#  Probing multi-particle bunching from intermittency analysis
in relativistic heavy-ion collisions

Valeria Zelina Reyna Ortiz<EMAIL_ADDRESS>Institute of Physics, Jan
Kochanowski University, 25-406 Kielce, Poland  Maciej Rybczyński
<EMAIL_ADDRESS>Institute of Physics, Jan Kochanowski University,
25-406 Kielce, Poland  Zbigniew Włodarczyk<EMAIL_ADDRESS>Institute of Physics, Jan Kochanowski University, 25-406 Kielce, Poland

###### Abstract

It has been demonstrated that intermittency, a self-similar correlation
regarding the size of the phase space volume, exhibits sensitivity to particle
bunching within a system situated in the two-dimensional cell. Based on recent
findings concerning fluctuations in charge particle density observed in
central Au + Au collisions at various energies ranging from $\sqrt{s_{NN}}$ =
7.7 to 200 GeV at RHIC/STAR, it is proposed that analyzing intermittency in
relativistic heavy-ion collisions could serve as a method to investigate
density fluctuations linked to correlation phenomena. Our understanding of
particle production mechanisms may be enriched by using intermittency analysis
to gain new information contained in them that was previously unavailable.

###### pacs:

XXX

##  I Introduction

The major goal of the intermittency analysis in relativistic heavy ion
collision programs is to explore the phase diagram of quantum chromodynamic
(QCD) matter. The critical end point (CEP) is a key feature of the QCD phase
diagram, representing the point where the first-order phase transition
boundary terminates [1, 2]. Many efforts have been made to search for the
possible CEP in heavy-ion collisions [3, 4].

Recently, there has been discussion regarding the potential of particle
density fluctuations in heavy-ion collisions to serve as a distinctive
indicator of the phase transition within the QCD phase diagram [5, 6].
Critical opalescence, a notable light scattering phenomenon observed in
continuous (or second-order) phase transitions in conventional quantum
electrodynamic matter [7, 8], is characterized by significant fluctuations
within the critical region. In the context of heavy-ion collisions, akin to
conventional critical opalescence, it is anticipated that the produced matter
would exhibit pronounced particle density fluctuations close to the CEP due to
the rapid increase in correlation length within the critical region [9, 10].
Such large density fluctuations are expected to persist through the kinetic
freeze-out stage if they can withstand final-state interactions during the
hadronic evolution of the system.

Upon approaching a critical point, the correlation length of the system
diverges and the system becomes scale invariant, or self-similar [11, 12, 13].
Based on the 3D-Ising universality class arguments [14, 15, 16, 17], the
density-density correlation function for small momentum transfer has a power-
law structure, leading to large density fluctuations in heavy-ion collisions
[14, 15, 16, 17]. Such fluctuations can be probed in transverse momentum phase
space within the framework of an intermittency analysis by utilizing the
scaled factorial moments [14, 15, 16, 17, 18, 19]. The methodology consists of
dividing the D-dimensional phase space into $M^{D}$ equal-sized cells and the
observable, $q$-th order scaled factorial moment $F_{q}\left(M\right)$, is
defined as follows [14, 20, 21, 22, 23, 24]:

$F_{q}\left(M\right)=\frac{\langle\frac{1}{M^{D}}\sum_{i=1}^{M^{D}}n_{i}\left(n_{i}-1\right)\ldots\left(n_{i}-q+1\right)\rangle}{\langle\frac{1}{M^{D}}\sum_{i=1}^{M^{D}}n_{i}\rangle^{q}},$
(1)

where $M^{D}$ is the number of cells in D-dimensional phase space and $n_{i}$
is the measured multiplicity of a given event in the $i$-th cell. The angle
bracket denotes an average over the events. The intermittency appears as a
power-law (scaling) behavior of scaled factorial moments [14, 20, 16, 23]. If
the system features density fluctuations, scaled factorial moments will obey a
power-law behavior of
$F_{q}\left(M\right)\propto\left(M^{D}\right)^{\phi_{q}}$, $M\gg 1$, where
$\phi_{q}$ is called the intermittency index quantifying the strength of
intermittency [14, 15, 16, 18, 21].

However, the consideration of ”intermittency” as a genuinely novel phenomenon
remained less apparent. Relation between Hanbury-Brown-Twiss effect and
intermitency phenomenon was discussed and advocate that they are perfectly
compatible [25, 26]. Has been shown that the multi-particle Bose-Einstein
correlations leads to the power-law like behavior of scaled factorial moments.
In Ref. [27], diverse datasets were analyzed, leading to the conclusion that
intermittency results from Bose-Einstein correlations, alongside a mechanism
driving power-law behavior. Conversely, in Ref. [28], the authors contend that
the observed intermittent-like patterns in moments of multiplicity
distributions are attributable to the quantum statistical properties of the
particle-emitting system, rather than serving as direct evidence of
intermittency.

Notice that implementing the ”mixed event method” effectively removes
contributions of multiplicity fluctuations (among others background
contributions) [18, 21, 22]. Mixed events are constructed by randomly
selecting particles from different original events, while ensuring that the
mixed events have the same multiplicity and momentum distributions as the
original events. Usually instead of $F_{q}(M)$ the following observable is
used:

$\Delta
F_{q}\left(M\right)=F_{q}\left(M\right)^{data}-F_{q}\left(M\right)^{mix},$ (2)

where the moments from mixed events are subtracted from the data.

Recently the STAR Collaboration published extensive data on $\Delta
F_{q}\left(M\right)$ in two dimensional ($D=2$) transverse momentum plane
($p_{Tx},p_{Ty}$) from central Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7,
11.5, 19.6, 27, 39, 62.4 and 200 GeV [29]. In the next sections, based on the
STAR experiment data, we provide a discussion concerning multi-particle
bunching and suggest that measuring the intermittency in relativistic heavy-
ion collisions could serve as a means to investigate density fluctuations
within a highly constrained number of cells. This observation will be our main
point for further discussion and calculations described in next sections.

The 0-5 $\%$ most central Au+Au collisions at $\sqrt{s_{NN}}=7.7$ GeV are
discussed in Sections II and III, while their energy dependence is considered
in Section IV. In Section V the scale-invariance as a possible origin of
power-law behavior of $\Delta F_{q}\left(M\right)$ is discussed. Section VI
summarizes and concludes our work.

##  II Multiplicity distributions of particles in cells

For large number of cells $M^{2}\gg N$ (where $N=\sum n_{i}$), the average
multiplicity in a cell $\langle n_{i}\rangle$ is small and usually one
observes zero or one particle in a given cell. In such a case the multiplicity
distributions of particles in a cell are indistinguishable. Considering two
borderline multiplicity distributions111For $n$-particle correlation function
$g_{\left(n\right)}=\left(n-1\right)!$ we have Bose-Einstein distribution,
while for Poisson distribution $g_{\left(n\right)}=0$. Between these
distributions we have negative binomial distribution and
$g_{\left(n\right)}=\left(n-1\right)!k^{-n+1}$ with real positive parameter
$k$. The $n!$ dependence is a consequence of Wick’s theorem [30] and its
validity was first demonstrated with thermal photons [31] and recently proved
for massive particles [32]., Poisson distribution:

$P\left(n_{i}\right)=\frac{\langle
n_{i}\rangle^{n_{i}}}{n_{i}!}\exp\left(-\langle n_{i}\rangle\right)$ (3)

and geometric (Bose-Einstein) distribution:

$P\left(n_{i}\right)=\frac{1}{1+\langle n_{i}\rangle}\left(\frac{\langle
n_{i}\rangle}{1+\langle n_{i}\rangle}\right)^{n_{i}},$ (4)

one can see that for small $\langle n_{i}\rangle$,
$P\left(0\right)=\exp\left(-\langle n_{i}\rangle\right)\simeq 1-\langle
n_{i}\rangle$ and $P\left(1\right)=\langle n_{i}\rangle\exp\left(\langle
n_{i}\rangle\right)\simeq\langle n_{i}\rangle$ for Poisson distribution, and
similarly $P\left(0\right)=1/\left(1+\langle n_{i}\rangle\right)\simeq
1-\langle n_{i}\rangle$ and $P\left(1\right)=\langle
n_{i}\rangle/\left(1+\langle n_{i}\rangle\right)^{2}\simeq\langle
n_{i}\rangle$ for geometric distribution [33].

Figure 1: Average multiplicity of charged particles in a bunch of particles as
a function of the number of cells $M^{2}$ evaluated from $\Delta F_{q}$ for
$\sqrt{s_{NN}}=7.7$ GeV. Figure 2: $\Delta F_{q}$ evaluated for corresponding
average multiplicity of particles in a bunch, in comparison with experimental
data [29].

Assuming that Bose-Einstein correlated bunch of particles appears only in one
cell, one have (for large number of cells): $F_{q}\simeq F_{q}^{PD}+\langle
f_{q}^{*}\rangle\left(\langle n_{i}\rangle^{q}M^{2}\right)^{-1}$ and finally

$\Delta F_{q}\simeq\langle
f_{q}^{*}\rangle\frac{\left(M^{2}\right)^{q-1}}{\langle N\rangle^{q}},$ (5)

where $\langle N\rangle=\sum n_{i}$ is the average multiplicity for $M^{2}=1$
and

$\langle f_{q}^{*}\rangle=\langle
n\left(n-1\right)\ldots\left(n-q+1\right)\rangle=\left<\frac{n!}{\left(n-q\right)!}\right>$
(6)

is the factorial moment averaged over multiplicity distribution in a cell with
bunch of particles. For geometric distribution (4) with the average
multiplicity $m$ one have $f_{q}^{*}=m^{q}q!$, and from Eq. (5) one can
evaluate

$m=\langle N\rangle\left(\frac{\Delta
F_{q}}{\left(M^{2}\right)^{q-1}q!}\right)^{1/q}.$ (7)

Fig. 1 shows average multiplicity in a bunch evaluated from experimental
$\Delta F_{q}$ values. Roughly $m=a\left(M^{2}/1000\right)^{b}$ with $a\simeq
3.32\cdot q^{-0.732}-5.33\cdot q^{-2}$ and $b=1.46\cdot q^{-1}-0.81$. Fig. 2
shows corresponding $\Delta F_{q}$ values compared to experimental data.
Without assumption on multiplicity distribution of particle’s bunch one can
evaluate factorial moments $\langle f_{q}^{*}\rangle$ (see Fig. 3) and
estimate multiplicity distribution $P\left(n\right)$ in a bunch. From Eq. (6)
one can write

$\langle
f_{q}^{*}\rangle=\sum_{n=q}^{n_{max}}\frac{n!}{\left(n-q\right)!}P\left(n\right)$
(8)

and

$P\left(n=q\right)=\frac{1}{q!}\left(\langle
f_{q}^{*}\rangle-\sum_{i=q+1}^{n_{max}}\frac{i!}{(i-q)!}P\left(i\right)\right),$
(9)

where we put in calculations $n_{max}=q_{max}+1$ and
$P\left(n_{max}\right)\simeq 0.1\cdot P\left(n_{max}-1\right)$. Estimated
multiplicity distributions in cells (for $\langle f_{q}^{*}\rangle$ taken at
$M^{2}=3962$ and $\sqrt{s_{NN}}=7.7$ GeV) is presented in Fig. 4. In one cell
we have particle bunch with $P\left(n\right)$ given by Eq. (9) and
multiplicity distributions in all the other cells are given by Poisson
distribution (3) with $\langle n_{i}\rangle=N/M^{2}$.

##  III Many bunches case

Figure 3: Average factorial moments $\langle f_{q}^{*}\rangle$ evaluated from
experimental $\Delta F_{q}$ at $\sqrt{s_{NN}}=7.7$ GeV. Fits show dependence
of $\langle f_{q}^{*}\rangle=\left(M^{2}/3962\right)^{b}$ with
$b=1.6-0.846\cdot q$. Figure 4: Multiplicity distribution of particles in one
cell ($M^{2}=3962$ and $\sqrt{s_{NN}}=7.7$ GeV).

For the case with $k$ bunches with multiplicity governed by geometric
distribution (4) one have negative binomial distribution (NBD)

$P\left(n\right)=\frac{\Gamma\left(n+k\right)}{\Gamma\left(n+1\right)\Gamma\left(k\right)}\left(1+\frac{k}{\langle
n\rangle}\right)^{-n}\left(1+\frac{\langle n\rangle}{k}\right)^{-k}$ (10)

and factorial moments

$\langle
f_{q}\rangle=\frac{\Gamma\left(k+q\right)}{\Gamma\left(k\right)}\frac{\langle
n\rangle^{q}}{k^{q}}.$ (11)

Correlations being a signature of Bose-Einstein statistics never reaches its
maximum allowed degree, the most natural interpretation is that the source is
not totally chaotic as would be expected from a purely thermalized system, but
that a certain degree of coherence ($1-\lambda$) is present in the boson
radiation [34]. Effective factorial moments are expected to be

$\langle f_{q}\rangle=\left(1-\lambda\right)\langle
f_{q}^{PD}\rangle+\lambda\langle f_{q}^{NBD}\rangle,$ (12)

where $\langle f^{NBD}\rangle$ (c.f. Eq. (11)) and $\langle
f^{PD}\rangle=\langle n\rangle^{q}$ are factorial moments for negative
binomial and Poisson distributions, respectively. In this case, one have

$\Delta
F_{q}\simeq\lambda\frac{\Gamma\left(k+q\right)}{\Gamma\left(k\right)k^{q}}-\lambda.$
(13)

Expected $\Delta F_{q}$ in comparison with experimental data are shown in Fig.
5. Comparing Eq. (13) with experimental values at $\sqrt{s_{NN}}=7.7$ GeV, we
estimate roughly $\lambda=0.02$ and $k=2.56\left(M^{2}\right)^{-0.42}$. For
small $k$-parameter, Eq. (13) leads to relation $\Delta F_{q}\propto k^{1-q}$
and for evaluated $k$-parameter we have dependence

$\Delta F_{q}\propto\left(M^{2}\right)^{0.42\left(q-1\right)}.$ (14)

Figure 5:  $\Delta F_{q}$ from Eq. (13) in comparison with experimental data
at $\sqrt{s_{NN}}=7.7$ GeV.

##  IV Energy dependence

Scaled factorial moments are weekly dependent on collision energy. In Figs. 6
and 7 we show ratios $\Delta
F_{4}^{1/4}\left(M^{2},\sqrt{s_{NN}}\right)/\Delta
F_{4}^{1/4}\left(M^{2},7.7~{}{\textrm{G}eV}\right)$
and $\Delta
F_{q}^{1/q}\left(M^{2},\sqrt{s_{NN}}=39~{}{\textrm{G}eV}\right)/\Delta
F_{q}^{1/q}\left(M^{2},7.7~{}{\textrm{G}eV}\right)$, respectively, for
different energies $\sqrt{s_{NN}}$ and different values of order $q$. Taking
into account Eq. (7), this ratio corresponds with ratios $m/\langle N\rangle$
at different energies. For large $M^{2}>1000$,
$\frac{m(\sqrt{s_{NN}})}{\langle
N(\sqrt{s_{NN}}\rangle)}=\frac{m(7.7~{}GeV)}{\langle
N(7.7~{}{\textrm{G}eV})\rangle}(0.985-0.0236\ln{\sqrt{s_{NN}}})$. Weak energy
dependence indicate that energy dependence $m(\sqrt{s_{NN}})$ follows energy
dependence of total multiplicity $\langle N(\sqrt{s_{NN}})\rangle$.

Figure 6: Ratios $\Delta F_{4}^{1/4}\left(M^{2},\sqrt{s_{NN}}\right)/\Delta
F_{4}^{1/4}\left(M^{2},7.7~{}{\textrm{G}eV}\right)$ for different
$\sqrt{s_{NN}}$ energies. Figure 7: Ratios $\Delta
F_{q}^{1/q}\left(M^{2},39~{}{\textrm{G}eV}\right)/\Delta
F_{q}^{1/q}\left(M^{2},7.7~{}{\textrm{G}eV}\right)$ for different $q$ values.

##  V Possible origin of power-law behavior

Intermittency appears as a power-law behavior of scaled factorial moments,
$\Delta F_{q}\propto(M^{2})^{\Phi_{q}}$. For a scenario in which a bunch of
particles appears only in one cell (or in a limiting number of cells),
following Eq. (5), we expect the intermittency index $\Phi_{q}=q-1$, for
multiplicity $m$ of particles in a bunch being independent on the number of
cells $M^{2}$. In reality, the division procedure of phase space changes
multiplicity $m$ in a cell. Number of particles in bunch of effective space
size $\delta$ in cell of size $\Delta\left(M\right)=\Delta\left(M=1\right)/M$
is expected to be $m=m(\delta/\Delta(M))=m(M)$.

Let’s consider a bunch of particles of transverse size $\delta$ (dispersion
with respect to the axis of a bundle) in the random direction
$(p_{Tx0},p_{Ty0})$ with a transverse momentum distribution

$f(p_{Tx},p_{Ty})=\frac{3}{\pi\delta^{2}}\exp\left(-\frac{\sqrt{6}\sqrt{(p_{Tx}-p_{Tx0})^{2}+(p_{Ty}-p_{Ty0})^{2}}}{\delta}\right).$
(15)

Numerical results of the $m\left(M^{2}\right)$ dependence are shown in Fig. 8.
For large $M^{2}$ we observe power-law dependence:

$m\left(M^{2}\right)\propto\left(M^{2}\right)^{-\kappa}$ (16)

which will result in

$\Delta F_{q}\left(M^{2}\right)\propto
m^{q}\left(M^{2}\right)\left(M^{2}\right)^{q-1}=\left(M^{2}\right)^{q\left(1-\kappa\right)-1}$
(17)

The intermittency index $\Phi_{q}=q\left(1-\kappa\right)-1$ is the result of
the division procedure, which leads to power-law dependence
$m\left(M^{2}\right)$ given by Eq. (16). It is known [35] that, if function
$m\left(M^{2}\right)$ follows a simple power law,
$m\left(M^{2}\right)\propto\left(M^{2}\right)^{-\kappa}$ then it is scale
invariant and

$m\left(\lambda M^{2}\right)=\mu m\left(M^{2}\right)$ (18)

with relation $\kappa=-\ln\mu/\ln\lambda$. Selecting different functions
$f(p_{Tx},p_{Ty})$ changes details but the scale invariant behavior is still
visible.

Figure 8:  The relative multiplicity in bunch
$m\left(M^{2}\right)/m\left(M^{2}=1\right)$ dependence on number of cells
$M^{2}$ for space size $\delta=0.04$ GeV and $\delta=0.06$ GeV. For large
$M^{2}$ we observe a power-law dependence
$m\left(M^{2}\right)\propto\left(M^{2}\right)^{-\kappa}$ with $\kappa$=0.46
and 0.49, respectively.

##  VI Conclusions

Density fluctuations in cells lead to the power-law-like behavior of scaled
factorial moments. Correlated bunch of particles which appears only in one
cell can be responsible for a ”intermittent-like” behavior of $\Delta
F_{q}\left(M^{2}\right)$. Multiplicity distribution $P\left(m\right)$ in a
bunch of particles and spatial distribution $f(p_{Tx},p_{Ty})$ determine
behavior of scaled factorial moments. For small size of cell $\Delta$,
multiplicity in a bunch with size $\delta$ is scale invariant (its form
follows a simple power law
$m\left(\Delta^{2}\right)\propto\left(\Delta^{2}\right)^{\kappa}$) which will
result in $\Delta
F_{q}\left(M^{2}\right)\propto\left(M^{2}\right)^{q\left(1-\kappa\right)-1}$.
Whether such a simple scenario is related to the broad spectrum of
experimental data further calculations from dynamical modeling of heavy-ion
collisions are required.

Acknowledgments

This research was supported by the Polish National Science Centre (NCN) Grant
2020/39/O/ST2/00277. In preparation of this work we used the resources of the
Center for Computation and Computational Modeling of the Faculty of Exact and
Natural Sciences of the Jan Kochanowski University of Kielce.

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