NonBessel–Gaussianity and flow harmonic finesplitting
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Abstract
Both collision geometry and eventbyevent fluctuations are encoded in the experimentally observed flow harmonic distribution \(p(v_n)\) and 2kparticle cumulants \(c_n\{2k\}\). In the present study, we systematically connect these observables to each other by employing a Gram–Charlier A series. We quantify the deviation of \(p(v_n)\) from Bessel–Gaussianity in terms of harmonic finesplitting of the flow. Subsequently, we show that the corrected Bessel–Gaussian distribution can fit the simulated data better than the Bessel–Gaussian distribution in the more peripheral collisions. Inspired by the Gram–Charlier A series, we introduce a new set of cumulants \(q_n\{2k\}\), ones that are more natural to use to study near Bessel–Gaussian distributions. These new cumulants are obtained from \(c_n\{2k\}\) where the collision geometry effect is extracted from it. By exploiting \(q_2\{2k\}\), we introduce a new set of estimators for averaged ellipticity \(\bar{v}_2\), ones which are more accurate compared to \(v_2\{2k\}\) for \(k>1\). As another application of \(q_2\{2k\}\), we show that we are able to restrict the phase space of \(v_2\{4\}\), \(v_2\{6\}\) and \(v_2\{8\}\) by demanding the consistency of \(\bar{v}_2\) and \(v_2\{2k\}\) with \(q_2\{2k\}\) equation. The allowed phase space is a region such that \(v_2\{4\}v_2\{6\}\gtrsim 0\) and \(12 v_2\{6\}11v_2\{8\}v_2\{4\}\gtrsim 0\), which is compatible with the experimental observations.
1 Introduction
It is a wellestablished picture that matter produced in a heavy ion experiment shows collective behavior. Based on this picture, the initial energy density manifests itself in the final particle momentum distribution. Accordingly, as the main consequence of this collectivity, the final particle momentum distribution has extensively been studied by different experimental groups in the past years. As a matter of fact, the experimental groups at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) measure the flow harmonics \(v_n\) [1, 2, 3, 4, 5, 6, 7, 8], the coefficients of the momentum distribution Fourier expansion in the azimuthal direction [9, 10, 11]. All these observations can be explained by several models based on the collective picture.
Finding the flow harmonics is not straightforward, because the reactionplane angle (the angle between the orientation of the impact parameter and a reference frame) is not under control experimentally. Additionally, the Fourier coefficients cannot be found reliably due to the low statistic in a single event. These issues enforce us to use an analysis more sophisticated than a Fourier analysis. There are several methods to find the flow harmonics experimentally, namely the eventplane method [12], multiparticle correlation functions [13, 14] and Lee–Yang zeros [15, 16]. The most recent technique to find the flow harmonics is using the distribution of the flow harmonic \(p(v_n)\). This distribution has been obtained experimentally by employing the unfolding technique [17, 18].
It is well known that the initial shape of the energy density depends on the geometry of the collision and the quantum fluctuations at the initial state. As a result, the observed flow harmonics fluctuate eventbyevent even if we fix the initial geometry of the collision. In fact, the eventbyevent fluctuations are encoded in \(p(v_n)\) and experimentally observed flow harmonics as well. It is worthwhile to mention that the observed eventbyevent fluctuations are a reflection of the initial state fluctuations entangled with the modifications during different stages of the matter evolution, namely collective expansion and hadronization. For that reason, exploring the exact interpretation of the flow harmonics is crucial to understand the contribution of each stage of the evolution on the fluctuations. Moreover, there has not been found a wellestablished picture for the initial state of the heavy ion collision so far. The interpretations of the observed quantities contain information about the initial state. This information can shed light upon the heavy ion initial state models too.
According to the theoretical studies, the flow harmonics \(v_n\{2k\}\) obtained from 2kparticle correlation functions are different, and their difference is due to nonflow effects [13, 14] and eventbyevent fluctuations [19]. We should point out that experimental observations show that \(v_2\{2\}\) is considerably larger than \(v_2\{4\}\), \(v_2\{6\}\) and \(v_2\{8\}\). Additionally, all the ratios \(v_2\{6\}/v_2\{4\}\), \(v_2\{8\}/v_2\{4\}\) and \(v_2\{8\}/v_2\{6\}\) are different from unity [20, 21]. Alternatively, the distribution \(p(v_n)\) is approximated by a Bessel–Gaussian distribution (corresponding to a Gaussian distribution for \(v_n\) fluctuations) as a simple model [10, 22]. Based on this model, the difference between \(v_2\{2\}\) and \(v_2\{4\}\) is related to the width of the \(v_2\) fluctuations. However, this model cannot explain the difference between the other \(v_2\{2k\}\).
In the past years, several interesting studies of the nonGaussian \(v_n\) fluctuations have been done [23, 24, 25, 26, 27, 28]. Specifically, it has been shown in Ref. [27] that the skewness of the \(v_2\) fluctuations is related to the difference \(v_2\{4\}v_2\{6\}\). Also, the connection between kurtosis of \(v_3\) fluctuations and the ratio \(v_3\{4\}/v_3\{2\}\) has been studied in Ref. [28]. It is worth noting that the deviation of \(v_n\) fluctuations from a Gaussian distribution immediately leads to the deviation of \(p(v_n)\) from a Bessel–Gaussian distribution. In [29], the quantities \(v_n\{4\}v_n\{6\}\) and \(v_n\{6\}v_n\{8\}\) for a generic narrow distribution are computed.
In the present work, we will introduce a systematic method to connect \(v_n\{2k\}\) to the distribution \(p(v_n)\). In Sect. 2, we have an overview of the known concepts of cumulants, flow harmonic distributions and their relation with the averaged flow harmonics \(\bar{v}_n\). Section 3 is dedicated to the Gram–Charlier A series in which we find an approximate flow harmonic distribution in terms of \(c_n\{2k\}\). Specifically for the second harmonics, we show that the deviation of \(p(v_2)\) from Bessel–Gaussianity is quantified by the finesplitting \(v_2\{2k\}v_2\{2\ell \}\) where \(k,\ell \ge 2\) and \(k\ne \ell \). These studies guide us to defining a new set of cumulants \(q_n\{2k\}\) where they depend on the eventbyevent fluctuations only. In Sect. 4, we use the new cumulants to introduce more accurate estimations for the average ellipticity. As another application of new cumulants, we use \(q_2\{2k\}\) to constrain the \(v_2\{4\}\), \(v_2\{6\}\) and \(v_2\{8\}\) phase space in Sect. 5. We show that the phase space is restricted to a domain where \(v_2\{4\}v_2\{6\}\gtrsim 0\)^{1} and \(12 v_2\{6\}11v_2\{8\}v_2\{4\}\gtrsim 0\). We present the conclusion in Sect. 6. The supplementary materials can be found in the appendices. We would like to emphasize that in Appendix C, we found a onedimensional distribution for \(p(v_n)\) which is different from that mentioned in Sect. 3. Additionally, an interesting connection between the expansion of \(p(v_n)\) in terms of the cumulants \(c_n\{2k\}\) and the relatively new concept of multiple orthogonal polynomials in mathematics is presented in Appendix D.
2 Flow harmonic distributions and 2kparticle cumulants
This section is devoted to an overview of already wellknown concepts concerning the cumulant and its application to study the collectivity in the heavy ion physics. We present this overview to smoothly move forward to the harmonic distribution of the flow and its deviation from Bessel–Gaussianity.
2.1 Correlation functions vs. distribution
According to the collective picture in the heavy ion experiments, the final particle momentum distribution is a consequence of the initial state after a collective evolution. In order to study this picture quantitatively, the initial anisotropies and flow harmonics are used extensively to quantify the initial energy density and final momentum distribution.
The initial energy (or entropy) density of a single event can be written in terms of the initial anisotropies, namely the ellipticity and triangularity. Specifically, ellipticity and triangularity (shown by \(\varepsilon _2\) and \(\varepsilon _3\), respectively) are cumulants of the distribution indicating how much it is deviated from a twodimensional rotationally symmetric Gaussian distribution [30].
The cumulants \(c_n\{2k\}\propto v_n^{2k}\{2k\}\) (see Eq. (12)) are indicating the characteristics of the distribution \(p(v_n)\) while the cumulants \({\mathcal {A}}_{mn}\) (or equivalently \({\mathcal {C}}_{m,n}\)) in Eq. (4) are shown the characteristics of \(p(v_{n,x},v_{n,y})\). The interconnection between \(v_n\{2k\}\) and \({\mathcal {A}}_{mn}\) have been studied previously in the literature. We point out that in order to define \(p(v_{n,x},v_{n,y})\), we considered \(\varPhi _{\text {RP}}=0\) for all events. In this case, the cumulant \({\mathcal {A}}_{30}=\langle (v_{n,x}\) \(\langle v_{n,x}\rangle )^3 \rangle \) is related to the skewness of \(p(v_{n,x},v_{n,y})\). For the case \(n=2\), this quantity is found for the first time in Ref. [27], and it is argued that \({\mathcal {A}}_{30}\propto v_2\{4\}v_2\{6\}\). In other words, \({\mathcal {A}}_{30}\) is related to the finesplitting between \(v_2\{4\}\) and \(v_2\{6\}\). In Ref. [28], the kurtosis of \(p(v_{3,x},v_{3,y})\) in the radial direction has also been calculated, and it is shown it is proportional to \(v_3^4\{4\}\)^{6}.
Equivalently, we can use the following argument: for the distribution \(\tilde{p}(v_{n,x},v_{n,y})\), one simply finds the only nonvanishing moments are \(\langle v_{n_x}^{2k} v_{n_y}^{2\ell } \rangle \). It means that in the polar coordinates only \(\langle v_n^{2(k+\ell )}\rangle \) are present. Additionally, by finding the twodimensional cumulants of \(\tilde{p}(v_{n,x},v_{n,y})\) in polar coordinates \({\mathcal {C}}_{m,n}\) (Eq. (4)), we find that the only nonzero cumulants are \({\mathcal {C}}_{2k,0}\propto c_n\{2k\}\) (see Appendix A).
As a result, in the presence of random reactionplane angle, \(c_n\{2k\}\)’s are all we can learn from the original \(p(v_{n,x},v_{n,y})\), whether we use 2kparticle correlation functions or obtain it from the unfolded distribution \(p(v_n)\) in principle. However, we should note that the efficiency of the two methods in removing single event statistical uncertainty and nonflow effects could be different, which leads to different results in practice.
Furthermore, the whole information about the fluctuations is not encoded in \(p(v_{n,x},v_{n,y})\). In fact, the most general form of the fluctuations are encoded in a distribution as \(p(v_{1,x},v_{1,y},v_{2,x},v_{2,y},\ldots )\). It is worth mentioning that the symmetric cumulants, which have been introduced in Ref. [33] and have been measured by the ALICE collaboration [34], are nonvanishing. Additionally, the eventplane correlations (which are related to the moments \(\langle \hat{v}_m^q (\hat{v}^*_{n})^{q\,m/n} \rangle \)) have been obtained by the ATLAS collaboration [35, 36]. They are nonzero too. These measurements indicate that \(p(v_{1,x},v_{1,y},v_{2,x},v_{2,y},\ldots )\) cannot be written as \(\prod _n p(v_{n,x},v_{n,y})\). In the present work, however, we do not focus on the joint distribution and leave this topic for studies in the future. Let us point out that moving forward to find a generic form for the moments of the harmonic distribution of the flow was already done in Ref. [37].
2.2 Approximated averaged ellipticity
A question arises now: how much information is encoded in \(p(v_n)\) from the original \(p(v_{n,x},v_{n,y})\)? In order to answer this question, we first focus on \(n=3\). Unless there is no net triangularity for spherical ion collisions, the nonzero triangularity at each event comes from the fluctuations. Hence, we have \(\bar{v}_3=0\) for such an experiment. In this case, the eventbyevent randomness of \(\varPhi _{\text {RP}}\) is similar to the rotation of the triangular symmetry plane due to the eventbyevent fluctuations. It means that \(p(v_{3,x},v_{3,y})\) itself is rotationally symmetric, and the main features of \(p(v_{3,x},v_{3,y})\) and \(\tilde{p}(v_{3,x},v_{3,y})\) are the same. As a consequence, \(p(v_3)\) or equivalently \(c_3\{2k\}\) can uniquely reproduce the main features of \(p(v_{3,x},v_{3,y})\).
However, it is not the case for \(n=2\) due to the nonzero averaged ellipticity \(\bar{v}_2\). The distribution \(p(v_{2,x},v_{2,y})\) is not rotationally symmetric and reshuffling \((v_{2,x},v_{2,y})\) leads to information loss from the original \(p(v_{2,x},v_{2,y})\). Therefore, there is at least some information in \(p(v_{2,x},v_{2,y})\) that we cannot obtain from \(p(v_2)\) or \(c_2\{2k\}\). Nevertheless, it is still possible to find some features of \(p(v_{2,x},v_{2,y})\) approximately. For instance, we mentioned earlier in this section that the skewness of this distribution in the \(v_{2,x}\) direction is proportional to \(v_2\{4\}v_2\{6\}\).

In order to relate \(\bar{v}_n\) to \(c_n\{2k\}\), one needs to estimate the shape of \(p(v_n)\) where \(\bar{v}_n\) is implemented in this estimation explicitly. We show this estimated distribution by \(p(v_n;\bar{v}_n)\).

One can check the accuracy of the estimated distribution by studying the finesplitting \(v_n\{2k\}v_n\{2\ell \}\) and comparing it with the experimental data.
3 RadialGram–Charlier distribution and new cumulants
In Sect. 2.2, we argued that the quantity \(\bar{v}_n\), which is truly related to the geometric features of the collision, can be obtained by estimating a function for \(p(v_{n,x},v_{n,y})\). We observed that the Dirac delta function choice for \(p(v_{n,x},v_{n,y})\) leads to \(\bar{v}_n=v_n\{2k\}\) for \(k>0\), while by assuming the distribution to be a 2D Gaussian located at \((\bar{v}_n,0)\) (the Bessel–Gaussian in one dimension) we find \(\bar{v}_n=v_n\{2k\}\) for \(k>1\). One notes that in modeling the harmonic distribution of the flow by a delta function or Gaussian distribution, the parameter \(\bar{v}_n\) is an unfixed parameter, which is eventually related to the \(v_n\{2k\}\). In any case, the experimental observation indicates that the \(v_n\{2k\}\) are split; therefore, the above two models are not accurate enough.
Instead of modeling \(p(v_{n,x},v_{n,y})\), we will try to model \(p(v_n)\) with an unfixed parameter \(\bar{v}_n\), namely \(p(v_n;\bar{v}_n)\). In this section, we introduce a series for this distribution such that the leading term in this expansion is the Bessel–Gaussian distribution. The expansion coefficients would be a new set of cumulants that specifies the deviation of the distribution from Bessel–Gaussianity. In fact, by using these cumulants, we would be able to model \(p(v_n;\bar{v}_n)\) more systematically.
It is well known that a given distribution can be approximated by a Gram–Charlier A series which approximates the distribution in terms of its cumulants (see Appendix B). Here we use this concept to find an approximation for \(p(v_n;\bar{v}_n)\) in terms of the cumulants \(c_n\{2k\}\). One of the formal methods of finding the Gram–Charlier A series is using orthogonal polynomials. In addition to this wellknown method, we will introduce an alternative method, which is more practical for finding the series of a onedimensional \(p(v_n;\bar{v}_n)\) around the Bessel–Gaussian distribution.
3.1 Gram–Charlier A series: 1D distribution with support \({\mathbb {R}}\)
Before finding the approximated distribution around the Bessel–Gaussian, let us have practice with the alternative method of finding Gram–Charlier A series by applying it to a onedimensional distribution p(x) with support \((\infty ,\infty )\).^{11} This method will be used in the next section to find the radialGram–Charlier distribution for arbitrary harmonics.
Now, consider an approximation for the original distribution where its cumulants are coincident with the original p(x) only for a few first cumulants. We show this approximated distribution by \(p_q(x)\) where the cumulants \(\kappa _n\) for \(1\le n \le q\) are the same as the cumulants of the original p(x).
3.2 RadialGram–Charlier distribution
It is worth noting that the concept of a 2D Gram–Charlier A series has been employed in heavy ion physics first in Ref. [30] by Teaney and Yan. They used this series to study the energy density of a single event.^{12} However, we use this to study the harmonic distribution of the flow in the present work.
Now let us consider a twodimensional Gram–Charlier A series for \(p(v_{n,x},v_{n,y})\). By this consideration, one can find a corresponding series for \(p(v_n)\) by averaging out the azimuthal direction. We should say that the results of this averaging for the second and third harmonics are different. For \(n=3\), the distribution \(p(v_{3,x},v_{3,y})\) is rotationally symmetric and, as we already remarked in the previous section, the whole information of the distribution is encoded in \(c_3\{2k\}\). As a result, we are able to rewrite the 2D cumulants \({\mathcal {A}}_{mn}\) in terms of \(c_3\{2k\}\). It has been done in Ref. [28], and an expansion for \(p(v_3)\) has been found. On the other hand, for \(n=2\) the whole information of \(p(v_{2,x},v_{2,y})\) is not in \(c_2\{2k\}\). Therefore, we are not able to rewrite all \({\mathcal {A}}_{mn}\) in terms of \(c_2\{2k\}\) after averaging out the azimuthal direction of a 2D Gram–Charlier A series.
For completeness, we study the azimuthal averaging of Eq. (23) in the most general case in Appendix C. In this appendix, we show that the distribution in Ref. [28] is reproduced only by assuming \({\mathcal {A}}_{10}={\mathcal {A}}_{01}=0\). Also, we discuss the information we find from the averaged distribution compared to the twodimensional one. However, the method which we will follow in this section is different from that pointed out in Appendix C. Consequently, the most general series we will find here is not coincident with the distribution obtained in Appendix C.
The expansion (24) together with Eq. (27) is exactly the series found in Ref. [28] which is true for any odd n. This approximated distribution is called the radialGram–Charlier (RGC) distribution in Ref. [28].
It should be noted that it is a series for the case that \(\bar{v}_n=0\). In the following, we will try to find a similar series for \(p(v_n;\bar{v}_n)\) where \(\bar{v}_n\) could be nonvanishing.
In order to find the Gram–Charlier A series for the distribution \(p(v_n,;\bar{v}_n)\) in the general case, we come back to the iterative method explained in Sect. 3.1 where we found the distribution (20) by considering the ansatz (18) and iteratively solving the equations in (17).

Its leading order corresponds to the Bessel–Gaussian distribution.

In the limit \(\bar{v}_n\rightarrow 0\), the distribution approaches (24).
In order to show in how far the distribution (39) is a good approximation, we need to have a sample for \(p(v_n)\) where its \(\bar{v}_n\) is known. To this end, we generate heavy ion collision events by employing a hydrodynamic based event generator which is called iEBEVISHNU [41]. The reactionplane angle is set to zero in this event generator. Thus, we can simply find \(p(v_{n,x}v_{n,y})\) and subsequently \(\bar{v}_n\). The events are divided into 16 centrality classes between 0 to 80 percent and at each centrality class we generate 14000 events. The initial condition model is set to be MCGlauber.
Let us recover the notation in Eq. (29) and assume that \(p_q(v_n;\bar{v}_n)\) is the distribution (39) where the summation is done up to \(i=q\). We first compute the \(c_2\{2k\}\) and \(\bar{v}_2\) from iEBEVISHNU output and plug the results in Eq. (39). After that we can compare the original simulated distribution \(p(v_n)\) with the estimated \(p_q(v_2;\bar{v}_2)\). The results are presented in Fig. 1 for the events in 65–70\(\%\), 70–75\(\%\) and 75–80\(\%\) centrality classes, in which we expect the distribution is deviated from the Bessel–Gaussian. In this figure, the black curve corresponds to the Bessel–Gaussian distribution (\(p_0(v_n;\bar{v}_n)\)) and the red, green and blue curves correspond to \(p_q(v_2;\bar{v}_2)\) with \(q=2\), 3 and 4, respectively. Recall that \(q=1\) has no contribution because \(\ell _2\) vanishes. As can be seen in the figure, the black curve shows that the distribution is deviated from Bessel–Gaussian and the distribution \(p_q(v_2;\bar{v}_2)\) with \(q\ne 0\) explains the generated data more accurately.
In order to compare the estimated distributions more quantitatively, we plotted \(\chi ^2/\text {NDF}\), comparing the estimated distribution \(p_q(v_2;\bar{v}_2)\) with iEBEVISHNU output. We plotted the results in Fig. 2 for \(q=0,2,3,4,5,6\) and 7 for the events in the 65–70\(\%\), 70–75\(\%\) and 75–80\(\%\) centrality classes. The value of \(\chi ^2/\text {NDF}\) associated with the Bessel–Gaussian distribution is much greater than the others. Therefore, we multiplied its value by 0.878 to increase the readability of the figure.
3.3 New cumulants

Referring to Eq. (13), all the cumulants \(q_n\{2k\}\) for \(k\ge 1\) are vanishing for the distribution \(\delta (v_{n,x}\bar{v}_n,v_{n,y})\).

Referring to Eq. (16), the only nonzero \(q_n\{2k\}\) for the Bessel–Gaussian distribution is \(q_n\{2\}=2\sigma ^2\).

In the limit \(\bar{v}_n\rightarrow 0 \), the cumulants \(q_n\{2k\} \) approach \(c_n\{2k\}\).
It is important to note that although we have found \(q_n\{2k\}\) by RGC distribution inspiration, we think it is completely independent of that and there must be a more direct way to find \(q_n\{2k\}\) independent of the RGC distribution.
Concerning the difference between \(c_n\{2k\}\) and \(q_n\{2k\}\) in terms of the Gram–Charlier expansion, we should mention that the cumulants \(c_n\{2k\}\) appear as the coefficients of the expansion when we expand the distribution \(p(v_n)\) around a radialGaussian distribution [see Eq. (24)] while \(q_n\{2k\}\) are those that appear in the expansion around the Bessel–Gaussian distribution. Now, if the distribution we study is more Bessel–Gaussian rather than the radialGaussian, we need infinitely many \(c_n\{2k\}\) cumulants to reproduce the correct distribution. For instance, for the second harmonics all \(v_2\{2k\}\) are nonzero and have approximately close values. It is because we are approximating a distribution which is more Bessel–Gaussian rather than radialGaussian. On the other hand for the third harmonics, we expect that the underling distribution is more radialGaussian, and practically we see a larger difference between \(v_3\{2\}\) and \(v_3\{4\}\) compared to the second harmonics [8]. Based on the above arguments, we deduce that the \(q_n\{2k\}\) are a more natural choice for the case that \(\bar{v}_n\) is nonvanishing.
Nevertheless, the cumulants \(q_n\{2k\}\) (unlike \(c_n\{2k\}\)) are not experimentally observable because of the presence of \(\bar{v}_n\) in their definition. However, they are useful to systematically estimate the distribution \(p(v_n)\) and consequently estimate the parameter \(\bar{v}_n\). This will be the topic of the next section.
4 Averaged ellipticity and harmonic finesplitting of the flow
In this section, we would like to exploit the cumulants \(q_n\{2k\}\) to find an estimation for \(\bar{v}_n\). Note that if we had prior knowledge about one of the \(q_2\{2k\}\) or even any function of them [for instance \(g(q_2\{2\},q_2\{4\},\ldots )\)], we could find \(\bar{v}_n\) exactly by solving the equation \(g(q_2\{2\},q_2\{4\},\ldots )\) \(=0\) in principle. Because the cumulants \(c_n\{2k\}\) are experimentally accessible, one would practically solve the equation \(g(\bar{v}_n)=0\). Unfortunately we have no such prior knowledge as regards \(q_2\{2k\}\), but we are still able to estimate \(\bar{v}_n\) approximately by assuming some properties for \(p(v_n)\).
Let us concentrate on \(n=2\) from now on. Recall that \(q_2\{4\}=q_2\{6\}=\cdots =0\) corresponds to a Bessel–Gaussian distribution. This choice of cumulants is equivalent to a distribution with \(v_2\{4\}=v_2\{6\}=\cdots \), which is not compatible with the splitting of \(v_2\{2k\}\) observed in the experiment. As we discussed at the beginning of this chapter, we find \(\bar{v}_2\) by estimating any function of cumulants \(q_2\{2k\}\). Here we use the most simple guess for this function, which is \(g(q_2\{2\},q_2\{4\},\ldots )=q_2\{2k\}\). Therefore, the equation \(q_2\{2k\}=0\) for each \(k \ge 1\) corresponds to a specific estimation for \(p(v_2)\).
For \(k=1\), we have \(q_2\{2\}=0\), which means \(\bar{v}_2=v_2\{2\}\). For this special choice, all the \(\varGamma _{2k2}\) in Eq. (37) diverge unless we set all other \(q_2\{2k\}\) to zero too. As a result, this choice corresponds to the delta function for \(p(v_{2,x},v_{2,y})\).
Note that the above \(q_2\{2k\}\) are characterizing an estimated distribution \(p(v_2;\bar{v}_2\{4\})\). For such an estimated distribution, \(q_2\{6\}\) is proportional to the skewness introduced in Ref. [27]. Interestingly, \(q_2\{8\}\) is proportional to \(\varDelta _2\{4,6\}11\varDelta _2\{6,8\}\), which has been considered to be zero in Ref. [27]. However, here we see that this combination can be nonzero and its value is related to the cumulant \(q_2\{8\}\). In fact, the same quantity can be computed for a generic narrow distribution [29]. In turns out that this quantity can be nonvanishing in the small fluctuation limit.
The equations in (42) indicate that by assuming \(q_2\{4\}=0\) all the other cumulants of \(p(v_2;\bar{v}_2\{4\})\) are written in terms of the finesplitting \(\varDelta _2\{k,\ell \}\). Therefore, the distribution \(p(v_2;\bar{v}_2\{4\})\) satisfies all the finesplitting structure of \(v_2\{2k\}\) by construction.
One can simply check in how far the estimation \(\bar{v}_2\{2k\}\) is accurate by using a simulation. We exploit again the iEBEVISHNU event generator to compare the true value of \(\bar{v}_2\) (\(\bar{v}_2^{\text {True}}\)) with \(\bar{v}_2\{4\}=v_2\{4\}\). The result is depicted in Fig. 4 by brown and green curves for \(\bar{v}_2^{\text {True}}\) and \(\bar{v}_2\{4\}\), respectively. As the figure illustrates, \(\bar{v}_2\{4\}\) is not compatible with \(\bar{v}_2^{\text {True}}\) for centralities higher than \(50\%\) where we expect that a Bessel–Gaussian distribution does not work well. It should be noted that all other \(v_2\{2k\}\) for \(k>2\) never are close to the true value of \(\bar{v}_2\) in higher centralities because all of them are very close to \(v_2\{4\}\).
By using iEBEVISHNU generated data, we can check the accuracy of the \(\delta _{2k}=1\) estimation by comparing different values of \(\delta _{2k}\) (for \(k=2,3,4,5\)) calculated from the simulation. The result is plotted in Fig. 5. This figure confirms the difference between the estimators \(\bar{v}_2\{2k\}\) discussed above. The quantity \(\delta _{4}\) has the greatest deviation from unity. Also, we see that \(\delta _{8}\) deviates from unity for centralities above \(55\%\) while \(\delta _6\) (and \(\delta _{10}\)) is closer to 1 up to \(65\%\) centrality. Moreover, \(\delta _6\) (\(\delta _{10}\)) is larger than \(\delta _8\) for all centralities. This can be considered as a reason for the fact that \(\bar{v}_2\{8\}\) is less accurate than \(\bar{v}_2\{6\}\) (and \(\bar{v}_2\{10\}\)).
Furthermore, let us mention that the cumulant \(q_2\{6\}\) changes its sign (see Figs. 3 and 5) for the centralities around 60–\(65\%\). It means it is exactly equal to 0 at a specific point in this range, and we expect that \(\bar{v}_2\{6\}\) becomes exactly equal to \(\bar{v}_2^{\text {True}}\) at this point. This can be seen also in Fig. 4 where the red curve (\(\bar{v}_2\{6\}\)) crosses the brown curve (\(\bar{v}_2^{\text {True}}\)).
The situation for \(q_2\{10\}\) is very similar to \(q_2\{6\}\). As a result, it is not surprising that we find the estimator \(\bar{v}_2\{10\}\) to be similar to \(\bar{v}_2\{6\}\). We have found \(\bar{v}_2\{10\}\) by solving \(q_2\{10\}=0\) numerically. The result is plotted by a black curve in Fig. 4. As can be seen, the results of \(\bar{v}_2\{6\}\) and \(\bar{v}_2\{10\}\) are approximately similar.
Now, we are in a position to estimate the \(\bar{v}_2\) of the real data by using \(\bar{v}_2\{2k\}\). According to the above discussions, we expect that \(\bar{v}_2\{6\}\), \(\bar{v}_2\{8\}\) and \(\bar{v}_2\{10\}\) are closer to the real value of the averaged ellipticity \(\bar{v}_2\) compared to \(v_2\{4\}\) (or any other \(v_2\{2k\}\) for \(k>2\)). The result is plotted in Fig. 6. In finding the estimated \(\bar{v}_2 \), we employed \(v_2\{2k\}\) reported by the ATLAS collaboration in Ref. [18]. The value of \(\bar{v}_2\{4\}\) is exactly equal to \(v_2\{4\}\), which is plotted by the green curve in the figure. By plugging experimental values of \(v_2\{2k\}\) into Eqs. (36c), (36d) and (36e) and setting them to 0, we have numerically found \(\bar{v}_2\{6\}\) (red curve), \(\bar{v}_2\{8\}\) (blue curve) and \(\bar{v}_2\{10\}\) (black curve), respectively.^{19} The errors of \(\bar{v}_2\{10\}\) are too large for the present experimental data, and a more precise observation is needed to find a more accurate estimation. Exactly similar to the iEBEVISHNU simulation, the value of \(\bar{v}_2\{8\}\) is between \(v_2\{4\}\) and \(\bar{v}_2\{6\}\).^{20} Therefore, we expect the true value of the averaged ellipticity to be close to the value of \(\bar{v}_2\{6\}\).^{21}
In this section, we introduced a method to estimate \(p(v_2,\bar{v}_2)\). By considering the cumulants \(c_2\{2k\}\) of the true distribution \(p(v_2)\), we estimated \(\bar{v}_2\) by assuming that the cumulant \(q_2\{2k\}\) of \(p(v_2,\bar{v}_2)\) is zero for a specific value of k. These estimations for \(q_2\{6\}=0\) and \(q_2\{8\}=0\) are presented analytically in Eqs. (44) and (46) and also with red and blue curves in Fig. 4 numerically. We exploited a hydrodynamic simulation to investigate the accuracy of our estimations. We found that \(\bar{v}_2\{6\}\) is more accurate than \(\bar{v}_2\{8\}\). Finally, we found the experimental values for \(\bar{v}_2\{6\}\), \(\bar{v}_2\{8\}\) and \(\bar{v}_2\{10\}\).
Until now, we considered the cumulants \(v_n\{2k\}\) as an input to find an estimation for \(\bar{v}_n\). In the next section, we try to restrict the phase space of the allowed region of \(v_n\{2k\}\) by using cumulants \(q_n\{2k\}\).
5 Constraints on the flow harmonics phase space
Referring to Eqs. (44) and (46), we see that these estimators lead to real values for \(\bar{v}_2\{4\}\) and \(\bar{v}_2\{6\}\) only if we have \(11 \varDelta _2\{6,8\}\ge \varDelta _2\{4,6\}\ge 0\). In this section, we would like to investigate these constraints and their validity range.
Alternatively, it is well known that the initial eccentricity point \((\varepsilon _{2,x},\varepsilon _{2,y})\) is bounded into a unit circle [25], and it leads to a negative skewness for \(p(\varepsilon _{2,x},\varepsilon _{2,y})\) in noncentral collisions. By considering the hydrodynamic linear response, the skewness in \(p(\varepsilon _{2,x},\varepsilon _{2,y})\) is translated into the skewness of \(p(v_{2,x},v_{2,y})\) and the condition \(v_2\{4\}>v_2\{6\}\) [27]. However, it is possible that the nonlinearity of the hydrodynamic response changes the order in the inequality to the case that \(v_2\{6\}\) is slightly greater than \(v_2\{4\}\). This is compatible with the result which we have found from a more general consideration.
Now, we concentrate on Eq. (47c). Due to the complications in finding the analytical allowed values of \(v_2\{2k\}\), we investigate it numerically. First, we consider the case that \(\delta _6=\delta _8=1\). In this case, we fix a value for \(v_2\{4\}\) and after that randomly generate \(v_2\{6\}\) and \(v_2\{8\}\) between 0 to 0.15. Putting the above generated and fixed values into Eq. (47c), we find \(\bar{v}_2\) numerically. If the equation has at least one real solution, we accept \((v_2\{6\},v_2\{8\})\), otherwise we reject it. The result is presented as scatter plots in Fig. 7. As can be seen from the figure, some region of the \(v_2\{2k\}\) phase space is not allowed. The condition \(11 \varDelta _2\{6,8\}\ge \varDelta _2\{4,6\}\) (see the square root in (46)) indicates that the border of this allowed region can be identified with \(v_2\{8\}=(12v_2\{6\}v_2\{4\})/11\) up to order \(\varDelta _2\{2k,2\ell \}\). This is shown by a red line in Fig. 7. Alternatively, the numerically generated border of the allowed region slightly deviates from the analytical border line. It happens for the region that \(v_2\{4\}\) is considerably different from \(v_2\{6\}\) and \(v_2\{8\}\). The reason is that \(\varDelta _2\{2k,2\ell \}\) is not small in this region, and the condition \(11 \varDelta _2\{6,8\}\ge \varDelta _2\{4,6\}\) is not accurate anymore.
Let us combine the constraint obtained from Eqs. (47b) and (47c). For a more realistic study, we use the ATLAS data for \(v_2\{4\}\) as an input. Instead of using a fixed value for \(v_2\{4\}\), we generate it randomly with a Gaussian distribution where it is centered around the central value of \(v_2\{4\}\), and we have a width equal to the error of \(v_2\{4\}\). The result for 40–\(45\%\) centralities is presented in Fig. 8a. For this case, we expect that the Bessel–Gaussian distribution works well. As a result, we assume \(\delta _6\simeq \delta _8\simeq 1\) (see Fig. 5). From the ATLAS results [18], we have \(v_2\{4\}=0.112\pm 0.002\) in 40–\(45\%\) centralities. The black star in the figure shows the experimental value of \((v_2\{6\},v_2\{8\})\) (the ellipse shows the one sigma error without considering the correlations between \(v_2\{6\}\) and \(v_2\{8\}\)). The width of the bands is due to the one sigma error of \(v_2\{4\}\). As the figure shows, the experimental result is compatible with the allowed region of \((v_2\{6\},v_2\{8\})\).
For more peripheral collisions, we expect a nonzero value for \(\delta _{2k}\). In the 65–\(70\%\) centrality class (the most peripheral class of events reported by ATLAS in Ref. [18]), we have \(v_2\{4\}\simeq 0.093\pm 0.002\). According to our simulation in this centrality class, we expect the values of \(\delta _6 \) and \(\delta _8\) to be 0.88 and 0.8, respectively. However, here we do not choose fixed values for \(\delta _6\) and \(\delta _8\). Instead, we generate a random number between 0.8 to 1 and assign the result to both \(\delta _6\) and \(\delta _8\). The result is presented in Fig. 8b. Referring to this figure, the allowed region is compatible with the experiment similar to the previous case. For nonzero values of \(\delta _6\) and \(\delta _8\), the allowed region can be identified by \(v_2\{6\}=\delta _6^{1/6} v_2\{4\}\) and \(v_2\{8\}=\delta _8^{1/8}(12v_2\{6\}v_2\{4\})/11\). These two constraints (similar to Fig. 8a) are presented by two bands in Fig. 8b. In this case, the width of the bands is due to the inaccuracy in \(\delta _6\) and \(\delta _8\) together with \(v_2\{4\}\).
By considering the correlation between \(v_2\{6\}\) and \(v_2\{8\}\), the experimental one sigma region of the \(v_2\{6\}\)–\(v_2\{8\}\) space would not be a simple domain. Nevertheless, for the present inaccurate case which is depicted in Fig. 8, we are able to restrict the one sigma domain by comparing it with the allowed region showed by the blue dots.
6 Conclusion and outlook
In the present work, we have employed the concept of the Gram–Charlier A series to relate the distribution \(p(v_n)\) to \(c_n\{2k\}\). We have found an expansion around the Bessel–Gaussian distribution where the coefficients of the expansion have been written in terms of a new set of the cumulants \(q_n\{2k\}\). We have shown that the corrected Bessel–Gaussian distribution can fit the actual distribution \(p(v_n)\) much better than the Bessel–Gaussian distribution. The new cumulants \(q_n\{2k\}\) were written in terms of \(c_n\{2k\}\) and the averaged flow harmonic \(\bar{v}_n\). Because the only nonvanishing new cumulants are \(q_n\{2\}\) for a Bessel–Gaussian distribution, they are a more natural choice to study the distributions near the Bessel–Gaussian case than \(c_n\{2k\}\).
By using the cumulants \(q_n\{2k\}\), we could systematically introduce different estimations for \(p(v_n)\) and consequently relate the averaged ellipticity \(\bar{v}_2\) to the flow harmonic finesplitting \(v_2\{2k\}v_2\{2\ell \}\) for \(k,\ell \ge 2\) and \(k\ne \ell \). As a specific example for the \(\bar{v}_2\) estimator, we have shown that \(\bar{v}_2\{6\}\simeq v_2\{6\}\sqrt{v_2\{6\}(v_2\{4\}v_2\{6\})}\). We have used the iEBEVISHNU event generator to compare the true value of the \(\bar{v}_2\) to the estimated one; also, we have shown that the estimator \(\bar{v}_2\{6\}\) is more accurate than \(v_2\{2k\}\) for \(k>1\). As another application of new cumulants, we have constrained the phase space of the flow harmonics \(v_n\{2k\}\) to the region \(v_2\{4\}v_2\{6\}\gtrsim 0\) and \(12 v_2\{6\}11v_2\{8\}v_2\{4\}\gtrsim 0\). It is experimentally confirmed that \(v_2\{4\}v_2\{6\}>0\). But we need a more accurate experimental observation for the quantity \(12 v_2\{6\}11v_2\{8\}v_2\{4\}\).
One should note that we have shown the compatibility of the allowed phase space of \(v_2\{2k\}\) with experimental results of (highmultiplicity) Pb–Pb collisions. Recently, the flow harmonics were measured for p–p, p–Pb and lowmultiplicity Pb–Pb collisions by ATLAS [42]. In the light of \(q_2\{2k\}\) cumulants, it would be interesting to study the similarity and difference between the splitting of \(v_2\{2k\}\) in these systems and examine the compatibility of the results with the allowed region comes from \(q_2\{2k\}\).
Furthermore, we have only focused on the distribution \(p(v_n)\) in the present study. However, based on the observation of symmetric cumulants and eventplane correlations, we expect that a similar systematic study for the distribution \(p(v_1,v_2,\ldots )\) can connect this joint distribution to the observations. Such a study would be helpful to relate the initial state eventbyevent fluctuations to the observation. This would be a fruitful area for further work.
Footnotes
 1.
In Ref. [27], the constraint \(v_2\{4\}>v_2\{6\}\) is deduced from the initial eccentricity, and the fact that the initial eccentricity is bounded to a unit circle.
 2.
In Ref. [30], \({\mathcal {A}}_{mn}\) has been shown by \(W_{n,ab}\) (\(n=1,2,\ldots \) and \(a,b\in \{x,y\}\)). Also \({\mathcal {C}}_{mn}\) has been found by \(W_{0,n}\), \(W^s_{0,n}\) and \(W^c_{0,n}\).
 3.
In general \(p(v_{n,x},v_{n,y})\) is not rotationally symmetric. For instance \(p(v_{2,x},v_{2,y})\) does not have this symmetry in a noncentral collision of spherically symmetric ions, while \(p(v_{3,x},v_{3,y})\) does, for the same collisions.
 4.
We simply use the notation \(\varphi \equiv n\psi _n.\)
 5.
We ignore the subscript 1D or 2D when it is not ambiguous.
 6.
 7.
It is an important question whether is it possible to determine \(p(v_n)\) uniquely from its moments [32] (see also Ref. [33])? Answering to this question is beyond the scope of the present work. Here we assume that \(p(v_n)\) is Mdeterminate which means we can find it from its moments \(\langle v_n^{2q}\rangle \) in principle.
 8.
We use the orthogonality relation \(\int _0^{\infty }k\,J_{\alpha }( k r)J_{\alpha }( k r') \,dk=\delta (rr')/r\).
 9.
 10.
We consider the reasonable assumption that the widths of the Gaussian distribution in the \(v_{2,x}\) and \(v_{2,y}\) directions are the same.
 11.
A standard method for finding the Gram–Charlier A series of p(x) is reviewed in Appendix B.1.
 12.
 13.
In Eq. (24), we chose the coefficient expansion as \(\frac{(1)^n\ell ^{\text {odd}}_{2n}}{n!}\) for convenience.
 14.
Refer to the footnote 15 and set \({\mathcal {A}}_{10}=0\).
 15.
Obviously, there is no onetoone correspondence between \(c_n\{2k\}\) and \({\mathcal {A}}_{mn}\) due to the loss of information by averaging. Specifically, one find \(c_n\{2\}={\mathcal {A}}_{10}^2+{\mathcal {A}}_{01}^2+{\mathcal {A}}_{20}+{\mathcal {A}}_{02}\) (see Eq. (84a)). Note that by assuming \(\varPhi _{\text {RP}}=0\) we have \({\mathcal {A}}_{01}=\langle v_{n,y}\rangle =0\) and \({\mathcal {A}}_{10}=\langle v_{n,x}\rangle =\bar{v}_n\). Also, it is a reasonable assumption that \({\mathcal {A}}_{20}\simeq {\mathcal {A}}_{02}\) [see Refs. [27, 28]]. By choosing \(\sigma =\sigma _x=\sigma _y={\mathcal {A}}_{20}\simeq {\mathcal {A}}_{02}\), one approximates \(p(v_{n,x},v_{n,y})\) around a symmetric Gaussian distribution located at \((\bar{v}_n,0)\) where its width is exactly similar to the distribution \(p(v_{n,x},v_{n,y})\). In this case, we find \(c_n\{2\}=\bar{v}_n^2+2\sigma ^2\).
 16.
This is an argument presented in Ref. [45]. In the same reference the convergence condition of Eq. (20) (which is not our main interest here) can be found. It is shown that if p(x) is a function of bounded variation in the range \((\infty ,\infty )\) and the integral \(\int _{\infty }^{\infty } \text {d}x\,\text {e}^{x^2/4}p(x)\) is convergent, then the series (20) is convergent. Otherwise it might diverge.
 17.
As an exception, the quantity \(\chi ^2/\text {NDF}\) increases slightly in moving from \(q=2\) to \(q=3\) for the events in the 65–\(70\%\) centrality class. However, the overall trend is decreasing in general.
 18.
It is important to note that, although we extracted the explicit collision geometry effect but its footprint still exists in the fluctuations implicitly; e.g. for \(\bar{v}_2\ne 0\), the distribution is skewed, while for \(\bar{v}_2= 0\) it is not.
 19.
In detail, all the Eqs. (36c)–(36e) were written in terms of the moments \(\langle v_2^{2k}\rangle \). Considering the reported experimental distribution \(p(v_2)\) in Ref. [18], we are able to produce the covariance matrix associated with statistical fluctuations of the moments \(\langle v_2^{2k}\rangle \). Using the covariance matrix, we generated 10,000 random numbers by using a multidimensional Gaussian distribution. Employing each random number, we solved Eqs. (36c)–(36e) numerically and found the estimated \(\bar{v}_2\). We obtained the standard deviation of the final \(\bar{v}_2\) distribution as the statistical error of the \(\bar{v}_2\{2k\}\).
 20.
We have computed the quantities \(\bar{\varepsilon }_2\{4\}\), \(\bar{\varepsilon }_2\{6\}\) and \(\bar{\varepsilon }_2\{8\}\) for an ellipticpower distribution [25] which is a simple analytical model for the initial state distribution. We have observed exactly the same hierarchy, and we have seen that \(\bar{\varepsilon }^{\text {True}}\) is closer to \(\bar{\varepsilon }_2\{6\}\). This is evidence that this behavior is generic for the distributions of interest in heavy ion physics.
 21.
Comparing Fig. 4 with Fig. 6, one finds that the values of \(\bar{v}_2\{2k\}\) from simulation are relatively smaller than that obtained from the real data. This deviation is due to the difference in \(p_T\) range. In Fig. 6, we used the data from Ref. [18] where \(p_T>0.5\,\text {GeV}\), while the output of the iEBEVISHNU is in the range \(p_T \lesssim 4\,\text {GeV}\). For a confirmation of iEBEVISHNU output, we refer the reader to Ref. [20, 21], where \(v_2\{4\}\) is reported for \(p_T\) below \(3\,\text {GeV}\). The order of magnitude of \(\bar{v}_2\{2k\}\) in our simulation is compatible with that mentioned in Refs. [20, 21].
 22.
Note that after scaling \(x\rightarrow x/\sigma \), we have \((\text {d}/\text {d}x)\rightarrow \sigma (\text {d}/\text {d}x)\). Additionally, \(\kappa _r\) scales to \(\kappa _r/\sigma ^r\). These two scalings cancel each other and we find Eq. (64).
 23.
If we had considered \({\mathcal {A}}_{20}=\sigma _x\) and \({\mathcal {A}}_{02}=\sigma _y\) in Eq. (69), we would have found the energy density distribution around a Gaussian distribution which is not rotationally symmetric.
 24.
Note that \({\mathcal {A}}_{mn}=\sigma ^{m+n}h_{mn}\) for \(m+n=3,4\) based on our assumptions in this section.
 25.
In this case, we have \(c_n\{4\}=4{\mathcal {A}}_{11}^2+{\mathcal {A}}_{40}+2{\mathcal {A}}_{22}+{\mathcal {A}}_{04}\).
 26.Another class of MOPs are called type II multiple orthogonal polynomial. In this class, a multiindexed monic polynomial \({\mathcal {P}}_{\mathbf {n}}\) satisfies the following r orthogonality conditions:where \(0\le i\le r\) [39]. Type II MOPs are not used here.$$\begin{aligned} \int x^m {\mathcal {P}}_{\mathbf {n}} w_i(x) \text {d}x,\qquad 0\le m < n_i, \end{aligned}$$
Notes
Acknowledgements
The authors would like to specially thank Hessamaddin Arfaei, HM’s supervisor, for useful discussions, comments and providing guidance over HM’s work during this project. We would like to thank JeanYves Ollitrault for useful discussions and comments during the “IPM Workshop on Collective Phenomena & Heavy Ion Physics”. We also thank Ali Davody for providing us with the iEBEVISHNU data and for discussions. We thank Mojtaba Mohammadi Najafabadi and Ante Bilandzic for useful comments. We would like to thank Navid Abbasi, Davood Allahbakhshi, Giuliano Giacalone, Reza Goldouzian and Farid Taghinavaz for discussions. We thank the participants of the “IPM Workshop on Collective Phenomena & Heavy Ion Physics”.
References
 1.K.H. Ackermann et al., [STAR Collaboration], Phys. Rev. Lett. 86, 402 (2001). arXiv:nuclex/0009011
 2.R.A. Lacey [PHENIX Collaboration], Nucl. Phys. A 698, 559 (2002). arXiv:nuclex/0105003
 3.I.C. Park et al., [PHOBOS Collaboration], Nucl. Phys. A 698, 564 (2002). arXiv:nuclex/0105015
 4.K. Aamodt et al., [ALICE Collaboration], Phys. Rev. Lett. 105, 252302 (2010). arXiv:1011.3914 [nuclex]
 5.K. Aamodt et al., [ALICE Collaboration], Phys. Rev. Lett. 107, 032301 (2011). arXiv:1105.3865 [nuclex]
 6.S. Chatrchyan et al., [CMS Collaboration], Phys. Rev. C 87, no. 1, 014902 (2013) arXiv:1204.1409 [nuclex]
 7.G. Aad et al., [ATLAS Collaboration], Phys. Lett. B 707, 330 (2012). arXiv:1108.6018 [hepex]
 8.G. Aad et al., [ATLAS Collaboration], Eur. Phys. J. C 74, no. 11, 3157 (2014) arXiv:1408.4342 [hepex]
 9.J.Y. Ollitrault, Phys. Rev. D 46, 229 (1992)ADSCrossRefGoogle Scholar
 10.J. Barrette et al., [E877 Collaboration], Phys. Rev. Lett. 73, 2532 (1994). arXiv:hepex/9405003
 11.J. Barrette et al., [E877 Collaboration], Phys. Rev. C 55, 1420 (1997). Erratum: Phys. Rev. C 56, 2336 (1997) arXiv:nuclex/9610006
 12.A.M. Poskanzer, S.A. Voloshin, Phys. Rev. C 58, 1671 (1998). https://doi.org/10.1103/PhysRevC.58.1671. arXiv:nuclex/9805001 ADSCrossRefGoogle Scholar
 13.N. Borghini, P.M. Dinh, J.Y. Ollitrault, Phys. Rev. C 63, 054906 (2001). arXiv:nuclth/0007063 ADSCrossRefGoogle Scholar
 14.N. Borghini, P.M. Dinh, J.Y. Ollitrault, Phys. Rev. C 64, 054901 (2001). arXiv:nuclth/0105040 ADSCrossRefGoogle Scholar
 15.R.S. Bhalerao, N. Borghini, J.Y. Ollitrault, Phys. Lett. B 580, 157 (2004). arXiv:nuclth/0307018 ADSCrossRefGoogle Scholar
 16.R.S. Bhalerao, N. Borghini, J.Y. Ollitrault, Nucl. Phys. A 727, 373 (2003). arXiv:nuclth/0310016 ADSCrossRefGoogle Scholar
 17.J. Jia, S. Mohapatra, Phys. Rev. C 88(1), 014907 (2013). https://doi.org/10.1103/PhysRevC.88.014907. arXiv:1304.1471 [nuclex]ADSCrossRefGoogle Scholar
 18.G. Aad et al., [ATLAS Collaboration], JHEP 1311, 183 (2013). arXiv:1305.2942 [hepex]
 19.M. Miller, R. Snellings, arXiv:nuclex/0312008
 20.A.M. Sirunyan et al., [CMS Collaboration], arXiv:1711.05594 [nuclex]
 21.S. Acharya et al., [ALICE Collaboration], arXiv:1804.02944 [nuclex]
 22.S.A. Voloshin, A.M. Poskanzer, A. Tang, G. Wang, Phys. Lett. B 659, 537 (2008). arXiv:0708.0800 [nuclth]ADSCrossRefGoogle Scholar
 23.R.S. Bhalerao, M. Luzum, J.Y. Ollitrault, Phys. Rev. C 84, 054901 (2011). arXiv:1107.5485 [nuclth]ADSCrossRefGoogle Scholar
 24.L. Yan, J.Y. Ollitrault, Phys. Rev. Lett. 112, 082301 (2014). arXiv:1312.6555 [nuclth]ADSCrossRefGoogle Scholar
 25.L. Yan, J.Y. Ollitrault, A.M. Poskanzer, Phys. Rev. C 90(2), 024903 (2014). arXiv:1405.6595 [nuclth]ADSCrossRefGoogle Scholar
 26.H. Grnqvist, J.P. Blaizot, J.Y. Ollitrault, Phys. Rev. C 94(3), 034905 (2016). arXiv:1604.07230 [nuclth]ADSCrossRefGoogle Scholar
 27.G. Giacalone, L. Yan, J. NoronhaHostler, J.Y. Ollitrault, Phys. Rev. C 95(1), 014913 (2017). arXiv:1608.01823 [nuclth]ADSCrossRefGoogle Scholar
 28.N. Abbasi, D. Allahbakhshi, A. Davody, S.F. Taghavi, arXiv:1704.06295 [nuclth]
 29.J. Jia, S. Radhakrishnan, Phys. Rev. C 92(2), 024911 (2015). arXiv:1412.4759 [nuclex]ADSCrossRefGoogle Scholar
 30.D. Teaney, L. Yan, Phys. Rev. C 83, 064904 (2011). arXiv:1010.1876 [nuclth]ADSCrossRefGoogle Scholar
 31.A. Bilandzic, CERNTHESIS2012018Google Scholar
 32.J. Stoyanov, Determinacy of Distributions by Their Moments, Proceedings for International Conference on Mathematics and Statistical Modeling (2006)Google Scholar
 33.A. Bilandzic, C.H. Christensen, K. Gulbrandsen, A. Hansen, Y. Zhou, Phys. Rev. C 89(6), 064904 (2014). arXiv:1312.3572 [nuclex]ADSCrossRefGoogle Scholar
 34.J. Adam et al., [ALICE Collaboration], Phys. Rev. Lett. 117, 182301 (2016). arXiv:1604.07663 [nuclex]
 35.J. Jia, [ATLAS Collaboration], Nucl. Phys. A 910911, 276 (2013). arXiv:1208.1427 [nuclex]
 36.G. Aad et al., [ATLAS Collaboration], Phys. Rev. C 90(2), 024905 (2014). arXiv:1403.0489 [hepex]
 37.A. Bilandzic, arXiv:1409.5636 [nuclex]
 38.M.G. Kendall, The advanced theory of statistics (Charles Griffin and Company, London, 1945)Google Scholar
 39.M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematica and its Applications vol. 98, Cambridge University Press (2005)Google Scholar
 40.E. Coussement, W. Van Assche, Constr. Approx. 19, 237 (2003)MathSciNetCrossRefGoogle Scholar
 41.C. Shen, Z. Qiu, H. Song, J. Bernhard, S. Bass, U. Heinz, Comput. Phys. Commun. 199, 61 (2016). [arXiv:1409.8164 [nuclth]]ADSMathSciNetCrossRefGoogle Scholar
 42.M. Aaboud, et al., [ATLAS Collaboration], Eur. Phys. J. C 77, no. 6, 428 (2017). https://doi.org/10.1140/epjc/s1005201749881
 43.V.A. Kalyagin, Russ. Acad. Sci. Sb. Math. 82, 199 (1995)Google Scholar
 44.W. Van Assche, Contemp. Math. 236, 325 (1999)CrossRefGoogle Scholar
 45.H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Nineteenth printing, 1999)Google Scholar
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