# The influence of hadronic rescatterings on the net-baryon number
fluctuations

Qian Chen  Guangxi Key Laboratory of Nuclear Physics and Technology, Guangxi
Normal University, Guilin, 541004, China  Key Laboratory of Nuclear Physics
and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University,
Shanghai, 200433, China  Shanghai Research Center for Theoretical Nuclear
Physics, NSFC and Fudan University, Shanghai, 200438, China  Rui Wen  School
of Physics, Dalian University of Technology, Dalian 116024, China  School of
Nuclear Science and Technology, University of Chinese Academy of Sciences,
Beijing, 100049, China  Shi Yin  School of Physics, Dalian University of
Technology, Dalian 116024, China  Wei-jie Fu<EMAIL_ADDRESS>School of
Physics, Dalian University of Technology, Dalian 116024, China  Shanghai
Research Center for Theoretical Nuclear Physics, NSFC and Fudan University,
Shanghai, 200438, China  Zi-Wei Lin<EMAIL_ADDRESS>Department of Physics, East
Carolina University, C-209 Howell Science Complex, Greenville, North Carolina
27858, USA  Guo-Liang Ma<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics
and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University,
Shanghai, 200433, China  Shanghai Research Center for Theoretical Nuclear
Physics, NSFC and Fudan University, Shanghai, 200438, China

###### Abstract

Fluctuations of conserved charges, such as the net-baryon number fluctuations,
are influenced by different dynamical evolution processes. In this paper, we
investigate the influence of hadronic rescatterings on different orders of
cumulants of the net-baryon number distribution. At the start of hadronic
rescatterings, we introduce net-baryon number distributions reconstructed
based on net-baryon cumulants of different orders obtained from computation in
functional renormalization group (FRG), where the distributions were
constructed using the maximum entropy method. This way we introduce the
critical fluctuations of Quantum Chromodynamics (QCD) into the AMPT model.
Firstly, we find that hadronic rescatterings have distinct effects on cumulant
ratios of different orders for the net-baryon number. Secondly, we observe
that the effect of hadronic rescatterings is more significant for critical
fluctuations than dynamical fluctuations, because the two-, three- and four-
particle correlation functions due to critical fluctuations are weakened more
significantly by hadronic rescatterings.

##  I Introduction

Investigating the phase structure of Quantum Chromodynamics (QCD) has been the
focus of research in past, present, and future experimental heavy-ion
collision programs Bzdak et al. (2020); Luo et al. (2020); Zhang et al.
(2023); Ko (2023). Because both the critical end point (CEP) and the first-
order phase transition are associated with the characteristics of
fluctuations, many experimental observables have been proposed to capture
these features. The cumulants of conserved charges have been proposed as a
promising observable Stephanov (2009); Athanasiou et al. (2010); Stephanov
(2011); Cheng et al. (2009); Gavai and Gupta (2011); Chen et al. (2021a), as
they are sensitive to the finer details of the phase structure in particular.
In a simplified scenario of a system in equilibrium with volume $V$ at
temperature $T$, the scaled susceptibilities of conserved charges of different
orders ($\chi_{ijk}^{BQS}$) are defined as derivatives of the pressure with
respect to the chemical potentials associated with the conserved charges
(baryon number $B$, electric charge $Q$, and strangeness number $S$)

$\displaystyle\chi_{ijk}^{BQS}=\frac{\partial^{(i+j+k)}\left[P/T^{4}\right]}{{\partial(\mu_{B}/T)^{i}}{\partial(\mu_{Q}/T)^{j}}{\partial(\mu_{S}/T)^{k}}}=\frac{C_{ijk}^{BQS}}{VT^{3}},$
(1)

which determine the cumulants ($C_{ijk}^{BQS}$) of the distribution of the
conserved charges in the grand canonical ensemble (GCE) for a system in
thermal equilibrium. The cumulants of conserved charges are associated with
powers of the correlation length, closely tied to susceptibility Stephanov
(2009). They provide valuable insights into potential phase transitions,
including remnants of chiral criticality at near-zero chemical potential
Friman et al. (2011).

From Eqs. (1), it is convenient for us to consider the ratios of cumulants,
because they are intensive and volume-independent in the thermodynamic limit.
The cumulant ratios are defined as follows Ding et al. (2015):

$\displaystyle\frac{\chi_{2}}{\chi_{1}}=\frac{C_{2}}{C_{1}}=\frac{\sigma^{2}}{M},\frac{\chi_{3}}{\chi_{2}}=\frac{C_{3}}{C_{2}}=S\sigma,\frac{\chi_{4}}{\chi_{2}}=\frac{C_{4}}{C_{2}}=\kappa\sigma^{2},$
(2)

where the cumulants of event-by-event conserved charge multiplicity
distributions are represented by $C_{n}$. In statistics, different cumulants
characterize different features of a probability distribution. In the context
of event-by-event particle multiplicity distribution, cumulants of the
distribution play a crucial role, providing a basis for calculating
correlation functions. The cumulants $C_{n}$ of multiplicity distributions of
various orders can be calculated as Luo et al. (2012, 2010); Luo (2011); Luo
and Xu (2017):

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle\left\langle
N\right\rangle,$ $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\left\langle\left(\delta N\right)^{2}\right\rangle,$
$\displaystyle C_{3}$ $\displaystyle=$ $\displaystyle\left\langle\left(\delta
N\right)^{3}\right\rangle,$ $\displaystyle C_{4}$ $\displaystyle=$
$\displaystyle\left\langle\left(\delta
N\right)^{4}\right\rangle-3\left\langle\left(\delta
N\right)^{2}\right\rangle^{2},$ (3)

where $N=N_{+}-N_{-}$ is the net number, and $N_{+}$ and $N_{-}$ denote the
numbers of particles and antiparticles on an event-by-event basis, $\delta
N=N-\langle N\rangle$ and $\left\langle\cdots\right\rangle$ represents an
event average.

Nonetheless, cumulants exhibit a limitation as they combine correlations of
different orders. Therefore, a more insightful approach involves the
examination of (integrated) multiparticle correlation functions Bzdak et al.
(2020, 2017); Kitazawa and Luo (2017). Utilizing the following relations
allows for the investigation of integrated $n$-particle correlation functions
$\kappa_{n}$, also referred to as factorial cumulants:

$\displaystyle\kappa_{1}$ $\displaystyle=$ $\displaystyle C_{1}=\left\langle
N\right\rangle,$ $\displaystyle\kappa_{2}$ $\displaystyle=$ $\displaystyle-
C_{1}+C_{2},$ $\displaystyle\kappa_{3}$ $\displaystyle=$ $\displaystyle
2C_{1}-3C_{2}+C_{3},$ $\displaystyle\kappa_{4}$ $\displaystyle=$
$\displaystyle-6C_{1}+11C_{2}-6C_{3}+C_{4}.$ (4)

The phase structure of QCD matter, governed by strong interactions, can be
unraveled through experiments involving heavy-ion collisions at various
collision energies Adams et al. (2005); Aggarwal et al. (2010); Arsene et al.
(2005); Friman et al. (2011). The extracted chemical freeze-out points from
heavy-ion experiments span a range from nearly vanishing baryon chemical
potential at the CERN Large Hadron Collider (LHC) to approximately
$\mu_{B}\simeq 750$ MeV at the BNL Relativistic Heavy Ion Collider (RHIC). The
QCD phase diagram has been extensively explored theoretically using various
approaches, including lattice QCD from first principles Ding et al. (2015);
Bazavov et al. (2014); Bellwied et al. (2013); Borsanyi et al. (2014); Ding et
al. (2023), functional renormalization group (FRG) in QCD Fu et al. (2020); Fu
(2022); Yin et al. (2023), effective theories Schaefer and Wambach (2005);
Herbst et al. (2013, 2014); Zhang et al. (2017); Grossi et al. (2021); Chen et
al. (2021b); Otto et al. (2022); Du et al. (2023), and effective models
Fukushima (2004); Skokov et al. (2011); Pisarski and Skokov (2016); Li et al.
(2018); Xu and Huang (2023). A crucial question involves establishing
connections between these quantities and experimental measurements. For
instance, lattice QCD faces certain challenges in its calculations, with the
emergence of the fermionic sign problem at finite baryon chemical potentials
$\mu_{B}$ Bazavov et al. (2012); Borsanyi et al. (2013, 2014); Ding et al.
(2015). The functional renormalization group (Mitter et al., 2015; Herbst et
al., 2014; Fu et al., 2020, 2021) enables the study of equations of state at
both high and low baryon chemical potentials. For small baryon chemical
potentials, the functional renormalization group calculations yield a phase
boundary curvature ($\kappa$) of $\kappa=0.0142(2)$ Fu et al. (2020), closely
aligning with recent lattice QCD computations, such as $\kappa=0.0145(25)$ in
Bonati et al. (2018), $\kappa=0.0153(18)$ in Borsanyi et al. (2020), and
$\kappa=0.0149(21)$ in Bellwied et al. (2015). Moreover, a recent estimate
from FRG in QCD indicates that the location of CEP is in a small region around
$(T_{CEP}$,$\mu_{B_{CEP}}$)$=$(107, 635) MeV. The FRG proves instrumental in
describing critical properties at high baryon chemical potentials, offering
valuable insights into locating the CEP in the QCD phase diagram.

The functional renormalization group calculations are confined to the
hadronization stage and neglect interactions between hadrons and decay
processes. In contrast, a multiphase transport model (AMPT) includes these
dynamic processes. In addition, the FRG incorporates the mechanism of critical
fluctuations, a feature not present in the AMPT model Chen and Ma (2022); Chen
et al. (2023a); Huang and Ma (2021); Chen et al. (2023b). The introduction of
the FRG into the AMPT model simultaneously addresses both of these issues.
Hence, in this study, we utilize the AMPT model to investigate the
fluctuations of the net-baryon multiplicity distributions in Au+Au collisions
at $\sqrt{s_{NN}}=7.7$ GeV. Our main emphasis is on studying the influence of
the hadronic rescatterings evolution stage on fluctuations in net-baryon
multiplicity distributions. Specifically, we aim to understand how
interactions between hadrons and decay processes in relativistic heavy-ion
collisions modify critical fluctuations.

The paper’s structure is outlined as follows. In Sec. II, we offer a brief
introduction to the AMPT model and clarify how the FRG reconstructs net-baryon
multiplicity distribution. In Sec. III, we demonstrate the significant effects
of hadronic interactions, and the critical fluctuation mechanism on the
cumulants of net-baryon multiplicity distributions. Finally, Sec. IV concludes
the paper.

##  II Model setup

###  II.1 The AMPT model

The AMPT (A Multi-Phase Transport) model with string melting mechanism,
extensively utilized in investigating relativistic heavy-ion collisions Lin et
al. (2005); Lin and Zheng (2021), encompasses four key components: initial
conditions, parton cascade, hadronization, and hadronic rescatterings. The
initial conditions establish the spatial and momentum distributions of minijet
partons through QCD hard processes and soft string excitations, leveraging the
HIJING model Wang and Gyulassy (1991); Gyulassy and Wang (1994). The parton
cascade delineates the evolution of partonic matter, featuring a quark-
antiquark plasma resulting from the melting of excited strings. Parton
scatterings are modeled by Zhang’s parton cascade (ZPC) Zhang (1998), which
currently incorporates two-body elastic parton scatterings using a
perturbative QCD cross section (3 mb) with a screening mass. When all partons
cease interactions, a spatial quark coalescence model combines nearby partons
into hadrons. The resonance decays and hadronic reactions in the hadronic
phase, involving baryon-baryon, baryon-meson, and meson-meson interactions,
are described by an extended version of a relativistic transport model (ART)
Li and Ko (1995). The AMPT model succeeds to describe multiple observables for
relativistic heavy-ion collisions at the RHIC Lin et al. (2002, 2005) as well
as the LHC Lin and Zheng (2021); Ma and Lin (2016) energies, including HBT
correlations Lin et al. (2002), dihadron azimuthal correlation Ma et al.
(2006); Wang et al. (2019), collective flows Bzdak and Ma (2014); Wang and Ma
(2022), and strangeness production Jin et al. (2018); Shao et al. (2020). It
has also been used to study the chiral magnetic effects Wang and Zhao (2018);
Zhao et al. (2019); Liu and Huang (2020); Wang et al. (2021). In this work, a
new version of the AMPT model, which ensures the conservation of various
charges (baryon number $B$, electric charge $Q$, and strangeness number $S$)
in each hadronic reaction channel during the hadronic phase, is used to study
the fluctuations of baryon numbers.

###  II.2 functional renormalization group

In the FRG approach, quantum and thermal fluctuations of different momentum
modes are successively integrated in with the evolution of the renormalization
group scale Wetterich (1993), see Refs. Berges et al. (2002); Pawlowski
(2007); Braun (2012); Dupuis et al. (2021); Fu (2022) for reviews. The FRG is,
therefore, well-suited for the studies of QCD phase transitions and QCD
thermodynamics at finite temperature and densities, cf. Fu (2022) for a recent
review. As we have discussed above, fluctuations of conserved charges, such as
the baryon number fluctuations, are sensitive to the chiral phase transition
as well as the critical dynamics pertinent to the CEP in the phase diagram.
High-order cumulants of net-proton number distributions are utilized to search
for the CEP in experiments, and recently a non-monotonic dependence of the
kurtosis on the collision energy is observed with $3.1\sigma$ significance by
the STAR collaboration Adam et al. (2021).

Table 1:  Baryon chemical potential $\mu_{B}$, pseudo-critical temperature for
the chiral phase transition $T_{\mathrm{pc}}$, and volume of the fireball $V$
for different centrality bins in Au+Au collisions at $\sqrt{s_{NN}}=7.7$ GeV,
where the three parameters are used to reconstruct FRG net-baryon multiplicity
probability distributions.

Centrality |  $\mu_{B}$ (MeV) |  $T_{\mathrm{pc}}$ (MeV) |  $V$ ($\mathrm{fm}^{3}$)
---|---|---|---
| 0–5%
---
| 399
---
| 139
---
| 980
---
| 5–10%
---
| 395
---
| 140
---
| 701
---
| 10–20%
---
| 391
---
| 140
---
| 508
---
| 20–30%
---
| 382
---
| 141
---
| 340
---
| 30–40%
---
| 376
---
| 143
---
| 212
---
| 40–60%
---
| 357
---
| 144
---
| 126
---
| 60–80%
---
| 337
---
| 146
---
| 48
---

In this work we utilize the cumulants of baryon number distributions
calculated in a QCD-assisted low energy effective theory within the FRG
approach in Fu et al. (2021, 2023). Then, the baryon number distributions are
reconstructed from the cumulants of different orders by means of the maximum
entropy method or Gaussian process regression, which is detailed in Huang et
al. (2023). In this work, the cumulants of the four orders are employed for
the reconstruction of baryon number distributions. Values of the baryon
chemical potential obtained in experimentsAdamczyk et al. (2017) for
corresponding to different centrality ranges at the collision energy
$\sqrt{s_{NN}}=7.7$ GeV are listed in Tab. 1. The temperature
$T_{\mathrm{pc}}$ listed in the third column is the corresponding pseudo-
critical temperature for every value of $\mu_{B}$, obtained in the
calculations of FRG Fu et al. (2021, 2023).

###  II.3 The AMPT model with FRG sampling

Figure 1: (Color online) Structure of the AMPT model with string melting and
FRG sampling.

In order to study the influence of hadronic rescatterings on the critical
fluctuations, we introduce the net-baryon probability distribution produced by
the FRG to the AMPT model before hadronic rescatterings. As illustrated in
Figure 1, we add the FRG sampling process to the AMPT model to introduce the
critical fluctuations. Our method is to calculate the net-baryons produced by
the AMPT hadronization in the chosen phase space and then select events so
that they satisfy the net-baryon probability distribution of the FRG; the
selected events then enter the next evolution stage of hadronic rescatterings.

Figure 2: (Color online) Upper panel: The probability distributions of net-
baryon multiplicity in 0–5% central Au+Au collisions at $\sqrt{s_{NN}}=7.7$
GeV. Lower panel: The ratio of the net-baryon multiplicity probability
distribution from the FRG (circles) to the sampled one based on the FRG (solid
curve) for the “before hadronic rescatterings (BHR)” stage.

In our calculations, we apply the same kinematic cuts as used in the STAR
experimental analysis Abdallah et al. (2021); Adam et al. (2021) to calculate
the aforementioned different cumulants of net-baryon multiplicity
distributions. The unstable strange baryons (e.g., $\Lambda$, $\Sigma$, $\Xi$,
and $\Omega$), which provide a feed-down contribution to protons and anti-
protons, are also considered in our analysis. We select net-baryon within a
transverse momentum range of $0.4<p_{T}<2.0$ GeV/$c$ and a midrapidity window
of $\left|y\right|<0.5$. We use the charged particle multiplicity distribution
to define centrality bins. To avoid self-correlation, we use the charged
particle multiplicity other than protons and antiprotons within the
pseudorapidity $\left|\eta\right|<1$. We apply the $\Delta$ theoretical
formula to estimate the statistical error calculation. More detailed
information on the error calculations can be found in the references Luo and
Xu (2017); Abdallah et al. (2021).

Figure 3: (Color online) The centrality dependences of the net-baryon cumulant
ratios of $C_{2}/C_{1}$, $C_{3}/C_{2}$ and $C_{4}/C_{2}$ from the FRG,
compared to the sampled one based on the FRG for the “before hadronic
rescatterings (BHR)” stage.

Figure 2 shows how we do the FRG sampling in detail. The upper panel displays
the probability distributions of net-baryon multiplicity in the chosen phase
space in 0–5% central Au+Au collisions at $\sqrt{s_{NN}}=7.7$ GeV, where the
circles represent the net-baryon multiplicity probability distribution from
the FRG method. The FRG distributions for different centrality bins are
obtained according to the parameters in Table 1. As we can see, the net-baryon
multiplicity distribution of “before hadronic rescatterings” from the AMPT
model (dashed curve) [labeled “BHR(W/O FRG)”] is very different from the FRG
distribution (circles). In order to introduce the FRG distribution in the
“before hadronic rescatterings” stage of the AMPT model, we sampled the net-
baryon multiplicity distribution in this stage according to the FRG
probability distribution. The sampled net-baryon multiplicity distribution
based on the FRG for the “before hadronic rescatterings” stage (solid curve)
[labeled “BHR(With FRG)”] almost coincides with the FRG probability
distribution. The lower panel in Fig. 2 shows the ratio of the FRG net-baryon
multiplicity probability distribution (circles in the upper panel) to the FRG
sampled net-baryon multiplicity probability distribution (solid curve in the
upper panel), where the ratio is mostly consistent with unity indicating
success of our sampling. As shown in Fig. 1, all hadrons in the selected AMPT
events that satisfy the FRG distribution participate in the following stage of
hadronic rescatterings. We will compare the net-baryon distribution labeled
“after hadronic rescatterings (With FRG)”(dash-dotted curve) with the FRG
distribution, as the difference between them reflects the effect of hadronic
rescatterings on the imported net-baryon fluctuations. However, the difference
between the “after hadronic rescatterings” stage based on FRG sampling (dash-
dotted curve) [labeled “AHR(With FRG)”] and the “after hadronic rescatterings”
stage (dotted curve) without sampling [labeled “AHR(W/O FRG)”] indicates the
contribution the introduced critical fluctuations.

To further verify that the net-baryon multiplicity distribution of the “before
hadronic rescatterings” stage based on FRG sampling essentially reproduces the
FRG probability distribution, we calculate the net-baryon cumulant ratios in
both cases. Figure 3 shows the centrality dependences of the net-baryon
cumulant ratios of $C_{2}/C_{1}$, $C_{3}/C_{2}$ and $C_{4}/C_{2}$ from the FRG
(solid circles), compared to the sampled one based on the FRG for the “before
hadronic rescatterings (BHR)” stage (solid square). We find these ratios from
the FRG sampling are mostly in good agreement with the results of FRG within
statistical errors, indicating that the sampling was done successfully.

From Table 1, FRG only provides the net baryon probability distributions for
seven centrality bins, i.e. 0–5%, 5–10%, 10–20%, 20–30%, 30–40%, 40–60% and
60–80%; therefore, AMPT results will be calculated for the seven centrality
bins. However, the net proton measurements in the STAR experiment contain
results for nine centrality bins in Au+Au collisions at $\sqrt{s_{NN}}=7.7$
GeV Abdallah et al. (2021). To compare our results with the net proton STAR
measurements Abdallah et al. (2021), we applied the same centrality bin width
correction (CBWC) to the following AMPT results as the STAR experiment in
order to eliminate the effect of volume fluctuations Luo (2011); Luo et al.
(2013); Luo and Xu (2017); Abdallah et al. (2021); Chen and Ma (2022); Chen et
al. (2023a); Huang and Ma (2021).

##  III Results and Discussions

Figure 4: (Color online) The centrality dependences of the net-baryon cumulant
ratios of $C_{2}/C_{1}$, $C_{3}/C_{2}$ and $C_{4}/C_{2}$ in different stages
in Au+Au collisions at $\sqrt{s_{NN}}=7.7$ GeV, compared to the net-proton
STAR measurements Abdallah et al. (2021). To facilitate plotting, the value of
$C_{4}/C_{2}$ based on FRG sampling for the stage of “before hadronic
rescatterings (BHR)” in the most centrality bin (0–5%) (triangle symbols) has
been scaled down by a factor of 2, indicated by the green box.

Figure 4 shows the centrality dependences of the net-baryon cumulant ratios of
$C_{2}/C_{1}$, $C_{3}/C_{2}$ and $C_{4}/C_{2}$ before and after hadronic
rescatterings in Au+Au collisions at $\sqrt{s_{NN}}=7.7$ GeV, compared to the
net-proton STAR measurements Abdallah et al. (2021). Without the FRG sampling,
we observe that the net-baryon cumulant ratios $C_{2}/C_{1}$ and $C_{3}/C_{2}$
for the “after hadronic rescatterings” stage (cross symbols) are larger than
the results for the “before hadronic rescatterings” stage (square symbols).
Similarly, we can also observe that the net-baryon cumulant ratios
$C_{2}/C_{1}$ and $C_{3}/C_{2}$ for the “after hadronic rescatterings” stage
based on FRG sampling (star symbols) are larger than the results for the
“before hadronic rescatterings” stage based on FRG sampling (triangle
symbols). This suggests that the process of hadronic rescatterings increases
the two cumulant ratios. We find that the cumulant ratios $C_{2}/C_{1}$ and
$C_{3}/C_{2}$ based on FRG sampling for the “before hadronic rescatterings”
and “after hadronic rescatterings” stages are smaller than the corresponding
ratios in the AMPT model without the FRG sampling. Although there are
relatively large errors, the cumulant ratios $C_{4}/C_{2}$ based on FRG
sampling for the stages of “after hadronic rescatterings” and “before hadronic
rescatterings” in the centrality bins of 0–5%, 5–10%, 10–20% and 20–30% seem
to be larger than the corresponding ratios in the AMPT model without the FRG
sampling. However, for the cumulant ratio $C_{4}/C_{2}$ for the other
centrality bins, the ordering is reversed. All the differences indicate the
impact of critical fluctuations on these cumulant ratios.

As shown in Fig. 3, we see that the cumulant ratios
$C_{2}/C_{1}$,$C_{3}/C_{2}$ and $C_{4}/C_{2}$ are the same for the FRG and
“before hadronic rescatterings” with the FRG sampling. However, Fig. 4 shows
that the cumulant ratios for the FRG are noticeably larger than those for the
“before hadronic rescatterings” with the FRG sampling because we have applied
the CBWC for the correction of volume fluctuations. This suggests that volume
fluctuations have an important impact on the cumulant ratios, which is
consistent with previous studies Chen and Ma (2022); Chen et al. (2023a);
Huang and Ma (2021). We observe that the cumulant ratios $C_{2}/C_{1}$ and
$C_{3}/C_{2}$ from the FRG significantly overestimate the STAR measurements
and display a different dependence on the centrality. This is expected since
the FRG calculations are done on the phase boundary as shown in Table 1,
rather than on the chemical freeze-out. Incorporating the FRG into the “before
hadronic rescatterings” stage of the AMPT model, our results underestimate the
STAR measurements but the centrality dependence is consistent with the
experimental measurements. Through the hadron rescattering evolution stage,
the values increase but still underestimate the STAR measurements. The
cumulant ratio $C_{4}/C_{2}$ from the FRG is consistent with STAR
measurements. After incorporating the FRG into the “before hadronic
rescatterings” stage of the AMPT model, we show that the value of
$C_{4}/C_{2}$ overestimates the STAR measurements in central collisions but
underestimates them in peripheral collisions. After hadronic rescatterings,
the $C_{4}/C_{2}$ value decreases for mid-central to central collisions but
increases for the most peripheral bin, bringing the model results with FRG
closer to the STAR measurements.

Figure 5: (Color online) The centrality dependences of the net-baryon cumulant
ratios of $C_{2}/C_{1}$, $C_{3}/C_{2}$ and $C_{4}/C_{2}$ with and without the
FRG sampling in the “after hadronic rescatterings (AHR)” stage of Au+Au
collisions at $\sqrt{s_{NN}}=7.7$ GeV, compared to the net-proton STAR
measurements Abdallah et al. (2021).

To study the effect of critical fluctuations on the net-baryon fluctuations,
we compare the AMPT results with and without the FRG sampling after hadronic
rescatterings. Figure 5 shows the centrality dependences of the net-baryon
cumulant ratios of $C_{2}/C_{1}$, $C_{3}/C_{2}$ and $C_{4}/C_{2}$ with and
without the FRG sampling in the “after hadronic rescatterings (AHR)” stage of
Au+Au collisions at $\sqrt{s_{NN}}=7.7$ GeV, where the original AMPT model
(square symbols) contains dynamical correlations without critical fluctuations
but the AMPT model with the FRG sampling (cross symbols) has both critical
fluctuations and dynamical correlations. The results for the cumulant ratio of
$C_{2}/C_{1}$ in both cases show a similar trend as the experimental
measurements, albeit with lower values. In both cases, the cumulant ratio of
$C_{3}/C_{2}$ follows the same trend as the experimental measurements. The
AMPT model without the FRG sampling roughly agrees with the experimental
measurements, while the AMPT model with the FRG sampling yields slightly lower
values. Overall, critical fluctuations reduce both ratios. The trend of
$C_{4}/C_{2}$ in the original AMPT model is different from the experimental
measurement, although the statistical errors are large. In contrast, the trend
in the AMPT with the FRG sampling looks more consistent with the experimental
measurement, because the inclusion of critical fluctuations increases the
ratio in central collisions and decreases the ratio in peripheral collisions.

Figure 6: (Color online) The net-baryon cumulant double ratio $\rm
Ratio_{C_{i}/C_{j}}$ of the “after hadronic rescatterings (AHR)” stage
${C_{i}/C_{j}(AHR)}$ to the “before hadronic rescatterings (BHR)” stage
${C_{i}/C_{j}(BHR)}$.

To see the different effects of hadronic rescatterings in the two cases, Fig.
6 shows the double ratio $\rm Ratio_{C_{i}/C_{j}}$ of the “after hadronic
rescatterings (AHR)” stage $\rm{C_{i}/C_{j}(AHR)}$ to the “before hadronic
rescatterings (BHR)” stage $\rm{C_{i}/C_{j}(BHR)}$. By comparing the double
ratios of the AMPT model with and without the FRG sampling, we observe the
effects of hadronic rescatterings on the cumulant ratios for the two different
cases. Both $\rm Ratio_{C_{2}/C_{1}}$ and $\rm Ratio_{C_{3}/C_{2}}$ are larger
than unity, which indicates that the effect of hadronic rescatterings
increases the cumulant ratios of $C_{2}/C_{1}$ and $C_{3}/C_{2}$. This is
consistent with our previous AMPT study on the effect of hadronic
rescatterings on dynamical fluctuations Chen and Ma (2022). Furthermore, the
double ratios for the AMPT model with the FRG sampling are larger than those
for the AMPT model without the FRG sampling, which indicates that hadronic
rescatterings more significantly enhance $C_{2}/C_{1}$ and $C_{3}/C_{2}$
caused by critical fluctuations. In terms of the double ratios of
$C_{4}/C_{2}$, the AMPT model without the FRG sampling shows results
consistent with unity; however, the AMPT model with the FRG sampling shows a
value smaller than unity in central collisions but larger than unity in
peripheral collisions. It indicates that hadronic rescatterings decrease the
cumulant ratio of $C_{4}/C_{2}$ in central collisions and increases the ratio
in peripheral collisions for critical fluctuations.

Figure 7: (Color online) The centrality dependences of the net-baryon
correlation functions of $\kappa_{2}$, $\kappa_{3}$ and $\kappa_{4}$ in the
“after hadronic rescatterings (AHR)” stage of Au+Au collisions at
$\sqrt{s_{NN}}=7.7$ GeV, compared to the net-proton STAR measurements Abdallah
et al. (2021).

As suggested in Refs. Bzdak et al. (2020), multiparticle correlation functions
are much cleaner than cumulant ratios. Figure 7 shows the centrality
dependences of the net-baryon correlation functions of $\kappa_{2}$,
$\kappa_{3}$ and $\kappa_{4}$ in the “after hadronic rescatterings (AHR)”
stage of Au+Au collisions at $\sqrt{s_{NN}}=7.7$ GeV, compared to the net-
proton STAR measurements Abdallah et al. (2021). As the yield of antibaryons
is much lower than that of baryons, the fluctuations of baryons are expected
to be similar to that of net baryons in Au+Au collisions at
$\sqrt{s_{NN}}=7.7$ GeV. Therefore, we calculated the correlation functions
using the method described in Eq. (4). It can be observed that the correlation
functions $\kappa_{2}$ and $\kappa_{3}$ follow the same trends as the
experimental measurements. In addition, we find that the strengths of the
correlation functions $\kappa_{2}$ and $\kappa_{3}$ in the AMPT model without
the FRG sampling are smaller than those in the AMPT model with the FRG
sampling. Although our results for $\kappa_{4}$ show large statistical errors
in both cases, the trend for the AMPT model without the FRG sampling seems to
be the opposite of the experimental measurement. However, the trend for the
AMPT model with the FRG sampling is in agreement with the experimental
measurement. This suggests that the inclusion of critical fluctuations will
change the four-particle correlation from negative to positive, which would be
more consistent with the current experimental measurement.

Figure 8: (Color online) The centrality dependences of the difference
$|\Delta\kappa_{n}|$ between the net-baryon correlation functions of the
“after hadronic rescatterings (AHR)” stage $|\kappa_{n_{(}AHR)}|$ and that of
the “before hadronic rescatterings (BHR)” stage $|\kappa_{n_{(}BHR)}|$.

In our previous study Chen and Ma (2022), we have found that multibaryon
correlations are weakened after the evolution of heavy ion collisions. We next
study the effect of hadronic rescatterings on the net-baryon correlation
functions in the two cases. Figure 8 shows the centrality dependences of the
difference $\Delta|\kappa_{n}|$ of net-baryon correlation functions between
the “after hadronic rescatterings (AHR)” stage $|\kappa_{n_{(}AHR)}|$ and the
“before hadronic rescatterings (BHR)” stage $|\kappa_{n_{(}BHR)}|$, i.e.
$\Delta|\kappa_{n}|$ = $|\kappa_{n_{(}AHR)}|$ \- $|\kappa_{n_{(}BHR)}|$. We
find that the value of $\Delta|\kappa_{n}|$ is consistently less than zero for
both cases, indicating that all two-, three- and four-particle correlation
functions are weakened by hadronic rescatterings. We also observe that the
values of $\Delta|\kappa_{n}|$ for the AMPT model with the FRG sampling are
more negative than those of the AMPT model without the FRG sampling, which
indicates that the critical fluctuation contributions to two-, three- and
four-particle correlation functions are weakened more significantly by
hadronic rescatterings. It should be noted that we focus on net-baryon
fluctuations in this work, but previous work has shown that the behavior of
net-proton fluctuations is similar to that of net-baryon fluctuations, with
only differences in magnitude Chen and Ma (2022); Vovchenko et al. (2022).

##  IV Summary

In summary, we incorporate critical fluctuations from the functional
renormalization group (FRG) into the AMPT model in order to reveal the
important effect of hadronic rescatterings on critical fluctuations. We
observed apparent influences of hadronic rescatterings on the cumulant ratios
of the conserved charge fluctuations. Hadronic rescatterings are found to
increase the cumulant ratios $C_{2}/C_{1}$ and $C_{3}/C_{2}$ for all
centrality bins but decrease $C_{4}/C_{2}$ in central collisions and increase
$C_{4}/C_{2}$ in peripheral collisions. The effect of hadronic rescatterings
is more significant for critical fluctuations than dynamical fluctuations.
This is because the two-, three- and four-particle correlation functions due
to critical fluctuations are weakened more significantly by hadronic
rescatterings.

## ACKNOWLEDGMENTS

We thank Prof. Xiao-Feng Luo for helpful discussions, and Dr. Chen Zhong for
maintaining the high-quality performance of the Fudan supercomputing platform
for nuclear physics. This work is supported by the National Natural Science
Foundation of China under Grants No. 12325507, No. 12147101, No. 11890714, No.
11835002, No. 11961131011, No. 11421505, No. 12105054, No. 12175030, the
Central Government Guides Local Scientific and Technological Development under
Grant No. Guike ZY22096024, the National Key Research and Development Program
of China under Grant No. 2022YFA1604900, the Strategic Priority Research
Program of Chinese Academy of Sciences under Grant No. XDB34030000, the
Guangdong Major Project of Basic and Applied Basic Research under Grant No.
2020B0301030008, and the National Science Foundation under Grant No. 2310021.

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