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Astudyofvorticityformationinhighenergynuclearcollisions F. Becattini,1,2 G. Inghirami,3,1 V. Rolando,4,5 A. Beraudo,6 L. Del Zanna,1,2,7 A. De Pace,6 M. Nardi,6 G. Pagliara,4,5 and V. Chandra8 1DipartimentodiFisicaeAstronomia,Universita` diFirenze,ViaG.Sansone1,I-50019SestoF.no(Firenze),Italy 2INFN - Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto F.no (Firenze), Italy 3FrankfurtInstituteforAdvancedStudies(FIAS),JohannWolfgangGoetheUniversity,FrankfurtamMain,Germany 4DipartimentodiFisicaeScienzedellaTerra,Universita` diFerrara,ViaSaragat1,I-44100Ferrara,Italy 5INFN - Sezione di Ferrara, Via Saragat 1, I-44100 Ferrara, Italy 6INFN - Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy 7INAF - Osservatorio Astrofisico di Arcetri, L.go E. Fermi 5, I-50125 Firenze, Italy 8Indian Institute of Technology Gandhinagar, Ahmedabad-382424, Gujrat, India (Dated:August18,2015) We present a quantitative study of vorticity formation in peripheral ultrarelativistic heavy ion collisions at 5 √ 1 sNN=200GeVbyusingtheECHO-QGPnumericalcode,implementingrelativisticdissipativehydrodynam- 0 icsinthecausalIsrael-Stewartframeworkin3+1dimensionswithaninitialBjorkenflowprofile. Weconsider 2 anddiscussdifferentdefinitionsofvorticitywhicharerelevantinrelativistichydrodynamics.Afterdemonstrat- ingtheexcellentcapabilitiesofourcode,whichprovestobeabletoreproduceGubserflowupto8fm/c,we g showthat, withtheinitialconditionsneededtoreproducethemeasureddirectedflowinperipheralcollisions u correspondingtoanaverageimpactparameterb=11.6fmandwiththeBjorkenflowprofileforaviscousQuark A GluonPlasmawithη/s = 0.1fixed,avorticityoftheorderofsome10−2 c/fmcandevelopatfreezeout. The ensuingpolarizationofΛbaryonsdoesnotexceed1.4%atmidrapidity.Weshowthattheamountofdeveloped 7 directedflowissensitivetoboththeinitialangularmomentumoftheplasmaanditsviscosity. 1 ] h I. INTRODUCTION ux = uy = uη = 0), henceforth denoted as BIC. It should be t pointed out from the very beginning that the purpose of this - cl Thehydrodynamicalmodelhasbynowbecomeaparadigm workistomakeageneralassessmentofvorticityattopRHIC u for the study of the QCD plasma formed in nuclear colli- energy and not to provide a precision fit to all the available n sionsatultrarelativisticenergies. Therehasbeenaconsider- data. Therefore,ourcalculationsdonottakeintoaccountef- [ ableadvanceinhydrodynamicsmodelingandcalculationsof fectssuchasviscouscorrectionstoparticledistributionatthe 3 these collisions over the last decade. Numerical simulations freezeout and initial state fluctuations, that is we use smooth v in2+1D[1]andin3+1D[2–7]includingviscouscorrections initialconditionsobtainedaveragingovermanyevents. 8 arebecomingthenewstandardinthisfieldandexistingcodes 6 arealsoabletohandleinitialstatefluctuations. 4 Aninterestingissueisthepossibleformationofvorticityin 4 peripheral collisions [8–10]. Indeed, the presence of vortic- 0 . ity may provide information about the (mean) initial state of A. Notations 1 thehydrodynamicalevolutionwhichcannotbeachievedoth- 0 erwise,anditisrelatedtotheonsetofpeculiarphysicsinthe 5 plasma at high temperature, such as the chiral vortical effect Inthispaperweusethenaturalunits,with(cid:126)=c= K =1. 1 TheMinkowskianmetrictensorisdiag(1,−1,−1,−1);forthe v: [to11p]o.lFaruirztahteiormnoorfe,piatrhtiacslebseienntshheowfinnatlhasttavteo,rtsicoittyhagtivee.gs.risΛe Levi-Civitasymbolweusetheconvention(cid:15)0123 =1. i Wewillusetherelativisticnotationwithrepeatedindicesas- X baryon polarization - if measurable - can be used to detect sumed to be summed over, however contractions of indices it [12, 13]. Finally, as we will show, numerical calculation r willbesometimesdenotedwithdots,e.g. u·T ·u≡u Tµνu . a of vorticity can be used to make stringent tests of numerical µ ν The covariant derivative is denoted as d (hence d g = 0), codes, astheT-vorticity(seesect.IIforthedefinition)isex- µ λ µν pectedtovanishthroughoutunderspecialinitialconditionsin theexteriorderivativebyd,whereas∂µistheordinaryderiva- tive. theidealcase. Lately, vorticity has been the subject of investigations in refs.[9,10]withpeculiarinitialconditionsincartesiancoor- dinates,idealfluidapproximationandisochronousfreezeout. Instead, in this work, we calculate different kinds of vorticity with our 3+1D ECHO-QGP 1 code [3], including dissi- II. VORTICITIESINRELATIVISTICHYDRODYNAMICS pativerelativistichydrodynamicsintheIsrael-Stewartformu- lation with Bjorken initial conditions for the flow (i.e. with Unlike in classical hydrodynamics, where vorticity is the curl of the velocity field v, several vorticities can be defined inrelativistichydrodynamicswhichcanbeusefulindifferent 1Thecodeispubliclyavailableatthewebsitehttp://theory.fi.infn.it/echoqgp applications(seealsothereview[14]). 2 A. Thekinematicalvorticity Theabove(6)isalsoknownasCarter-Lichnerowiczequa- tion[14]foranidealunchargedfluidanditentailsconserva- Thisisdefinedas: tion properties which do not hold for the kinematical vorticity. Thiscanbebetterseeninthethelanguageofdifferential ω = 1(d u −d u )= 1(∂ u −∂ u ) (1) forms,rewritingthedefinitionoftheT-vorticityastheexterior µν 2 ν µ µ ν 2 ν µ µ ν derivativeofathevectorfield(1-form)Tu,thatisΩ=d(Tu). Indeed,theeq.(6)implies-throughtheCartanidentity-that where u is the four-velocity field. This tensor includes both theLiederivativeofΩalongthevectorfielduvanishes,that the acceleration A and the relativistic extension of the angu- is lar velocity pseudo-vector ω in the usual decomposition of µ an antisymmetric tensor field into a polar and pseudo-vector LuΩ=u·dΩ+d(u·Ω)=0 (7) fields: 1 becauseΩisitselftheexternalderivativeofthevectorfieldTu ωµν =(cid:15)µνρσωρuσ+ (Aµuν−Aνuµ) anddd=0.Theeq.(7)statesthattheT-vorticityisconserved 2 A =2ω uν =uνd u ≡ Du alongtheflowand,thus,ifitvanishesataninitialtimeitwill µ µν ν µ µ remain so at all times. This can be made more apparent by 1 ω =− (cid:15) ωρσuτ (2) expandingtheLiederivativedefinitionincomponents: µ µρστ 2 (L Ω)µν = DΩµν−∂ uµΩσν−∂ uνΩσµ =0 (8) where(cid:15) istheLevi-Civitasymbol.Usingofthetransverse u σ σ µνρσ (tou)projector: TheaboveequationisinfactadifferentialequationforΩpre- cisely showing that if Ω = 0 at the initial time then Ω ≡ 0. ∆µν ≡gµν−uµuν, Thereby,theT-vorticityhasthesamepropertyastheclassical vorticity foran ideal barotropic fluid, such as theKelvin cir- andtheusualdefinitionoftheorthogonalderivative culationtheorem,sotheintegralofΩoverasurfaceenclosed ∇ ≡∆αd =d −u D, byacircuitcomovingwiththefluidwillbeaconstant. µ µ α µ µ OnecanwritetherelationbetweenT-vorticityandkinemat- where D = uαd , it is convenient to define also a transverse icalvorticitybyexpandingthedefinition(5): α kinematicalvorticityas: 1(cid:104) (cid:105) Ω = (∂ T)u −(∂ T)u +Tω ω∆ =∆ ∆ ωρσ = 1(∇ u −∇ u ) (3) µν 2 ν µ µ ν µν µν µρ νσ 2 ν µ µ ν implyingthatthedouble-transverseprojectionofΩ: Usingtheabovedefinitioninthedecomposition(2)itcanbe ∆ ∆ Ωρσ ≡Ω∆ =Tω∆ shownthat: µρ νσ µν µν ω∆ =(cid:15) ωρuσ (4) Hence,thetensorω∆ sharesthesameconservationproperties µν µνρσ ofΩ∆, namelyitvanishesatalltimesifitisvanishingatthe thatisω∆isthetensorformedwiththeangularvelocityvector initialtime. Conversely,themixedprojectionofthekinemat- only. Aswewillshowinthenextsubsection,onlyω∆ shares icalvorticity: the “conservation” property of the classical vorticity for an 1 idealbarotropicfluid. uρωρσ∆σν = Aσ 2 doesnot. Itthenfollowsthatforanidealunchargedfluidwith B. TheT-vorticity ω∆ =0attheinitialtime,thekinematicalvorticityissimply: 1 Thisisdefinedas: ω = (A u −A u ) (9) µν µ ν ν µ 2 1(cid:104) (cid:105) Ω = ∂ (Tu )−∂ (Tu ) (5) µν ν µ µ ν 2 C. Thethermalvorticity and it is particularly useful for a relativistic uncharged fluid, suchastheQCDplasmaformedinnuclearcollisionsatvery Thisisdefinedas[13]: high energy. This is because from the basic thermodynamic relations when the temperature is the only independent ther- (cid:36) = 1(∂ β −∂ β ) (10) modynamicvariable,theidealrelativisticequationofmotion µν 2 ν µ µ ν (ε+ p)A = ∇ p can be recast in the simple form (see e.g. µ µ where β is the temperature four-vector. This vector is de- [15]): fined as (1/T)u once a four-velocity u, that is a hydrody- 1 namical frame, is introduced, but it can also be taken as a uµΩµν = (TAν−∇νT)=0 (6) primordial quantity to define a velocity through u ≡ β/(cid:112)β2 2 3 [16]. Thethermalvorticityfeaturestwoimportantproperties: ifthetransverselyprojectedvorticitytensorω∆ initiallyvan- it is adimensional in natural units (in cartesian coordinates) ishes, so will the transverse projection Ω∆ and (cid:36)∆ and the and it is the actual constant vorticity at the global equilib- kinematical and thermal vorticities will be given by the for- rium with rotation [17] for a relativistic system, where β is mulae (9) and (14) respectively. Indeed, the T-vorticity Ω aKillingvectorfieldwhoseexpressioninMinkowskispace- will vanish throughout because also its longitudinal projec- timeisβ = b +(cid:36) xν beingband(cid:36)constant. Inthiscase tionvanishesaccordingtoeq.(6). Thisispreciselywhathap- µ µ µν the magnitude of thermal vorticity is - with the natural con- pens for the usually assumed BIC for the flow at τ , that is 0 stantsrestored-simply(cid:126)ω/k T whereωisaconstantangular ux = uy = uη = 0,whereonehasω∆ = 0atthebeginningas B velocity. In general, (replacing ω with the classical vorticity itcanbereadilyrealizedfromthedefinition(1). Ontheother defined as the curl of a proper velocity field) it can be read- hand,foraviscousunchargedfluid,transversevorticitiescan ily realized that the adimensional thermal vorticity is a tiny developeveniftheyarezeroatthebeginning. number for most hydrodynamical systems, though it can be It should be noted though, that even if the space-space significantfortheplasmaformedinrelativisticnuclearcolli- components (x,y,η indices) of the kinematical vorticity ten- sions. sor vanish at the initial Bjorken time τ , they can develop at Furthermore,thethermalvorticityisresponsibleforthelo- 0 later times even for an ideal fluid if the spatial parts of the cal polarization of particles in the fluid according to the for- acceleration and velocity fields are not parallel, according to mula[12]: eq. (9). The equation makes it clear that the onset of spatial 1 pτ components of the vorticity is indeed a relativistic effect as, Πµ(x,p)=−8(cid:15)µρστ(1−nF) (cid:36)ρσm (11) withtheproperdimensions,itgoeslike(a×v)/c2. whichappliestospin1/2fermions,n beingtheFermi-Dirac- In the full longitudinally boost invariant Bjorken picture, Juttnerdistributionfunction. F thatisuη =0throughoutthefluidevolution,intheidealcase, as ω∆ = 0, the only allowed components of the kinematical n = 1 (12) vorticity are ωτx,ωτy and ωxy from the first eq. (2). The ωxy F eβ(x)·p−µ/T +1 component,atη=0,becauseofthereflectionsymmetry(see fig.1)inboththexandyaxes,canbedifferentfromzerobutit Similarlytotheprevioussubsection,onecanreadilyobtain oughttochangesignbymovingclockwise(orcounterclock- therelationbetweenT-vorticityandthermalvorticity: wise)totheneighbouringquadrantofthexyplane;forcentral 1 (cid:104) (cid:105) 1 collisionsitsimplyvanishes. (cid:36) = (∂ T)u −(∂ T)u + Ω (13) µν 2T2 µ ν ν µ T2 µν However,intheviscouscase,morecomponentsofthevor- ticities can be non-vanishing. Furthermore, in more realistic Again, the double transverse projection of (cid:36) is proportional totheoneofΩ: 3+1 D hydrodynamical calculations, a non-vanishing uη can developbecauseoftheasymmetriesoftheinitialenergyden- ∆ ∆ (cid:36)ρσ ≡(cid:36)∆ = 1 Ω∆ = 1ω∆ sity in the x−η and y−η planes at finite impact parameter. µρ νσ µν T2 µν T Theasymmetryisessentialtoreproducetheobserveddirected flowcoefficientv (y)ina3+1Didealhydrodynamiccalcula- 1 whereasthemixedprojectionturnsouttobe,usingeq.(13) tionwithBIC,asshownbyBozek[18],andgivestheplasma 1 Aν atotalangularmomentum,asitwillbediscussedlateron. uρ(cid:36) ∆σν = ∇νT + ρσ 2T2 2T Inthiswork,wecalculatethevorticities,andespeciallythe thermalvorticity(cid:36)byusingbasicallythesameparametriza- Again,foranidealunchargedfluidwithω∆ = 0attheinitial tion of the initial conditions in ref. [18]. Those initial con- time,byusingtheequationsofmotion(6),onehastheabove ditions are a modification of the usual BIC to take into ac- projectionisjustAν/T andthatthethermalvorticityissimply: countthattheplasma,inperipheralcollisions,hasarelatively largeangularmomentum(seeAppendixA).Theyareamin- 1 (cid:36) = (A u −A u ) (14) imalmodificationsoftheBICinthattheinitialflowvelocity µν µ ν ν µ T Bjorkencomponentsarestillzero,buttheenergydensitylon- A common feature of the kinematical and thermal vorticity gitudinal profile is changed and no longer symmetric by the is that their purely spatial components can be non-vanishing reflection η → −η. They are summarized hereinafter. Given if the acceleration and velocity field are non-parallel, even theusualthicknessfunctionexpression: thoughvelocityisvanishingatthebeginning. (cid:90) ∞ (cid:90) ∞ n T(x,y)= dzn(x,y,z)= dz √ 0 III. HIGHENERGYNUCLEARCOLLISIONS −∞ −∞ 1+e( x2+y2+z2−R)/δ (15) Innuclearcollisionsatverylargeenergy,theQCDplasma where n = 0.1693fm−3, δ = 0.535fm and R = 6.38fm 0 is an almost uncharged fluid. Therefore, according to pre- are the nuclear density, the width and the radius of the nu- vious section’s arguments, in the ideal fluid approximation, clear Fermi distribution respectively, the following functions 4 and  y f+(η)=01η2+ηmηm η−ηη<>m−η≤ηmη≤ηm m x Finally, the initial proper energy density distribution is as- sumedtobe: ε(x,y,η)=ε W(x,y,η)H(η), (20) 0 wherethetotalweightfunctionW(x,y,η)isdefinedas: (1−α)W (x,y,η)+αn (x,y) W(x,y,η)= N BC (cid:12) . (21) (1−α)W (0,0,0)+αn (0,0)(cid:12)(cid:12) z N BC (cid:12)b=0 and: J   H(η)=exp−2η˜σ22θ(η˜) η˜ =|η|−ηflat/2 (22) η In the eq. (21) n (x,y) is the mean number of binary colli- BC sions: nBC(x,y)=σinT+(x,y)T−(x,y) (23) Figure1: Collidingnucleiandconventionalcartesian andαisthecollisionhardnessparameter,whichcanvarybe- referenceframe. Alsoshowntheinitialangularmomentum tween0and1. vector. This parametrization, and especially the chosen forms of the functions f±, are certainly not unique as a given angular momentum can be imparted to the plasma in infinitely many aredefined: ways. Nevertheless, as has been mentioned, it proved to re- T1 =T+ (cid:32)1−(cid:18)1− σAT−(cid:19)A(cid:33) (16) pcaroldcuaclceuclaotriroenctolyfptheeripdhireercatleAdufl-oAwuicnoalli3s+io1nDsahtyhdirgohdyennaemrgiy- T2 =T− (cid:32)1−(cid:18)1− σT+(cid:19)A(cid:33) (17) [th1i8s],intihtuiaslwcoentdoiotikonitwaisllabgeobordiesfltyardtiinscgupsoseindt.inAsevcat.riVatIiIo.nBoe-f A sides,theparametrization(20)essentiallyrespectsthecausal- whereσistheinelasticNNcrosssection,Athemassnumber ityconstrai√ntthattheplasmacannotextendbeyondη=ybeam. ofthecollidingnuclei,and: Indeed,at sNN =200GeVybeam (cid:39)5.36whilethe3σpoint inthegaussianprofileineq.(22)liesatη=η /2+3σ (cid:39)4.4. flat η T+(xT)=T(xT +b/2) T−(xT)=T(xT −b/2) (18) The free parameters have been chosen following ref. [19], wheretheywereadjustedtoreproducethedatainAu-Aucol- √ where xT = (x,y) is the vector of the transverse plane coor- lisionsat sNN =200GeV.TheyarereportedintableI. dinates and b is the impact parameter vector, connecting the We have run the ECHO-QGP code in both the ideal and centersofthetwonuclei. Inourconventionalcartesianrefer- viscousmodeswiththeparametersreportedintableIandthe enceframe, thebvectorisorientedalongthepositive x axis equationofstatereportedinref.[20]. Theimpactparameter and the two nuclei have initial momentum along the z axis valueb = 11.57waschosenas,intheopticalGlaubermodel, (whencethereactionplaneisthexzplane)andtheirmomenta itcorrespondstothemeanvalueofthe40-80%centralityclass aredirectedsoastomaketheinitialtotalangularmomentum (9.49<b<13.42fm[21])usedbytheSTARexperimentfor oriented along the negative y axis (see fig. 1). The wounded the directed flow measurement in ref. [22]. The initial flow nucleonsweightfunctionWN isthendefined: velocities ux,uy,uη were set to zero, according to BIC. The freezeout hypersurface - isothermal at T = 130 MeV - is WN(x,y,η)=2 (T1(x,y)f−(η)+T2(x,y)f+(η)) (19) determinedwiththemethodsdescribedinforefs.[3,23]. where:  f−(η)=01−η2+ηmηm −ηηη><m−η≤ηmη≤ηm ofTthoIeVsh.moawQtteUthrAapLtroIoFudIruCccAoeTddeIiOnisNhweOeaFlvlyTs-uHioitEnedEcCotolHlmiOsioo-dQnesGlPathnCedOehvDeonElucetiotno m 5 Par√ameter Value ofthegridresolution(theboundariesin x,y,ηbeingfixed)2 sNN 200GeV Asitisexpected, thenormalizedT-vorticitydecreasesasthe α 0. resolutionimproves. (cid:15) 30GeV/fm3 0 Because of the relation (13), the residual value at our best σ 40mb in spatialresolutionof0.15fmcanbetakenasanumericalerror τ 0.6fm/c 0 forlatercalculationsofthethermalvorticity. η 1 flat σ 1.3 η T 130MeV fo b 11.57fm 0.0016 η ideal 3.36 τx η mviscous 2.0 0.0014 τy m τη η/s 0.1 0.0012 xy yη 0.001 ηx TableI:Parametersdefiningtheinitialconfigurationofthe 0.0008 fluidinBjorkencoordinates. Thelasttwoparametervalues havebeenfixedforthelastphysicalrun. 0.0006 0.0004 0.0002 carryoutourstudyonthedevelopmentofvorticityinsuchan environment, we have performed two calculations, referring 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 toanidealandviscousscenariorespectively,providingavery Grid resolution (fm) stringentnumericaltest. Beforedescribingthesetests, itshouldbepointedoutthat Figure2: (coloronline)Meanoftheabsolutevalueof thevorticitiescomponentsaretobecalculatedinBjorkenco- T-vorticitycomponents,dividedbyT2,atthefreeze-outasa ordinates, whose metric tensor is g = diag(1,−1,−1,−τ2), functionofthegridresolution. µν hence they do not all have the same dimension nor they are adimensionalasitisdesirable(exceptthethermalvorticity,as ithasbeenemphasizedinSect.II).Forapropercomparison itisbettertousetheorthonormalbasis,whichinvolvesafac- tor τ when the η components are considered. Moreover, the 0.0006 4 cumulative contribution of all components is well described 0.0004 bytheinvariantmodulus,which,foragenericantisymmetric tensorAµν is: 2 0.0002 0 A2 = AµνAµν =2[A2xy−A2τx−A2τy+(A2ηx+A2ηy−A2ητ)/τ2]. (24) x (fm) 0 -0.0002 Furthermore,wehavealwaysrescaledtheT-vorticityby1/T2 -0.0004 inordertohaveanadimensionalnumber.SincetheT-vorticity -2 -0.0006 has always been determined at the isothermal freezeout, in ordertogetitsactualmagnitude,onejustneedstomultiplyit -0.0008 byT2 . -4 fo -0.001 -4 -2 0 2 4 η A. T-vorticityforanidealfluid Figure3: (coloronline)ContourplotofΩ /τT2atthe xη freeze-outhypersurfaceaty=0. SincethefluidisassumedtobeunchargedandtheinitialT- vorticityΩisvanishingwiththeBIC,itshouldbevanishing throughout,accordingtothediscussioninsect.II).However, the discretization of the hydrodynamical equations entails a numerical error, thus the smallness of Ω in an ideal run is a 2It should be pointed out that, throughout this work, by mean values of thevorticitieswemeansimpleaveragesofthe(possiblyrescaledby1/τ) gauge of the quality of the computing method. In fig. 2 we Bjorkencomponentsoverthefreezeouthypersurfacewithoutgeometrical show the mean of the absolute values of the six independent cellweighting.Therefore,theplottedmeanvalueshavenophysicalmean- Bjorkencomponentsatthefreeze-outhypersurface,oftheT- ingandtheyshouldbetakenasdescriptivenumberswhicharerelatedto vorticitydividedbyT2tomakeitadimensional,asafunction theglobalfeaturesofvorticitycomponentsatthefreeze-out. 6 B. Gubserflow mostgeneralformactuallyadmitsfurthertermsthatwerede- rivedforasystemofmasslessparticlesinrefs.[28,29]),valid A very useful test for the validation of a numerical code forthecaseofaconformalfluidwithε=3p: of relativistic dissipative hydrodynamics is the explicit solution of Israel-Stewart theory of a Bjorken flow with an az- DT + θ − πµνσµν =0 (28a) imuthally symmetric radial expansion [24–27], the so-called T 3 3sT (cid:32) (cid:33) Gubser flow. Indeed, this solution provides a highly non- τ ∆µ∆νDπαβ+ 4πµνθ +πµν =2ησµν. (28b) trivialtheoreticalbenchmark. π α β 3 Forthesakeofclarity,webrieflysummarizethemainsteps leading to the analytical solution, to be compared with the InthecaseoftheGubserflowinEq.(28),duetothetraceless numerical computation. In the case of a conformal fluid, andtransverseconditionsπˆµµ=0anduˆµπˆµν=0,onehassimply with p = (cid:15)/3 EOS, the invariance for scale transformations tosolvethetwoequations(π¯ηη ≡πˆηη/sˆTˆ) sets the terms entering the second-order viscous hydrodynamic equations. The additional requests of azimuthal and 1 dTˆ + 2tanhρ= 1π¯ηηtanhρ (29) longitudinal-boost invariance, constrain the solution of the Tˆ dρ 3 3 hydrodynamic equations, which has to be invariant under SO(3) ⊗ SO(1,1) ⊗ Z transformations. To start with, one and(ηˆ/sˆ=η/s,beingtheratiodimensionless) q 2 defines a modified space-time metric as follows (with usual (cid:34) (cid:35) dπ¯ηη 4 4 ηˆ Bjorkencoordinates,ηbeingthespacetimerapidity): τˆ + (π¯ηη)2tanhρ +π¯ηη = tanhρ. (30) (cid:32) (cid:33) R dρ 3 3sˆTˆ dτ2−dx2 ds2 =τ2dsˆ2 ≡τ2 −dη2 τ2 The solution can be then mapped back to Minkowski space (cid:32) (cid:33) throughtheformulae: dτ2−dr2−r2dφ2 =τ2 −dη2 , τ2 1 ∂xˆα∂xˆβ T =Tˆ/τ, π = πˆ . (31) whichcanbeviewedasarescalingofthemetrictensor: µν τ2 ∂xµ ∂xν αβ ds2 −→dsˆ2 ≡ds2/τ2 ⇐⇒ g −→gˆ ≡g /τ2. Infig.4weshowthecomparisonbetweentheGubseranalyt- µν µν µν ical solution and our numerical computation for the temper- Itcanbeshownthatdsˆ2 istheinvariantspacetimeintervalof ature T and the components πxx, πxy and πηη of the viscous dS ⊗R, where dS is the three-dimensional de Sitter space stresstensorrespectively,atdifferenttimes.Theinitialenergy 3 3 and R refers to the rapidity coordinate. It is then convenient densityprofilewastakenfromtheexactGubsersolutionatthe to perform a coordinate transformation (q is an arbitrary pa- time τ = 1 fm/c. The simulation has been performed with a rametersettinganenergyscaleforthesolutiononceonegoes gridof0.025fminspaceand0.001fmintime. Theshearvis- backtophysicaldimensionfulcoordinates) cositytoentropydensityratiowassettoη/s = 0.2,whilethe shearrelaxationtimeisτ = 5η/(ε+ p). Theenergyscaleis R 1−q2(τ2−r2) 2qr settoq=1fm−1.Asitcanbeseen,theagreementisexcellent sinhρ≡− , tanθ≡ , (25) 2qτ 1+q2(τ2−r2) uptolatetimes. afterwhichtherescaledspacetimeelementdsˆ2reads C. T-vorticityforaviscousfluid dsˆ2 =dρ2−cosh2ρ(dθ2+sin2θdφ2)−dη2. (26) Thefullsymmetryoftheproblemisnowmanifest. SO(1,1) Unlikeforanidealunchargedfluid,T-vorticitycanbegen- and Z refer to the usual invariance for longitudinal boosts erated in a viscous uncharged fluid even if it is initially van- 2 and η → −η inversion, while SO(3) reflects the spherical ishing. Thus,theT-vorticitycanbeusedasatooltoestimate q symmetryoftherescaledmetrictensorinthenewcoordinates. the numerical viscosity of the code in the ideal mode by ex- InGubsercoordinatesthefluidisatrest: trapolatingtheviscousruns. Acommentisinorderhere. Ingeneral,inadditiontostan- uˆ =1, uˆ =uˆ =uˆ =0. (27) dard truncation errors due to finite-difference interpolations, ρ θ φ η all shock-capturing upwind schemes are known to introduce ThecorrespondingflowinMinkowskispacecanbeobtained numericalapproximationsthatbehaveroughlyasadissipative taking into account both the rescaling of the metric and the effects, especially in the simplified solution to the Riemann changeofcoordinates problems at cell interfaces [30]. It is therefore important to ∂xˆν check whether the code is not introducing, for a given reso- u =τ uˆ , lution, numerical errors which are larger than the effects in- µ ∂xµ ν ducedbythephysics. Werefertotheglobalnumericalerrors where xˆµ = (ρ,θ,φ,η) and xµ = (τ,r,φ,η). Other quantities genericallyasnumericalviscosity. suchasthetemperatureortheviscoustensorsrequiretheso- WehavethuscalculatedtheT-vorticityfordifferentphysi- lution of the following set of hydrodynamic equations (their calviscosities(infactη/sratios),inordertoprovideanupper 7 0.06 0.000 τ=4 fm/c 0.05 −0.001 V) mperature (Ge00..0034 τ=8 fm/c xx3π (GeV/fm) −−00..000032 τ=8 fm/c (3x) e T −0.004 0.02 −0.005 τ=4 fm/c 0.01 −10 −5 0 5 10 −10 −5 0 5 10 x (fm) x (fm) 0.000 τ=4 fm/c 0.0015 −0.001 3V/fm) −0.002 3eV/fm) 0.0010 τ=8 fm/c (10x) e G xyπ (G −0.003 τ=8 fm/c (3x) 2ηη (τπ 0.0005 −0.004 τ=4 fm/c 0.0000 −10 −5 0 5 10 −10 −5 0 5 10 x (fm) x (fm) Figure4: (coloronline)Comparisonbetweenthesemi-analyticsolutionoftheGubserviscousflow(solidlines)andthe numericalECHO-QGPcomputation(dots). boundforthenumericalviscosityofECHO-QGPintheideal pends of course on the initial conditions, particularly on the mode. The mean value of the T-vorticity is shown in fig. 5 parameterη (seeSect.III),asshowninfig.6. Thedirected m anditsextrapolationtozerooccurswhen|η/s|(cid:46)0.002which flowalsodependsonη/sasshowninfig.7andcouldthenbe isaverysatisfactoryvalue,comparablewiththeoneobtained usedtomeasuretheviscosityoftheQCDplasmaalongwith in ref. [4]. The good performance is due to the use of high- other azimuthal anisotropy coefficients. It should be pointed orderreconstructionmethodsthatareabletocompensatefor out that, apparently, the directed flow can be reproduced by thehighlydiffusivetwo-waveRiemannsolveremployed[3]. ourhydrodynamicalcalculationonlyfor−3<y<3. Thede- pendenceofv (y)onη andη/smakesitpossibletoadjustthe 1 m η parameterforagivenη/svalue.Thisadjustmentcannotbe m V. DIRECTEDFLOW,ANGULARMOMENTUMAND properlycalledaprecisionfitbecause,aswehavementioned THERMALVORTICITY intheIntroduction,severaleffectsinthecomparisonbetween dataandcalculationshavebeendeliberatelyneglectedinthis WiththeinitialconditionsreportedattheendoftheSect.III work. However, since our aim was to obtain a somewhat re- we have calculated the directed flow of pions (both charged alisticevaluationofthevorticities, wehavechosenthevalue states) at the freezeout and compared it with the STAR data of ηm for which we obtain the best agreement between our for charged particles collected in the centrality interval 40- calculated pion v1(y) and the measured for charged particles 80% [22]. Directed flow is an important observable for sev- in the central rapidity region. For the fixed η/s = 0.1 (ap- eral reasons. Recently, it has been studied at lower energy proximatelytwicetheconjectureduniversallowerbound)the [31]√with a hybrid fluid- transport model (see also ref. [32]). correspondingbestvalueofηmturnsouttobe2.0(seefig.8). At s = 200 GeV, it has been calculated with an ideal NN 3+1DhydrocodefirstbyBozek[18]. Herein,weextendthe It is worth discussing more in detail an interesting rela- calculationtotheviscousregime. tionship between the value of the parameter η and that of m Theamountofgenerateddirectedflowatthefreezeoutde- a conserved physical quantity, the angular momentum of the 8 0.1 0.04 ηm=1.5 0.03 ηm=2.0 ηm=2.5 0.01 0.02 ηm=3.0 ηm=3.5 STAR c=40-80% 0.01 0.001 V1 0 τx -0.01 τy τη 0.0001 -0.02 xy yη ηx -0.03 1e-05 -0.04 0 0.04 0.08 0.12 0.16 -4 -3 -2 -1 0 1 2 3 4 η/s Y (rapidity) 0.008 Figure6: (coloronline)Directedflowofpionsfordifferent valuesofη parameterwithη/s=0.1comparedwithSTAR m 0.007 τx data[22]. τy 0.006 τη xy 0.005 yη 0.04 ηx 0.004 0.03 0.003 0.02 0.002 0.01 0.001 V1 0 unviscid 0 -0.01 η/s=0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 η/s=0.04 η/s -0.02 η/s=0.08 η/s=0.10 η/s=0.12 Figure5: (coloronline)Meanoftheabsolutevaluesof -0.03 η/s=0.16 Ω /T2componentsatthefreeze-outhypersurfaceasa STAR c=40-80% µν -0.04 functionofη/s. NotethattheΩxη,Ωyη,Ωτηhavebeen -4 -3 -2 -1 0 1 2 3 4 multipliedby1/τ. Upperpanel: logscale. Lowerpanel: Y (rapidity) magnificationoftheregionaroundzeroviscosity. Figure7: (coloronline)Directedflowofpionsfordifferent valuesofη/swithη =2.0comparedwithSTARdata[22]. m plasma,which,forBICisgivenbytheintegral(seeAppendix Aforthederivation): sharpspheres. Inourconventionalreferenceframe,theinitial (cid:90) Jy =−τ dxdydη xε(x,y,η) sinhη (32) angular momentum of the nuclear overlap region is directed 0 alongtheyaxiswithnegativevalueandcanbewrittenas: √ (cid:90) Sinceη controlstheasymmetryoftheenergydensitydistri- s bution imn the η− x plane, one expects that Jy will vary as a Jy = dxdyw(x,y)(T+−T−)x 2NN (33) function of η . Indeed, if the energy density profile is sym- m metricinη,theintegralineq.(32)vanishes.Yet,foranyfinite whereT±arethethicknessfunctionslikeineq.18and ηm (cid:44) 0,theprofile(20)isnotsymmetricand Jy (cid:44) 0(looking min(n(x+b/2,y,0),n(x−b/2,y,0)) atthedefinitionof f+and f−itcanberealizedthatonlyinthe w(x,y)= max(n(x+b/2,y,0),n(x−b/2,y,0)) limitη →∞theenergydensityprofilebecomessymmetric). m Thedependenceoftheangularmomentumonη withallthe is the function which extends the simple product of two θ m initialparameterskeptfixedisshowninfig.9. Forthevalue functionsusedfortheoverlapoftwosharpspheres. Notethat η =2.0itturnsouttobearound3.18×103in(cid:126)units. the ω˜(x,y) is 1 for full overlap (b=0) and implies a vanish- m Itisalsointerestingtoestimateanupperboundonthean- ingangularmomentumforverylargeb(seefig.10)(seealso gularmomentumoftheplasmabyevaluatingtheangularmo- ref.[33]). mentumoftheoverlapregionofthetwocollidingnuclei.This At b = 11.57 fm the above angular momentum is about can be done by trying to extend the simple formula for two 3.58 × 103 in (cid:126) units. This means that, with the current 9 0.04 90000 0.03 80000 70000 0.02 60000 0.01 s) nit 50000 V1 -0.0 01 y-J| ( uh 40000 | 30000 -0.02 20000 -0.03 10000 -0.04 0 -4 -3 -2 -1 0 1 2 3 4 0 2 4 6 8 10 12 14 Y (rapidity) b (fm) Figure8: Directedflowofpionsatη/s=0.1andη =2.0 Figure10: (coloronline)Estimatedangularmomentum(in(cid:126) m comparedwithSTARdata[22]. units)oftheoverlapregionofthetwocollidingnuclei(solid line)andtotalangularmomentumoftheplasmaaccordingto theparametrizationoftheinitialconditions(dashedline),as afunctionoftheimpactparameter. 3600 3400 3200 0.25 s) 3000 nit 0.2 y-|J| ( uh 22680000 0.15 2400 0.1 2200 τx τy 2000 τη 1 1.5 2 2.5 3 3.5 4 0.05 xy η yη m ηx Figure9: Angularmomentum(in(cid:126)units)oftheplasmawith 0 0 0.04 0.08 0.12 0.16 Bjorkeninitialconditionsasafunctionoftheparameterηm. η/s Figure11: (coloronline)Meanoftheabsolutevalueof parametrization of the initial conditions, for that impact pa- thermalvorticitycovariantcomponentsatthefreeze-outasa rameter about 89% of the angular momentum is retained by functionofη/s. Notethatthe(cid:36)xη,(cid:36)yη,(cid:36)τηhavebeen the hydrodynamical plasma while the rest is possibly taken multipliedby1/τ. awaybythecoronaparticles. With the final set of parameters, we have calculated the thermal vorticity (cid:36). As it has been mentioned in Sect. II, out, for a fixed value of the y coordinate y = 0, is shown this vorticity is adimensional in cartesian coordinates) and it in fig. 13 where it can be seen that it attains a top (negative) is constant at global thermodynamical equilibrium [17], e.g. value of about 0.05 corresponding to a kinematical vorticity, foragloballyrotatingfluidwitharigidvelocityfield. Inrel- atthefreezeouttemperatureof130MeV,ofabout0.033c/fm ativistic nuclear collisions we are far from such a situation, (cid:39) 1022s−1. In this respect, the Quark Gluon Plasma would neverthelesssomethermalvorticitycanbegenerated,bothin be the fluid with the highest vorticity ever made in a terres- the ideal and viscous case. This is shown in figs. 11 and 12. trial laboratory. However, the mean value of this component It can be seen that the generated amount of thermal vor- at the same value of η/s = 0.1 is of the order of 5.4×10−3, ticity has some non-trivial dependence on the viscosity. Par- that is about ten times less than its peak value, as shown in ticularly, as it is apparent from fig. 12, the (cid:36) component - fig.12. Thismeanthermalvorticityistheconsistentlylower xη which is directed along the initial angular momentum - has thantheoneestimatedinref.[13](about0.05)withthemodel a non-vanishing mean value whose magnitude significantly described in refs. [9, 10] implying an initial non-vanishing increases with increasing viscosity. Its map at the freeze- transversekinematicalandthermalvorticity(cid:36)∆. Thisreflects 10 interesting feature of this relation is that it makes it possible 0.001 to obtain an indirect measurement of the mean thermal vor- 0 ticity at the freezeout by measuring the polarization of some hadron. For instance, the polarization of Λ baryons, as it is -0.001 wellknown,canbedeterminedwiththeanalysisoftheangu- -0.002 lardistributionofitsdecayproducts,becauseofparityviola- -0.003 tion. The polarization pattern depends on the momentum of thedecayingparticle,asitisapparentfromeq.(34). -0.004 τx The formula (34) makes sense only if the components of τy -0.005 τη the integrand are Minkowskian, as an integrated vector field -0.006 xy yieldsavectoronlyifthetangentspacesarethesameateach yη -0.007 ηx point. Before summing over the freezeout hypersurface we havethentransformedthecomponentsofthethermalvortic- -0.008 ity from Bjorken coordinates to Minkowskian by using the 0 0.04 0.08 0.12 0.16 known rules. The thus obtained polarization vector Π(p) is η/s theoneinthecollisionframe. However,thepolarizationvec- torwhichismeasurableistheoneinthedecayingparticlerest Figure12: (coloronline)Meanvaluesofthermalvorticity frame which can be obtained by means of the Lorentz trans- componentsatthefreeze-outasafunctionofη/s. Notethat formations: the(cid:36) ,(cid:36) ,(cid:36) havebeenmultipliedby1/τ. xη yη τη (cid:15) p·Π Π0 = Π0− 0 m m 4 0.03 Π =Π− p·Π p (35) 0.02 0 (cid:15)(m+(cid:15)) 2 0.01 Infigure14weshowtheΛpolarizationvectorcomponents, aswellasitsmodulus,asafunctionofthetransversemomen- m) 0 tum pT for pz = 0 expected under the assumptions of local x (f 0 -0.01 thermodynamicalequilibriumforthespindegreesoffreedom maintained till kinetic freezeout. It can be seen that the po- -0.02 larization vector has quite an assorted pattern, with an over- -2 -0.03 allmagnitude(seefig.14, panel(a))hardlyexceeding1%at -0.04 momentaaround4GeV.Asexpected,theycomponentispre- -4 dominantly negative, oriented along the initial angular mo- -0.05 mentumvectorandamagnitudeoftheorderof0.1%. Indeed, -4 -2 0 2 4 themaincontributiontothepolarizationstemsfromthelongi- η tudinalcomponentΠz,withamaximumandminumumalong 0 thebisector|px|=|py|. Figure13: (coloronline)Contourplotof1/τ-scaledηx Theobtainedpolarizationvaluesare-asexpected-consis- covariantcomponentofthethermalvorticity,(cid:36) /τoverthe ηx tently smaller than those estimated in ref. [13] (of the order freeze-outhypersurfacefory=0,η/s=0.1,ηm=2.0. ofseveralpercentwithatopvalueof8-9%)withthealready mentioned initial conditions used in refs. [9, 10]. This is a consequenceofthemuchlowervalueoftheimpliedthermal inaquitelowvalueofthepolarizationofΛbaryons,asitwill vorticity, as discussed in the previous section. Also, the Πy beshowninthenextsection. patternisremarkablydifferent,withdifferentlocationofmax0- imaandminima. VI. POLARIZATION VII. CONCLUSIONS,DISCUSSIONANDOUTLOOK AsithasbeenmentionedintheIntroduction,vorticitycan result in the polarization of particles in the final state. The Tosummarize,wehavecalculatedthevorticitiesdeveloped relation between the polarization vector of a spin 1/2 parti- √ inperipheral(b = 11.6fm)nuclearcollisionsat s = 200 cle and thermal vorticity in a relativistic fluid was derived in NN GeV(b=11.6fm)withthemostcommonlyusedinitialcon- ref.[12]andreads: ditions in the Bjorken hydrodynamical scheme, by using the (cid:82) Πµ(p)= 81m ΣdΣλpλnF(cid:82)(1d−ΣnpFλ)npσ(cid:15)µνρσ∂νβρ (34) ctiovdiseticECdiHssOip-aQtiGvePhiymdproledmyneanmtinicgs.sAecnonexdt-eonrdsievre, tceasutisnagl,orfetlhae- Σ λ F highaccuracyandverylownumericaldiffusionpropertiesof wheren istheFermi-Dirac-Juttnerdistributionfunction(12) thecodehasbeencarriedout,followedbylong-timesimula- F andtheintegrationisoverthefreeze-outhypersurfaceΣ. The tions(uptoτ=8fm/c)oftheso-calledviscousGubserflow,a