About the Author

Frithjof Karsch is a Senior Scientist in the Physics Department at Brookhaven National Laboratory. He is head of the Lattice Gauge Theory group that was established in 2005 to study properties of strongly interacting matter in large scale computer simulations.

The Equation of State that Controls the Expansion of Matter Created at RHIC

QCDOC and NYBlue Join Forces

By Frithjof Karsch

The expansion of hot and dense matter created in a heavy ion collision at RHIC is controlled by an equation of state which describes the dependence of the pressure in the medium on its energy density. Knowledge of the equation of state is, for instance, indispensable for a correct hydrodynamic modeling of the expansion of the transient form of matter created in a gold-gold collision at RHIC.

The calculation of the equation of state of strongly interacting matter has been a central goal of numerical studies of Quantum Chromo Dynamics (QCD) in large scale computer simulations ever since the pioneering application of Monte Carlo techniques for this purpose at BNL in 1980 (M. Creutz, Phys. Rev. D21 (1980) 2308). In fact, the first lattice calculation of the equation of state in a quark gluon plasma was attempted at Bielefeld University only one year later (J. Engels et al., Phys. Lett. B101 (1981) 89). While these first calculations neglected the contribution of quarks altogether and also replaced the SU(3) color group of QCD by the simpler SU(2) group, much more realistic calculations became possible through the steady increase in computing power and the refinement of numerical techniques.

A big step towards a final determination of the equation of state of strongly interacting matter was recently completed at BNL. Making use of the computing resources at BNL and Bielefeld University the RIKEN-BNL-Columbia-Bielefeld collaboration performed the first calculation of the equation of state of QCD with a nearly physical quark mass spectrum that covers the entire energy density regime accessible in heavy ion collisions at RHIC and even further up to energy densities of about 1 TeV/fm3. (1 fm is 10-15 m, roughly the dimensions of a proton or neutron.) Energies of that order of magnitude will soon become accessible in heavy ion collisions at the Large Hadron Collider (LHC) at CERN in Geneva. These calculations were carried out over a period of more than a year on the two 10 Teraflops QCDOC computers installed at BNL, one funded by DOE the other owned by the RIKEN/BNL Research Center. In addition, a 5 Teraflops special purpose computer developed for lattice calculations (apeNEXT) was used at Bielefeld University, Germany (1 Teraflops = a trillion floating point operations per second). With the arrival of the 100 Teraflops BlueGene/L (NYBlue) earlier this year at the newly founded New York Center for Computational Science (NYCCS), hosted by BNL, the calculations could be completed more speedily. First results are now reported in (RBC-Bielefeld Collaboration, arXiv:0710.0354 [hep-lat]).

The new results on the equation of state (see figure) show features familiar from earlier approximate calculations; a rapid rise of the energy density in the transition region at about 1 GeV/fm3 or a temperature of about 190 MeV and a comparatively slow increase of the pressure which comes close to that of a quasi-free quark-gluon gas only at temperatures of about 600 MeV. In fact, even at these high temperatures it stays about 10% below the ideal gas limit, which is known to be reached eventually at extremely high temperatures.

The new set of lattice calculations has been performed with a nearly physical spectrum of light and strange quarks; the light quarks are still about a factor 2 too heavy while the strange quark mass is adjusted to its physical value. The high temperature behavior of the difference between energy density and three times the pressure was analyzed for the first time for three different choices of the lattice cut-off, which allowed to control systematic errors in this regime. Most important, however, is that these calculations allow one to present with confidence results on the equation of state on an absolute temperature scale. This is expected to change only on the 10% level once the next generation of supercomputers performs calculations with physical quark mass values even closer to the continuum limit.

Most notably the equation of state shows that the ideal gas limit, where the energy density equals three times the pressure, may only be reached at energy densities beyond 100 GeV/fm3. This region may soon become accessible at the LHC. The entire evolution of dense matter created at RHIC, however, happens in a regime where deviations from the simple ideal gas equation of state are significant. The rapid change of pressure in units of energy density close to the transition back to ordinary hadronic matter leads to a rapid drop of the velocity of sound and thus slows down the expansion of the medium. This also is reflected in the behavior of the bulk viscosity of the medium which will grow large in this region (D. Kharzeev and K. Tuchin, arXiv:0705.4280 [hep-ph]).

Experimental results on relative abundances of various hadron species created in gold-gold collisions at RHIC and their comparison with particle abundances realized in a simple hadron resonance gas (HRG) suggest that the transition back to ordinary hadrons, the so-called chemical freeze out, occurs at temperatures of about (160-170) MeV. Whether this freeze-out temperature is in any way related to the transition temperature in QCD is an important question that currently is studied extensively in large scale computer simulations. The simulations of the RBC-Bielefeld Collaboration performed on the QCDOC and apeNEXT computers hint at a larger transition temperature of about 190 MeV (RBC-Bielefeld Collaboration, arXiv:hep-lat/0608013). This, however, requires further confirmation through calculations on larger lattices which currently are being performed by a new US-wide lattice collaboration (hotQCD) on BlueGene/L computers at BNL and LLNL. First results of this project have been presented at this years Lattice conference (C. DeTar and R. Gupta, arXiv:0710.1655 [hep-lat]). Hopefully, a complete analysis of this project will be ready for presentation at Quark Matter 2008....stay tuned.

Figure 1. Energy density (e) and pressure (p) of Quantum Chromo Dynamics as function of temperature calculated in lattice simulations on two different 4-dimensional lattices (upper plot) and the equation of state (p/e) from which the velocity of sound (cs) is deduced (lower plot). For low energy densities the lattice calculations are compared to the equation of state of a hadron resonance gas (HRG).