Kaon HBT radii from perfect fluid dynamics using the Buda-Lund model M. Csana´d 8 E¨otv¨os University, H-1117 Budapest, P´azm´any P´eter s. 1/A, Hungary 0 and 0 2 T. Cso¨rgo˝ n a MTA KFKI RMKI, H-1525 Budapest 114, P.O.Box 49, Hungary J 5 ] h In this paper we summarize the ellipsoidally symmetric Buda-Lund t - model’s results on HBT radii. We calculate the Bose-Einstein correlation l c functionfromthemodelandderiveformulasforthetransversemomentum u dependenceofthecorrelationradiiintheBertsch-Prattsystemofout,side n √ and longitudinal directions. We show a comparison to s = 200GeV [ NN RHICPHENIXtwo-pioncorrelationdataandmakepredictiononthesame 1 observable for different particles. v 0 0 1. Perfect fluid hydrodynamics 8 0 Perfect fluid hydrodynamics is based on local conservation of entropy . 1 and four-momentum. The fluid is perfect if the four-momentum tensor is 0 diagonal in the local rest frame. The conservation equations are closed by 8 the equation of state, which gives the relationship between energy density (cid:15), 0 : pressure p. Typically (cid:15)−B = κ(p+B), where B stands for a bag constant v (B = 0 in the hadronic phase, non-zero in a QGP phase), and κ may be a i X constant, but can be an arbitrary temperature dependent function. r There are only a few exact solutions for these equations. One (and a historically the first) is the famous Landau-Khalatnikov solution discovered morethan50yearsago[1,2,3]. Thisisa1+1dimensionalsolution, andhas realistic properties: it describes a 1+1 dimensional expansion, does not lack acceleration and predicts an approximately Gaussian rapidity distribution. Another renowned solution of relativistic hydrodynamics is the Hwa- Bjorken solution [4, 5, 6], which is a simple, explicit and exact, but acceler- ationless solution. This solution is boost-invariant in its original form, but (1) 2 csanad˙ismd07 printed on February 5, 2008 this approximation fails to describe the data [7, 8]. However, the solution allowed Bjorken to obtain a simple estimate of the initial energy density reached in high energy reactions from final state hadronic observables. Therearesolutionswhichinterpolatebetweentheabovetwosolutions[9, 10], are explicit and describe a relativistic acceleration. 2. The Buda-Lund model We focus here on the analytic approach in exploring the consequences of the presence of such perfect fluids in high energy heavy ion experiments in Au+Au collisions at RHIC. Such exact analytic solutions were published recently in refs. [9, 10, 11, 12, 13]. A tool, that is based on the above listed exact, dynamical hydro solutions, is the Buda-Lund hydro model of refs. [14, 15]. TheBuda-LundhydromodelsuccessfullydescribesBRAHMS,PHENIX, PHOBOSandSTARdataonidentifiedsingleparticlespectraandthetrans- verse mass dependent Bose-Einstein or HBT radii as well as the pseudora- pidity distribution of charged particles in central Au+Au collisions both at √ √ s = 130 GeV [16] and at s = 200 GeV [17] and in p+p collisions at √ NN NN s = 200GeV[18],aswellasdatafromPb+PbcollisionsatCERNSPS[19] and h+p reactions at CERN SPS [20, 21]. The model is defined with the help of its emission function; to take into account the effects of long-lived resonances, it utilizes the core-halo model [22]. It describes an expanding fireball of ellipsoidal symmetry (with the time-dependent principal axes of the ellipsoid being X, Y and Z). 3. HBT from the Buda-Lund model Letuscalculatethetwo-particleBose-Einsteincorrelationfunctionfrom the Buda-Lund source function of the Buda-Lund model as a function of q = p −p , the four-momentum difference of the two particles. The result 1 2 is C(q) = 1+λe−q02∆τ∗2−qx2R∗2,x−qy2R∗2,y−qz2R∗2,z. (1) withλbeingtheinterceptparameter(squareoftheratioofparticlesemitted from the core versus from the halo [22]), and 1 1 m d2 t = + , (2) ∆τ2 ∆τ2 T τ2 ∗ 0 0 (cid:16) (cid:17)−1 R2 = X2 1+m (a2+X˙2)/T , (3) ∗,x t 0 (cid:16) (cid:17)−1 R2 = Y2 1+m (a2+Y˙2)/T , (4) ∗,y t 0 csanad˙ismd07 printed on February 5, 2008 3 (cid:16) (cid:17)−1 R2 = Z2 1+m (a2+Z˙2)/T , (5) ∗,z t 0 with X˙,Y˙,Z˙ being the time-derivative of the principal axes, m the average t transverse mass of the pair. T is the central temperature at the freeze-out, 0 ∆τ is the mean emission duration and τ is the freeze-out time. Further- 0 more, a and d are the spatial and time-like temperature gradients, defined (cid:68) (cid:69) (cid:68) (cid:69) as a2 = ∆T and d2 = ∆T . From the mass-shell constraint one finds T ⊥ T τ q = β q +β q +β q , if expressed by the average velocity β. Thus we 0 x x y y z z can rewrite eq. 1 with modified radii to   C(q) = 1+λ∗exp− (cid:88) Ri2,jqiqj, where (6) i,j=x,y,z R2 = R2 +β2∆τ2, and R2 = β β ∆τ2, (7) i,i ∗,i i ∗ i,j i j ∗ From this, we can calculate the radii in the Bertsch-Pratt frame [23] of out (o, pointing towards the average momentum of the actual pair, rotated from x by an azimuthal angle ϕ), longitudinal (l, pointing towards the beam direction) directions and side (s, perpendicular to both l and o) di- rections. The detailed calculations are described in ref. [24]. These include azimuthally sensitive oscillating cross-terms. However, due to space limita- tions, the angle dependent radii are not shown here. If one averages on the azimuthal angle, and goes into the LCMS frame (where β = β = 0), the l s Bertsch-Pratt radii are: R2 = (R−2 +R−2)−1+β2∆τ2, (8) o ∗,x ∗,y o ∗ R2 = (R−2 +R−2)−1, (9) s ∗,x ∗,y R2 = R2 . (10) l ∗,z These can be fitted then to the data [25] as in ref. [26], see fig. 1. This allows us to predict the transverse momentum dependence of the HBT radii of two-kaon correlations as well: if they are plotted versus m , the data of t all particles fall on the same curve. This is also shown for kaons on fig. 1. REFERENCES [1] L. D. Landau, Izv. Akad. Nauk SSSR Ser. Fiz. 17, 51 (1953). [2] I. M. Khalatnikov, Zhur. Eksp. Teor. Fiz. 27, 529 (1954). [3] S. Z. Belenkij and L. D. Landau, Nuovo Cim. Suppl. 3S10, 15 (1956). [4] R. C. Hwa, Phys. Rev. D10, 2260 (1974). 4 csanad˙ismd07 printed on February 5, 2008 Fig.1. HBT radii from the axially Buda-Lund model from ref. [26], compared to data of ref [25]. We also show a prediction for kaon HBT radii on this plot: these overlap with that of pions if plotted versus transverse mass m . t [5] C.B.Chiu,E.C.G.Sudarshan,andK.-H.Wang,Phys.Rev.D12,902(1975). [6] J. D. Bjorken, Phys. Rev. D27, 140 (1983). [7] B. B. Back et al., Phys. Rev. Lett. 87, 102303 (2001). [8] I. G. Bearden et al., Phys. Rev. Lett. 88, 202301 (2002). 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