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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 [1–4]. 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

[13–16] and off-diagonal [17–19], as well as higher-order

ﬂuctuation measures [11,20–22].

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 [23–32]. 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 [35–37], re-

quiring large corrections to the grand-canonical distributions.

Other mechanisms include volume ﬂuctuations [38–40], 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 [49–52].

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 [53–56]. More advanced procedures incorporate

exact conservation of the QCD conserved charges and/or

energy-momentum [57–61], 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,η

s).

Here τ=t2−r2

zand ηs=1

2ln t+rz

t−rzare 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,μ

Q,μ

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 μi−pμuμ(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.

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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>η

min

j. Furthermore, one has ηmin

j=

ηmax

j−1for 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

j,ηmax

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,μ)∼eVj[or, equivalently,

ln ZiHRG (T,Vj,μ)∼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 T→Tiand μ→μi:

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

j,ηmax

j]

dσμ(x)pμfj,i(x,p) (15)

with

fj,i(x,p)=diλint

i(Tj,μj)

(2π)3exp μj,i−pμuμ(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 [70–72] 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

4/χB

2and the hyperkurtosis

χB

6/χB

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,μ)=∞

N=0

[(V−bN )φ(T)eμ/T]N

N!θ(V−bN ).

(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,μ)=[(V−bN )φ(T)eμ/T]N

N!θ(V−bN ).

(23)

In the thermodynamic limit, N→∞, the particle density

nev (T,μ)=Nev /Vis determined by the maximum term in

Eq. (21). Maximizing ˜

Pev with respect to Ngives a transcen-

dental equation deﬁning nev (T,μ):

bnev

1−bnev ebnev

1−bnev =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

1−bnev =Wbφ(T)eμ/T,(25)

or

nev (T,μ)=W[bφ(T)eμ/T]

b{1+W[bφ(T)eμ/T]}.(26)

The pressure reads

pev(T,μ)=Tn

ev

1−bnev =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,μ) and the excluded volume parameter bby

two dimensionless quantities, namely a reduced volume ˜

V≡

V/band a parameter κ≡bφ(T)eμ/Tthat characterizes the

strength of repulsive interactions. The probability distribution

(23) then takes the form

˜

Pev (N;˜

V,κ)=[( ˜

V−N)κ]N

N!θ(˜

V−N).(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)k−1(k−1)! Bn,k(N,...,Nn−k+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 κ→κet:

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)[1−2W(κ)]

[1 +W(κ)]5,(36)

κev

4[N]=˜

VW(κ){1−8W(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,μ), which we take to be a Poisson

distribution centered around Nev:

˜

Paux (N;T,V,μ)=(Nev )N

N!θ(V−bN ).(38)

Here Nev ≡nev (T,μ)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 V−bN <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 n≡N/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)=1−bn

1−bn

ev ebnev

1−bnev N

,(40)

whereweusedEq.(24). Nmax is determined from

∂w(N)/∂N=0. One obtains an equation

ln 1−bn

max

1−bn

ev ebnev

1−bnev =bnmax

1−bnmax .(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

1−bnev Nev .(42)

In numerical calculations it is more convenient to work

directly with normalized weights:

˜w(N)≡w(N)

wmax

=1−bn

1−bn

ev N

exp bnev

1−bnev (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/κ

κ/κ

FIG. 2. The behavior of cumulant ratios κ2/κ1(black), κ4/κ2

(blue), and κ6/κ2(red) in a single component grand-canonical ex-

cluded volume model as a function of the reduced volume ˜

V≡V/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)=˜

V−N

˜

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 κ2/κ1, kurtosis κ4/κ2, and hyperkurtosis κ6/κ2

(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,μ),(46)

pB(¯

B)=Tn

id

B(¯

B)(T,μ)exp−bp

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

(b→0):

nid

M,B,¯

B(T,μ)=

i∈M,B,¯

B

nid

i(T,μ

i),(48)

nid

i(T,μ

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,μ).(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,μ

i),i∈M,(51)

Niev =V

b

W[κB(¯

B)]

1+W[κB(¯

B)]

nid

i(T,μ

i)

nid

B(¯

B)(T,μ),

=Vn

id

i(T,μ

i)W[κB(¯

B)]

κB(¯

B){1+W[κB(¯

B)]},i∈B(¯

B).

(52)

Here κB(¯

B)≡bn

id

B(¯

B)(T,μ). 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, i∈M, one has λiλint

i(T,μ)=1. For

(anti)baryons

λint

i(T,μ)=W[κB(¯

B)]

κB(¯

B){1+W[κB(¯

B)]},i∈B(¯

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 κ4/κ2and hyperkurtosis

κ6/κ2of 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

4/χB

2and (b) χB

6/χB

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,μ)=⎛

⎝

i∈B(¯

B)

{[˜

V−NB(¯

B)]κi}Ni

Ni!⎞

⎠θ[˜

V−NB(¯

B)],

(54)

where, as before, ˜

V=V/band

NB(¯

B)=

i∈B(¯

B)

Ni,(55)

κi=bn

id

i(T,μ

i).(56)

The auxiliary envelope distribution for the sampling is a

cut multi-Poisson distribution:

˜

Paux

B(¯

B)({Ni};T,μ)=⎡

⎣

i∈B(¯

B)

(Niev )Ni

Ni!⎤

⎦θ[˜

V−NB(¯

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})=

i∈B(¯

B)1−bn

B(¯

B)

1−bn

ev

B(¯

B)Ni

×exp bn

ev

B(¯

B)

1−bn

ev

B(¯

B)Ni−Nev

i

=1−bn

B(¯

B)

1−bn

ev

B(¯

B)NB(¯

B)

×exp bn

ev

B(¯

B)

1−bn

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<η

s<η

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/dyηs≈400 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 Vi≈400 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, βr∝rn

⊥. This corresponds to a ﬂow

proﬁle uμ(x)=(cosh ρcosh ηs,sinh ρ

e⊥,cosh ρsinh ηs),

where ρ=tanh−1βr, and βr=βsζnis the transverse ﬂow

velocity proﬁle. Here ζ≡r⊥/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(y−η)1

0

ζdζ

×e−mTcosh ρcosh(y−η)

TI0pTsinh ρ

T,(62)

Here mT=p2

T+m2is the transverse mass, y=1

2log ωp−pz

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, y≡ηs.

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

κ2/κSkellam

2(top), κ4/κ2(middle), and κ6/κ2(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 =1−α, 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]/κ2[B−¯

B]. In the GCE this quantity coincides with

the corresponding ratio χB

4/χB

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

4/χB

2=0.65 ±0.03) [82] and Wuppertal-

Budapest (χB

4/χB

2=0.69 ±0.03) [81] Collaborations, taken

at the same temperature T=160 MeV.

The rapidity acceptance dependence of κ4[B−¯

B]/κ2[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

2−3αβ χB

3

χB

22

.(67)

Here β≡1−α. At LHC energies one has χB

3/χB

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]/κ2[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]/κ2[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]/κ2[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

6/χB

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

6/χB

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

6/χB

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

6/χB

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 =[1−5αβ(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]/κ2[B−¯

B])binom

LHC =1−15αβ(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]/κ2[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 κ2/κSkellam

2,κ4/κ2, and κ6/κ2calculated 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]/κ2[B−¯

B]atYacc 1 for the chiral crossover

interpretation to be valid.

044903-12

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 κ2/κSkellam

2

(top), κ4/κ2(middle), and κ6/κ2(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