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STATUS OF LATTICE GAUGE THEORY\footnote*{This work was 
supported in part by funds provided by the U.~S.~Department of
Energy under contract \#DE-ACO2-76ERO3069.}
\vskip 20pt

Benjamin SVETITSKY
\vskip 10pt
\item{}Center for Theoretical Physics, Laboratory for Nuclear Science and 
Department of Physics, Massachusetts Institute of Technology, Cambridge, 
Massachusetts\ \ 02139\ \ USA
\vskip 20pt
\item{}I review recent
results in lattice gauge theory which pertain to the physics
of high temperature QCD and quark matter.
Topics covered include:
the transition temperature in the pure gauge theory,
scaling and finite temperature behavior in QCD with quarks,
problems with non-zero chemical potential,
the Debye length in the high temperature plasma,
and quantities relevant to thermodynamics and hydrodynamics
in relativistic heavy ion collisions.

\baselineskip=.208in	% roughly 15pt
% \vskip 15pt %  ??

\section{1.}PREFACE

Two years ago in Helsinki, Tom DeGrand summarized the state of
the art in lattice gauge theory$^1$.
He gave tantalizing hints of what
the future may bring, and exhorted lattice theorists to work harder.
I will present a picture of lattice gauge theory today which
will show that theorists have indeed been working hard,
and that the future has begun to arrive.
Ambitious research programs are starting to give
answers to some of the questions we care about the most, and
much of the early confusion in lattice results is yielding to
systematic analysis.
I will be forced, however, to temper my boastful optimism with
some bitter realism, and warn you that further progress with
current algorithms will
require new generations of supercomputers.

I will discuss only results relevant to the physics of quark matter,
and I will concentrate on progress since Helsinki.
First I will deal with a nagging loose end in the study of pure
gauge theory (quarkless QCD), namely, what is the transition
temperature $\Ttr$?
The problem here has been the general one of taking the continuum
limit of lattice results.
I will describe the essence of the problem, and the results of
recent calculations.

Next I will summarize the rather startling progress that has
been accomplished in simulations of full QCD, quarks and all.
Most surprising is the preliminary evidence that continuum
scaling is being seen on the rather small lattices accessible to
current calculations.
I will show Monte Carlo data for the
finite-temperature chiral symmetry phase transition temperature;
if the tentative scaling behavior is the real
thing, then the calculated value of the transition temperature
may not be totally fanciful.
The phenomenological interest in
this transition is of course obvious; equally interesting would
be information about the phase structure of QCD at non-zero
chemical potential (i.e., non-zero baryon density).
I will speak briefly of the formal problems which prevent
analysis of this regime.

While the numerical studies of QCD are glamorous and impressive,
they are close to yielding information only about a bare minimum
of physically interesting quantities.
The phenomenology of relativistic heavy ion collisions will, on the
other hand, require rather detailed predictions.
To begin with, the high-temperature quark-gluon plasma is now
believed to be considerably more complex than its electrodynamic
counterpart.
I will describe some of the conundrums the theory poses, and
show some Monte Carlo data (in the pure gauge theory) which
address them.
Finally, I will give publicity to crude but audacious
attempts to calculate quantities relevant to the
phenomenological hydrodynamical theories of high energy
collisions.

\section{2.}LATTICE GAUGE THEORY IN A NUTSHELL

The conferences in this series have benefitted from excellent review
talks on lattice gauge theory by Helmut Satz$^2$, John Kogut$^3$,
and Tom DeGrand$^1$.
I presume, however, that there are some people in the audience who
have successfully resisted being turned into experts in the field.
Thus I begin with a brief description of what the lattice approach is
and what its basic problems are$^4$.

Much of quantum field theory can be boiled down to the problem of how
to deal with the Feynman path integral formula for the
thermodynamic partition
function,
$$
Z = \int [dA_\mu][d\psi][d\bar\psi]\, \exp \left[-S(A_\mu,\psi,\bar\psi)
\right]\,,
\equ1
$$
where (in the case of QCD) $S$ is the Euclidean
action written in terms of the glue field
$A_\mu$ and the quark fields $\psi$ and $\bar\psi$.
The fields are defined in a funny space-time, the temporal extent of which
is finite and given by the inverse temperature $\beta \equiv 1/T$,
and periodic boundary conditions are imposed at $t=0$ and $t=\beta$.
The partition function is the integral over all space-time configurations
of the fields.

The first step, necessary for {\it any} calculation in quantum field
theory, is to impose a short distance cutoff so that intermediate results
of calculations will be finite.
We choose to do this via the introduction of a space-time lattice
with $N_t \times N_s^3$ sites and lattice spacing~$a$:
the fields now live on the sites and links of the lattice, and the
inverse temperature is given by the temporal extent of the lattice via
$\beta=N_ta$.

{\baselineskip 10pt\midinsert
\vskip 1.75in
\cfigcapt1{The space-time lattice}
\endinsert}

The next step---the hardest---is to evaluate expectation values of operators
in the thermodynamic ensemble.
The general definition is
$$
\eval{\cal O} \equiv Z^{-1} \int [dA_\mu][d\psi][d\bar\psi]\,
	e^{-S} \,{\cal O},
\equ2
$$
where $\cal O$ is some function of the field variables.
Eschewing perturbation theory and other analytic approximations,
let us contemplate a direct numerical calculation.
I will not dwell on how many variables there are to integrate over in
(1) and (2), or on how many times the age of the universe it would
take to do those integrals on a computer.
Suffice it to say that even if we could do it in only, say, a tenth the age of
the universe, we would probably be too late to make concrete predictions
before RHIC comes on line.
Thus one is forced to adopt a Monte Carlo strategy.
We store in the computer's memory a single space-time configuration
of the fields, and we give the computer a rule for generating another
configuration from it.
This rule is rigged so that the sequence of configurations generated
turns out to be distributed (in the long run) as $e^{-S}$.
We calculate the value of $\cal O$ in each configuration, and
we estimate the expectation value $\eval{\cal O}$ by 
averaging the values of $\cal O$ in the sequence.

The only input parameter to the computer calculation is the dimensionless
coupling constant $g$.
Simple dimensional analysis shows that
$\eval{\cal O}$ is necessarily of the form
$$
\eval{\cal O} = a^{-d}f(g),
\equ3
$$
where $d$ is the dimension of $\cal O$ (in mass units) and $f(g)$
is the (dimensionless) function of $g$ given by the computer output.
The final step in deriving a physical result is to ask the
obvious question, what is the lattice spacing $a$?
In the end, of course, $a$ must go to zero, giving the continuum
limit and the physics of the continuum theory via renormalization.
An operational prescription for extracting this physics is the following.
$a$ is a function of the coupling, taking the form
$$
a=\Lambda^{-1}_L R(g),
\equ4
$$
where $\Lambda_L$ is an arbitrary scale with dimensions of mass
and $R$ is a dimensionless function.
$R$ must have the property that the $a\to 0$ limit is the same
as $g\to 0$, in accordance with asymptotic freedom (since $g$ is
the effective coupling at the scale of the lattice spacing).
When $g$ is small, perturbation theory may be used to derive the
expression
$$
R(g) = \left( {11g^2 \over 16\pi^2} \right)
	\exp \left(-{8 \pi^2 \over 11 g^2}\right)
\equ5
$$
for the pure gauge theory.
(The coefficients depend on the number
of quark flavors.)
Inserting (4) into (3) gives
$$
\eval{\cal O} = \Lambda_L^d \,R(g)^{-d}\,f(g).
\equ6
$$
Thus if $\eval{\cal O}$ is to have a finite limit then $f(g)$
had better behave as $f \sim R^d$ as $g\to0$.
The final result of the calculation is then that $\eval{\cal O}$
takes the value $f/R^d$ in units of the lattice scale $\Lambda_L$.
Note that $R(g)$ is a {\it universal} function, not dependent on the
observable being calculated;
$f(g)$ is of course different for each observable.

Now consider the
temperature $T$.
It is given by $T = \left[ N_ta(g)\right] ^{-1}
= \Lambda_L \left[ N_t R(g)\right]^{-1}$.
Maintaining a finite non-zero temperature in the limit $a\to0$
means that $N_t\to\infty$:
calculations near the continuum limit demand ever larger lattices.
The {\it transition} temperature $\Ttr$ is given by
$\Ttr = \left[ N_ta(g_{\rm tr})\right] ^{-1}$,
where $g_{\rm tr}$ is the coupling at which the transition occurs
on a lattice of fixed size
(it varies with $N_t$).
Using (4),
$$
\Ttr = \Lambda_L \left[ N_t R(g_{\rm tr})\right]^{-1},
\equ7
$$
and this combination should approach a finite limit as
$g$ (and $a$) approach zero.

\section{3.}THE TRANSITION TEMPERATURE IN THE PURE GAUGE THEORY

Now we can discuss recent progress in calculating $\Ttr$ in the
pure gauge theory as an exercise in taking the continuum limit.
According to (7), one calculates $g_{\rm tr}$ on lattices with
various values of $N_t$, and then uses (when $g$ is sufficiently small)
the formula (5) for the effective lattice spacing to obtain
the dimensionless ratio $\Ttr/\Lambda_L$.
Recent Monte Carlo results$^5$ for this ratio are shown in Fig.~2.
The points to the right of $6/g^2=6.0$ show no drift as the
continuum $(g\to0)$ limit is approached.
In years past, only points to the left of $6/g^2=5.7$ were available,
and it was thought$^6$ that the ratio had settled down to a
continuum limit in the neighborhood of $\Ttr/\Lambda_L\sim76$.
It is hoped that the move to the new value is a final one.
(A competing group$^7$ agrees that $\Ttr/\Lambda_L$ is constant for
$6/g^2>6.0$, but reports a value 10\% lower than that shown in Fig.~2.
The discrepancy is due to systematics.)

{\baselineskip 10pt\midinsert
\vskip 3.17in
\figcapt2{Ratio of the transition temperature $\Ttr$ to the lattice scale
parameter $\Lambda_L$ in the pure SU(3) gauge theory.
Monte Carlo data of Gottlieb {\it et al.\/}$^5$, with $\Ttr$ calculated
via two-loop scaling.
The line is the result of extrapolating to a constant.}
\endinsert}

The lattice scale $\Lambda_L$ may not look familiar, but the ratio
$\Lambda_L/\Lambda_{\overline{MS}}$ is calculable in
perturbation theory$^8$, so that the scale of the theory may be
related to that of the real world.
Unfortunately, $\Lambda_{\overline{MS}}$ in the real world
pertains to {\it full} QCD, not to a quarkless version;
moreover, it is only known to within a factor of 2.
Thus it has become customary to relate results in the pure gauge
theory to the string tension $\sigma$.
The string tension is calculated at $T=0$, and
the formula (3)
takes the form
$$
\sigma=a^{-2}f_\sigma(g),
\equ8
$$
where $f_\sigma$ is the output of the computer calculation.
Equation (4) allows us to define the dimensionless ratio
$\sigma/\Lambda_L^2$, which should approach a finite continuum
limit.
This behavior is seen in the Monte Carlo results$^9$ shown in Fig.~3,
in the points to the right of $6/g^2=6.0$.
(The point at $6/g^2=6.6$ is believed to suffer from systematic
errors and should be ignored.)

{\baselineskip 10pt\midinsert
\vskip 3.17in
\figcapt3{Ratio of the string tension $\sigma$ to the square of
the lattice scale
parameter $\Lambda_L$ in the pure SU(3) gauge theory.
Monte Carlo data of Barkai, Moriarty, and Rebbi$^9$, with $\sigma$ calculated
via two-loop scaling.
The line is the result of extrapolating to a constant.}
\endinsert}

Now we can eliminate $\Lambda_L$ and quote a value for the ratio
$\Ttr/\sqrt{\sigma}$.
This is an improvement over $\Ttr/\Lambda_{\overline{MS}}$
because $\sigma$ is known quite well from
quarkonium spectra and from the Regge slope [$\sigma\sim\rm(400\ MeV)^2$],
and one can argue that it should not change much as one adds
quarks to the pure gauge theory.
The result, taking account of the conflicting Monte Carlo data,
is
$$
\Ttr = (200 - 230) {\rm\ MeV}.
\equ9
$$

A historical note:
When $\Ttr/\Lambda_L$ was thought to have reached a continuum
limit in the region $6/g^2<5.7$, the ratio $\sigma/\Lambda^2_L$
appeared to have settled down as well$^{10}$.
This was not fortuitous: it was due to the fact that the
theory's $\beta$ function$^{11}$ passed through its perturbative value
before dipping below it in the region $5.7<6/g^2<6.0$.
What was fortuitous was that the ratio $\Ttr/\sigma$ took the
same value then as is quoted now, yielding the same result (9).
Thus the bloody battles fought over the last two years have
resulted in no ground gained by either side, but with high hopes
for a lasting peace at the current cease-fire lines.
A note of caution, however, is being sounded by Philippe de
Forcrand, who tells me he has results for $\sigma/\Lambda^2_L$
which show a new drift for values of $6/g^2$
to the right of the points shown in Fig.~3.
Oh, well.
See his talk for details.
For yet more recent results on $\Ttr/\Lambda_L$, see Tony Kennedy's talk.

It is important to note that lattice calculations this deep in the
scaling regime require supercomputer power, even though we are
still talking about the pure gauge theory.
The early days, when VAX-scale calculations were interesting, are no more.

\section{4.}SIMULATIONS OF FULL QCD

When quarks are included in the dynamics of the lattice gauge
theory, even rough first calculations cannot be attempted
without supercomputers.
Two years ago, reliable results were thought to lie in the dim future.
This is still the case if one takes a strict definition of the word
``reliable'';
nevertheless, I can report exciting progress in simulating full
QCD.
As will be clear, we have entered a period in the study of QCD
which is roughly comparable to the early days of
the study of pure gauge theories.

4.1.\quad Scaling and the continuum limit

The simplest fermionic operator to evaluate in QCD is the scalar
bilinear $\bar\psi\psi$.
This quantity is non-zero (at $T=0$)
in the limit of zero quark mass
because of the spontaneous breaking of chiral symmetry:
the origin of
the nearly massless pion, soft pion theorems, and the rest of PCAC.
Equation (3) takes the form
$$
\eval{\bar\psi\psi} = a^{-3}f(g),
\equ{10}
$$
and a finite limit as $a\to0$ requires\footnote*{Not quite.
$\bar\psi\psi$ actually does {\it not} have a finite continuum limit,
but rather scales with an anomalous dimension.
This is not discernible in the Monte Carlo results because
of large error bars.}
that
$f(g)\propto a(g)^3$.
In Fig.~4
I show Monte Carlo data obtained at great expense by Kogut$^{12,13}$.
The theory is QCD with 4 flavors of quarks, and the points are extrapolations
to the $m=0$ limit.
The last three points lie, within error bars, on a scaling curve,
which is a straight line if one ignores the power-law factor in (5).
These are the {\it first} results showing scaling behavior in full QCD;
they are the product of large amounts of supercomputer time expended on
running the ``hybrid microcanonical'' algorithm$^{14}$.

{\baselineskip 10pt\midinsert
\vskip 4.8in
\figcapt4{Logarithm of the chiral order parameter, extrapolated to
zero quark mass, in 4-flavor QCD with Susskind fermions;
Monte Carlo data of Kogut$^{13}$.
The line is the behavior expected from one-loop scaling in the continuum
limit for a theory with four flavors, ignoring anomalous dimensions.}
\endinsert}

4.2.\quad High temperature

In Fig.~5
I show Kogut's$^{13}$ data for $\eval{\bar\psi\psi}$ in
finite-temperature QCD.
Again, this is the theory with 4 degenerate light flavors, but the
points shown this time are actual Monte Carlo
data for a small non-zero quark mass.
Recall that varying the coupling $g$ means varying the effective
lattice spacing $a$ via (4), and hence gives a way of adjusting
the temperature via $T=(N_ta)^{-1}$.
It is easy to verify that increasing $6/g^2$ means increasing $T$.

{\baselineskip 10pt\midinsert
\vskip 4.1in
\figcapt5{The chiral order parameter $\eval{\bar\psi\psi}$ and
the Wilson line $W\!L$ (the confinement order parameter)
in 4-flavor QCD with light Susskind fermions at $T\not=0$;
Monte Carlo data of Kogut$^{13}$.}
\endinsert}

The region in Fig.~5 marked by the double arrow around
$6/g^2=5.17$ is the finite-temperature ``phase transition.''
To the left of this region, $\eval{\bar\psi\psi}$ takes large
values not very different from its value at $T=0$.
To the right, $\eval{\bar\psi\psi}$ is small, and would probably
be zero if the quark mass were zero:
this is chiral symmetry restoration, which is expected to be
diagnostic of the high-temperature phase in QCD with light quarks.
Also plotted in Fig.~5 is the Wilson line (denoted $W\!L$) which
tests confinement physics.
It rises steeply in the vicinity of the ``transition,''
indicating dramatic variation in the $q\bar q$ potential.\footnote\dag{
Unlike the situation in the pure gauge theory, $\eval{W\!L}$ is
non-zero even in the low temperature phase: there is no absolute quark
confinement even for low $T$ because of $q\bar q$ pair production.}
Assuming scaling according to (7) gives
$$
T_{\rm``tr\hbox{''}}/\Lambda_{\overline{MS}} = (2.14 \pm .10)
\equ{11}
$$
for QCD with four flavors.

The reason I have put ``phase transition'' in quotes is that the
data in Fig.~5 do not show one!
There is no sign, at any sufficiently small (but non-zero) quark mass,
of critical fluctuations or phase metastability as $T$ is raised
through the relevant regime; at most, there may be a very weak
first-order transition hiding in the apparently smooth variation
of $\eval{\bar\psi\psi}$ and $\eval{W\!L}$.
Kogut's data, derived from the hybrid microcanonical algorithm,
agree in detail with results obtained by Gavai$^{15}$ using a
pseudofermion Metropolis algorithm (on a somewhat smaller lattice);
see Rajiv Gavai's talk for a summary.

The conclusion that there is {\it no} phase transition between
high- and low-temperature regimes contradicts the earlier
results of Fucito {\it et al.\/}$^{16}$ (also from the pseudofermion
Metropolis method), who claimed to have encountered a strong
first-order transition for all small and intermediate
values of the quark mass.\footnote*{
This work dealt with QCD with {\it three} flavors, reduced from four
via an approximation valid in perturbation theory.}
Gavai and Karsch$^{17}$ have given detailed
criticism of the earlier work, the gist of which is that Fucito {\it et al.}
didn't simulate for long enough to let the lattice come to equilibrium.
Thus the hysteresis did not originate in a true first-order transition.

On the other hand, Fukugita and Ukawa$^{18}$, calculating with a
Langevin algorithm, also claim to have seen a first-order transition
for sufficiently light quark masses.
One may argue with the interpretation of their data;
indeed, here also one may argue that the simulations never reached
equilibrium.
In any case,
reconciliation of their results with those of the other groups
will take some time.

To be fair, those who see first-order transitions may reply to their
detractors that the other simulations were {\it too} long, and that
any theory on a lattice of finite size will show rounding of a phase
transition, given enough time.
There are always ambiguities in analyzing Monte Carlo results from
finite lattices.
Nevertheless, the burden of proof seems at present to be upon
the advocates of the transition.

The current overall picture of finite-temperature QCD is
summed up in Fig.~6.
At the top of the diagram, at $m_q=\infty$, is the pure gauge
theory, where there is a first-order confinement transition.
Work done with the hopping parameter expansion$^{19}$, which is an
expansion about the large-$m_q$ limit, shows that the transition
persists into the phase diagram, and that eventually a critical
point is reached beyond which the high- and low-temperature
phases are continuously connected.
The simplest interpretation of
Kogut's data is that there is a critical point (i.e., a
continuous transition) at $m_q=0$, with no singularities at all
encountered as the temperature is varied at small but finite
quark mass.

{\baselineskip 10pt\midinsert
\vskip 2.75in
\figcapt6{Summary of our knowledge of the phase diagram of QCD in the
temperature--quark-mass plane.
The line is a first-order transition, and the crosses are
critical points.}
\endinsert}

Unfortunately, there is a real problem with this picture.
General arguments$^{20}$ lead to the prediction that there should be a
{\it first} order transition for $m_q=0$ when the number of flavors
is three or greater.
Three possible resolutions come to mind.
One is that there is actually a weak first order transition
hiding in the finite resolution of Kogut's
Monte Carlo data.
Another possibility is that there is a first order transition at
$m_q=0$ but that it
vanishes quickly upon moving to $m_q\not=0$, so that it cannot
be seen in the finite-mass Monte Carlo simulations.
In either case, the likely {\it true} phase diagram of QCD would
have a first-order transition at $m_q=0$ which continues into
the diagram and ends in a critical point, much like the behavior
of the $m_q=\infty$ transition.

The third possibility is somewhat more sinister.
It rests upon the observation that all simulations done to date at $T\not=0$
have defined the lattice quark fields with the Susskind
prescription.
Within this scheme, the statement that there are four flavors
has meaning only close to the continuum limit.
The lattice theory is not symmetric under the chiral $\rm SU(4)
\times SU(4)$ symmetry group of the continuum theory, but only
under an Abelian subgroup.
It is quite possible that the theory with
the Abelian symmetry undergoes a continuous transition, which
will become first order as the continuum limit is approached and
the full symmetry restored.
Such worries can be laid to rest either by working closer to the
continuum limit (that is, on larger lattices), or by doing
parallel studies with Wilson fermions.
See Akira Ukawa's talk for an example of the latter.


For further discussion of Kogut's results, I refer you of course
to John Kogut's talk this afternoon.

4.3. \quad The problem of finite chemical potential

Simulating QCD with non-zero chemical potential coupled to the
baryon number is obviously very desirable for the study of quark
matter.
The major formal problem standing in the way of such
work has nothing directly to do with the lattice cutoff, and is
easy to explain.
Starting from the path integral expression (1) for the partition
function of QCD, we may integrate over the quark field variables
$\psi$ and $\bar\psi$ to get
$$
Z=\int[dA_\mu] \exp\left({-{1\over4}\int d^4x\,F^a_{\mu\nu}F^a_{\mu\nu}}\right)
\mathop{\rm Det}(\thru D+m+\mu\gamma_0)^{n_f}\ .
\equ{12}
$$
The argument of the determinant is the Dirac operator (including
the gauge coupling and the chemical potential $\mu$),
and $n_f$ is the number of flavors.
Since only the gauge field $A_\mu$ is left in (12), one may
define an effective action, writing
$$
Z=\int[dA_\mu] \exp\left[{-S_{\rm eff}(A_\mu)}\right]\ .
\equ{13}
$$
So far everything is well defined.
In particular, the fact that the determinant in (12) may make
$S_{\rm eff}$ complex is no cause for worry.

The catch comes when one contemplates doing Monte Carlo
simulations.
The steps leading to (12) and (13) are in fact the first steps
in all numerical calculations:
One always integrates out the fermions analytically and leaves the gauge
field for numerical work.
All Monte Carlo methods depend on being able to treat
$\exp[{-S_{\rm eff}(A_\mu)}]$ as a (positive-definite) probability
measure for the configuration of $A_\mu$.
Thus these methods cannot even be formulated when
$\exp({-S_{\rm eff}})$  is complex.
Unfortunately, this is exactly the case when $\mu\not=0$.

Two approaches to this problem have been taken recently.
The first is the most straightforward:
{\it Throw away} the imaginary part of $S_{\rm eff}$.
I show in Fig.~7
the rough results of Berg {\it et al.}$^{21}$
They worked with a hopping parameter expansion of the fermion
determinant, which is valid for large quark masses, and
calculated such quantities as the Wilson line, $\eval{\bar\psi\psi}$, and
the thermodynamic energy density
$\epsilon$ as functions of $T$~and~$\mu$.
The points shown in the figure mark regions of rapid variation
in these quantities, and do {\it not} necessarily indicate a phase
transition.
The neglect of ${\rm Im}\,S_{\rm eff}$, the hopping parameter
expansion, and the method of locating the ``phase boundary'' are
all reasons to feel that these results are no more than
qualitatively descriptive of the situation in real QCD.
J.~Engels will discuss this work further in his talk.

{\baselineskip 10pt\midinsert
\vskip 3.75in
\figcapt7{Results of Berg {\it et al.}$^{21}$ for the boundary between
confining and plasma regimes in the
temperature--chemical potential plane.}
\endinsert}

The second approach, which may gain more popularity in the
future, is to use a Langevin process to simulate the partition
function in place of standard Monte Carlo methods.
The Langevin method allows continuing hitherto real parameters
to complex values$^{22}$.
Theoretical justification of this continuation is still lacking
(at least, outside of perturbation theory), but that is of
course not a fatal shortcoming.
The method has been applied$^{23}$ to a model resembling QCD with
$\mu\not=0$, but not yet to QCD itself.
Presumably nobody wants to invest considerable amounts of
computer time in a questionable algorithm.

For further news (and further problems$^{24}$) regarding finite
chemical potential studies, I refer you to Frithjof Karsch's
talk this afternoon.

\section{5.}PHYSICS OF HIGH-TEMPERATURE QCD

Now I shall leave the glamorous world of incredibly expensive
simulations of full QCD, and discuss more mundane questions of
physics.
These questions are posed well in the pure gauge theory, and
answers may be pursued by the methods of lattice gauge theory.
It is likely that both the questions and the methods to address
them will pass on to the full theory when adequate computer
power is available.

5.1. \quad What {\it is} the quark-gluon plasma?

The fact that high-temperature QCD is far more complicated than
high-temperature QED has been appreciated for quite some time$^{25}$.
I would begin by posing the following riddle (suggested by Janos
Polonyi):

Consider the energy density of a black body:
$$
\epsilon = \gamma {\pi^2 \over 15} T^4,
\equ{14}
$$
where $\gamma=1$ for photons and $\gamma=8$ for free gluons.
An old argument states that since QCD is asymptotically free,
the coupling goes to zero logarithmically with increasing
temperature, and thus gluons should indeed be free in the
high-temperature limit.
Indeed, Monte Carlo results for the pure gauge theory$^{26}$ have
confirmed the validity of (14) for high-temperature glue.

Let us reexamine the old argument, however.
An effective theory may be derived$^{27}$ for gluons in the
high-temperature limit.
This theory is defined in {\it three} Euclidean
dimensions, and the action takes the form
$$
S = {1\over4} \int d^3x \, {1\over g^2T} F^a_{ij}F^a_{ij}
+ (A_0 \rm\ terms),
\equ{15}
$$
where I have picked out the self-interaction of the spacelike
glue fields.
(Recall that in a non-Abelian theory the field strength
$F^a_{ij}$ is non-linear in the gauge potential $A^a_\mu$.)
This self-interaction is precisely that of three-dimensional
QCD, which is believed to {\it confine} color:
There should be no free gluons!
To be precise, the spectrum of states that contribute to the
thermodynamic partition function should bear no resemblance to
a set of eight massless, propagating, vector particles.
What then is the meaning of the Monte Carlo results?

The resolution$^{28}$ of this particular paradox is fairly
straightforward (once Nathan Weiss explained it to me).
High-temperature QCD is a system with several scales.
$\epsilon$ is dominated by physics at an energy scale on
the order of $T$.
Electric screening (Debye screening) occurs at the
Debye scale, given perturbatively by $gT$.
Three-dimensional confinement physics sets in at the scale
$g^2T$, 
which for high temperatures (small effective coupling) is much smaller
than $T$.
Thus a different number of degrees of freedom is seen at different
scales, and the high-momentum modes which dominate the energy density
should be unaffected by the three-dimensional non-Abelian dynamics.

If $\epsilon$ is dominated by high-momentum degrees of freedom,
can it be calculated order by order in perturbation theory?
The answer$^{25}$ is a resounding {\it no}.
The non-Abelian dynamics sneak into perturbative calculations
and introduce serious infrared divergences, even in the large
$T$ limit.
Furthermore, perturbation theory gives no guidance regarding
separation of physical scales as $T\to\Ttr$, since then the
effective coupling $g(T)\sim 1$, implying $gT\sim g^2T\sim T$.

The same problems$^{29}$ appear in the Debye screening length
$\lambda_D$.
In QED, the one-loop result $\lambda_D\sim (eT)^{-1}$ is not seriously
affected by higher-order terms;
in particular, there is no mixing of the screened electric field
with the unscreened magnetic field.
This is almost certainly {\it not} the case in QCD.
I show in Fig.~8
Monte Carlo data$^{30}$ for
the screening mass $\mu_D\equiv1/\lambda_D$ in the pure gauge theory.
The data are not really good enough to differentiate
among $\mu_D\sim T$,
$\mu_D\sim gT$, or $\mu_D\sim g^2T$;
the fitted curve is $\mu_D\propto gT$, but with a coefficient some 30\%
smaller than the one-loop perturbation theory result.
It is apparent that the high $T$ limit where perturbative calculations
are relevant is approached slowly, if at all.
Earlier work$^{31}$ led to a similar conclusion at high $T$, but showed
markedly different behavior as $T\to\Ttr$ from above;
presumably the later results, obtained with much longer runs, are
more reliable.

{\baselineskip 10pt\midinsert
\vskip 4in
\figcapt8{The electric screening mass
in the pure SU(3) gauge theory,
calculated by DeGrand and DeTar$^{30}$.
The curve is $M\propto g(T)T$.}
\endinsert}

5.2.\quad Thermodynamics and hydrodynamics

The calculation of thermodynamic properties of QCD was an early aim
of lattice Monte Carlo simulations.
I will show you recent results for
the energy, the pressure, the entropy, and the speed of sound in hadronic
and quark matter, and for ratios relevant to hydrodynamical models
of high energy nucleus-nucleus collisions.

The energy density $\epsilon$ and the pressure $P$ are of obvious physical
interest;
they can be written in terms of local operators whose measurement via Monte
Carlo is straightforward. 
The entropy density $s$ is
related in phenomenological theories to the total particle multiplicity;
it turns out that it can be calculated via numerical integration of Monte
Carlo data.
Finally, the speed of sound must be calculated from the formula
$$
v_s^2={dP\over d\epsilon}
\equ{16}
$$
which requires numerical {\it differentiation}, a grisly prospect.

The results I show in Figs.~9--12 are from the work of Redlich and Satz$^{32}$
who reanalyzed existing Monte Carlo output$^{33}$.
The effects of dynamical quarks were included qualitatively via a low-order
hopping parameter expansion.
I remind you that this is an expansion about the $m_q=\infty$ limit;
Redlich and Satz decline to supply statistical error bars for their
results because the systematic errors in the expansion are likely to
be severe.

$P$, $\epsilon$, and $s$ are shown in Fig.~9.
The steep rise is the would-be phase transition, where color degrees
of freedom are liberated.
(There is no phase transition for sufficiently light quarks in the
hopping parameter expansion$^{19}$.)
I believe the statistical errors are not large.


{\baselineskip 10pt\midinsert
\vskip 3.8in
\figcapt9{Pressure $P$, energy density $\epsilon$, and entropy density $s$
of hot QCD, calculated via Monte Carlo
in the hopping parameter (large quark mass)
expansion$^{32}$.}
\endinsert}

Perhaps more interesting is the speed of sound $v_s$, shown in Fig.~10.
If there is a first order transition, $v_s$ should be zero there
but non-zero, discontinuously, on either side.
At a continuous transition one expects a smooth drop to zero
from either side, and if
there is no phase transition at all, there ought to be a residual
dip.
This is indeed seen in the Monte Carlo results, but the error bars
from numerical differentiation are probably large:
In the earlier work of Gavai and Gocksch$^{34}$ on the pure gauge
theory, the error bars were almost as large as the dip.


{\baselineskip 10pt\midinsert
\vskip 4in
\cfigcapt{10}{The speed of sound in hot QCD}
\endinsert}

Finally, I show in Figs.~11~and~12 two quantities used in hydrodynamical
models.
The ratio $P/(P+\epsilon)$ has been related to the collective transverse
expansion of the central rapidity region$^{35}$, and it is seen to
take a dip in the transition region.
This is a tantalizing hint of slowed expansion, and hence longer lifetime,
in collisions where plasma has been produced.
The ratio $\epsilon/s$ is proportional$^{36}$ to the average transverse
momentum $p_T$ of emerging secondaries, and is in turn a measure of
temperature, since $\epsilon\sim T^4$ and $s\sim T^3$ in ideal gases.
Note the change in slope in the transition region.

{\baselineskip 10pt\midinsert
\vskip 3.6in
\cfigcapt{11}{The ratio $P/(P+\epsilon)$, from the data in Fig.~9}
\endinsert}

{\baselineskip 10pt\midinsert
\vskip 3.75in
\cfigcapt{12}{The ratio of energy to entropy in hot QCD, from the data in
Fig.~9.}
\endinsert}

The approximations used in this work are rather crude, and the statistics
are not sufficient;
it is indicative of the possibilities for
future applications of the more sophisticated Monte Carlo algorithms.

\section{6.}SUMMARY AND PROSPECTS

Concerning the status of lattice gauge theory, I have brought you both good
news and bad news.
\item{$\bullet$}
The phase transition in the pure gauge theory seems to be under control.
That's the good news.
The bad news is that it took 6 years and the CYBER 205 to beat it into shape.
Amid, and indeed before, the euphoria surrounding the early Monte Carlo results,
Wilson$^{37}$ warned that lattice gauge theory would not yield believable
results until a difficult
renormalization group program had come to fruition.
Events seem to have borne him out$^{38}$.

\item{$\bullet$}
Full QCD is giving numbers sooner than was expected two years ago.
The Monte Carlo data for full QCD are about as good as the earliest data for the
pure gauge theory.
That's good news.
But whereas Monte Carlo methods for the pure gauge theory allowed
anybody with a CDC 6600, or even a VAX, to contribute to further
progress,
for full QCD the simplest calculations require supercomputer power,
and plenty of it.
It will take a while before any numbers settle down to hard predictions.
One can hope for (and indeed, expect) dramatic advances in supercomputer
technology and in algorithms, but for now, QCD is a rich man's game.

\item{$\bullet$}
High temperature QCD is still a mess.
Sorry, here I started with the bad news.
But there {\it is} no good news yet.
Neither perturbation theory nor Monte Carlo work has yet given a reliable
picture of the excitations of the high-temperature phase.
The problem is, indeed, becoming urgent:
What kind of quark-gluon plasma are we to look for at RHIC?

\noindent
I will end on an optimistic note.
While current lattice results for various phenomenologically interesting
quantities are extremely rough, they do open the door towards more complete
calculations.
There {\it is} good work to be done in the near future.
We can look forward to applications of state-of-the-art QCD programs
to questions of interest for RHIC.
Even if the first results will not be the last word, we will begin
to define the parameters of the quark-gluon plasma.

\vskip15pt\noindent
ACKNOWLEDGEMENTS

I thank Tom DeGrand and Carleton DeTar for giving me
graphs of their results in advance of their general distribution.
They also, along with Larry McLerran, Janos Polonyi, and Nathan
Weiss, called relevant references to my attention and provided
enlightenment on a variety of matters.

\def\PRD{Phys. Rev. D}
\def\PRL{Phys. Rev. Lett. }
\def\PL{Phys. Lett. }
\def\NP{Nucl. Phys. }
\def\ibid{{\it ibid.}, }
\frenchspacing
\baselineskip .139in
\vskip15pt
\noindent
REFERENCES
\ref1
T.~DeGrand, Lattice Monte Carlo calculations of finite temperature QCD,
in: Quark Matter '84, Proceedings of the Fourth International Conference
on Ultra-Relativistic Nucleus-Nucleus Collisions, ed. K.~Kajantie
(Springer-Verlag, Berlin, 1985), pp.~17--38.

\ref2
H.~Satz, \NP A418 (1984) 447c.

\ref3
J.~Kogut, \NP A418 (1984) 381c.

\ref4
J.~Kogut, Rev. Mod. Phys. 55 (1983) 775;\hfill\break
H.~Satz, Ann. Rev. Nucl. Part. Sci. 35 (1985) 245;\hfil\break
J.~Cleymans, R.~V.~Gavai, and E.~Suhonen, Phys. Rep. 130 (1986) 217;\hfil\break
B.~Svetitsky, \ibid 132 (1986) 1.

\ref5
S.~A.~Gottlieb, J.~Kuti, D.~Toussaint,
A.~D.~Kennedy, S.~Meyer, B.~J.~Pendleton, and
R.~L.~Sugar,
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\ref6
T.~\c Celik, J.~Engels, and H.~Satz, \PL 129B (1983) 323; see also Ref.~2.

\ref7
N.~H.~Christ and A.~E.~Terrano, \PRL 56 (1986) 111.

\ref8
A.~Hasenfratz and P.~Hasenfratz, \PL B93 (1980) 165;\hfil\break
R.~Dashen and D.~J.~Gross, \PRD23 (1981) 2340;\hfil\break
P.~Weisz, \PL 100B (1981) 331;\hfil\break
W.~Celmaster and D.~J.~Maloof, \PRD24 (1981) 2730; \ibid 24 (1981) 3208;
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H.~Kawai, R.~Nakayama, and K.~Seo, \NP B189 (1981) 40.

\ref9
D.~Barkai, K.~J.~M.~Moriarty, and C.~Rebbi, \PRD30 (1984) 1293.

\ref{10}
See references in Ref.~9.

\ref{11}
A.~Hasenfratz, P.~Hasenfratz, U.~Heller, and F.~Karsch, \PL 143B (1984) 193.

\ref{12}
J.~B.~Kogut, \PL 168B (1986) 367.

\ref{13}
J.~B.~Kogut, Towards the continuum limit of lattice gauge theory with
dynamical fermions, Illinois preprint ILL-(TH)-86-\#19, February, 1986
(unpublished).

\ref{14}
J.~Pol\'onyi and H.~W.~Wyld, \PRL 51 (1983) 2257;\hfil\break
J.~Pol\'onyi, J.~Kogut, H.~W.~Wyld, J.~Shigemitsu, and D.~K.~Sinclair,
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S.~Duane, \NP B257 [FS14] (1985) 652;\hfil\break
S.~Duane and J.~Kogut, \PRL 55 (1985) 2774.

\ref{15}
R.~V.~Gavai, Simulations of gauge theories with many fermions using the
pseudo-fermion method, Brookhaven preprint BNL 37214 (unpublished, 1985).

\ref{16}
F.~Fucito and S.~Solomon, \PL 140B (1984) 387;\hfil\break
F.~Fucito, C.~Rebbi, and S.~Solomon, \NP B248 (1984) 615;
\PRD31 (1985) 1460;\hfil\break
F.~Fucito, R.~Kinney, and S.~Solomon, \NP B253 (1985) 727.

\ref{17}
R.~V.~Gavai and F.~Karsch, \NP B261 (1985) 273.

\ref{18}
M.~Fukugita and A.~Ukawa, Deconfining and chiral transitions of
finite-temperature quantum chromodynamics in the presence of dynamical
quark loops, Kyoto preprint RIFP-642, January, 1986 (unpublished).

\ref{19}
P.~Hasenfratz, F.~Karsch, and I.~Stamatescu \PL 133B (1983) 221;\hfil\break
T.~\c Celik, J.~Engels, and H.~Satz, \ibid 133B (1983) 427;
\NP B256 (1985) 670.

\ref{20}
R.~Pisarski and F.~Wilczek, \PRD29 (1984) 338.

\ref{21}
B.~Berg, J.~Engels, E.~Kehl, B.~Waltl, and H.~Satz, Critical behavior in
baryonic matter, Bielefeld preprint BI-TP 86/05, January, 1986 (unpublished).

\ref{22}
G.~Parisi, \PL 131B (1983) 393;\hfil\break
J.~Klauder, Phys. Rev. A29 (1984) 2036.

\ref{23}
F.~Karsch and H.~W.~Wyld, \PRL55 (1985) 2242.

\ref{24}
I.~Barbour, N.-E.~Behilil, E.~Dagotto, F.~Karsch, A.~Moreo, M.~Stone,
and H.~W.~Wyld, Problems with finite density simulations of lattice QCD,
Illinois preprint ILL-(TH)-86-\#23, March, 1986 (unpublished).

\ref{25}
A.~D.~Linde, Rep. Prog. Phys. 42 (1979) 389;\hfil\break
D.~Gross, R.~Pisarski, and L.~Yaffe, Rev. Mod. Phys. 53 (1981) 43.

\ref{26}
I.~Montvay and E.~Pietarinen, \PL 110B (1982) 148; \ibid 115B (1982) 151;
\hfil\break
U.~Heller and F.~Karsch, \NP B257 [FS13] (1985) 254.

\ref{27}
T.~Appelquist and R.~Pisarski, \PRD23 (1981) 2305;\hfil\break
S.~Nadkarni, \PRD27 (1983) 917.

\ref{28}
C.~DeTar, Phys. Rev D32 (1985) 276.

\ref{29}
S.~Nadkarni, Non-Abelian Debye screening: I. The color-averaged potential,
Rutgers preprint RU-86-01, January, 1986 (unpublished).

\ref{30}
T.~A.~DeGrand and C.~E.~DeTar, Static screening lengths in the gluon
plasma, Colorado preprint COLO-HEP-113 (unpublished, 1986).

\ref{31}
M.~Fukugita, T.~Kaneko, and A.~Ukawa, \PL 154B (1985) 185.

\ref{32}
K.~Redlich and H.~Satz, Critical behavior near deconfinement, Bielefeld
preprint BI-TP 85/33, November, 1985 (unpublished).

\ref{33}
T.~\c Celik, J.~Engels, and H.~Satz, \NP B256 (1985) 670.

\ref{34}
R.~V.~Gavai and A.~Gocksch, \PRD33 (1986) 614.

\ref{35}
E.~V.~Shuryak and O.~V.~Zhirov, \PL 89B (1980) 253.

\ref{36}
J.~D.~Bjorken, Phys. Rev. D27 (1983) 1940.

\ref{37}
K.~G.~Wilson, Quantum chromodynamics on a lattice, in: New Developments
in Quantum Field Theory and Statistical Mechanics, Carg\`ese 1976,
eds. M.~L\'evy and P.~Mitter (Plenum, New York, 1977), p. 143;
Monte Carlo calculations for the lattice gauge theory, in: Recent
Developments in Gauge Theories, Carg\`ese 1979, eds. G.~'t~Hooft
{\it et al.} (Plenum, New York, 1980), p. 363.

\ref{38}
R.~Gupta, Monte Carlo renormalization group: a review, talk given at
Lattice Gauge Theory---A Challenge in Large Scale Computing, Wuppertal,
November 1985, 
Los Alamos preprint LA-UR-86-1116 (1986), to appear in
proceedings.

\bye
