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\def \R#1{$\lbrack#1\rbrack$}
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\pageno 0
{\baselineskip 18pt
\rightline{TAUP 1728-89}
\rightline{May 1989}
\vskip.5in
\centerline{\bf MULTIPLE HADRONIZATION AND}
\centerline{\bf THE MOMENTUM OF CHARMED PARTICLES:}
\centerline{\bf A SIGNAL OF THE QUARK-GLUON PLASMA}
\vskip\baselineskip
\centerline{Benjamin Svetitsky}
\vskip\baselineskip
{\it
\centerline{Center for Theoretical Physics}
\centerline{Massachusetts Institute of Technology}
\centerline{Cambridge, MA 02139 USA}
\centerline{\rm and}
\centerline{School of Physics and Astronomy}
\centerline{Sackler Faculty of Exact Sciences}
\centerline{Tel Aviv University}
\centerline{Tel Aviv 69978 Israel
\rm\footnote*{Current address}}}
\vskip 1in
}
%\vfill eject
%\pageno 2
\noindent{\it ABSTRACT.}
The transverse momentum of charmed particles should be degraded by their
passage through the mixed-phase fluid existing during the course of the
quark-hadron phase transition.
The mechanism is multiple hadronization, the repeated conversion
of quark to meson and back as phase boundaries are crossed.
The effect can be reduced by thermalization via ordinary collisions with
particles of the plasma or the hadron gas, but passage through a
homogeneous dense medium cannot duplicate the momentum degradation.
The effect, if observed, thus constitutes a simple signal of the
confinement phase transition in ultrarelativistic heavy ion collisions.

\vfill\eject
\pageno 2

In the continuing experimental effort to create a quark-gluon plasma in
ultrarelativistic heavy ion collisions \R1, a simple
signal of plasma creation remains elusive.
Tools suggested to date include the spectrum of continuum dileptons \R2,
the yield of resonances of the $\psi$ family \R3, the ratio of strange to
non-strange particles \R4, and the pattern of two-pion correlations \R5.
All of these currently suffer from theoretical or practical difficulties.
The first is heavily contaminated by backgrounds stemming from direct
Drell-Yan emission and emission from the late hadron gas phase \R2;
in the second, the effect of plasma suppression may be simulated
by prosaic QCD effects and by scattering in the hadron gas \R6;
the third is obscured by considerable processing
(i.e., collision and recombination) in the course of the collision \R7;
and the fourth mainly probes the structure of the collision region
at the breakup stage, illuminating only weakly the intermediate
stages of the process.
At a more basic, conceptual level, the first three proposals are
deficient in that they assume models for the quark-gluon plasma based
(usually) on kinetic theory in a weakly interacting
gas.  Thus unforeseen
complexity in the structure of the plasma, which is, after all,
a non-linear, strongly interacting system \R8, may vitiate the
predictions.
The fourth proposal, pion interferometry, is attractive in that it
stands on the most basic features of hydrodynamic flow in the presence
of a phase transition.
Unfortunately, it cannot yet be tested because of insufficient statistics
in the light-ion CERN experiments.

In this note, I propose the study of charmed particles as likely to
yield a signal which is sensitive to features of the early
evolution of a collision while relatively insensitive to the detailed
structure of the produced matter.
In particular, the transverse momentum distribution should be profoundly
affected by the mere fact of a phase transition, irrespective of any but
the grossest features of the high-temperature phase.
Charmed quarks can only be produced in the initial hard collisions
of partons present in the incoming nuclei (as long as the initial
temperature is not above, say, 400 MeV \R9).
Before they emerge, they must traverse the produced
fluid, and they do so rather slowly, i.e., non-relativistically.
Whatever the other characteristics of the high-temperature plasma,
the charmed quarks are not confined; confinement only
sets in when they hadronize in the course of the phase transition.
The phase transition, however, is a complex kinetic process, which
proceeds through the nucleation, growth, and coalescence of bubbles
of the low-temperature phase---the true QCD vacuum, populated with
a gas of hadrons.
This ``mixed phase'' is very long-lived, with perhaps ten times the
lifetime of the original plasma.
During this long time, a charmed quark may emerge again and again into
bubbles of confining vacuum, only to reenter the plasma each time
(see Fig.~1).
Each emergence requires hadronization.
In each hadronization, the charmed quark loses energy; this
process of {\it multiple hadronization} exerts a relatively large
drag force on the quark, which will be reflected in the momentum
of the charmed meson that is ultimately emitted.
This drag thus offers
a signal of the existence of a mixed phase in the course
of the evolution of produced matter.

For a rough estimate of the drag force due to multiple hadronization,
consider a charmed quark traveling through a region of mixed phase.
On entering a bubble of vacuum, the charmed quark is forced to pull
a light antiquark along to form a meson, losing energy in the process.
(More precisely, the charmed quark stretches a QCD flux tube into the
vacuum, and the tube breaks either by sucking in an existing antiquark
or by creation of a light $q\bar q$ pair \R{10}.)
Upon reentering the plasma, the light antiquark is stripped off and
diffuses
away, forcing the heavy quark to find another partner in the next
hadronization.
We estimate the rate of energy loss (the drag force) via
$$
{dE\over dx} = {\eval{\Delta E} \over\lambda},
\eqno{(1)}
$$
where
$\eval{\Delta E}$ is the mean energy loss in a single passage through a
bubble and $\lambda$ is the mean free path for quark-bubble collisions.
The latter is given by
$$
\lambda^{-1}=\rho\sigma,
\eqno{(2)}
$$
where $\rho$ is the density of bubbles
and $\sigma$ the cross-sectional area
of a bubble, assumed the same for all bubbles at a given time.

A lower bound on $\eval{\Delta E}$ comes from the kinematics of the
$c\to D\to c'$ hadronization/ionization process.
Suppose the $c\to D$ step occurs at its kinematic limit, with no energy
loss to the plasma or to the production of other particles.
Then we have $E_D=E_c$, and the heavy quark is slowed because of the
greater mass of the meson.
Suppose similarly that the heavy quark loses no more momentum in the
$D\to c'$ step, i.e., that it keeps its velocity as the light quark
is stripped off.
Then $E_c'/m_c=E_D/m_D$, and thus
$$
\eqalignno{
\Delta E &\equiv E_c'-E_c=-\left(1-{m_c\over m_D}\right) E_c \cr
&\approx -{1\over6}E_c. &(3)\cr}
$$
The minimal value for this energy loss is obtained by setting $E_c'=m_c
\approx 1.5\ {\rm GeV}$, namely, $|\Delta E|> 300\ {\rm MeV}$.
Thus even one extra hadronization causes substantial energy loss.

Creation of a $D^*$ instead of a $D$, or fragmentation into $D\pi$,
will more than double the energy loss. 
When the $c$ quark is sufficiently energetic that threshholds
are unimportant, these events are much more likely than $D$ creation.

Returning to Eq.~(2), we require estimates of $\rho$~and~$\sigma$.
For these, we shall adopt a simple model of the phase transition
kinetics.
Begin with a plasma which cools to the
transition temperature at proper time $\tau_p$.
(We work in the framework of the Bjorken scaling expansion \R{11}, a
one-dimensional expansion with fluid flow invariant under longitudinal
boosts.)
We suppose that at this instant there is enough fluctuation in the fluid
to allow instant nucleation of bubbles of vacuum, with some density
$\rho_0$; furthermore we assume that these bubbles can grow quickly
enough to prevent any supercooling which would bring about additional
nucleation at later times \R{12}.
Thus the number of bubbles is conserved, and the one-dimensional scaling
expansion gives
$$
\rho=\rho_0 {\tau_p\over\tau}.
\eqno{(4)}
$$

Since all the bubbles grow in parallel,
they present identical geometrical
cross-sections to the charmed quark, and we shall take $\sigma =
\pi r^2$ to be the effective cross section in (2).
$r(\tau)$ may be calculated from
$$
{4\over3}\pi r^3\rho = 1-R,
\eqno{(5)}
$$
where $R(\tau)$ is the volume fraction of plasma in the mixed phase,
which drops from unity (at $\tau_p$) to zero (when the plasma
disappears).
$R$ in turn can be calculated via entropy conservation.
Assuming that the entropy density of the plasma and of the hadron gas
can be written as $s_p=c_pT_c^3$ and $s_h=c_hT_c^3$, respectively,
and noting that the scaling expansion gives
$s=Rs_p+(1-R)s_h=s_p\tau_p/\tau$,
we obtain
$$
1-R={c_p\over c_p-c_h}\left(1-{\tau_p\over\tau}\right),
\eqno{(6)}
$$
whence
$$
\sigma\rho\propto (\tau-\tau_p)^{2/3} \tau^{-1}.
\eqno{(7)}
$$
$c_p$ and $c_h$ are proportional to the effective number of degrees of
freedom in the respective phases.
If we take ideal gas thermodynamics of quarks and gluons on the one hand
and pions on the other, we have $c_p/c_h=37/3$.

This relation is only valid, of course, as long as the bubbles don't
touch.
After the bubbles begin to coalesce, the surfaces join in such a way as
to
lower the total surface area.
In that case Eq.~(7) provides an upper bound on the total area.
Coalescence {\it must} occur by the point at which
$1-R=0.74$, which is the packing
fraction for a face-centered-cubic lattice of dense-packed bubbles
(an admittedly fanciful possibility).

After appreciable coalescence has taken place, we have an inverted
picture:
Now there are shrinking droplets of plasma surrounded by the hadron gas.
The passage of the charmed quark through the medium at this stage is
however similar to that discussed above, namely, a series of
hadronizations alternating with ionizations.
We continue using Eqs.~(1)--(2), where now $\rho$ represents the density
of droplets and $\sigma$ the droplet cross section.
Let us suppose that the number of droplets is roughly equal to the
original
number of bubbles.
(Recall that these droplets are formed from the interstices of the
lattice of bubbles.)
Then we have as before $\rho=\rho_0\tau_p/\tau$, while (5) is replaced by
$$
{4\over3}\pi r^3 \rho = R.
\eqno{(8)}
$$
Using (6), we obtain
$$
\sigma\rho\propto (\tau_h-\tau)^{2/3} \tau^{-1},
\eqno{(9)}
$$
where $\tau_h=c_p\tau_p/c_h$ is the time at which the phase transition
ends, leaving a pure hadron gas.

The scaled variable $\rho\sigma/\rho_0^{1/3}$ is plotted against the
scaled time $\tau/\tau_p$ in Fig.~2, for both vacuum bubbles in
the plasma and plasma droplets in the vacuum.
Coalescence of bubbles will give an interpolation between the two
curves.
There is a period of peak drag lasting a time of perhaps $4\tau_p$ from
the onset of the phase transition.
Inserting the threshold value of $\Delta E = 300\ \rm MeV$, and guessing
$\rho_0=1\ \rm fm^{-3}$, we obtain $|dE/dx|> 180\ \rm MeV/fm$
during this period.
Note that for an initial temperature of $T_0= 1.5\,T_c$
at $\tau_0=1\ \rm fm$, the time scale $\tau_p=3.4\ \rm fm$, so the period
of peak drag is long indeed.

The bubble/droplet radius is shown in Fig.~3.
The bubbles grow rapidly past a radius of 0.5~fm, which is large enough
to accomodate the width of a flux tube and also its length (see below).
The droplets created at coalescence stay above this size for
a time period longer than the period of peak drag noted above,
so here too the ionization/hadronization process can safely take
place.

The effect of the drag force will be seen in the $p_\bot$ distribution of
the final $D$ mesons, but not in the longitudinal rapidity distribution.
The reason is that the longitudinal motion is unaffected by the drag.
The charmed quarks are created near $t=z=0$ by the hard collisions of
the incoming partons, and thus their trajectories are given by $z=v_zt$.
This is also the equation governing fluid flow in the one-dimensional
scaling expansion, and hence there is no initial relative motion between
the charmed quarks and the fluid.

To discuss the $p_\bot$ distribution, let us begin with $pp$ collisions,
where the distribution of $D$ mesons is measured to be \R{13}
$$
\left({d\sigma\over d p_\bot^2}\right)_{pp\to DX}
\propto e^{-p_\bot^2/p_{\bot0}^2},
\eqno{(10)}
$$
with $p_{\bot0}=1.0\ \rm GeV$.
We shall assume that this distribution is independent of longitudinal
rapidity $\eta$.
A $D$ meson traveling with a typical $p_\bot$ of 1~GeV  has a
velocity
of $0.5\,c$ in the frame where its $\eta=0$, and $E=2.06$~GeV in that frame.
We take this to be indicative of the energy of the parent $c$ quark,
and also of the typical $c$ quark created in nucleus--nucleus collisions.
Such a quark thus has $p_\bot=1.4\ \rm GeV$ and a kinetic energy of 560~MeV.
The collision region in a central Pb--Pb collision is some 15~fm in
diameter, so it is clear that any quark created not too near the surface
will stay within the collision region well into the mixed-phase stage,
and furthermore will lose most of its initial momentum.

It is significant that this energy loss arises through interaction
with phase interfaces, and not with constituent particles (quarks,
gluons, or mesons).
If the latter were the case, the drag force would be proportional to
the momentum of the $c$ quark, and the fluctuation-dissipation theorem
would take effect \R{14}.
The theorem states, simply, that a large drag force will
be accompanied by rapid diffusion.
The diffusion {\it in momentum space} would rapidly bring the charmed
quark into thermal equilibrium.
Then the observed $p_\bot$ distribution of $D$ mesons would be thermal,
$$
\left({d\sigma\over d p_\bot^2}\right)_{AA\to DX}
\propto e^{-p_\bot^2/2m_DT}.
\eqno{(11)}
$$
If the $D$ were to escape during the mixed phase, and if we assume
a transition temperature of 200~MeV,
we would have $2m_DT=(775\ \rm MeV)^2$, not very different from
the coefficient in (10).

This points the way to a possible mechanism for avoiding the effect,
namely, re-thermalization either by the plasma or by the hadron gas,
either during the phase transition or after.
The plasma drag coefficient $\gamma_p$
was calculated perturbatively in \R{14},
giving a result which may be approximated by
$$
\gamma_p=c\,{T^2\over m_c},
\eqno{(12)}
$$
with $c\simeq 0.3$.
The corresponding time scale is given by $\tau_{\rm diff}\equiv 
\gamma_p^{-1}$.
(The diffusion time scale is the same as the drag time scale by the
fluctuation-dissipation theorem.)
Using this result, perhaps without justification, at $T_c=200\ \rm MeV$,
we get an equilibration time of more than 20~fm, {\it without} taking
into account the dilution of the plasma in the mixed phase.
Of course, higher-order and non-perturbative effects could make this time
much shorter when $T\sim T_c$;
furthermore, the corresponding coefficient for a charmed meson moving
through the hadron gas has not been calculated.
It is thus not ruled out that re-thermalization may prevent degradation
of the charmed meson momentum.

A question of perhaps greater importance, however, is whether scattering
by a dense hadron gas (or other dense medium)
may cause $p_\bot$ degradation
similar to the multiple hadronization mechanism.
Here the fluctuation-dissipation theorem indicates that the answer
is no.
As discussed above, large drag due to collisions with thermalized
particles
would be accompanied by rapid thermalization, hence yielding
perhaps a 20\% reduction in the mean $p_\bot$.
Only drag due to bubbles, which move hydrodynamically rather than
thermally, can give a dramatic momentum degradation.

Still, if $T_c$ is much less than 200 MeV, we must entertain the possibility
that the charmed particle re-thermalizes at a lower temperature, ending
up with a low mean $p_\bot$.
Likewise, if the charmed particle stays in the hadron gas after the
transition concludes, will it cool with the medium?
I believe both concerns can be dismissed.
The plasma drag coefficient (13), even if it is somewhat larger
because of strong-coupling effects, will decrease as $T^2$,
where the exponent comes from a combination of the density of massless
quanta ($\propto T^3$), their mean momentum ($\propto T$), and
the Debye-screened scattering cross section ($\propto T^{-2}$).
Thus the diffusion time scale will increase as $T^{-2}$.
The drag coefficient in the hadron gas will drop as a larger power
(because of the absence of Debye screening, and because of PCAC)
or even exponentially (as the pion mass begins suppressing the
equilibrium density of pions).
Thus the charmed particle will decouple rapidly as the system cools,
preserving in its momentum a relic of earlier times.

Finally, it is instructive to note the contrast between the multiple
hadronization process and the perturbation of light-quark jets by
the plasma \R{15}.
The essential difference is the amount of time needed to break the
flux tube into hadrons in the two cases.
In common flux-tube hadronization pictures,
such as the Lund scheme \R{16}, hadronization
typically occurs at a proper time $\tau_0\sim 1\ \rm fm$.
Thus the length of the flux tube at the time the leading hadron is
formed is $z=\beta\gamma\tau_0=(p/m)\tau_0$.
For the slow-moving $D$ meson discussed above, $z<0.7\ \rm fm$, and
hadronization can easily be completed within the confines of a
bubble in the mixed phase.
For a high-energy jet, on the other hand, $z$ can be several hundred
fm, leading to hadronization well outside
the collision region and allowing a perturbative treatment of
scattering of the leading quark by the plasma.
\vskip\baselineskip
\centerline{\bf ACKNOWLEDGMENTS}
\vskip\baselineskip
I thank Tetsuo Matsui for seminal discussions,
Shmuel Nussinov for criticism, and the Weizmann Institute
of Science for hospitality.
This work was supported by the U.S. Department of Energy under
Contract \#DE-AC02-76-ER03069, and by the Center for Absorption
in Science of the Israel Ministry of Immigrant Absorption.
Further support was provided by the German-Israel Foundation.
\vskip\baselineskip
\centerline{\bf REFERENCES}
{\frenchspacing
\vskip\baselineskip
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H.~Satz, H.~Specht, and R.~Stock, eds., {\it Quark Matter,}
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{\bf 18} (1988) Nos.~1/2.
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K.~Kajantie, M.~Kataja, L.~McLerran, and P.~V.~Ruuskanen, Phys.
Rev. D{\bf 34} (1986) 811 and references therein.
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T.~Matsui and H.~Satz, Phys. Lett. {\bf 178B} (1986) 416.
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J.~Rafelski and B.~M\"uller, Phys. Rev. Lett. {\bf 48} (1982) 1066.
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}
\vfill\eject
\centerline{\bf FIGURE CAPTIONS}
\vskip\baselineskip
\noindent
FIG. 1.
(a) Passage of a charmed quark through the mixed phase early in the
phase transition.
The $c$ quark enters a bubble of vacuum and hadronizes into a $D$ meson,
which then reenters the plasma and dissociates, leaving quark $c'$ with
lower energy than $c$.
(b) Later in the phase transition, we have a $D$ meson which enters
a droplet of plasma and dissociates.
The resulting $c$ quark then reemerges into the vacuum and hadronizes
into meson $D'$.

\noindent
FIG. 2.
Inverse mean free path for $c$ quark in the mixed phase, scaled by the
density of bubbles at nucleation.
The solid curve corresponds to scattering by bubbles,
the endpoint due to bubbles at their maximum
packing fraction.
The dashed curve corresponds
to scattering by droplets thereafter.

\noindent
FIG. 3.
Radius of bubbles before coalescence (solid curve) and of droplets
afterwards (dashed curve).

\bye
