Extraction of the specific shear viscosity of quark-gluon plasma from two-particle transverse momentum correlations
Victor Gonzalez, Sumit Basu, Pedro Ladron de Guevara, Ana Marin, Jinjin Pan, Claude A. Pruneau
EExtraction of the specific shear viscosity of quark-gluon plasma fromtwo-particle transverse momentum correlations
Victor Gonzalez a, ∗ , Sumit Basu b, ∗∗ , Pedro Ladron de Guevara c,d , Ana Marin e, , Jinjin Pan f , Claude A. Pruneau a, a Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA b Lund University, Department of Physics, Division of Particle Physics, Box 118, SE-221 00, Lund, Sweden c Universidad Complutense de Madrid, Spain d Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, CP 04510, CDMX, Mexico e GSI Helmholtzzentrum f¨ur Schwerionenforschung, Research Division and ExtreMe Matter Institute EMMI, Darmstadt, Germany f Cyclotron Institute, Texas A & M University, College Station, Texas 77843, USA
Abstract
The specific shear viscosity, η/ s , of the quark-gluon plasma formed in ultrarelativistic heavy-ion collisions at RHICand LHC is estimated based on the progressive longitudinal broadening of transverse momentum two-particle correla-tors, G , reported as a function of collision centrality by the STAR and ALICE experiments. Estimates are computedas a function of collision centrality using the Gavin ansatz which relates the G longitudinal broadening to the specificshear viscosity. Freeze out times required for the use of the ansatz are computed using a linear fit of freeze out timesreported as a function of the cubic root of the charged particle pseudorapidity density (d N ch / d η ) / . Estimates of η/ s based on ALICE data exhibit little to no dependence on collision centrality at LHC energy, while estimates obtainedfrom STAR data hint that η/ s might be a function of collision centrality at top RHIC energy. Keywords:
Heavy-ion collisions, QGP, specific shear viscosity, transverse momentum correlations
1. Introduction
A key focus of the ultrarelativistic heavy-ion colli-sion programs conducted at the Large Hadron Collider(LHC) and the Relativistic Heavy-Ion Collider (RHIC)involves precision measurements of the properties ofthe quark-gluon plasma (QGP) formed in high-energynucleus-nucleus collisions. Of particular interest are themagnitude and temperature dependence of the specificshear viscosity of the QGP, expressed as the ratio η/ s ofthe shear viscosity η to the entropy density s of the mat-ter produced in the collisions. Shear viscosity charac-terizes the ability of a medium to transport momentumand carry deformations. Transverse particle anisotropypatterns, quantified in terms of anisotropic flow coe ffi -cients, measured in mid central heavy-ion collisions atRHIC and LHC are rather large and were, from the on-set, relatively well reproduced by viscous free hydrody-namical calculations thereby suggesting the QGP might ∗ [email protected] ∗∗ [email protected]@[email protected] be a perfect fluid, i.e., a fluid with vanishing or negligi-ble shear viscosity [1, 2, 3, 4, 5, 6]. The possibility thatthe high temperature, high density systems formed inthe midst of heavy-ion collisions might be a perfect fluidthus generated ‘quite’ a bit of excitement [7]. Consid-erable experimental and theoretical e ff orts were conse-quently expanded to determine the specific shear viscos-ity of the matter produced at RHIC and more recently atthe LHC [8, 9] based on measurements of anisotropicflow in the collision transverse plane. Although theo-retical e ff orts have been quite successful in reducing therange of η/ s compatible with state of the art measure-ments of flow anisotropies, there still remains a certaindegree of ambiguities owing to several technical di ffi -culties. One of these technical di ffi culties involves thelack of knowledge on the initial conditions of the sys-tems produced in A–A collisions at RHIC and LHC[5, 10, 11] . For instance, CGC inspired initial con-ditions yield larger initial spatial anisotropy than MCGlauber type initial conditions and thus require a some-what larger level of shear viscosity to match the ob-served flow coe ffi cients when used as input to viscoushydrodynamics simulations. While e ff orts to reduce the Preprint submitted to Elsevier December 22, 2020 a r X i v : . [ nu c l - e x ] D ec nitial conditions ambiguity based on measurements ofsymmetric cumulants [12, 13], in particular, have hadsome success, it remains of interest to identify tech-niques that might enable measurements of specific shearviscosity that are less susceptible to uncertainties asso-ciated with initial conditions. Such a technique exists.Proposed by Gavin et al. already more than a decadeago [14], it involves measurements of the longitudinalbroadening of a transverse momentum two particle cor-relators, now dubbed G , with increasing collision cen-trality. The correlator G , defined in Ref. [14, 15], isdesigned to be proportional to the covariance of mo-mentum currents and is as such sensitive to dissipativeviscous forces at play during the transverse and longitu-dinal expansion of the matter formed in A–A collisions.Gavin et al. showed these forces lead to a longitudinalbroadening of G measured as a function of the pseu-dorapidity di ff erence of measured charged particles. Asthe matter expands, neighboring fluid cells drag one an-other. Fast fluid cells tend to slow down whereas slowfluid cells accelerate. This has the e ff ect of dampeningthe expansion and produces a progressive broadening ofthe G correlator with time. The longer the system lives,the longer viscous e ff ects play a role, and the broaderthe G correlator becomes. Gavin et al. showed thebroadening, characterized in terms of the di ff erence ofthe variance of the correlator observed in most centraland most peripheral collisions, should be proportionalto η/ s and given by the following formula herein calledthe Gavin ansatz σ − σ = T c η s (cid:32) τ − τ c , f (cid:33) , (1)where σ c is the longitudinal width of the correlator mea-sured in most central collisions whereas σ is the longi-tudinal width of the correlator at formation time τ . T c and τ c , f are respectively the critical temperature and thefreeze-out time in most central collisions.We first briefly review, in sec. 2, prior e ff orts to de-termine η/ s based on the longitudinal broadening of the G correlator in A – A collisions. The method and re-sults of this work are presented in sec. 3 and discussedin sec. 4. Our conclusions are presented in sec. 5.
2. Prior estimates of η/ s based on transverse mo-mentum correlations A first estimate of the QGP viscosity based on theGavin ansatz was reported several years ago by theSTAR collaboration using a measurement of the chargeindependent correlator G CI2 in Au – Au collisions at √ s N N = . G correlator grows considerably frommost peripheral to most central Au – Au collisions.Given the observed broadening might arise in part fromother dynamical e ff ects, STAR used the Gavin ansatzto estimate an upper limit and reported η/ s to be inthe range 0.06 - 0.21 [16]. More recently, the AL-ICE collaboration reported precise measurements of theevolution of the longitudinal and azimuthal widths ofcharge independent and charge dependent two-particletransverse momentum correlators, G CI2 and G CD2 , respec-tively, as a function of the centrality of Pb – Pb col-lisions at √ s N N = .
76 TeV [17]. Examining specif-ically the overall change of the correlator longitudi-nal width from most peripheral to most central colli-sions, the collaboration concluded that their observa-tions favour small values of η/ s , that is, values closeto the KSS bound of 1 / π [18].The η/ s estimates reported by the ALICE and STARcollaborations focused on the overall change of the lon-gitudinal width G CI2 from peripheral to central collisionsbut did not utilize correlator widths observed in mid-central collisions. This omission resulted in large partfrom the lack of precise estimates of the system’s lifetime in mid-central collisions. E ff ectively, STAR andALICE did not consider the possibility that the viscositymight evolve with collision centrality and thus did notapply the Gavin ansatz to intermediate ranges of colli-sion centralities. We note, however, that the viscositymight in fact become a function of the collision central-ity if, in particular, the temperature or density of the pro-duced system or other conditions a ff ecting the viscosityevolve with centrality. It is also conceivable that otheraspects of the collision dynamics, not related to viscouse ff ects, could impact the broadening of the G vs. cen-trality. It is thus of interest to consider what the evolu-tion of the G correlator observed by STAR and ALICEimplies. Two specific questions arise. The first is con-cerned purely with the experimental technique used toestimate η/ s while the second concerns a possible evo-lution of the e ff ective shear viscosity of the system withcollision centrality.Let us first consider the experimental technique onits own merits. Is the technique sound? Are there ex-perimental artifacts that can bias or skew the evaluationof η/ s based on the Gavin ansatz? Indeed, the ansatzrequires estimates of a critical temperature T c , as wellas initial (formation) and freeze-out times τ and τ c , f ,respectively. These quantities are not evaluated in thecontext of the G measurement and thus require exter-nal inputs. They may thus be subjected to systematicbias of their own and independent of the STAR and AL-2CE measurements of the G correlator. Additionally,estimation of the broadening of the correlator might per-haps be biased by the finite acceptance or other artifactsof the measurement process. One might wonder, in par-ticular, whether the width observed in most central col-lisions could be underestimated because of the finite ra-pidity width of the acceptance of the measurements. Inthis context, it becomes of interest to study what pro-gressive changes of the width might imply about thestrength of the specific shear viscosity, and whether, inparticular, the evolution of the widths with centrality isself-consistent, that is, whether changes of the widthfrom one fractional cross section to the next are con-sistent with the overall change from most peripheral tomost central collisions.The second set of concerns is of greater interest,from a physical standpoint, but perhaps more di ffi cultto elucidate. Are viscous e ff ects strictly proportionalto the system lifetime? Can the correlator be a ff ectedby other physical e ff ects, such as, possibly, the radialand anisotropic expansion of the collision system? Isthe characteristic temperature used in the ansatz truly aconstant independent of the collision centrality? Andperhaps, most interestingly, could the e ff ective shearviscosity extracted from the measurement be a func-tion of collision centrality? Theoretical considerationssuggest η/ s is likely a function of the QGP tempera-ture [19, 20, 21, 13, 22, 23, 24, 25]. Is it then pos-sible that collisions at di ff erent impact parameter yieldsystems at di ff erent temperatures with slightly di ff erenttime evolution of the shear viscosity, thereby resultingin e ff ective or time-averaged shear viscosity that mightdepend on the collision centrality? Conceivably, an-swers to these questions may require more and betterdata than those available, but it is nonetheless of inter-est to consider what the available data can say about apossible evolution of η/ s with collision centrality andsystem temperature. It is thus the primary objective ofthis work to explore how η/ s values obtained with theGavin ansatz evolve with collision centrality.
3. Evolution of η/ s with system size We proceed with the evaluation of η/ s as a functionof (d N ch / d η ) / , based on the G CI2 longitudinal widthsalready reported by the STAR and ALICE collabora-tions [16, 17] in Au – Au collisions at √ s N N = . √ s N N = .
76 TeV, respectively, as afunction of the collision centrality using Eq. (1). How-ever, we also need estimates of the lifetimes τ f of thesystem with collision centrality. Estimates of freeze-out times reported as a function of measured charged particle densities, d N ch / d η , in [26], are used. Valuesof τ f are obtained from two-pion Bose-Einstein mea-surements from AGS to LHC energies [26]. Freeze-outtimes relevant for each of the centrality ranges consid-ered in this work are obtained by fitting a first degreepolynomial to estimated values τ f according to τ f = A · (d N ch / d η ) / . (2)Freeze-out times and associated uncertainties are listedin Tab. 1 and plotted in Fig. 1. Statistical and systematicerrors reported by the E895, CERES, NA49, PHOBOS,STAR and ALICE collaborations [26, 27, 28, 29, 30, 31,32, 33, 34, 35, 36] are used in the least square fit proce-dure. The driving criteria for the fit has been to repro-duce the results at 0.2 and 2.76 TeV in the best approx-imate way. Not a single linear fit complies with that forboth energies. The adopted fit procedure has been thento extract di ff erent fits and to evaluate the significance ofthe di ff erences of using each of them and assign them asa systematic uncertainty for the fit. The first fit consid-ers all the available results. Then for each energy of thetwo of interest two fits are carried suppressing the re-sult corresponding to the other energy, one forcing thepass through the origin and other without forcing any-thing. All fits yield a chi-square per degrees of freedomabout χ / ndf =
2. For the STAR energy, 0.2 TeV, the fitthat better reproduces the published result does not gothrough the origin and gives A = .
72 while for the AL-ICE energy, 2.76 TeV, is the one that goes through it andgives A = .
88. Fig. 1 shows the extrapolated decou-pling times for the charged particle density correspond-ing to the centrality classes on each of both energies.Error bands show the systematic uncertainty introducedby the fit procedure.Computation of the Gavin ansatz is accomplishedusing the canonical values T c = ± τ = . ± . / c for the critical temperature andformation time, respectively [37]. The value of σ isestimated by extrapolating the width of the correlator to (cid:104) N part (cid:105) =
2. Values of (d N ch / d η ) / for a given centralityclass are taken from [38, 36]. The STAR and ALICEcollaborations estimated shear viscosities, using Eq. 1,based exclusively on most central and most peripheralcollisions. In this letter, the collision centrality depen-dencies of the longitudinal widths reported by both ex-periments are used to investigate whether η/ s exhibits adependence on (d N ch / d η ) / . The longitudinal broaden-ing of the G correlator, defined below, is expected tobe insensitive to initial state density fluctuations in thetransverse plane. As such, it powerfully complementsstudies of η/ s based on measurements of anisotropicflow that su ff er in part from such a dependence [25].3easurements of G correlators additionally have a dif-ferent sensitivity to non-flow e ff ects which make theman invaluable tool in the understanding of the dynam-ics of A – A collisions, and as such, provide additionaltesting grounds of hydrodynamical and other types oftheoretical models.The ALICE measurements were reported in terms ofa dimensionless variant of the G correlator [14, 15] de-fined as G ( η , ϕ , η , ϕ ) = (cid:104) p T , (cid:105)(cid:104) p T , (cid:105) × (cid:34) S ( η , ϕ , η , ϕ ) (cid:104) n (cid:105)(cid:104) n (cid:105) − (cid:104) p T , (cid:105)(cid:104) p T , (cid:105) (cid:35) (3)with S ( η , ϕ , η , ϕ ) = (cid:42) n (cid:88) i n (cid:88) j (cid:44) i p T , i p T , j (cid:43) (4)where n ≡ n ( η , ϕ ) and n ≡ n ( η , ϕ ) are the num-ber of charged particle tracks detected, in each event,within bins centered at η , ϕ and η , ϕ , respectively.Sums are carried over particle transverse momenta p T , i , i ∈ [1 , n ], and p T , j , j (cid:44) i ∈ [1 , n ], respectively. Thebracket notation (cid:104) O (cid:105) is used to represent event ensem-ble averages computed within the bins η i , ϕ i , i = , (cid:104) n i (cid:105) and (cid:104) p T , i (cid:105) represent average number of par-ticles and average transverse momenta in bin η i , ϕ i , re-spectively. The reported ALICE measurement was lim-ited to charged particles with transverse momenta in therange 0 . ≤ p T < / c and pseudorapidities within | η | < . G C I correlator dependence on ∆ η and ∆ ϕ was parametrizedwith a two-component model defined as F ( ∆ η, ∆ ϕ ) = B + (cid:88) n = a n × cos ( n ∆ ϕ ) + A γ ∆ η ω ∆ η Γ (cid:18) γ ∆ η (cid:19) e − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ ηω ∆ η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ ∆ η γ ∆ ϕ ω ∆ ϕ Γ (cid:18) γ ∆ ϕ (cid:19) e − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ ϕω ∆ ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ ∆ ϕ , (5)where B and a n describe the long-range mean corre-lation strength and azimuthal anisotropy, respectively,while the bidimensional generalized Gaussian, whoseshape is determined by the parameters A , ω ∆ η , ω ∆ ϕ , γ ∆ η and γ ∆ ϕ , is used to model the correlation signal of in-terest. The ALICE collaboration reported longitudinalwidths σ ∆ η computed as the standard deviation of thegeneralized Gaussian σ ∆ η = (cid:115) ω ∆ η Γ (3 /γ ∆ η ) Γ (1 /γ ∆ η ) (6) along ∆ η [17]. These values are plotted as a functionof the number of collision participants estimated fromGlauber models [39] in Fig. 2.Instead of using a fitting procedure, the STAR col-laboration estimated the longitudinal width of measuredcorrelators by computing the rms of one-dimensionalprojections of G correlators onto the ∆ η axis [16].These rms width values are plotted vs. the number ofcollision participants in Fig. 2.All parameters (d N ch / d η , G CI2 widths and τ f ) used inthe computation of η/ s as well as the extracted values of η/ s at RHIC and LHC energies are listed in Tab. 1 as afunction of collision centrality expressed in terms of thefractional cross section: the range 0–5% corresponds tomost central collisions while ranges 5–10%, 10–20%,etc, represent collisions with increasingly larger impactparameters. Our analysis, based on STAR and ALICE islimited to quasi-peripheral collisions up to the range 70–80%, beyond which the applicability of Gavin’s modelmight be put into question. Estimates of η/ s computedwith Eq. 1 based on the above widths and freeze-outtimes are listed, for both RHIC and LHC energies, inthe two right-most columns of Tab. 1 . The values of η/ s vs (d N ch / d η ) / are plotted in Fig. 3. Statistical andsystematic uncertainties, the last ones incorporating theuncertainties from the fit procedure for freeze-out timesextraction, are displayed with vertical bars and rectan-gular boxes, respectively. æh /d ch N d Æ ) c ( f m / f t E895 2.7E895 3.3E895 3.8E895 4.3NA49 8.7NA49 12.5NA49 17.3CERES 17.3 STAR 200.0PHOBOS 62.4PHOBOS 200.0STAR 62.4ALICE 2760.0
ALICE extrapolatedSTAR extrapolated
Figure 1: Open symbols: Compilation of decoupling (freeze-out)times, τ f , plotted as a function of the cubic root of the charged par-ticle density, (d N ch / d η ) / , observed in various collision systems andfor a wide range of beam energies [26]. Filled symbols: extrapolatedvalues of τ f corresponding to each of the centrality classes used inour computation of η/ s based on Au – Au collisions at √ s NN = . √ s NN = .
76 TeV, measured by theSTAR and ALICE collaborations, respectively. Red / blue solid linesrepresent systematic uncertainties of the polynomial fits to the data. Computed values of η/ s range from 0 . ± . sys to4entrality dN ch / d η G CI σ ∆ η τ f (fm / c ) η/ s LHC0– 5% 1601.00 ± ± sta + sys − sys . ± . ± . fit . ± . sta ± . sys ± ± sta + sys − sys . ± . ± . fit . ± . sta ± . sys ± ± sta + sys − sys . ± . ± . fit . ± . sta ± . sys ± ± sta + sys − sys . ± . ± . fit . ± . sta ± . sys ± ± sta + sys − sys . ± . ± . fit . ± . sta ± . sys ± ± sta + sys − sys . ± . ± . fit . ± . sta ± . sys ± ± sta + sys − sys . ± . ± . fit . ± . sta ± . sys ± ± sta + sys − sys . ± . ± . fit . ± . sta ± . sys ± ± sta + sys − sys . ± . ± . fit . ± . sta ± . sys RHIC0– 5% 691.00 ± ± sta ± sys . ± . ± . fit . ± . sta ± . sys ± ± sta ± sys . ± . ± . fit . ± . sta ± . sys ± ± sta ± sys . ± . ± . fit . ± . sta ± . sys ± ± sta ± sys . ± . ± . fit . ± . sta ± . sys ± ± sta ± sys . ± . ± . fit . ± . sta ± . sys ± ± sta ± sys . ± . ± . fit . ± . sta ± . sys ± ± sta ± sys . ± . ± . fit . ± . sta ± . sys ± ± sta ± sys . ± . ± . fit . ± . sta ± . sys Table 1: Compilation of measured charged particle densities, dN ch / d η , and longitudinal widths, σ ∆ η , of the G CI correlator, interpolated freeze-outtimes, τ f , and computed values of η/ s as a function of the centrality of Pb – Pb collsions at √ s NN = .
76 TeV [40, 26, 17, 38] and Au – Au at √ s NN =
200 GeV [36, 16] æ part N Æ hDs C I G = 2.76 TeV NN s Pb-Pb at = 0.2 TeV NN s Au-Au at
Figure 2: Longitudinal width, σ ∆ η , of the G CI2 correlator vs. theestimated number of participants measured in Au – Au collisionsat √ s NN =
200 GeV [16] and in Pb – Pb collisions at √ s NN = .
76 TeV [17], reported by the STAR and ALICE collaborations, re-spectively. Error bars and error boxes represent statistical and system-atic uncertainties, respectively. . ± . sys and from a value compatible with 0 to0 . ± . sta ± . sys for LHC energies and RHIC en-ergies, respectively. One observes that values extractedfrom Pb – Pb collisions at the LHC exhibit a weak de-pendence on (d N ch / d η ) / , while those from Au – Aucollisions, measured at RHIC, show a rising trend with increasing (d N ch / d η ) / , albeit with large uncertainties.Values of η/ s obtained from Pb – Pb collisions are closebut somewhat lower than the KSS bound of 1 / π , whilethose obtained from Au – Au collisions are compatiblewith vanishing viscosities in the range (d N ch / d η ) / < N ch / d η ) / ≈ η/ s values derived from STAR and ALICEdata are compatible with one another at the one σ levelat all values of (d N ch / d η ) / .
4. Discussion
A compilation of η/ s values obtained in this work aswell as those reported in theoretical and phenomenolog-ical calculations is presented in Fig. 4. Somewhat oldercompilations have also been reported [58]. A compara-tive analysis of the results presented in this compilationis complicated in part by the fact that viscous e ff ects arelikely to accumulate throughout a system’s evolution.However, the shear and bulk viscosities may depend onthe temperature, matter density, the presence of mag-netic fields, and possibly other system conditions, thatevolve as the QGP expands and goes through a transi-tion into a hadron phase. The model used in this workand several of the calculations listed in the compilationneglect such a time / temperature dependence and repre-5 æh /d ch N d Æ s / h = 2.76 TeV NN s Pb-Pb at = 0.2 TeV NN s Au-Au at
Figure 3: Values of the shear viscosity per unit of entropy density, η/ s , computed in this work, as a function of the cubic root of thecharged particle density, (d N ch / d η ) / , measured in Pb – Pb collisionsat √ s NN = .
76 TeV [17] and in Au – Au collisions at √ s NN =
200 GeV [16]. Error bars and error boxes represent statistical andsystematic uncertainties, respectively. sent the viscosity as a single e ff ective value, while oth-ers attempt to account for time and temperature depen-dencies using various prescriptions. The horizontal spanof the lines displayed in Fig. 4 is thus meant to repre-sent either the range of e ff ective η/ s values constrainedby comparisons with experimental data or the ranges of η/ s values considered in the models and yielding a goodrepresentation of the measured data. It should thus notbe considered as a statement of the precision achievedin the studies included in this compilation.Comparative studies of hydrodynamics and mea-sured data arguably culminated with studies based ona Bayesian estimation of the properties of the QGP [25,41, 42, 43] yielding most probable η/ s values in rathergood agreement with the results of this work. Esti-mates of η/ s obtained in this work are also in quan-titative agreement with QCD inspired calculations in-cluding, for instance, estimates based on nonperturba-tive gluon spectral functions at finite temperature inquenched QCD with the maximum entropy method [23]and calculations based on the Kubo formula in Yang-Mills theory [24], Results of this work are also inqualitative agreement with estimates based on latticeQCD (LQCD) calculations of the QGP transport coe ffi -cients [44] and perturbative QCD calculations at almostNLO [45], as well as recent calculation based on theMUSIC framework that used a temperature dependent η/ s computed with a QCD based approach [46].Taking the estimates of η/ s shown in Fig. 3 at facevalue, it is interesting to consider whether they mighthave any implications concerning the nature and prop- -
10 1 ) s / h ( T i m e e v o l u t i on K SS L i m i t H e li u m ( H e ) H o l og r a ph i c B ound s [2] (Ideal Hydro)[56] (Lattice QCD)[53] (Light Quark Flow)[49] (Gluon Gas) U l t r ac o l d F e r m i g as [50] [11] (Hydro) [14] (Correlation & Fluctuation) [57] (Gluon Plasma)[52] (Heavy Quark Flow)[5] (Viscous Hydro+MC KLN) RHICLHC [8] (IP-Glasma) [54] (Hadron Gas)
LHC (This Work) O ) W a t e r ( H RHIC (This Work)/s(T)) h [25,44] (Bayesian [42,43] (JETSCAPE) [45] (Lattice QCD)/s(T) QCD Based) h [23,24] ([46] (NLO QCD)/s(T) + bulk (T)) h [47] ( = M e V c a t T = M e V c a t T Figure 4: Comparison of η/ s obtained in this work with a collection ofvalues published since 2000 [25, 41, 42, 43, 23, 24, 44, 45, 46, 47, 48,49, 50, 51, 52, 53, 54, 55, 56, 57], including both theoretical valuesand experimental values obtained from direct comparisons of theoryto data. Some more details are given in the text. erties of hot QCD matter produced in heavy-ion colli-sions at RHIC and LHC. First consider that estimates ofthe initial temperature reached in central Pb – Pb colli-sions at LHC suggest it is of the order of 300 MeV, i.e.,30% larger compared to that achieved at RHIC in cen-tral Au – Au collisions [59, 60, 61, 62]. Also considerthat the fireball formed in Pb – Pb collisions at the LHChave been estimated to live approximately 40% longerthat those produced in Au – Au collisions at RHIC [26].This implies that shear viscous forces have more time tooperate in central Pb – Pb collisions at LHC than in Au –Au at RHIC. For systems of equal η/ s and temperature,one would expect to observe a large longitudinal broad-ening of the G correlator in Pb – Pb but the observedbroadening is in fact smaller than that seen in centralAu – Au collisions. Taken at face value, this suggeststhat the e ff ective shear viscosity per unit of entropy issmaller in Pb - Pb at 2.76 TeV. We should stress, how-6ver, that the extracted values of η/ s reflect the completeevolution of systems formed in A – A collisions. It isconsequently incorrect to associate and use a particularsystem temperature to evaluate the shear viscosity. In-deed, estimates should account for possible evolution of η/ s with temperature explicitly or be based on an ap-propriate e ff ective, time averaged, system temperature.The interpretation of the data is further complicated bythe likely presence of kinematic narrowing associatedto radial flow. The average transverse momentum, (cid:104) p T (cid:105) ,is found to be approximately 10% larger at LHC ener-gies compared to RHIC. This increase may in part resultfrom faster radial flow at the TeV energy scale. It is wellestablished that strong radial flow produces a sizablenarrowing of two particle correlators, such as balancefunctions B , as well as generic number and transversemomentum correlators R and P , respectively. A sim-ilar narrowing is thus expected also for G and has infact been found to occur in Pb – Pb collisions: the G CD2 correlator, in particular, exhibits a significant narrowingfrom peripheral to central Pb – Pb collisions reportedby the ALICE collaboration [17]. While this narrow-ing is most easily and explicitly observed for unlike-charge particle pairs, it should also be occurring for like-sign pairs contributing to the G CI2 correlator. Narrowinge ff ects associated with kinematic focusing might thenpartially counterbalance the broadening due to viscousforces, and thus e ff ectively reduces values of η/ s ex-tracted from both the ALICE and STAR data. But giventhe radial flow is likely somewhat stronger at LHC, thatcould imply the di ff erence seen between central Au –Au and Pb – Pb collisions is in part due to the presenceof extra focusing at LHC energy.Additional theoretical calculations [63] suggest that η/ s should increase with decreasing collision energywithin the RHIC energy domain in part as a result ofan explicit dependence on the matter baryochemicalpotential µ B [64]. Studies of relative yields of pro-duced hadrons indicate that the baryochemical potentialis nearly vanishing at central rapidities in Pb – Pb col-lisions, with values of order µ B ∼ µ B ∼
20, were extracted based on Au – Au collisions atRHIC top energy [67]. Di ff erences of η/ s observed bySTAR and ALICE collaborations might thus also resultin part from this change of the baryochemical potential.While we note that the precision of the data is clearly in-su ffi cient to establish any firm conclusion on such a de-pendence, we stress that precise measurements of the G correlator in the context of the second RHIC beam en-ergy scan (BES-II) might in fact provide better groundsto seek evidence of this dependence. Studies of the G correlator with RHIC beam energy scan data are thusindeed of high interest.Other considerations are also of interest. Collisionsof large nuclei at ultra-high energy, both at RHIC andLHC, are expected to produce very large magnetic fieldsand have been predicted to induce large vorticity andglobal polarization e ff ects as well as finite out of planecharge separation associated with the chiral magnetice ff ect (CME). While the existence of the CME remainsto be established, both STAR and ALICE collaborationshave reported observations of global polarization of Λ -baryons [68, 69, 70] believed to result from the presenceof large vorticity in Au – Au and Pb – Pb collisions. Ithas been suggested that the presence of large magneticfields might also have an impact on viscous e ff ects [71]as they might strongly suppress momentum di ff usion inthe reaction plane or impart a “paramagnetic squeezing”e ff ect capable of altering pressure gradients. Given thestrength of the magnetic field should change consider-ably between peripheral and central A – A collisions,it might imply finite changes in the e ff ective di ff usiv-ity and viscosity of the QCD matter produced in thesecollisions. The magnitude of the e ff ect is as of yet un-known but nonetheless worthy of additional investiga-tions given the current uncertainties in η/ s values do notallow to conclude about a possible di ff erence at the twoenergies.Additionally, in order to make progress on a full char-acterization of η/ s as a function of temperature, col-lision energy, baryochemical potential, etc., additionaland more precise measurements of the centrality of the G CI2 longitudinal width at di ff erent collision energies arenecessary; for example, as already mentioned, from theBES-II at RHIC, or at the highest LHC energies. Fur-thermore, supporting theoretical studies in the frame-work of relativistic hydrodynamics will also be greatlybeneficial. One needs, in particular, to establish the in-fluence of the temperature and viscosity of the di ff er-ent stages of the collision (QGP, phase transition region,hadronic phase) on the longitudinal broadening of G CI2 .The role of resonance decays and charge conservationmust also be clarified in association with quantitativestudies of the radial flow velocities imparted to the mat-ter produced in A – A collisions. Ideally, these studiesshould be conducted for several system sizes and col-lision energies. In light of observations of Λ -baryonglobal polarization already mentioned, it shall also be ofinterest to examine whether the strong magnetic fieldspresent at the onset of A – A collisions can persist longenough to have a quantitatively measurable impact onthe shear viscosity in general, and on the longitudinalbroadening of transverse momentum correlators in par-7icular.
5. Conclusion
We presented an evaluation of the collision centralitydependence of the shear viscosity per unit of entropy, η/ s of the Quark Gluon Plasma produced in A – A col-lisions at RHIC and LHC based on measurements of the G correlator by the STAR and ALICE collaborationsusing the Gavin ansatz embodied in Eq.1. Freeze-outtimes required to carry out the calculations were deter-mined as a function of the cubic root of the charged par-ticle multiplicity (or collision centrality) from two-pionBose-Einstein measurements. Values of η/ s obtainedin Pb – Pb collisions, based on ALICE data, indicatethe shear viscosity per unit of entropy is of the order ofthe KSS bound and essentially independent of collisioncentrality at LHC energy. By contrast, the STAR dataare consistent with vanishing η/ s values in peripheralcollisions and values exceeding the KSS bound in morecentral collisions. However, given the large system-atic uncertainties of these data, one cannot exclude η/ s might be invariant with collision centrality. Likewise,one cannot readily exclude that values of η/ s might alsobe invariant with beam energy.The precision of our estimates of the dependence of η/ s , particularly at RHIC, are limited by the accuracyof the STAR measurement of G . Uncertainties arealso largely determined by the various caveats associ-ated with the Gavin ansatz discussed above, most par-ticularly the choice of characteristic temperature usedin the calculation. Clearly, a more detailed calculationalong the lines of Ref. [72] are needed to improve onthis work.Finally, we stress that although measurements of G are challenging, owing in particular to their sensitiv-ity to p T dependent e ffi ciency corrections, they arenonetheless possible as demonstrated by the recent AL-ICE measurement. Precise studies of the evolution ofthe G CI2 correlator with collision centrality thus stand tobecome a discriminating gauge of non only the aver-age magnitude of the shear viscosity per unit of entropy, η/ s , but its temperature dependence also. As such, theymight provide new and valuable inputs to multi-systemBayesian constraints methods. Ideally, this will requiremeasurements of G be completed based, for instance,on the beam energy scan at RHIC and at higher LHCenergies ( √ s NN = Acknowledgements
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