequence of $a_{D}= 0.$

QCD corrections to annihilation decays
are at present available  for hadronic  ($gg-$),
for  $\gamma\gamma-$ 
\cite{Barbieri}
 and for the $t\bar{t}-$decay modes. 
\begin{eqnarray}
\Gamma_{\gamma\gamma}&=&
\Gamma_{\gamma\gamma}^{Born}
\left[1+\frac{\alpha_{s}(\mu)}{\pi}
\left(\frac{\pi^{2}}{3}-\frac{20}{3}\right)\right]\nonumber
\\[2mm]
\Gamma_{had}&=&\Gamma_{had}^{Born}
\left[1+\frac{\alpha_{s}(\mu)}{\pi}
\left\langle \left(11-\frac{2}{3}n_{f}\right)
\ln\frac{\mu}{M}
+\frac{159}{6}-\frac{31}{24}\pi^{2}
-\frac{8}{9}n_{f}
\right\rangle\right]
\label{cott}
\\[2mm]
\Gamma_{t\bar{t}}&=&\Gamma_{t\bar{t}}^{Born}
\left(1-4\frac{\alpha_{s}(\mu)}{\pi}\right)
\left(1+\frac{4}{3}\frac{\alpha_{s}(\mu)}{\pi}
\frac{1}{\beta_{t}}
\left[A(\beta_{t})+P(\beta_{t})\ln\frac{1+\beta_{t}}{1-\beta_{t}}
+Q(\beta_{t})
\right]\right)\nonumber
\end{eqnarray}
where $n_{f}=6$ denotes the number of light flavours (including top).
In the last equation
of (\ref{cott}) the functions $A(\beta_{t}), P(\beta_{t})$ and
$Q(\beta_{t})$  are given by
\begin{eqnarray}
A(\beta_{t})&=& (1+\bt^2)\left[
\frac{\pi^{2}}{6}+\ln\frac{1+\bt}{1-\bt}\,
\ln\frac{1+\bt}{2}
+2 \mbox{Li}_{2}\left(\frac{1-\bt}{1+\bt}\right)
+2 \mbox{Li}_{2}\left(\frac{1+\bt}{2}\right)\right.
       \nonumber\\
&-&\hspace{-2mm} \left.2 \mbox{Li}_{2}\left(\frac{1-\bt}{2}\right)
   -4 \mbox{Li}_{2}(\bt)
   +\mbox{Li}_{2}(\bt^{2})\right]
   +3\bt\ln\frac{1-\bt^{2}}{4\bt}-\bt\ln \bt .\\[2mm]
P(\beta_{t})&=& 
\frac{19}{16}+\frac{2}{16}\bt^{2}+\frac{3}{16}\bt^{4}\\[2mm]
Q(\beta_{t})&=&\frac{21}{8}\bt-\frac{3}{8}\bt^{3}
\label{qbeta}
\end{eqnarray}
where
$\beta_{t}=\sqrt{1-4m_{t}^{2}/M^{2}}$.
The first factor in the corrections  for $t\bar{t}$
has its origin in the $\eta_{Q}$ vertex correction, the second
factor is given in \cite{Reinders}.
Finally, the scale $\mu$ in $\alpha_{s}$ is adopted to M
 in our numerical calculation.
The QCD corrections to the $\gamma\gamma$ decay rate are negative
at the level of about -10\% whereas the corrections to the hadronic 
decay increases the decay rate by about +35\%.
The QCD corrections to $\Gamma_{t\bar{t}}$ depend on the mass 
of the boundstate and vary from +40\% for $M_{\eta}=300$ GeV
to -20\% for $M_{\eta}=2000$ GeV.


Since $\gamma\gamma, \gamma Z,ZZ$ and $WW$ can be identified in 
the limit $M_{Z},M_{W}<<2m_{b',D}$
 after appropriate adjustment of the gauge 
coupling, the QCD corrections can be assumed to cancel to a 
large extent in the ratios listed in eqs. (\ref{dgamz}, \ref{dzz},
\ref{dww}).
Corrections for 
the HZ mode are missing and we shall simply adopt eq. (\ref{dhz}) for  
the present purpose. 
% The normalized decay rates $R_{ab}$  are 
%shown in figs. 4a and b for $\eta_{b^{\prime}}$ and $\eta_{D}$
%respectively, the resulting branching ratios 
% $\Gamma_{ab}/\sum\Gamma_{ab}$
The branching ratios 
$\Gamma_{ab}/\sum\Gamma_{ab}$
for
$\eta_{b^{\prime}}$ and $\eta_{D}$
are displayed 
%in figs. 4c and d.  The pattern is significantly different for 
in figs. 4a and b.  The pattern is significantly different for 
the two cases: 

The branching ratio of $\eta_{b^{\prime}}$ into $HZ$ is sizable and 
 dominant for a large mass range.
For $M\approx 400$ GeV the QCD corrections 
given in eqs. (\ref{cott}-\ref{qbeta}) reduce the
branching ratio for the $HZ$ mode by about 30\%, whereas 
the corrections for $M$ above 600 GeV are below 5\%.
  Also the $WW$ mode is 
fairly important, amounting to about 1\%  of the hadronic 
mode, which is presumably difficult to detect 
experimentally.
The $\gamma\gamma,ZZ$ and $\gamma Z$ 
modes are all relatively small
with their ratio basically governed by their relative 
couplings and the statistical factor 1/2 for 
$\gamma\gamma$ and $ZZ$.
\begin{equation}
\Gamma_{WW}\,:\,\Gamma_{ZZ}\,:\,\Gamma_{\gamma Z}\,:\,
                    \Gamma_{\gamma\gamma}
\approx
1 \,:\, 0.462 \,:\, 0.064 \,:\, 0.021
\end{equation}
This expectation for asymptotic quark mass  
values is well met above 
$m_{b^{\prime}}> 150 \,\mbox{GeV}$.
One thus anticipates a  large  production rate for $HZ$ and 
even $WW$ will become fairly important.

This is in contrast to the situation for $\eta_{D}$ decays: 
 $HZ$ and 
$WW$ are absent, and the asymptotic relation reads
\begin{equation}
\Gamma_{\gamma\gamma}\,:\,\Gamma_{\gamma Z}\,:\, \Gamma_{ZZ}
\approx
1 \,:\,  0.60 \,:\, 0.091
\end{equation}
In
this case $\gamma\gamma,\,\gamma Z$
and $ZZ$ constitute the dominant visible 
modes with a  
branching ratio of about $10^{-4}$.

  The resulting product of 
production cross sections times branching ratio
for SSC and LHC energies
 are displayed 
in figs. 5a and b for several interesting 
$\eta_{b^{\prime}}$ channels, indicating 
large rates in particular for $HZ$.  
Rates for the continuum $HZ$ production are discussed
in \cite{Eichten,barger}. The total cross section 
for this background
amounts to about 5 pb or below for $m_{H}>100$ GeV.
Since the invariant mass of the $HZ$-system is fixed in our case this
background will not pose a serious problem.
The production and decay of $\eta_{D}$ is shown in figs. 5c and d
indicating that $\eta_{D}$  could be 
observed in the  $\gamma\gamma,\, \gamma Z$ and $ZZ$ modes
 with
characteristic signals.
\\[5mm]
To summarize:\\
Production and decay rates for bound states of exotic
quarks can be predicted unambigously.
The uncertainty from the QCD potential, $\alpha_{s}$
and the structure functions appears to be well
under control. The rates for $pp\rightarrow \eta_{b^{\prime}}
(\rightarrow HZ)\, + \,X$ are particularly
promising  as a consequence of the large branching ratio.
The branching rations into $WW,\,ZZ,\,\gamma Z,$ or $\gamma\gamma$
are significantly smaller but in any case larger than the
corresponding branching ratios of a toponium resonance.

Throughout this paper the SQD mode has been ignored completely,
a clearly oversimplifying assumption.
The mixing of both $b'$ and $D$ with other quarks
is presumably small and it seems difficult to make
any hypothesis.
Instead we compare the total annihilation rate (for
$V_{J}$ with $\Lms{4}=300$ MeV) with the single quark
decay rate (through the charged current)
in fig. 6
 for an assumed mixing angle
of $10^{-2}$. This illustrates that annihilation decays would 
be clearly dominant for $b'$ under this assumption.
For a complete treatment of $\eta_{D}$ we would have to include
FCNC single quark decays $D\rightarrow qZ$.
In view of the completely unknown mixing angle we refrain from a 
detailled analysis. It is evident from fig. 6
that for mixing angles of $O(10^{-2})$ or less and $m_{D}$ below
$\sim 500$ GeV the main conclusions remain unaffected.


\def\app#1#2#3{{\it Act. Phys. Pol. }{\bf B #1} (#2) #3}
\def\apa#1#2#3{{\it Act. Phys. Austr.}{\bf#1} (#2) #3}
\def\lhc{Proc. LHC Workshop, CERN 90-10}
\def\npb#1#2#3{{\it Nucl. Phys. }{\bf B #1} (#2) #3}
\def\plb#1#2#3{{\it Phys. Lett. }{\bf B #1} (#2) #3}
\def\prd#1#2#3{{\it Phys. Rev. }{\bf D #1} (#2) #3}
\def\prl#1#2#3{{\it Phys. Rev. Lett. }{\bf #1} (#2) #3}
\def\prc#1#2#3{{\it Phys. Reports }{\bf C #1} (#2) #3}
\def\cpc#1#2#3{{\it Comp. Phys. Commun. }{\bf #1} (#2) #3}
\def\nim#1#2#3{{\it Nucl. Inst. Meth. }{\bf #1} (#2) #3}
\def\pr#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3}
\def\sovnp#1#2#3{{\it Sov. J. Nucl. Phys. }{\bf #1} (#2) #3}
\def\jl#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3}
\def\jet#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3}
\def\zpc#1#2#3{{\it Z. Phys. }{\bf C #1} (#2) #3}
\def\ptp#1#2#3{{\it Prog.~Theor.~Phys.~}{\bf #1} (#2) #3}
\def\nca#1#2#3{{\it Nouvo~Cim.~}{\bf #1A} (#2) #3}
%
\newpage
\sloppy
\raggedright
\begin{thebibliography}{99}
\bibitem{Rubbia}G. Pancheri, J.-P. Revol and C. Rubbia,
                \plb{277}{518}{1992}.
%\bibitem{Barbieri} R. Barbieri, M. Caffo and E. Remiddi,
%                 \npb{192}{1981}{61}; \plb{61}{1980}{93}.
%                 E. Remiddi,
%                 Int. School of Physics E. Fermi (Varenna).
\bibitem{letter} J.H. K\"uhn
        and E. Mirkes,  \plb{296}{1992}{425}.
\bibitem{rev} J.H. K\"uhn
        and E. Mirkes, Karlsruhe preprint TTP92-32,
                  to appear in \prd{}{1993}{}.
\bibitem{acta} J.H.~K\"uhn, \app{12}{1981}{347}.\\
               J.H.~K\"uhn, \apa{Suppl.XXIV}{1982}{203}.\\
\bibitem{barger} V. Barger et al., \prd{35}{1987}{3366}.\\
                 V. Barger et al., \prl{57}{1987}{1672}.
\bibitem{kurep} J.H.~K\"uhn and P. Zerwas, \pr{167}{1988}{323}.
\bibitem{Igi} K. Igi and S. Ono, \prd{33}{1986}{3349}.
\bibitem{bemerkung} For a review on quarkonium potentials
         see  \cite{kurep}.
\bibitem{MT} J. Morfin and Wu-Ki Tung, \zpc{52}{1991}{13}.
\bibitem{Barbieri} R. Barbieri, C. Curci, E. d'Emilio
        and E. Remiddi, \npb{154}{1979}{535}.
\bibitem{Reinders} L.J. Reinders, H. Rubinstein
        and S. Yazaki, \pr{127}{1985}{1}.
\bibitem{MRS} A. Martin,
        R. Roberts and  J. Stirling, \prd{43}{1991}{3645}.
\bibitem{KMRS} J. Kwieci\'{n}ski,
        A. Martin, R. Roberts and  J. Stirling, \prd{42}{1990}{3645}.
\bibitem{GRV} M. Gl\"uck, E. Reya and A. Vogt, DO-TH 91/07: see also
            \zpc{53}{1992}{127}.
\bibitem{Eichten} E. Eichten, I. Hinchliffe, K. Lane and C. Quigg,
         {\it Rev. Mod. Phys.} {\bf 56} (1984) 579.
\end{thebibliography}
\vspace{1cm}
\noindent
{\bf Figure captions}\\[2mm]
%------------------------
\begin{itemize}
\item[{\bf Fig. 1}]
 Predictions for
$R(0)^2/M^3$ of the S--wave ground state 
as a  function of $M$ for the potential $V_{J}$ with 
$\Lms{4}$=200, 300 and 500 MeV (dotted, solid and dashed lines).
\item[{\bf Fig. 2}] 
Ratio between the radiatively corrected production
 cross section
 for $pp\rightarrow \eta_{b^{\prime},D}+X$
and the lowest order result
for $\sqrt{S}$=16 and 40 TeV.
 We use MT set B1 parton distributions with
$\Lms{4}$=194 MeV  and work in the DIS factorization
scheme.
\item[{\bf Fig. 3}] 
a) Production
 cross section
 for $pp\rightarrow \eta_{b^{\prime},D}+X$ in NLO
for $\sqrt{S}$=16 and 40 TeV.
 We use MT set B1 parton distributions with
$\Lms{4}$=194 MeV  and work in the DIS factorization
scheme (potentials as in fig. 1).\\
b) Dependence of the cross section for $\sqrt{S}=40$ GeV
   on the parton distribution
   functions. Shown are the results for
   MT set SN-DIS \cite{MT} (dotted),  MRS set B200 \cite{MRS}
   (short-dash-dotted),
   KMRS set B \cite{KMRS} (long-dash-dotted)  and 
   GRV \cite{GRV} (dashed) normalized to
   the predictions obtained with MT set B1-DIS \cite{MT}.
   Note that the parametrizations in
   \cite{MRS,KMRS,GRV} are defined in
   the $\overline{MS}$-scheme.
\item[{\bf Fig. 4}] 
%Ratios $R_{ab}=\Gamma_{ab}/ \Gamma_{\gamma\gamma}$
% ($ab=HZ,t\bar{t},gg,WW,ZZ,\gamma Z$) 
Branching ratios for
$\Gamma_{ab}$ ($ab=HZ (m_{H}=100
\mbox{GeV}),t\bar{t} (m_{t}=140 \mbox{GeV}),
gg,WW,ZZ,\gamma Z,\gamma\gamma$) 
of $\eta_{b^{\prime}}$ (fig. a) and
 $\eta_{D}$ (fig. b) 
 as functions of $M$.
% and corresponding branchings ratios (figs. c and d)
%for a top mass of 140 GeV in $\Gamma_{t\bar{t}}$.
%a) Branching ratios
% $\Gamma_{ab}/ \sum_{ab}\Gamma_{ab}$
%($ab= HZ,\,gg,\,WW,\,ZZ,\,\gamma Z,\gamma\gamma$)
%of $\eta_{b^{\prime}}$ decays
% as functions of $M=2m_{b^{\prime}}$.
%\item[{\bf Fig. 7}] 
%Branching ratios 
% $\Gamma_{ab}/ \sum_{ab}\Gamma_{ab}$
%($ab=gg,\,\gamma Z,\,ZZ,\gamma\gamma$ )
%of $\eta_{D}$ decays
%as functions of $M=2m_{D}$.
\item[{\bf Fig 5.}] 
 Cross section for $\eta_{b^{\prime}}$ (fig. a,b)  and
$\eta_{D}$ (fig. c,d)  production 
including NLO corrections
multiplied by the  branching ratios
% into 
%$HZ,\,gg,\,WW,\,ZZ,\,\gamma Z$ and $\gamma\gamma$
at $\sqrt{S}=40$ TeV and 16 TeV.
 We use MT set B1 parton distributions with
$\Lms{4}$=194 MeV  and work in the DIS factorization
scheme (potentials as in fig. 1).
\item[{\bf Fig 6.}] 
Total widths for $\eta_{b', D}$ from annihilation (see fig. 4)
compared with the widths from single quark decays into
$b+W$.
\end{itemize}
\end{document}

