# Conserved charge fluctuations in the relativistic mean-field hadron
resonance gas model: constraints on hadronic repulsive interactions

Somenath Pal  School of Physical Sciences, National Institute of Science
Education and Research Bhubaneswar, HBNI, Jatni 752050, Odisha, India
Guruprasad Kadam  Department of Physics and Material Science and Engineering,
Jaypee Institute of Information Technology, A-10, Sector-62, Noida, UP-201307,
India<EMAIL_ADDRESS>Abhijit Bhattacharyya  Department of
Physics, University of Calcutta, 92, A.P.C. Road, Kolkata-700009, India

###### Abstract

We investigate the effect of repulsive interaction between hadrons on the
susceptibilities of conserved charges, namely baryon number (B), electric
charge (Q) and strangeness (S). We estimate second and fourth order
susceptibilities of conserved charges, their differences, ratios and
correlations within ambit of mean-field hadron resonance gas (MFHRG) model. We
consider repulsive mean-field interaction among meson pairs, anti-meson pairs,
baryon pairs and anti-baryon pairs separately and constrain them by
confronting MFHRG results of various susceptibilities with the recent lattice
QCD (LQCD) data. We find that the repulsive interactions between baryon-baryon
pairs and antibaryon-antibaryon pairs are sufficient to describe the
thermodynamics of hadronic matter at temperatures below QCD transition
temperature. Very weak mesonic repulsive interaction is needed only to
describe electric charge susceptibilities and can be neglected in the
description of other susceptibilities. We finally conclude that the repulsive
interaction between hadrons play a very important role in describing the
thermodynamic properties of hadronic matter especially near quark-hadron phase
transition temperature ($T_{c}$). The mean-field parameter for baryons
($K_{B}$) should be constrained to the range $0.40\leq K_{B}\leq 0.450$
$\text{GeV.fm}^{3}$ to get a good agreement with the LQCD results.

##  I Introduction

Currently accepted gauge theory of strongly interacting matter is quantum
chromodynamics (QCD) in which the fundamental constituents of strongly
interacting matter are (coloured) quarks and gluons. If we have a macroscopic
system composed of quarks and gluons then its thermodynamic state is specified
by four external control parameters, namely temperature (T), and three
chemical potentials ($\mu_{B},\mu_{Q},\mu_{S}$) corresponding to conservation
of baryon number (B), electric charge (Q) and strangeness quantum number (S).
In a plot of temperature ($T$) against baryon chemical potential
($\mu_{B}$)-QCD phase diagram, every point corresponds to an equilibrium
state. QCD phase diagram is largely conjectured one and understanding it,
theoretically as well as experimentally, is one of the main objectives of
elementary particle physics research todayDing  _et al._ (2015a); Bzdak  _et
al._ (2020); Ratti and Bellwied (2021).

The heavy-ion collision experiments (HIC), namely Large Hadron Collider (LHC),
the Relativistic Heavy-Ion Collider (RHIC) etc., has been very successful in
probing a part of the QCD phase diagram where the QCD coupling constant is
large. In these experiments the thermodynamic state of QCD matter as it is
existed at the time of hadronization is probed. The thermodynamic parameters
specific to hadronization stage in the evolution can be extracted either from
the particle yields or from higher order cumulants characterizing the
distribution of produced hadrons at freeze-out. These observables are compared
with the statistical thermal model calculations, namely hadron resonance gas
model(HRG). In this model the equilibrium thermodynamic state of low
temperature and low density hadronic matter can be effectively described by an
ideal gas of point-like hadrons and resonancesDashen  _et al._ (1969); Dashen
and Rajaraman (1974); Welke  _et al._ (1990); Venugopalan and Prakash (1992).
HRG model has been tremendously successful in describing low temperature
hadronic phase of QCDBraun-Munzinger  _et al._ (2004). Some studies have
discussed the effect of magnetic field on hadronic equation of state within
ambit of HRG modelBhattacharyya  _et al._ (2016); Kadam  _et al._ (2020);
Pradhan  _et al._ (2023). It is also very successfully describe the hadron
yields as measured in HIC experimentsBraun-Munzinger  _et al._ (1995, 1996,
1999); Cleymans  _et al._ (1997); Cleymans and Redlich (1999); Becattini  _et
al._ (2006); Braun-Munzinger  _et al._ (2001); Andronic  _et al._ (2006,
2009), as well as reproduce the QCD equation of state (EoS) calculated in
lattice QCD (LQCD) simulations at zero as well an finite $\mu_{B}$Karsch  _et
al._ (2003a, b); Tawfik (2005); Huovinen and Petreczky (2010); Alba  _et al._
(2015); Huovinen and Petreczky (2018); Vovchenko  _et al._ (2015).

Fluctuations and correlations of conserved charges, namely baryon number (B),
electric charge (Q) and strangeness (S) have been sensitive probes of
deconfinementBazavov  _et al._ (2013a). There have been several studies in
this regardBhattacharyya  _et al._ (2014, 2019, 2015); Fukushima (2008);
Roessner  _et al._ (2007); Sasaki  _et al._ (2007); Ratti  _et al._ (2007);
Fukushima (2009); Schaefer and Wambach (2007); Schaefer  _et al._ (2010,
2007); Wambach  _et al._ (2010); Motornenko  _et al._ (2021); Pal  _et al._
(2021a, b). They can also be used to calculate thermodynamic quantities at
finite baryon density via Taylor series expansionPetreczky (2012); Ding  _et
al._ (2015a). Despite its success recent studies show that the differences
between actual QCD calculations and that of ideal HRG model become appreciable
in the properties of higher order cumulants. At higher temperature the
particle densities are higher and virial expansion only upto second order
coefficient may not be sufficient to describe the thermodynamic properties.
The validity of HRG model at higher temperature can be justified by detailed
comparison with the recent LQCD results. The fluctuations and correlation of
conserved charges have been studied in the LQCDBorsanyi  _et al._ (2010,
2012); Bazavov  _et al._ (2012, 2013b, 2013a, 2014a, 2014b); Borsanyi  _et
al._ (2014); Bellwied  _et al._ (2015); Ding  _et al._ (2015b); Bazavov  _et
al._ (2016); D’Elia  _et al._ (2017); Bazavov  _et al._ (2017); Borsanyi  _et
al._ (2018) and HRG models has been used as a benchmark results. It is found
that the higher order cumulants estimated in ideal HRG model shows significant
deviations from LQCD results near $T_{c}$. The prominent reason for this
discrepancy is the exclusion of short-range repulsive interactions between
hadrons as shown in the recent studies which include repulsive interaction in
the ideal HRG modelKarthein  _et al._ (2021). These repulsive interactions are
especially important at higher densities and make appreciable changes in the
EoS as described by hadron resonance gas model. Various approaches to account
for the short range repulsive interactions has been proposed, namely excluded-
volume approachRischke  _et al._ (1992), relativistic mean-field
approachKapusta and Olive (1983); Olive (1981), relativistic virial
expansionHuovinen and Petreczky (2018) etc.

The parameters that specify the repulsive interactions in a given model can be
constrained by confronting the model results with the LQCD calculations.
Second order cumulants are especially important observables to constrain the
QCD effective models. Recently, such studies has been carried out to constrain
the excluded volume parameter ($b$) of excluded-volume HRG (EVHRG) model
Bollweg  _et al._ (2021) . The repulsive interactions can be constrained by
constraining the excluded volume parameter $b$ . It was found that
$b>1\>\text{fm}^{3}$ to get a reasonable agreement of EVHRG results with the
LQCD results at vanishing chemical potential for second order cumulants. In
the present work, we do similar analysis, however, for relativistic mean-field
HRG model (MFHRG). In this model, repulsive interactions are accounted via
density dependent mean-fields. The shift in a single particle energy is
$U=kn$, where $n$ is the number density and $K$ is the parameter determining
the strength of the repulsive interaction. We shall discuss the validity of
MFHRG model by comparing it with the LQCD results for the second order
cumulants. This analysis, as we will see, also constrains the mean-field
parameter $K$ which is related to the mean-field potential for the repulsive
interaction between hadrons. We shall also confront MFHRG results with the
LQCD results for the ratios and differences of various susceptibilities.

This paper is organised as follows: In section II, we present a brief
description of hadron resonance gas model with repulsive mean-field
interactions and calculation of various susceptibilities. In section III we
shall discuss the results, and finally in section IV we summarise our
findings.

##  II  Mean-field Hadron Resonance Gas (MFHRG) model

Iddeal HRG model takes into account only the attractive interactions between
hadrons by including the heavy resonance states in the partition function.
Repulsive interactions between the hadrons can be incorporated in this model
via mean-field approach. In this approach the single particle energies are
shifted by a term proportional to the particle number density as

$\varepsilon_{a}=\sqrt{p^{2}+m_{a}^{2}}+U(n)=E_{a}+U(n)$ (1)

where $n$ is the total hadron number density. The potential energy $U$
represents repulsive interaction between hadrons. We assume that the potential
energy is proportional to the number density $n$ asKapusta and Olive (1983)

$U(n)=Kn$ (2)

Here, $K$ is a constant parameter which characterise the strength of repulsive
interaction. We assume different repulsive interaction parameters for baryons
$(K_{B})$ and mesons $(K_{M})$ such that

$U(n_{B\\{\bar{B}\\}})=K_{B}n_{B\\{\bar{B}\\}}\hskip
28.45274pt(\text{baryons(antibaryons)})$ (3)

$U(n_{M})=K_{M}n_{M}\hskip 28.45274pt(\text{mesons})$ (4)

Where $n_{M}$ and $n_{B}$ are total number densities for mesons and baryons
respectively. For baryons,

$n_{{B}}(T,\mu_{B},\mu_{Q},\mu_{S})=\sum_{a\in B}\int
d\Gamma_{a}\>\frac{1}{e^{\frac{(E_{a}-{\mu_{\text{eff},B}})}{T}}+1}$ (5)

where $\mu_{\text{eff},{B}}=c_{i}\mu_{i}-K_{B}n_{B}$ and
$c_{i}=(B_{i},Q_{i},S_{i}),{\mu_{i}}=(\mu_{B},\mu_{Q},\mu_{S})$ and
$d\Gamma_{a}\equiv\frac{g_{a}d^{3}p}{(2\pi)^{3}}$, $g_{a}$ is the degeneracy
of $a^{\text{th}}$ hadronic species. Similarly, the number density of
antibaryons is

$n_{{\bar{B}}}(T,\mu_{B},\mu_{Q},\mu_{S})=\sum_{a\in\bar{B}}\int
d\Gamma_{a}\>\frac{1}{e^{\frac{(E_{a}-{\mu_{\text{eff},\bar{B}}})}{T}}+1}$ (6)

where $\mu_{\text{eff},{\bar{B}}}=\bar{c_{i}}{\mu_{i}}-K_{B}n_{\bar{B}}$. Note
that we further assume same repulsive mean-field parameter for baryons and
anti-baryons. For mesons,

$n_{{M}}(T)=\sum_{a\in M}\int
d\Gamma_{a}\>\frac{1}{e^{\frac{(E_{a}-{\mu_{\text{eff},M}})}{T}}-1}$ (7)

where $\mu_{\text{eff},{M}}={c_{i}}\mu_{i}-K_{M}n_{M}$. Eqs. (5)-(7) are self
consistent equations and can be numerically solved. The contributions to the
pressure from baryons and mesons are then given, respectively, as

$\displaystyle P_{B\\{\bar{B}\\}}(T,\mu_{B},\mu_{Q},\mu_{S})$ $\displaystyle=$
$\displaystyle T\sum\limits_{a\in B\\{\bar{B}\\}}\int
d\Gamma_{a}\text{ln}\bigg{[}1$ (8) $\displaystyle+$ $\displaystyle
e^{-(\frac{E_{a}-\mu_{\text{eff}}\\{\bar{\mu}_{\text{eff}}\\}}{T})}\bigg{]}$
$\displaystyle-$ $\displaystyle\phi_{B\\{\bar{B}\\}}(n_{B\\{\bar{B}\\}})$
$\displaystyle P_{M}(T)$ $\displaystyle=$ $\displaystyle-T\sum_{a\in M}\int
d\Gamma_{a}\text{ln}\bigg{[}1-{e^{\frac{(E_{a}-{\mu_{\text{eff},M}})}{T}}}\bigg{]}$
(9) $\displaystyle-$ $\displaystyle\phi_{M}(n_{M})$

where,

$\phi_{B}(n_{B\\{\bar{B}\\}})=-\frac{1}{2}K_{B}n_{B\\{\bar{B}\\}}^{2}$ (10)

and

$\phi_{M}(n_{M})=-\frac{1}{2}K_{M}n_{M}^{2}$ (11)

For a typical phenomenological values of mean-field parameter $K$ quantity
$\beta U$ turns out to be very small in the temperature range in which we are
interested. Therefore, we can expand the exponential in the equations for
$n_{B\\{\bar{B}\\}}$ to first order in $K$. Similar approximation can be
carried out in the expression for pressure (Eq. 8). The total pressure of
baryons and antibaryons in this leading order approximation is given by

$P^{\text{tot}}_{(B,\bar{B})}(T,\mu_{B},\mu_{Q},\mu_{S})=T(n_{B}^{\text{id}}+n^{\text{id}}_{\bar{B}})-\frac{K_{B}}{2}\bigg{[}(n_{B}^{\text{id}})^{2}+(n^{\text{id}}_{\bar{B}})^{2}\bigg{]}$
(12)

Here, $n_{B}^{\text{id}}$ is the ideal HRG model number density given by

$n_{B}^{\text{id}}(T,\mu_{B},\mu_{Q},\mu_{S})=\sum_{j}n_{j}^{\text{id}}=\frac{T}{2\pi^{2}}\sum_{j}g_{j}m_{j}^{2}\>\mathcal{K}_{2}(\beta
m_{j})\>\text{Exp}\bigg{(}\frac{c_{i}\mu_{i}}{T}\bigg{)}$ (13)

where $\mathcal{K}_{n}$ is the $n^{\text{th}}$ order modified Bessel’s
function of second kind.

Fluctuations and correlation of conserved charges are defined as

$\chi^{mn}_{ij}=\frac{\partial^{m+n}(P(T,\mu_{i},\mu_{j})/T^{4})}{\partial\left({\mu_{i}/T}\right)^{m}\partial\left({\mu_{j}/T}\right)^{n}}$
(14)

where $i,j=B,Q,S$ corresponding to baryon number, electric charge and
strangeness respectively. In the leading order approximation of the pressure
(Eq.12) the baryon number fluctuation, baryon-strangeness correlation and
baryon-electric charge correlations are respectively given byHuovinen and
Petreczky (2018)

$\chi_{B}^{n}=(\chi_{B}^{\text{id}})^{n}-2^{n}\beta^{4}(n_{B}^{\text{id}})^{2}\hskip
28.45274pt(\text{even}\>n)$ (15)

$\chi_{BS}^{n1}=(\chi_{BS}^{\text{id}})^{n}+2^{n+1}\beta^{5}K_{B}n_{B}^{\text{id}}\sum_{j}B_{j}S_{j}P_{j}^{\text{id}}\hskip
28.45274pt(\text{odd}\>n)$ (16)

$\chi_{BQ}^{n1}=(\chi_{BQ}^{\text{id}})^{n}+2^{n+1}\beta^{5}K_{B}n_{B}^{\text{id}}\sum_{j}B_{j}Q_{j}P_{j}^{\text{id}}\hskip
28.45274pt(\text{odd}\>n)$ (17)

where, $P_{j}^{\text{id}}=Tn_{j}^{\text{id}}$ is the partial pressure of
$j^{\text{th}}$ species.

##  III Results and Discussion

Recently, different set of hadron list has been used to confront the
thermodynamics of HRG model with LQCD results:

  1. 1.

PDG2016/ PDG2020: List of only well known states (**** and ***) from Particle
Data Booklet.

  2. 2.

PDG2016+: List of well established and unconfirmed states (** and *).

  3. 3.

Quark model: List of hadrons predicted by the quark model.

  4. 4.

Quark model HRG (QMHRG): List of well known hadrons (2016 and 2020 list) which
is augmented with a list of QM states in the strange and non-strange baryons.

In present work we shall use QMHRG list with PDG2020 hadrons. Resulting HRG
model has been named as QMHRG2020Bollweg  _et al._ (2021).

MFHRG model is characterised by two parameters, namely $K_{M}$ and $K_{B}$. We
first confront the MFHRG results of second order cumulants with LQCD results
to establish the best fit values of these parameters. In the present work we
adopt an approach in which mean-field interactions are included only for
baryon-baryon pair and antibaryon-antibaryon pairs. This corresponds to
minimal mean-field extension of HRG model that does not affect meson-meson and
meson-baryon interactions, which are presumed to be dominated by resonance
formation, which is the basis of ideal HRG model by construction. For the
baryon–antibaryon system the short-range repulsive interactions are unlikely
due to annihilation processes, which are included in the hadron resonance gas
at equilibrium . The absence of short-range repulsion in the baryon–antibaryon
system leads to only a small correction, since mesons dominate at small
$\mu_{B}$ and since there are very few antibaryons at large $\mu_{B}$.

|  |
---|---|---

Figure 1: (Color online) Temperature dependence of second order
susceptibilities of conserved charges (at $\mu_{B}=\mu_{Q}=\mu_{S}=0$) for
different values of baryon mean-field parameter $K_{B}$ and for vanishing
meson mean-field parameter $K_{M}$. Lattice data is taken from Ref. Bollweg
_et al._ (2021). QMHRGH2020 model corresponds to particle list compiled from
PDG2020 list augmented with quark model states in the strange and non-strange
baryonsBollweg  _et al._ (2021). Note that the mesons do not contribute to the
baryon susceptibilities.

.

Figure 1 shows the results of second order susceptibilities estimated within
HRG and MFHRG model. Dashed red color corresponds to HRG model estimated
which, when compared with LQCD data, overestimates $\chi_{B}^{2}$ after $T\sim
150$ MeV. In case of MFHRG we fix $K_{M}=0$ and vary $K_{B}$ from 0.4 to 0.5
$\text{GeV.fm}^{3}$ in the steps of 0.05. We note that as we increase the
value of $K_{B}$ $\chi_{B}^{2}$ (top panel) is suppressed at high temperature
as compared to ideal HRG model. The best fit is observed for $K_{B}=0.450$
$\text{GeV.fm}^{3}$ (purple curve). However, with $K_{M}=0$, MFHRG
overestimates $\chi_{Q}^{2}$ (middle panel). As we will see later in this
section, we need a small but non-zero value of the meson mean-field parameter
to get a better agreement with the lattice data as . In case of $\chi_{S}^{2}$
ideal HRG results are in better agreement with the LQCD results upto $T\sim
160$ MeV. MFHRG model slightly underestimates $\chi_{S}^{2}$, however,
reproduce the general behaviour observed in LQCD. This may indicate a need of
including hitherto unknown strange resonances in the particle spectrum of HRG
model as discussed in Ref.Bazavov  _et al._ (2014a).

|
---|---
Figure 2: (Color online) Temperature dependence of $\chi_{Q}^{2}$ and
$\chi_{S}^{2}$ (at $\mu_{Q}=\mu_{S}=0$) for different values of meson mean-
field parameter $K_{M}$ and for a fixed value of baryon mean-field parameter
$K_{B}$. Lattice data is taken from Ref. Bollweg  _et al._ (2021). Note that
we have excluded $\chi_{B}^{2}$ because variation in $K_{M}$ does not affect
it.

Figure 2 shows the results of second order susceptibilities, $\chi_{Q}^{2}$
and $\chi_{S}^{2}$, with $K_{B}=0.450$ $\text{GeV.fm}^{3}$ and for for
different values of $K_{M}$ ranging from o to 0.15 $\text{GeV.fm}^{3}$ in the
steps of 0.05. Note that mesons do not contribute to $\chi_{B}^{2}$ and hence
the variation in $K_{M}$ won’t have any effect on this susceptibility. As
noted earlier, both HRG model and MFHRG model with $K_{M}=0$ overestimates
$\chi_{Q}^{2}$ over all the temperature range under consideration. However, if
we assign small but non-zero value of meson mean-field parameter $K_{M}=0.05$
$\text{GeV.fm}^{3}$ we get good agreement with the LQCD data upto $T\sim 160$
MeV. If we increase $K_{M}$ further even slightly MFHRG underestimate the LQCD
data due to stronger suppression from the repulsive interactions. In case of
$\chi^{2}_{S}$, HRG model results already being in good agreement with LQCD,
assignment of small mean-field value to mesons strongly suppress it. However,
again, the inclusion of hitherto unknown strange resonances in the baryonic
sector might improve the agreement of MFHRG with the LQCD data.

Above discussion indicates that we should consider a minimal extension of HRG
model to discuss the charge susceptibilities in HRG model. In this minimally
extended HRG model we include the repulsive interactions (via mean-field
approach) only for baryon-baryon and antibaryon-antibaryon pairs. In this
case, the repulsive interactions between meson-meson pairs and meson-baryon
pairs is neglected. In fact, such interactions are dominated by resonance
formation which render the attractive interactions already included in HRG
model.

|
---|---
Figure 3: (Color online) temperature dependence of baryon-charge correlation
$\chi_{BQ}^{11}$ and baryon-strangeness correlation $\chi_{BS}^{11}$. Lattice
data is taken from .

Fig.3 shows BQ correlation $\chi_{BQ}^{11}$ and BS correlation
(-)$\chi_{BS}^{11}$. We note that ideal HRG model overestimates conserved
charge correlations and we need to include repulsive interactions to supress
these quantities and get reasonable agreement with the LQCD data. However, in
case of $\chi_{BQ}^{11}$, inclusion of repulsive interaction in the ideal HRG
model improves the agreement with LQCD results upto $T\sim 150$ MeV. At higher
temperature a substantial deviation cannot be ignored. This deviation indicate
missing charged heavy-baryon states that need to be included in the HRG model.
In case of $\chi_{BS}^{11}$ ideal HRG is agreement with the LQCD results upto
$T\sim 150$ and deviates at higher temperature. MFHRG model underestimates
$\chi_{BS}^{11}$ and we need to include hitherto unknown strange baryonic
states to improve the description.

|
---|---
|

Figure 4: (Color online) Top panel shows the ratios
$\chi_{BQ}^{11}/\chi_{B}^{2}$ and $\chi_{BS}^{11}/\chi_{B}^{2}$ estimated
within MFHRG model. In the bottom panel we compare HRG model estimates with
two different lists of hadrons, namelly QMHRG2020 and HRG2016. Lattice data
has been taken from Ref. Karthein  _et al._ (2021)

.

We next consider another interesting set of observable quantities, namely
$\chi_{BS}^{11}/\chi_{B}^{2}$ and $\chi_{QS}^{11}/\chi_{B}^{2}$ as shown if
Fig.4. We note that the effect of repulsive interaction accounted via mean-
field interaction cancels; i.e these ratios are independent of repulsive
interactions and only attractive interaction accounted through the inclusion
of resonances in the partition function have any effect. Such cancellation has
also been observed where the repulsive interactions are accounted via excluded
volume formulation of HRG (EVHRG)Karthein  _et al._ (2021).

|
---|---

Figure 5: (Color online) Fourth order susceptibilities of conserved charge vs
temperature at $\mu_{B}=\mu_{S}=0$. Lattice data has been taken from Ref.
Borsanyi  _et al._ (2018)

.

Fig.5 shows fourth order susceptibilities. We note that MFHRG estimates of
$\chi_{B}^{4}$ (top panel) are in remarkable agreement with LQCD results over
wide range of temperatures. In case of $\chi_{Q}^{4}$ (middle panel), mean-
field interactions between baryons are not sufficient; one should also include
repulsive interactions between mesons as well. Further, unlike $\chi_{Q}^{2}$,
a relatively large value of $K_{M}$ is required to get a reasonable agreement
with the LQCD results. However, HRG model and its extension underestimates
$\chi_{S}^{4}$ (bottom panel) which again indicates the necessity of unknown
strange baryons.

|
---|---

Figure 6: (Color online) Temperature dependence of ratios of second to fourth
order susceptibilities of conserved charge at $\mu_{B}=\mu_{S}=0$. LQCD data
of $\chi_{B}^{4}/\chi_{B}^{2}$ is taken from the Ref.Bellwied  _et al._ (2015)
and that of $\chi_{S}^{4}/\chi_{S}^{2}$ is taken from the Ref. Borsanyi  _et
al._ (2018).

.

Fig.6 show ratios of fourth and second order susceptibilties of various
conserved chages. In particular, the ratio $\chi_{B}^{4}/\chi_{B}^{2}$ can be
approximately given by

$\frac{\chi_{B}^{4}}{\chi_{B}^{2}}\simeq
1-12\frac{K_{B}T^{2}}{2}(\beta^{3}n_{B}^{\text{id}})$ (18)

In case of ideal HRG this ratio is equal to one as shown by red dashed curve.
However, presence of repulsive interaction reduces this ratio. In MFHRG model
the deviation from HRG model results proportional to mean field parameter
$K_{B}$ and we get good agreement with the LQCD results.

|
---|---
Figure 7: (Color online)Temperature dependence of difference of second to
fourth order susceptibilities of conserved charge at $\mu_{B}=\mu_{S}=0$. LQCD
data of $\chi_{B}^{2}-\chi_{B}^{4}$ is taken from the Ref.Bellwied  _et al._
(2015) and that of $\chi_{S}^{2}-\chi_{S}^{4}$ is taken from the Ref. Borsanyi
_et al._ (2018).

Separating the effects of repulsive interactions from other medium effects,
namely in medium mass modifications and broadening of spectral width etc., is
very difficult. It turns out that these effects are removed if we take
differences of fluctuations and correlations. Fig.7 shows differences of
second and fourth order charge susceptibilities. In case of baryon
susceptibilities, it is easy to show that $\chi_{B}^{2}-\chi_{B}^{4}$ is
approximately given by

$\chi_{B}^{2}-\chi_{B}^{4}\simeq
12\frac{K_{B}T^{2}}{2}(\beta^{3}n_{B}^{\text{id}})$ (19)

where, $n_{B}^{\text{id}}$ is the given by Eq.(). In the absence of repulsive
interaction ($K_{B}=0$) $\chi_{B}^{2}-\chi_{B}^{4}$ vanishes which is expected
in ideal HRG model. Effect of repulsive interaction is to increase this
difference. MFHRG model with $K_{M}=0$ and $K_{B}=0.45$ $\text{GeV.fm}^{3}$
reproduce LQCD results quite remarkably all the way upto $T=170$ MeV. MFHRG
model is also in remarkable agreement with LQCD results of
$\chi_{S}^{2}-\chi_{S}^{4}$ .

##  IV Conclusion

We discussed the effect of repulsive interaction on the fluctuations and
correlation of conserved charges, namely baryon number (B), electric charge
(Q) and strangeness (S). We included the repulsive interaction in ideal HRG
model via relativistic mean-field approach (MFHRG). In our study, we
considered QMHRG2020 version of HRG model in which the partition function
includes a list of well known hadrons (2016 and 2020 list) augmented with a
list of QM states in the strange and non-strange baryons. We found that ideal
HRG model is not sufficient to describe the LQCD data of conserved charge
fluctuations and correlations especially near QCD transition temperature
($T_{c}$). We get good agreement with LQCD results if we include repulsive
interactions whose role becomes more important near $T_{c}$. In our analysis
we found that we have to consider a minimal extension of ideal HRG model in
which we include the repulsive interactions (via mean-field approach) only for
baryon-baryon and antibaryon-antibaryon pairs. In this case, the repulsive
interaction between meson-meson pairs and meson-baryon pairs is neglected. In
fact, interaction between meson-meson pairs and meson-baryon pairs is
dominated by resonance formation which render the attractive interactions
already included in HRG model. However, a small but finite repulsion between
meson-meson pairs in necessary to describe $\chi_{Q}^{n}$ and
$\chi_{BQ}^{11}$.

Ratios of second to fourth order susceptibilities estimated within MFHRG model
are also found to be in good agreement with LQCD results. Specifically, the
ratio $\chi_{B}^{4}/\chi_{B}^{2}$ (Eq.18) deviates from its HRG model value
($=1$) as the temperature increases. This result is in agreement with the LQCD
results especially near $T_{c}$. We also found that, although the correlations
$\chi_{BS}^{11}$ and $\chi_{BQ}^{11}$ depends on the repulsive interactions,
the ratios $\chi_{BS}^{11}/\chi_{B}^{2}$ and $\chi_{BQ}^{11}/chi_{B}^{2}$ are
independent of repulsive interactions. These later quantities only depends on
the spectrum of hadrons included in the partition function of ideal HRG model.
Further, in order to separate the effects of repulsive interactions from other
medium effects, we considered the differences in second and fourth order
susceptibilities. Difference $\chi_{B}^{2}-\chi_{B}^{4}$ vanishes for ideal
HRG and increases in we include the repulsive interaction (Eq.(19)) in
agreement with the LQCD results.

We finally conclude that the role of repulsive interaction in the description
of QCD matter is very important especially near QCD transition temperature.
The susceptibilities, correlations, their ratio and differences estimated
within ambit of MFHRG are in good agreement with the LQCD results if we
constrain mean-field parameter $K_{B}$ characterising the repulsive
interaction between baryon-baryon pairs and antibaryon-antibaryon pairs to
$0.40\leq K_{B}\leq 0.450$.

##  V Acknowledgement

GK is financially supported by the DST-INSPIRE faculty award under the grant
number
DST/INSPIRE/04/2017/002293.

## References

  * Ding  _et al._ (2015a) H.-T. Ding, F. Karsch, and S. Mukherjee, Int. J. Mod. Phys. E 24, 1530007 (2015a), arXiv:1504.05274 [hep-lat] .
  * Bzdak  _et al._ (2020) A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov, and N. Xu, Phys. Rept. 853, 1 (2020), arXiv:1906.00936 [nucl-th] .
  * Ratti and Bellwied (2021) C. Ratti and R. Bellwied,  _The Deconfinement Transition of QCD: Theory Meets Experiment_, Lecture Notes in Physics, Vol. 981 (2021).
  * Dashen  _et al._ (1969) R. Dashen, S.-K. Ma, and H. J. Bernstein, Phys. Rev. 187, 345 (1969).
  * Dashen and Rajaraman (1974) R. F. Dashen and R. Rajaraman, Phys. Rev. D 10, 694 (1974).
  * Welke  _et al._ (1990) G. M. Welke, R. Venugopalan, and M. Prakash, Phys. Lett. B 245, 137 (1990).
  * Venugopalan and Prakash (1992) R. Venugopalan and M. Prakash, Nucl. Phys. A 546, 718 (1992).
  * Braun-Munzinger  _et al._ (2004) P. Braun-Munzinger, K. Redlich, and J. Stachel, “Quark gluon plasma 3,” (2004).
  * Bhattacharyya  _et al._ (2016) A. Bhattacharyya, S. K. Ghosh, R. Ray, and S. Samanta, EPL 115, 62003 (2016), arXiv:1504.04533 [hep-ph] .
  * Kadam  _et al._ (2020) G. Kadam, S. Pal, and A. Bhattacharyya, J. Phys. G 47, 125106 (2020), arXiv:1908.10618 [hep-ph] .
  * Pradhan  _et al._ (2023) G. S. Pradhan, D. Sahu, S. Deb, and R. Sahoo, J. Phys. G 50, 055104 (2023), arXiv:2106.14297 [hep-ph] .
  * Braun-Munzinger  _et al._ (1995) P. Braun-Munzinger, J. Stachel, J. P. Wessels, and N. Xu, Phys. Lett. B 344, 43 (1995), arXiv:nucl-th/9410026 .
  * Braun-Munzinger  _et al._ (1996) P. Braun-Munzinger, J. Stachel, J. P. Wessels, and N. Xu, Phys. Lett. B 365, 1 (1996), arXiv:nucl-th/9508020 .
  * Braun-Munzinger  _et al._ (1999) P. Braun-Munzinger, I. Heppe, and J. Stachel, Phys. Lett. B 465, 15 (1999), arXiv:nucl-th/9903010 .
  * Cleymans  _et al._ (1997) J. Cleymans, D. Elliott, H. Satz, and R. L. Thews, Z. Phys. C 74, 319 (1997), arXiv:nucl-th/9603004 .
  * Cleymans and Redlich (1999) J. Cleymans and K. Redlich, Phys. Rev. C 60, 054908 (1999), arXiv:nucl-th/9903063 .
  * Becattini  _et al._ (2006) F. Becattini, J. Manninen, and M. Gazdzicki, Phys. Rev. C 73, 044905 (2006), arXiv:hep-ph/0511092 .
  * Braun-Munzinger  _et al._ (2001) P. Braun-Munzinger, D. Magestro, K. Redlich, and J. Stachel, Phys. Lett. B 518, 41 (2001), arXiv:hep-ph/0105229 .
  * Andronic  _et al._ (2006) A. Andronic, P. Braun-Munzinger, and J. Stachel, Nucl. Phys. A 772, 167 (2006), arXiv:nucl-th/0511071 .
  * Andronic  _et al._ (2009) A. Andronic, P. Braun-Munzinger, and J. Stachel, Phys. Lett. B 673, 142 (2009), [Erratum: Phys.Lett.B 678, 516 (2009)], arXiv:0812.1186 [nucl-th] .
  * Karsch  _et al._ (2003a) F. Karsch, K. Redlich, and A. Tawfik, Eur. Phys. J. C 29, 549 (2003a), arXiv:hep-ph/0303108 .
  * Karsch  _et al._ (2003b) F. Karsch, K. Redlich, and A. Tawfik, Phys. Lett. B 571, 67 (2003b), arXiv:hep-ph/0306208 .
  * Tawfik (2005) A. Tawfik, Phys. Rev. D 71, 054502 (2005), arXiv:hep-ph/0412336 .
  * Huovinen and Petreczky (2010) P. Huovinen and P. Petreczky, Nucl. Phys. A 837, 26 (2010), arXiv:0912.2541 [hep-ph] .
  * Alba  _et al._ (2015) P. Alba, R. Bellwied, M. Bluhm, V. Mantovani Sarti, M. Nahrgang, and C. Ratti, Phys. Rev. C 92, 064910 (2015), arXiv:1504.03262 [hep-ph] .
  * Huovinen and Petreczky (2018) P. Huovinen and P. Petreczky, Phys. Lett. B 777, 125 (2018), arXiv:1708.00879 [hep-ph] .
  * Vovchenko  _et al._ (2015) V. Vovchenko, D. V. Anchishkin, and M. I. Gorenstein, Phys. Rev. C 91, 024905 (2015), arXiv:1412.5478 [nucl-th] .
  * Bazavov  _et al._ (2013a) A. Bazavov, H. T. Ding, P. Hegde, F. Karsch, C. Miao, S. Mukherjee, P. Petreczky, C. Schmidt, and A. Velytsky, Phys. Rev. D 88, 094021 (2013a), arXiv:1309.2317 [hep-lat] .
  * Bhattacharyya  _et al._ (2014) A. Bhattacharyya, S. Das, S. K. Ghosh, R. Ray, and S. Samanta, Phys. Rev. C 90, 034909 (2014), arXiv:1310.2793 [hep-ph] .
  * Bhattacharyya  _et al._ (2019) A. Bhattacharyya, S. K. Ghosh, S. Maity, S. Raha, R. Ray, K. Saha, S. Samanta, and S. Upadhaya, Phys. Rev. C 99, 045207 (2019), arXiv:1708.04549 [hep-ph] .
  * Bhattacharyya  _et al._ (2015) A. Bhattacharyya, R. Ray, and S. Sur, Phys. Rev. D 91, 051501 (2015), arXiv:1412.8316 [hep-ph] .
  * Fukushima (2008) K. Fukushima, Phys. Rev. D 77, 114028 (2008), [Erratum: Phys.Rev.D 78, 039902 (2008)], arXiv:0803.3318 [hep-ph] .
  * Roessner  _et al._ (2007) S. Roessner, C. Ratti, and W. Weise, Phys. Rev. D 75, 034007 (2007), arXiv:hep-ph/0609281 .
  * Sasaki  _et al._ (2007) C. Sasaki, B. Friman, and K. Redlich, Phys. Rev. D 75, 074013 (2007), arXiv:hep-ph/0611147 .
  * Ratti  _et al._ (2007) C. Ratti, S. Roessner, and W. Weise, Phys. Lett. B 649, 57 (2007), arXiv:hep-ph/0701091 .
  * Fukushima (2009) K. Fukushima, Phys. Rev. D 79, 074015 (2009), arXiv:0901.0783 [hep-ph] .
  * Schaefer and Wambach (2007) B.-J. Schaefer and J. Wambach, Phys. Rev. D 75, 085015 (2007), arXiv:hep-ph/0603256 .
  * Schaefer  _et al._ (2010) B.-J. Schaefer, M. Wagner, and J. Wambach, Phys. Rev. D 81, 074013 (2010), arXiv:0910.5628 [hep-ph] .
  * Schaefer  _et al._ (2007) B.-J. Schaefer, J. M. Pawlowski, and J. Wambach, Phys. Rev. D 76, 074023 (2007), arXiv:0704.3234 [hep-ph] .
  * Wambach  _et al._ (2010) J. Wambach, B.-J. Schaefer, and M. Wagner, Acta Phys. Polon. Supp. 3, 691 (2010), arXiv:0911.0296 [hep-ph] .
  * Motornenko  _et al._ (2021) A. Motornenko, S. Pal, A. Bhattacharyya, J. Steinheimer, and H. Stoecker, Phys. Rev. C 103, 054908 (2021), arXiv:2009.10848 [hep-ph] .
  * Pal  _et al._ (2021a) S. Pal, A. Bhattacharyya, and R. Ray, Nucl. Phys. A 1010, 122177 (2021a), arXiv:2006.08985 [hep-ph] .
  * Pal  _et al._ (2021b) S. Pal, G. Kadam, H. Mishra, and A. Bhattacharyya, Phys. Rev. D 103, 054015 (2021b), arXiv:2010.10761 [hep-ph] .
  * Petreczky (2012) P. Petreczky, J. Phys. G 39, 093002 (2012), arXiv:1203.5320 [hep-lat] .
  * Borsanyi  _et al._ (2010) S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo (Wuppertal-Budapest), JHEP 09, 073 (2010), arXiv:1005.3508 [hep-lat] .
  * Borsanyi  _et al._ (2012) S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. Szabo, JHEP 01, 138 (2012), arXiv:1112.4416 [hep-lat] .
  * Bazavov  _et al._ (2012) A. Bazavov _et al._ (HotQCD), Phys. Rev. D 86, 034509 (2012), arXiv:1203.0784 [hep-lat] .
  * Bazavov  _et al._ (2013b) A. Bazavov _et al._ , Phys. Rev. Lett. 111, 082301 (2013b), arXiv:1304.7220 [hep-lat] .
  * Bazavov  _et al._ (2014a) A. Bazavov _et al._ , Phys. Rev. Lett. 113, 072001 (2014a), arXiv:1404.6511 [hep-lat] .
  * Bazavov  _et al._ (2014b) A. Bazavov _et al._ , Phys. Lett. B 737, 210 (2014b), arXiv:1404.4043 [hep-lat] .
  * Borsanyi  _et al._ (2014) S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo, Phys. Rev. Lett. 113, 052301 (2014), arXiv:1403.4576 [hep-lat] .
  * Bellwied  _et al._ (2015) R. Bellwied, S. Borsanyi, Z. Fodor, S. D. Katz, A. Pasztor, C. Ratti, and K. K. Szabo, Phys. Rev. D 92, 114505 (2015), arXiv:1507.04627 [hep-lat] .
  * Ding  _et al._ (2015b) H. T. Ding, S. Mukherjee, H. Ohno, P. Petreczky, and H. P. Schadler, Phys. Rev. D 92, 074043 (2015b), arXiv:1507.06637 [hep-lat] .
  * Bazavov  _et al._ (2016) A. Bazavov _et al._ , Phys. Rev. D 93, 014512 (2016), arXiv:1509.05786 [hep-lat] .
  * D’Elia  _et al._ (2017) M. D’Elia, G. Gagliardi, and F. Sanfilippo, Phys. Rev. D 95, 094503 (2017), arXiv:1611.08285 [hep-lat] .
  * Bazavov  _et al._ (2017) A. Bazavov _et al._ , Phys. Rev. D 95, 054504 (2017), arXiv:1701.04325 [hep-lat] .
  * Borsanyi  _et al._ (2018) S. Borsanyi, Z. Fodor, J. N. Guenther, S. K. Katz, K. K. Szabo, A. Pasztor, I. Portillo, and C. Ratti, JHEP 10, 205 (2018), arXiv:1805.04445 [hep-lat] .
  * Karthein  _et al._ (2021) J. M. Karthein, V. Koch, C. Ratti, and V. Vovchenko, Phys. Rev. D 104, 094009 (2021), arXiv:2107.00588 [nucl-th] .
  * Rischke  _et al._ (1992) D. H. Rischke, J. Schaffner, M. I. Gorenstein, A. Schaefer, H. Stoecker, and W. Greiner, Z. Phys. C 56, 325 (1992).
  * Kapusta and Olive (1983) J. I. Kapusta and K. A. Olive, Nucl. Phys. A 408, 478 (1983).
  * Olive (1981) K. A. Olive, Nucl. Phys. B 190, 483 (1981).
  * Bollweg  _et al._ (2021) D. Bollweg, J. Goswami, O. Kaczmarek, F. Karsch, S. Mukherjee, P. Petreczky, C. Schmidt, and P. Scior (HotQCD), Phys. Rev. D 104 (2021), 10.1103/PhysRevD.104.074512, arXiv:2107.10011 [hep-lat] .