Tables for the Quark-Hadron-Crossover 2018 (QHC18)– Users’ Manual v.1.0 (Jan.1, 2018) –Abstract
This is a short note on the equation of state (EoS) tables, Quark-Hadron-Crossover in 2018 (QHC18),
which covers the EoS from the crust to quark matter domains. The QHC18 is a zero temperature EoS
for neutron star matter in β-equilibrium. The physics behind QHC18 is discussed in the review paper(we call it the QHC18 paper),“From hadrons to quarks in neutron stars: a review” (arXiv:1707.04966 [astro-ph.HE]),
G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song and T. Takatsuka,(Reports on Progress in Physics, to be published).
This manual includes more remarks on numerics which may be important for some EoS users working
on numerical simulations. The fitting forms of the numerical tables can be also found in Appendix Cof the QHC18 review paper.
1. Tables
1.1. Thermodynamic quantities in the tables
The current version of the QHC equation of state (EoS), QHC18, includes numerical tables for1) µB: baryon chemical potential [MeV]

2) P: pressure [MeV fm−3 ]3) nB /n0 : baryon density/saturation density (n0 = 0.16 fm−3 )4) ε: energy density [MeV fm−3 ]5) cs /c: sound velocity/light velocity(1)

These tables are computed at T = 0 and β-equilibrium. The speed of sound was computed numericallyusing P and ε after preparing the tables for 1)-4).
1.2. The data files for QHC18 EoS
At present there are several versions of the QHC18 depending on the parameters (gV , H) used forthe quark matter EoS. We present our tables in text files
Preprint submitted to Elsevier

January 4, 2018

eos HQC18 gv***H***.txt
where *** indicates the values for (gV , H). For instance, the quark EoS with the label “gv080H150”
means (gV , H) = (0.80, 1.50)Gs , where Gs is the scalar coupling of the Nambu–Jona-Lasinio (NJL)
model for quark matter. The corresponding QHC18 EoS should be labelled as “QHC18-gv080H150”.
Unless otherwise stated, we choose the ”QHC18-gv080H150” as representative, and it will be simply
called “QHC18”. When we use other parameters, we will write “gv***H***” explicitly.
1.3. The data files for quark matter EoS
In addition to the tables of QHC18, we separately present the tables for quark matter EoS. The
corresponding data files have the nameeos Q18 gv***H***.txt
For the quark matter EoS tables, we cover the range of (gV , H) for gV /Gs = 0.70, 0.80, 0.90, 1.00
and H/Gs = 1.40, 1.50, 1.60. The range of the tables is from nB ' 4n0 to ' 10n0 .
1.4. Constants
In our tables we use natural units c = ~ = 1. The constants used in our tables are
1 fm−1 = 197.326 MeV ,π = 3.14159265 .

(2)

The proton, neutron, electron, and muon masses are
mp = 938.272 MeV ,mn = 939.565 MeV ,me = 0.511 MeV ,2mµ = 105.6 MeV .

(3)

2. Unified equations of state: remarks on numerics (details)
The equations of state are divided into four distinct domains: the crust, nuclear liquid, hadron-quarkcrossover, and quark matter domains. For each domain we assign an equation of state as
Crust : Togashi EoS[10−9 n0 ≤ nB ≤ 0.26n0 ]Nuclear liquid : APR98 EoS[0.26n0 ≤ nB ≤ 2n0 ]Crossover : QHC18[2n0 ≤ nB ≤ 5n0 ]Quark matter : QHC18[5n0 ≤ nB ≤ 10n0 ](4)

The description of each EoS is given in the following:i) Togashi EoS for the crust is explained in details in the paper:
H. Togashi, K. Nakazato, Y. Takehara, S. Yamamuro, H. Suzuki and M. Takano,Nucl. Phys. A 961, 78 (2017) ,and the data tables can be found from the website [http://user.numazu-ct.ac.jp/∼sumi/eos/#manybody] where our manual is located. The reason we use this EoS is that this is a microscopic calculationconsistent with the Akmar-Pandheripande-Ravenhall (APR) EoS (which we use for the nuclear liquidpart), and therefore the crust and nuclear liquid parts join smoothly, in a way consistent with thethermodynamic relations. For the density below the neutron drip regime the EoS correctly reproduces
the results obtained from laboratory nuclei.
The tables presented by these authors give µB , P , nB /n0 , and ε. We use them as given in our EoStables. The relation P = µB nB − ε is satisfied to very good accuracy.
ii) APR EoS for the nuclear liquid is explained in the paper:
A. Akmal, V. R. Pandharipande and D. G. Ravenhall,
Phys. Rev. C 58 (1998) 1804 .
We created the data tables from the fit functions for εN (energy density for nuclear part) presented inthis paper. The fit is given in the form εN (nB , xp ), where xp is the proton fraction. The total energy iscalculated by adding lepton contributions, εl (nB , xp ). The requirement of charge neutrality determines
xp as a function of nB , so the total energy density is ε(nB ) = εN (nB , xp (nB )) + εl (nB , xp (nB )). From
the analytic expression of ε(nB ), one can calculate the analytic expression for µB = ∂ε(nB )/∂nB fromwhich the expression of P can be also derived by using the thermodynamic relation P = nB µB − ε.
The numerical tables are obtained by substituting the values of nB and x(nB ) (determined numerically)3

into the analytic expressions for ε, µB , and P . Alternatively, we may take the numerical derivatives toevaluate µB and P , but the quality of the thermodynamic relations worsens.
The fitting function for APR contains the nucleon mass mN . While we were not able to find out
which values of mN were used in their calculations, in creating our tables we used the average of theproton and neutron mass, mN = (mp + mn )/2 = 938.919 MeV.
iii) QHC18 EoS for the hadron-quark crossover region:
The physics behind the crossover domain is explained in the QHC18 paper. To construct the EoS
for the crossover domain, it is necessary to first construct the quark matter EoS at nB ≥ 5n0 which
will be explained below. After calculating a particular quark EoS, we interpolate the APR and quark
matter EoS by polynomial functions for P (µB ). We used the sixth order polynomials
P (µB ) =5X

cn µnB .

(5)

n=0

From this analytic expression one can readily derive the analytic expression for nB and ε as functionsof µB . The coefficients cn are determined by demanding that the crossover EoS match the APR andquark EoS’s smoothly at nB = 2n0 and 5n0 , up to the 2nd order derivatives. The tables were obtainedby simultaneously substituting µB into the expressions of P , nB , and ε.
iv) QHC18 EoS for the quark matter region:
The quark matter EoS in the QHC18 was calculated using the NJL model within the mean fieldapproximation. As variable parameters, we choose gV and H which quantify the strength of therepulsive density-density interaction and the attractive paring-interaction between quarks, respectively. As mentioned before, the quark EoS with the label “gv080H150” uses the parameter values(gV , H) = (0.80, 1.50)Gs .
Firstly, the EoS is required to be stiff enough to be able to produce neutron stars of at least 2M .
Secondly the EoS must be such that, when used as the boundary condition at nB = 5n0 for theinterpolation, the interpolating curve has a speed of sound 0 ≤ cs ≤ 1. These conditions severely
restrict the ranges of (gV , H).
There are analytic expressions for P , nB , and ε as functions of µB , with a number of dynamicalquantities related to the condensates which must be determined numerically. Once the condensates arefixed, all the thermodynamic variables are determined at once. This procedure yields the tables for thequark matter EoS.

4

3. Contact
We would appreciate it very much if you could give us suggestions on these EoSs. If you find anyerror or strange behavior, please contact
Toru Kojo
Central China Normal University
Wuhan, Hubei, China
E-mail: torujj@mail.ccnu.edu.cn5