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Particlization of an interacting hadron resonance gas with global conservation laws for event-by-event fluctuations in heavy-ion collisions

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We revisit the problem of particlization of a QCD fluid into hadrons and resonances at the end of the fluid dynamical stage in relativistic heavy-ion collisions in a context of fluctuation measurements. The existing methods sample an ideal hadron resonance gas, and therefore, they do not capture the non-Poissonian nature of the grand-canonical fluctuations, expected due to QCD dynamics such as the chiral transition or QCD critical point. We address the issue by partitioning the particlization hypersurface into locally grand-canonical fireballs populating the space-time rapidity axis that are constrained by global conservation laws. The procedure allows to quantify the effect of global conservation laws, volume fluctuations, thermal smearing, and resonance decays on fluctuation measurements in various rapidity acceptances and can be used in fluid dynamical simulations of heavy-ion collisions. As a first application, we study event-by-event fluctuations in heavy-ion collisions at the Large Hadron Collider (LHC) using an excluded volume hadron resonance gas model matched to lattice QCD susceptibilities, with a focus on (pseudo)rapidity acceptance dependence of net baryon, net proton, and net charge cumulants. We point out large differences between net proton and net baryon cumulant ratios that make direct comparisons between the two unjustified. We observe that the existing experimental data on net-charge fluctuations at the LHC shows a strong suppression relative to a hadronic description.
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PHYSICAL REVIEW C 103, 044903 (2021)
Particlization of an interacting hadron resonance gas with global conservation laws
for event-by-event fluctuations in heavy-ion collisions
Volodymyr Vovchenko and Volker Koch
Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
(Received 6 January 2021; accepted 18 March 2021; published 12 April 2021)
We revisit the problem of particlization of a QCD fluid into hadrons and resonances at the end of the fluid
dynamical stage in relativistic heavy-ion collisions in a context of fluctuation measurements. The existing
methods sample an ideal hadron resonance gas, and therefore, they do not capture the non-Poissonian nature
of the grand-canonical fluctuations, expected due to QCD dynamics such as the chiral transition or QCD critical
point. We address the issue by partitioning the particlization hypersurface into locally grand-canonical fireballs
populating the space-time rapidity axis that are constrained by global conservation laws. The procedure allows
to quantify the effect of global conservation laws, volume fluctuations, thermal smearing, and resonance decays
on fluctuation measurements in various rapidity acceptances and can be used in fluid dynamical simulations
of heavy-ion collisions. As a first application, we study event-by-event fluctuations in heavy-ion collisions at
the Large Hadron Collider (LHC) using an excluded volume hadron resonance gas model matched to lattice
QCD susceptibilities, with a focus on (pseudo)rapidity acceptance dependence of net baryon, net proton, and net
charge cumulants. We point out large differences between net proton and net baryon cumulant ratios that make
direct comparisons between the two unjustified. We observe that the existing experimental data on net-charge
fluctuations at the LHC shows a strong suppression relative to a hadronic description.
DOI: 10.1103/PhysRevC.103.044903
I. INTRODUCTION
Event-by-event fluctuations in relativistic heavy-ion colli-
sions have long been considered sensitive experimental probes
of the QCD phase structure [14]. At the highest collision
energies achievable at the Large Hadron Collider (LHC) and
Relativistic Heavy Ion Collider (RHIC) they can be used to
analyze the QCD chiral crossover transition at small baryon
densities [5]. The equilibrium fluctuations of the QCD con-
served charges in the grand-canonical ensemble (GCE) have
been computed at μB=0 from first principles, via lattice
gauge theory simulations [6,7]. An appropriately performed
comparison between experimental measurements and lattice
QCD predictions can, in principle, establish whether a locally
equilibrated QCD matter is indeed created in experiment.
At lower collision energies, the fluctuations are used in the
experimental search for the hypothetical QCD critical point
and the first-order phase transition at finite baryon density.
This is motivated by the fact that fluctuations, in particular
the net proton cumulants of higher order, are increasingly
sensitive to the proximity of the critical point [8,9]. The
corresponding measurements are in the focus of several exper-
imental programs, including beam energy scans performed at
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RHIC [10,11] and CERN-SPS [12]. The experimental data in
the literature includes second-order cumulants, both diagonal
[1316] and off-diagonal [1719], as well as higher-order
fluctuation measures [11,2022].
A proper theoretical modeling is crucial for interpreting
the experimental data. It is not uncommon in the literature to
directly compare the theoretical fluctuations evaluated in the
GCE with experimental measurements [2332]. Such com-
parisons, however, have several important drawbacks. For
one thing, the experimental measurements are performed in
momentum space, whereas the theoretical approaches op-
erate in configuration space. Cuts in the momentum space
may be identified with the coordinate space if strong space-
momentum correlations are present, for instance due to
Bjorken flow, but even in this case a degree of smearing will
be present because of the thermal motion [33,34]. Event-by-
event fluctuations, especially the high-order cumulants, are
strongly affected by global conservation laws [3537], re-
quiring large corrections to the grand-canonical distributions.
Other mechanisms include volume fluctuations [3840], finite
system size [41], as well as nonequilibrium dynamics such as
memory effects [42] or hadronic phase evolution [43]. Proper
modeling of these effects is thus required for analyzing the
experimental data quantitatively.
The standard approach to describe the evolution of strongly
interacting QCD matter created in heavy-ion collisions is rela-
tivistic fluid dynamics [44,45]. The hydrodynamic description
terminates at a so-called particlization stage [46], where the
QCD fluid is transformed into an expanding gas of hadrons
and resonances. This picture forms the basis of the hybrid
models of heavy-ion collisions [47,48] and it works quite well
2469-9985/2021/103(4)/044903(22) 044903-1 Published by the American Physical Society
VOLODYMYR VOVCHENKO AND VOLKER KOCH PHYSICAL REVIEW C 103, 044903 (2021)
in describing the spectra and flow of bulk hadrons measured
in a broad range of collision energies [4952].
Event-by-event fluctuations of hadron yields, on the other
hand, are seldom analyzed in the hydro picture. The yields
of hadrons and resonances are usually sampled in each fluid
element from a Poisson distribution. Because the Poisson
distribution is additive, this means that the yields of all hadron
species in the full space follow the Poisson distribution as
well. This picture corresponds to the multiplicity distribu-
tion of an ideal Maxwell-Boltzmann hadron resonance gas
(HRG) in the GCE. Most hydro simulations use this type
of sampling [5356]. More advanced procedures incorporate
exact conservation of the QCD conserved charges and/or
energy-momentum [5761], however, these procedures are
still restricted to the equation of state of an ideal HRG. The
existing methods, therefore, are not suitable to analyze the
fluctuation signals of any effect that goes beyond the physics
of an ideal hadron gas.
Interacting HRG models, on the other hand, offer more
flexibility. For instance, an HRG model with excluded volume
corrections can describe the lattice QCD cumulants of net
baryon distribution in vicinity of the chemical freeze-out at
μB=0[62,63], which the ideal HRG model cannot. An-
other example is HRG model with van der Waals interactions,
which captures the physics of nuclear liquid-gas transition at
large μB[31,64]. It is the purpose of this work to formulate
a particlization routine appropriate to describe event-by-event
fluctuations encoded in the equation of state of such interact-
ing HRG models.
The paper is organized as follows. In Sec. II we introduce
a method for sampling an interacting HRG at particlization
stage of heavy-ion collisions that we call subensemble sam-
pler. Section III describes the technical details of sampling
an excluded volume HRG model that we study this work as
an example. In Sec. IV the subensemble sampler is applied
for the description of event-by-event fluctuations in heavy-ion
collisions at LHC energies. Discussion and summary in Sec. V
close the article.
II. SUBENSEMBLE SAMPLER
Consider the particlization stage of heavy-ion collisions at
the end of the ideal hydrodynamic evolution. This stage is
characterized by a hypersurface σ(x), where the space-time
coordinate xis taken in the Milne basis, x=(τ,rx,ry
s).
Here τ=t2r2
zand ηs=1
2ln t+rz
trzare the longitudinal
proper time and space-time rapidity, respectively, rx,ry, and
rzare the Cartesian coordinates. The QCD matter is assumed
in local thermodynamic equilibrium at each point xon this
hypersurface.1As the fluid is converted into hadrons at this
stage, the equation of state is described by hadron and res-
onance degrees of freedom, i.e., this has to be a variant of
the hadron resonance gas model matched to the actual QCD
equation of state at each point on the hypersurface.
1In a more general case the deviations from local equilibrium are
described using viscous corrections.
Let us denote ZHRG (T,V,μ) as the grand partition
function of a hadron resonance gas at temperature T,vol-
ume V, and chemical potentials μ=(μB
Q
S), and
PHRG ({Ni}f
i=1;T,V,μ) as the corresponding multiplicity dis-
tribution for all hadron species. Here fis the number of
different hadron species. In case of the commonly used ideal
HRG model PHRG has a form of a multi-Poisson distribution
where the Poisson means correspond to the mean multiplici-
ties of primordial hadrons and resonances. Most particlization
routines work with the multi-Poisson distribution of the ideal
HRG model. However, PHRG will differ from the multi-
Poisson distribution in a more general case of a nonideal
HRG. Thus, in the present work we generalize the particliza-
tion routine for arbitrary hadron multiplicity distributions.
A. Uniform fireball
Let us first consider a case of the grand-canonical en-
semble, where the global conservations laws are enforced on
average. Later we will relax this assumption to incorporate
exact global conservation.
If we further assume for the time being that the intensive
thermal parameters T,μB,μQ, and μSare the same across
the entire fireball, the partition function of the entire system
coincides with the grand partition function ZiHRG of a uniform
interacting HRG (iHRG):
Zgce,unif
tot =ZiHRG (T,V,μ).(1)
Here
ZiHRG (T,V,μ)=
Q
eμ·QZiHRG (T,V,Q)(2)
with ZiHRG (T,V,Q) being the canonical partition function of
the HRG model with Q=(B,Q,S), and
V=σ
dσμ(x)uμ(x)(3)
is the effective system volume at particlization. Here the
integral is over the particlization hypersurface σand the
space-time points without any matter are omitted from σ.
The single-particle momentum distribution function is
given by the Cooper-Frye formula [65]:
Ep
dNi
d3p=dσμ(x)pμfi(x,p).(4)
Here fi(x,p) is the single-particle distribution function. In
the following we neglect quantum statistics and viscous cor-
rections but take into account the possibility of interactions
between hadrons. We assume that the distribution function
takes the following general form2
fi(x,p)=diλint
i(T,μ)
(2π)3exp μipμuμ(x)
T.(5)
Here μi=biμB+qiμQ+siμS,uμ(x) is the flow velocity
profile, diis the degeneracy factor, and λint
i(T,μ)isacor-
rection factor which describes deviations from the ideal gas
2Here we neglect the possible modifications of the momentum
distribution due to interactions.
044903-2
PARTICLIZATION OF AN INTERACTING HADRON PHYSICAL REVIEW C 103, 044903 (2021)
...
1,1,1 2,2,2, ,
1 2
FIG. 1. A schematic view of the partition of the space-time rapid-
ity axis at particlization into Nlocally grand-canonical subvolumes,
each characterized by values of the local temperature Tj, the chemical
potential μj, and the volume Vj.
distribution function induced by interactions. The explicit
form of this factor depends on the interacting HRG model un-
der consideration. The mean particle number Niis obtained
by integrating Eq. (4) over the momenta:
Ni=λint
i(T,μ)dim2
iT
2π2K2(mi/T)eμi/TV,(6)
=λint
i(T,μ)Niid.(7)
The full space hadron multiplicity distribution is given by
the multiplicity distribution of the grand-canonical HRG:
Pgce,unif {Ni}f
i=1=PiHRG ({Ni}f
i=1;T,V,μ).(8)
B. Partition in rapidities
Let us now split the hypersurface into sslices along the
space-time rapidity axis (see Fig. 1).3The boundaries of each
slices are ηmin
jand ηmax
j
min
j. Furthermore, one has ηmin
j=
ηmax
j1for j>1, and ηmin
1=−ηmax and ηmax
s=ηmax, where
ηmax is the global maximum value of the space-time rapidity.
One could, for instance, identify ηmax with the beam rapidity.
The subvolume characterizing the physical size of slice j
is
Vj=x[ηmin
jmax
j]
dσμ(x)uμ(x).(9)
The key assumption in the following is that each subvol-
ume Vjis sufficiently large for it to be in the thermodynamic
limit. Or in other words, Vjξ3for each iwhere ξis any
relevant correlation length. If that is the case, one can neglect
the surface effects, namely the interactions between parti-
cles from different subvolumes. Mathematically speaking,
this implies a scaling ZiHRG (T,Vj)eVj[or, equivalently,
ln ZiHRG (T,Vj)Vj]forVjξ3. Also, the total partition
function factorizes into a product of partition functions for
each of the subvolumes:
Zgce,unif
tot
s
j=1
ZiHRG (T,Vj,μ),Vjξ3,(10)
ln Zgce,unif
tot
s
j=1
ln ZiHRG (T,Vj,μ),Vjξ3.(11)
3If the partition in ηsleads to several disconnected hypersurfaces
in a single slice, these should all be treated as separate subvolumes.
More generally, the partition should always be performed into con-
tiguous subvolumes.
The form of Eq. (11) allows us to relax the assumption of
the constancy of thermal parameters. Let us now assume that
the intensive thermal parameters depend on the space-time
rapidity ηs. This implies that each of the rapidity slices is char-
acterized by its own set of values of the thermal parameters,
i.e., in Eqs. (10) and (11) one has TTiand μμi:
Zgce
tot
s
j=1
ZiHRG (Tj,Vj,μj),Vjξ3,(12)
ln Zgce
tot
s
j=1
ln ZiHRG (Tj,Vj,μj),Vjξ3.(13)
Let us denote the hadron multiplicities in a subvolume j
by ˆ
Nj={Nj,i}f
i=1. The multiplicity distribution ˆ
Njis given
by the corresponding multiplicity distribution of the HRG
model with thermal parameters of the given subvolume, i.e.,
Pgce (ˆ
Nj)=PiHRG (ˆ
Nj;Tj,Vj,μj). Due to the fact that we ne-
glected all correlations between particles from the different
subvolumes, the multiplicity distribution of ˆ
Njis independent
of the multiplicity distributions in all other subvolumes. The
probability distribution for multiplicities {ˆ
Nj}s
j=1across all
subvolumes thus factorizes as follows:
Pgce{ˆ
Nj}s
j=1=
s
j=1
PiHRG (ˆ
Nj;Tj,Vj,μj).(14)
The factorization in Eq. (14) will no longer hold once we
introduce exact global conservation of conserved charges.
The momentum distribution of hadron species iemitted
from a rapidity slice jreads
Ep
dNj,i
d3p=x[ηmin
jmax
j]
dσμ(x)pμfj,i(x,p) (15)
with
fj,i(x,p)=diλint
i(Tj,μj)
(2π)3exp μj,ipμuμ(x)
T.(16)
Here μj,i=biμB,j+qiμQ,j+siμS,j.
C. Exact global conservation laws
Let us now incorporate the effect of exact global conserva-
tion of conserved charges. As we work in the thermodynamic
limit, Vjξ3, the exact conservation will not affect the mean
multiplicities due to the thermodynamic equivalence of statis-
tical ensembles. However, as the thermodynamic equivalence
does not extend to fluctuations, the fluctuation observables
will be affected by the exact conservation, no matter how large
the system is.
The total values of the globally conserved baryon number,
electric charge, and strangeness coincide with the GCE mean
values due to the thermodynamic equivalence of ensembles:
Qtot =
s
k=1Qkgce.(17)
To enforce the global conservation laws on the level of
multiplicity distributions one has to project out all microstates
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VOLODYMYR VOVCHENKO AND VOLKER KOCH PHYSICAL REVIEW C 103, 044903 (2021)
that violate the global conservations laws from the grand-
canonical partition function. This is achieved by introducing
a Kronecker delta into the grand partition function (12)ofthe
entire system:
Zce
tot
s
j=1
Qj
eμj·QjZiHRG (Tj,Vj,Qj)δQtot
s
k=1
Qk.
(18)
The presence of the delta function in Eq. (18) breaks the
factorization of multiplicity distributions in different rapidity
slices. The joint multiplicity distribution reads
Pce{ˆ
Nj}s
j=1=
s
j=1
PiHRG (ˆ
Nj;Tj,Vj,μj)δQtot
s
i=k
Qk,
(19)
Qk=
f
i=1
qiNk,i.(20)
Here qi=(bi,qi,si) is a vector of conserved charge values
carried by hadron species j.
D. Sampling the multiplicity distribution
Here we present a general method for sampling the joint
multiplicity distribution Eq. (19) of hadron numbers in all the
subsystems. The method is based on rejection sampling and
it assumes that it is known how to sample the multiplicity
distribution of the grand-canonical variant of the HRG model
used. To generate a configuration from the distribution (19)
(1) Sample ˆ
Njfor j=1···sindependently for each
subsystem from the grand-canonical variant of an in-
teracting HRG model characterizing each subsystem.
(2) Compute s
k=1Qkvia Eq. (20). Accept the config-
uration if Qtot =s
k=1Qk, or go back to step (1)
otherwise.
The method is general in the sense that it does not assume
anything about the specific HRG model used. It will work
both for an ideal and interacting HRG. It should be noted,
however, that the algorithm may become inefficient if the
acceptance rate in step (2) becomes low. This can happen for
large systems and multiple conserved charges. More efficient
algorithms can be devised for specific versions of the HRG
model, see, e.g., a multistep method of Becattini and Ferroni
in Ref. [58]. We do employ this method in our Monte Carlo
simulations in Sec. IV.
E. Thermal smearing
The algorithm in the previous section allows to sample
hadron multiplicity distributions differentially in space-time
rapidity. The experiments, however, perform measurements
in momentum rather than coordinate space, therefore, a tran-
sition to momentum space is necessary. In some cases, such
as the Bjorken flow scenario at the highest collision energies,
it is possible to identify the space-time rapidity ηswith the
momentum rapidity Y, allowing to study rapidity-dependent
hadron distributions without the transition to the momentum
space. Even in this case, however, a degree of smearing be-
tween ηsand Yis present due to thermal motion. The boost
invariance breaks down at lower collision energies and the
problem of space-momentum correlations becomes even more
severe. For these reasons it is necessary to assign each of
the hadrons a three-momentum. Furthermore, if a subsequent
afterburner stage is to be included into the modeling, one has
to generate both the spatial and momentum coordinates for
each hadron.
The procedure to generate the momenta of all the hadrons
is fairly straightforward. Once the multiplicity distributions
{ˆ
Nj}s
j=1for all the rapidity slices have been sampled, the
coordinates and momenta of all the hadrons can be gener-
ated through the standard Cooper-Frye momentum sampling,
applied independently to each hadron in each of the rapidity
slices. Several implementations for this task are available, see,
e.g., Refs. [53,54,66]. The sampled hadrons should then be
provided as input into a hadronic afterburner like UrQMD
[67,68]orSMASH[69], if one is used, or a cascade of
resonance decays performed to obtain the final state particles
that are measured experimentally. The comparison with data
can then be done in the standard way, by computing the
observables in a given acceptance as statistical averages.
III. EXCLUDED VOLUME MODEL FOR
NET BARYON FLUCTUATIONS
To illustrate the developed formalism we shall apply it to
net proton and net baryon fluctuations in heavy-ion collisions
at energies reachable at LHC and RHIC. In this section we
describe the motivation and the technical details behind an
excluded volume HRG model that we use for the analysis. A
reader interested only in the final heavy-ion results may skip
to Sec. IV where these are presented and discussed.
The typical chemical freeze-out temperatures, Tch 155–
160 MeV at the LHC [7072] and Tch 160–165 MeV at
the top RHIC energies [73], are close to the pseudocritical
temperature of the QCD crossover transition determined by
lattice QCD, which is Tpc 155–160 MeV [74,75]atμB=
0. Lattice QCD predicts that the high-order net baryon cu-
mulants, namely the kurtosis χB
4B
2and the hyperkurtosis
χB
6B
2ratios deviate significantly from the Skellam distribu-
tion baseline of the ideal HRG model, where these ratios are
equal to unity. The hyperkurtosis in particular turns negative
around Tpc which is thought to be related to the remnants
of the chiral criticality [5] at vanishing light quark masses.
It would certainly be of great interest to verify this theory
prediction of a negative χB
6experimentally, which may serve
as an experimental evidence for the chiral crossover transition.
The measurement of higher-order net proton fluctuations is
planned in future runs at the LHC [76].
In our previous work [77] we studied this question an-
alytically, in the framework of the subensemble acceptance
method (SAM). There, the sensitivity of measurements to
the equation of state was predicted to be not overshadowed
if the measurements are performed in acceptance spanning
one to two units of rapidity. However, the entire argument in
Ref. [77] has been done in the configuration space, relying
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PARTICLIZATION OF AN INTERACTING HADRON PHYSICAL REVIEW C 103, 044903 (2021)
on perfect momentum-space correlations due to Bjorken flow.
Here we would like to determine how the results will be
distorted by the thermal smearing and resonance decays.
To apply the formalism of Sec. II we need to employ an
interacting HRG model that matches the lattice QCD equation
of state and be able to sample the grand-canonical multiplicity
distribution of such a model. Here we take an HRG model
with excluded volume interactions in the baryonic sector—the
EV-HRG model—which was formulated in Refs. [62,64] and
shown to describe well the lattice data on the diagonal net-
baryon susceptibilities at μB=0 at temperatures up to and
even slightly above Tpc .
A. Single-component EV model
Before discussing the full model let us first consider a
single-component excluded volume model in order to intro-
duce the multiplicity sampling procedure. The grand partition
function at fixed temperature T, volume V, and chemical
potential μreads
Zev (T,V)=
N=0
[(VbN )φ(T)eμ/T]N
N!θ(VbN ).
(21)
Here
φ(T)=dm
2T
2π2K2(m/T) (22)
is an ideal gas density of particle species with degeneracy d
and mass mat vanishing chemical potential. K2is the modified
Bessel function of the second kind.
Equation (21) defines the multiplicity distribution of the
EV model, giving the following (unnormalized) probability
function:
˜
Pev (N;T,V)=[(VbN )φ(T)eμ/T]N
N!θ(VbN ).
(23)
In the thermodynamic limit, N→∞, the particle density
nev (T)=Nev /Vis determined by the maximum term in
Eq. (21). Maximizing ˜
Pev with respect to Ngives a transcen-
dental equation defining nev (T):
bnev
1bnev ebnev
1bnev =bφ(T)eμ/T.(24)
The solution to Eq. (24) is given in terms of the Lambert W
function (see Ref. [78] for details):
bnev
1bnev =Wbφ(T)eμ/T,(25)
or
nev (T)=W[bφ(T)eμ/T]
b{1+W[bφ(T)eμ/T]}.(26)
The pressure reads
pev(T)=Tn
ev
1bnev =T
bW[bφ(T)eμ/T].(27)
1. Dimensionless form
In the EV model it is possible to replace the three thermal
parameters (T,V) and the excluded volume parameter bby
two dimensionless quantities, namely a reduced volume ˜
V
V/band a parameter κbφ(T)eμ/Tthat characterizes the
strength of repulsive interactions. The probability distribution
(23) then takes the form
˜
Pev (N;˜
V,κ)=[( ˜
VN)κ]N
N!θ(˜
VN).(28)
The mean particle number reads
Nev =˜
VW(κ)
1+W(κ).(29)
This reduced form implies that the multiplicity distribution
is fully specified if the values of parameters ˜
Vand κare
known.
2. Cumulants of particle number distribution
Cumulants of the particle number distribution in the EV
model can be evaluated from the probability distribution func-
tion (28). The nth moment reads
Nn=˜
V
N=0Nn˜
Pev (N)
˜
V
N=0˜
Pev (N)
.(30)
The sums over Nare finite due to the presence of the θ
function in Eq. (28). Thus, for finite ˜
V, they can be carried
out explicitly. The cumulants can be expressed in terms of the
moments as
κn[N]=
n
k=1
(1)k1(k1)! Bn,k(N,...,Nnk+1).
(31)
Here Bn,kare the partial Bell polynomials.
Explicit expressions for κn[N] can be obtained in the
thermodynamic limit, ˜
V→∞. This is achieved through the
cumulant generating function
GN(t)lnetN.(32)
The t-dependent mean value Nev(t) is obtained from
Eq. (29) by a substitution κκet:
N(t)ev =˜
VW(κet)
1+W(κet).(33)
Equation (33) corresponds to the first cumulant. The higher-
order cumulants are obtained by differentiating N(t)ev with
respect to t. The results up to fourth-order read
κev
1[N]=˜
VW(κ)
1+W(κ),(34)
κev
2[N]=˜
VW(κ)
[1 +W(κ)]3,(35)
κev
3[N]=˜
VW(κ)[12W(κ)]
[1 +W(κ)]5,(36)
κev
4[N]=˜
VW(κ){18W(κ)+6[W(κ)]2}
[1 +W(κ)]7.(37)
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VOLODYMYR VOVCHENKO AND VOLKER KOCH PHYSICAL REVIEW C 103, 044903 (2021)
It follows that all cumulant ratios in the EV model depend
exclusively on the value of a single parameter κin the ther-
modynamic limit.
B. Sampling the excluded volume model
To sample particle numbers from the probability distribu-
tion (23) of the EV model we will use a rejection sampling
technique. First we sample Nfrom an auxiliary envelope
distribution ˜
Paux (N;T,V), which we take to be a Poisson
distribution centered around Nev:
˜
Paux (N;T,V)=(Nev )N
N!θ(VbN ).(38)
Here Nev nev (T)Vwith nev defined by Eq. (26). The
theta function ensures that the packing limit is not violated,
i.e., if for a value Nsampled from the Poisson distribution
one has VbN <0, this value is rejected.
To correct for the difference between ˜
Paux (N) and ˜
Pev (N)
we apply rejection sampling for each value of Nsampled
from ˜
Paux. First, we define a weight factor w(N) as the ratio
between the true and auxiliary multiplicity distributions:
w(N)˜
Pev (N)
˜
Paux (N)=(1 bn)φ(T)eμ/T
nev N
.(39)
Here nN/V. The number Nsampled from ˜
Paux (N) shall
be accepted if η<w(N)/wmax where wmax is the maximum
possible value of w(N) and ηis a random number uniformly
distributed in an interval [0,1].
To determine wmax w(Nmax) let us rewrite Eq. (39)as
w(N)=1bn
1bn
ev ebnev
1bnev N
,(40)
whereweusedEq.(24). Nmax is determined from
w(N)/∂N=0. One obtains an equation
ln 1bn
max
1bn
ev ebnev
1bnev =bnmax
1bnmax .(41)
The solution to the above equation is nmax =nev, i.e., the
weight is maximized at the mean value of Nin the thermo-
dynamic limit.4wmax reads
wmax =exp bnev
1bnev Nev .(42)
In numerical calculations it is more convenient to work
directly with normalized weights:
˜w(N)w(N)
wmax
=1bn
1bn
ev N
exp bnev
1bnev (N−Nev).(43)
4Note that w(N)/∂N=0 may generally correspond either to
a minimum or a maximum of w(N). The particular case can be
clarified by analyzing the second derivative of w(N) with respect
to N. We checked that w(N)/∂N=0 corresponds to the maximum
of w(N)ifb>0. Thus, ˜
Paux (N) is an envelope of ˜
Pev (N).
κ
κ
κ
FIG. 2. The behavior of cumulant ratios κ21(black), κ42
(blue), and κ62(red) in a single component grand-canonical ex-
cluded volume model as a function of the reduced volume ˜
VV/b.
Calculations are performed through a Monte Carlo sampling of 108
events (symbols) and analytically via Eqs. (30)and(31) (solid lines).
The horizontal dashed lines correspond to cumulant ratios in the
thermodynamic limit.
The reduced weight in terms dimensionless variables ˜
Vand
κreads
˜w(N)=˜
VN
˜
V−Nev N
exp Nev N−Nev
˜
V−Nev ,(44)
where Nev is given by Eq. (29).
The sampling procedure described here is similar to the
Monte Carlo EV model analysis performed in Ref. [79],
with one distinction. In Ref. [79] an importance sampling
technique was employed, where each generated event was
accepted with a weight ˜w(N). Here, instead, all accepted
events have the same weight, but their sampling involves an
additional rejection step with respect to the weights ˜w(N).
1. Testing the sampling procedure
To test the sampling procedure described above we take
κ=0.03 and perform Monte Carlo sampling for different
values of ˜
V. The choice κ=0.03 is motivated by the fact that
this value is obtained in the EV-HRG model with baryonic ex-
cluded volume b=1fm
3at T=160 MeV and μB=0[62],
therefore, the exercise approximately corresponds to sampling
the baryon multiplicity distribution in the vicinity of the QCD
chiral crossover transition where the EV-HRG model approxi-
mates well the QCD cumulants of the net baryon distribution.
We sam p l e 1 0 8numbers at each value of ˜
Vand calculate
cumulants of the resulting particle number distribution up
to κ6. Figure 2depicts the resulting ˜
Vdependence of the
scaled variance κ21, kurtosis κ42, and hyperkurtosis κ62
(symbols). The solid lines correspond to an analytic cal-
culation of these ratios via a direct summation over all
probabilities [Eqs. (30) and (31)]. The Monte Carlo calcula-
tions agree with the analytic expectations at all studied values
of ˜
V, validating the sampling method.
Figure 2allows also to establish when the condition V
ξ3is reached. This is signalled by the approach of the
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PARTICLIZATION OF AN INTERACTING HADRON PHYSICAL REVIEW C 103, 044903 (2021)
cumulant ratios to their expected values in the thermodynamic
limit [Eqs. (34)–(37)], shown in Fig. 2by the horizontal
dashed lines. Cumulant of a higher order generally requires
larger values of ˜
Vto reach the thermodynamic limit, reflect-
ing the fact that higher cumulants are more sensitive to the
correlation length ξ. We observe that cumulant ratios up sixth
order are within few percentages or less of the thermodynamic
limit for ˜
V20. The cumulants then scale linearly with the
volume for larger values of ˜
V.Thevalue ˜
V20 thus es-
tablishes a lower bound on the physical volume of a single
rapidity slice for the subensemble sampler in Sec. II to be
applicable.
C. EV-HRG model
Having established the baryon multiplicity sampling pro-
cedure in a single-component case, we now turn to the
full model. Quantitative applications to heavy-ion fluctuation
observables require an equation of state with hadron and res-
onance degrees of freedom matched to first-principle lattice
QCD equation of state. For the purposes of net baryon and
net proton fluctuations studied here we employ a variant of
an excluded volume hadron resonance gas (EV-HRG) model
introduced in Refs. [64,80]. The repulsive EV interactions are
introduced for all baryon-baryon and antibaryon-antibaryon
pairs in the EV-HRG model, with a common value bof the
EV parameter for all these pairs.
The pressure in the EV-HRG model is partitioned into a
sum of meson, baryon and antibaryon contributions
p=pM+pB+p¯
B.(45)
Here
pM=Tn
id
M(T),(46)
pB(¯
B)=Tn
id
B(¯
B)(T)expbp
B(¯
B)
T,(47)
where nid
Mand nid
B(¯
B)correspond to cumulative number den-
sities of mesons and (anti)baryons in the ideal HRG limit
(b0):
nid
M,B,¯
B(T)=
iM,B,¯
B
nid
i(T
i),(48)
nid
i(T
i)=dim2
iT
2π2K2(mi/T)eμi/T.(49)
Here μi=qi·μis the chemical potential of particle
species i.
The expression (47) can be rewritten in terms of the
Lambert Wfunction in close to analogy to Eq. (23)ofthe
single-component EV model:
pB(¯
B)=T
bWbn
id
B(¯
B)(T).(50)
The particle number densities of individual hadrons species
are calculated as derivatives of the pressure with respect to the
corresponding chemical potential nev
i=p/∂μi. The mean
multiplicities Niev Vn
ev
iin the grand-canonical EV-HRG
model read
Niev =Vn
id
i(T
i),iM,(51)
Niev =V
b
W[κB(¯
B)]
1+W[κB(¯
B)]
nid
i(T
i)
nid
B(¯
B)(T),
=Vn
id
i(T
i)W[κB(¯
B)]
κB(¯
B){1+W[κB(¯
B)]},iB(¯
B).
(52)
Here κB(¯
B)bn
id
B(¯
B)(T). The mean multiplicities of
mesons coincide with the ideal HRG model baseline. The
multiplicities of (anti)baryons, on the other hand, are sup-
pressed relative to ideal HRG due to EV interactions. This is
quantified by a factor in the r.h.s of Eq. (52). For κB0.03, a
value corresponding to T=160 MeV and μ=0 (see below),
the yields of baryons are suppressed by about 5%.
Equations (51) and (52) define the factor λint
i(T,μ) en-
tering the single-particle distribution functions fi(x,p)for
particle species iin the Cooper-Frye formula, Eqs. (4)
and (15). For mesons, iM, one has λiλint
i(T,μ)=1. For
(anti)baryons
λint
i(T)=W[κB(¯
B)]
κB(¯
B){1+W[κB(¯
B)]},iB(¯
B).(53)
The EV-HRG model has been studied in Refs. [62,63]in
the context of lattice QCD results on diagonal net baryon
susceptibilities and Fourier coefficients of net baryon density
at imaginary chemical potentials. Reasonable description of
these observables at temperatures close to Tpc has been ob-
tained for b=1fm
3, corresponding to κB0.03. We employ
this value of bin the present analysis. Figure 3depicts the
temperature dependence of kurtosis κ42and hyperkurtosis
κ62of net baryon fluctuations at vanishing chemical poten-
tials. The calculations are compared with the lattice data of
Wuppertal-Budapest (blue bands) [81] and HotQCD (green
bands and symbols) [82] Collaborations. The model is in
quantitative agreement with the lattice data for these two
quantities up to T180 MeV. This implies that net-baryon
distribution of the EV-HRG model in this temperature range
closely resembles that of QCD, at least on the level of sixth
leading cumulants. And while this does not necessarily im-
ply that EV interactions is the correct physical mechanism
behind the behavior of net baryon susceptibilities, we view
the EV-HRG model to be an appropriate tool for the purpose
of analysis net baryon and net proton cumulants in heavy-ion
collisions.
The sampling procedure in Sec. III B can be generalized
for the EV-HRG model that has multiple hadron components.
We note that the system in the EV-HRG model is parti-
tioned into three independent subsystems, mesons, baryons,
and antibaryons, see Eq. (45). Therefore, the sampling of
the grand-canonical multiplicities proceeds independently for
each of the three subsystems. The multiplicities of the nonin-
teracting mesons are sampled from the Poisson distribution,
in the same manner as in the ideal HRG. The joint probability
distribution of numbers of all the (anti) baryon species, on the
044903-7
VOLODYMYR VOVCHENKO AND VOLKER KOCH PHYSICAL REVIEW C 103, 044903 (2021)
χ χ
μ
χ χ χ χ
χ χ
μ
FIG. 3. Temperature dependence of net baryon susceptibility ratios (a) χB
4B
2and (b) χB
6B
2(right) evaluated at μB=0intheEV-
HRG model. The blue and green bands/symbols depicts lattice QCD results of Wuppertal-Budapest [81] and HotQCD [82] Collaborations,
respectively.
other hand, reads
˜
Pev
B(¯
B)({Ni};T)=
iB(¯
B)
{[˜
VNB(¯
B)]κi}Ni
Ni!
θ[˜
VNB(¯
B)],
(54)
where, as before, ˜
V=V/band
NB(¯
B)=
iB(¯
B)
Ni,(55)
κi=bn
id
i(T
i).(56)
The auxiliary envelope distribution for the sampling is a
cut multi-Poisson distribution:
˜
Paux
B(¯
B)({Ni};T)=
iB(¯
B)
(Niev )Ni
Ni!
θ[˜
VNB(¯
B)].(57)
The theta function is introduced to avoid exceeding the pack-
ing limit.
Finally, the normalized weight for the rejection sampling
step reads
˜wB(¯
B)({Ni})=
iB(¯
B)1bn
B(¯
B)
1bn
ev
B(¯
B)Ni
×exp bn
ev
B(¯
B)
1bn
ev
B(¯
B)NiNev
i
=1bn
B(¯
B)
1bn
ev
B(¯
B)NB(¯
B)
×exp bn
ev
B(¯
B)
1bn
ev
B(¯
B)NB(¯
B)Nev
B(¯
B).(58)
Here nB(¯
B)NB(¯
B)/Vand nev
B(¯
B)≡NB(¯
B)ev/V.
The algorithm for sampling the multiplicity distribution of
the EV-HRG model is the following:
(1) Sample the multiplicities {Ni}of all baryons from the
cut multi-Poisson distribution (57).
(2) Generate a random number ηfrom the uniform distri-
bution on the unit interval (0, 1). If η< ˜wB({Ni}), then
go to the next step. Otherwise, go back to step (1).
(3) Repeat steps 1 and 2 in the same fashion to sample the
multiplicities of antibaryons.
(4) Sample multiplicities of mesons from the multi-
Poisson distribution of the ideal HRG model.
The procedure for generating the multiplicity distribution
in the EV-HRG model in various rapidity slices that are
constrained by global conservation of conserved charges, as
described in Secs. II and III, is implemented in an extended
version of the Thermal-FIST package [83]. We use this pack-
age in all our calculations.
IV. FLUCTUATIONS IN HEAVY-ION COLLISIONS
AT LHC ENERGIES
A. The setup
We apply our formalism to study the rapidity acceptance
dependence of fluctuation observables in heavy-ion collisions.
To proceed we need to specify the partition of the space-time
rapidity ηsaxis into fireballs as well as the ηsdependence of
thermal parameters and volume.
Let us consider Pb-Pb collisions at sNN =2.76
TeV. At midrapidity the chemical freeze-out is character-
ized by vanishing chemical potentials, temperature values
T155–160 MeV and freeze-out volume per rapidity unit
dV/dy 4000–5000 fm3[70,84]. The simplest possibility
then is to assume boost invariance across the entire space-time
rapidity range. In this scenario, the mean total number of
particles of given kind in full space, say charged multiplicity
N4π
ch or number of (anti)baryons N4π
B(¯
B), is then simply given by
multiplying the rapidity density at y=0 by the total (space-
time) rapidity coverage ηmax
s
s
max
s, for example
N4π
ch =2ηmax
sdNch /dy|y=0(59)
for the charged multiplicity.
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PARTICLIZATION OF AN INTERACTING HADRON PHYSICAL REVIEW C 103, 044903 (2021)
The question is how to determine ηmax
s. One possibil-
ity is to equate this quantity to the beam rapidity ybeam
ln[sNN/(2mN)] 8. However, such an estimate is too crude
and will overestimate the actual N4π
ch . The rapidity density
of charged multiplicity measured at sNN =2.76 TeV by
the ALICE Collaboration [85] is consistent with a Bjorken
plateau only in a rapidity range |y|2, whereas at higher
rapidities dNch /dy drops. The entire measured rapidity depen-
dence of dNch /dy is described well by Gaussian with a width
σ=3.86 ±0.05 [85]. We can use this fact to relate N4π
ch and
dNch /dy|y=0in an empirical way:
N4π
ch =dy exp y2
2σ2dNch /dy|y=0
(9.68 ±0.13) dNch/dy|y=0.(60)
Here the error is due to the uncertainty in the value of σ.
Comparing Eq. (60) with (59) one obtains
ηmax
s=(4.84 ±0.07) for sNN =2.76 TeV.(61)
We shall use a value ηmax
s=4.8 for Pb-Pb collisions at
sNN =2.76 TeV in the following. We take T=160 MeV
and μ=0 for all rapidities. With this choice the model ac-
curately reproduces the rapidity densities of various hadron
species at |y|2, where the Bjorken plateau is observed in
the data [85], and provides an accurate estimate of the total
hadron multiplicities in full phase space. As we have assumed
boost invariance across the entire space-time rapidity range,
the model does not describe rapidity distributions at |y|2
and thus should not be applied to calculate observables at
large rapidities. However, given the fact that the model does
reproduce the 4πcharged multiplicity, it is suitable to describe
the influence of global conservation laws on observables com-
puted around midrapidity, |y|2. In the following we focus
on these regions around midrapidity. In a more general study
the assumption of boost invariance can be relaxed to incor-
porate a more accurate description of the forward-backward
rapidity regions.
Our model yields a vanishing total net baryon number in
the full space. Essentially, this means that we neglect baryons
from the fragmentation regions. This is similar to a recent
study [86] performed in the framework of the ideal HRG
model. There it was estimated that the effect of fragmentation
baryons at the LHC does not exceed 6% for the sixth-order net
proton cumulant. We therefore expect the possible influence
of the fragmentation region baryons on our results to be small.
We partition the space-time rapidity axis uniformly into
slices of width ηs=0.1. With ηmax
s=4.8 this implies a
total of 96 slices. The volume of a single slice in 5% most
central collisions is Vi=dV/dyηs400 fm3. This value
is sufficiently large to ensure that the thermodynamic limit
is reached in each of the subvolumes and thus the require-
ments for the validity of the sampling procedure described in
Sec. II satisfied. This also implies that all intensive quantities,
such as cumulant ratios, are independent of the value of Vi
in this regime, i.e., Vican be scaled up and down as long
as Viξ3. This feature is very useful for the Monte Carlo
sampling procedure. Indeed, as the statistical error in higher-
order cumulants increases with the volume, this error can
be minimized by choosing the volume as small as possible.
According to Fig. 2,avalueVi=20 fm3is sufficiently large
to capture all the relevant physics for cumulants up to sixth
order. For this reason we take Vi=20 fm3in our Monte Carlo
simulations and then linearly scale up the resulting cumulants
to match the volume Vi400 fm3in 0–5% Pb-Pb collisions.
We take the EV-HRG model with b=1fm
3. As discussed
in Sec. III B, this model provides a reasonable description of
high-order net baryon susceptibilities from lattice QCD. The
grand-canonical distribution of hadron multiplicities can be
efficiently sampled following the rejection sampling based
algorithm described in Sec. III B.WetakeTi=160 MeV
and vanishing chemical potentials, μi=0, uniformly for all
subvolumes along the rapidity axis.
For the net baryon cumulants we shall take into account
only the exact conservation of baryon number, which is ex-
actly vanishing, B=0, in all events. In principle, one should
also take into account the exact conservation of electric charge
and strangeness. However, as discussed in Refs. [77,87], the
influence of these conserved charges on net baryon cumulants
is negligible at LHC energies. Their influence on net proton
cumulants is more sizable [87] but still expected to be sublead-
ing compared to baryon number conservation. Neglecting the
exact conservation of electric charge and strangeness allows
to significantly speed up the Monte Carlo event generation,
as this strongly reduces the rejection rate associated with
exact conservation of multiple conserved charges and allows
to gather enough statistics within a reasonable time period
to accurately evaluate cumulants up to sixth order. We do
analyze the influence of electric charge and strangeness con-
servations on second-order cumulants of various net-particle
distributions in Sec. IV F.
Once the joint hadron multiplicity distribution from all
the subvolumes has been sampled, we generate the hadron
momenta, independently for each hadron. To that end we em-
ploy the blast-wave model [88], which provides a reasonable
description of bulk particle’s pTspectra at LHC [89]. The
model corresponds to a particlization of a cylindrically shaped
fireball (r2
x+r2
yrmax), at a constant value of the longitu-
dinal proper time τ=τ0. The longitudinal collective motion
obeys the Bjorken scaling while the radial velocity scales with
the transverse radius, βrrn
. This corresponds to a flow
profile uμ(x)=(cosh ρcosh ηs,sinh ρ
e,cosh ρsinh ηs),
where ρ=tanh1βr, and βr=βsζnis the transverse flow
velocity profile. Here ζr/rmax is a normalized transverse
radius. The momentum distribution of hadron species with
mass memerging from a jth space-time rapidity subvolume
is given by
dN
pTdp
Tdy mTηmax
j
ηmin
j
dηcosh(yη)1
0
ζdζ
×emTcosh ρcosh(yη)
TI0pTsinh ρ
T,(62)
Here mT=p2
T+m2is the transverse mass, y=1
2log ωppz
ωp+pz
is the longitudinal rapidity, and I0is a modified Bessel
function.
044903-9
VOLODYMYR VOVCHENKO AND VOLKER KOCH PHYSICAL REVIEW C 103, 044903 (2021)
β
FIG. 4. The pTspectrum of protons in 5% most central Pb-Pb
collisions at sNN =2.76 TeV at midrapidity (|y|<0.5), as mea-
sured by the ALICE Collaboration (red symbols) [89] and given
by the blast-wave model (T=160 MeV, βs=0.77, n=0.36) with
(dashed line) or without (solid line) pTshape modification due to
resonance decays. The normalization factor of the blast-wave model
distribution has been fitted to the data.
The sampling of momenta from the distribution (62)is
readily implemented in the Thermal-FIST package that we
employ. We are only left with specifying the values of the
blast-wave model parameters βsand n. For this purpose we
make use of the result of a recent study [90], where the blast-
wave model was fitted to experimental data of the ALICE
Collaboration with account for modification of pTspectra
due to resonance decays. For most central Pb-Pb collisions
at sNN =2.76 TeV one has βs=0.77 and n=0.36, which
gives a reasonable description of bulk hadron pTspectra.5One
should note that Ref. [90] has extracted a temperature value
of T=149 MeV from the pTspectra fits rather rather than
the T=160 MeV value that we use here for fluctuations.
However, the T=160 MeV value shows a similarly good
agreement of the blast-wave model proton pTspectrum with
the data, as the one shown in Ref. [90]forT=149 MeV.
Figure 4compares the shape of the pTspectrum of protons as
observed in the data (red symbols) [89] and predicted by the
blast-wave model [Eq. (62)] with T=160 MeV, βs=0.77,
and n=0.36. The dashed line in Fig. 4corresponds to blast-
wave model spectrum which includes the modification of the
proton pTspectrum due to resonance decays. This effect,
computed here via Monte Carlo simulations of decays, only
slightly modifies the momentum distribution.
In the final step of the Monte Carlo event generation
procedure we perform all strong and electromagnetic decays
until only stable hadrons are left. We generate 1010 events in
total6and study the rapidity dependence of various fluctuation
observables. As our analysis only concerns the baryons, to
5One notable exception here are low-pTpions that are significantly
underestimated by the blast-wave model. These pions have no influ-
ence on the net baryon fluctuations that we study here.
6Such a large number of events is needed to compute cumulants of
sixth order with a sufficiently small statistical uncertainty.
speed-up the Monte Carlo procedure we omit all the primor-
dial mesonic species [step (4) in the algorithm of Sec. III B],
as these do not affect the behavior of (anti)baryons in any way
within the EV-HRG model that we use.
B. Rapidity acceptance dependence of net baryon cumulants
We start with the rapidity acceptance dependence of net
baryon number cumulants. First, we look at the second cu-
mulant of net baryon fluctuations normalized by the Skellam
distribution baseline, κ2[B¯
B]/B+¯
B. This type of ratio
has been extensively studied at LHC energies by the ALICE
Collaboration [15] for net protons. This ratio equals unity
for the case of a grand-canonical ideal HRG model at any
temperature and chemical potentials. The ratio, however, does
exhibit small deviations from unity in the EV-HRG model
that we use. For instance, at T=160 MeV and μ=0the
grand-canonical value reads
κ2[B¯
B]
B+¯
Bev,gce
0.94.(63)
We note that it is currently challenging to directly com-
pute κ2[B¯
B]/B+¯
Bin lattice QCD, as the denominator
B+¯
Bis not a conserved quantity. Given the good agreement
of the EV-HRG model with lattice QCD for the higher-order
cumulants, however, we expect QCD to have a similar value
to the one given by Eq. (63). An interesting question now is to
determine if and how the grand-canonical value in Eq. (63)is
reflected in heavy-ion data.
The top panel of Fig. 5depicts the rapidity acceptance
Yacc dependence of κ2[B¯
B]/B+¯
Bthat results from
the Monte Carlo sampling within the subensemble sampler.
Here the acceptance is centered at midrapidity, i.e., particles
with rapidity |y|<Yacc/2 are accepted. The red symbols
depict the full result which includes the distortion of hadron
momenta due to thermal smearing at particlization and sub-
sequent resonance decays. The black symbols, on the other
hand, correspond to the case when these effects are neglected,
i.e., the final kinematical rapidity is taken to be equal to
the space-time rapidity coordinate at particlization, yηs.
Comparing the two allows to establish the effect of thermal
smearing and resonance decays. We observe that the Monte
Carlo results in the no-smearing case agree with the analytic
expectations of the SAM (black lines). The SAM baseline for
κ2[B¯
B]/B+¯
Bis given by [77,87]
κ2[B¯
B]
B+¯
BSAM
=(1 α)κ2[B¯
B]
B+¯
Bev,gce
.(64)
Here αis a fraction of the total volume which corresponds to
the acceptance |ηS|<Yacc/2:
α=Yacc
2ηmax
s
.(65)
The agreement of the Monte Carlo points with the SAM is
the expected result and serves as a validation of the sampling
procedure.
Notable differences between the red (momentum rapidity)
and black (space-time rapidity) points in Fig. 5appear when
044903-10
PARTICLIZATION OF AN INTERACTING HADRON PHYSICAL REVIEW C 103, 044903 (2021)
κ 〈 〉
σ
κ κ
κ κ
Δ
FIG. 5. Rapidity acceptance dependence of cumulant ratios
κ2Skellam
2(top), κ42(middle), and κ62(bottom) of net baryon
distribution in 0–5% central Pb-Pb collisions at the LHC in an ex-
cluded volume HRG model matched to lattice QCD. The symbols
depict the results of the Monte Carlo event generator, the full black
squares correspond to neglecting the momentum smearing, the open
red triangles include the thermal smearing at particlization, and the
full red circles incorporate the smearing due to both the thermal
motion and resonance decays. The dashed black lines correspond
to the predictions of the SAM framework [77]. The solid red lines
correspond to adding a Gaussian rapidity smearing on top of the
SAM. The dashed blue lines correspond to the binomial acceptance,
which describes the effects of baryon number conservation in the
ideal HRG model limit.
the acceptance is sufficiently small, Yacc 1. This is a con-
sequence of the dilution of momentum-space correlations due
to thermal motion. For a very small acceptance, Yacc 1,
the results converge to the baseline given by the binomial dis-
tribution, (κ2[B¯
B]/B+¯
B)binom =1α, shown in Fig. 5
by the dashed blue line. The binomial distribution corresponds
to an independent acceptance for all (anti)particles and de-
scribes the cumulants of net baryon distribution in the ideal
HRG model, where the global baryon conservation constitutes
the only source of correlations between baryons [37,40,91].
The additional momentum smearing due to decays of
baryonic resonances is virtually negligible, being completely
overshadowed by the thermal smearing. This is true not only
for the variance, but also for the kurtosis and hyperkurtosis,
as seen by comparing the red points (thermal smearing +
resonance decays) with the open red triangles (thermal smear-
ing only) in all three panels of Fig. 5. To understand this
behavior one can consider, e.g., decays Nπ. In such a
decay the released momentum is split evenly between the two
decay products in the resonance center-of-mass frame. This
leads to a larger velocity (rapidity) smearing of the lighter
decay product—the pion—whereas the velocity (rapidity) of
nucleon is less affected. We conclude that the smearing of
baryon fluctuations due to resonance decays can be safely
neglected. Note that this statement does not extend to (net-
)particle fluctuations involving lighter hadrons such as pions
or kaons. There the effect of resonance decays should be
carefully taken into account.
In the Appendix we develop a simplified analytic model
to take into account the momentum smearing in net baryon
cumulants. There we assume that the shift in kinematical
rapidity relative to the space-time rapidity is described for
all baryon species by a Gaussian smearing. The red lines
in Fig. 5exhibit the results of such a simplified calculation.
For a Gaussian width of σy=0.3 the simplified calculations
agree very well with the full Monte Carlo results. Therefore,
this model can be used to predict the rapidity dependence
of pT-integrated net baryon cumulants without invoking the
time-consuming Monte Carlo event generator.
Net-baryon fluctuations in a sufficiently large rapidity ac-
ceptance Yacc 1 are accurately described by the analytical
SAM baseline (64). This conclusion is important, because
the SAM makes the connection between the grand-canonical
susceptibilities and cumulants constrained by global conser-
vation laws without any additional assumptions regarding the
equation of state. In our previous work [77] where the SAM
is introduced, we argued that the SAM is reliable for rapidity
acceptances Yacc 1, where the distortion due to thermal
smearing is expected to be subleading. The results obtained
in the present work using the EV-HRG model explicitly con-
firm this. The grand-canonical (κ2[B¯
B]/B+¯
B)ev,gce can
therefore be extracted from data by fitting the αdependence of
net-baryon fluctuations measured in sufficiently large rapidity
acceptance via Eq. (64).
We turn now to the kurtosis of net baryon fluctuations,
κ4[B¯
B]2[B¯
B]. In the GCE this quantity coincides with
the corresponding ratio χB
4B
2of the susceptibilities. The
EV-HRG model at LHC energies yields the following value
χB
4
χB
20.66.(66)
This is in agreement with lattice QCD continuum estimates
of HotQCD (χB
4B
2=0.65 ±0.03) [82] and Wuppertal-
Budapest (χB
4B
2=0.69 ±0.03) [81] Collaborations, taken
at the same temperature T=160 MeV.
The rapidity acceptance dependence of κ4[B¯
B]2[B
¯
B] is depicted in the middle panel of Fig. 5. The qualitative
behavior of the kurtosis largely mirrors that of the variance. In
the absence of momentum smearing, the Monte Carlo results
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VOLODYMYR VOVCHENKO AND VOLKER KOCH PHYSICAL REVIEW C 103, 044903 (2021)
agree with the analytical SAM baseline of Ref. [77]:
κ4[B¯
B]
κ2[B¯
B]SAM
=(1 3αβ)χB
4
χB
23αβ χB
3
χB
22
.(67)
Here β1α. At LHC energies one has χB
3B
2=0, thus,
the second term in Eq. (67) does not contribute.
With thermal smearing and resonance decays included, the
kurtosis deviates from the SAM baseline for Yacc 1 and for
Yacc 1 tends to the binomial distribution baseline, which
at the LHC energies reads (κ4[B¯
B]2[B¯
B])binom
LHC =1
3αβ.ForYacc 1 the full result is described well by the
SAM (67).
Finally, we look at the behavior of the hyperkurtosis,
κ6[B¯
B]2[B¯
B]. Lattice QCD predicts a sign change of
the grand-canonical hyperkurtosis at μ=0 in the vicinity
of the pseudocritical temperature (Fig. 3). This qualitative
feature has been argued to be a signature of the QCD
chiral crossover transition [39]. If this interpretation is cor-
rect, a corresponding measurement of κ6[B¯
B]2[B¯
B]
in heavy-ion collisions at the LHC can potentially serve as the
first experimental signature of that transition. The EV-HRG
model reproduces the available lattice QCD data for χB
6B
2
and gives the following value at T=160 MeV:
χB
6
χB
2−0.27.(68)
This agrees within errors with the continuum estimate of the
Wuppertal-Budapest Collaboration, χB
6B
2=−0.26 ±0.17
[81]aswellaswithNτ=8 results of the HotQCD Collabora-
tion [82] shown in Fig. 3. It should be noted that the EV-HRG
model itself does not incorporate the chiral crossover transi-
tion but predicts the negative hyperkurtosis at T160 MeV
as a consequence of repulsive interactions between baryons.
Here we do not discuss whether the negative χB
6B
2seen in
lattice QCD is indeed a signature of the chiral crossover but
rather use the EV-HRG model to establish how a negative
χB
6B
2would be reflected in heavy-ion observables.
The lower panel of Fig. 5shows the rapidity acceptance
dependence of the hyperkurtosis. In the absence of momentum
smearing, the Monte Carlo results are described by the analyt-
ical SAM baseline, which for LHC energies, i.e., for μ=0,
reads [77]
κ6[B¯
B]
κ2[B¯
B]SAM
LHC =[15αβ(1 αβ )]χB
6
χB
2
10α(1 2α)2βχB
4
χB
22
.(69)
The hyperkurtosis, in the absence of momentum smearing,
is sensitive to the grand-canonical value (68) in acceptances
up to Yacc 1.5. For larger acceptances baryon conserva-
tion dominates, making it difficult to disentangle between
the EV-HRG model and the binomial baseline, given by
(κ6[B¯
B]2[B¯
B])binom
LHC =115αβ(1 3αβ ). This was
already pointed out in our previous study [77]. The thermal
smearing distorts the signal at small acceptances, Yacc 0.5,
where the hyperkurtosis is closer to the binomial distribution
baseline than it is to the SAM. At 0.5Yacc 1.5, on the
other hand, κ6[B¯
B]2[B¯
B] is overshadowed neither by
the thermal smearing nor by the baryon number conservation.
We, therefore, argue that a measurement of a hyperkurtosis,
which is negative over this entire range may be interpreted
as a signal of the chiral crossover, provided that the negative
grand-canonical hyperkurtosis seen in lattice QCD is indeed a
consequence of chiral criticality.7
C. Net baryon vs net proton fluctuations
Our discussion has so far been restricted to cumulants
of net baryon distribution. However, experiments typically
cannot measure all baryons, in particular the measurement of
neutrons is extremely challenging. For this reason one usually
uses net protons as a proxy for net baryons. It is natural to
expect net protons to carry at least some information about
net baryon fluctuations. In fact, as shown by Kitazawa and
Asakawa [92,93], under the assumption of isospin randomiza-
tion at late stages of heavy-ion collisions, one can reconstruct
the cumulants of net baryon distribution from the measured
factorial moments of proton and antiproton distributions.
However, these considerations do not imply that ratios
of proton cumulants can be used directly in place of the
corresponding ratios of baryon cumulants, something which
has nevertheless been employed in a number of works in the
literature [28,29,94]. The proton and baryon cumulant ratios
do coincide in the free hadron gas limit, where they both
trivially reduce to the Skellam baseline, but this does not hold
in general case.
Large differences between net proton and net baryon cu-
mulant ratios were reported earlier in Ref. [31] for the van der
Waals HRG model in the GCE. Here we study these differ-
ences in the framework of the EV-HRG model constrained to
lattice data and include effects of global baryon conservation
and momentum smearing.
Figure 6depicts the rapidity acceptance dependence of
net baryon (black squares) and net proton (blue symbols)
cumulant ratios κ2Skellam
2,κ42, and κ62calculated us-
ing Monte Carlo sampling within the SAM. The calculations
incorporate the thermal smearing and resonance decays. The
results reveal large differences between net proton and net
baryon cumulants ratios. Net proton cumulant ratios are con-
siderably closer to the Skellam baseline of unity. This can
be understood in the following way. By taking only a subset
of baryons—the protons—one dilutes the total signal due to
baryon correlations. This leads to a smaller deviation of cumu-
lants from Poisson statistics—the limiting case of vanishing
correlations.
The large difference between net proton and net baryon cu-
mulants clearly indicates that direct comparison between the
two is not justified. It is interesting that net proton cumulant
ratios cross the grand-canonical value of the corresponding
7We note that at Yacc 1 baryon number conservation leads to
a negative hyperkurtosis also in the case of the ideal HRG, see
the dashed blue line in Fig. 5. Thus it is essential to establish a
negative κ6[B¯
B]2[B¯
B]atYacc 1 for the chiral crossover
interpretation to be valid.
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PARTICLIZATION OF AN INTERACTING HADRON PHYSICAL REVIEW C 103, 044903 (2021)
κ κκ κ
κ κ
Δ
FIG. 6. Rapidity acceptance dependence of net baryon (red
squares) and net proton (blue circles) cumulant ratios κ2Skellam
2
(top), κ42(middle), and κ62(bottom) in 0–5% central Pb-Pb
collisions at the LHC in an excluded volume HRG model matched
to lattice QCD. The open blue diamonds correspond to net proton
cumulants evaluated from net baryon cumulants using a binomial-
like method of Kitazawa and Asakawa [92,93]. The solid lines
correspond to the analytical predictions of the SAM framework with
(solid) and without (dashed) Gaussian rapidity smearing.
net baryon ratios in the grand-canonical limit (horizonal lines
in Fig. 6) for a sufficiently large acceptance. This, for instance,
takes place at Yacc 1.4for