Pion interferometry at 200 GeV using anisotropic hydrodynamics
PPion interferometry at 200 GeV using anisotropic hydrodynamics
Mubarak Alqahtani
Department of Basic Sciences, College of Education,Imam Abdulrahman Bin Faisal University, Dammam 34212, Saudi Arabia
Michael Strickland
Department of Physics, Kent State University, Kent, OH 44242 United States (Dated: July 9, 2020)
Abstract
In this paper, we continue our phenomenological studies of heavy-ion collisions using 3+1danisotropic hydrodynamics (aHydro). In previous works, we compared quasiparticle aHydro (aHy-droQP) with ALICE 2.76 TeV Pb-Pb and RHIC 200 GeV Au-Au collision results. At both energies,the agreement was quite good between aHydroQP and the experimental data for many observables.In this work, we present comparisons of the Hanbury Brown–Twiss (HBT) radii and their ratiosdetermined using π + π + pairs produced in 200 GeV Au-Au collisions. We first present comparisonswith STAR results for the HBT radii and their ratios. We then present comparisons with PHENIXresults for the HBT radii and their ratios. In both cases, we find reasonable agreement betweenaHydroQP predictions and available experimental results for the ratios of HBT radii. At the levelof the radii themselves, in some cases quantitative differences on the order of 10-20% remain, whichdeserve further study. Keywords: Quark-gluon plasma, Relativistic heavy-ion collisions, Anisotropic hydrodynamics, femtoscopyradii, Boltzmann equation a r X i v : . [ nu c l - t h ] J u l . INTRODUCTION Heavy-ion collision experiments at the Relativistic Heavy Ion Collider (RHIC) and LargeHadron Collider (LHC) create and study the quark-gluon plasma (QGP) under very ex-treme conditions. Relativistic dissipative hydrodynamics has been quite successfully used todescribe the collective behavior seen in these experiments [1–7]. A decade ago, anisotropichydrodynamics was introduced as a new approach which takes into account the fact thatthe QGP is a highly momentum-space anisotropic plasma at early times and near its lon-gitudinal/transverse edges [8–12]. It was only recently that anisotropic hydrodynamics wasextended to include both shear and bulk viscous effects and a 3+1d code was developedwhich enabled practioners to perform phenomenological comparisons [13–15].In recent years, using our quasiparticle anisotropic hydrodynamics code (aHydroQP),we presented phenomenological comparisons with data available at different collision ener-gies. We first showed comparisons with 2.76 TeV Pb-Pb collision data from the ALICEcollaboration [13, 14]. We presented comparisons of charged-hadron multiplicity, identified-particle spectra, identified-particle average transverse momentum, charged-particle ellipticflow, identified-particle elliptic flow, the integrated elliptic flow vs pseudorapidity, and theHBT radii. These comparisons showed that the agreement is quite good between our modeland the experimental data. Subsequently, we presented comparisons with 200 GeV Au-Aucollision data from the RHIC experiments [15]. In this prior work, we presented comparisonsof the identified particle spectra, charged particle multiplicity versus pseudorapidity, iden-tified particle multiplicity versus centrality, identified particle elliptic flow versus transversemomentum, and charged particle elliptic flow as a function of transverse momentum andrapidity.In this paper, we will present aHydroQP predictions for Hanbury Brown-Twiss (HBT)radii in 200 GeV Au-Au collisions. HBT interferometry studies the correlations of pairs ofparticles obeying Bose-Einstein statistics and is based on quantum statistical interference.The understanding of these correlations is crucial to study the system’s dynamics and thespace-time structure of the emitting sources formed at freeze-out [16–18]. We present com-parisons of the HBT radii and their ratios for π + π + pairs in different centrality classes asa function of k T (pair mean transverse momentum). The dependence of the HBT radii on k T provides information about the size of the regions over which the system emits particles2ith similar momentum (region of homogeneity). We first present comparisons with STARresults for the HBT radii and their ratios. We then show comparisons with PHENIX resultsfrom 2015 for the HBT radii and their ratios. In both cases, we find reasonable agreementbetween our model and the experimental results, however, quantitative differences remain.One should note that there are prior works which were able to describe results from pioninterferometry at 200 GeV Au-Au collisions [19–26]. In this paper, we want to investigatethe effect of the aHydro anisotropic distribution function on the HBT radii using our pre-viously obtained parameter set, since the HBT correlations are sensitive to the freeze-outconditions.The structure of the paper is as follows. In Sec. II, we review the basics of anisotropichydrodynamics and the 3+1d dynamical equations. In Sec. III, we summarize the maincomponents of the 3+1d aHydroQP model, which was used previously and will be used inthis work. In Sec. IV, for Au-Au collisions at 200 GeV, we show comparisons of the HBTradii and their ratios obtained using the 3+1d aHydroQP model with experimental data.Sec. V contains our conclusions and an outlook for the future. II. QUASIPARTICLE ANISOTROPIC HYDRODYNAMICS
The evolution of massive relativistic quasiparticle systems is governed by the Boltzmannequation [27] p µ ∂ µ f ( x, p ) + 12 ∂ i m ∂ i ( p ) f ( x, p ) = − C [ f ( x, p )] , (1)where the mass ( m ) is a function of temperature obtained from lattice QCD calculationsand C [ f ( x, p )] is the collisional kernel which is taken to be in relaxation-time approximation(RTA) [27].In anisotropic hydrodynamics, the distribution function in the local rest frame is givenby f ( x, p ) = f eq (cid:32) λ (cid:115)(cid:88) i p i α i + m (cid:33) , (2)where the α i parameters encode the momentum-space anisotropy of the medium. This formreduces back to the equilibrium distribution function with temperature T when α i = 1 andthe temperature like parameter, λ , is taken to be λ = T. Once the form of the distribution3 - % - STAR R ou t [ f m ] ( a ) - % - aHydro R s i de [ f m ] ( b ) R l ong [ f m ] ( c ) - % - STAR R ou t [ f m ] - % - aHydro R s i de [ f m ] R l ong [ f m ] - % - STAR R ou t [ f m ] - % - aHydro R s i de [ f m ] R l ong [ f m ] - % - STAR k T [ GeV ] R ou t [ f m ] - % - aHydro k T [ GeV ] R s i de [ f m ] k T [ GeV ] R l ong [ f m ] FIG. 1. HBT radii are as a function of k T for π + π + in the 0-5%, 5-10%, 10-20%, and 20-30%centrality classes. The left, middle, and right columns show R out , R side , and R long , respectively.All results are for 200 GeV Au-Au collisions and data shown are from the STAR collaboration [30].The p T cuts used were 0 . < p T < . function is specified, the dynamical equations can be obtained using the first and secondmoments of the Boltzmann equation. Detailed derivation of the dynamical equations foraHydroQP can be found in Refs. [12, 27–29].4 II. 3+1D AHYDROQP MODEL
In this section, we highlight the main ingredients and assumptions used in the 3+1daHydroQP code used herein. First, we used smooth Glauber initial conditions, neglecting theeffects of fluctuations. We also used momentum-space isotropic initial conditions α i ( τ ) = 1and assumed that η/s = const [15]. For modeling the primordial hadron production andsubsequent hadronic decays, we use a customized version of THERMINATOR 2 which allowsfor momentum-space anisotropies at freeze-out [31]. To determine the free parameters ( η/s and T ), we fit the pion, kaon, and proton spectra in the 0-5% and 30-40% centrality classes.Once the parameters were determined [15], they can be used to compute other observablessuch as the elliptic flow and the HBT radii. We note that our code is publicly available andcan be obtained using Ref. [32]. IV. PHENOMENOLOGICAL COMPARISONS
In this section, we present comparisons of the HBT radii predicted by aHydroQP with √ s NN = 200 GeV Au-Au collision data. In this work, we continue our previous work [15]and use exactly the same model parameters determined therein. It is worth mentioning thatthe parameters obtained from [15] are τ = 0 .
25 fm/c, T = 455 MeV, η/s = 0 . T FO = 130 MeV. For more details about the aHydroQP model and comparisons to RHIChadron spectra, v , etc., we refer the reader to Ref. [15].The correlation function used to obtain the HBT radii is parametrized in terms of threeGaussians (the 3-D Bertsch-Pratt parameterization) [33, 34] C ( q, k ) = 1 + λ exp[ − R q − R q − R q ] . (3)where λ is the incoherence parameter. The longitudinal component ( R long ) is parallel to thebeam axis (the z-axis), the out component ( R out ) is chosen parallel to the mean transversemomentum of the pair (the x-axis), and the side component ( R side ) is perpendicular toboth R long and R out (the y-axis). In a similar way, the relative momentum ( q = p - p ) isdecomposed into three components ( q long , q out , q side ) [35, 36]. We have ignored final stateinteractions such as the Coulomb repulsion between the similar charged pion pairs.Now, let us turn to the comparisons between our model and the experimental results. In5igs. 1-4, we show the k T dependence of the HBT radii and their ratios. As can be seen inFigs. 1 and 3, the HBT radii tend to decrease with increasing k T due to the change in thesize of the homogeneity regions. In Figs. 1-2, we show comparisons with STAR experimentalresults in four k T bins [30]. In Fig. 1, we show the predicted HBT radii by aHydroQP (solidlines) and the experimental results in four different centrality classes, 0-5%, 5-10%, 10-20%,and 20-30%. In the left, middle, and right columns we show R out , R side , and R long as afunction of the mean transverse momentum of the pair π + π + . In the left column, we seethat our model overestimates the experimental data, particularly at k T ∼ R long , wesee that the agreement is clearly not as good as the other two columns especially at low k T .Next, in Figs. 2 we compare the ratios of the HBT radii. In this set of figures, in the left,middle, and right columns we show R out /R side , R out /R long , and R side /R long , respectively, as afunction of k T . As can be seen from this Figure, we find quite good agreement between ourmodel and the experimental results for HBT radii ratios. This is true for all ratios shown andin all centrality classes. We note that our model slightly underestimates the ratio R side /R long due to overestimating R long in Fig. 1. The agreement with data found herein is similar towhat was found in previous works using dissipative hydrodynamics, however, in this workwe present comparisons in a larger range of centrality classes and compare with both STARand PHENIX data [19–25].Next, we show comparisons with PHENIX experimental results obtained from [37] wheremore k T bins are presented. In Fig. 3, we show the comparisons of the HBT radii predictedby aHydroQP and the experimental results in 0-10% and 10-20% centrality classes. As inFig. 1, the left, middle, and right columns show R out , R side , and R long as a function of themean transverse momentum of the pair π + π + . In a similar way to what we have seen in thecomparisons with STAR results, we see that our model overestimates the data especially atlarge k T . As an example, in column (a), we see that there is an approximately 30% differencebetween our model prediction and the experimental results at large k T ∼ k T ∼ R long and in the bottom row for R out and R long . However,as can be seen from column (b), in both centrality classes, we see that better agreement6 - % - STAR R ou t / R s i de ( a ) - % - aHydro R ou t / R l ong ( b ) R s i de / R l ong ( c ) - % - STAR R ou t / R s i de - % - aHydro R ou t / R l ong R s i de / R l ong - % - STAR R ou t / R s i de - % - aHydro R ou t / R l ong R s i de / R l ong - % - STAR k T [ GeV ] R ou t / R s i de - % - aHydro k T [ GeV ] R ou t / R l ong k T [ GeV ] R s i de / R l ong FIG. 2. Ratios of HBT radii as a function of k T for π + π + in the same centrality classes shownin Fig 1. The left, middle, and right columns show R out /R side , R out /R long , and R side /R long , re-spectively. All results are for 200 GeV Au-Au collisions and the data shown are from the STARcollaboration [30]. between our model prediction and the experimental results for R side .In Fig. 4, we show the HBT radii ratios reported by PHENIX as a function of the pairmean transverse momentum for π + π + in the 0-10% and 10-20% centrality classes. As canbe seen from these panels, the agreement is quite good up to k T ∼ - % - PHENIX R ou t [ f m ] ( a ) - % - aHydro R s i de [ f m ] ( b ) R l ong [ f m ] ( c ) - % - PHENIX k T [ GeV ] R ou t [ f m ] - % - aHydro k T [ GeV ] R s i de [ f m ] k T [ GeV ] R l ong [ f m ] FIG. 3. HBT radii as a function of k T for π + π + in the 0-10% and 10-20% centrality classes. Theleft, middle, and right columns show R out , R side , and R long , respectively. All results are for 200GeV Au-Au collisions and the data shown are from the PHENIX collaboration [37]. The k T cutsused were 0 . < k T < . column, which shows R out /R side , one sees that after k T ∼ R out /R long , one sees that the agreement in the 0-10% centrality class is quitegood, while in the 10-20% centrality class one sees that our model underestimates the dataat large k T . Finally, in the right column, which shows ( R side /R long ), one can see the modelunderestimates the data at large k T . Despite these quantitative differences, we note thatoverall the model does a good job in reproducing the systematic trends seen in the data.From our comparisons with both STAR and PHENIX data, we see that our model over-estimates R long by on the order of 10-20%. This means that the longitudinal expansion ofthe QGP in aHydro is stronger than what is seen experimentally. This is most likely dueto the fact that aHydro was taken to have a strong initial boost-invariant Bjorken flow. Itwould be interesting to study the impact of having a slightly weaker initial longitudinal flowprofile. We note here that the agreement between model and data presented here is quitesimilar to our prior comparisons between aHydroQP and ALICE results for 2.76 TeV Pb-Pbcollisions [14], however, in the case of the PHENIX data, in particular, the experimental8 - % - PHENIX R ou t / R s i de ( a ) - % - aHydro R ou t / R l ong ( b ) R s i de / R l ong ( c ) - % - PHENIX k T [ GeV ] R ou t / R s i de - % - aHydro k T [ GeV ] R ou t / R l ong k T [ GeV ] R s i de / R l ong FIG. 4. Ratios of HBT radii as a function of k T for π + π + in the 0-10% and 10-20% centrality classes.The left, middle, and right columns show R out /R side , R out /R long , and R side /R long , respectively. Allresults are for 200 GeV Au-Au collisions and data shown are from the PHENIX collaboration [37]. error bars are quite small which presents a challenge for the model. V. CONCLUSIONS AND OUTLOOK
In previous works we have presented comparisons of 3+1d aHydroQP and relativisticheavy-ion collision experimental data obtained at different collision energies. These paststudies allowed us to fix a set of model parameters such as initial central temperature, shearviscosity, etc. for use in predicting additional observables. In this paper, using those pre-viously determined parameters, we presented predictions of the aHydroQP model for thefemtoscopic HBT radii and their ratios at RHIC energies. We showed comparisons of modelpredictions with data from both the STAR and PHENIX collaborations obtained usingcharged pion correlations. We found reasonable agreement overall with experimental resultsin several centrality classes and as a function of the pair mean transverse momentum, inparticular with respect to trends seen in the data. There were, however, quantitative dif-ferences which were evident, for example, in R long . These quantitative differences deservefurther study. In particular, it would be interesting to see how sensitive this observable9s to the assumed initial longitudinal flow of the QGP. Herein, for continuity with priorwork, we assumed the initial longitudinal flow profile to be Bjorken flow, which is extremal.Perhaps a modest reduction in the initial longitudinal flow would improve agreement withthe experimental data beyond what has already been achieved. Additionally, we have usedsmooth (non-fluctuating) Glauber initial conditions which is definitely too simplistic. Byincluding fluctuating initial conditions, one can expect to see reductions in the size regionsof homogeneity, which may help to bring model predictions into better agreement with ex-perimental data. Looking to the future, we are working on comparisons with experimentaldata from 5.02 TeV Pb-Pb collisions, including HBT radii. We also are working on in-cluding off-diagonal terms in the anisotropy tensor [39], using more realistic (fluctuating)initial conditions, and using collisional kernels that go beyond the relaxation-time approxi-mation [40, 41]. ACKNOWLEDGMENTS
M. A. is supported by the Deanship of Scientific Research at the Imam Abdulrahman BinFaisal University under grant number 2020-080-CED. M. Strickland was supported by theU.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award No.DE-SC0013470. This research utilized Imam Abdulrahman Bin Faisal (IAU)’s Bridge HPCfacility, supported by IAU Scientific and High Performance Computing Center [42]. [1] P. Huovinen, P. F. Kolb, U. W. Heinz, P. V. Ruuskanen, and S. A. Voloshin, Phys. Lett.
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