Summary of Recent Work (last update 1999)

The main motivation of the investigations of high energy nuclear collisions is to study the quark-gluon substructure of nuclear matter and the possibility of a phase transition from a hadronic matter to a quark-gluon plasma at extremely high energy densities. The ultrarelativistic nuclear collision program in has been designed to study both the fundamental and phenomenological problems in this subfield. We have develped programs in particular to assist the analysis and understanding of both the existing and future experimental data, to find a promising, reliable set of signatures for the phase transition, and to predict new results from fundamental theories under the extreme conditions.

This program covers many areas from theoretical field theory to phenomenology of current and future high energy hadron and nuclear collisions. Some aspects of the formal theoretical work is directly related to the problems that we encounter in developing phenomenological models. The ultimate goal is to include all major features of the fundamental theories into a model which can describe ultrarelativistic heavy ion reactions from low to high energies. Such a model can also guide experimentalists to design their detectors for the forthcoming RHIC collider ( and proposed LHC) to best take into account of the new physics in that energy regime.

1. Perturbative Approach to QGP Formation

As the experimentalists are preparing for their experiments of ultrarelativistic heavy-ion collisions at RHIC(BNL) and LHC(CERN), the most pressing question theorists face is the initial condition of the dense matter which is created in the reactions and later on materialize into hadrons. Since the whole purpose of the RHIC (and LHC) program is to create and detect a quark-gluon plasma in high-energy heavy-ion collisions, it is important to know theoretically whether such an initial condition is formed. In the last few years, there have been enormous developments in a perturbative QCD based description of high energy heavy-ion collisions. Currently there are two Monte Carlo models of such description, Parton Cascade Model by K. Geiger and B. M\"uller and HIJING by X.-N. Wang and M Gyulassy [review1, pub_91.3]. Both models predict a large number of initial partons produced in a very short time (less than 1 fm/c). Such a dense parton system has been shown to be in the deconfined state [pub_92.8]. However, it is still not clear whether such a deconfined parton system has actually achieved thermal and chemical equilibrium. If so, how the system evolves toward the full equilibration and what measurements can be used to study the evolution. These questions are the focus of our study in the partonic description of ultrarelativistic heavy-ion collisions in the 1995. The long range objective is to develop a space-time parton cascade program that utilizes HIJING as the initial condition prior to hadronization. The space-time cascade can only be done after a proper understanding of how subtle interference phenomena, e.g. the Landau-Pomeranchuk-Migdal (LPM) effect, can be included in a classical cascade simulation, and how hard partons affect the underlining soft interactions.

1.1 Parton thermalization in an expanding system

We have studied thermalization in an expanding parton plasma within the framework of Boltzmann equation [pub_96.1, pub_95.3]. In particular, we study the time-dependence of the relaxation time to the lowest order in finite temperature QCD and how such time-dependence affects the thermalization of an expanding parton plasma. Because of Debye screening and Landau damping at finite temperature, the relaxation time (or transport rates) is free of infrared divergencies in both longitudinal and transverse interactions. The resultant relaxation time decreases with time in an expanding plasma like $1/\tau^\beta$, with $\beta<1$. We prove in this case that thermal equilibrium will eventually be established given a long life-time of the system. However, a fixed momentum cut-off in the calculation of the relaxation time gives rise to a much stronger time dependence which will slow down thermal equilibrium. It is also demonstrated that the ``memory effect'' of the initial condition affects the approach to thermal equilibrium and the final entropy production.

1.2 LPM effect and radiative energy loss

We have continued a systematic study of multiple scatterings and radiation within the frame work of pQCD. Due to the unique feature of QCD, the interference pattern of multiple radiation induced by multiple scatterings is very different from QED. We have confirmed the validity of LPM effect in QCD [pub_94.1, pub_94.12]. Our results show that the radiative energy loss could be much larger than the elastic contribution. Because of the LPM effect however, it is highly sensitive to the infrared cutoff scale. We have investigated whether that extra sensitivity could be exploited as a tool to "measure" that unknown scale in the vicinity of the QGP transition [pub_95.1]. For this purpose, jet quenching and mono-jet production were proposed. We found that a singular variation of the color screening mass near the QCD phase transition will lead to a unique energy denpendence of the mono-jet rate. The same effect has also been included in the calculation of gluon equilibration rate.

1.3 Parton equilibration via gluon multiplication

Based on the initial condition given by HIJING Monte Carlo model which was developed at LBL, the subsequent evolution of the dense partonic gas to a thermally and chemically equilibrated quark-gluon plasmas is also investigated [pub_93.2, pub_95.2], including gluon multiplication via radiative processes. Medium effects such as color screening, multiple scattering, and interference have also been investigated and incorporated. We find that gluons can achieve equilibrium when their initial density is high enough, but quarks cannot reach chemical equilibrium within the life-time of the partonic system which has serious consequences for detecting QGP. Due to the consumption of energy by the additional parton production, the effective temperature of the parton plasma cools down considerably faster than the ideal Bjorken's scaling solution. Therefore, the life time of the plasma is reduced to 4 - 6 fm/$c$ before the temperature drops below the QCD phase transition temperature. However, this picture is very sensitive to the initial parton density. If we very the initial density by a factor of 4, gluons can achieve equilibrium much faster and the life-time of the system is also much longer.

1.4 Screening of initial parton production

One of the sources which contribute to the uncertainties in the initial parton density and temperature is the soft parton production below a cut-off $p_0=2$ GeV which is imposed on the model calculations to regulate the infrared divergence of pQCD. Such a cut-off is necessary for an effective model. However, for high-energy collisions of very heavy nuclei, partons with large $p_T$ are produced in large number, so that a medium of large $p_T$ partons is formed. Such a medium should then screen the soft interactions with smaller transverse momenta. The screening mass then provide a self-consistent regularization of the infrared divergence and the parton scattering cross section is calculable for very small $p_T$ as far as the screening mass is much larger than the QCD scale $\Lambda$. We have studied screening of initial parton production due to the presence of on-shell partons in high-energy heavy-ion collisions [pub_95.7]. We have shown that for sufficiently high collision energies and large nuclei the divergent cross sections in the calculation of parton production can be regulated dynamically and self-consistently without a constant transverse momentum cut-off. It is also shown that the resultant parton production and transverse energy production rates become finite. Extrapolate to small transverse momentum, we find that the initial parton density can be 3 or 4 time higher than the estimate from HIJING model.


The same kind of hard processes which lead the system to equilibration can also be used as direct probes of the early parton dynamics and the evolution of the quark-gluon plasma. Among these hard probes, electromagnetical signals, like direct photons and dileptons, are considered more direct since they can escape the dense matter without further interaction. They can thus reveal the dynamics of initial parton production and evolution. Similarly, open charm production, jet quenching due to energy loss and $J/\psi$ suppression can all provide us information about parton scattering and thermalization inside a parton plasma. In particular, preequilibrium $J/\psi$ suppression can reveal evidence of the deconfinement of the parton gas since the $J/\psi$ dissociation cross section with a deconfined parton gas is very different from that with a hadronic gas.

2.1 Open charm production

Unlike strange quarks, charm quarks cannot be easily produced during the mixed and hadronic phases of the dense matter since the charm mass is much larger than the corresponding temperature scale. The only period when charm quarks can be easily produced is during the early stage of the parton evolution when the effective temperature is still high. At this stage, the parton gas is still not fully equilibrated yet so that the temperature is only an effective parameter describing the average momentum scale. By measuring this pre-equilibrium charm production, one can thus probe the initial parton density in phase space and shed light on the equilibration time. We calculated open charm production during the equilibration of a gluon dominated parton plasma [pub_94.13], with both the time-dependent temperature and parton densities given by a set of rate equations. Including pre-thermal production, the total enhancement of open charm production over the initial gluon fusion depends sensitively on the initial parton density and the effective temperature. Especially, if the initial parton density is 4 times higher than the original HIJING estimate, the thermal production could double the total charm yield.

2.2 Dilepton and photon production

Similar to open charm production, dilepton and photon can also be produced from parton scatterings during the equilibration processes. Since dileptons and photons are not subject to final state scatterings due to their week interaction with the medium their momentum distribution will remain intact throughout the lifetime of the system, and thus can directly reveal the evolution history. Their production rates are directly related to the quark content of the parton system. Due to the small production cross section and chemical equilibration rate, quarks are far below chemical equilibrium. Therefore, we found that the dilepton production during the equilibration is well below the initial Drell-Yan. Only when the initial parton system is in complete thermal and chemical equilibrium, can the thermal dilepton overcome the Drell-Yan background in a small window of their invariant mass.

2.3 $J/\psi$ suppression

Unlike thermal open charm and dilepton production, $J/\psi$ suppression is less sensitive to the initial parton density and more sensitive to the deconfinement of the parton system. A study of charmonium dissociation cross section shows that a prerequisite for the break-up of a charmonium is the large momentum gluons which a hadronic gas at a normal temperature cannot provide. However, an initially produced parton system, though not in equilibrium, can be in a deconfined state. In the pre-equilibrium stage, the average parton transverse momentum is sufficiently large to break up a $J/\psi$, provided the partons are deconfined. The dissociation of $J/\psi$ will continue during the whole equilibration process until the effective temperature drops below a certain value or the beginning of hadronization, whichever takes place first. We have employed short-distance QCD is to calculate the $J/\psi$ survival probability in an equilibrating parton gas pub_95.8]. We found the break-up caused by gluon scattering during the evolution of the parton gas gives rise to a substantial $J/\psi$ suppression at both RHIC and LHC energies, using estimates of the initial parton densities. The transverse momentum dependence of the suppression is also shown to be sensitive to the initial conditions and the evolution history of the parton plasma.

2.4 Jet quenching and monojet production

Complementary to $J/\psi$ suppression, study of high $p_T$ jets and their propagation inside the medium can also probe the structure of the dense matter and possibly the phase transition, since these high $p_T$ jets are produced on a very short time scale as compared to the soft processes and their production rates and spectrum can be reliably calculated via pQCD. What jets could probe in high energy heavy ion collisions is the stopping power, $dE/dz$, of the dense matter for high energy quarks and gluons [pub_94.12]. That stopping power in turn is controlled by the color screening mass $\mu_D$ in that medium. A possible rapid change of $\mu_D$ near the phase transition point could lead to a variation of jet quenching phenomena which may serve as signatures of QGP formation [9]. The energy loss, $dE/dz$, of partons through interaction is also closely related to the thermalization and equilibration of partonic system as we have discussed. Using a recent estimate of the radiative energy loss of a fast parton inside a quark-gluon plasma, we calculate the suppression of high $E_T$ jets and the monojet production rates in high energy nuclear collisions [pub_95.1]. The sensitivity of these rates to the energy loss, especially to a possible sudden change during the QCD phase transition is studied. For a model of the initial energy density of the matter based on an estimate of the soft and the semi-hard contributions, it is demonstrated that a dip in the monojet rate as a function of the collision energy may constitute a signature of the phase transition. A rapid cross-over from quark to gluon jets at RHIC energies may lead to a shoulder in the energy dependence of the dijet suppression ratio at RHIC energies, which could provide a first check of the theoretical prediction that the energy loss of gluons in dense matter is about twice as large as the energy loss of quarks.


There are a few arguments against the proposed probes as the unambiguous signatures of a quark-gluon plasma formed in heavy-ion collisions. The size of the plasma volume is expected to be small (a few fm in diameter) and it does not live very long (between 2 and 10 fm/c). Furthermore, all signals from the quark-gluon plasma have background from the hot hadronic phase that follows the QCD phase transition. Therefore, it is important to understand the physics in hot hadronic phase in order to understand the experimental measurements. In addition, hadron properties in the hot hadronic matter is often closely related to QCD phase transitions. For example, the restoration of spontaneously broken chiral symmetry, and partial restoration of $U(1)_A$ symmetry, will affect hadron properties. The most often studied signatures of the the chiral symmetry restoration is strangeness enhancement. Recently, it has been suggested that a rapid cooling of the hot hadronic matter in heavy-ion collisions could lead to the evolution of the chiral order parameter to a non-equilibrium state. As a result, domains of the ``unconventionally'' oriented vacuum configurations allowed by the chiral symmetry may be formed. Detection of this so-called disoriented chiral condensate (DCC), would provide valuable information on the vacuum structure of the strong interaction and the nature of chiral phase transition.

3.1 Formation of disoriented chiral condensate

We have continued our study on the formation of DCC in the context of a linear sigma model on lattice [pub_94.11]. Assuming boost invariance and including both transverse and longitudinal expansion, we automatically incorporated the cooling and relaxation of the thermal fluctuations, and thus dynamically included the chiral phase transition in our calculation. To study the formation of DCC's under different initial conditions. we investigated two scenarios: quenching and annealing which correspond to non-equilibrium and equilibrium initial conditions. We found that under the assumed longitudinal and transverse expansion for a system with 6 fm in diameter, only the nonequilibrium initial condition can lead to formation of any sizable domains of DCC. Collaborations with Asakawa (now at Columbia University) and Huang (now at University of Arizona) will continue on Fourier study of the simulation, the pion momentum spectrum, iso-spin correlation and fluctuations. Studies of the low momentum enhancement of the pion fields, two-particle correlations due to the squeezed state of the chiral fields, and the thermalization properties in the effective linear sigma-model are still underway.

3.2 Wavelet analysis of DCC

The most often proposed signatures of the DCC is the distribution of neutral pion fraction $f$, predicted to be $P(f)=1/2\sqrt f$. However, the characteristic probability distribution in $f$ emitted from a disoriented region may be lost due to combinatorial statistics. This would follow from the Central Limit Theorem in statistics. The integrated probability distribution would be Gaussian instead of the proposed one if there are many uncorrelated domains. This would make the experimental search for the DCC signal a rather difficult task. We have proposed a novel method to study the DCC domain structure which features both space and scale localities. It is a multiresolution analysis performed by a discrete wavelet transformation (DWT). We demonstrate that the DWT proves to be very useful in identifying and measuring the DCC domain structures {\em simultaneously} in terms of their size (in scale) and location (in space). Since it is likely that there are other physical scales accompanying the typical DCC domain scale in a physical process, the multiresolution feature of the DWT is essential for identification of the structures of interest. We have conducted a preliminary study [pub_95.9] using the wavelet method to analyze the fluctuation of the neutral pion fraction $f$ in the spatial rapidity in numerical simulations of a classical linear $\sigma$-model in 1+1 dimensions. The probability distributions of the neutral pion fraction for various rapidity-bin sizes have distinctive shapes in the case of a DCC and exhibit a delay in approaching the Gaussian distribution required by the Central Limit Theorem. We also find the wavelet power spectrum for a DCC, to exhibit a strong dependence on the scale while an equilibrium system and the standard dynamical models such as HIJING have a flat spectrum. We would like to apply the wavelet method to analyze the experimental data when they become available. It seems promising that the DCC domain structure in heavy-ion collisions may be revealed by observing a {\em Delayed Central Limit} in probability distribution of wavelet transformed neutral pion fraction and a rise in wavelet power spectrum.

3.3 Partial $U(1)_A$ restoration and $\eta$ enhancement

At the Lagrangian level, QCD has, in addition to $SU(N_f)\times SU(N_f)$ chiral symmetry, an approximate $U(1)_A$ symmetry. It is well known that the $U(1)_A$ symmetry is violated by the axial anomaly present at the quantum level and thus cannot give rise to the Goldstone boson which would occur when $U(N_f)\times U(N_f)$ chiral symmetry is spontaneously broken. The $U(1)_A$ particle, known as $\eta '(958)$ in the $N_f=3$ case, acquires an additional mass through the quantum tunneling effects mediated by instantons. The $\eta (547)$ particle also acquires an additional mass through the mixing with $\eta '$. If the instanton effect becomes partially suppressed at a temperature lower than the chiral phase transition temperature, as indicated by the lattice calculation, both $\eta$ and $\eta'$ mass will drop. $\eta$ mass will decrease significantly because of the lightness of $u$ and $d$ quarks. As a consequence, $\eta$ particle production at small and intermediate transverse momenta will be enhanced. However, the final yield of the $\eta$ particles and their $p_t$ distributions both depend crucially on the chemical and thermal equilibrating processes involving the $\eta$. We have computed the thermal cross sections for various processes related to $\eta$ chemical equilibrium [pub_95.6]. The calculation is based on models which explicitly incorporate the $U(1)_A$ anomaly. An exponential suppression of the $U(1)_A$ anomaly due to the Debye-type screening of the instanton effect is parameterized according to the lattice calculation, which leads to the temperature dependence of the $\eta$ and $\eta '$ masses. It is found that the chemical equilibrium between $\eta$ and $\pi$ breaks up considerably earlier than the thermal equilibrium. Two distinct scenarios (scenarios A and B) for the $\eta$ freeze-out are discussed. We have predicted a modest enhancement of thermal $\eta$ production as a signal for the relic of $U(1)_A$ restoration.


4.1 High $p_T$ particles

The phenomenological aspect of our relativistic nuclear physics program tries not only to provide constraints for the development of a realistic model at both low and high energies but also to explain the current experimental data and propose possible measurements for future experiments. We plan to continue investigating a variety of physics topics with HIJING including $pA\rightarrow$ dihadron reactions and back-to-back two-particle correlations. Large $p_T$ particle production and suppression, especially baryons and kaons can be used as a measurement of the jet propagation inside a dense matter. Since a quark jet is more like to produce a leading proton or $K^+$, whereas a gluon can produce both a leading proton ($K^+$) or anti-proton ($K^-$), the relative suppression of proton (K+) vs anti-proton ($K^-$) can be used to measure the relative energy loss of a gluon vs a quark inside a QGP. Detailed analysis using HIJING is underway.

4.2 Strangeness production in p+A and S+A interactions at SPS energies

To understand the recent data on strangeness production at SPS energies, we used HIJING and VENUS models to study the systematics of strangeness enhancement and compared to recent data on $pp$, $pA$ and $AA$ collisions [pub_95.4]. The HIJING model is used to perform a {\em linear} extrapolation from $pp$ to $AA$. VENUS is used to estimate the effects of final state cascading and possible non-conventional production mechanisms. This comparison shows that the large enhancement of strangeness observed in $S+Au$ collisions, interpreted previously as possible evidence for quark-gluon plasma formation, has its origins in non-equilibrium dynamics of few nucleon systems. Strangeness enhancement is therefore traced back to the change in the production dynamics from $pp$ to minimum bias $pS$ and central $SS$ collisions. A factor of two enhancement of $\Lambda^{0}$ at mid-rapidity is indicated by recent $pS$ data, where on the average {\em one} projectile nucleon interacts with only {\em two} target nucleons. There appears to be another factor of two enhancement in the light ion reaction $SS$ relative to $pS$, when on the average only two projectile nucleons interact with two target ones.

5. Collaborations and Outreach Programs

Workshop on Pre-equilibrium Parton Dynamics

Since 1993, we have begun a new program to investigate parton production and equilibration in ultrarelativistic nuclear collisions and the pre-equilibrium parton dynamics. Our program, including the summer workshop on Preequilibrium Parton Dynamics at LBL organized by X.~N.~Wang, M. Gyulassy and B. M\"uller in 1993 has played a pivotal role in stimulating discussions and international collaborations in this area of study. ( There are still a few copies of the proceddings left. If you want a copy, please send me your request by email ) Several workshops (CERN, LBL, Trento) have been held in the US and Europe since then. Another one rganized by L. McLerran and M. Gyulassy will be held at INT in seattle.

Hard Probes Collaboration

This is our second year of organizing and participating in the Hard Probe Collaboration, an international theoretical collaboration on Hard Probes of Heavy Ion Collisions. Members of the collaboration are: J. Cleymans(Cape Town), K. J. Eskola (Helsinki), R. Gavai(Bombay), S. Gavin(BNL), S. Gupta(Bombay), D. Kharzeev (Moscow), E. Quack(Heidelberg), V. Ruuskanen(Helsinki), K. Redlich(Wroclaw), H. Satz(CERN), G. Schuler(CERN), D. Srivastava(Calcutta), R. Thews(Tucson), R. Vogt(LBL), X.-N. Wang (LBL).

The main purpose of this collaboration is to study hard processes in $pp$, $pA$ and $AA$ collisions systematically and explore their potential as probes of the dense matter formed in ultrarelativistic heavy ion collisions. Three workshops has been held at CERN, LBL and Trento. What the collaboration has concentrated on in the past three workshops is to compile systematic estimates of the interaction rates of various hard processes, on the nucleon-nucleon level, using the most detailed QCD calculations and the most recent parton distribution functions available. These calculations will serve as a baseline for measurements at RHIC and LHC. The compilation edited by H. Satz and X.-N. Wang has been published in refereed journal [proc2]. We also made LBL reports and they are widely distributed. The report was welcomed by the community with enthusiasm. It can be used as a handbook of different hard interaction rates for both theorists and experimentalists. The next goal of this working group is to consider nuclear effects on these hard processes.