Bose-Einstein or HBT correlations in high energy reactions
aa r X i v : . [ h e p - ph ] J a n Bose-Einstein or HBT correlations in high energy reactions
T. Cs¨org˝o ∗ MTA KFKI RMKI, H-1525 Budapest 114, P.O.Box 49, Hungary
Abstract
Concepts of thermalization and hydrodynamical behavior are appliedfrom time to time to e + + e − , hadron+hadron and heavy ion collisions.These applications are scrutinized paying attention to particle multi-plicities, spectra, and Bose-Einstein correlations in particular. Can hy-drodynamics describe these data? In 2008, the speakers of the International Symposium on Multiparticle Dynamics were given aquiz of 18 questions, that were compiled by Hannes Jung and G ¨osta Gustafson, the Chair and theCo-Chair of this meeting [1]. My goal is to discuss three of these problems:1. Can thermal & hydrodynamical models describe e + + e − , h + p and A + B reactions?
2. What heavy ion physics can learn from e + + e − , h − + p and p + p collisions?3. How can correlations be used to determine the size of the interaction region and the char-acteristics of phase transitions?These questions are related to Bose-Einstein correlations, that appear due to the sym-metrization of hadronic final states for the interchange of identical bosons, and are also knownby other names, for example Hanbury Brown – Twiss or HBT correlations in heavy ion colli-sions, intensity interferometry, or intensity correlations [2]. These correlations are also tools offemtoscopy, because they are used to measure length scales on the femtometer scale [3–5]. e + e − collisions at LEP In e + e − collisions, Bose-Einstein correlations were used to record the fastest film ever made: theformation of a ring-like, non-thermal source in the transverse plane of jet production, a processthat ends in less than − seconds [6, 7]. Can thermal models describe multiplicities, spectraand correlations in these collisions?A number of recent papers consider the possibility of thermal particle production in e + e − reactions. Two recent, interesting examples are refs. [8] and [9] , that present thermal model fitsto these data with similar level of statistical significance but with very different physics conclu-sions. A model cannot be excluded with the help of mathematical statistics if its confidence levelis CL ≥ . , thus the probability that the model describes the data is at least one in thousand.Fig. 1 of ref. [8] is a very beautiful plot indicating intriguing similarities between particleabundances in e + e − at √ s NN = 91 GeV and thermal model calculations. The fit quality ischaracterized by a χ /NDF = 631/30. The corresponding confidence level is CL = 1.1 10 − %. ∗ e-mail: [email protected] Instead of the originally given e − +p problem, let me discuss soft hadron-proton collisions, for clarity. his confidence level is an extremely small positive number and so the probability that the ther-mal particle production describes this data set is practically zero [8] . These authors also observeand point out correctly that the statistical or thermodynamical description of these data fails com-pletely at the high level of present experimental precision. Approximate qualitative agreementbetween thermal particle production and data can only be obtained if the relative errors on thesedata are magnified to about 10 % [8]. Their conclusion can be contrasted to other manuscripts,that claim that a thermodynamical or statistical description of particle multiplicities in e + e − reactions is possible. For example, the same data set was analyzed in ref. [9] , using a slightlydifferent thermal model description and the quality of their fit is given in their Table V as χ / NDF= 215./27 . The corresponding confidence level is CL = 3.4 10 − % hence the probability thatthis thermal model describes particle abundances in electron-positron annihilation is practicallyzero. When the analysis is restricted to include only those 15 resonances in the fitting, whosewidth is less than 10 MeV, the same thermal model description yields χ /N DF = 39./12 . Theconfidence level of this fit is still CL = 1.1 10 − % , which is many orders of magnitude improve-ment, but still an order of magnitude less than the conventional threshold of acceptance, CL =0.1 %. Thus thermal particle production models do not describe the multiplicities of elementaryparticles of e + e − at √ s NN = 91 GeV in an acceptable manner.Two important and well known features of hadronic spectra also disagree with a thermal,statistical picture of particle production. The observation of jets (2 and 3 jet events at this ener-gies) can be contrasted to the lack of preferred direction in the initial conditions and in a thermalpicture of particle production. Perhaps the thermal picture can be limited to the transverse di-rection? The power-law tail the transverse momentum spectra, which can be explained in termsof perturbative QCD processes and jets decaying to jets to jets and in particular the correlationsamong these jets are inconsistent with a thermal and/or a hydrodynamical interpretation, that leadtypically to exponential spectra. Furthermore, generalized thermal models that describe the spec-tra cannot naturally interpret the correlation structures observed in two and three jet events whichare basically energy momentum conservation laws and have a trivial interpretation in partonicpicture, the emission of quark and gluon jets in perturbative QCD.Bose-Einstein correlations are more subtle features of two-particle distributions. Theycarry information on the space-time structure and on the chaotic or coherent nature of particleemitting sources. Recently measured Bose-Einstein correlations disagree qualitatively with thehypothesis that the produced particles are emitted from a thermal or hydrodynamical source in e + + e − reactions, because in thermal models the two-particle Bose-Einstein correlation functionis always given by a 1 + positive definite function, and this constraint is violated by a recentanalysis of L3 data [7,10].With other words, there is no region of two-particle relative momentumspace, where a chaotic (or thermal) picture of particle production would lead to anti-correlations.However, recently analyses L3 data as detailed in ref. [7, 10] indicate very clearly the existenceof a region of anti-correlation: if the correlation functions are measured as a function of theLorentz invariant relative momentum variable Q = p − ( k − k ) , where k i stands for thefour-momentum of particle i , a dip is found experimentally in the region of 0.6 GeV < Q < C ( Q ) =1 + λ exp( − Q R ) , χ /N DF = 234 ./ and the corresponding confidence level is practically ig. 1: Comparison of Gaussian (left), Edgeworth (middle) and L´evy fits to L3 Bose-Einstein correlation functions.The 1+ positive definite forms, Gaussian and L´evy do not have a statistically acceptable level, CL < . %. TheEdgeworth expansion has an acceptable CL = 1 % and describes the dip using a 1+ non-positive definite expression. zero. A generalization of the Gaussians is given by the symmetric L´evy form C ( Q ) = 1 + λ exp( − Q α R α ) , χ /N DF = 148 ./ , CL = 0.04 %. This is only a factor of 2.5 below theconventional domain of acceptable results, but the chance that this form represents the data is only4 in 10000. This form however is 1 + a positive definite function. The right panel also indicates alinear fit to the long range, Q > . GeV correlations, shown with a dot-dashed line. It describesthe data in the fitted
Q > . GeV region, and it clearly cuts into the ”dip” region of the data,located at . < Q < . GeV . When a L´evy fit form is enforced, these long range correlationsget distorted, pushed below the dip region by the fit, as indicated by the dashed line in the rightpanel, and the overall fit quality is decreased below the limit of acceptability, CL < . %. Thebest fit is achieved using an Edgeworth expansion, C ( Q ) = 1 + λ exp( − Q R )[1 + κ H ( QR )] ,where H ( x ) is the third order Hermite polynomial, see ref. [7] for details. This Edgeworth fithas a statistically acceptable CL = 1 % and describes the dip using 1 + a non-positive definiteexpression, in a model and interpretation independent manner.The τ -model of ref. [11] also predicted the existence of such anti-correlated regions, basedon the assumption that e + + e − annihilations indeed correspond to point-like collisions hence theproduced particles with a given momentum k µ appear in a direction parallel to their momentum, x µ ∝ k µ , however with a broad proper-time distribution H ( τ ) . This model leads to simplefitting forms, that improve the description of the data as compared to the model-independentEdgeworth expansion method, CL is increased from 1 % to 40 % , and when the fit parametersare required to satisfy the model constraints, CL is slightly increased to 42 % , see ref. [7] fordetails. The parameters of the proper-time distribution are determined from detailed fits to the L3Bose-Einstein correlation functions. This way the proper-time evolution of particle production isreconstructed in these reactions, and the following points can be made: in 2-jet events, particleemission starts just after the collision, so that the most probable value for τ is 0.3 fm/c, butthis one-sided proper-time distribution has a power-law tail, corresponding to a one-sided L´evydistribution with an index of stability of α = 0 . ± . . Using a recently developed methodbased on the τ -model [12], even a movie of the space-time evolution of particle emission can bereconstructed. This movie – the shortest film ever recorded – practically ends in about 0.3 fm/c. ig. 2: Left panel: A snapshot picture from the reconstructed video of jet formation in the transverse plane of 2-jetevents in e + + e − annihilation at LEP at τ = 0 . fm/c. A non-thermal, expanding ring is observed, the amplitudeof the ring diminishes very quickly, while its radius grows nearly with the speed of light. Right panel: The recon-structed transverse part of the particle emission function in h+p reactions at CERN SPS as inferred from Bose-Einsteincorrelations and single particle spectra as measured by the NA22 collaboration, describing a tiny ring of fire. h + p and p + p collisions In hadron-proton collisions, Bose-Einstein correlations have been used to make a snapshot pic-ture of the smallest ring of fire, ever detected: the diameter is less than 1 fm or − m, but thesource seems to be thermal. The ring formation here is a hydrodynamical effect, the temperaturedrops from T ≈ MeV in the center to nearly zero within about 1 fm radial distance, hencea strong pressure gradient builds up. However, the experimentally seen transverse flow is tooweek to move the matter away from the surface, hence a pile-up at the surface, a fire-ring isfound [2, 13]. A similarly hydrodynamical ring of fire formation due to large temperature gra-dients and small transverse flows can be inferred from a simultaneous analysis of single particlespectra of pions, kaons, protons and STAR preliminary Bose-Einstein or HBT correlation radiiof pion pairs in √ s NN = 200 GeV p+p collisions at RHIC [14]. Au + Au collisions at RHIC In Au+Au collisions at √ s NN = 200 GeV at the RHIC accelerator at BNL, Bose-Einsteincorrelation measurements also yield snapshot pictures of the hottest and most perfect fluid, evermade in a laboratory experiment.The following milestones lead to this important discovery: PHENIX was the first to ob-serve a new phenomena in 0-10 % central Au+Au collisions at √ s NN = 130 GeV at RHIC:the suppression of particle production with high transverse momentum, the first RHIC discoverythat made it to the cover page of The Physical Review Letters in January 2002. However, it wasnot clear initially if this effect is due to the nuclear modification of the structure functions (initialconditions) at such a high energies, or if this is indeed a hadronic final state effect. As a control, d + Au measurement was performed and all the four RHIC collaborations: BRAHMS, PHENIX,HOBOS and STAR reported the absence of suppression in these reactions. This discovery im-plied that the suppression in Au+Au reaction is a final state effect, due to the formation of a newform of matter , that also made its way to the cover page of the Physical Review Letters in August2003. The third milestone was the publication of the so called “White Papers” or review papersby all the four RHIC experiments. After several year’s worth of high energy collisions, and froma detailed analysis of the elliptic flow data, a consensus interpretation emerged that the fireballmade in Au+Au collisions at RHIC behaves like a liquid of strongly interacting constituents,also known as “the perfect fluid”. This discovery became also known as the Top Physics Storyfor 2005 by the American Institute of Physics. This discovery has been considerably sharpenedwhen STAR and PHENIX pointed out that the observed elliptic flow patterns scale with the num-ber of constituent quarks and strange and even charm quarks participate in the flow. Althoughthe theoretical interpretation of this effect is still open for discussions in particular because theunsolved problem of quark confinement in QCD prevents the application of first principle QCDcalculations for this phenomena, in my opinion the experimental evidence is very clear, it is ir-refutable that quark degrees of freedom are active and the perfect fluid seen in Au+Au collisionsis a fluid of quarks [16]. (The role of gluons is less clear and less directly measurable from theexperimental point of view.) The fifth milestone was the quantification, how perfect is the per-fect fluid at RHIC? Answers were obtained by measuring the so called kinematic viscosity η/s ,which is the ratio of the shear viscosity to the entropy density. Two theoretical analyses werepublished in 2007 based on elliptic flow patters, a third measurement was based on the transversemomentum correlations, while PHENIX studied the energy loss and flow of heavy (charmed)quarks and based on a charm diffusion picture, found that η/s = (1 . − . π [16]. Even morerecently, PHENIX was able to put a lower limit on the initial temperature of the fireball at RHICfrom the analysis of direct photon data [17], T i > MeV. These numbers can be comparedto similar characteristics of other known fluids, like water, liquid nitrogen or helium, see Fig. 3,based on refs. [18, 19].Note that He becomes superfluid at extremely low temperatures and its kinematic viscos-ity η/s reaches a minimum at the onset of superfluidity, so for superfluid He η/s ≥ π . Thematter created in Au+Au collisions at RHIC has temperatures larger than 2 Terakelvin, neverthe-less its kinematic viscosity is the lowest value ever produced in laboratory: it is at least a factorof 4 smaller than that of superfluid He. We may thus refer this property of the matter created inAu+Au collisions at RHIC as high temperature superfluidity [20]: the matter created in Au+Aucollisions at RHIC is the most perfect fluid ever made by humans.We gain information on the type of transition from hadronic matter to quark matter withthe help of the Bose-Einstein correlations. By now, circumstantial evidence is obtained that thistransition is either a cross-over or, a non-equilibrium transition. This consensus opinion is basedon important and highly selective constraints given by Bose-Einstein correlations and particleinterferometry data in Au+Au collisions at RHIC [5].
After having discussed e + e − , hadron-proton and heavy ion reactions one after the other, andbased on the presented evidence let me attempt to give answers to the questions discussed in theIntroduction, keeping in mind that these answers were worked out predominantly from the point ig. 3: Comparison of the properties of the perfect fluid created in √ s NN = 200 GeV Au+Au collisions with lessextraordinary materials like water, nitrogen or helium. of view of Bose-Einstein correlations and their models in these reactions.
Can thermal & hydrodynamical models describe e + + e − , h + p and A + B reactions? Itseems that thermal models cannot interpret particle multiplicities, spectra and Bose-Einsteincorrelations in e + + e − reactions at the present level of experimental precision, and using theconventional threshold for acceptable confidence levels (99.9 % ≥ CL ≥ p t < . GeV)hadron-proton, proton-proton and heavy ion reactions. At higher values of the transverse mo-menta, jet physics and interaction of jets and the hydrodynamical medium opens up new researchdirections at the intersection of particle and nuclear physics.
What heavy ion physics can learn from e + + e − , h − + p and p + p collisions? One lessonthat I presented was to take statistical analysis and confidence level determinations seriously.Based on detailed and precision analysis of Bose-Einstein correlations in two-jet events and asimultaneous analysis of single particle spectra, the time evolution, a movie or a video like filmof particle emission has been already reconstructed in e + e − reactions at LEP. In heavy ionphysics, only snapshot like pictures can be reconstructed at present. Further developments ofthe femtoscopic tools are needed to allow for a video like reconstruction of the time evolutionof particle emission in heavy ion reactions. Based on Bose-Einstein data in e + + e − reactions,it seems that the most probable value of the proper-time parameter of particle production is τ =0 . fm/c, a surprisingly short value. It would be interesting to consider the phenomenologicalconsequences of this number in heavy ion reactions, and if possible, to extract similar numbersfor jets that are produced in a nuclear medium. How can correlations be used to determine the size of the interaction region and the char-acteristics of phase transitions?
Of course a complete answer to this question goes well beyondthe scope of this conference contribution. Let me just emphasize here, that correlations are rou-tinely used to take a snapshot picture of the interaction region [2–5]. The resolution of thesenapshot pictures has been increased recently and more detailed information about structures(like a ring of fire) or heavy tails (non-Gaussian behavior) are seen in all kind of reactions [5].Recent progress even allowed for the determination of the time evolution of the region of particleproduction in e + e − reactions, based on a non-thermal description. Similar techniques are notyet developed for soft hadron-proton, proton-proton and heavy ion collisions, where the thermo-dynamical and hydrodynamical models can readily be applied. However, in heavy ion reactionsmatter formation and also a transition to a perfect fluid of quarks has been experimentally proven(although with open theoretical issues). Bose-Einstein correlations have been proven to con-strain models in an extremely efficient manner. At present, models with a strong first order QCDphase transition or with a second order phase transition point disagree with Bose-Einstein corre-lation data in heavy ion collisions at RHIC, however, models with a cross-over transition or withnon-equilibrium rehadronization scenario cannot be excluded at present [5, 21]. Acknowledgments:
It is my pleasure to thank the Organizers for a most professionallyorganized conference. This research was supported by the OTKA grants NK73143, T49466 aswell as by a Senior Scholarship Award of the Hungarian-American Enterprise Scholarship Fund.
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