QCD Equation of State

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Parametrization of the Equation of State

(Pasi Huovinen and Peter Petreczky, arXiv:0912.2541)

E-3p para.png

The trace anomaly of the recent Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): calculations (Bazavov et al, Phys. Rev. D80, 014504 (2009)) can be parametrized as Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.):

and the trace anomaly of a chemically equilibrated hadron resonance gas with resonances up to 2 GeV mass (see the list of included resonances and their properties) can be parametrized as

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where Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeVFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeVFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeVFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeVFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , in a temperature interval Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): MeV.

The fit to the lattice data can be constrained in several ways. We require that the parametrized lattice trace anomaly connects smoothly to the trace anomaly of the hadron resonance gas, i.e. the trace anomaly and its first and second derivative with respect to temperature are continuous. We have made the fit to the lattice data above Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): MeV temperature, and constrained the entropy density at Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): MeV temperature to be either 90% or 95% of the Stefan-Boltzmann value (parametrizations Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , respectively). Furthermore we may add one "datapoint" to the fit to make the peak in the trace anomaly higher (label Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): ), fix the temperature where we switch from HRG to lattice (label Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): ), or have no further constraints (label Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , like "normal"). The parameter values for three different fits are

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (GeVFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): ) Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (GeVFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): ) Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (GeVFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): ) Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (GeVFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): ) Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (MeV)
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.):
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.):
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.):

Note that in the above plot the solid line depicts EoS Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , the dashed Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , and the dotted Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): .

The EoS can be obtained from this parametrization by integrating over temperature

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where Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): MeV and

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): .

Alternatively, the EoS is provided in a tabulated form here: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): .

Chemically frozen hadron gas

We have not parametrized the trace anomaly, since a full EoS would require parametrizing chemical potential for each conserved particle species. Instead we provide an EoS in a tabulated form for Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): MeV, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -PCE-Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): MeV, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -PCE165-Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): .

We plan to have the program to calculate these EoSs publicly available.


Parametrization of chemically frozen hadron gas

Tom Riley, Chun Shen, and Ulrich Heinz at Ohio State University have provided the following parametrization for the chemically frozen EoS Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -PCE165-Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): as a function of energy density.

Pressure

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where Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeV/fmFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeV/fmFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeV/fmFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeV/fmFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): .


Entropy

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where Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeV/fmFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeV/fmFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeV/fmFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): GeV/fmFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): .


Temperature

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In the above, the units for energy density and pressure are GeV/fmFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , for entropy density the unit is fmFailed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): , and for temperature GeV.

Note that to calculate the final spectra, one still needs the chemical potentials from the EoS tables provided here.


EoS for hybrid models

In hybrid model calculations it is easier to conserve energy and momentum at the boundary of the fluid dynamical and molecular dynamics descriptions if the degrees of freedom are the same in both descriptions. For this purpose Hirano et al. used in arXiv:1010.6222 an EoS where the hadron resonance gas contained the same hadrons and resonances as JAM hadron cascade. The details of this EoS, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): will be shown here soon. Meanwhile, if interested, contact the author at huovinen(meow)th. physik. uni-frankfurt. de