QCD Equation of State

Parametrization of the Equation of State

(Pasi Huovinen and Peter Petreczky, arXiv:0912.2541)

The trace anomaly of the recent $N_{\tau }=8$ calculations (Bazavov et al, Phys. Rev. D80, 014504 (2009)) can be parametrized as ${\frac {\epsilon -3P}{T^{4}}}={\frac {d_{2}}{T^{2}}}+{\frac {d_{4}}{T^{4}}}+{\frac {c_{1}}{T^{n_{1}}}}+{\frac {c_{2}}{T^{n_{2}}}},$

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

${\frac {\epsilon -3P}{T^{4}}}=a_{1}T+a_{2}T^{3}+a_{3}T^{4}+a_{4}T^{10},$

where $a_{1}=4.654$ GeV$^{-1}$, $a_{2}=-879$ GeV$^{-3}$, $a_{3}=8081$ GeV$^{-4}$ and $a_{4}=-7039000$ GeV$^{-10}$, in a temperature interval $70<T<190$ 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 $T>250$ MeV temperature, and constrained the entropy density at $T=800$ MeV temperature to be either 90% or 95% of the Stefan-Boltzmann value (parametrizations $s90$ and $s95$, respectively). Furthermore we may add one "datapoint" to the fit to make the peak in the trace anomaly higher (label $p$), fix the temperature where we switch from HRG to lattice (label $f$), or have no further constraints (label $n$, like "normal"). The parameter values for three different fits are

	$d_{2}$ (GeV$^{2}$)	$d_{4}$ (GeV$^{4}$)	$c_{1}$ (GeV$^{n1}$)	$c_{2}$ (GeV$^{n2}$)	$n_{1}$	$n_{2}$	$T_{0}$ (MeV)
$s95p$-$v1$	$0.2660$	$2.403\cdot 10^{-3}$	$-2.809\cdot 10^{-7}$	$6.073\cdot 10^{-23}$	$10$	$30$	$183.8$
$s95n$-$v1$	$0.2654$	$6.563\cdot 10^{-3}$	$-4.370\cdot 10^{-5}$	$5.774\cdot 10^{-6}$	$8$	$9$	$171.8$
$s90f$-$v1$	$0.2495$	$1.355\cdot 10^{-2}$	$-3.237\cdot 10^{-3}$	$1.439\cdot 10^{-14}$	$5$	$18$	$170.0$

Note that in the above plot the solid line depicts EoS $s95p$, the dashed $s90f$, and the dotted $s95n$.

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

${\frac {p(T)}{T^{4}}}-{\frac {p(T_{\mathrm {low} })}{T_{\mathrm {low} }^{4}}}=\int _{T_{\mathrm {low} }}^{T}dT'{\frac {\epsilon -3P}{{T'}^{5}}},$

where $T_{\mathrm {low} }=70$ MeV and

$P(T_{\mathrm {low} })/T_{\mathrm {low} }^{4}=0.1661$.

Alternatively, the EoS is provided in a tabulated form here: $s95p$-$v1$, $s95n$-$v1$, and $s90f$-$v1$.

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 $T_{chem}=150$ MeV, $s95p$-PCE-$v1$, and $T_{chem}=165$ MeV, $s95p$-PCE165-$v0$.

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 $s95p$-PCE165-$v0$ as a function of energy density.

Pressure

$P(\epsilon )=\left\{{\begin{array}{rcl}0.3299(e^{0.4346\epsilon }-1)&:&\epsilon <\epsilon _{1}\\1.024\cdot 10^{-7}e^{6.041\epsilon }+0.007273+0.14578\,\epsilon &:&\epsilon _{1}<\epsilon <\epsilon _{2}\\0.30195\cdot e^{0.31308\epsilon }-0.256232&:&\epsilon _{2}<\epsilon <\epsilon _{3}\\\left.{\begin{array}{r}0.332\,\epsilon -0.3223\,\epsilon ^{0.4585}-0.003906\,\epsilon \cdot e^{-0.05697\epsilon }\\+0.1167\,\epsilon ^{-1.233}+0.1436\,\epsilon \cdot e^{-0.9131\epsilon }\end{array}}\right\}&:&\epsilon _{3}<\epsilon <\epsilon _{4}\\0.3327\,\epsilon -0.3223\,\epsilon ^{0.4585}-0.003906\,\epsilon \cdot e^{-0.05697\epsilon }&:&\epsilon >\epsilon _{4}\end{array}}\right.$

where $\epsilon _{1}=0.5028563305441270$ GeV/fm$^{3}$, $\epsilon _{2}=1.62$ GeV/fm$^{3}$, $\epsilon _{3}=1.86$ GeV/fm$^{3}$, and $\epsilon _{4}=9.9878355786273545$ GeV/fm$^{3}$.

Entropy

$s(\epsilon )^{4/3}=\left\{{\begin{array}{rcl}12.2304\,\epsilon ^{1.16849}&:&\epsilon <\epsilon _{1}\\11.9279\,\epsilon ^{1.15635}&:&\epsilon _{1}<\epsilon <\epsilon _{2}\\0.0580578+11.833\,\epsilon ^{1.16187}&:&\epsilon _{2}<\epsilon <\epsilon _{3}\\\left.{\begin{array}{r}18.202\,\epsilon -62.021814-4.85479\,e^{(-2.72407\cdot 10^{-11}\epsilon ^{4.54886})}\\+65.1272\,\epsilon ^{-0.128012}e^{(-0.00369624\,\epsilon ^{1.18735})}-4.75253\epsilon ^{-1.18423}\end{array}}\right\}&:&\epsilon _{3}<\epsilon <\epsilon _{4}\\\left.{\begin{array}{r}18.202\,\epsilon -63.0218-4.85479\,e^{(-2.72407\cdot 10^{-11}\epsilon ^{4.54886})}\\+65.1272\,\epsilon ^{-0.128012}\,e^{(-0.00369624\,\epsilon ^{1.18735})}\end{array}}\right\}&:&\epsilon >\epsilon _{4}\end{array}}\right.$

where $\epsilon _{1}=0.1270769021427449$ GeV/fm$^{3}$, $\epsilon _{2}=0.446707952467404$ GeV/fm$^{3}$, $\epsilon _{3}=1.9402832534193788$ GeV/fm$^{3}$, and $\epsilon _{4}=3.7292474570977285$ GeV/fm$^{3}$.

Temperature

$T(\epsilon )=\left\{{\begin{array}{rcl}0.203054\epsilon ^{0.30679}&:&\epsilon <0.5143939846236409{\frac {\mathrm {GeV} }{\mathrm {fm} ^{3}}}\\{\frac {\epsilon +P}{s}}&:&\epsilon >0.5143939846236409{\frac {\mathrm {GeV} }{\mathrm {fm^{3}} }}\end{array}}\right.$

In the above, the units for energy density and pressure are GeV/fm$^{3}$, for entropy density the unit is fm$^{-3}$, 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, $s95p$-$v1.1$ will be shown here soon. Meanwhile, if interested, contact the author at huovinen(meow)th. physik. uni-frankfurt. de