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Monte Carlo Glauber model with meson cloud: predictions for 5.44 TeV Xe + Xe collisions

  • B. G. Zakharov
Open Access
Regular Article - Theoretical Physics

Abstract

We study, within the Monte-Carlo Glauber model, centrality dependence of the midrapidity charged multiplicity density \(dN_{ch}/d\eta \) and of the anisotropy coefficients \(\varepsilon _{2,3}\) in Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV and in Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV. Calculations are performed for versions with and without nucleon meson cloud. The fraction of the binary collisions, \(\alpha \), has been fitted to the data on \(dN_{ch}/d\eta \) in Pb + Pb collisions. We obtain \(\alpha \approx 0.09(0.13)\) with (without) meson cloud. The effect of meson cloud on the \(dN_{ch}/d\eta \) is relatively small. For Xe + Xe collisions for 0–5% centrality bin we obtain \(dN_{ch}/d\eta \approx 1149\) and 1134 with and without meson cloud, respectively. We obtain \(\varepsilon _2(\mathrm {Xe})/\varepsilon _2(\mathrm {Pb})\sim 1.45\) for most central collisions, and \(\varepsilon _2(\mathrm {Xe})/\varepsilon _2(\mathrm {Pb})\) close to unity at Open image in new window . We find a noticeable increase of the eccentricity in Xe + Xe collisions at small centralities due to the prolate shape of the Xe nucleus. The triangularity in Xe + Xe collisions is bigger than in Pb + Pb collisions at Open image in new window . We obtain \(\varepsilon _3(\text{ Xe })/\varepsilon _3(\text{ Pb })\sim 1.3\) at Open image in new window .

1 Introduction

It is believed that production of soft particles in AA collisions at RHIC and LHC energies occurs via formation of the quark-gluon plasma (QGP) that expands hydrodynamically as a near-ideal liquid [1, 2]. Hydrodynamic models have been successfully used for description of the data from RHIC and LHC on centrality dependence of hadron multiplicities and flow effects in AA collisions. The currently available LHC data on AA collisions have been obtained for Pb + Pb collisions at \(\sqrt{s}=2.76\) and 5.02 TeV. Recently at the LHC there has been performed a run for Xe + Xe collisions (\(A=129\)) at \(\sqrt{s}=5.44\) TeV. The data from this run will allow to study variation of the A-dependence of soft hadron production in AA collisions. One can expect that the flow effects for Xe + Xe collisions should be stronger than for Pb + Pb collisions because fluctuations in the initial entropy density should increase for nuclei with a smaller nucleon number. For this reason the data on Xe + Xe collisions are of great interest for testing the hydrodynamic picture of the QGP fireball evolution. The Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV have been discussed recently in Refs. [3, 4].

The hydrodynamic simulations of AA collisions require imposing the initial conditions for the entropy/energy distribution at the QGP production time \(\tau _0\sim 0.5-1\) fm [5, 6]. One of the popular approach for setting the initial conditions for the QGP fireball in AA collisions is the Monte-Carlo Glauber (MCG) wounded nucleon model [7, 8, 9, 10]. In Refs. [11, 12] we have developed a version of the MCG wounded nucleon model for nucleons with meson cloud. The meson-baryon Fock components of the nucleon play an important role in the flavor dependence of nucleon parton distribution functions (PDFs) in deep inelastic scattering [13], and allow to explain the violation of the Gottfried sum rule [13]. It is important that, similarly to the wounded nucleon model with constituent quarks [14, 15, 16, 17, 18, 19], the meson degrees of freedom lead to a nonlinear increase of \(dN_{ch}(AA)/d\eta \) with the number of the wounded nucleons [12]. This effect should emerge independently of the specific mechanism of inelastic interactions. It is important that, contrary to the MCG models with the quark subnucleon degrees of freedom, the interaction of the meson components is better understood, say, within the quark-gluon string model [20, 21]. Similarly to the ordinary two-component MCG model without meson cloud [9] the model [11, 12] accounts for the contributions from soft interaction (participant wounded nucleons) and from hard binary collisions [22]. However, the results of Refs. [11, 12] show that in the presence of the meson-baryon Fock components the required fraction of the binary collisions, \(\alpha \), becomes smaller. The results of Ref. [12] show that the meson cloud may improve somewhat agreement with the data on the dependence of the elliptic flow on the charged multiplicity for very small centralities defined via the ZDCs signals for collisions of the deformed uranium nuclei at \(\sqrt{s}=0.193\) TeV [23]. In the present paper we apply the model of Refs. [11, 12] to obtain predictions for Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV. To fix the parameters of the model we use the data on pp and Pb + Pb collisions.

The plan of the paper is as follows. In Sect. 2 we outline the theoretical framework. In Sect. 3 we present the numerical results. We give conclusions in Sect. 4.

2 Outline of the model

In this section we briefly sketch our MCG scheme. We refer the reader to Ref. [12] for more details.

We represent the physical nucleon wave function \( |N_{phys}\rangle \) in the infinite-momentum frame (IFM) as the Fock-state composition of a bare nucleon \(|N\rangle \) and an effective two-body meson-baryon \(|MB\rangle \) state
$$\begin{aligned} |N_{phys}\rangle =\sqrt{1-n_{MB}}|N\rangle + \int dxd{\mathbf{k}}\Psi _{MB}(x,{\mathbf{k}})|MB\rangle \,, \end{aligned}$$
(1)
where x is the fractional longitudinal meson momentum in the physical nucleon, \({\mathbf{k}}\) is the transverse meson momentum, \(\Psi _{MB}\) is the IMF (light-cone) wave function of the MB Fock state, and
$$\begin{aligned} n_{MB} =\int dxd{\mathbf{k}}|\Psi _{MB}(x,{\mathbf{k}})|^2 \end{aligned}$$
(2)
is the total weight of the MB Fock state. The previous analyses of the meson cloud effects in deep inelastic scattering (for a review, see Ref. [13]) show that the total weight of the meson-baryon Fock states in the nucleon is \(\sim 40\)%. Due to this, we take for the probability of the effective MB state \(n_{MB}=0.4\). Since the meson-baryon component is dominated by the pion-nucleon \(\pi N\) state [13], we calculate the IMF distribution of the effective MB state for the \(\gamma _5\) spin vertex. As in the analyses of the deep inelastic scattering [13], we introduce a phenomenological vertex formfactor, F, to account for the internal structure of the hadrons. We take it in the form [13]
$$\begin{aligned} F=\left( \frac{\Lambda ^2+m_N^2}{\Lambda ^2+M^{2}_{\pi N}}\right) ^2\,, \end{aligned}$$
(3)
where \(M_{\pi N}\) is the invariant mass of the \(\pi N\) system. We take \(\Lambda =1.3\) GeV, supported by the analysis [24] of the data on the process \(pp\rightarrow nX\). This value, at the same time, is also supported by the data on the violation of the Gottfried sum rule for the nucleon PDFs [13]. Note, however, that our results are not very sensitive to the value of \(\Lambda \).
In our model, as in the well known version of the MCG wounded nucleon model at the elementary nucleon-level GLISSANDO [9], the entropy is deposited in the soft sources from participants (related to soft interactions) and in the hard sources from the binary collisions, related to hard reactions, between the colliding particles. However, in our scheme the inelastic interaction of the physical nucleons from the colliding objects may occur as \(N+N\), \(N+MB\), \(MB+N\) and \(MB+MB\) collisions. Because at high energy the midrapidity multiplicity density for all baryons and mesons should be similar, we assume that the constituents M and B in the MB effective Fock state interact like a pion and a nucleon, respectively. We assume that the inelastic cross sections for the bare baryon and meson states obey the constituent quark counting rule \(4\sigma ^{NN}_{in}=6\sigma ^{MB}_{in}=9\sigma ^{MM}_{in}\). We use the Gaussian impact parameter profile for the probability of ab inelastic interaction of the bare constituents
$$\begin{aligned} P_{ab}(\rho )=\exp \left( -\pi \rho ^2/\sigma _{in}^{ab}\right) \,. \end{aligned}$$
(4)
The parameter \(\sigma _{in}^{NN}\) has been adjusted to fit the experimental inelastic pp cross section for non-single-diffractive (NSD) events. The use of the data on the NSD pp events is reasonable because the diffractive events do not contribute to the multiplicity in the midrapidity region that we consider.
We assume isentropic evolution of the QGP fireball. Then the initial entropy rapidity density produced in an AA collision is proportional to the final charged multiplicity pseudorapidity density
$$\begin{aligned} dS/dy=C dN_{ch}/d\eta \,, \end{aligned}$$
(5)
where \(C\approx 7.67\) [25]. For this reason in our calculations we consider the soft and the hard sources of the entropy as direct sources of the multiplicity density. We will consider the charged multiplicity density \(dN_{ch}/d\eta \) at the central pseudorapidity \(\eta =0\) defined as \(N_{ch}\) in the unit pseudorapidity window \(|\eta |<0.5\). We assume that the sources generated in all possible collisions of the bare constituents (i.e., for NN, MN, and MM collisions) have the same intensity. This approximation is supported by the calculations within the quark-gluon string model [20, 21] which show that the difference between the midrapidity multiplicity density generated in NN, MN and MM interactions is small.
We model the fluctuations of the charged particle density generated by the sources by the Gamma distribution
$$\begin{aligned} \Gamma (n,\langle n\rangle )= \left( \frac{n}{\langle n\rangle }\right) ^{\kappa -1} \frac{\kappa ^\kappa \exp \left[ -\,n\kappa /\langle n\rangle \right] }{\langle n\rangle \Gamma (\kappa )}\,, \end{aligned}$$
(6)
which is widely used in the MCG simulations. For each soft source corresponding to a wounded constituent the contribution to the multiplicity density is given by \((1-\alpha )\Gamma /2\), and for a hard source from a binary collision it is simply \(\Gamma \). However, for each pair of wounded particles the probability of a hard binary collision is suppressed by \(\alpha \). The parameters \(\langle n\rangle \) and \(\kappa \) have been adjusted to reproduce the experimental pp data on the mean charged multiplicity and its variance in the unit pseudorapidity window \(|\eta |<0.5\), and the value of \(\alpha \) has been fitted from the data on Pb + Pb collisions (see below).
For calculations of the multiplicity density \(dN_{ch}/d\eta \) one can use the approximation of the point-like sources. But the smearing of the sources may be important in calculations of the initial anisotropy coefficients \(\varepsilon _n\) of the QGP fireball, which in terms of the spacial entropy distribution \(\rho _s=dS/dyd\varvec{\rho }\) read [26, 27]
$$\begin{aligned} \varepsilon _n=\frac{\left| \int d\varvec{\rho }\rho ^n e^{in\phi }\rho _s(\varvec{\rho })\right| }{\int d\varvec{\rho }\rho ^n\rho _s(\varvec{\rho })}\,, \end{aligned}$$
(7)
where the transverse vectors \(\varvec{\rho }\) are calculated in the c.m. frame, i.e., \(\int d\varvec{\rho }\varvec{\rho }\rho _s(\varvec{\rho })=0\). To model the smearing of the sources we use a Gaussian source distribution
$$\begin{aligned} \exp {\left( -\varvec{\rho }^2/\sigma ^2\right) }/\pi \sigma ^2\,. \end{aligned}$$
(8)
We perform calculations for \(\sigma =0.7\) and 0.4 fm. The results for the anisotropy coefficients become sensitive to \(\sigma \) only for very peripheral AA collisions.
We perform calculations for the Woods–Saxon nuclear distributions with the hard-core repulsion. Following to Ref. [9], we take for the hard-core radius \(d=0.9\) fm. For the one-body Woods–Saxon distributions for \(^{129}\)Xe we use the \(\theta \)-dependent nuclear density
$$\begin{aligned} \rho _{A}(r,\theta )= & {} \frac{\rho _0}{1+\exp [(r-R_A(\theta )/a]}\,, \end{aligned}$$
(9)
$$\begin{aligned} R_A(\theta )= & {} R[1+\beta _2Y_{20}(\theta )+\beta _4Y_{40}(\theta )] \end{aligned}$$
(10)
with \(Y_{20}\) and \(Y_{40}\) the spherical harmonics, with \(\beta _2=0.162\), and \(\beta _4=-\,0.003\) [28]. For the \(^{208}\)Pb nucleus we use the ordinary spherically symmetric Woods–Saxon formula with a \(\theta \)-independent (\(\beta _{02}=\beta _{04}=0\)) radius \(R_A\). For the nucleus radii we use the formula \(R=(1.1A^{1/3}-0.656/A^{1/3})\) fm, and take \(a=0.459\) fm borrowed from Ref. [9]. To understand the role of the prolate shape of the Xe nucleus, we also perform calculations for Xe + Xe collisions using the spherically symmetric Woods–Saxon formula. As will be seen from our results the prolate shape of the Xe nucleus increases noticeably the ellipticity \(\varepsilon _2\) at small centralities.

3 Results

The direct pp data on the charged multiplicity and inelastic cross section at \(\sqrt{s}=5.02\) and 5.44 TeV are absent. We obtained \(dN_{ch}/d\eta \) for these energies with the help of the power law interpolation \(dN_{ch}/d\eta \propto s^{\delta }\) between the ALICE data [29] at \(\sqrt{s}=2.76\) TeV (\(dN_{ch}/d\eta \approx 4.63\)) and at \(\sqrt{s}=7\) TeV (\(dN_{ch}/d\eta =\,\approx 5.74\)) for the charged multiplicity in NSD events. It gives \(dN_{ch}/d\eta [\sqrt{s}=5.02\,,5.44\,\mathrm {TeV}] \approx [5.32,\,5.42] \). We used a similar procedure to obtain the NSD pp inelastic cross sections at \(\sqrt{s}=5.02\) and 5.44 TeV from the ALICE [30] results for \(\sqrt{s}=2.76\) TeV (\(\sigma _{in}^{NSD}\approx 50.24\) mb) and 7 TeV (\(\sigma _{in}^{NSD}\approx 58.56\) mb). We obtained the values: \(\sigma _{in}^{pp}[\sqrt{s}=5.02\,,5.44\,\text{ TeV }] \approx [54.44,\,56.18]\) mb.

Since the difference in the energy between Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV and Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV is relatively small, we use the same fraction of the binary collisions \(\alpha \) for \(\sqrt{s}=5.02\) and 5.44 TeV. We determine \(\alpha \) from fits to the ALICE [31] data on the centrality dependence of the midrapidity charged multiplicity density \(N_{ch}\) at \(|\eta |<0.5\) in Pb + Pb collisions \(\sqrt{s}=5.02\) TeV. As in Ref. [12], we use a two step procedure. First, we fitted \(\langle n\rangle \) and \(\kappa \) for a broad set of \(\alpha \) to the \(N_{ch}\) in pp collisions imposing the condition \(N_{ch}/D=1\), which is well satisfied for \(|\eta |<0.5\) window [29, 32]. Then, we used the values of \(\langle n\rangle \) and \(\kappa \) to fit the parameter \(\alpha \) from the Pb + Pb data. This procedure gives \(\alpha \approx 0.09\) and \(\alpha \approx 0.13\) for the scenarios with and without meson cloud, respectively. The parameters of the Gamma distribution (6) obtained from the fit with meson cloud to the pp data for the above optimal value \(\alpha =0.09\) read: \(\langle n\rangle \approx 4.69(4.8)\), \(\kappa \approx 0.516(0.524)\) for \(\sqrt{s}=5.02(5.44)\) TeV. For the scenario without meson cloud for the optimal value \(\alpha =0.13\) we obtained \(\kappa \approx 0.56(0.56)\) for \(\sqrt{s}=5.02(5.44)\) TeV, (in the version without meson cloud the value of \(\langle n\rangle \) is simply equal to the experimental \(N_{ch}\) for pp collisions).

In Fig. 1 we show the results of our fit to the ALICE data on the centrality dependence of the midrapidity charged multiplicity density in Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV [31] obtained by Monte Carlo generation of \(\sim 2\cdot 10^6\) events for the scenarios with and without meson cloud. The scenario with meson cloud gives somewhat better agreement with the data (\(\chi ^2/\text{ d.p }\approx 0.1\)) as compared to the version without meson cloud (\(\chi ^2/\text{ d.p }\approx 0.3\)).
Fig. 1

Centrality dependence of midrapidity \(dN_{ch}/d\eta \) for Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV. Left: MCG simulation for the scenario with meson cloud at \(\alpha =0.09\). Right: MCG simulation for the scenario without meson cloud at \(\alpha =0.13\). Data are from ALICE [31]

Fig. 2

Centrality dependence of midrapidity \(dN_{ch}/d\eta \) for Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV obtained for the scenarios with (solid) meson cloud at \(\alpha =0.09\) and without (dashed) meson cloud at \(\alpha =0.13\)

Fig. 3

Centrality dependence of the rms \(\varepsilon _2\) for the Gaussian source distribution (8) for \(\sigma =0.7\) fm for Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV (solid) and Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV for the \(\theta \)-dependent (dashed) and symmetric Woods–Saxon distribution (dotted). Left: the version with meson cloud at \(\alpha =0.09\). Right: the version without meson cloud at \(\alpha =0.13\)

In Fig. 2 we show our predictions for centrality dependence of the charged multiplicity in Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV obtained with and without meson cloud. The difference between two version is relatively small. For intermediate centrality region the meson cloud increases \(dN_{ch}/d\eta \) by \(\sim 5\)%. For the 0–5% centrality \(dN_{ch}/d\eta \approx 1149\) and 1134 for the versions with and without meson cloud, respectively. It is smaller by \(\sim 6-7\)% than the charged multiplicity density obtained in Ref. [4]. From Figs. 1 to 2 one sees that as compared to Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV for Xe + Xe 0-5% central collisions at \(\sqrt{s}=5.44\) TeV, \(dN_{ch}/d\eta \) becomes smaller by a factor of \(\sim 1.7\). That corresponds to decrease of the initial QGP temperature by a factor of \(\sim 1.2\). A remark is in order here. The curves in Fig. 2 are obtained under assumption of an isentropic flow. But we fitted the parameters of the model from the data on Pb + Pb collisions also ignoring the non-isentropic effects. Since these effects are weak, possible errors in the extrapolation of the results from Pb + Pb to Xe + Xe collisions should be small.
Fig. 4

Same as in Fig. 3 but for \(\sigma =0.4\) fm

Fig. 5

Centrality dependence of the rms \(\varepsilon _3\) for the Gaussian source distribution (8) for \(\sigma =0.7\) fm for Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV (solid) and Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV for the \(\theta \)-dependent Woods–Saxon distribution (dashed). Left: the version with meson cloud at \(\alpha =0.09\). Right: the version without meson cloud at \(\alpha =0.13\)

Fig. 6

Centrality dependence of the rms \(\varepsilon _{2,3}\) for the version with meson cloud with the Gaussian source distribution (8) for \(\sigma =0.7\) fm in Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV (left) and in Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV (right) for the \(\theta \)-dependent Woods–Saxon distribution with (solid) and without (dashed) the hard-core repulsion

Fig. 7

Centrality dependence of \(v_{2}\{2\}\) (filled circles) and \(v_3\{2\}\) (filled triangles) for Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV obtained from \(\varepsilon _{2,3}\) with meson cloud for \(\sigma =0.7\) (left) and 0.4 fm (right) with the help of the linear response approximation (see text for explanations) using the results of the hydrodynamic simulations of Ref. [3]. Open circles and triangles are \(v_{2,3}\{2\}\) from ALICE [41]

Fig. 8

Same as in Fig. 7 but for Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV

In Figs. 3 and 4 we present the results for the rms \(\varepsilon _2\) (the rms \(\varepsilon _n\) is often denoted \(\varepsilon _n\{2\}\), for clarity, we omit \(\{2\}\)) versus centrality for Pb + Pb at \(\sqrt{s}=5.02\) TeV and Xe + Xe at \(\sqrt{s}=5.44\) TeV for two version of the model. The anisotropy coefficients depend on the smearing parameter \(\sigma \) in (8). We present the results for two values of the Gaussian width of the sources \(\sigma =0.7\) and 0.4 fm. From Figs. 3 to 4 one sees that for small centralities the results with and without meson cloud are close to each other. For intermediate centralities the version with meson cloud gives a little smaller \(\varepsilon _2\). For very peripheral collisions with centrality Open image in new window the model without meson cloud gives bigger \(\varepsilon _2\). From comparing the results in Figs. 3 and 4 one can see that the value of the smearing width becomes important at Open image in new window , where the eccentricity grows with decreasing \(\sigma \). From Figs. 3 to 4 one sees that for the most central collisions Open image in new window , and \(\varepsilon _s(\mathrm {Xe})/\varepsilon _2(\mathrm {Pb})\) becomes close to unity at Open image in new window . For very peripheral collisions with Open image in new window becomes a little smaller than \(\varepsilon _2(\mathrm {Pb})\). From Figs. 3 to 4, by comparing the curves for Xe + Xe collisions obtained for the asymmetric and symmetric Woods–Saxon distribution, one can see that the effect of the prolate shape of the Xe nucleus on the eccentricity becomes important at Open image in new window . For most central collisions the asymmetric Woods–Saxon distribution increases the eccentricity by \(\sim 12\)%. This is due to the highly overlapping body-body collisions of the nucleus ellipsoids, that lead naturally to an asymmetric entropy deposition.

In Fig. 5 we present the results for the rms triangularity \(\varepsilon _3\) for \(\sigma =0.7\) fm. One can see that both for Pb + Pb and Xe + Xe collisions the triangularity for the versions with and without meson cloud are very similar for small and intermediate centralities. For very peripheral collisions Open image in new window for the version without meson cloud becomes bigger. Our calculations show that for Xe + Xe collisions the effect of the prolate form of the Xe nucleus is very small. For this reason we do not plot in Fig. 5 the results for Xe + Xe collisions for the symmetric Woods–Saxon distribution. From Fig. 5 one sees that \(\varepsilon _3(\mathrm {Xe})/\varepsilon _3(\mathrm {Pb})\sim 1.3\) for most central collisions, and decreases to unity at \(c\sim 70\)%.

The results shown in Figs. 1, 2, 3, 4 and 5 have been obtained for the Monte-Carlo sampling with the nuclear distribution with the hard-core radius \(d=0.9\) fm. It is evident that the presence of the hard-core repulsion should reduce the fluctuations of the nuclear matter density in the colliding nuclei, that can suppress the effect of the fluctuations in the initial entropy deposition. To study the role of the hard-core repulsion we also performed the calculations for \(d=0\). In this case we also use the parameters of the Woods–Saxon distribution from the analysis [9]: \(R=(1.12A^{1/3}-0.86/A^{1/3})\) fm, and \(a=0.54\) fm. Calculations in this version show that the effect of the hard-core repulsion on the centrality dependence of the charged multiplicity density is small. But the effect is stronger for the anisotropy coefficients. It is illustrated in Fig. 6 where we plot \(\varepsilon _{2,3}\) versus centrality obtained with (solid) and without (dashed) the hard-core repulsion for the version with meson cloud at \(\sigma =0.7\) fm. From Fig. 6 one sees that the effect of the hard-core is bigger for the purely fluctuation-driven quantity \(\varepsilon _3\). At Open image in new window the hard-core reduces \(\varepsilon _3\) by \(\sim 10\)% both for Pb + Pb and Xe + Xe collisions. For the eccentricity \(\varepsilon _2\) the effect of the hard-core is noticeable for small centralities. In this region \(\varepsilon _2\) is also dominated by fluctuations, and the hard-core repulsion reduces \(\varepsilon _2\) by \(\sim 10\)% both for Pb + Pb and Xe + Xe collisions.

Our results for the initial \(\varepsilon _{2,3}\) for Pb + Pb and Xe + Xe collisions do not differ strongly from that of Refs. [3, 4]. The analysis [3] is based on the TRENTO Monte-Carlo model [33, 34], and [4] is based on the so called EKRT [35] model with mini-jet parton production. For intermediate centralities \(c\sim 50\)% our \(\varepsilon _2\) for \(\sigma =0.7\) fm is smaller than that of Refs. [3, 4] by \(\sim 10-15\)%. But for small centralities our predictions for \(\varepsilon _{2,3}\) are somewhat bigger. Note that in Refs. [3, 4] the calculations for Xe + Xe collisions have been performed only with the symmetric Woods–Saxon nuclear distribution. For this reason the grows of the ratio \(\varepsilon _2(\mathrm {Xe})/\varepsilon _2(\mathrm {Pb})\) at \(c\rightarrow 0\) in Refs. [3, 4] is not so strong as in our results.

The event by event hydrodynamic modeling of AA collisions shows that, except for very peripheral collisions, to a good approximation \(v_n\approx k_n\varepsilon _n\) for \(n=2,~3\) [36, 37, 38, 39] (for a comprehensive review, see, e.g., Ref. [40]). This linear response approximation works better for \(v_2\) [38, 39]. In the present paper we do not perform the hydrodynamic simulation of Pb + Pb and Xe + Xe collisions that are necessary for accurate calculations of \(v_n\). To obtain predictions for the flow coefficients via our MCG results for \(\varepsilon _n\) we use the above linear response relation with \(k_{2,3}\) defined via the ratio of \(v_{2,3}\{2\}\) to rms \(\varepsilon _{2,3}\) obtained for Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV and for Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV in the hydrodynamic calculations with the shear viscosity over entropy ratio \(\eta /s=0.047\) in Ref. [3].1 In Fig. 7 we compare \(v_{2,3}\{2\}\) obtained in this way for Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV with the results from ALICE [41]. We used our anisotropy coefficients \(\varepsilon _{2,3}\) obtained with meson cloud for the Woods–Saxon distribution with the hard-core repulsion for the smearing width \(\sigma =0.7\) and 0.4 fm. From Fig. 7 one sees that for \(n=2\) the theoretical results underestimate the data by \(\sim 10-20\)% at \(c\sim 30-60\)%, but for \(n=3\) the agreement is quite reasonable. The theoretical predictions for \(v_{2,3}\{2\}\) in Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV are shown in Fig. 8. The relation between our predictions for \(v_n\{2\}\) to that of Ref. [3] similar to the situation with predictions for \(\varepsilon _n\). For most central collisions, due to bigger values of \(\varepsilon _{2,3}\), we predict somewhat bigger \(v_{2,3}\{2\}\) (by \(\sim 5\)% for \(\sigma =0.7\) fm and by \(\sim 10\)% for \(\sigma =0.4\) fm).

The predictions in Figs. 7 and 8 correspond to the version with the hard-core repulsions. From the curves in Fig. 6 one sees that the hard-core repulsion reduces the initial anisotropy coefficients due some reduction of the the nuclear density fluctuations. However, one should bear in mind that the dynamical effects, which can affect the long range fluctuations of the nuclear density, are completely ignored in our Monte-Carlo scheme. For this reason the predictions for the initial anisotropy coefficients may be questioned (especially for the fluctuation driven quantity \(\varepsilon _3\)). Indeed, it is well known [42, 43] that the dynamical long range fluctuations of the nuclear matter in heavy nuclei are dominated by the giant resonances, e.g., by the giant dipole, monopole, and quadrupole resonances. In Ref. [44] we have demonstrated that for the dipole mode of the \(^{208}\)Pb nucleus the classical treatment based on the Monte-Carlo simulation with the Woods–Saxon nuclear density overestimates the fluctuations of the dipole moment squared by a factor of \(\sim 5\). The situation with the dynamical quantum effects for the monopole and quadrupole collective modes, that potentially may also be important, remains unclear. It would be of great interest to study the role of the dynamical quantum effects due to the giant resonances on the fluctuations of the entropy deposition in AA collisions (of course, these effects may also be important for calculations in other approaches, say, in the EKRT model [35], TRENTO model [34], or in the color glass condensate scheme [45, 46]). We leave this for future work.

4 Conclusions

We have studied the centrality dependence of the charged midrapidity multiplicity density in Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV and in Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV within the MCG model with and without meson cloud developed in Ref. [12]. The parameters of the model have been fixed to the ALICE data [31] on \(dN_{ch}/d\eta \) in Pb + Pb collisions. We obtained the fraction of the binary collisions \(\alpha \approx 0.09(0.13)\) with (without) meson cloud. With these parameters we give predictions for future LHC data on Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV. We find that the effect of the meson cloud on the \(dN_{ch}/d\eta \) is relatively small. For Xe + Xe collisions the meson cloud increases \(dN_{ch}/d\eta \) by \(\sim 5\)% in the intermediate centrality region. For the 0-5% centrality bin we obtained \(dN_{ch}/d\eta \approx 1149\) and 1134 with and without meson cloud, respectively. As compared to Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV for Xe + Xe 0–5% central collisions at \(\sqrt{s}=5.44\) \(dN_{ch}/d\eta \) becomes smaller by a factor of \(\sim 1.7\). It corresponds to decrease of the initial QGP temperature by a factor of \(\sim 1.2\).

Both for Pb + Pb and Xe + Xe collisions we do not find a significant effect of the meson cloud on the \(\varepsilon _{2,3}\) at Open image in new window . But the meson cloud reduces \(\varepsilon _{2,3}\) for very peripheral collisions. We find that the ratio of the eccentricity in Xe + Xe collisions to that for Pb + Pb collisions is close to unity at Open image in new window , but it becomes bigger than unity at Open image in new window . We obtained \(\varepsilon _2(\text{ Xe })/\varepsilon _2(\text{ Pb })\sim 1.45\) for most central collisions ( Open image in new window ). We predict a noticeable increase of the eccentricity in Xe + Xe collisions at small centralities due to the prolate shape of the Xe nucleus. This effect gives \(\sim 50\)% to the difference between the eccentricity in most central Xe + Xe and Pb + Pb collisions.

We find that at Open image in new window the triangularity in Xe + Xe collisions is bigger than in Pb + Pb collisions. We obtain \(\varepsilon _3(\text{ Xe })/\varepsilon _3(\text{ Pb })\sim 1.3\) at Open image in new window and \(\varepsilon _3(\text{ Xe })/\varepsilon _3(\text{ Pb })\sim 1.1\) at \(c\sim 50\)%. We have investigated the effect of the hard-core repulsion in the Monte-Carlo sampling of the nuclear distributions. We found that the hard-core repulsion gives a relatively small effect on the charged multiplicity density. But its effect is sizeable for the anisotropy coefficients \(\varepsilon _{2,3}\). For \(\varepsilon _2\) the effect of the hard-core is noticeable for small centralities where \(\varepsilon _2\) is dominated by fluctuations. For most central collisions the hard-core repulsion reduces \(\varepsilon _2\) by \(\sim 10\)% both for Pb + Pb and Xe + Xe collisions. The triangularity \(\varepsilon _3\) is reduced by the hard-core repulsion by \(\sim 10\)% at Open image in new window both for for Pb + Pb and Xe + Xe collisions.

To obtain predictions for the flow coefficients via our MCG results for \(\varepsilon _n\) we have used the linear response relation \(v_n\approx k_{n}\varepsilon _n\) with \(k_n\) defined via the ratio of \(v_{2,3}\{2\}\) to rms \(\varepsilon _{2,3}\) obtained in the recent hydrodynamic analysis [3]. The results for \(v_{2,3}\{2\}\) obtained in this way for Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV are in reasonable agreement with the data from ALICE [41].

Footnotes

  1. 1.

    In principle, it is clear that this procedure should be quite accurate even beyond the linear response picture (say, when the cubic terms become important [39]) because the difference between our predictions for \(\varepsilon _{2,3}\) and that of Ref. [3] is not very large.

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© The Author(s) 2018

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Authors and Affiliations

  1. 1.L.D. Landau Institute for Theoretical PhysicsMoscowRussia

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