%Title:  Fragmentation production of doubly heavy baryons
%Authors:  M. A. Doncheski, J. Steegborn, and M. L. Stong
%Published: <EM>Phys. Rev.</EM> <B>D53</B> (1996) 1247-1252 
%
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\begin{document}
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\bigskip
{\Large\bf
\centerline{Fragmentation production of doubly heavy baryons}}
\bigskip
\normalsize
\bigskip
\centerline{Michael A. Doncheski}
\centerline{\sl Ottawa-Carleton Institute for Physics, Department of Physics,}
          \centerline{\sl Carleton University, Ottawa, Ontario K1S 5B6, Canada}
\bigskip
\centerline{and}
\bigskip
\centerline{J.\ Steegborn and M.\ L.\ Stong}
\centerline{\sl Institut f\"ur Theoretische Teilchenphysik, Universit\"at, }
\centerline{\sl D-76128 Karlsruhe, Germany}
\bigskip
\bigskip
%
\begin{abstract}
%
Baryons with a single heavy quark are being studied experimentally at present.
Baryons with two units of heavy flavor will be abundantly produced not only at
future colliders, but also at existing facilities.  In this paper we study the
production via heavy quark fragmentation of baryons containing two heavy quarks
at the Tevatron, the LHC, HERA, and the NLC.  The production rate is woefully
small at HERA and at the NLC, but significant at $pp$ and $p\bar{p}$ machines.
We present distributions in various kinematical variables in addition to the
integrated cross sections at hadron colliders.
%
\end{abstract}
\newpage\setcounter{page}{1}
%

The constituent quark model has been remarkably successful in describing the 
observed hadronic states.  This includes mesons with one heavy and one light 
quark ({\it i.e.}, $B$ and $D$ mesons), baryons with one heavy and two light 
quarks, and the $J/\psi$ and $\Upsilon$ mesons, which contain a heavy quark and 
heavy antiquark.  Recent experiments have made great progress in the 
observations of the $c$ and $b$ baryons\cite{bar_a,bar_c} and their 
decays\cite{bar_dec_a,bar_dec_c}.  The constituent quark model predicts the 
existence of baryons containing two heavy quarks ($cc$, $bc$, or $bb$) and one 
light quark, and we are on the verge of obtaining the necessary experimental 
sensitivity to observe these states.  The energies necessary to produce these 
particles are already reached; the difficulty remaining is in their 
reconstruction.  These states have in general a large number of decay modes, so 
that their observation and a measurement of their properties will require a 
large number of them to be produced.  This difficulty is increased by the fact 
that the production rates at $e^+ e^-$ colliders are extremely small, so that 
the identification of these particles must take place in the messier 
environment of hadronic collisions.

Were the heavy quark of infinite mass, the two heavy quarks would be bound in a 
point-like diquark, and the light degrees of freedom would ``see'' this diquark 
as a static antitriplet color source.  In this limit, the interactions of the 
light degrees of freedom with the heavy diquark are quite similar to 
interactions of the light degrees of freedom with the $\bar{b}$ quark in a $B$ 
meson\cite{savagewise,whitesavage}.  For realistic heavy quark masses, this 
simple picture is not yet valid.  In particular, the hyperfine splittings of 
these baryons are not yet well described by the heavy-mass limit\cite{ml}.  The 
states may nonetheless be accurately treated as a combination of light degrees 
of freedom and heavy although not-pointlike diquark.  The interactions between 
the two heavy quarks are analogous to the $Q\overline{Q}$ system familiar from 
$J/\psi$ and $\Upsilon$ spectroscopy.  At short distances, the interaction will 
be dominated by one-gluon-exchange, although with a color factor 2/3 from the 
$QQ$ interaction rather than 4/3 from $Q\overline{Q}$, and at long distances it 
will be confining\cite{fleck}.

The production of these states is reliably calculable as a hard process.  One
takes the full set of Feynman diagrams for the production of two pairs of heavy
quarks and heavy antiquarks and requires that the momenta of the two heavy
quarks be nearly equal, then projects out the correct quantum numbers for the
relevant $Q_1Q_2$ state.  This is similar to the calculation of the hard
production of heavy quarkonium states\cite{baierruckl}, though more complicated
as an additional $Q\overline{Q}$ pair is needed.  On the other hand, because
the quarks are heavy, fragmentation functions for a heavy quark to produce a
doubly heavy diquark can be predicted in perturbative QCD\cite{falk,qfrag}.
The fragmentation description of the $Q_1Q_2$ production process is not valid
where the masses of the heavy quarks become important, that is, for small
$p_{_T}$ or small energies, but for high-energy colliders such as LEP or
Tevatron it should be applicable.  The calculation is similar to that described
in \cite{qfrag} for $J/\psi$ production.  The form of the matrix element is
different in the two cases, requiring a somewhat changed method of projection
onto the proper spin states.  For the production of a $Q\overline{Q}$ bound
state, the standard procedure is to replace the heavy quark production matrix
element $\bar{u}{\cal O}v = \Tr ({\cal O}v\bar{u})$ with the trace $\Tr ({\cal
O}P_{SS_z})$, where $P_{SS_z}$ is the proper projection onto the spin of the
$Q\overline{Q}$ state\cite{kuehn}.  This procedure is, however, not simply
applicable to the $Q_1Q_2$ bound states due to the form of the matrix element,
which contains not $u$ and $\bar{v}$ as in the production of $Q\overline{Q}$
but $u_1$ and $u_2$.  The projection procedure described above is nothing more
than a clever method of performing the sum over the quark spins.  This sum can
also be performed by summing over helicity amplitudes with the proper
Clebsch-Gordan coefficients.  Although the procedure is somewhat different, the
calculations are identical in their essentials, and the fragmentation function
is simply proportional to that for a $Q\overline{Q}$ pair.  The differences
reflect the changed color and statistical factors.  The fragmentation function
for $Q_1$ to produce a spin-1 diquark $(Q_1Q_2)$ is
%
\begin{eqnarray}
& &D_{Q_1\to(Q_1Q_2)}(z,\mu=(m_1+2m_2)) = 
{2N_{12} \over 9 \pi}{|R_{(Q_1Q_2)}(0)|^2 \over m_2^3}
\alpha_s^2(2m_2) F(z), \label{eq:D_lowmom} \\
& &F(z) =  {rz (1-z)^2 \over 4(1 - z + rz)^6} 
\left[ 2 - 2z(3-2r) + 3z^2(3-2r+4r^2) \right. \nonumber \\
&& \hphantom{F(z) = dum} \left. 
-2z^3(1-r)(4-r+2r^2) - z^4(1-r)^2(3-2r+2r^2) \right], \label{eq:F_spin1} 
\end{eqnarray}
%
where $r = m_2/(m_1+m_2)$.  The factor $N_{12} = (1)2$ for (un)equal quark 
masses, and reflects the presence of identical fermions in the final state.  
For production of a spin-0 state, $(Q_1Q_2)^\prime$, the function $F(z)$ must 
be replaced by 
%
\begin{eqnarray}
F^\prime(z) & = & {rz (1-z)^2 \over 12 (1 - z + rz)^6}\left[ 6 - 18 z(1 - 2 r) 
+ z^2(21-74r+68r^2) \right. \nonumber \\ && \hphantom{dum}\left. 
- 2z^3(1-r)(6-19r+18r^2) + 3z^4(1-r)^2(1-2r+2r^2) \right]. \label{eq:F_spin0}
\end{eqnarray}
%
As in the case of hard quarkonium production, the requirement of nearly equal 
momenta and the projection onto the proper $Q_1Q_2$ quantum numbers will modify 
the $p_{_T}$ spectrum of the diquark; it will fall more rapidly than that of 
single heavy quark production.  For heavy quarkonium production, the faster 
fall-off in $p_{_T}$ of the hard production process means that fragmentation 
production will dominate for sufficiently large $p_{_T}$ despite the 
suppression of the fragmentation production by powers of $\alpha_s$\cite{us1}.  
For the $Q_1Q_2$ states, no hard production processes of lower order in 
$\alpha_s$ exist.  We assume that the doubly heavy diquarks will always 
hadronize into baryons with two heavy quarks, so that the fragmentation 
function $D_{Q_1 \to {\rm baryon}} = D_{Q_1 \to (Q_1Q_2)}$.

A $Q_1Q_2$ diquark in the ground state has a symmetric spatial wavefunction.  
Because the $Q_1Q_2$ must combine with a light quark to produce a colorless 
baryon, the diquark must be a color antitriplet (antisymmetric) state.  For 
identical quarks the wavefunction must satisfy Fermi-Dirac statistics, and thus 
the $cc$ and $bb$ ground state diquarks have spin 1.  The $bc$ states are not 
restricted by statistics, and we denote the spin singlet (triplet) state by 
$bc^\prime$ ($bc$).  The spin triplet can hadronize into a spin-1/2 or -3/2 
baryon, $\Xi_{bc}$ and $\Xi_{bc}^*$, respectively; the spin singlet can produce 
only the spin-1/2 $\Xi'_{bc}$.  The probabilities for the fragmentation of 
heavy quarks into these baryons are: $c \to \Xi_{cc},\Xi^*_{cc}$, about 
$2 \times 10^{-5}$; $b \to \Xi'_{bc}$, about $4 \times 10^{-5}$; 
$b \to \Xi_{bc}, \Xi^*_{bc}$, about $5 \times 10^{-5}$.  The remaining 
probabilities are suppressed by $\sim (m_c/m_b)^3$ and therefore approximately 
two orders of magnitude smaller\cite{falk}.

In addition to these probabilities, it is desirable to have predictions of the 
$p_{_T}$ (or other kinematical variables) distributions for the production of 
these states.  The fragmentation functions given above at low momentum scale 
$\mu_0$ must be evolved to higher scales $\mu$ in order to be of use in 
calculations for high energy colliders:
%
\begin{equation}
d\sigma(A + B \to (Q_1Q_2) + X) = \sum_i \int_0^1 dz \;
d\sigma(A + B \to i(\frac{p_{_T}}{z},\eta)+X, \mu^2) \; 
D_{i\to(Q_1Q_2)}(z,\mu^2).
\label{sig}
\end{equation}
%
The evolution occurs via an Altarelli-Parisi equation: 
%
\begin{equation}
\mu^2 \frac{\partial}{\partial \mu^2} D_{i \to (Q_1Q_2)}(z, \mu^2) =
\frac{\alpha_s}{2\pi}
\sum_{j} \int^{1}_{z} \frac{dy}{y} \; P_{ij}(z/y)
\; D_{j \to (Q_1Q_2) }(y, \mu^2).
\label{ap}
\end{equation}
%
which does not change the overall probabilities given above.  In the evolution 
of the fragmentation functions via (\ref{ap}), we include only the $P_{QQ}$ 
splitting function.  The fragmentation of gluons to heavy diquarks is 
suppressed relative to that of heavy quarks by $\alpha_s$, and may be 
neglected.  A separate numerical evolution of the fragmentation functions to 
each $z$ and $\mu$ needed in the course of a numerical simulation would be an 
extremely computer-intensive approach; instead we generate values of the 
fragmentation function for a reasonably-spaced set of $\mu$ and $z$ values and 
interpolate to the desired points.  The unevolved fragmentation functions 
depend on $|R(0)|^2$, the non-relativistic radial wavefunction at the origin, 
the heavy quark masses, and $\alpha_s(\mu_0)$.  These input parameters are 
given in Table~\ref{tab:frag_par}.

\begin{table}[h]
\begin{center}
\begin{tabular}{lccc}
\hline \hline
                    &  $\alpha_s$  & $m_Q$   &    $|R(0)|^2$  \\ \hline
$c \to (cc)$        & 0.238        & 1.5 GeV & (0.65 GeV)$^3$ \\ 
$c \to (bc)$        & 0.173        & 4.5 GeV & (0.80 GeV)$^3$ \\ 
$c \to (bc)^\prime$ & 0.173        & 4.5 GeV & (0.80 GeV)$^3$ \\ 
$b \to (bb)$        & 0.173        & 4.5 GeV & (1.20 GeV)$^3$ \\ 
$b \to (bc)$        & 0.238        & 1.5 GeV & (0.80 GeV)$^3$ \\
$b \to (bc)^\prime$ & 0.238        & 1.5 GeV & (0.80 GeV)$^3$ \\ \hline \hline
\end{tabular}
\end{center}
\caption{Parameters used to calculate the unevolved fragmentation functions}
\label{tab:frag_par}
\end{table}

The $J/\psi$ and $\Upsilon$ decays to lepton pairs give the wavefunctions 
$|R(0)|^2$ for these states relatively accurately\cite{schuler} and 
independently of phenomenological potential models.  The QCD-corrected 
expression for the leptonic decay of heavy quarkonium states
%
\begin{equation}
\Gamma(V \to \ell^+ \ell^-) = \frac{4 \alpha^2 e_Q^2}{M_V^2} |R_V(0)|^2 \left[
1 - \frac{16}{3 \pi} \alpha_s(m_Q) \right]
\label{wfn_data}
\end{equation}
%
is used to extract the $\Upsilon$ and $J/\psi$ radial wavefunction at the 
origin.  There is no information on the $B_c$ meson to determine the 
wavefunction for the $b\bar{c}$ states, and the annihilation of this state into 
lepton pairs would not exist in any case to provide the measurement.  However, 
there is an empirical relationship among leptonic decays of the $\phi$, 
$J/\psi$ and $\Upsilon$ mesons, 
$\Gamma(V \to e^+e^-) \simeq 12 e_Q^2 \; {\rm keV}$, which indicates that 
$|R(0)|^2/\mu^2$ is approximately constant.

The one-gluon exchange contribution for $Q_1Q_2$ differs from that for
$Q_1\overline{Q}_2$ by a relative color factor 1/2.  In a Coulomb potential,
therefore, the relation $|R_{J/\psi}(0)|^2 = 8 |R_{cc}(0)|^2$ must hold.  The
heavy quarks are, however, not in a pure Coulomb potential, so that we must
rely on phenomenological potentials to better estimate the radial wavefunction.
While many functional forms for the potential provide quite good fits to the
heavy quarkonium data, the fitted potentials all have very similar shapes in
the region of $r \sim 1 \; {\rm GeV}^{-1}$.  We use 1/2 the $Q_1\overline{Q}_2$
potential for the $Q_1Q_2$ potential, and find the wavefunction at the origin
using a numerical solution of the Schr\"odinger equation.  We compare in
Table~\ref{tab:wfn_orig} the results of this calculation for two familiar
potentials with the values extracted from the $J/\psi$ and $\Upsilon$ decay
rates.

\begin{table}[h]
\begin{center}
\begin{tabular}{rccc}\hline \hline
& Indiana\cite{indiana} & Richardson\cite{richardson} & data\cite{pdg}\\ \hline 
$c\bar{c}$ & (0.95 GeV)$^3$ & (0.93 GeV)$^3$ & (0.98 GeV)$^3$ \\ 
$cc$       & (0.67 GeV)$^3$ & (0.65 GeV)$^3$ &                \\ 
$b\bar{c}$ & (1.18 GeV)$^3$ & (1.18 GeV)$^3$ &                \\ 
$bc$       & (0.82 GeV)$^3$ & (0.81 GeV)$^3$ &                \\ 
$b\bar{b}$ & (1.72 GeV)$^3$ & (1.88 GeV)$^3$ & (1.96 GeV)$^3$ \\ 
$bb$       & (1.15 GeV)$^3$ & (1.23 GeV)$^3$ &                \\ \hline
\end{tabular}
\end{center}
\caption{Radial wavefunctions at the origin, $|R(0)|^2$, for 
$Q_1\overline{Q}_2$ and $Q_1Q_2$ states for two interquark potentials compared 
to values extracted from data, see (6).}
%Eqn.~\ref{wfn_data}.}
\label{tab:wfn_orig}
\end{table}

Doubly heavy baryons will be produced at all present and future accelerators 
where there is sufficient energy.  The results presented here include both 
baryon and antibaryon production.  We consider first the $e^+e^-$ colliders 
LEP, LEPII, and NLC.  At LEP, the only subprocess for the production of a heavy 
quark $Q$ is $e^+e^- \to Z$, with the $Z$ decaying to $Q\overline{Q}$.  The 
production rates in this case are quite small, a few events per year.  For 
LEPII and NLC the additional processes $e^+e^- \to W^+W^-$ where one $W$ decays 
to a $c$ quark and $e^+e^- \to ZZ$ where one $Z$ decays to $c\bar{c}$ or 
$b\bar{b}$ also contribute (depending on the LEPII energy, of course).  Despite 
the additional subprocesses, of which $W^+W^-$ dominates for NLC, the event 
rates remain woefully small.  At LEPII, only a few events/year can be expected, 
and a 60~fb$^{-1}$/yr, 500 GeV linear collider will produce only of order 30 
events per year.  The production rates at $\gamma\gamma$ colliders and 
$e\gamma$ colliders were also calculated, including backscattered and 
Weizs\"acker-Williams photons and the contributions from resolved-photon 
processes.  The cross sections for production of these particles at these 
colliders are also hopelessly small.

The results at HERA are similar.  The main subprocesses here are 
$\gamma g \to Q\overline{Q}$ and $\gamma Q \to gQ$.  Assuming the design 
luminosity of 200~pb$^{-1}$/yr, we again find that the event rate is rather 
small, of order 30 events/year.  The resolved photon processes are likewise
negligible.

The situation is considerably more hopeful at hadron colliders.  The relevant
subprocesses are $gg \to Q\overline{Q}$, $q\bar{q} \to Q\overline{Q}$, $gQ \to
gQ$, and $qQ \to qQ$ (in the latter two subprocesses, $q$ and $Q$ stand for
both quarks and antiquarks), although the main contribution is from the
gluon-fusion subprocess.  For the Tevatron, with an integrated luminosity of
100~pb$^{-1}$/yr, we anticipate of order $8\times10^4$ events/year ($p_{_T} >
5$~GeV, $|\eta| < 0.5$), while the LHC, with 100~fb$^{-1}$/yr, should produce
of order $1.3\times10^8$ events per year($p_{_T} > 10$ GeV, $|\eta| < 0.5$).
Total production cross sections per unit rapidity for the various diquarks at
the Tevatron and the LHC (operating at both 10~TeV and 14~TeV) are given in
Table~\ref{tab:tot_sig}.  The rates for production of the $bc$-type states are
similar to those expected for the $B_c$ meson, where rates of 10$^4$ $B_c$'s
have been predicted at the Tevatron ($p_{_T}>10$~GeV) and 10$^7$
at LHC ($p_{_T}>20$~GeV)\cite{kol2}.  On the other hand, the
production rate for $cc$-type states is significantly smaller than that for
$J/\psi$ production at the Tevatron which is expected to be $5\times 10^6$
($p_{_T}>5$~GeV)\cite{us1}.

\begin{table}[h]
\begin{center}
\begin{tabular}{lrrr}
\hline \hline 
                         & Tevatron & LHC (10 TeV) & LHC (14 TeV) \\ \hline
 $\Xi_{cc} + \Xi_{cc}^*$ & 430 pb & 330 pb & 470 pb \\
 $\Xi_{bc}^\prime$       & 145 pb & 220 pb & 330 pb \\
 $\Xi_{bc} + \Xi_{bc}^*$ & 215 pb & 350 pb & 490 pb \\
 $\Xi_{bb} + \Xi_{bb}^*$ &  16 pb & 27 pb  &  36 pb \\   \hline
\end{tabular}
\end{center}
\caption{Production cross sections per unit rapidity in the central region.}
\label{tab:tot_sig}
\end{table}

% figures 1 and 2 --- Tevatron results
\begin{figure}[hp]
\begin{center}\mbox{
\epsfig{file=hhfig1.eps,height=0.40\textheight,width=0.80\linewidth}}
\end{center}
\caption{$p_{_T}$ distributions (at small rapidity) for doubly heavy baryon
production at the Tevatron.}
\label{fig:pt_tev} 
\vspace{0.3cm}
%
\begin{center}\mbox{
\epsfig{file=hhfig2.eps,height=0.40\textheight,width=0.80\linewidth}}
\end{center}
\caption{Pseudorapidity distributions for doubly heavy baryon production at
the Tevatron.}
\label{fig:eta_tev}
\end{figure}

% figures 3 and 4 --- LHC!
\begin{figure}[hp]
\begin{center}\mbox{
\epsfig{file=hhfig4.eps,height=0.40\textheight,width=0.80\linewidth}}
\end{center}
\caption{$p_{_T}$ distributions (at small rapidity) for doubly heavy baryon
production at the LHC, 14 TeV.}
\label{fig:pt_lhc14} 
\vspace{0.25cm}
%
\begin{center}\mbox{
\epsfig{file=hhfig4.eps,height=0.40\textheight,width=0.80\linewidth}}
\end{center}
\caption{$p_{_T}$ distributions (at small rapidity) for doubly heavy baryon
production at the LHC, 10 TeV.}
\label{fig:pt_lhc10}
\end{figure}

Some results are given in Figs.~\ref{fig:pt_tev} and \ref{fig:eta_tev}.  
Fig.~\ref{fig:pt_tev} shows $d\sigma/dp_{_T}/d\eta|_{\eta=0}$ {\it vs.} 
$p_{_T}$.  A kinematical cut of 5 GeV for the baryon $p_{_T}$ has been imposed 
here.  Fig.~\ref{fig:eta_tev} gives the rapidity distribution of $Q_1Q_2$ 
baryons, again with $p_{_T}>5$ GeV.  The rapidity distributions are rather flat 
in the central region, so that $d\sigma/dp_{_T}$ may be estimated from 
$d\sigma/dp_{_T}/d\eta|_{\eta=0}$ by multiplying with the $\eta$-region desired.

Figs.~\ref{fig:pt_lhc14} and \ref{fig:pt_lhc10} show 
$d\sigma/dp_{_T}/d\eta|_{\eta=0}$ {\it vs.} $p_{_T}$ for the LHC operating at 
14~TeV (Fig.~\ref{fig:pt_lhc14}) and 10~TeV (Fig.~\ref{fig:pt_lhc10}).  A 
kinematical cut of $p_{_T} > 10$ GeV is placed on the $Q_1Q_2$ baryon.  The 
production rate at the lower center-of-mass energy is slightly lower, but even 
at the lower energy, the LHC will produce $Q_1Q_2$ baryons copiously.  The 
rapidity distributions are again relatively flat in the central region 
($|\eta|<3$), and event rates for any specific detector can be estimated by 
simply multiplying the results in Table~\ref{tab:tot_sig} by the $\eta$ 
coverage of the detector.

In order to estimate the uncertainties due to the choice of factorization and
fragmentation scales, we vary these scales (all chosen to be equal) by a factor
of 2 from our nominal choice $\mu$ = $p_{_T}$.  The effect of varying the scale
on $d\sigma/dp_{_T}/d\eta|_{\eta=0}$ {\it vs.} $p_{_T}$ can be seen in
Fig.~\ref{fig:tev_pt_mu}, and the effect on $d\sigma/d\eta$ {\it vs.} $\eta$
can be seen in Fig.~\ref{fig:tev_eta_mu}.  The processes shown are
representative of all the processes.  The behavior seen in
Fig.~\ref{fig:tev_eta_mu} is somewhat counter-intuitive in that one expects the
rate with $\mu = p_{_T}/2$ to be larger than that for $\mu = p_{_T}$.  However,
the fragmentation functions are cut off at low $\mu_{frag}$.  This causes the
drop at low $p_{_T}$ for this choice seen in Fig.~\ref{fig:tev_pt_mu} and the
lower cross sections in Fig.~\ref{fig:tev_eta_mu}.  We use MRSA\cite{mrsa}
parton distribution functions.  The effect of changing the parton distribution
set choice is negligible --- far smaller than the effect of varying the choice
of $\mu$.  Other sources of theoretical uncertainty include the parameters used
for the unevolved fragmentation functions, primarily $|R(0)|^2$ where different
phenomenological potentials give about 10\% differences in the result, unknown
QCD corrections to the parton-level cross sections, the fragmentation function
and the evolution equations, and unknown relativistic corrections to the
initial fragmentation functions.

% figures 5 and 6 --- TeV, mu dependence
\begin{figure}[hp]
\begin{center}\mbox{
\epsfig{file=hhfig5.eps,height=0.40\textheight,width=0.80\linewidth}}
\end{center}
\caption{$p_{_T}$ distributions (at small rapidity) for $bb$ and $bc$ 
production at the Tevatron for different choices of scale $\mu$.}
\label{fig:tev_pt_mu}
\vspace{0.3cm}
%
\begin{center}\mbox{
\epsfig{file=hhfig6.eps,height=0.40\textheight,width=0.80\linewidth}}
\end{center}
\caption{Pseudorapidity distributions for $bb$ and $bc$ production at 
the Tevatron for different choices of scale $\mu$.}
\label{fig:tev_eta_mu} 
\end{figure}

Recently there has been some concern that the fragmentation approximation may
not work as well as anticipated for $B_c$ production.  This assertion is based
on a comparison between the full calculation and the fragmentation
approximation for $B_c$ in $\gamma\gamma$ collisions\cite{kol1}.  There is, of
course, a significant difference between $\gamma\gamma$ and $gg$ initial
states, and the convolution of the gluon distributions with the parton-level
subprocess can have a large effect.  The calculation of $gg \to B_c X$ is much
more complex than that of $\gamma \gamma \to B_c X$, but is necessary to
understand the validity of the fragmentation approximation.  The authors of
\cite{kol1} went on to study $B_c$ production at hadron colliders\cite{kol2}.
They found that the fragmentation approximation still differs from the full
calculation at the parton level ($\hat{\sigma}[g g \to b \bar{b}(\to B_c X)]$
{\it vs.}  $\hat{\sigma}[g g \to B_c b \bar{c}]$) except at high $p_{_T}$, but
after the convolution of $\hat{\sigma}$ with the gluon distribution functions,
the full calculation and the fragmentation approximation agree quite well for
$p_{_T} > 10$~GeV both at Tevatron and LHC energies.  Fig.~6 of
\cite{kol2} shows that the agreement between the full calculation and the
fragmentation approximation is acceptable down to $p_{_T}^{min} = 5$~GeV at the
Tevatron and $p_{_T}^{min} = 10$~GeV at LHC.  As the fragmentation functions
and the production mechanisms are extremely similar for $B_c$ and heavy diquark
production, our calculation of the production of doubly heavy baryons should
also be accurate in this $p_{_T}$ range.

The decay modes of doubly charmed baryons have been studied in \cite{rox} 
using SU(3) flavor symmetry.  This approach relates the decays of these 
particles, and could be used to find relations among the decays of the $bb$ 
baryons and among the $bc$ baryons.  The transitions between the doubly heavy 
baryons\cite{whitesavage} and some decays of the $bc$ baryons\cite{sanchis} 
have been discussed in the heavy quark limit.  The lifetimes of $c$ and $b$ 
quarks are not very different ($\tau_{_{\Lambda_c}} \sim 0.2 \times 10^{-12}$~s 
while $\tau_{_{\Lambda_b}} \sim 1.07 \times 10^{-12}$~s\cite{pdg}), so that the 
weak decays of both $b$ and $c$ are important when considering the decays of 
the $bc$ baryons.  $bc$ baryons will decay weakly (both hadronically and 
semi-leptonically) to single $b$ quark baryons as well as doubly charmed 
baryons.  Both of these possibilities have been discussed in the 
literature\cite{rox,fleckrich,datta}.  Either a single $b$ baryon or a 
doubly charmed baryon traced back to a displaced vertex using a vertex detector 
will be a clear sign of doubly heavy baryon production\cite{baur}.  In the case 
of $bb$ baryons, the production of a doubly charmed baryon (possibly with same 
sign di-lepton!), with both $b$-quark decays tagged by a vertex detector, would 
be an impressive signal.  Unfortunately, the $bb$ baryons are produced less 
frequently than the other doubly heavy baryons, and two semi-leptonic decays 
will severely reduce the event rate.

In conclusion, the production via fragmentation of baryons containing two heavy 
quarks has been calculated in the fragmentation approximation.  The 
fragmentation approximation reproduces well the full calculation of $B_c$ 
production at hadron colliders for the $p_{_T}$ studied here\cite{kol2}; it is 
expected that the fragmentation approximation will work well for doubly heavy 
diquark production as well.  The production rates of these baryons at $e^+e^-$ 
colliders and at the $ep$ collider HERA are found to be negligible.  The 
situation is much better at hadron colliders, with approximately $8\times10^4$ 
events/yr expected at Tevatron and $1.3\times10^8$ events/year at LHC.  In 
addition to predictions for total production cross sections ($|\eta|<0.5$ and 
$p_{_T}>5(10)$~GeV at the Tevatron (LHC)), the distributions 
$d\sigma/dp_{_T}/d\eta|_{\eta=0}$ and $d\sigma/d\eta$ for $p_{_T}>5(10)$~GeV at 
the Tevatron (LHC) are studied.  The detection of these particles and 
measurements of their properties will provide an experimental challenge due to 
the large number of their decay modes, but will be a rich testing ground, 
{\it e.g.}, for HQET and QCD potential models.

\vspace*{0.4cm}
\noindent{\Large {\bf Acknowledgments}} \\
The work of M.A.D. was supported by the Natural Sciences and Engineering 
Research Council (NSERC) --- Canada.

\newpage
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\end{document}

