A viscous blast-wave model for relativistic heavy-ion collisions
AA viscous blast-wave model for relativistic heavy-ion collisions
Amaresh Jaiswal
GSI, Helmholtzzentrum f¨ur Schwerionenforschung, Planckstrasse 1, D-64291 Darmstadt, Germany
Volker Koch
Lawrence Berkeley National Laboratory, Nuclear Science Division, MS 70R0319, Berkeley, California 94720, USA (Dated: August 25, 2015)Using a viscosity-based survival scale for geometrical perturbations formed in the early stages ofrelativistic heavy-ion collisions, we model the radial flow velocity during freeze-out. Subsequently, weemploy the Cooper-Frye freeze-out prescription, with first-order viscous corrections to the distribu-tion function, to obtain the transverse momentum distribution of particle yields and flow harmonics.For initial eccentricities, we use the results of Monte Carlo Glauber model. We fix the blast-wavemodel parameters by fitting the transverse momentum spectra of identified particles at the LargeHadron Collider (LHC) and demonstrate that this leads to a fairly good agreement with transversemomentum distribution of elliptic and triangular flow for various centralities. Within this viscousblast-wave model, we estimate the shear viscosity to entropy density ratio η/s (cid:39) .
24 at the LHC.
PACS numbers: 25.75.-q, 24.10.Nz, 47.75+f
I. INTRODUCTION
High-energy heavy-ion collision experiments at theRelativistic Heavy Ion Collider (RHIC) [1, 2] and theLarge Hadron Collider (LHC) [3–5] have conclusively es-tablished the formation of a strongly interacting Quark-Gluon Plasma (QGP). The QGP formed in these colli-sions exhibit strong collective behaviour, and thereforecan be studied within the framework of relativistic hy-drodynamics. The hydrodynamical analyses of the flowdata suggests that the QGP behaves like a nearly perfectfluid with an extremely small shear viscosity to entropydensity ratio η/s [6, 7]. Local pressure gradients due todeformations and inhomogeneities in the the initial stagesof the collision results in anisotropic fluid velocity. Theseanisotropies subsequently translate into flow harmonicsdescribing the momentum asymmetry of produced par-ticles [8, 9]. The η/s of the fluid governs the conversionefficiency, which results in a suppression of elliptic flowand higher order flow coefficients [10–23].Apart from hydrodynamics, the observables pertainingto collective behaviour of QGP can also be studied usingthe so-called blast-wave model. Using a simple functionalform for the phase-space density at kinetic freezeout,Schnedermann et. al. [24] approximated hydrodynamicalresults with boost-invariant longitudinal flow [25]. Theyused this blast-wave model to successfully fit the trans-verse momentum spectra with only two parameters: a ki-netic temperature, and a radial flow strength. However,this model was only valid for central collisions at midra-pidity. In order to make it applicable for non-centralcollisions, Huovinen et. al. [26] generalized this model toaccount for the anisotropies in the transverse flow pro-file by introducing an additional parameter. This newparameter controlled the difference between the strengthof the flow in and out of the reaction plane. This leadto a fairly good fit with the measured elliptical flow as afunction of transverse momentum. However, the STAR Collaboration achieved better fits when they generalizedthe model even further by introducing a fourth param-eter to account for the anisotropic shape of the sourcein coordinate space [27]. Teaney made the first attemptto estimate the effect of viscosity on elliptic flow usinga variant of the blast-wave model model [28]. However,the centrality dependence of the fit parameters left themodel with little predictive power.In this paper we generalize the blast-wave model to in-clude viscous effects by employing a viscosity-based sur-vival scale for geometrical anisotropies, formed in theearly stages of relativistic heavy-ion collisions, in theparametrization of the radial flow velocity. The presentmodel has five parameters, including η/s , which has to befitted only for one centrality. In the Cooper-Frye freeze-out prescription for particle production, we consider thefirst-order viscous corrections to the distribution function[28]. In essence, we provide a model which incorporatesthe important features of viscous hydrodynamic evolu-tion but does not require to do the actual evolution. Weuse this viscous blast-wave model to obtain the trans-verse momentum distribution of particle yields and flowharmonics for LHC. The blast-wave model parametersare fixed by fitting the transverse momentum spectra ofidentified particles. Subsequently, we show that this leadsto fairly good agreement with transverse momentum dis-tribution of elliptic and triangular flow for various cen-tralities as well as centrality distribution for integratedflow. We estimate the shear viscosity to entropy densityratio η/s (cid:39) .
24 at the LHC, within the present model.
II. BLAST WAVE MODEL
The blast-wave model has been used extensively to fitexperimental data and it provides good description ofspectra and elliptic flow observed in relativistic heavy-ioncollisions [26–30]. The previously used blast-wave mod- a r X i v : . [ nu c l - t h ] A ug els employ a simple parametrization for the flow velocityof boost invariant ideal hydrodynamics. The most im-portant feature is the parametrization of the transversevelocity which is assumed to increase linearly with re-spect to the radius. This parametrization is found to bein agreement with hydro results [31, 32]. This essentiallyleads to an exponential expansion of the fireball in thetransverse direction, hence the term blast-wave. Apartfrom boost invariance, the model also assumes rotationalinvariance. In the following, we quickly outline the keyfeatures of the blast-wave model.In order to consider a boost invariant framework, it iseasier to work in the Milne co-ordinate system where, τ = (cid:112) t − z , (1) η s = tanh − ( z/t ) , (2) r = (cid:112) x + y , (3) ϕ = atan2( y, x ) . (4)The metric tensor for this co-ordinate system is g µν =diag(1 , − τ , − , − r ). Boost invariance and rotationalinvariance implies u ϕ = u η s = 0, whereas linearly risingtransverse velocity flow profile leads to u r ∼ r . The blast-wave model further assumes that the particle freeze-outhappens at a proper time τ f having a constant temper-ature T f and uniform matter distribution, in the trans-verse plane. In summary, the hydrodynamic fields areparametrized as [28] T = T f Θ( R − r ) , (5) u r = u rR Θ( R − r ) (6) u ϕ = u η s = 0 , (7) u τ = » u r ) , (8)where R is the transverse radius of the fireball at freeze-out. The expression for u τ is obtained by requiring thatthe fluid four-velocity satisfy the condition u µ u µ = 1.The hadron spectra can be obtained using the Cooper-Frye prescription for particle production [33] dNd p T dy = 1(2 π ) (cid:90) p µ d Σ µ f ( x, p ) , (9)where d Σ µ is the oriented freeze-out hyper-surface and f ( x, p ) is the phase-space distribution function of theparticles at freeze-out. The distribution function can bewritten in terms of the equilibrium and non-equilibriumparts, f = f + δf . The equilibrium distribution functionis given by f = 1exp( u µ p µ /T ) + a , (10)where a = +1 for baryons and a = − δf (cid:28) f ,we use the Grad’s 14-moment approximation for the non-equilibrium part [34, 35] δf = f ˜ f (cid:15) + P ) T p α p β π αβ , (11) where ˜ f = 1 − af and π αβ is the shear stress tensor.Approximating the shear stress tensor with its first-orderrelativistic Navier-Stokes expression, π αβ = 2 η ∇ (cid:104) α u β (cid:105) ,the expression for the 14-moment approximation reducesto δf = f ˜ f T (cid:16) ηs (cid:17) p α p β ∇ (cid:104) α u β (cid:105) . (12)Here η is the coefficient of shear viscosity, s = ( (cid:15) + P ) /T is the entropy density and the angular brackets denotetraceless symmetric projection orthogonal to the fluidfour-velocity [36]. The form of p α p β ∇ (cid:104) α u β (cid:105) in the caseof blast-wave model is calculated in Appendix A.The anisotropic flow is defined as v n ( p T ) ≡ (cid:90) π − π dφ cos[ n ( φ − Ψ n )] dNdy p t dp T dφ (cid:90) π − π dφ dNdy p t dp T dφ , (13)where Ψ n is the n -th harmonic event-plane angle. In thepresent case, we do not consider event-by-event fluctua-tions and therefore Ψ n = 0. Up to first order in viscosity[28], v n ( p T ) = v (0) n ( p T ) Ü − (cid:90) dφ dN (1) dy p t dp T dφ (cid:90) dφ dN (0) dy p t dp T dφ ê + (cid:90) dφ cos[ n ( φ − Ψ n )] dN (1) dy p t dp T dφ (cid:90) dφ dN (0) dy p t dp T dφ , (14)where the superscript ‘(0)’ denotes quantities calculatedusing the ideal distribution function, Eq. (10), and ‘(1)’denotes those obtained using the first-order viscous cor-rection, Eq. (12). III. VISCOUS BLAST-WAVE MODEL
The definition of the participant anisotropies, ε n , viathe Fourier expansion for a single-particle distribution is f ( ϕ ) = 12 π (cid:34) ∞ (cid:88) n =1 ε n cos[ n ( ϕ − ψ n )] (cid:35) , (15)where ψ n are the angles between the x axis and the ma-jor axis of the participant distribution. The geometricalanisotropies in the initial particle distribution, ε n , even-tually converts to anisotropies in the radial fluid velocity, u r = u rR (cid:34) ∞ (cid:88) n =1 u n cos[ n ( ϕ − ψ n )] (cid:35) . (16) Rn = 2 Rn = 3 Rn = 4 FIG. 1: The harmonics form standing waves on the fireballcircumference having radius R at freezeout. In the following, we determine the conversion efficiencyof the initial eccentricity to anisotropy in the radial fluidvelocity, u n /ε n .Using the well known dispersion relation for sound ina viscous medium [35], ω = c s k + ik T Å ηs ã , (17)the authors of Ref. [37] introduced a viscosity-based sur-vival scale which all structures formed by point like per-turbations should attain at freeze-out. In the above equa-tion, η is the coefficients of shear viscosity and c s is thespeed of sound in the medium. In the present work, weignore the contribution due to bulk viscosity. Using aplane-wave Fourier ansatz, exp( iωt − ikx ), we observethat the amplitudes of the stress tensor harmonics withmomentum k are attenuated by a factor δT µν ( t, k ) = exp ï − Å ηs ã k tT ò δT µν (0 , k ) , (18)where we have suppressed the oscillatory pre-factor. Wenote that the presence of momentum squared in the ex-ponent leads to enhanced effect of viscosity for the higherharmonics. We expect the same qualitative behaviour forthe radial flow velocity as will be explained in the follow-ing.First and foremost, we notice that each harmonicsis essentially a damped oscillator with wave-vector k .Moreover, throughout the evolution the harmonics formstanding waves on the fireball circumference, as shownin Fig. 1, whose amplitude is progressively damped duethe viscous effects. Therefore, the fireball circumferenceis an integer multiple of the wavelength with wave-vector k , i.e., 2 πR = n πk , (19)where R is the transverse radius of the fireball at freeze-out. Hence, at the freeze-out time t f , the wave amplitudereaction is given by δT µν | t = t f δT µν | t =0 = exp ï − n Å ηs ã t f R T f ò , (20) where T f is the freeze-out temperature.In absence of viscosity, the initial geometrical pertur-bations in the fluid will result in the development of ra-dial flow velocity and the conversion efficiency will re-main the same for all harmonics. In the case of a viscousmedium, however, the conversion efficiency of the initialgeometrical perturbation to radial fluid velocity must beproportional to the wave amplitude reaction, u n ε n = α exp ï − n Å ηs ã t f R T f ò , (21)where α is the constant of proportionality. The suddenstopping of the damped oscillator at the freeze-out timemay lead to certain phases due to the oscillatory pre-factor. These phases can, in general, lead to secondarypeaks in the power spectrum of higher harmonics. How-ever, as no secondary peaks has been observed in thespectrum of relativistic heavy-ion collisions, we will con-tinue to ignore these phase factors.We emphasize that the acoustic damping should beapplied to the hydrodynamic variables, such as the mo-ments of the flow velocity, u n , rather then to the finalstate observables such as v n , as done in Ref. [38]. InRef. [38], Shuryak and Zahed (S-Z) proposed that the ra-tio of the initial eccentricity ε n to the final p T -integrated v n should be proportional to the wave amplitude reac-tion, i.e., the r.h.s. of Eq. (21) should be equal to v n /ε n .However, ε n is the eccentricity in the configuration spacewhereas v n is the momentum anisotropy of the parti-cles after freeze-out. The momentum space asymmetriesare not hydrodynamic variables and are only affected in-directly via damping. On the other hand, the acous-tic damping should be applicable to hydrodynamic vari-ables and it should only capture the viscous effects of thehydrodynamic evolution. This assumption also missesthe additional effect of viscosity at freeze-out using theCooper-Frye formula. Moreover, it does not provide theopportunity to study the p T dependence of anisotropicflow and hence to estimate the viscosity of the expand-ing medium. IV. INITIAL CONDITIONS
In this section, we set-up the initial conditions of thecollisions in order to reduce the number of free param-eters in the blast-wave model. To this end, we evaluatethe parameters corresponding to the initial geometry ofthe collisions. We also approximate the subsequent trans-verse expansion of the fireball by using the radial velocityparametrization in the blast-wave model.We consider the collision of two identical nuclei withmass number A . The radius of each nucleus is givenby R = 1 . A / fm and the impact parameter is b ,as shown in Fig. 2. The shaded region in Fig. 2 is theoverlap zone of the colliding nuclei. We draw a circleof radius r , with its centre coinciding with that of theoverlap zone, in such a way that the boundary of the b r R R FIG. 2: Initial transverse geometry of the collision of twoidentical nuclei with radius R and impact parameter b . Theshaded region represents the the overlap zone of the collidingnuclei. The circle with radius r is drawn such that it equallydivides the boundary of the overlap zone in four parts, i.e., ˆ
123 = ˆ
345 = ˆ
567 = ˆ overlap zone is equally divided in four parts, i.e., ˜
123 = ˜
345 = ˜
567 = ˜ ε asthe second harmonics of initial geometrical fluctuations,analogous to the n = 2 case as shown in Fig. 1. Theradius r is therefore the initial transverse radius of theexpanding fireball and is given by r = 12 (cid:32) b − bR bR + 4 R (cid:33) / , (22)which reduces to r = R for head-on collisions ( b =0). Since ε is the most prominent eccentricity for non-central collisions, all other geometrical eccentricities aretreated as boundary perturbations to this circle.The subsequent transverse expansion of the fireball isobtained by employing the radial velocity parametriza-tion of the blast-wave model. Using the perturbation-freeexpression for the transverse velocity, Eq. (6), we get u r ≡ drdτ = u rR ⇒ (cid:90) Rr drr = (cid:90) τ f u R dτ. (23)After performing the straightforward integration, we ob-tain a transcendental equation for the freeze-out radius, R , R = r exp (cid:16) u τ f R (cid:17) , (24)which can be solved for R given the isotropic expansionvelocity u and the freeze-out time τ . In the follow-ing, using Bjorken’s scaling solution for one-dimensionalboost-invariant expansion, we obtain an expression todetermine the freeze-out times for non-central collisionsonce it is fixed for the central one. Centrality (%) ε i / ε i , τ f / τ f τ f / τ f0 ε i / ε i0 R (f m ) Pb + Pb @ √ s NN = 2.76 TeVGlauber Model FIG. 3: (Color online) Centrality dependence of the initialenergy density (cid:15) i and freeze-out time τ f for Pb+Pb collisionsat √ s NN = 2 .
76 TeV scaled by the corresponding values in0 −
5% central collisions. The inset shows centrality depen-dence of the freeze-out radius of the fireball R , obtained usingEq. (24). For the ideal hydrodynamic evolution of relativisticfluid, in the one-dimensional boost-invariant scenario, theevolution of the energy density follows (cid:15) ∝ τ − / . Assum-ing the initial thermalization time and the final freeze-outenergy density (i.e., the freeze-out temperature) to besame for all collisions, we get τ f = τ f Å (cid:15) i (cid:15) i ã / . (25)Here τ f is the freeze-out time for most central collisions,which has to be fixed by fitting the corresponding trans-verse momentum spectra. The freeze-out times for othercentralities can then be obtained using the above equa-tion and therefore they are not free parameters. Theratio (cid:15) i /(cid:15) i is the initial central energy density scaled byits corresponding value in most central collisions.Figure 3 shows (cid:15) i /(cid:15) i for various centralities obtainedusing the Glauber model calculations [39, 40] in the caseof Pb–Pb collisions at √ s NN = 2 .
76 TeV at the LHC.We observe that (cid:15) i /(cid:15) i (and hence the initial temper-ature) decreases for non-central collisions compared tocentral ones. Therefore according to Eq. (25), freeze-outhappens earlier in peripheral collisions which is also re-flected in Fig. 3 for τ f /τ f . The inset of Fig. 3 shows thecentrality dependence of the freeze-out radius of the fire-ball R , obtained using Eq. (24). We find a rather largetransverse radius of the fireball at freeze-out. Finally, theparameters that we need to fix within the viscous-blastwave model to fit the spectra are the freeze-out temper-ature T f , the freeze-out time for central collision τ f andthe unperturbed maximum radial flow velocity u . An in-terplay of the coefficient of proportionality α in Eq. (21)and η/s will be crucial to reproduce the flow harmonics. p T (GeV) ( / π p T ) d N / ( dp T dy ) [ c / G e V ] p T (GeV) (a) (b) Pb+Pb@ √ s NN = 2.76 TeV ALICE data: symbols π + K + p π + K + p0 - 5% 20 - 30% FIG. 4: (Color online) Transverse momentum distribution ofparticle multiplicities in Pb+Pb collisions at √ s NN = 2 . π + , K + , and p in two centralityranges, (a): 0 −
5% and (b): 20 − V. RESULTS AND DISCUSSIONS
In this section, we show our results for Pb+Pb colli-sions at √ s NN = 2 .
76 TeV and compare it with experi-mental data measured at LHC by the ALICE and ATLAScollaborations.Figure 4 shows the transverse momentum distributionof pions, kaons, and protons spectra for 0 −
5% and20 −
30% central Pb+Pb collisions at √ s NN = 2 .
76 TeVmeasured at LHC by the ALICE collaboration (symbols)at midrapidity [41] and calculated using the viscous blast-wave model (lines). The results are obtained using rootmean square values of eccentricities ε and ε in a Monte-Carlo Glauber model, with a shear viscosity to entropydensity ratio η/s = 0 .
24. We observe that the spectrafor π + and K + from the viscous blast-wave model are ingood overall agreement with the experimental data for afreeze-out temperature of 120 MeV. On the other hand,within the viscous blast-wave model, the proton yield fora freeze-out temperature of 120 MeV is severely under-estimated. To obtain an overall fair agreement with theexperimental data, the freeze-out temperature for pro-tons is considered to be 135 MeV. The freeze-out timefor 0 −
5% most central collision was found to be 8 fm.For other centralities, the freeze-out time was obtainedby using Eq. (25) and results shown in Fig. 3.Figure 5 shows our results for the v n ( p T ), in compar-ison with the ATLAS data [4] for various centralities.We find overall fair agreement with the data for elliptic( v ) and triangular ( v ) flow, at all centralities. This isachieved by choosing a single fixed value η/s = 0 .
24 toobtain the required suppression (relative to ideal blast-wave results) of v and v , for all centralities. Apart from η/s , flow also depends on the constant of proportional-ity α appearing in Eq. (21). It controls the conversion p T (GeV) v n ( p T ) p T (GeV) v n ( p T ) v v (a) (b)(c) (d) Pb+Pb@ √ s NN = 2.76 TeV ATLAS data: symbols η /s = 0.24 FIG. 5: (Color online) Transverse momentum dependence ofthe anisotropic flow coefficients v n ( p T ) of charged hadrons, for n = 2 and 3, calculated at various centralities in Pb+Pb col-lisions at √ s NN = 2 .
76 TeV in the viscous blast-wave model(lines) with η/s = 0 .
24 as compared to the ATLAS data [4](symbols). efficiency of the initial eccentricity to final fluid veloc-ity. We consider the initial eccentricity ε n to be the rmsvalues of the eccentricities obtained in the MC-Glaubermodel, as given in Ref. [42]. A large value of α meanslarger conversion of eccentricity leading to increased flowvelocity and hence higher v and v . On the other hand,as is well known, an increase in η/s leads to suppressionof v and v . Large (small) value of α can be compen-sated by choosing a higher (lower) value of η/s up to acertain extent. However, beyond a certain range of η/s ,the relative behaviour of v and v is destroyed.In order to match the elliptic and triangular flow data,we find the most suitable parameter values for α = 0 . η/s = 0 .
24. Moreover, we find that v n is insensi-tive to a for transverse velocity of the form v r ∼ ( r/R ) a .This may be attributed to the fact that the exponent a controls the rate of isotropic transverse expansion. Theanisotropic flow originates from the initial eccentricitywhich translates into final flow. Therefore v n is sensi-tive to u n which depends on α and η/s , as is apparentfrom Eq. (21). On the other hand, it should be notedthat the slope of the particle spectra is sensitive to theexponent a and could be tuned to get a better fit withthe experimental data. However, in the present work, weare interested in the anisotropic flow and therefore we set a = 1 and do not fit it to match the particle spectra.In Fig. 6(a), p T -integrated values of v and v obtainedfrom the viscous blast-wave model are compared withALICE data [3], as a function of centrality. We observethat using the same constant η/s = 0 .
24 for all central- v n / ε n Centrality (%) v / v VBWS-Z v n (a) Pb+Pb@ √ s NN = 2.76 TeV (b)(c) n = 2 v v n = 3ALICE: Symbols ALICE: SymbolsViscous Blast Wavep T > 0.2 GeV FIG. 6: (Color online) (a): Centrality dependence of the p T integrated anisotropic flow coefficients v n of charged hadronsin Pb+Pb collisions at √ s NN = 2 .
76 TeV calculated in theviscous blast-wave model (lines) with η/s = 0 .
24, as comparedto ALICE data [3] (symbols). (b): Centrality dependence ofthe ratio v n /ε n in the viscous blast-wave model where ε n is the rms values of the eccentricities obtained in the MC-Glauber model [42]. In panels (a) and (b), we show resultsfor n = 2 and 3. (c): Centrality dependence of the ratio v /v for ALICE data (symbols), using the viscous blast-wave model (solid line) and from the Shuryak-Zahed estimate(dashed line). ities, the model shows a good agreement with the data.Since v is driven mostly by the initial spatial anisotropy,it exhibits a strong centrality dependence compared to v .Figure 6(b) shows the conversion efficiency of the initialspatial anisotropy into the final momentum anisotropy.A linear relation between v n and ε n , as observed in somehydrodynamic calculations [23], is not obtained withinthe viscous blast-wave model presented here. On theother hand, in view of the non-linear nature of the hy-drodynamic equations a linear relation between v n and ε n is not obvious. Indeed other calculations [43] as wellas a recent analysis of the LHC data [44] also result ina similar centrality dependence as obtained in our anal-ysis In Fig. 6(c), we show the ratio v /v as a functionof centrality for the ALICE data (symbols), the presentviscous blast-wave model (blue solid line) and the esti-mate due to Shuryak and Zahed (red dashed line). Wesee that the viscous blast-wave model provides a betteragreement with ALICE data compared to the S-Z esti-mate. However, we should keep in mind that the freeze-out parameters used in the S-Z estimate is the same asthat of the viscous blast-wave fit values. VI. CONCLUSIONS AND OUTLOOK
In this paper we have generalized the blast-wave modelto include viscous effects by employing a viscosity-basedsurvival scale for geometrical anisotropies formed in theearly stages of relativistic heavy-ion collisions. This vis-cous damping is introduced in the parametrization ofthe radial flow velocity. The viscous blast wave modelpresented here involved five parameters, including η/s ,which has to be fitted for only one centrality. Thismodel therefore incorporates the important features ofviscous hydrodynamic evolution but does not require todo the actual evolution. We have used this viscous blast-wave model to obtain the transverse momentum distri-bution of particle yields and anisotropic flow harmonicsfor LHC. The blast-wave model parameters were fixedby fitting the transverse momentum spectra of identifiedparticles. We demonstrated that a fairly good agreementwas achieved for transverse momentum distribution of el-liptic and triangular flow for various centralities as wellas centrality distribution for the integrated flow. Withinthe present model, we estimated the shear viscosity toentropy density ratio η/s (cid:39) .
24 at the LHC.One of the drawbacks of the present model is thatwe have employed root mean squared eccentricity overnumber of events, which is analogous to “single shot”hydrodynamic evolution. On the plus side, the presentmodel could also be implemented on an event-by-eventbasis. Another difficulty that we encountered was thatin order to fit the proton spectra, we had to consider adifferent freeze-out temperature (135 MeV) compared tothat for pions and Kaons (120 MeV). This problem couldbe addressed by parametrizing the transverse velocity inthe form v r ∼ ( r/R ) a and fitting a for different particlespecies, separately. We leave these problems for a futurework. Acknowledgments
A.J. acknowledges useful discussions with Subrata Pal,Krzysztof Redlich, Edward Shuryak and Derek Teaney.A.J. thanks Victor Roy for helpful comments and pro-viding data from Glauber model. A.J. was supportedby the Frankfurt Institute for Advanced Studies (FIAS),Germany.
Appendix A: Viscous stress tensor
In this Appendix, we calculate the viscous tensor ∇ (cid:104) α u β (cid:105) and therefore obtain the viscous corrections tothe distribution function at freeze-out. We work in Milneco-ordinate system, Eqs. (1)-(4) with the metric ten-sor g µν = diag(1 , − τ , − , − r ). Therefore, the inversemetric tensor is g µν = diag(1 , − /τ , − , − /r ), its de-terminant g is √− g = τ r and the non-vanishing Christof-fel symbols are Γ τη s η s = τ , Γ η s τη s = 1 /τ , Γ rϕϕ = − r , andΓ ϕrϕ = 1 /r . Using the parametrization of the fluid veloc-ity given in Eqs. (6)-(8), we get∆ rϕ = 0 , ∆ ϕϕ = − r , ∆ rr = − − ( u r ) , (A1) ∂ r u r = u r r , ∂ ϕ u r = − u rR ∞ (cid:88) n =1 n u n sin[ n ( ϕ − ψ n )] . (A2)where ∆ µν ≡ g µν − u µ u ν is the projection operator or-thogonal to the fluid four-velocity.To fix the time derivatives of the fluid velocity, we as-sume that if the particles are freezing-out, they are freestreaming, which means that Du µ = 0. Here D ≡ u µ d µ is the co-moving derivative and d µ is the covariant deriva-tive. With this prescription, we have ∂ τ u ϕ = 0 ,∂ τ u r = − v ∂ r u r = − ( u r ) ru τ ,∂ τ u τ = v ∂ τ u r = − ( u r ) r ( u τ ) (A3)where v = u r /u τ is the radial velocity. The expansionscalar is given by1 √− g ∂ µ ( √− gu µ ) = u τ τ + u r r + ∂ ϕ u ϕ + ∂ r u r + ∂ τ u τ , = u τ τ + 2 u r r − ( u r ) r ( u τ ) . (A4)Assuming boost invariance, the spatial components of theviscous tensor are given by r ∇ (cid:104) r u ϕ (cid:105) = − r ∂ r u ϕ − r ∂ ϕ u r − r u r Du ϕ − r u ϕ Du r − r ∆ rϕ √− g ∂ µ ( √− gu µ )= u R ∞ (cid:88) n =1 n u n sin[ n ( ϕ − ψ n )] , (A5) r ∇ (cid:104) ϕ u ϕ (cid:105) = − ∂ ϕ u ϕ − u r r − r u ϕ Du ϕ − r ∆ ϕϕ √− g ∂ µ ( √− gu µ )= 13 ï u τ τ − u r r − ( u r ) r ( u τ ) ò , (A6) ∇ (cid:104) r u r (cid:105) = − ∂ r u r − u r Du r −
13 ∆ rr √− g ∂ µ ( √− gu µ )= 13 ï ( u τ ) τ − u r r + ( u r ) r ò , (A7) where we have used the fact that ( u τ ) = 1 + ( u r ) . τ ∇ (cid:104) η s u η s (cid:105) = − u τ τ + 13 1 √− g ∂ µ ( √− gu µ )= 13 ï u r r − u τ τ − ( u r ) r ( u τ ) ò , (A8) ∇ (cid:104) r u η s (cid:105) = ∇ (cid:104) ϕ u η s (cid:105) = 0 . (A9)To obtain the temporal components of the viscousstress energy tensor, we use the Landau frame condition, ∇ (cid:104) α u β (cid:105) u β = 0. ∇ (cid:104) τ u τ (cid:105) u τ + ∇ (cid:104) τ u r (cid:105) u r = 0 ⇒ ∇ (cid:104) τ u τ (cid:105) = v ∇ (cid:104) τ u r (cid:105) , (A10) ∇ (cid:104) η s u τ (cid:105) u τ + ∇ (cid:104) η s u r (cid:105) u r = 0 ⇒ ∇ (cid:104) τ u η s (cid:105) = 0 , (A11) ∇ (cid:104) r u τ (cid:105) u τ + ∇ (cid:104) r u r (cid:105) u r = 0 ⇒ ∇ (cid:104) τ u r (cid:105) = v ∇ (cid:104) r u r (cid:105) , (A12) ∇ (cid:104) ϕ u τ (cid:105) u τ + ∇ (cid:104) ϕ u r (cid:105) u r = 0 ⇒ ∇ (cid:104) τ u ϕ (cid:105) = v ∇ (cid:104) r u ϕ (cid:105) . (A13)Therefore, from Eqs. (A10) and (A12), we see that ∇ (cid:104) τ u τ (cid:105) = v ∇ (cid:104) τ u r (cid:105) = v ∇ (cid:104) r u r (cid:105) = 13 ï ( u r ) u τ τ − ( u r ) r ( u τ ) + ( u r ) r ( u τ ) ò . (A14)Next, in order to verify our algebra, we confirm that theviscous stress tensor is traceless, i.e., g µν ∇ (cid:104) µ u ν (cid:105) = 0.Using Eqs. (A6), (A7), (A8) and (A14) g µν ∇ (cid:104) µ u ν (cid:105) = ∇ (cid:104) τ u τ (cid:105) − τ ∇ (cid:104) η s u η s (cid:105) − ∇ (cid:104) r u r (cid:105) − r ∇ (cid:104) ϕ u ϕ (cid:105) = 13 ï ( u r ) u τ τ − ( u r ) r ( u τ ) + ( u r ) r ( u τ ) ò − ï u r r − u τ τ − ( u r ) r ( u τ ) ò − ï ( u τ ) τ − u r r + ( u r ) r ò − ï u τ τ − u r r − ( u r ) r ( u τ ) ò = 0 . (A15)For a particle at the space-time point ( τ, η s , r, ϕ )with the four momentum p µ = ( E, p x , p y , p z ) =( m T cosh y, p T cos ϕ p , p T sin ϕ p , m T sinh y ), we get p τ = m T cosh( y − η s ) ⇒ p τ = m T cosh( y − η s ) ,τ p η s = m T sinh( y − η s ) ⇒ p η s = − τ m T sinh( y − η s ) ,p r = p T cos( ϕ p − ϕ ) ⇒ p r = − p T cos( ϕ p − ϕ ) ,r p ϕ = p T sin( ϕ p − ϕ ) ⇒ p ϕ = − r p T sin( ϕ p − ϕ ) . (A16)The oriented freeze-out hyper-surface is d Σ µ =( τ dη s rdr dϕ, , , p µ d Σ µ = m T cosh( y − η s ) τ dη s rdr dϕ. (A17)The viscous correction to the equilibrium distributionfunction is proportional to p µ p ν ∇ (cid:104) µ u ν (cid:105) = p τ ∇ (cid:104) τ u τ (cid:105) + p η s ∇ (cid:104) η s u η s (cid:105) + p r ∇ (cid:104) r u r (cid:105) + p ϕ ∇ (cid:104) ϕ u ϕ (cid:105) + 2 p τ p r ∇ (cid:104) τ u r (cid:105) + 2 p r p ϕ ∇ (cid:104) r u ϕ (cid:105) + 2 p τ p ϕ ∇ (cid:104) τ u ϕ (cid:105) . (A18)The final form of f = f + δf obtained from Eq. (12) using the above equation is required to evaluate the spectragiven by d Nd p T dy = 1(2 π ) (cid:90) R r dr (cid:90) π dϕ (cid:90) ∞−∞ τ dη s m T cosh( y − η s ) f. (A19) [1] J. Adams et al. [STAR Collaboration], Nucl. Phys. A , 102 (2005).[2] K. Adcox et al. [PHENIX Collaboration], Nucl. Phys. A , 184 (2005).[3] K. Aamodt et al. [ALICE Collaboration], Phys. Rev.Lett. , 032301 (2011).[4] G. Aad et al. [ATLAS Collaboration], Phys. Rev. C ,014907 (2012).[5] S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. C , 044906 (2014).[6] P. Romatschke and U. Romatschke, Phys. Rev. Lett. ,172301 (2007).[7] H. Song and U. W. Heinz, Phys. Rev. C , 064901(2008).[8] U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci. ,123 (2013).[9] C. Gale, S. Jeon and B. Schenke, Int. J. Mod. Phys. A , 1340011 (2013).[10] M. Luzum and P. Romatschke, Phys. Rev. C , 034915(2008). [Erratum ibid. C , 039903(E) (2009)].[11] H. Song and U. Heinz, Phys. Lett. B658 , 279 (2008);Phys. Rev. C , 024902 (2008);[12] J. Steinheimer, M. Bleicher, H. Petersen, S. Schramm,H. Stocker and D. Zschiesche, Phys. Rev. C , 034901(2008).[13] K. Dusling and D. Teaney, Phys. Rev. C , 034905(2008).[14] D. Molnar and P. Huovinen, J. Phys. G , 104125(2008).[15] P. Bozek, Phys. Rev. C , 034909 (2010).[16] A. K. Chaudhuri, J. Phys. G , 075011 (2010).[17] Z. Xu, C. Greiner, and H. St¨ocker, Phys. Rev. Lett. ,082302 (2008).[18] H. Holopainen, H. Niemi and K. J. Eskola, Phys. Rev. C , 034901 (2011).[19] Z. Qiu and U. W. Heinz, Phys. Rev. C , 024911 (2011).[20] H. Song, S. A. Bass, U. Heinz, T. Hirano and C. Shen,Phys. Rev. Lett. , 192301 (2011) [Erratum-ibid. ,139904 (2012)].[21] B. Schenke, S. Jeon and C. Gale, Phys. Rev. Lett. ,042301 (2011); Phys. Rev. C , 024901 (2012). [22] Z. Qiu, C. Shen and U. Heinz, Phys. Lett. B , 151(2012).[23] R. S. Bhalerao, A. Jaiswal and S. Pal, Phys. Rev. C ,014903 (2015).[24] E. Schnedermann, J. Sollfrank and U. W. Heinz, Phys.Rev. C , 2462 (1993).[25] J. D. Bjorken, Phys. Rev. D , 140 (1983).[26] P. Huovinen, P. F. Kolb, U. W. Heinz, P. V. Ruuskanenand S. A. Voloshin, Phys. Lett. B , 58 (2001).[27] C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. , 182301 (2001).[28] D. Teaney, Phys. Rev. C , 034913 (2003).[29] Z. Tang, Y. Xu, L. Ruan, G. van Buren, F. Wang andZ. Xu, Phys. Rev. C , 051901 (2009).[30] X. Sun, H. Masui, A. M. Poskanzer and A. Schmah, Phys.Rev. C , 024903 (2015).[31] D. Teaney, J. Lauret and E. V. Shuryak, nucl-th/0110037.[32] M. Habich, J. L. Nagle and P. Romatschke, Eur. Phys.J. C , 15 (2015).[33] F. Cooper and G. Frye, Phys. Rev. D , 186 (1974).[34] H. Grad, Comm. Pure Appl. Math. , 331 (1949).[35] P. Romatschke, Int. J. Mod. Phys. E , 1 (2010).[36] R. S. Bhalerao, A. Jaiswal, S. Pal and V. Sreekanth,Phys. Rev. C , 054903 (2014).[37] P. Staig and E. Shuryak, Phys. Rev. C , 034908 (2011).[38] E. Shuryak and I. Zahed, Phys. Rev. C , 044915 (2013).[39] P. F. Kolb, U. W. Heinz, P. Huovinen, K. J. Eskola andK. Tuominen, Nucl. Phys. A , 197 (2001).[40] V. Roy and A. K. Chaudhuri, Phys. Lett. B , 313(2011).[41] B. Abelev et al. [ALICE Collaboration], Phys. Rev. C , 044910 (2013).[42] E. Retinskaya, M. Luzum and J. Y. Ollitrault, Phys. Rev.C , 014902 (2014).[43] H. Niemi, G. S. Denicol, H. Holopainen and P. Huovinen,Phys. Rev. C , 054901 (2013).[44] L. Yan, J. Y. Ollitrault and A. M. Poskanzer, Phys. Lett.B742