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Citation: Sinyukov, Y.; Shapoval, V.;
Adzhymambetov, M. Space–Time
Structure of Particle Emission and
Femtoscopy Scales in Ultrarelativistic
Heavy-Ion Collisions. Universe 2023,
9, 433. https://doi.org/10.3390/
universe9100433
Academic Editors: Khusniddin
Olimov, Fu-Hu Liu and Kosim
Olimov
Received: 29 April 2023
Revised: 12 July 2023
Accepted: 26 September 2023
Published: 28 September 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
universe
Article
Space–Time Structure of Particle Emission and Femtoscopy
Scales in Ultrarelativistic Heavy-Ion Collisions
Yuri Sinyukov 1,2,*, Volodymyr Shapoval 1and Musfer Adzhymambetov 1
1
Department of High-Density Energy Physics, Bogolyubov Institute for Theoretical Physics, 14b Metrolohichna
Street, 03143 Kyiv, Ukraine; shapoval@bitp.kiev.ua (V.S.); adzhymambetov@gmail.com (M.A.)
2Faculty of Physics, Warsaw University of Technology, 75 Koszykowa Street, 00-662 Warsaw, Poland
*Correspondence: sinyukov@bitp.kiev.ua
Abstract:
The analysis of the spatiotemporal picture of particle radiation in relativistic heavy-ion
collisions in terms of correlation femtoscopy scales, emission, and source functions allows one to
probe the character of the evolution of the system created in the collision. Realistic models, such
as the integrated hydrokinetic model (iHKM), used in the present work, are able to simulate the
entire evolution process of strongly interacting matter produced in high-energy nuclear collisions.
The mentioned model describes all the stages of the system’s evolution, including thermalisation
and hydrodynamisation, which can help researchers figure out the specific details of the process
and better understand the formation mechanisms of certain observables. In the current paper, we
investigated the behaviour of the pion and kaon interferometry radii and their connection with
emission functions in ultrarelativistic heavy-ion collisions at the Large Hadron Collider within iHKM.
We focused on the study of the emission time scales at different energies for both particle species
(pions and kaons) aiming to gain deeper insight into relation of these scales and the peculiarities of
the mentioned system’s collective expansion and decay with the experimentally observed femtoscopy
radii. One of our main interests was the problem of the total system’s lifetime estimation based on
the femtoscopy analysis.
Keywords: kaon; pion; femtoscopy radius; emission function; emission time; particlisation
1. Introduction
The investigation of the evolution process character of the extremely small, hot, dense,
and rapidly expanding systems created in relativistic heavy-ion collisions has always been
of great interest to researchers since the first studies of these phenomena started. The
comprehensive analysis demonstrates that, soon after the smashing of the two colliding
nuclei (each one of them can be initially thought of as a set of nucleons), the strongly coupled
quark–gluon system is formed. Being at first non-equilibrium, this system some time later
becomes thermalised and continues expanding as a whole liquid-like piece of matter [
1
–
4
].
Its expansion at this stage can be described within a relativistic hydrodynamics approach.
As the system expands, it gradually cools down, becomes dedensified, and begins losing
equilibrium. Below some temperature or energy density (usually the temperatures close
to 150–160 MeV are spoken of), the system disintegrates, and its parts one by one are
transformed into sets of hadrons. However, the produced hadrons do not likely become
free immediately after the fireball decay. The latter would imply a sudden switching from
a near-zero mean free path length to a near-infinity one, which seems hardly feasible. The
particles would rather continue interacting intensively and evolving collectively, as single
hadronic medium for some time. Therefore, to estimate the overall lifetime of the system
created after the collision, one should in theory account for the hadronic phase of the
matter evolution and try to find out the time scale (at least approximate), during which the
hadronic medium stays connected enough to be considered a unified system.
Universe 2023,9, 433. https://doi.org/10.3390/universe9100433 https://www.mdpi.com/journal/universe
Universe 2023,9, 433 2 of 17
In the literature, however, there are two different approaches to the stated problem.
The first one supposes that, right after the hadronisation (and maybe a very short hadronic
stage), the so-called “freeze-out” takes place, so that the chemical composition of the sys-
tem and, almost simultaneously, the particle momentum spectra are frozen and no longer
change, except for decays of resonances. Such an assumption goes back to the pioneer-
ing works by Fermi, Pomeranchuk, and Landau [
5
–
7
] and is typical for thermal models
with statistical hadronisation [
8
–
12
], hydro-inspired freeze-out parametrisations, e.g., Cra-
cow [
13
,
14
], Buda–Lund [
15
], and blast wave [
16
–
18
], and hydrodynamics models with
statistical hadronisation [
19
–
32
]. The latter usually employ the Cooper–Frye formalism [
33
]
or its generalisations [
34
] to hadronise the system at the freeze-out hypersurface. There are
various computer codes, developed for the calculations of statistical hadronisation in differ-
ent models, e.g., SHARE [
35
,
36
], THERMUS [
37
], used for the extraction of thermodynamic
parameters from the fits to the ratios of particle yields, or THERMINATOR [
38
,
39
], used
as an event generator, producing particles based on the thermodynamic parameter values
specified at a certain freeze-out hypersurface.
The second approach suggests the existence of a long “afterburner” stage of the
collision (lasting about 5–10 fm/
c
or until the temperature drops to 80–120 MeV), which
plays an important role in the measured observables’ formation. The corresponding
theoretical description of heavy-ion collisions in this case is carried out within various
hybrid (hydro+cascade) and full evolutionary models, aiming to simulate the entire process
of the created system’s evolution, from the initial state formation to the final hadron
cascade stage [
40
–
51
]. The study presented in this paper was carried out in the most-
developed complete model—the integrated hydrokinetic model (iHKM)—which includes
as one of the stages of the system’s evolution also a description of the process of the
thermalisation/hydrodynamisation of initially non-thermal matter [52].
The existing experimental data on
pT
spectra do not allow giving a clear preference to
one of these approaches and yield the estimates for the kinetic freeze-out temperature in a
very wide range (such as 100–140 MeV [
29
]), depending on the particle species selected
for the analysis, momentum cuts, the inclusion/exclusion of resonance decays, and so on.
The results on
K∗
resonance yields and
K∗/K
yields’ ratios [
53
–
55
] speak in favour of the
afterburner scenario; the experimental
K∗/K
ratio values decrease when one goes from
peripheral to central collisions, in contrast with almost flat dependency (keeping close to
the measured values for peripheral events) obtained in the models, which do not account
for the hadron re-scattering stage of collision (e.g., thermal model, blast wave model).
These models also overestimate the measured
K∗
yield values. A nearly flat dependency
on centrality is actually observed in experiments for the
φ(
1020
)/K
ratio, as the
φ
’s decay
products almost do not interact with the dense hadronic medium due to a large lifetime of
this resonance. The observed suppression of the short-lived
K∗
yield ratio to kaon yield
is likely due to re-scattering of
K∗
decay products during the afterburner collision stage,
and this effect is stronger in central events because of the higher number of produced
particles and longer hadronic stage duration in this case. However, the system’s lifetime
determined based on these results depends on the cross-sections and reaction channels used
in the model for the hadronic stage. The correlation femtoscopy data are also considered
as a potential source of information on the duration of the matter collective motion in
high-energy A + A collisions. For instance, in the ALICE Collaboration’s experimental
analysis [
56
], the “freeze-out time” value for the 2.76 TeV Pb + Pb collisions at the LHC
is defined from the fit to the measured longitudinal femtoscopy radii dependency on
mT
(the result is 10–11 fm/
c
), but the fitting formula does not account for the strong transverse
collective flow that should develop in such collisions.
In our previous papers [
57
–
59
], a simple method for the estimation of the times of
pion and kaon maximal emission, based on the particle
pT
spectra and the dependencies
of longitudinal femtoscopy radii
Rlon g
on the pair transverse mass
mT
, was proposed
and tested in simulations within the different versions of the (integrated) hydrokinetic
model [
52
,
60
]. Soon after the first of the mentioned publications, the proposed prescription
Universe 2023,9, 433 3 of 17
was used by the ALICE Collaboration in their experimental analysis of kaon femtoscopy for
the 2.76 TeV Pb + Pb collisions at the LHC [
61
]. It provided estimates for the kinetic freeze-
out temperature,
Tkin =
144 MeV, and the maximal emission times,
τπ=
9.5
±
0.2 fm/
c
and
τK=
11.6
±
0.1 fm/
c
, close to those obtained in our paper [
60
] (see the details below).
A clear influence of the post-hydrodynamic phase on the soft physics results was also
noted in our analysis of different particle number ratios [
62
,
63
] at the LHC energies. These
studies include a comparison of the values calculated in the full iHKM simulation regime
with those obtained in a reduced regime without a hadronic cascade stage and also with
the thermal model results [8,9].
According to simulation analysis presented in [
64
,
65
], a large fraction (about 60% or
even more) of identified
K∗(
892
)
resonances at the LHC are produced in the hadronic
phase of collision, while primary particles coming from the QGP hadronisation cannot be
observed due to rescattering of their decay daughters. By contrast, primary
φ(
1020
)
mesons
with a lifetime of
≈
50 fm/
c
can be identified without problems, apparently because they
decay after the finishing of the intensive hadron rescattering phase (however, the iHKM
simulation results in [
64
] do not exclude that the hadronic stage may lead to an additional
increase in the identified
φ(
1020
)
number due to the
KK
correlation effect). The recent
ALICE Collaboration data on the
K∗/K
and
φ/K
yield ratios in 5.02 TeV Pb + Pb collisions
at the LHC confirm “the dominance of the rescattering effect in the hadronic phase” [
55
]
for K∗results and its small significance in the φcase.
All these results seem to support the conception of continuous (rather than sudden)
freeze-out and a prolonged afterburner stage. In accordance with this evolution picture,
particles possessing different momenta and originating from different sources (primary
particles and those leaving the system after elastic/inelastic reactions with the hadronic
medium at the afterburner stage of collision) can be expected to radiate from the system at
different times.
In the current paper, we tried to investigate the process of pion and kaon emission
considered in [
57
,
59
] in more detail, clarify the relationship between the extracted emission
times and the femtoscopy radii, and if possible, propose a way for the estimation of the full
system’s lifetime, including the afterburner stage, based on our analysis of ultrarelativistic
heavy-ion collisions within the integrated hydrokinetic model.
2. Research Motivation
In our studies [
57
–
59
], the transverse momentum spectra and femtoscopy radii for the
A + A collisions at the top RHIC and the two LHC energies were fit with simple analytical
formulas, containing parameters describing the system’s effective temperature, the strength
of the collective flow, and the time of maximal emission for particles of each sort. The fitting
formulas were obtained in approximation, suggesting that all the particles with comparably
low momenta (
pT.
0.5 GeV/
c
) are emitted from the system at the hypersurface of constant
proper time
τm.e.
, limited in the transverse direction. Of course, the real emission picture
for both pion and kaon mesons is much more complicated, so that the extracted maximal
emission times correspond only to some effective values, approximately reflecting the time
scales when the emission process for a given particle species is the most intensive.
To cross-check the fitting results and to obtain a more-detailed picture of the particle
emission process in [
57
–
59
], we additionally built plots of averaged emission functions
in coordinates
(rT
,
τ)
for particles with small
pT
(in accordance with the earlier applied
approximation). The figures show that the emission function maximum for pions should be
close to the particlisation time for the centre of the expanding system, which is in agreement
with the previously obtained fitting result. As for the kaon emission function, it had two
apparent maxima (one close to the pionic one and the other lying about 4 fm/
c
higher), such
that the
τm.e.
value obtained from the fit was lying between them and could be associated
with the “mean” maximal emission time with respect to the two maxima on the plot. The
second maximum of the kaon emission function, causing the extracted maximal emission
times for kaons to be about 2–3 fm/
c
larger than those for pions, is likely connected with
Universe 2023,9, 433 4 of 17
the decays of
K∗(
892
)
resonance (its lifetime is about 4 fm/
c
), producing an additional
noticeable portion of kaons after particlisation of the created fireball.
To investigate the emission process in more detail, we constructed the emission time
distributions for both considered particle sorts (see Figures 1and 2), which can be obtained
by integrating the previously analysed emission function histograms over transverse radius
rT
. An interesting thing one can notice about these distributions is that most particles of
both species seem to be produced in the late, hadronic stage of the collision, so that the total
number of pions and kaons emitted close to the hadronisation hypersurface is noticeably
smaller than that of mesons radiated later. Near the time of hadronisation in the centre of
the system (9–10 fm/
c
), one can see only local
τ
distribution maxima on the presented plots.
slice_py_of_hS
Entries 6275992
Mean 16.05
RMS 6.236
(fm/c)τ
0 5 10 15 20 25 30
Number of Entries
0
20
40
60
80
100
120
140
160
3
10 slice_py_of_hS
Entries 6,275,992
Mean 16.05
RMS 6.236
Figure 1.
The time of emission distribution for pions in central Pb + Pb collisions (
c
= 0–5%) at the
LHC energy √sNN =2.76 TeV simulated within iHKM, 0.2 <pT<0.3 GeV/c,|y|<0.5.
slice_py_of_hS
Entries 307018
Mean 18.7
RMS 5.768
(fm/c)τ
0 5 10 15 20 25 30
Number of Entries
0
1000
2000
3000
4000
5000
6000
7000
slice_py_of_hS
Entries 307,018
Mean 18.7
RMS 5.768
Figure 2. The same as in Figure 1for kaons.
The found peculiarity of
τ
distributions is in agreement with our previous conclu-
sions about the continuous character of particle spectra freeze-out; however, it is not clear
why the extracted values of
τm.e.
, obtained from the fits to single-particle
pT
spectra and
femtoscopy scales, do not reflect the apparent prevalence of the late post-hydrodynamic
stage contribution in the total amount of emitted particles, following from the
τ
distribu-
tion figures.
Universe 2023,9, 433 5 of 17
Another interesting fact about the extracted maximal emission times is that their
values are very close for the two considered LHC collision energies, 2.76 TeV and 5.02 TeV
(the same applies to the times of particlisation of the corresponding systems). At first
glance, it seems that, at higher collision energy, the created system should live longer
and produce more particles, so that the corresponding times of maximal emission should
be larger. Instead, at the two energies in our analysis for the most-central collisions
(c= 0–5%),
we have comparable
τm.e.
for both pions and kaons,
τπ=
8.97
±
0.04 fm/
c
and
τK=
12.73
±
0.12 fm/
c
at
√sNN =
5.02 TeV and
τπ=
10.34
±
0.06 fm/
c
(or even
τπ=
9.44
±
0.02 fm/
c
; see [
57
] for details) and
τK=
12.65
±
1.58 fm/
c
at
√sNN =
2.76 TeV.
The corresponding temperatures, entering the fitting formulas, are
T=
138 MeV and
T=144 MeV, respectively.
In Section 4, we try to investigate in detail the above-stated issues.
3. Materials and Methods
As was already mentioned, the analysis of relativistic heavy-ion collisions for this work
was carried out based on the results of computer simulations within the integrated hydroki-
netic model (iHKM) [
52
,
60
,
66
,
67
]. The model successfully reproduces the experimental data
on bulk hadronic observables (including femtoscopy radii) for the high-energy collisions at
the LHC and RHIC [
52
,
58
,
63
,
68
]. It also allows achieving a good agreement with the data
on direct photon production for the two mentioned collider experiments [69–71].
One of the most-important distinct features of the iHKM, as compared to other hybrid
models used for the simulation of A + A collisions, is the presence of the pre-thermal
dynamics description for the strongly interacting matter created as a result of the collision (it
is implemented using the energy–momentum transport approach based on the Boltzmann
equation in the relaxation time approximation—see the papers [
52
,
66
,
67
] for the details).
During the corresponding evolution stage, the system is gradually transformed from an
essentially non-equilibrium state it has at the very initial times right after the overlapping
of the two colliding nuclei to a nearly equilibrated state, which can be further described in
the approximation of relativistic viscous hydrodynamics [
72
,
73
]. The initial energy density
distribution, serving as a starting point for the pre-thermal stage, is generated with the
GLISSANDO code [74]:
e(b,τ0,rT) = e0(τ0)(1−α)Nw(b,rT)/2 +αNbin(b,rT)
(1−α)Nw(b=0, rT=0)/2 +αNbin(b=0, rT=0). (1)
Here, the parameter
e0
defines the initial energy density in the centre of the system,
and the parameter
α
regulates the proportion between the contributions from the binary
collision and wounded nucleon models to the
e(b
,
τ0
,
rT)
in (1). These two parameters are
used to adjust the model to the simulation of the concrete collision experiment.
For the initial momentum distribution, we take the anisotropic momentum distribu-
tion, inspired by the colour glass condensate effective gluon field theory:
f0(p) = gexp
−v
u
u
t
(p·U)2−(p·V)2
λ2
⊥
+(p·V)2
λ2
k
, (2)
where
Uµ= (cosh η
, 0, 0,
sinh η)
,
Vµ= (sinh η
, 0, 0,
cosh η)
,
η
is space–time rapidity, and
Λ=λ⊥/λk=
100 is the model parameter, describing the initial momentum anisotropy
(see [52,66,67] for the details).
The hydrodynamic stage of the matter expansion lasts until the temperature in the sys-
tem drops to the particlisation value Tp(depending on the equation of state (EoS) utilised
for the continuous quark–gluon matter; in the iHKM, we use the Laine–Schroeder [
75
] and
the HotQCD Collaboration [
76
] EoSs). Below the particlisation temperature, the contin-
uous medium transforms into the system of hadrons, which keeps on expanding, while
the hadrons interact intensively with each other, so that numerous particle creation and
Universe 2023,9, 433 6 of 17
annihilation processes take place. This post-hydrodynamic stage of the matter evolution is
described with the help of the UrQMD hadron cascade model [77,78].
The analytical formulas for spectra and femtoscopy radii fitting in [
57
–
59
] are obtained
from the following considerations.
One assumes that the particlisation occurs at the hypersurface
τ=const =τm.e.
,
limited in the direction transverse to the beam axis (
rT≤rmax
T
), which should be reasonable
at least for the particles with not very high momenta (
pT.
0.5 GeV/
c
). Accordingly, the
Wigner function for soft bosons can be written as follows:
fl.eq.(x,p) = 1
(2π)3[exp(βp·u(τm.e.,rT)−βµ)−1]−1ρ(rT), (3)
where
β
is the inverse temperature,
uµ(x)=(cosh ηLcosh ηT
,
rT
rTsinh ηT
,
sinh ηLcosh ηT)
is the collective four-velocity (here, in the
uµ
definition,
ηL=arctanh vL
and
ηT=
arctanh vT(rT)
are longitudinal and transverse rapidities, which correspond to the re-
spective velocities
vL
and
vT
), and
ρ(rT)
is the cutoff factor, narrowing the particlisation
hypersurface in the transverse direction:
ρ(rT) = exp[−α(cosh ηT(rT)−1)]. (4)
The parameter
α
here describes the collective flow intensity, in such a way that the
lower is the value of
α
, the stronger is the flow (in the case of absent flow,
α→∞
; for more
details, see [57,59]).
Given Equation (3) for the Wigner function, one can obtain the approximate formulas
for the momentum spectra and correlation functions. Applying the modified Cooper–Frye
prescription (it suggests that the overall particlisation hypersurface is built as a set of
points
(t(r
,
p)
;
r)
, corresponding to the maximal emission of particles with momentum
p
;
see [
60
] for the details) at the previously defined particlisation hypersurface and using the
saddle point method, one arrives at the following expressions for the
pT
spectra and the
longitudinal femtoscopy radii [79]:
p0
d3N
d3p∝exp [−(mT/T+α)(1−¯
v2
T)1/2], (5)
R2
long(mT) = τ2λ21+3
2λ2. (6)
Here,
mT=qm2+k2
T
,
kT
is the pair mean transverse momentum,
T=Tm.e.
is the
temperature at the maximal emission hypersurface
τ=τm.e.
,
¯
vT=kT/(mT+αT)
is the
transverse collective velocity at the saddle point, and
λ2=T
mT
(1−¯
v2
T)1/2 (7)
is the squared ratio of the longitudinal homogeneity length
λlong
to
τm.e.
(as usual in
femtoscopy studies, the
long
direction coincides with that of the beam axis,
out
corre-
sponds to the pair transverse momentum vector, and
side
is orthogonal to the two others).
Equation (6)
is valid for the transverse flow of an arbitrary profile and strength; however,
it was derived under the assumption of small
qlong
, corresponding to the top part of the
correlation function peak.
According to the method, originally proposed in [
57
], in order to estimate the
τm.e.
values for pions and kaons, one needs to find the values of
T
and
α
first from the combined
fit to the pion and kaon
pT
spectra based on Equation (5) and, after that, determine the
maximal emission times
τπ
and
τK
from the femtoscopy radii
Rlong(mT)
fitting (at already
fixed Tand α) using Equation (6).
Universe 2023,9, 433 7 of 17
4. Results and Discussion
4.1. Emission Time Distributions
As was already mentioned, the emission time distributions for both pions and kaons,
shown in Figures 1and 2for the case of 2.76 TeV Pb + Pb central collisions at the LHC,
suggest that a large number of particles leave the system noticeably later than the time
of full particlisation, which is about 9–10 fm/
c
for the considered LHC energies (rather
similar plots were obtained for 5.02 TeV collisions as well). The mean
τ
values that can
be calculated based on the presented model histograms are 16.05 fm/
c
and 18.70 fm/
c
for
pions and kaons, respectively, and an interesting question is why these time scales do not
match with
τm.e.
estimated based on the femtoscopy radii
mT
dependencies (i.e., 10.3 fm/
c
and 12.7 fm/c).
Here, one should consider several different factors that could possibly lead to such a
result and try to find which of them played the major role in this situation.
One of the relevant factors could be the non-Gaussian shape of the corresponding
particle emission functions and the correlation functions, which are used to determine
the femtoscopy radii. In [
80
], the non-Gaussian character of pion and kaon emission was
observed in our studies of the corresponding source functions (which can be defined as
time-integrated distributions of distances between the particles forming pairs in femtoscopy
analysis), calculated in the HKM (a previous version of the iHKM) for the case of Pb + Pb
collisions at the LHC energy
√sNN =
2.76 TeV. In Figure 3, we demonstrate the projections
of the mentioned meson source functions together with the Gaussian fits to them. As one
can see, the actual shape of the source functions, especially in the out and long directions,
includes noticeable non-Gaussian tails. The effect looks even more pronounced for pions.
In our method for the maximal emission time estimation, we fit
Rlon g
interferometry radii
dependencies on
mT
, and as follows from Equation (6), the larger are the
Rlon g
values,
the larger can be the resulting
τm.e.
. The radii are extracted from the Gaussian fits to the
correlation functions
C(q)
, which are connected with the source functions
S(r)
through the
so-called Koonin integral equation [81]:
C(q∗) = 1+Zd3r∗S(r∗)K(r∗,q∗), (8)
where
K(r∗
,
q∗)
is the integral transform kernel, reflecting the correlation mechanism(s)
supposed to exist between the emitted particles, and the asterisk denotes the pair rest frame.
Accordingly, the pairs from the non-Gaussian tail of the source function, characterised
by the large distances between emitted particles, will not contribute directly to the value of
the femtoscopy radius and, eventually, to the
τm.e.
value. In such a way, some amount of
particles that escape from the system at the late stage of the collision can be missed by the
femtoscopy analysis, but still be visible in the overall τdistribution.
In the correlation function plots, such pairs, containing, e.g., particles coming from the
long-lived resonance decays or liberated from the system after a series of elastic/inelastic
reactions with other hadrons constituting the expanding hadronic medium (so that the
distance between the two particles in the pair can be fairly large), typically form a practically
invisible sharp and narrow peak [
82
], which can reduce the intercept
λ
value, determined
from the fit to the correlation function, but does not really affect the radii.
In [
57
], we tried to reduce the effects stemming from the non-Gaussianity of the
correlation functions by fitting them in a narrow
q
range (0
<qlong <
0.04 GeV/
c
), which
also better corresponds to the assumptions made when deriving Formula (6). As a result, we
observed the more-uniform behaviour of the pion and kaon radii, closer to
mT
scaling (still
broken because of the collective flow and afterburner rescattering effects), with common
T
and
α
parameters corresponding to the combined pion and kaon
pT
spectra fitting; however,
only a hardly noticeable increase of the femtoscopy radii (about 4%) could be reached.
Universe 2023,9, 433 8 of 17
(fm)
out
r
0 10 20 30 40 50 60 70
)
-3
fm
-7
) (10
out
S(r
10
2
10
(a)
(fm)
side
r
0 10 20 30 40 50 60 70
)
-3
fm
-7
) (10
side
S(r
10
2
10
(b)
(fm)
long
r
0 10 20 30 40 50 60 70
)
-3
fm
-7
) (10
long
S(r
10
2
10
(c)
(fm)
out
r
0 10 20 30 40 50 60 70
)
-3
fm
-7
) (10
out
S(r
10
2
10
3
10
(d)
(fm)
side
r
0 10 20 30 40 50 60 70
)
-3
fm
-7
) (10
side
S(r
10
2
10
3
10
(e)
(fm)
long
r
0 10 20 30 40 50 60 70
)
-3
fm
-7
) (10
long
S(r
10
2
10
3
10
(f)
Figure 3.
The source function projections calculated in the iHKM for pions (Panels (
a
–
c
)) and
kaons (Panels (
d
–
f
)) produced in central Pb + Pb collisions at the LHC energy
√sNN =
2.76 TeV,
0.2
<pT<
0.36 GeV/
c
,
|y|<
0.5. Squares represent the model output, and lines show the
corresponding Gaussian fits.
The presence of a post-hydrodynamic hadronic cascade stage in the model certainly
increases the resulting femtoscopy radii; in our analysis, carried out during the work in
the papers [
57
,
83
], we found for the case of the LHC 2.76 TeV Pb + Pb collisions that the
switching off of the cascade stage reduced the radii values by 12% for pions and by 25–30%
for kaons. However, the maximal emission time extraction we are currently interested in
was performed based on the full calculation mode results, i.e., with the hadron cascade
turned on, so the above observation can hardly help us solve the stated problem.
A more-thorough analysis of the procedure we applied for the
τm.e.
extraction, how-
ever, suggests the following possible solution. Although the analytical Formulas (5) and (6),
used for the spectra and the femtoscopy radii fitting, were derived under the assumption
of isochronous particle emission from the hypersurface
τ=const =τm.e.
fragment with
rT<rmax
T
, typical for particles with not very high transverse momenta, the actual data
fitting included also the region of comparably high
pT
(0.5
<pT<
1.0 GeV/
c
) and
kT
(0.3
<kT<
1.1 GeV/
c
). This apparently implies that the emission time distributions,
actually corresponding to the applied fitting scheme, should as well include particles from
a wide momentum region (still not exceeding, however, the region of hydrodynamics ap-
proximation applicability). Therefore, in Figures 4and 5, we show the pion
τ
distributions
constructed with a momentum cut 0.5
<pT<
2.0 GeV/
c
for central Pb + Pb collisions at
the LHC energy
√sNN =
2.76 TeV (analogous distributions for kaons demonstrate a similar
tendency as compared to the previous Figure 2). In Figure 5, an additional
rT
constraint,
rT<10 fm, was applied.
As one can see, the presented histograms correspond much better to the
τm.e.
esti-
mates we previously obtained from the spectra and radii fitting, so that the distribution
maxima are close to the full particlisation time (
≈
10 fm/
c
), the tails corresponding to the
hadronic phase look much smaller and do not dominate in the overall distributions, and
the mean emission time values are 12.5 fm/
c
and 10.4 fm/
c
for the
rT
-unconstrained and
rT-constrained cases, respectively.
Universe 2023,9, 433 9 of 17
slice_py_of_hS
Entries 251792
Mean 12.47
RMS 6.057
(fm/c)τ
0 5 10 15 20 25 30
Number of Entries
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
slice_py_of_hS
Entries 251,792
Mean 12.47
RMS 6.057
Figure 4. The same as in Figure 1, but for a wide pTregion, 0.5 <pT<2.0 GeV/c.
slice_py_of_hS
Entries 173887
Mean 10.44
RMS 5.127
(fm/c)τ
0 5 10 15 20 25 30
Number of Entries
0
1000
2000
3000
4000
5000
6000
7000
8000
slice_py_of_hS
Entries 173,887
Mean 10.44
RMS 5.127
Figure 5. The same as in Figure 4, but with the constraint rT<10 fm.
The obtained wide-momentum-range
τ
distributions seem to be more relevant for
characterising the emission process from the system as a whole. However, what about the
low-momentum distributions shown before in Figures 1and 2and depicting a quite rich
and prolonged hadronic stage of the matter evolution? Could we use these distributions to
obtain estimates of the afterburner phase duration, corresponding to the “maximal emission
times”, but now determined based on low-momentum spectra and radii? In principle,
such an estimate would be in agreement with the modified Cooper–Frye prescription [
60
],
used in [
57
,
59
] for the derivation of Formulas (5) and (6) and suggesting that the particles
with different momenta
p
should be emitted from separate pieces
σp
of the complex
hadronisation hypersurface, composed as a united set of all such fragments.
Therefore, to test such an approach, we used the described method for the
τm.e.
estimation, but applying the low-momentum cuts on
pT
and
kT
. The combined
π
and
K
spectra fitting was performed for the
pT
range 0.25
<pT<
0.55 GeV/
c
and gave the values
T=
106
±
16 MeV,
απ=
2.8
±
0.5,
αK=
1.2
±
0.7, which were then fixed at the
Rlong(mT)
fitting. The latter was performed for
kT<
0.6 GeV/
c
and resulted in the maximal emission
time values τπ=13.65 ±0.44 fm/cand τK=16.73 ±0.49 fm/c.
Universe 2023,9, 433 10 of 17
Comparing these times with Figures 1and 2, one can see that the new
τm.e.
values
correspond to the maxima of the shown low-momentum emission time distributions and
can be possibly interpreted as evidence of a prolonged afterburner stage and used as
“upper-limit” estimates for the overall system’s lifetime (in the sense that they are defined
based on low-momentum data and particles with low
pT
are expected to leave the system
later than those with high
pT
). The new
τm.e.
value for kaons is also in agreement with
the emission picture presented in Figure 2 of our paper [
64
], which implies that active
K∗
production at the hadronic stage of collision lasts until
≈
15–20 fm/
c
, while
K∗
resonance
decays serve as an extra source of kaons (see also Figure 6of this article).
(fm)
T
r
0 5 10 15 20 25 30
(fm/c)τ
0
5
10
15
20
25
30
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
(a) no UrQMD
(fm)
T
r
0 5 10 15 20 25 30
(fm/c)τ
0
5
10
15
20
25
30
0
0.0002
0.0004
0.0006
0.0008
0.001
(b) UrQMD
Figure 6.
Emission functions
g(τ
,
rT)
constructed in the iHKM for
K+π−
pairs, coming from
K∗(
892
)
decays [
64
], in two regimes: (
a
) free streaming of hadrons created at the particlisation stage (plus
those coming from resonance decays), and (
b
) full iHKM calculation including UrQMD cascade
as the final afterburner stage. The graphs correspond to the LHC Pb + Pb collisions at
√sNN =
2.76 TeV,
c
= 5–10%; the transverse momentum and rapidity ranges are 0.3
<kT<
5 GeV/
c
and
|y|<0.5, respectively.
4.2. Particlisation Times at Different Collision Energies
In the previous subsection, we focused on the particle emission picture including long
and intensive emission from the hadronic stage of collision and on the estimation of the
effective time scales characterising the emission duration. However, a detailed analysis of
this problem cannot bypass the question of the hydrodynamic stage duration, defining the
moment, when one should switch from the description of the system’s evolution in terms
of a continuous medium to the description in terms of particles.
Conventionally, one assumes that the system’s hadronisation takes place at some
critical temperature or energy density (whose concrete values depend on the applied
equation of state for quark–gluon matter). As was mentioned, in the iHKM, we use two such
equations of state, namely the Laine–Schroeder [
75
] EoS and the HotQCD Collaboration [
76
]
one. The former is associated with the particlisation temperature
Tp=
165 MeV and
the corresponding energy density
ep=
0.5 GeV/fm
3
, while for the latter, the values
Tp=
156 MeV and
ep=
0.27 GeV/fm
3
are taken. The stated switching criterion implies
that the particlisation process is not isochronous for different parts of the system, but starts
quite early at the periphery and gradually reaches the centre. Thus, the system can be
considered fully hadronised when the hadronisation has completed in its centre.
As we already remarked in the Introduction, an interesting fact about the particlisation
times in ultrarelativistic heavy-ion collisions at the top RHIC and the LHC energies is that,
in the iHKM, they are very close to each other (about 8–10 fm/
c
), despite great differences
in the collision energies (and, apparently, the corresponding initial energy densities). This
feature is also reflected in the close maximal emission times obtained for pions in our
studies related to the corresponding collision experiments [57–59].
To examine this issue, we built plots of the time evolution for the energy density in
the centre of hydrodynamic grid
e(τ)
within iHKM (see Figures 7and 8) in the case of the
LHC Pb + Pb collisions at the two energies,
√sNN =
2.76 TeV and
√sNN =
5.02 TeV. The
Universe 2023,9, 433 11 of 17
Laine–Schroeder EoS was used in the simulations for both energies with the critical energy
density
ep=
0.5 GeV/fm
3
. For comparison, we also placed the curves corresponding to
the ideal relativistic hydrodynamics three-dimensional Gubser solution [84]:
e(rT,τ) = e0
τ4/3
(2q)8/3
[1+2q2(τ2+r2
T) + q4(τ2−r2
T)2]4/3 , (9)
and the Bjorken solution
e(τ) = e0τ−4/3
for purely longitudinal expansion, in the pre-
sented figures. The initial maximal
e
values for the Gubser and the Bjorken flows corre-
spond to those in the iHKM. Furthermore, in the Gubser flow case, the parameter
q
value,
q=0.15 fm−1
, defining the transverse width of the energy density profile was chosen
based on the fit to the initial iHKM e(rT)profile.
(fm/c)τ
1 10
)
3
(GeV/fm∈
1
10
2
10
iHKM, 5.02 ATeV
iHKM, 2.76 ATeV
276
0
∈=
max
∈Gubser flow,
502
0
∈=
max
∈Gubser flow,
3
=0.5 GeV/fm∈
Figure 7.
The energy density
e
at the centre of the system depending on proper time
τ
. The
iHKM curves for the two LHC energies,
√sNN =
2.76 TeV and
√sNN =
5.02 TeV, together with
the 3D Gubser solutions (9) of the relativistic hydrodynamics equations. The parameter
q
value,
q=
0.15 fm
−1
, is defined from the Gubser fit to the initial iHKM
e(rT)
profiles; the initial
e
values
correspond to the iHKM ones.
From the comparison with the 1D Bjorken flow case (see Figure 8), one can conclude
that the presence of strong transverse flow along with the longitudinal one is an important
factor leading to the observed small difference in the particlisation times for the two
collision energies (about 9.2 fm/
c
and 9.8 fm/
c
for the iHKM in the shown figure), defined
as the moments when the respective energy densities drop to
ep=
0.5 GeV/fm
3
. Indeed,
we see that, at the times after 3 fm/
c
, the Bjorken expansion is much slower than in the
iHKM, the ratio between the two
e(τ)
(about 1.5) does not change with time, so that
the respective particlisation times (about 26 fm/
c
and 36 fm/
c
) are much higher than in
our model, the ratio between these times,
(e0,2/e0,1 )3/4
, is about 1.35, and therefore, the
difference between them is quite large (about 10 fm/c).
The Gubser flow, accounting for the transverse matter expansion, results in an
e(τ)
behaviour more similar to the iHKM curves at
τ>
3 fm/
c
and particlisation times much
closer to those from the iHKM (about 6.7 fm/
c
and 7.3 fm/
c
), also with a small difference
between the two values. The analysis of the particlisation time
τp
dependency on the initial
energy density
e0(τ0)
in the iHKM shows that, for the considered RHIC and LHC energy
range in central collisions at a fixed EoS and initial time
τ0
, the
τp
in the iHKM grows slowly
with
e0
, approximately as
e1/s
0
, where
s
is close to 7. In our opinion, such a behaviour can
Universe 2023,9, 433 12 of 17
be explained by the intensive 3D expansion of the system, which takes place at large
τ
in
high-energy A + A collisions and leads to a fast subsequent decay of the created fireball.
The suggestion about the breakdown of the hydrodynamic description as a result of the
intensive 3D expansion of the systemsformed in high-energy hadron collisions was made
already in 1953 by Landau in his pioneering paper concerning the hydrodynamic approach
in high-energy physics [7].
(fm/c)τ
1 10
)
3
(GeV/fm∈
1
10
2
10
iHKM, 5.02 ATeV
iHKM, 2.76 ATeV
4/3
τ/
0,1
∈)=τ(∈Bjorken flow,
4/3
τ/
0,2
∈)=τ(∈Bjorken flow,
3
=0.5 GeV/fm∈
Figure 8.
The same as in Figure 7, but 1D Bjorken solutions,
e(τ) = e0τ−4/3
, are shown for compari-
son with the iHKM curves.
The energy density behaviour, somewhat similar to that shown in Figures 7and 8,
can be also observed in the following non-relativistic analytical solution of both ideal hy-
drodynamics and Boltzmann equations (see, e.g., [
85
,
86
]). The corresponding distribution
function can be written as
f(t,x,v) = N
(2πR0)3m
T03/2
exp −mv2
2T0−(x−vt)2
2R2
0!, (10)
where
N
is total particle number,
m
is the particle mass,
R0
is the initial Gaussian radius of
the system, and T0is the initial temperature.
Given the distribution function (10), one easily finds the corresponding dependency
of the energy density at x=0 on time:
e(x=0, t) = AT4
0(mR2
0/T0)5/2
(t2+mR2
0/T0)5/2 , (11)
where Ais a constant.
In Figure 9, we present the illustration corresponding to the solution (11). The black
line corresponds to a higher initial energy density, regulated by the
T0
parameter, which in
this case has the value
T0=
350 MeV. For the blue line,
T0=
300 MeV, and the initial energy
density is about two-times lower than for the black line. In both cases, the Gaussian radius
of the initial fireball is
R0=
6 fm, and the mass of the particles in the gas is
m=
0.5 GeV/
c2
.
The black curve in the figure is shifted 1 fm/
c
left, so that, after
t=
8 fm/
c
, we have the
same energy density for both evolving systems. Therefore, similar to Figures 7and 8, the
same “particlisation” conditions are reached at nearly the same expansion times for the
systems with the two different initial energy densities/temperatures, when the system’s
expansion becomes essentially three-dimensional.
Universe 2023,9, 433 13 of 17
(fm/c)τ
0 2 4 6 8 10
)
3
(GeV/fm∈
0
0.2
0.4
0.6
0.8
1
1.2
=350 MeV, shifted 1 fm/c left
0
T
=300 MeV
0
T
Figure 9.
The energy density
e
at the centre of the system dependencies on time
τ
in non-relativistic
hydrodynamics and the Boltzmann equations’ solution (11) with the two different initial
e
values
reaching the value e=0.5 GeV/fm3almost simultaneously (see the text for the details).
Although we consider the presented results quite interesting and worth attention
in the context of heavy-ion collision studies, of course, there are certain aspects of the
investigated phenomena left outside the scope of our research, so that additional analysis
combined with further improvement of the utilised approaches should bring an even
more-comprehensive and -detailed understanding of the collision dynamics. In particular,
so far, we did not address the issue of light nuclei, molecules, and other loosely bound
objects’ production in A + A collisions, while the corresponding observables were actively
analysed in recent experimental and theoretical studies. In our study of the
K∗
resonance
production for 2.76 TeV LHC Pb + Pb collisions, we obtained underestimated values of the
K∗/K
ratio in peripheral events. This problem could be re-analysed in more detail, and
the corresponding calculations for the 5.02 TeV case could be carried out. Other possible
ways of extending our analysis could be accounting for high-
pT
phenomena (like jets),
consideration of the physics outside the midrapidity region, and adjusting the model for
the description of planned ultra-high- and intermediate-energy (corresponding to the Beam
Energy Scan program) experiments. The viscous hydrodynamics part of our model can be
additionally improved by the implementation of the bulk viscosity and heat conductivity
effects, as well as switching from boost-invariant to full-(3 + 1)D simulations.
5. Conclusions
We considered the two paradoxical femtoscopic observations in ultrarelativistic heavy-
ion collisions. The first one is the closeness of the observed maximal emission times at the
quite different collision energies. It can be explained by the intensive 3D (!) expansion of
the system, which leads, starting from some time, to a fast decay of the formed continuous
medium for a wide range of high collision energies. Therefore, all the corresponding
systems reach the decay energy almost simultaneously.
Another paradoxical effect is that, despite the sufficient duration of the post-hydrodyna-
mic/afterburner cascade stage, the observed times of the maximal emission are close to the
particlisation times. The key to the answer lies in different pictures of particle radiation
in the narrow soft
pT
range and in a more wide one, typical for the femtoscopy analysis.
For the latter, the times of maximal emission extracted from the corresponding fits nearly
coincide with the particlisation times. For the former, the times are significantly larger and
reflect the duration of the afterburner stage at this momentum interval.
Universe 2023,9, 433 14 of 17
Author Contributions:
Conceptualisation, Y.S.; data curation, M.A. and V.S.; funding acquisition,
Y.S.; investigation, M.A. and V.S.; methodology, Y.S.; project administration, Y.S.; supervision, Y.S.;
visualisation, M.A. and V.S.; writing, original draft, V.S. All authors have read and agreed to the
published version of the manuscript.
Funding:
The research was carried out within the NAS of the Ukraine Targeted Research Program
“Collaboration in advanced international projects on high-energy physics and nuclear physics”,
Agreement No. 7/2023 between the NAS of Ukraine and BITP of the NAS of Ukraine. The work was
also supported by a grant from the Simons Foundation (Grant Number 1039151, Y.S., V.S., and M.A.).
The author Yu.S. is supported by the Excellence Initiative Research University grant of the Warsaw
University of Technology.
Data Availability Statement:
All the necessary data are presented in the tables and figures within
the article. The model results demonstrated in the graphs can also be obtained from the authors in
tabular form.
Acknowledgments:
Y.S. is grateful to J. Schukraft, J. Stachel, P. Braun-Munzinger, and A. Dainese for
stimulating and fruitful discussions. Y.S. is also grateful to the ExtreMe Matter Institute (EMMI) at
GSI in Darmstadt, where he was affiliated at the beginning of the work on the current paper, and to
Hanna Zbroszczyk for her invitation to continue these studies in Warsaw University of Technology
(WUT) as the Visiting Professor.
Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design
of the study; in the collection, analyses, or interpretation of the data; in the writing of the manuscript;
nor in the decision to publish the results.
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