Cent. Eur. J.Phys.• 1-4 Authorversion Central European Journal of Physics Bulk viscosity in heavy ion collision Research Article Victor Roy ∗ and A.K. Chaudhuri 2 VariableEnergyCyclotronCentre, 1 1/AF,Bidhannagar,Kolkata-700064,India 0 2 n Abstract: Theeffectofatemperaturedependentbulkviscositytoentropydensityratio(ζ/s)alongwithaconstant a shear viscosity to entropy density ratio (η/s) on the space time evolution of the fluid produced in high J energy heavy ion collisions have been studied in a relativistic viscous hydrodynamics model. The boost 0 invariantIsrael-Stewarttheoryofcausalrelativisticviscoushydrodynamicsisusedtosimulatetheevolution 2 ofthefluidin2spatialand1temporaldimension. Thedissipativecorrectiontothefreezeoutdistribution forbulkviscosityiscalculatedusingGrad’sfourteenmomentmethod. Fromoursimulationweshowthat ] themethodisapplicableonlyforζ/s<0.004. h t PACS (2008): 12.38.Mh,47.75.+f,25.75.Ld - l c Keywords: BulkViscosity• Relativistichydrodynamics• Grad’smoment u © VersitaWarsawandSpringer-VerlagBerlinHeidelberg. n [ 1 v 0 1. Introduction 3 2 4 Recentexperimentsinhighenergynuclearcollisionsatrelativisticheavyioncollider(RHIC)confirmstheexistence . 1 ofanewstateofmatterknownasQuarkGluonPlasma(QGP)[1]. TheproductionofQGPinheavyioncollision 0 2 anditssubsequentcollectiveevolutionprovideustheuniqueopportunitytostudythetransportpropertiesofthis 1 : mostfundamentalformofmatter. Relativisticviscoushydrodynamicssimulationsofobservableslikeellipticflow v i X (v2) and transverse momentum (pT) spectra have been compared to experimental data to extract the QGP η/s. r Most studies show that the estimated value of η/s lies between 1−4×(1/4π). However to correctly extract the a η/softheQGPfluid,itisimportanttoknowtheeffectoffinitebulkviscosityonthefluidevolution. Theoretical calculationsbasedonpQCD[2]andlatticeQCD[3]showsthatthebulkviscosityisnon-zeroforthetemperature rangeapplicableintheheavyioncollision. Inthisworkweuseatemperaturedependentformofζ/stostudythe effect of bulk viscosity in fluid evolution. The dissipative correction to the freezeout distribution function bulk viscosity has also been considered using Grad’s 14 moment method. ∗ E-mail: [email protected] 1 Bulkviscosityinheavyioncollision Figure 1. (Color online) ζ/s as a function of temperature. Red dashed line is the η/s=1/4π. 2. Viscous hydrodynamic model Thespacetimeevolutionofthefluidwassimulatedbysimultaneouslysolvingtheenergymomentumconservation equation∂ Tµν =0,alongwiththerelaxationequationforshearandbulkstress. HereTµν =((cid:15)+p+Π)uµuν− µ (p+Π)gµν +πµν is the energy momentum tensor and (cid:15),p are energy density, pressure of the fluid,gµν is the metrictensor; Πandπµν arebulkandshearstresstensorrespectively. AccordingtotheIsrael-Stewarttheoryof causal viscous hydrodynamics [4], the shear and bulk viscosity obey the following relaxation equations Dπµν = 1 [2η∇<µuν> −πµν]−(cid:0)uµπνλ+uνπµλ(cid:1)Du ; and DΠ = − 1 [Π+ζ∇ uµ + 1ζTΠ∂ (τΠuµ)]. Here D is the τπ λ τΠ µ 2 µ ζT convective derivative,τ and τ are the relaxation time for shear and bulk stresses respectively. We assume that π Π the fluid achieve near local thermalization at proper time 0.6 fm. Initial transverse velocity (v ) is assumed to T bezero. TheinitialenergydensityprofileintransverseplaneiscalculatedfromatwocomponentGlaubermodel withacentralenergydensity(cid:15) =30GeV/fm3. Initialvalueofπµν andΠwassettotheircorrespondingNavier- 0 Stokesestimate. Weassumethefluidfreezesoutwhenanelementofitcoolsdownbelowaconstanttemperature T =130 MeV. The freezeout procedure was carried out by using Cooper-Frey algorithm. In the present study, fo wehaveusedanequationofstate(EoS)wheretheWuppertal-Budapestlatticecalculation[3]forthedeconfined phaseissmoothlyjoinedatcrossovertemperature174MeV,withhadronicresonancegasEoScomprisingallthe resonances below mass m =2.5 GeV. ζ/s and η/s are inputs to viscous hydrodynamics simulation. Figure 1 res showstheζ/s(T),whereζ/sintheQGPphaseisobtainedbyusingpQCDformulaζ/s=15η(T)(1/3−c2(T))2, s s the squared speed of sound c2 was calculated from lattice data [5]. In the hadronic phase ζ/s is parametrized s from [6]. The red dashed line in figure 1 is η/s. 3. Results and discussion We first discuss the change in pion p spectra and v due to bulk and shear viscosity in the fluid evolution only. T 2 √ In the left panel of figure 2 temporal evolution of spatially averaged transverse velocity (cid:104)(cid:104)v (cid:105)(cid:105)= (cid:104)(cid:104)γ vx2+vy2 (cid:105)(cid:105) is T (cid:104)(cid:104)γ(cid:105)(cid:105) 2 VictorRoy andA.K.Chaudhuri Figure 2. (Coloronline)Theleftplotshowsthetemporalevolutionoffluidtransversevelocityforideal(red),bulk(dasheddot) and shear viscous (dotted) evolution. The right plot is the π− invariant yield as a function of pT for ideal (red) ,bulkdasheddotandshearviscous(dotted)evolution. Theinsetfigureshowstherelativecorrectiontoinvariantyield duetothebulkviscosityincomparisontoidealfluid. Figure 3. (Coloronline)Theleftplotisthetemporalevolutionofmomentumanisotropyofthefluidforideal(red),bulk(dashed dot)andshearviscous(dotted)evolution. Rightplotisthecorrespondingellipticflow(v2)ofπ−. Theinsetplotshows therelativechangeinv2 duetobulkviscositycomparedtoidealfluid. shown for ideal, shear and bulk viscous fluid. Here the angular bracket denotes space average and γ = √1 . 1−v2 Becauseofthereducedpressureinbulkviscousevolution,(cid:104)(cid:104)v (cid:105)(cid:105)isreducedincomparisontoidealfluidevolution. T Whereasshearviscosityincreasethepressureinthetransversedirection,asaresultthe(cid:104)(cid:104)v (cid:105)(cid:105)islargercompared T to ideal fluid. The effect of the changed fluid velocity in viscous evolution is reflected in the slope of the p T spectraofπ− shownintherightpaneloffigure2. Therelativechangeintheπ− invariantyield(δN/N ,δN = ideal N −N )duetothebulkviscosityincomparisontoidealfluidisshownintheinsetofrightplotoffigure2. bulk ideal The relative correction is within ∼10%. Thetemporalevolutionofmomentumspaceanisotropy(cid:15) = (cid:82)dxdy(Txx−Tyy) isshownintheleftpaneloffigure3. p (cid:82)dxdy(Txx+Tyy) Viscositytriestodiminishanyvelocitygradientpresentinthefluid,asaresultofthat(cid:15) issmallerforbothshear p and bulk viscous evolution compared to ideal fluid. In a hydrodynamic model v is proportional to (cid:15) hence a 2 p reduction in (cid:15) will result in a reduced v . Elliptic flow of π− as a function of p is shown for ideal, shear and p 2 T bulk viscous evolution in the right plot of figure 3. The inset shows the relative correction to v due to bulk 2 viscosity. The relative correction to v is within ∼3%. 2 3 Bulkviscosityinheavyioncollision Figure4. (Coloronline)Theinvariantyieldofπ−asafunctionofpT foridealandbulkviscousevolutionwithfourdifferentζ/s valuesareshownintheupperleftplot. Thelowerleftplotshowstherelativecorrectiontotheπ− yieldasafunction ofpT forfourdifferentζ/s. Seetextfordetails. Therightsideplotissameastheleftbutforv2. We have employed Grad’s fourteen-moment method for calculating the dissipative correction to the freezeout distribution function as described in [7]. The details of the implementation of this method to our viscous code ”‘AZHYDRO-KOLKATA”’ can be found in [8, 9]. The top left panel of figure 4 shows the p spectra T of pions for ideal(black solid line) and for four different values of ζ/s. The corresponding relative correction to thep spectraisshowninthebottomleftpanel. Thev ofpionandtherelativecorrectionisshownintheright T 2 panel of figure 4. Freeze-out correction in Grad’s moment method is obtained under the assumption that the non-equilibrium correction to the distribution function is small than the equilibrium distribution function. It is thenimpliedthattherelativecorrectionδN/N issmallforGrad’smethodtobeapplicable. Theshadedbandin eq thebottomleftpanelcorrespondstotherelativecorrectionof50%. Ifweconsiderhereacorrectionofmagnitude greater than 50% indicates the breakdown of the the freezeout correction procedure then our study shows that the Grad’s method will be applicable if the ζ/s has value less than 0.01 times the present form considered here. References [1] J. Adams et al. [STAR Collaboration],Nucl. Phys. A 757, 102 (2005). [2] S. Weinberg,Astrophys. J. 168, 175 (1971). [3] H. B. Meyer,Phys. Rev. Lett. 100, 162001 (2008). [4] W. Israel,Annals of Physics 100,310-331 (1976). [5] S. Borsanyi et al.,JHEP 1011, 077 (2010). [6] J. Noronha-Hostler, J. Noronha and C. Greiner,Phys. Rev. Lett. 103, 172302 (2009). [7] A. Monnai and T. Hirano,Phys. Rev. C 80, 054906 (2009). [8] V. Roy, A. K. Chaudhuri, submitted to Phys. Rev. C. [9] A. K. Chaudhuri,arXiv:0801.3180 [nucl-th]. 4