# Charm degrees of freedom in hot matter from lattice QCD

A. Bazavov  D. Bollweg  O. Kaczmarek  F. Karsch  Swagato Mukherjee  P.
Petreczky  C. Schmidt  Sipaz Sharma  Department of Computational Mathematics,
Science and Engineering and Department of Physics and Astronomy, Michigan
State University, East Lansing, MI 48824, USA  Physics Department, Brookhaven
National Laboratory, Upton, New York 11973, USA  RIKEN-BNL Research Center,
Brookhaven National Laboratory, Upton, New York 11973, USA  Fakultät für
Physik, Universität Bielefeld, D-33615 Bielefeld, Germany

###### Abstract

We study the nature of charm degrees of freedom in hot strong interaction
matter by performing lattice QCD calculations of the second and fourth-order
cumulants of charm fluctuations, and their correlations with net baryon
number, electric charge and strangeness fluctuations. We show that below the
chiral crossover temperature thermodynamics of charm can be very well
understood in terms of charmed hadrons. Above the chiral transition charm
quarks show up as new degrees of freedom contributing to the partial charm
pressure. However, up to temperatures as high as 175 MeV charmed hadron-like
excitations provide a significant contribution to the partial charm pressure.

###### keywords:

QCD thermodynamics , charm degrees of freedom , deconfinement

††journal: Physics Letters B

##  1 Introduction

It is now well established that strong interaction matter at vanishing baryon
chemical potential undergoes restoration of the spontaneously broken chiral
symmetry via a crossover transition since the small yet non-vanishing up and
down quark masses result also in the explicit breaking of the
${SU(2)_{L}\times SU(2)_{R}}$ chiral symmetry group. This chiral crossover
transition occurs at a pseudo-critical temperature, ${T_{pc}=156.5\pm 1.5}$
MeV [1]. However, the deconfinement aspect of the transition is not well
understood in QCD with light dynamical quarks. Ultimately, the deconfinement
can be related to the nature of the underlying degrees of freedom in the hot
matter. We expect that with increasing temperatures hadronic excitations
become broader and may have masses different from the vacuum ones. This is the
case for the pseudo-scalar mesons [2] and the vector mesons, e.g. the
$\rho$-meson [3, 4]. There is some evidence that also charmed hadrons get
modified with increasing temperature [5, 6, 7]. At sufficiently high
temperatures the hadronic excitations will become too broad and not well
defined, and partonic excitations will be the dominant ones.

Fluctuations of conserved charges and correlations among different conserved
charges can be used to understand the relevant degrees of freedom in the hot
matter. Precision lattice QCD calculations showed that the Hadron Resonance
Gas (HRG) [8] provides a good description of the fluctuations and correlations
of conserved charges in the light and strange quark sector below the chiral
crossover [9, 10]. There are indications that the same is true for charm
fluctuations and charm baryon number correlations [11]. However, because of
the large statistical errors no firm conclusion could be drawn in the earlier
studies about the onset for the presence of charm quark degrees of freedom
[11]. At high temperature the fluctuations and correlations of conserved
charges can be well understood in terms of a quark gas [9, 11, 12, 13].

For understanding of the spectrum and elliptic flow of charmed hadrons it is
important to know how these hadrons are formed and whether charmed hadron
states can exist also above the chiral crossover temperature [14, 15]. Using
previously obtained lattice QCD data on charm baryon number correlations [11]
it has been argued that charmed hadron-like excitations can exist above
$T_{pc}$ [16].

The aim of this letter is to clarify the nature of charm degrees of freedom in
the vicinity of $T_{pc}$ using high precision calculations of charm
fluctuations and correlations. We test to what extent the HRG model can
describe the thermodynamics of charm below the chiral crossover. Furthermore,
we probe the onset of deconfinement and establish the existence of charmed
hadronic excitations above $T_{pc}$.

##  2 Details of the lattice QCD calculations

In order to study the nature of the charm degrees of freedom we calculate so-
called generalized susceptibilities, i.e. derivatives of the QCD pressure
($P$) with respect to chemical potentials of net baryon number ($B$), electric
charge ($Q$), strangeness ($S$) and charm ($C$),

${\chi^{BQSC}_{klmn}=\dfrac{\partial^{(k+l+m+n)}\;\;[P\;(\hat{\mu}_{B},\hat{\mu}_{Q},\hat{\mu}_{S},\hat{\mu}_{C})\;/T^{4}]}{\partial\hat{\mu}^{k}_{B}\;\;\partial\hat{\mu}^{l}_{Q}\;\;\partial\hat{\mu}^{m}_{S}\;\;\partial\hat{\mu}^{n}_{C}}}\bigg{|}_{\vec{\mu}=0}\;,$
(1)

in lattice QCD.

We performed lattice QCD calculations in (2+1)-flavor QCD using the Highly
Improved Staggered Quark (HISQ) action [17] for physical strange quark mass,
$m_{s}$, and light quark mass, $m_{l}=m_{s}/27$. The latter corresponds to a
pion mass of $140$ MeV in the continuum limit. We consider the temperature
range from 145 MeV to 175 MeV. To fix the lattice spacing the $f_{K}$ scale
setting from Ref. [10] is used. The values of lattice strange quark mass are
taken from Ref. [18]. We use a set of gauge field configurations generated on
lattices of size $32^{3}\times 8$ and used in earlier studies of the HotQCD
collaboration [10]. The charm quarks have been treated in quenched
approximation, which can be justified since the charm quark mass, $m_{c}$, is
quite large, and earlier lattice calculations showed that the influence of
dynamical charm quarks can be neglected in the temperature range of interest
for our current analysis [19]. The HISQ action is very well suited for the
study of charm quarks [17]. Discretization effects related to the charm quarks
can be reduced by using a mass-dependent coefficient in the HISQ action which
eliminates ${\cal O}((am_{c})^{4})$ tree level lattice artifacts [17, 20].

We calculated all generalized susceptibilities involving charm up to fourth
order. The calculation of $\chi^{BQSC}_{klmn}$ involves derivatives of the
pressure and on the lattice this is achieved by the stochastic estimation of
various traces – consisting of inversions and derivatives of the fermion
matrices ($D$) – using the random noise method. In particular, $500$ random
vectors have been used to calculate various traces per configuration, except
for Tr ${\big{(}D^{-1}\frac{\partial D}{\partial\mu}\big{)}}$ – which turned
out to be particularly noisy. Therefore, $2000$ random vectors have gone into
its calculation. We used two different lines of constant physics (LCPs) to
tune the charm-quark mass. The first LCP corresponds to keeping the spin-
averaged charmonium mass, ${(3m_{J/\psi}+m_{\eta_{c\bar{c}}})/4}$ fixed to its
physical value. We calculated the $J/\psi$ and $\eta_{c}$ masses using the
zero temperature lattices generated for the study of chiral crossover
temperature and equation of state [18]. As in Ref. [11], we fitted the
corresponding values of the bare charm quark mass $am_{c}$ with
renormalization group inspired form to obtain $am_{c}$ as function of the
inverse gauge coupling, $\beta=10/g_{0}^{2}$. In Table 1 we give the values of
the charm quark masses obtained through this procedure used in our
calculations for different $\beta$ as well as the temperature values. The
second LCP is defined by the physical charm to strange quark mass ratio,
$m_{c}/m_{s}=11.76$ [21]. Results based on the above two LCPs will henceforth
contain subscripts [a] and [b], respectively. Results without any of these
subscripts will correspond to LCP[b].

|  | $am_{c}$
---|---|---
$\beta$ |  $T$[MeV] | LCP[a] | LCP[b]
6.315 | 145.1 | 1.04112 | 0.892231
6.354 | 151.1 | 0.97025 | 0.857304
6.390 | 156.9 | 0.91534 | 0.816144
6.423 | 162.4 | 0.87069 | 0.787450
6.445 | 166.1 | 0.84320 | 0.765223
6.474 | 171.2 | 0.80920 | 0.742996
6.500 | 175.8 | 0.78059 | 0.723946
Table 1:  The lattice gauge coupling $\beta=10/g_{0}^{2}$, the corresponding
temperature values and the bare charm quark masses for LCP[a] and LCP[b].

As stated above our aim is the high statistics lattice QCD calculations of the
generalized susceptibilities involving charm. The generalized susceptibilities
involving charm have been first studied in Ref. [11] using $32^{3}\times 8$
lattices in (2+1)-flavor. We extend this study as well as the analysis of
charmed degrees of freedom [16] in two significant ways. We included in the
analysis two temperatures below the crossover temperature and increased the
statistics by a factor (60-70) in the vicinity of $T_{pc}$ and a factor 20 at
$T\simeq 175$ MeV. The smallest temperature used in Ref. [22] was $T=157$ MeV,
which is too high to test the range of validity of HRG model calculations. We
have used approximately one-third of the available (2+1)-flavor HISQ
configurations generated by the HotQCD collaboration [10]. At the highest two
temperatures, i.e., at $T=171.6$ MeV and $T=176.7$ MeV we also performed
calculations on $48^{3}\times 12$ lattices. At these two temperatures we used
36078 and 39080 gauge configurations, respectively.

Figure 1: The ratios of different baryon-charm fluctuations as functions of
temperature. The open symbols represent the results from Ref. [11]. The yellow
band represents $T_{pc}$ with its uncertainty. The red solid line is the ideal
charm quark gas limit of the ratio $\chi_{13}^{BC}/\chi_{22}^{BC}$.

##  3 Generalized charm susceptibilities and hadron resonance gas model

In a non-interacting HRG model the QCD pressure can be written as the sum of
the partial pressures of hadrons carrying open charm degrees of freedom and
the partial pressure of hadrons with no charm. Furthermore, the partial
pressure of charmed hadrons can be written as the sum of partial pressures of
charmed mesons and charmed baryons.

${P^{C}(T,\vec{\mu})=P_{M}^{C}(T,\vec{\mu})+P_{B}^{C}(T,\vec{\mu})}\text{ .}$
(2)

As the masses of charmed mesons and baryons are much larger than the
temperature range of interest, one can use Boltzmann statistics and write
$P_{M}^{C}$ and $P_{B}^{C}$ in the following form [11]:

$\displaystyle\begin{aligned}
P_{M}^{C}(T,\vec{\mu})&=\dfrac{1}{2\pi^{2}}\sum_{i\in\text{C-mesons}}g_{i}\bigg{(}\dfrac{m_{i}}{T}\bigg{)}^{2}K_{2}(m_{i}/T)\\\
&\times\text{cosh}(Q_{i}\hat{\mu}_{Q}+S_{i}\hat{\mu}_{S}+C_{i}\hat{\mu}_{C})\text{
,}\\\
P_{B}^{C}(T,\vec{\mu})&=\dfrac{1}{2\pi^{2}}\sum_{i\in\text{C-baryons}}g_{i}\bigg{(}\dfrac{m_{i}}{T}\bigg{)}^{2}K_{2}(m_{i}/T)\\\
&\times\text{cosh}(B_{i}\hat{\mu}_{B}+Q_{i}\hat{\mu}_{Q}+S_{i}\hat{\mu}_{S}+C_{i}\hat{\mu}_{C})\text{
.}\end{aligned}$ (3)

Here ${K_{2}(x)}$ is a modified Bessel function of the second kind; the
summation is over all charmed mesons/baryons with masses given by ${m_{i}}$;
$g_{i}$ denotes the degeneracy factors of states with identical mass and
quantum numbers. We also introduced a dimensionless notation for chemical
potentials, ${\hat{\mu}_{X}=\mu_{X}/T}$, with $X\in\\{B,Q,S,C\\}$.

Figure 2: The ratio $\chi_{13}^{BC}/\chi_{4}^{C}$ as a function of temperature
obtained for LCP[a] and LCP[b]. Also shown is the result obtained in PDG-HRG
and QM-HRG model calculations. The yellow band represents $T_{pc}$ with its
uncertainty.

For large value of the argument, $K_{2}(m_{i}/T)$ is exponentially suppressed.
Therefore, the contribution of multi-charm baryons is exponentially small, and
effectively only the $|C|=1$ sector contributes to the pressure. This means
that $\chi_{2}^{C}=\chi_{n}^{C}=P^{C}(T,\vec{\mu})$, for $n$ even, and
$\chi_{11}^{BC}=\chi_{1m}^{BC}=P_{B}^{C}$, for $m$ odd. We find that these
relations are indeed well satisfied in the temperature range used in our
calculations as shown in Fig. 1. Furthermore, in a hadron gas only $|B|=1$
sector contributes to $\chi_{nm}^{BC}$, and therefore, the ratio
$\chi_{13}^{BC}/\chi_{22}^{BC}$ should be unity up to the crossover
temperature. We see from Fig. 1 that this is indeed true. The hadron resonance
gas model approximation thus appears to be valid. In Fig. 1 we also show the
earlier lattice QCD results as open symbols for $T>176$ MeV [11]. At lower
temperatures our results agree with those of Ref. [11] within the large errors
of the latter. The present results have much smaller errors. Just above the
chiral crossover temperature the HRG description breaks down and the ratio
approaches a value which at 330 MeV is only 10% below the value of an ideal
charm quark gas.

In Fig. 2 we show the ratio $\chi_{13}^{BC}/\chi_{4}^{C}$, which is a proxy
for the ratio of the charmed baryon pressure and the total charm pressure. We
compare our lattice QCD results with HRG model predictions where we include
all charmed hadrons listed by the Particle Data Group [21] (PDG-HRG). As can
be seen, the PDG-HRG under-predicts the lattice data significantly. This is
not surprising since it was pointed out already in Ref. [11] that many charmed
baryons predicted in quark model [23] as well as lattice QCD calculations [24]
are missing in PDG tables. There are also missing charmed meson states in PDG-
HRG. However, their number is significantly smaller [11]. When including all
the missing hadron states, using the spectrum obtained in quark model
calculations [25, 23, 26], in the HRG model, we obtain very good agreement
between the lattice QCD results and the quark model extended hadron resonance
gas (QM-HRG) for $\chi_{13}^{BC}/\chi_{4}^{C}$ at the three lowest
temperatures. Furthermore, as one can also see in Fig. 2 the ratio
$\chi_{13}^{BC}/\chi_{4}^{C}$ is not very sensitive to the choice of the LCP
for the charm mass. Thus our present findings are in agreement with the
observations of Ref. [11] that additional charmed baryon states are needed to
explain the lattice QCD results on generalized susceptibilities, but now this
claim is on more solid numerical footings. The apparent agreement between the
lattice QCD results and HRG also for $T>T_{pc}$, that can be seen in Fig. 2,
is somewhat accidental. It is related to the fact that at high temperatures
$\chi_{13}^{BC}/\chi_{4}^{C}$ approaches 1/3. As discussed above, the HRG
description description breaks down for $T>T_{pc}$, and for $T_{pc}<T<250$ MeV
the ratio $\chi_{13}^{BC}/\chi_{22}^{BC}$ moves from the hadron gas to the
quark gas expectations. In the next section we discuss the implication of this
finding for effective charm degrees of freedom.

Figure 3: Partial pressures of charmed mesons, charmed baryons and charm
quarks as functions of temperature. The dashed lines show corresponding
results obtained from the QM-HRG model. The open symbols show the results for
$N_{\tau}=12$ lattices, see text. The yellow band represents $T_{pc}$ with its
uncertainty.

Figure 4: Constraints $c_{1}$ and $c_{4}$ on the quasi-particle model given by
Eq. 4, see text and Eqs. 8 and Eq. 11, respectively. The dashed lines indicate
that these constraints are explicitly fulfilled in non-interacting HRG models.
The yellow band represents $T_{pc}$ with its uncertainty.

##  4 Charm degrees of freedom below and above $T_{pc}$

As shown in the previous section the simple hadron gas model, given by Eqs. 2
and 3, breaks down for $T>T_{pc}$. Therefore, following Ref. [16] we extend
this model by allowing the presence of partial charm quark pressure and by
treating the charmed baryon and charmed meson sectors as quasi-particle
excitations,

$\displaystyle
P^{C}(T,\vec{\mu})=P_{M}^{C}(T,\vec{\mu})+P_{B}^{C}(T,\vec{\mu})+$
$\displaystyle
P_{q}(T)\cosh\bigg{(}\dfrac{2}{3}\hat{\mu}_{Q}+\dfrac{1}{3}\hat{\mu}_{B}+\hat{\mu}_{C}\bigg{)}\;,$
(4)

where the last term corresponds to the charm quark partial pressure. Here we
explicitly give the dependence on chemical potentials, using Boltzmann
approximation, and $P_{q}(T)$ being a function of temperature only. At very
high temperature we can also write
$P_{q}(T)=\frac{3}{\pi^{2}}\bigg{(}\frac{m_{c}}{T}\bigg{)}^{2}K_{2}(m_{c}/T)$
with $m_{c}$ being the pole mass of the charm quark. Using Eqs. (3) and (4) we
can express the partial pressures of quasi-particles with quantum numbers of
charm quarks, charmed baryons and charmed mesons, respectively. For
$\vec{\mu}=0$ we express these partial pressures in terms of generalized
susceptibilities as follows,

$\displaystyle P_{q}^{C}$
$\displaystyle=9(\chi^{BC}_{13}-\chi^{BC}_{22})/2\;,$ (5) $\displaystyle
P_{B}^{C}$ $\displaystyle=(3\chi^{BC}_{22}-\chi^{BC}_{13})/2\;,$ (6)
$\displaystyle P_{M}^{C}$
$\displaystyle=\chi^{C}_{4}+3\chi^{BC}_{22}-4\chi^{BC}_{13}\;.$ (7)

Using lattice QCD results on the generalized susceptibilities in the above
expressions we estimate the different partial pressures and normalize them by
dividing with the total partial charm pressure, $P_{C}=\chi^{C}_{4}$. Results
are shown in Fig. 3. Corresponding results obtained in QM-HRG model
calculations are shown as dashed lines in this figure. Obviously the quark
pressure is zero in this model. As can be seen, the HRG works well up to
$T_{pc}$. Above $T_{pc}$, however, the charmed baryon and charmed meson
partial pressures drop below the HRG results. At the same time the quark
pressure becomes non-zero just above $T_{pc}$. These results may be taken as
evidence for partial melting of the hadron-like states and the liberation of
quark degrees of freedom at $T_{pc}$. For the highest two temperatures we also
show our results for the charm quark pressure obtained on $N_{\tau}=12$
lattices, which agree with the $N_{\tau}=8$ results within errors, indicating
that our main conclusion is not affected by lattice artifacts.

As was pointed out in Ref. [16], if the model given by Eq. (4) takes care of
all relevant degrees of freedom below and above $T_{pc}$, then there are four
constraints that the generalized susceptibilities up to fourth order must
satisfy:

$\displaystyle c_{1}\equiv\chi_{13}^{BC}-4\chi_{22}^{BC}+3\chi_{31}^{BC}=0,$
(8) $\displaystyle c_{2}\equiv
2\chi_{121}^{BSC}+4\chi_{112}^{BSC}+\chi_{22}^{SC}-2\chi_{13}^{SC}+\chi_{31}^{SC}=0,$
(9) $\displaystyle c_{3}\equiv
3\chi_{112}^{BSC}+3\chi_{121}^{BSC}-\chi_{13}^{SC}+\chi_{31}^{SC}=0,$ (10)
$\displaystyle c_{4}\equiv\chi_{211}^{BSC}-\chi^{BSC}_{112}=0.$ (11)

The lattice results in Ref. [11] were consistent with these constraints.
However, because of the large errors it was not possible to test the model for
$T<200$ MeV. With our new lattice data one can show that these constraints are
clearly satisfied in the vicinity of $T_{pc}$ within errors. As an example in
Fig. 4 we show the constraints $c_{1}$ (left) and $c_{4}$ (right). The
observables defined by Eq. 8 and Eq. 11 are in fact related to the partial
pressures of charm diquarks, $P_{d}^{C}$, and charm-strange diquarks,
$P_{ds}^{C}$. Namely we can write $P_{d}^{C}=-9c_{1}/2$ and
$P_{ds}^{C}=9c_{4}/2$. While the current data on $c_{1}$ and $c_{4}$ are
consistent with not having a diquark contribution to the pressure at
$T>T_{pc}$, the current bounds coming from the data on $c_{1}$ and $c_{4}$ are
still too weak to rule out the existence of diquarks above $T_{pc}$
completely.

Additional insight into the charm degrees of freedom above $T_{pc}$ can be
obtained by considering partial charm pressures corresponding to different
electric charge sectors. Again using Eqs. (3) and (4) we can determine the
partial charm pressures in different $Q$ and $B$ sectors using the following
expressions:

$\displaystyle P_{C}^{Q=0}$
$\displaystyle=\frac{1}{4}\big{[}4\chi^{C}_{4}-12\chi^{QC}_{13}+11\chi^{QC}_{22}-3\chi^{QC}_{31}\big{]}$
(12) $\displaystyle P_{C}^{Q=1}$
$\displaystyle=-4\chi^{QC}_{13}+8\chi^{QC}_{22}-3\chi^{QC}_{31}$ (13)
$\displaystyle P_{C}^{Q=2}$
$\displaystyle=\frac{1}{8}\big{[}2\chi^{QC}_{13}-5\chi^{QC}_{22}+3\chi^{QC}_{31}\big{]}$
(14) $\displaystyle P_{C}^{Q=2/3}$
$\displaystyle=\frac{1}{8}\big{[}54\chi^{QC}_{13}-81\chi^{QC}_{22}+27\chi^{QC}_{31}\big{]}$
(15) $\displaystyle P_{C}^{B=1,Q=2}$
$\displaystyle=\frac{1}{4}\big{[}-\chi^{BQC}_{211}+2\chi^{BQC}_{121}-\chi^{BQC}_{112}\big{]}$
(16) $\displaystyle P_{C}^{B=1,Q=1}$
$\displaystyle=2\chi^{BQC}_{211}-\chi^{BQC}_{121}$ (17) $\displaystyle
P_{C}^{B=1,Q=0}$
$\displaystyle=\frac{1}{2}[2\chi^{BC}_{22}-13\chi^{BQC}_{112}+\chi^{BQC}_{121}+10\chi^{BQC}_{211}]$
(18) $\displaystyle P_{C}^{B=1/3,Q=2/3}$
$\displaystyle=\frac{27}{4}\big{[}\chi^{BQC}_{112}-\chi^{BQC}_{211}]$ (19)

We expect the partial pressures for $|Q|=2/3$ and/or ($|Q|=2/3$, $|B|=1/3$)
sectors will agree with $P^{C}_{q}$. Using the lattice QCD results for the
generalized susceptibilities we estimated $P_{C}^{Q=2/3}$ and
$P_{C}^{B=1/3,Q=2/3}$. The results are shown in Fig. 5. As expected these
partial pressures vanish below $T_{pc}$ and agree with the charm quark
pressure $P_{q}^{C}$. This again shows that charm quark degrees of freedom
appear just above the chiral crossover temperature. The partial charm
pressures with integer values of $Q$ correspond to combinations of charmed
baryon and charmed meson pressures. We find that all these partial pressures
differ significantly from zero above the chiral crossover temperature. In
particular, the partial charm pressure with $|Q|=2$ comes from charmed baryon
contributions. In Fig. 6 we show the corresponding lattice QCD results. As can
be seen, the partial charm pressures $P_{C}^{|Q|=0,1,2}$ are non-zero above
$T_{pc}$ implying again that charmed hadron-like excitations still exist at
$T\simeq 175$ MeV. At the same time these partial pressures show significant
deviations from QM-HRG.

Figure 5: The partial pressures of quasi-particles carrying (i) baryon number
$1/3$ ($P_{q}^{C}$), (ii) electric charge $Q=2/3$ ($P_{C}^{Q=2/3}$), and (iii)
($B=1/3,\ Q=2/3$) ($P_{C}^{B=1/3,Q=2/3}$), respectively. All three observables
have been normalized to the total partial charm pressure. The figure confirms
that these three observables project onto the same quasi-particle sector. The
yellow band represents $T_{pc}$ with its uncertainty. Note that the
T-coordinates of case (ii) and case (iii) are shifted by $\pm 0.51$ MeV
respectively.

##  5 Summary and conclusions

In this paper we studied the nature of charm degrees of freedom across the
chiral crossover transition using high statistics lattice QCD calculations on
generalized charm susceptibilities. We showed that below the chiral crossover
transition the generalized susceptibilities agree with HRG model calculations,
although in order to achieve this agreement additional charmed baryon states
need to be included that are not yet included in the PDG tables, but are
expected to exist based on quark model and lattice QCD calculations. We thus
confirm and corroborate earlier assertions about the existence of additional
charmed hadrons [11]. The high statistics results of generalized
susceptibilities also allow to clarify the nature of the charm degrees of
freedom above $T_{pc}$. We find that a charm quark contribution to the
pressure appears as new degree of freedom just above $T_{pc}$. However, charm
quarks become the dominant degree of freedom only at $T>175$ MeV. Charmed
meson- and baryon-like excitations exist above $T_{pc}$ and make up half of
the contribution to the charmed pressure at temperature $T\simeq 175$ MeV.

The lattice QCD calculations presented in this paper have been performed at a
single value of lattice spacing in temperature units that correspond to
$N_{\tau}=8$ lattices. We find that cutoff effects in the generalized
susceptibilities are insignificant in ratios of different generalized
susceptibilities. These cutoff effects can be estimated by considering
different prescriptions for fixing the lines of constant physics for the charm
quark mass. While the choice of LCP has a large influence on the absolute
values of generalized susceptibilities the effect cancels in ratios of
generalized susceptibilities. Thus our conclusions will not be affected by
discretization errors.

Figure 6: Partial charm pressures for $Q=0,1$ and $2$ sectors normalized by
the total charm pressure as functions of temperature. Dashed lines show the
prediction of the QM-HRG model. The yellow band represents $T_{pc}$ with its
uncertainty.

## Acknowledgements

This material is based upon work supported by The U.S. Department of Energy,
Office of Science, Office of Nuclear Physics through Contract No. DE-
SC0012704, and within the frameworks of Scientific Discovery through Advanced
Computing (SciDAC) award Fundamental Nuclear Physics at the Exascale and
Beyond and the Topical Collaboration in Nuclear Theory Heavy-Flavor Theory
(HEFTY) for QCD Matter as well as by the U.S. National Science Foundation
under award PHY-2309946. This work was supported by The Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - Project number
315477589-TRR 211, ”Strong interaction matter under extreme conditions” and
project number 460248186 (PUNCH4NFDI).

The authors gratefully acknowledge the computing time and support provided to
them on the high-performance computer Noctua 2 at the NHR Center PC2 under the
project name: hpc-prf-cfpd. These are funded by the Federal Ministry of
Education and Research and the state governments participating on the basis of
the resolutions of the GWK for the national high-performance computing at
universities (www.nhr-verein.de/unsere-partner). Numerical calculations have
also been performed on the GPU-cluster at Bielefeld University, Germany. We
thank the Bielefeld HPC.NRW team for their support.

All computations in this work were performed using SIMULATeQCD code [27].

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