## Parametrization of the Equation of State ##
Auvinen, Eskola, Huovinen, Niemi, Paatelainen and Petreczky, [arXiv:2006.12499][1], submitted to PRC.
Those who know what they are looking for, can find the EoS tables here. Everyone else, read further:
| Chemical equilibrium: | [$s83s_{18}$][2] | [$s87h_{04}$][3] | [$s88h_{18}$][4] |
|---------------------------|--------------------|--------------------|--------------------|
|$T_\mathrm{chem}=150$ MeV: | [$s83s_{18}$-PCE150][5] | [$s87h_{04}$-PCE150][6] | [$s88h_{18}$-PCE150][7] |
|$T_\mathrm{chem}=153$ MeV: | | | [$s88h_{18}$-PCE153][8] |
|$T_\mathrm{chem}=154$ MeV: | [$s83s_{18}$-PCE154][9] | [$s87h_{04}$-PCE154][10] | [$s88h_{18}$-PCE154][11] |
|$T_\mathrm{chem}=155$ MeV: | [$s83s_{18}$-PCE155][12] | | |
We base our Equations of State (EoSs) on parametrized lattice QCD trace anomaly at large temperatures, and on hadron resonance gas (HRG) at low temperatures:
Of our three parametrizations, $s87h_{04}$ and $s88h_{18}$ are based on lattice results obtained using the HISQ discretization scheme at fixed lattice spacing (Bazavov et al., [Phys. Rev. D 90, 094503 (2014)][13] [\[arXiv:1407.6387\]][14] and Bazavov, Petreczky & Weber, [Phys. Rev. D 97, 014510 (2018)][15] [\[arXiv:1710.05024\]][16]), whereas $s83s_{18}$ is based on continuum extrapolated stout action results (Borsanyi et al., [JHEP 1011, 077 (2010)][17] [\[arXiv:1007.2580\]][18] and Borsanyi et al., [Phys. Lett. B 730, 99 (2014)][19] [\[arXiv:1309.5258\]][20]).
The HRG part is either based on [$m<2$ GeV resonances][21] in the summary tables of the 2004 Particle Data Book ($s87h_{04}$) or on [all strange and non-strange states][22] in the summary tables of the 2018 Particle Data Book ($s83s_{18}$ and $s88h_{18}$). The summary tables contain 3- and 4-star baryon resonances, and all the meson resonances not marked "omitted from summary table" in full listings.
For further details, see the above mentioned paper, Auvinen et al., arXiv:2006.00000.
The lattice QCD trace anomaly can be parametrized as
$$ \frac{\epsilon -3p}{T^4}
= d_0 + \frac{d_1}{T^2} + \frac{d_2}{T^4} +
\frac{d_3}{T^{n_3}} + \frac{d_4}{T^{n_4}} +
\frac{d_5}{T^{n_5}},
$$
where the parameter values are
| | $d_0$ | $d_1$[GeV$^2$] | $d_2$[GeV$^4$] | $d_3$[GeV$^{n_3}$] | $d_4$[GeV$^{n_4}$] | $d_5$[GeV$^{n_5}$] | $n_3$ | $n_4$ | $n_5$ | $T_0$[MeV] |
|-------|-------|-----|------|------|------|------|-----|------|------|-------|
| $s83s_{18}$ | $5.688\cdot 10^{-3}$| $0.3104$ | $-6.217\cdot 10^{-3}$| $-6.680\cdot 10^{-32}$| $1.071\cdot 10^{-32}$| -- | $41$ | $42$ | -- | $166$ |
| $s87h_{04}$ | $5.669\cdot 10^{-2}$| $0.2974$ | $-4.184\cdot 10^{-3}$| $-5.146\cdot 10^{-8}$ | $1.420\cdot 10^{-33}$| -- | $10$ | $42$ | -- | $172$ |
| $s88h_{18}$ | $4.509\cdot 10^{-2}$| $0.3082$ | $-5.136\cdot 10^{-3}$| $-1.150\cdot 10^{-10}$| $2.076\cdot 10^{-32}$| $-3.021\cdot 10^{-33}$| $13$ | $41$ | $42$ | $155$ |
Of these parameters, $T_0$ is the temperature where the parametrization matches the HRG trance anomaly.
Correspondingly, within the temperature interval $70 \mathrm{MeV} < T < T_{\rm high}$, the HRG trace anomaly can be parametrized as
$$ \frac{\epsilon-3p}{T^4} = a_1T^{m_1} + a_2T^{m_2} + a_3T^{m_3} + a_4T^{m_4}, $$
where the parameter values are
| | $a_1$[GeV$^{-m_1}$] | $a_2$[GeV$^{-m_2}$] | $a_3$[GeV$^{-m_3}$] | $a_4$[GeV$^{-m_4}$] | $m_1$ | $m_2$ | $m_3$ | $m_4$ | $T_{\rm high}$[MeV] |
|-------------|---------------------|---------------------|---------------------|---------|--------|-----|------|-----|------|
| $s83s_{18}$ | $0.1850$ | $1.985\cdot 10^4$ | $1.278\cdot 10^5$ | $-1.669\cdot 10^7$ | $0$ | $5$ | $7$ | $10$ | $170$ |
| $s87h_{04}$ | $4.654$ | $-879$ | $8081$ | $-7.039\cdot 10^6$ | $1$ | $3$ | $4$ | $10$ | $190$ |
| $s88h_{18}$ | $0.1844$ | $2.043\cdot 10^4$ | $8.550\cdot 10^5$ | $-2.434\cdot 10^7$ | $0$ | $5$ | $8$ | $10$ | $169$ |
EoS can be obtained from these parametrization by integrating over temperature
$$\frac{p(T)}{T^4}-\frac{p(T_{\rm low})}{T^4_{\rm low}}=\int_{T_{\rm low}}^{T} dT' \frac{\epsilon(T)-3p(T)}{{T'}^5},$$
where $T_{\rm low} = 70$ MeV and $p(T_{\rm low})/T_{\rm low}^4 = 0.1661$.
Alternatively, the EoSs are provided as tables at the top of this wiki.
We have not parametrized the trace anomaly of chemically frozen hadron gas, since a full EoS would require parametrizing chemical potential for each conserved particle species. Instead we provide the EoS as tables, see the links at the top of this wiki,
<br />
For historical reasons we also provide the old $s95p$, $s95n$ and $s90f$ parametrizations and EoS tables [here][23].
[1]: https://arxiv.org/abs/2006.12499
[2]: https://osf.io/pcnsa/download
[3]: https://osf.io/4wy7h/download
[4]: https://osf.io/ezyt6/download
[5]: https://osf.io/skxrf/download
[6]: https://osf.io/q7m5r/download
[7]: https://osf.io/3ypn5/download
[8]: https://osf.io/q9gn4/download
[9]: https://osf.io/fq58g/download
[10]: https://osf.io/tfy4r/download
[11]: https://osf.io/fka93/download
[12]: http://osf.io/n8jzs/download
[13]: https://dx.doi.org/10.1103/PhysRevD.90.094503
[14]: https://arxiv.org/abs/1407.6387
[15]: https://dx.doi.org/10.1103/PhysRevD.97.014510
[16]: https://arxiv.org/abs/1710.05024
[17]: https://dx.doi.org/10.1007/JHEP11%282010%29077
[18]: https://arxiv.org/abs/1007.2580
[19]: https://dx.doi.org/10.1016/j.physletb.2014.01.007
[20]: https://arxiv.org/abs/1309.5258
[21]: https://osf.io/rhb9p/download
[22]: https://osf.io/x9sjg/download
[23]: https://osf.io/yseq3/wiki/home
