# Anisotropic flows and the shear viscosity of the QGP within an event-by-event massive parton transport approach

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## Abstract

We have developed an event-by-event relativistic kinetic transport approach to study the build up of the anisotropic flows \(v_{n}(p_T)\) for a system at fixed \(\eta /s(T)\). The partonic approach describe the evolution of massless partons which imply \(\epsilon =3p\) as Equation of State (EoS). We extend previous studies to finite partonic masses tuned to simulate a system that expand with an EoS close to the recent lQCD results. We study the role of EoS and the effect of \(\eta /s(T)\) ratio on the build up of \(v_n(p_T)\) up to \(n=5\) for two beam energies: RHIC energies at \(\sqrt{s}=200\) GeV and LHC energies at \(\sqrt{s}=2.76\) TeV. We find that for the two beam energies considered the suppression of the \(v_n(p_T)\) due to the viscosity of the medium have different contributions coming from the cross over or QGP phase. We shows that in ultra-central collisions (0–0.2%) the \(v_n(p_T)\) have a stronger sensitivity to the T dependence of \(\eta /s\) that increases with the order of the harmonic n. Finally, we discuss the results for the integrated flow harmonics \(\langle v_{n} \rangle \) in ultra-central collisions pointing-out how the relative strength of \(\langle v_{n} \rangle \) depend on the colliding energies as well as on the freeze-out dynamics.

## 1 Introduction

Ultra-relativistic heavy-ion collisions (uRHICs) experiments conducted at the relativistic heavy-ion collider (RHIC) and at large hadron collider (LHC) are the only tools to access experimentally the properties of hot strongly interacting matter. In the last decades it has been reached a general consensus confirmed by several experimental signatures and supported by several theoretical calculations that the matter created in these collisions consists of a strongly interacting quark-gluon plasma (QGP) rather than hadronic matter. One of the most important measurement was the large elliptic flow \(v_2(p_T)= \langle (p_x^2-p_y^2)/(p_x^2+p_y^2) \rangle \) observed [1, 2]. The elliptic flow \(v_2\) is a measure of the anisotropy in momentum space of the emitted particles. It is an observable that encodes information about the Equation of State (EoS) and transport properties of the matter created in these collisions [3, 4, 5]. The comparison between theoretical calculations and experimental results first within the viscous hydrodynamics framework [6, 7, 8] and in the recent years also within kinetic transport approach [9, 10, 11, 12, 13] have shown that the large value of \(v_2\) is consistent with a matter with a very low shear viscosity to entropy density ratio \(4\pi \eta /s \sim 1-2\). This value is close but larger then the conjectured lower bound for a strongly interacting system, \(\eta /s=1/4\pi \) [14].

In the recent years it has been possible to measure the event-by-event angular distribution of emitted particles. These measurements have made possible to extend this analysis to higher order harmonics \(v_n (p_T)\) [15, 16, 17]. The origin of these high order harmonics flows is attributed to the fluctuations in the initial geometry [18, 19, 20, 21, 22, 23, 24]. In the recent years most of the theoretical efforts has been focused on the development of the event-by-event viscous hydrodynamics simulations. The comparison between event-by-event viscous hydrodynamical calculations and the experimental results for \(v_n(p_T)\) seems to confirm a finite but not too large average value of \(\eta /s\) with \(4 \pi \langle \eta /s \rangle \sim 1-3\) [21, 22]. Early works, most hydrodynamic simulations of heavy ion collisions, have assumed a constant \(\eta /s\) but a small value of \(\langle \eta /s \rangle \) is not an evidence of the creation of a QGP phase formation. On the other hand, it is known that the \(\eta /s\) of the QGP is expected to have a temperature dependence with a minimum close to the cross over region [25, 26, 27, 28, 29]. A phenomenological estimation of its temperature dependence could give further information if the matter created in these collisions undergoes a phase transition [30, 31, 32].

In the recent years, the CMS, ATLAS, and ALICE collaborations have been able to measure the anisotropic flow coefficients \(\langle v_n \rangle \) in ultra-central heavy ion collisions [17, 33, 34]. These experiments have shown that the triangular flow \(n=3\) appears to be comparable or even larger of the elliptic flow \(n=2\). Such a result poses a problem to hydrodynamical approach that predict a significantly larger \(\langle v_2 \rangle \) w.r.t. \(\langle v_3 \rangle \) in ultra-central collisions. This discrepancy is even more intriguing considering that it occurs for ultra-central collisions where hydrodynamics is expected to work at best. In the recent years, several theoretical calculations within hydrodynamic simulations have shown that in ultra-central collisions the correlations that takes place between the initial eccentricities \(\langle \epsilon _n \rangle \) and the corresponding final anisotropic flows \(\langle v_n \rangle \) are the highest than other centralities. On the other hand, recently within a transport approach it has been pointed-out that at LHC energies and for ultra-central collisions a quite large correlation between \(\langle \epsilon _n \rangle \) and \(\langle v_n \rangle \) similar to viscous hydrodynamics is present up to \(n=5\) at variance with RHIC energies and/or other centralities [24]. At the same time a much larger sensitivity to \(\eta /s(T)\) has been spot. Therefore the study of the anisotropic flows in ultra-central collision offer a powerful tool infer information about the initial geometry of the fireball.

In this paper we study the role of the Equation of State and \(\eta /s (T)\) on the build up of \(v_n(p_T)\) up to the order 5 by using a kinetic transport approach with initial state fluctuations. The paper is organized as follows. In Sect. 2, we introduce the transport approach at fixed shear viscosity to entropy density \(\eta /s\). In Sect. 3, the implementation of the initial state fluctuations in the transport approach. In Sect. 4, we study the role of the EoS on the anisotropic flows and the time evolution of the anisotropic flows \(\langle v_n \rangle \). Finally in Sect. 5 we study the effect of the \(\eta /s(T)\) on the differential \(v_n(p_T)\). In this paper we will show results on \(v_n(p_T)\) for \(n=2,\ldots ,5\) for two different systems: \(Au+Au\) collisions at \(\sqrt{s}=200\) GeV and \(Pb+Pb\) collisions at \(\sqrt{s}=2.76\) TeV at different centralities.

## 2 Transport approach at fixed \(\eta /s\)

In this work the evolution of the matter created in these collisions is studied employing the kinetic transport theory. This study was performed using a relativistic transport code developed to perform studies of the dynamics of heavy-ion collisions at both RHIC and LHC energies [9, 11, 12, 35, 36, 37, 38, 39, 40]. Recently this transport code has been extended to include the initial state fluctuations in order to study the elliptic flow \(v_2\) and high order harmonics \(v_n(p_T)\) with \(n > 2\). For a more detailed discussion about the implementation of the initial state fluctuation see [24]. The development of the partonic transport equation has been done only for massless partons [9, 11, 12]. This implies that the matter simulated has an equation of state (EoS) \(\epsilon -3p=0\) which is different to the one evaluated by lQCD [41, 42] which clearly exhibit a large trace anomaly (\(T_{\mu }^{\mu } = \epsilon -3p \ne 0\)) and consequently a sound velocity \(c_{S}^2(T)\) that is significantly smaller than 1 / 3 also in the initial stages of a Heavy Ion Collisions reaching about 1 / 10 at \(T \approx T_C=155 \, MeV\). In this paper we extend the solution of the Relativistic Boltzmann Transport (RBT) equation to massive partons which allows to simulate a fluid with an EoS similar to the recent lQCD calculations.

*f*(

*x*,

*p*,

*t*) is given by solving the following RBT equation:

*C*[

*f*] is the Boltzmann-like collision integral. In this paper we have considered only the \(2 \leftrightarrow 2\) processes. The

*C*[

*f*] can be written as,

*f*(

*x*,

*p*) [37, 38, 47]. Furthermore, as will be discussed below, the kinetic freeze-out can be determined self-consistently by increasing \(\eta /s(T)\) at low temperature which gives a smooth switching-off of the scattering rates. The current disadvantage of the present approach is that hadronization has not yet been included and it will be the main topics of forthcoming studies. In the transport calculations shown in this paper we use massive particles providing a soften equation of state respect to the massless case with a decreasing speed of sound when the crossover region is approached similar to the one of lQCD calculations [41, 42].

*f*(

*z*) is given by

In Fig.1 it is shown a collection of different theoretical calculations about the T dependence of the \(\eta /s\) ratio. In literature there are several indications that \(\eta /s\) should have a particular behaviour with the temperature [25, 26, 27, 30, 31, 52]. As shown in Fig.1 in general \(\eta /s\) should have a typical behaviour of phase transition with a minimum close to the critical temperature \(T_C\) [30, 31, 32]. Estimations of \(\eta /s\) within chiral perturbation theory for a meson gas [25, 26], have shown that at temperature lower than the critical temperature the \(\eta /s\) is a decreasing function with the temperature, see down-triangles in Fig.1. Similar results have been obtained from extrapolation of heavy-ion collisions data at intermediate energies, see HIC-IE diamonds in Fig.1. At higher temperature \(T>T_c\) lQCD calculation on pure gauge have shown that in general \(\eta /s\) becomes an increasing function with the temperature [27, 28, 29], see up-triangles, circles and squares in Fig.1. However due to the large error bars in the lQCD results for \(\eta /s\) at moment it is not possible to infer a clear temperature dependence in the QGP phase.

In Fig.1 we show the three different T dependencies of \(\eta /s\) studied in this paper. In particular we have considered the following cases: one of constant \(4\pi \eta /s=1\) during all the evolution of the system shown by dot dashed line in Fig.1 another one of \(4\pi \eta /s=1\) at higher temperature in the QGP phase and an increasing \(\eta /s\) in the cross over region towards the estimated value for hadronic matter \(4\pi \eta /s \approx 6\) [26, 53] and shown by solid line. The third one is shown in Fig.1 by the dot dashed line. In this case we consider the increase of \(\eta /s\) at higher temperature with a linear temperature dependence and a minimum close to the critical temperature with a temperature dependence similar to that expected from general considerations as suggested by quasi particle models [51, 54] or by recent calculations on pure gauge [29]. Notice that in our approach an increase of \(\eta /s\) at lower temperature \(0.8 T_C \le T \le 1.2 T_C\) permits to account for a smooth realistic kinetic freeze-out (f.o.) because at lower T the mean free path \(\lambda \) goes like \(\lambda \propto \frac{\eta }{s} \frac{1}{T}\) and the total cross section goes like \(\sigma \propto (\eta /s)^{-1}\). In the following discussion the term f.o. means that we take into account the increase of \(\eta /s\) at \(T < 1.2 T_C\).

## 3 Initial conditions

*k*is an overall normalization factor fixed in order to reproduce the final hadron multiplicity \(dN_{ch}/dy\). In the results shown in this paper we have fixed the Gaussian width of the fluctuations to \(\sigma _{xy} = 0.5 \, fm\). In our calculation we assume a longitudinal boost invariant distribution from \(y=-2.5\) to \(y=2.5\).

*T*(

*x*,

*y*) is evaluated by using the standard thermodynamical relation for massive particles \(\rho _T(x,y)=\gamma T^3 z^2 K_{2}(z)/(2 \pi ^{2})\) with \(z=m/T\) and \(\gamma =2 \times (N_c^2-1) + 2 \times 2 \times N_c \times N_f=40\) with \(N_c=3\) and \(N_f=2\). In Fig. 3 it is shown the initial temperature profile in the transverse plane for a given event and for \(Au+Au\) collisions at \(\sqrt{s_{NN}}=200\) GeV (upper panel) at initial time of \(\tau _0=0.6\) fm/c and for \(Pb+Pb\) collisions at \(\sqrt{s_{NN}}=2.76\) TeV (lower panel) at initial time \(\tau _0=0.3\) fm/c. These plots are at mid rapidity and for central collisions.

While for partons with \(p_T > p_0\) we have assumed the spectrum of non-quenched minijets according to standard NLO-pQCD calculations with a power law shape [56, 57]. The initial transverse momentum of the particles is distributed uniformly in the azimuthal angle. We fix the starting time of the simulation to \(\tau _0=0.6\) fm/c for RHIC and \(\tau _0=0.3\) fm/c for LHC as commonly done also in hydrodynamical approaches.

## 4 Role of the equation of state on the \(v_n\)

*n*is related to the fact that higher harmonics have larger formation time and therefore they develop in a region where the \(p/\epsilon \) or the speed of sound is smaller. In fact for example the second coefficient \(\langle v_{2}\rangle \) is almost completely developed within 3–4 fm/c where the \(p/\epsilon \approx 0.3\) and close to the conformal limit and the effect increases for large

*n*because higher harmonics develop later at lower temperature where \(p/\epsilon < 0.3\).

In Fig. 6 we compare the final \(v_n(p_T)\) for both massless (dashed lines) and massive case (solid lines). These results are for \(Pb+Pb\) collisions at \(\sqrt{s_{NN}}=2.76\) TeV and for (20–30)% centrality collisions. We observe that for the elliptic flow \(v_2\) a mass ordering at low transverse momentum where larger is the mass smaller is the corresponding elliptic flow which is typical of hydrodynamic expansion that boost strongly massive particles [4, 60]. The explanation of the mass ordering is due to the fact that at low \(p_T\) the elliptic flow is \(v_{2}(p_T) \propto p_T -\langle \beta _{T}\rangle m_T\) therefore for light particles \(m_T = \sqrt{p_T^2+m^2} \approx p_T\) and \(v_2(p_T) \propto p_T\) while for massive particle the coupling of \(m_T\) with the radial flow gives at the same \(p_{T}\) a smaller \(v_2(p_T)\) and it is translated in a non linear behaviour at low \(p_T\). Moreover a similar mass ordering it is shown also for higher harmonics \(v_n(p_T)\). At low \(p_T\) the effect is to give a non linear behaviour for the second and third coefficients. Moreover we observe that the sensitivity to the EoS of the \(v_n(p_T)\) increases with the increasing of the harmonic *n* where for example the reduction is less of 10% for the third harmonic and about 15% for \(v_{4}(p_T)\) and 25 % for \(v_{5}(p_T)\). The enhancement of the sensitivity with the increase of the order of the harmonic is again related to the different formation time where for large *n* the corresponding \(v_n(p_T)\) develops later where the speed of sound \(c_{s}^2\) is smaller compared to the one in the conformal limit with \(p/\epsilon \approx 0.25\) at \(t \approx \, 3 fm/c\), see Fig. 4.

## 5 Effects of \(\eta /s(T)\) on the \(v_n(p_T)\)

In the recent years, it has been possible to access experimentally to the ultra-central collisions. In ultra-central collisions the initial asymmetry in coordinate space measured by \(\epsilon _n\) comes from the fluctuations in the initial geometry and there is no coupling with the global overlap shape. Moreover, it has been shown within event-by-event ideal and viscous hydrodynamics calculations [55, 61, 62] that for these collision centralities the final integrated anisotropic flows coefficient \(\langle v_{n}\rangle \) are strongly correlated to the initial eccentricities \(\langle \epsilon _{n}\rangle \) with a linear correlation coefficient \(C(\epsilon _n,v_n)\approx 1\). Similar results have been obtained also within event-by-event transport approach [24].

The comparison between RHIC and LHC highlights the role played by the EoS and the role of the collision energy in the build-up of \(\langle v_{n} \rangle \) in ultra central collisions. In ultra-central collisions, anisotropic flows coefficients are generated by density fluctuations in the initial state rather than by geometric overlap effects. On the other hand, the predicted anisotropic flow coefficients for charged hadrons with viscous hydrodynamics calculations with both the Monte Carlo Glauber (MC-Glb) and Monte Carlo Kharzeev–Levin–Nardi (MC-KLN) models produce initial fluctuations that gives final anisotropic flow spectrum as a decreasing function with the order of the harmonic n. On the other hand the CMS Collaboration has measured the anisotropic flow coefficients of charged hadrons in ultra-central Pb+Pb collisions at \(\sqrt{s}=2.76\) TeV corresponding to 0–0.2% centrality. The experimental measurement have shown that the triangular flow \(n=3\) appears to be comparable or even larger of the elliptic flow \(n=2\). Such a result poses a problem to hydrodynamical approach that predict a larger \(\langle v_2 \rangle \) w.r.t. \(\langle v_3 \rangle \) in ultra-central collisions. In Fig. 9 it is shown \(\langle v_{n} \rangle \) as a function of the order of the harmonic *n* for ultra-central collisions and for \(Au+Au\) collisions at \(\sqrt{s_{NN}}=200\) GeV (left panel) and \(Pb+Pb\) collisions at \(\sqrt{s_{NN}}=2.76\) TeV (right panel) As shown in the right panel of Fig. 9 by comparing circles with diamonds the effect of a soften Equation of State is to reduce the build-up of the \(\langle v_{n} \rangle \). In particular we observe a greater suppression for higher harmonics where for example for the massless case we get that \(\langle v_{4} \rangle \approx \langle v_{2} \rangle \) while for the massive case \(\langle v_{4} \rangle < \langle v_{2} \rangle \) with a suppression of about 15%. On the other hand the role of the collision energy is to reduce the production of the anisotropic flows and moreover it can play a role on the peak for \(n=3\). The different behaviour between RHIC and LHC is due to the fact that the degree of correlation between the initial \(\langle \epsilon _n \rangle \) and the final anisotropic flows produced \(\langle v_{n} \rangle \) is different. In fact the linear correlation coefficient between them \(C(\epsilon _n,v_m)\) is a decreasing function with the collision energy and impact parameter as shown for massless partons in [24]. This implies that for ultra-central collisions and at LHC energies the degree of correlation is maximum with \(C(\epsilon _n,v_m)\approx 1\) for \(n=2,\ldots ,4\). Therefore at higher energies like at LHC energies the \(\langle v_n \rangle \) keep more information about the initial anisotropies in the coordinate space and in particular they tend to keep the initial ordering of the eccentricities where for ultra-central collisions we have \(\langle \epsilon _2 \rangle< \langle \epsilon _3 \rangle < \langle \epsilon _4 \rangle \cdots \), see Fig. 2. At LHC it has been observed a \(\langle v_{2} \rangle \lesssim \langle v_{3} \rangle \) for ultra-central collisions. Our simulation show that such a feature should disappear at RHIC because of the impact of the increasing \(\eta /s(T)\) in the cross-over region. On the other hand at LHC a \(\langle v_{2} \rangle \lesssim \langle v_{3} \rangle \) survives because the freeze-out dynamics implied by \(\eta /s(T)\) does not affect the \(\langle v_{n} \rangle \) at high energy.

## 6 Conclusions

In this paper we have investigated the build up of the anisotropic flows \(v_{n}(p_T)\) for \(n=2,3,4\) and 5 within an event-by-event transport approach. We have studied the role of the EoS, by using a finite partonic mass, on the time evolution of \(\langle v_n \rangle \) and the initial eccentricities \(\langle \epsilon _n \rangle \) and on the differential \(v_{n}(p_T)\). We have found that the effect of the mass is to give a non linear behaviour at low \(p_T\) for the elliptic and triangular flow. In general the mass has the effect to reduce the final \(v_{n}(p_T)\) produced. The system is less efficient in converting the initial anisotropy in coordinate space into final anisotropic flows. Due to the different formation time of the harmonics the efficiency is lower for higher harmonics that start to develop at smaller speed of sound. Moreover, we have also studied the effect the temperature dependence of the \(\eta /s\) ratio on \(v_{n}(p_T)\) for two different beam energies: at RHIC for \(Au+Au\) collisions at \(\sqrt{s}=200\) GeV and at LHC for \(Pb+Pb\) collisions at \(\sqrt{s}=2.76\) TeV. We have found that at RHIC the \(v_n(p_T)\) are more affected by the value of \(\eta /s\) in the cross over region (\(T<1.2 T_C\)) and the sensitivity increases with the order of the harmonics. At LHC energies we get that almost all the \(v_n(p_T)\) develop in the QGP phase and they have less contamination of the value of \(\eta /s\) in the cross-over region and we observe a weaker sensitivity to the T dependence of the \(\eta /s\). These results are in qualitative agreement with the results obtained for the massless case. Such a scenario changes for ultra-central collisions. In particular in these collision an enhancement of the sensitivity to the value of \(\eta /s\) at lower temperature for RHIC is observed while at LHC the \(v_n(p_T)\) are more sensitive to the value at high temperature. In general we found an enhancement of the sensitivity of the \(v_n(p_T)\) for \(n=2,\ldots ,5\) that can reaches about a 30–35%.

## Notes

### Acknowledgements

I would like to thank V. Greco and F. Scardina for the fruitful and stimulating discussions.

### Data Availability Statement

This is a theoretical paper and this manuscript has non associated data. The data points of the figures can be provided by the author.

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