Coalescence Models For Hadron Formation From Quark Gluon Plasma
aa r X i v : . [ nu c l - t h ] J u l Hadronization by Coalescence Coalescence Models For Hadron FormationFrom Quark Gluon Plasma
Rainer Fries , ([email protected])Vincenzo Greco , ([email protected])Paul Sorensen ([email protected]) Texas A&M University, College Station, Texas, RIKEN/BNL Research Center, Upton, New York, Istituto Nazionale di Fisica Nucleare –INFN-LNS, Catania, Italy, Department of Physics and Astronomy, University of Catania, Italy Brookhaven National Laboratory, Upton, New York
Key Words quark gluon plasma, recombination, coalescence, hadronization,elliptic flow, heavy-ion collisions
Abstract
We review hadron formation from a deconfined quark gluon plasma (QGP) via coa-lescence or recombination of quarks and gluons. We discuss the abundant experimental evidencefor coalescence from the Relativistic Heavy Ion Collider (RHIC) and compare the various coa-lescence models advocated in the literature. We comment on the underlying assumptions andremaining challenges as well as the merits of the models. We conclude with a discussion of somerecent developments in the field.
CONTENTS nnu. Rev. Nucl. Part. Sci. 2008 72
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hadronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Approaches to Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Challenges at RHIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulations of Hadronization by Recombination . . . . . . . . . . . . . . . . . . . Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different Implementations of Recombination . . . . . . . . . . . . . . . . . . . . . . . . Competing Mechanisms of Hadron Production . . . . . . . . . . . . . . . . . . . . . . . Elliptic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Approximations and Assumptions . . . . . . . . . . . . . . . . . . . . . Data from Elementary and Heavy-ion collisions . . . . . . . . . . . . . . . . . . . . Hadron Spectra and Baryon to Meson Ratios . . . . . . . . . . . . . . . . . . . . . . . Elliptic Flow and Quark Number Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . Heavy Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle Correlations and Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . Beam Energy Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Challenges and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Space-Momentum Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . adronization by Coalescence Collisions between heavy nuclei are used to probe the properties of nuclear mat-ter at high temperature and density. Lattice QCD calculations indicate that ifnuclear matter is heated above a critical temperature T c ≈
185 MeV, quark andgluon degrees of freedom will be liberated and a deconfined quark-gluon plasma(QGP) forms (1, 2). Unambiguous signatures of quark gluon plasma formationin heavy-ion collisions have been sought for decades. Recently, experiments atthe Relativistic Heavy Ion Collider (RHIC) have presented evidence that such anew state of matter has finally been found in collisions of Au atoms at a centerof mass energy of √ s = 200 GeV per nucleon-nucleon pair (3, 4).The hot QGP phase formed in nuclear collisions at RHIC with a core temper-ature in excess of 300 MeV only lasts for an extremely short time. It quicklyexpands due to the large pressure and cools on the way. Eventually, the quarkand gluon constituents need to combine into color-neutral objects and hadronshave to be formed when the temperature reaches T c . The process of hadroniza-tion from a QGP may be quite different from hadronization in other cases, suchas hadronization of hard scattered parton in elementary collisions where no ther-malization is reached and no bulk of partons is formed. In this review we discussa model of QGP hadronization by coalescence or recombination of quarks andgluons. The models discussed here have had success in describing many salientfeatures of hadron production in heavy-ion collisions.The emergence of recombination models was largely motivated by several unex-pected observations (3) which were discussed as “the baryon puzzle” for a while.This was referring to measurements of baryon production in the intermediatetransverse momentum region (1 . < p T < c ) (5, 6). Both the yield and adronization by Coalescence p T = 3 GeV/c, only one baryon is produced for every three mesons (1:3),reflecting the larger mass and the requirement of a non-zero baryon number toform the baryon. In Au+Au collisions at RHIC however, baryons and mesons arecreated in nearly equal proportion (1:1) despite those differences. In the same p T -region, the elliptic anisotropy ( v ) of baryons is also 50% larger than that formesons. Therefore, baryon production is particularly enhanced in the directionof the impact vector between the colliding nuclei (in-plane) (6, 7).The large baryon v eliminates several possible alternative solutions put for-ward for the baryon puzzle. The most common explanations for the baryonanomaly at RHIC were • coalescence or recombination — Multi-quark or gluon processes duringhadron formation (8, 9, 10, 11, 12). • baryon junctions — Gluon configurations that carry baryon number (13). • flow — Collective motion that populates the higher p T -regions of phasespace for the more massive baryons, as described by hydrodynamics (14,15, 16).Only coalescence models have survived the tests imposed by an impressive amountof data taken after the original discovery of the baryon enhancement. They areparticularly attractive because they seem to provide a natural explanation for thevalence quark-number scaling that has been observed in v measurements. Theyalso relate hadronic observables to a pre-hadronic stage of interacting quarksand gluons. As such, they touch on questions central to the heavy-ion physicsprogram: deconfinement and chiral symmetry restoration.This review is organized as follows. In the remainder of this section we discuss adronization by Coalescence Hadronization has always been a challenging aspect of quantum chromodynamics(QCD), the fundamental theory of the strong force. QCD bound-states are non-perturbative in nature and a first-principle description of their formation has yetto be obtained. In this subsection we briefly discuss two approaches to deal withhadronization which are routinely used in nuclear and particle physics; both ofthem have connections to the recombination model discussed in this review.Light cone wave functions are used to describe the structure of hadrons relevantfor exclusive processes (17). Exclusive here means that they deal with a full setof partons with the quantum numbers of the hadron. Exclusive processes at highmomentum transfer are naturally dominated by the few lowest Fock states. For-mally, light cone wave functions are matrix elements of the set of parton operatorsbetween the vacuum and the hadron state in the infinite momentum frame, e.g. φ p ∼ h | uud | p i , schematically, for a proton p . They describe the decompositionof the hadron in longitudinal momentum space in terms of partons with momen-tum fractions x i . From theory these wave functions are only constrained by verygeneral arguments like Lorentz-covariance and approximate conformal symmetry.Direct measurements are difficult, but estimates have become available in recent adronization by Coalescence initial state → h + X , at large momentum transfers in which a single coloredparton a has to hadronize into the hadron h . For this purpose fragmentation or“parton decay” functions D a → h ( z ) have been defined (20). They give the proba-bility to find the hadron h in parton a with a momentum fraction z , 0 < z < e + + e − , lepton-hadron orhadron-hadron collisions can then be written as a convolution σ H = σ a ⊗ D a → h (1)of the production cross section σ a for parton a with the fragmentation function D a → h ( z ) (21, 22). Fragmentation functions are not calculable in a reliable wayfrom first principles in QCD. However, they are observables and can be measuredexperimentally. Parameterizations using data mostly from e + + e − collisions areavailable from several groups (23). Physically, the fragmentation of a singleparton happens through the the creation of q ¯ q pairs (through string breaking orgluon radiation and splitting) which subsequently arrange into color singlets, andeventually form hadrons.Both examples above apply to processes with a large momentum transfer, i.e.with a perturbative scale µ ≫ Λ QCD . They are based on the concept of QCDfactorization which separates the long and short distance dynamics. Such aperturbative scale is absent for the hadronizing bulk of partons in a heavy ioncollision and neither technique, fragmentation nor exclusive wave functions, can Note that we have neglected the scale dependence in the notation for wave functions andfragmentation functions for simplicity. A discussion of the scale dependence can be found in theoriginal references given. adronization by Coalescence T c . Theexact degree of thermalization is not clear a priori, but we will see below, thatcomplete thermalization might not be necessary.Rather, the crucial point seems to be that partons have a certain abundancein phase space such that there is no need for the creation of additional partonsthrough splitting or string breaking. The most naive expectation for such a sce-nario is a simple recombination of the deconfined partons into bound states. In-deed, there is experimental evidence that this is the correct picture for hadroniza-tion even long before a thermal occupation of parton phase space is reached. Recombination models have first been suggested shortly after the invention ofQCD in the 1970s. They successfully described hadron production in the veryforward region of hadronic collisions (24). The observed relative abundances ofhadrons clearly deviate from expectations from fragmentation in this region. Thisis known as the leading particle effect (25). E.g. a clear asymmetry between D − and D + mesons was found in fixed target experiments with π − beams on nucleiby the FNAL E791 collaboration (26). The measured D − / D + asymmetry goes to1 in the very forward direction, while fragmentation predicts that this asymmetryis very close to 0. This result can be explained by recombination of the ¯ c from a c ¯ c adronization by Coalescence d valence quark from the beam π − remnants.This mechanism is enhanced compared to the c + ¯ d recombination which involvesonly a sea quark from the π − (27). There is no thermalized parton phase in thisexample, which strongly backs our argument at the end of the last subsection.We are led to the important conclusion that the presence of any reservoir ofpartons leads to significant changes in hadronization. Vacuum fragmentation isno longer a valid picture in this situation. The reservoir of partons in the caseof the leading particle effect is the soft debris from the broken beam hadron. Inheavy ion collisions it is the distribution of thermal partons. First applicationsof the coalescence picture to nuclear collisions appeared in the early 1980s (28).This eventually led to the development of the ALCOR coalescence model in the1990s (29, 30, 31). ALCOR focuses on hadron multiplicities and was successfullyapplied to hadron production at RHIC and the lower energies at the CERN SPS. Results from the first years of RHIC triggered a revival for recombination modelsapplied to heavy ion collisions in an unexpected region. Three measurements inparticular, taken in the intermediate p T range (1.5 GeV/ c < p T < c ),have defied all other explanations. This region is outside of what was thoughtto be the “bulk” of hadron production ( p T < . c ) whose features shouldbe described by thermalization and hydrodynamic collective motion (ALCORdescribes bulk hadronization). Rather, the intermediate p T region was expectedto be dominated by fragmentation of QCD jets, after it was confirmed that thiswas the case for pion production in p + p collisions at RHIC (32). However, theresults from RHIC clearly pointed towards a strong deviation from the fragmen- adronization by Coalescence p T in central Au+Au collisions. The three keyobservables were • the enhanced baryon-to-meson ratios (5, 33). • the nuclear modification factors R AA and R CP — i.e. the ratio of yieldsin central Au+Au collisions compared to peripheral Au+Au ( R CP ) or p + p ( R AA ) collisions scaled by the number of binary nucleon-nucleon col-lisions (5, 6). • the anisotropy of particle production in azimuthal angle relative to thereaction plane — i.e. the elliptic flow parameter v (6, 35, 34, 7).Fig. 1 shows the measured anti-proton/pion (5) and Λ/ K S (33) ratios as afunction of p T for various centralities and collision systems. At intermediate p T ,a striking difference is observed between the baryon-to-meson ratios in centralAu+Au collisions and those in e + + e − (36) or p + p collisions (37). The mea-surements in Fig. 1 indicate that the process by which partons are mapped ontohadrons are different in Au+Au collisions and in p + p collisions. Changes solelyto the parton distributions prior to hadronization are not likely to lead to suchdrastic changes in the relative abundances.Fig. 2 shows the nuclear modification factor R CP measured at RHIC for variousidentified hadrons. If the centrality dependence of particle yields scales with thenumber of binary nucleon-nucleon collisions, R CP will equal one. A suppressionat high p T is taken as a signature for the quenching of jets in the bulk matterformed in central collisions. However, baryons (Λ + Λ, Ξ + Ξ, and Ω + Ω) (33, 38)systematically show less suppression than mesons (kaons or φ ) (39, 40)). Thesame behavior was found for protons and pions (41). This key result shows thatthe mass of a hadron is less important for its behavior at intermediate p T than adronization by Coalescence φ mesons which have the same mass but adifferent valence quark content: φ mesons behave like other, lighter mesons, notlike protons (40, 42).In non-central nucleus-nucleus collisions, the overlap region of the nuclei is el-liptic in shape. Secondary interactions can convert this initial coordinate-spaceanisotropy into an azimuthal anisotropy of the final momentum-space distribu-tion. That anisotropy is commonly expressed in terms of the coefficients froma Fourier expansion of the azimuthal dependence of the invariant yield (43), seeSec. 2. The second component (the “elliptic flow” parameter v ) is large due tothe elliptic shape of the overlap region. Fig. 3 shows the measured values for v as a function of p T for pions, kaons, protons and Lambda hyperons (6,35). In thebulk region ( p T < . v is increasing with p T (44). In this region the v values for different hadrons are ordered by their mass with more massive particleshaving smaller v values (6,35,45). This mass ordering is qualitatively understoodin hydrodynamic models of the expansion of the bulk of the fireball (14). Somehydrodynamic calculations are also shown in the figure. For p T > . v seems tosaturate, as predicted by parton cascades (46), and the particle-type dependencereverses: v values for the more massive baryons are larger than those for mesons. v can also be generated if jets are quenched in the quark gluon plasma (47).However, such calculations grossly underestimate the measured values of v , inparticular when they simultaneously have to explain values of R CP close to one. adronization by Coalescence R CP values nearunity, their maximum v values exceed those of pions and kaons by approximately50%. Taken together, the particle-type dependence of v and R CP provide verystringent tests of various models for particle production and have ruled out purejet fragmentation or simple hydrodynamics as models for hadron production atintermediate p T . Coalescence or recombination of particles is a very general process that occurs ina wide array of systems from the femtometer scale to astrophysics. In all thesefields a first approach is to discard the details of the dynamical process in favorof exploiting an adiabatic approximation in which a projection of the initial stateonto the final clusterized state is considered. In the specific case of recombinationof partons, most work found in the literature uses an instantaneous projectionof parton states onto hadron states. The expected number of hadrons h from apartonic system characterized by a density matrix ρ is given by N h = Z d P (2 π ) h h ; P | ρ | h ; P i . (2)Instantaneous here means that the states are defined on a hypersurface whichis typically either taken to be at constant time, t = const., or on the light-cone t = ± z . In this case information about the hadron bound state is schematicallyencoded in a wave function or Wigner function. As we will see this approachleads to very simple math, but it has the conceptual disadvantage that onlythree components of the four momentum are conserved in such a 2 → → adronization by Coalescence P is (49) dN M d P = X a,b Z d R (2 π ) d qd r (2 π ) W ab (cid:18) R − r , P − q ; R + r , P q (cid:19) Φ M ( r , q ) . (3)Here M denotes the meson and a , b are its coalescing valence partons. W ab and Φ M are the Wigner functions of the partons and the meson respectively, P and R are the momentum and spatial coordinate of the meson, and q and r are related to the relative momentum and position of the quarks. The sum runsover all possible combinations of quantum numbers of the quarks in the hadron,essentially leading to a degeneracy factor C M .Note that coalescence, just as its counterpart in exclusive processes, is basedon the assumption of valence quark dominance, i.e. the lowest Fock states arethe most important ones. The corresponding formula for baryons containing 3valence quarks can easily be written down as well. It is also straightforward togeneralize Eq. (3) to include more partons which would be gluons or pairs of seaquarks, accounting for the next terms in a Fock expansion (51).For a meson consisting of two quarks its Wigner function is formally definedas Φ M ( r , q ) = Z d se − i s · q ϕ M (cid:18) r + s (cid:19) ϕ ∗ M (cid:18) r − s (cid:19) (4)where the 2-quark meson wave function in position space ϕ M can be represented adronization by Coalescence h r ; r | M ; P i = e − i P · ( r + r ) / ϕ M ( r − r ) (5)The Wigner function of the partons can be defined in a similar way from thedensity matrix ρ (49).To evaluate Eq. (3) expressions for the hadron wave functions and for thedistribution of partons have to be used as input. We discuss the different imple-mentations in the next subsection. Let us emphasize two common features of allimplementations. For one, the Wigner function for the multi-parton distributionis usually approximated by its classical counterpart, the phase space distributionof the partons on the hypersurface of hadronization. Secondly, Eq. (3) is madeexplicitly Lorentz-covariant to account for the relativistic kinematics. Different manifestations of Eq. (3) have been used in the literature (52). Clos-est to the master formula is the implementation by Greco, Ko and L´evai [GKL](11, 53). In this approach the full overlap integral in Eq. (3) over both relativeposition and momentum of the partons is calculated. On the other hand, sev-eral groups, (e.g. Fries, M¨uller, Nonaka and Bass [FMNB] (10, 49, 50); Hwa andYang [HY] (54, 55) and Rapp and Shuryak [RS] (56) simplify the situation byintegrating out the information about position space. This leads to a formula-tion solely in momentum space in which the information about the hadron isfurther compressed into a squared (momentum space) wave function (also calleda recombination function by some authors).The implementation by Greco, Ko and L´evai was originally motivated by a rel-ativistic extension of the formalism for the coalescence of nucleons into deuterons adronization by Coalescence N M = C M Z Y i = a,b ( p · dσ ) i d p i δ ( p i − m i ) W ab ( r a , p b ; r b , p b )Φ M ( r ; q ) . (6)The relative phase space coordinates r = r b − r a and q = p b − p a are the four-vector versions of the vectors r and q in Eq. (3). dσ is a volume element of aspace-like hypersurface. The hypersurface of coalescing partons is usually fixedby GKL through the condition of equal longitudinal proper time τ = √ t − z .In the GKL formalism the full phase space overlap of the coalescing particlesis calculated. For mesons this leads to a 6-dimensional phase space integralwhich is computed using Monte-Carlo techniques (53). This has the advantageto avoid some of the more restrictive approximations employed by other groups.In addition, the numerical implementation of the 6D-phase space integral canbe applied directly to a quark phase which has been extracted from a realisticdynamic modeling of the phase space evolution in the collision. Soon after thefirst implementation of GKL, similar techniques were used for hadronization inthe partonic cascade approach by Molnar (59).The hadron Wigner function for light quarks used by GKL is a simple productof spheres in position and momentum spaceΦ M ( r ; q ) = 9 π h ∆ r − r i × Θ h ∆ p − q + ( m − m ) i . (7)The radii ∆ r and ∆ p in the Wigner formalism obey the relation ∆ p = ∆ − r ,motivated by the uncertainty principle. The parameter ∆ p is taken to be differentfor baryons and mesons and is of the order of the Fermi momentum. GaussianWigner functions are used if heavy quarks are involved (60) (see also Fig. 9). adronization by Coalescence p T also provide a bulk of the partonic fireball which is consistentwith what can be inferred from hydrodynamics and experimental data. E.g., theradial flow, parameterized as β = β r/R exhibits a slope parameter β = 0 . ǫ = 0 . ,which is very close to what is expected from lattice QCD calculations (1, 2). Inaddition, the entropy is found to be dS/dy ≈ adronization by Coalescence | P | of the hadron is much larger than the mass M . This allowsone to treat the hadron as being on the light cone with a large + -momentum, P + ≫ P − (the z -axis in the lab frame is here taken to point into the direction ofthe hadron; this is called the hadron light cone frame in (49)). The momenta ofthe partons inside the hadron can be parameterized by light cone fractions x i of P + (0 < x i <
1) and transverse momenta k i orthogonal to the hadron momen-tum P . The momenta k i are usually integrated as well in a trivial way, leavinga single longitudinal momentum integration.In the absence of any perturbative scale the light cone wave functions are notknown from first principles. But, again, the coalescence from partons thermallydistributed in phase space is not very sensitive to the shape of the wave functions.For the lowest Fock state of a meson the squared wave function or recombinationfunction is usually parameterized as (49)Φ M ( x , x ) = Bx α x α δ ( x + x − . (8)Here the α i are powers which determine the shape, and the constant B is fixedto normalize the integral over Φ M to unity. The yield of mesons with momentum P can then be expressed as dN M d P = C M Z Σ dσ · P (2 π ) Z dx dx Φ M ( x , x ) W ab ( x P ; x P ) (9)where dσ is the hypersurface of hadronization. In many cases the emission integralover the hypersurface is not calculated explicitly, but replaced by a normalizationfactor proportional to the volume of the hadronization hypersurface.Several choices for the powers α i can be found in the literature. Asymptoticlight cone distribution amplitudes suggest α i = 2 for light valence quarks. Forheavy-light mesons the relative size of the powers has to be adjusted such that the adronization by Coalescence D meson, values α c = 5, α u,d = 1 are used by Rapp and Shuryak(56). It is sometimes useful to look at the extreme case α i → ∞ with the ratio ofthe α i fixed. For two light quarks this implies Φ M ( x , x ) = δ ( x − / δ ( x − / p T in heavyion collisions usually assume a thermal distribution of partons at hadronization.For such a system it seems to be sufficient to neglect correlations between partonsand to use a factorization into single-particle phase space distributions W ab ( r a , p a ; r b , p b ) = f a ( r a , p a ) f b ( r b , p b ) . (10)With thermal one-particle distributions f this gives good results for single inclu-sive spectra, hadron ratios, etc. at RHIC. Observables dealing with correlationsof hadrons are more sensitive to correlations among partons. Results from RHICseem to suggest that jet-like correlations between bulk partons exist down to in-termediate p T and have to be taken into account (62). This will be addressed inmore detail below. Coalescence applied to non-thermal systems, e.g. to partonshowers (63, 64), or hadronic collisions (56), require more sophisticated modelsfor multi-parton distributions.As we have mentioned above, recombination of partons in a thermal ensemblehas the interesting property that the process is largely independent of the shapeof the hadron wave function. This can be most easily seen using the FMNBformalism with the partons coming from the tail of a Boltzmann distribution adronization by Coalescence e − p/T . For a meson with large momentum P the integral in (9) is ∼ Z dx a dx b Φ M ( x a , x b ) e − x a P/T e − x b P/T ∼ e − P/T Z dx a dx b Φ M ( x a , x b ) , (11)independent of the shape of Φ M . Moving to lower hadron P T or using the fullphase-space overlap, as in GKL, make this argument less rigorous. But evenin those cases the results are only weakly dependent of the shape of the wavefunction unless very extreme choices are made. From this small exercise wecan read off another important fact, which is tantamount to solving the baryonpuzzle at RHIC: both mesons and baryons would lead to the same Boltzmanndistribution ∼ e − P/T (for sufficiently large momentum P ), which is very differentfrom the suppression for baryons expected from fragmentation. In order to compute realistic hadron spectra that can be compared to data mea-sured in heavy-ion collisions, other important mechanisms of hadron productionhave to be considered as well. QCD factorization theorems state that leading-twist hard parton scattering with fragmentation is the dominant mechanism ofhadron production at asymptotically high momentum transfer (22). This canalso be seen from the simple analytic formulas discussed in the previous subsec-tion. Let us again consider the tail of a thermal parton distribution f th ∼ Ae − p/T and compare it to a power-law distribution f jet ∼ Bp − α for large p . Power-lawdistributions are typical for partons coming from single hard scatterings. Bothrecombination and fragmentation preserve the basic shapes of the underlyingparton distribution.As we have already argued above, recombination of n thermal partons leads toa thermal distribution for the resulting hadrons with the same slope ∼ A n e − p/T , adronization by Coalescence n hard partons would steepen to ∼ B α p − nα (note that these n hard partons would come from n different jets!).On the other hand, fragmentation from a single hard parton just leads to a shiftin the slope of the power law ∼ Bp − α − δ . Given that α ≈ . . . p and lead to a power-law spectrum of hadrons from fragmentation off jets at verylarge p , in accordance with perturbative QCD.Hence from very basic considerations we expect a transition from a domaindominated by recombination of thermal partons at intermediate p T to a regimedominated by fragmentation of jets at very high p T . It is also clear that thistransition happens at higher values of p T for baryons compared to mesons, sincerecombination produces baryons and mesons with roughly the same abundance,while baryons are suppressed in jet fragmentation.This dual aspect of hadron production and the transition region are treated indifferent ways in the literature. • In publications by the FMNB group thermal recombination is supplementedby a perturbative calculation including jet quenching and fragmentation.The two components of the spectrum are simply added (49). No mixingof the thermal and hard partons is included, leading to a rather sharptransition between the two regions. • The GKL group allows coalescence between soft and hard partons as well.For mesons this would correspond to a term ∼ f th ( p a ) f jet ( p b )Φ M ( p a − p b ) . (12) adronization by Coalescence • A technically very different approach is used by Hwa and Yang (63). Insteadof fragmenting hard partons directly, they define the parton contents of ajet (initiated by a hard parton), the so-called shower distributions. Theyare given by non-perturbative splitting functions S ij ( z ) which describe theprobability to find a parton of flavor j with momentum fraction z in a jetoriginating from a hard parton i . The parton content of a single jet canthen recombine and the resulting hadron spectrum has to match the resultfrom jet fragmentation. Hwa and Yang fit the shape of the parton showerdistributions to describe the known fragmentation functions for pions, pro-tons and kaons (63). The power of this approach lies in the fact that thefragmentation part of the hadron spectrum is computed with the same for-malism. It is then very natural to also coalesce shower partons with thermalpartons (55).The HY approach is also well-suited to discuss medium corrections to fragmen-tation in much more dilute systems like p + A collisions (64). It was found thatthe hadron-dependent part of the Cronin effect in d + Au collisions at RHIC canbe attributed to coalescence of jet partons with soft partons from the underlyingevent. A more rigorous definition of parton showers and a discussion of the scaledependence can be found in the work by Majumder, Wang and Wang (65). In the momentum-space formulation it is straight forward to predict the particle-type dependence of the elliptic flow v of hadrons (43) coming from coalescence.This derivation, repeated below, has been criticized as being too simplistic (66,67). However, the scaling law holds numerically to very good approximation in adronization by Coalescence a just before hadroniza-tion is given by an anisotropy v a ( p T ) at mid-rapidity ( y = 0). The phase spacedistribution of partons a can then be written in terms of the azimuthal angle φ as f a ( p T ) = ¯ f a ( p T ) (1 + 2 v a ( p T ) cos 2 φ ) , (13)where odd harmonics are vanishing due to the symmetry of the system and higherharmonics are neglected. ¯ f is the distribution averaged over the azimuthal angle φ . A general expression for the elliptic flow of hadrons coalescing from thesepartons can be derived as a function of the parton elliptic flow. For a meson withtwo valence partons a and b and for small elliptic flow v ≪ v M ( p T ) = R dφ cos(2 φ ) dN M /d p T R dφdN M /d p T (14) ∼ Z dx a dx b Φ M ( x a , x b ) h v a ( x a p T ) + v b ( x b p T ) i . The full expressions including corrections for large elliptic flow can be found in(49). In the case of a very narrow wave function in momentum space ( α → ∞ )this leads to the expression v M ( p T ) = v a ( x a p T ) + v b ( x b p T ) . (15)with fixed momentum fractions x a and x b ( x a + x b = 1).Thus for hadrons consisting of light quarks which exhibit the same elliptic flowbefore hadronization we arrive at a simple scaling law with the number of valencequarks n : v h ( p T ) = nv a ( p T /n ) . (16) adronization by Coalescence D mesons (69, 60). The treatment hasalso been extended to harmonics beyond the second order. Generalized scalinglaws for the 4th and 6th order harmonics have been derived in Ref. (70). The main features and the overwhelming success of the coalescence models ofhadronization are shared by all the approaches discussed here. However, despitethe agreement on the general properties and their ability to describe baryonand meson spectra, there are different approximations and assumptions involved.Some have already been mentioned in the previous paragraphs. We want todiscuss some additional points in more detail here.One important difference not mentioned thus far is the mass of the quarksin the parton phase. GKL and FMNB use effective masses that are roughly ofthe size of the constituent quark masses in the hadrons formed (i.e. m u,d ≈ m s ≈
475 MeV). This can be justified by the fact that coalescence does notexplicitly include all the interactions. A part of the non-perturbative physics isencoded in the dressing of quarks, leading to a finite mass. This is also consistentwith the requirement of (at least approximate) energy conservation. Furthermorequasi-particle descriptions of the thermodynamics properties of the quark gluonplasma estimate thermal masses of about 400 MeV (71, 72). However, the exactrelation between masses in a chirally broken phase and thermal masses above T c remains to be an open question. On the other hand, in the HY approach adronization by Coalescence p > m , masses do not play a too important role, wherefore a good description ofmeasured spectra can be obtained with both assumptions.The missing position-space information is a weakness of the FMNB and HYimplementations. In principle, very complex space-momentum correlations mightexist in the parton phase before hadronization, and they might be importantto describe elliptic flow in an appropriate fashion (66, 67, 68). However, in theactual GKL computations, the spatial distribution is taken to be uniform, similarto the assumption used in pure momentum-space implementations. The onlyspace-momentum correlation in GKL are those coming from radial flow and nosystematic tests of more complicated space-momentum correlations are availablein this formalism.On the other hand, GKL has the advantage to be able to easily accommodateresonance formation and decay (53). Direct observations of baryon anomaly andelliptic flow scaling are available only for stable hadrons so far. But stable hadronscan contain a large feed-down contribution from resonance decays, especially thepions (73, 74, 75). At intermediate p T the role of resonance decays is somewhatreduced, which justifies neglecting resonances as done by FMNB and HY. Theviolation of the v scaling law is generally mild, which emphasizes this point.However, GKL shows that by including resonance decays both p T -spectra and v exhibit better agreement with data towards lower p T (76). A schematic studyof the elliptic flow of resonances themselves has been conducted in the FMNBformalism. It was found that elliptic flow is sensitive to the amount of resonancesformed in the hadronic phase vs resonances emerging directly from hadronization(77). adronization by Coalescence In this section we compare various coalescence model calculations to the avail-able data and present predictions for future measurements. We start with singleinclusive measurement, in particular spectra, hadron ratios and nuclear modifi-cation factors. We then proceed to discuss elliptic flow, particle correlations, andfluctuations. While all of these observables naturally focus at RHIC data takenduring runs at √ s = 62 .
4, 200 GeV, we conclude by giving an overview of thesituation at different energies.
In Fig. 4 we show results from a coalescence model calculation of identified particlespectra using the FMNB method (49). The spectra of neutral pions, kaons,protons and hyperons for central Au+Au collisions at 200 GeV are compared todate from RHIC (38, 41). The salient features of the spectra are an exponentialfall-off at intermediate p T with a transition to a harder power-law shape at higher p T , as we predicted above. The transition from an exponential shape to a power-law shape happens at a higher p T for baryons than it does for mesons, again inaccordance with the predictions from simple underlying principles.Fig. 5 shows two baryon-to-meson ratios: anti-protons vs pions (left panel), andΛ-baryons vs K S -mesons (right panel). Results from the GKL model for p/π − (53) and Λ / K s (78) and from the FMNB model (49, 79) are compared to datafrom RHIC. Both calculations describe a baryon enhancement at intermediate adronization by Coalescence p T that diminishes until the spectra are dominated by fragmentation at higher p T . The GKL model appears to provide a better description of the data buta comprehensive analysis of the systematic uncertainties in the models has notbeen presented. More baryon-to-meson ratios can be found in Fig. 12. Early identified particle measurements at RHIC showed that for p T < v at a given p T is smaller for more massive hadrons and that when plotted vs m T − m , the v for different species fell on a single curve. With higher statistics,measurements began to reveal that at higher p T the mass ordering breaks andmore massive baryons exhibit larger v values (80, 81). This observation led tothe first speculation about hadron formation from coalescence and scaling of v with quark number. These speculations then culminated in detailed calculationsthat we show in this subsection.Fig. 6 shows data on v scaled by the number n of valence quarks in a givenhadron as a function of p T /n for several species of identified hadrons at √ s NN = 200 GeV (40, 82). A polynomial function has been fit to the scaled values of v . To investigate the quality of agreement between hadron species, the datafrom the top panel are scaled by the fitted polynomial function and plotted inthe bottom panel. Best agreement with scaling is found for p T /n > . v /n is ordered by mass.By combining m T − m scaling and quark number scaling, one can achievea better scaling across the whole momentum range (42). Fig. 7 shows v /n vs( m T − m ) /n for several species of mesons and baryons. The scaling at low m T − m holds with an accuracy of 5-10%. At higher m T − m , a violation of the adronization by Coalescence v /n is scaledby a polynomial fit to the meson v /n only. The ratio of the data to the fit showsthat baryon v /n tends to lie below the meson v /n .The break-down of the simple quark number scaling was predicted by severalauthors (76, 66, 67, 68, 69) on the grounds of numerous arguments On the otherhand, no clear consensus has emerged on whether the kinetic energy scalingat intermediate p T is just a consequence of p T scaling (since m T − m → p T with increasing p T ) or whether it offers genuine new insights. Fig. 8 presents acomparison of data with the predictions for scaling violations. The bottom panelshows the ratio ( B − M ) / ( B + M ), where B is v /n for baryons and M is v /n formesons. In this figure K S serve as an example for mesons and Λ + Λ for baryons.The pion and proton v have been shown to be, within errors, consistent with the v of K S and Λ + Λ, respectively (84, 83). Two predictions for scaling violationsare shown as well.Three possible sources of violations of quark number scaling have been studiedwithin the GKL and FMNB implementations. One source are realistic wave func-tions with finite width (as opposed to the limit α = 0 needed for the derivationof scaling). Both theoretical curves shown in Fig. 8 use realistic wave functions(note however, that in GKL in addition the quarks don’t have to be collinear).Another correction is expected from higher Fock states which should scale withhigher weights n + 1, n + 2, etc. A study within the FMNB framework showedthat while thermal spectra are almost unaltered, there are visible effects for v .However, those are numerically surprisingly small (51). Fig. 8 also contains aprediction including a 50% admixture of a state with one additional gluon. Athird breaking of scaling is expected from resonance decays studied in (76). adronization by Coalescence v /n to fall below the quark v values.The reduction is larger for baryons so that the naive scaling is broken. Thepredicted violation (51, 76) are in fairly good agreement with the data (84). For p T > − . B − M ) / ( B + M ) should relax to the value − . v and depend only weakly on p T . The fragmentationcontribution is not included in either theoretical calculation in Fig. 8. We discussfurther arguments against v -scaling in Sec. 4.2.The RHIC program has also confirmed, for the first time, the existence of non-vanishing higher azimuthal anisotropies, beyond elliptic flow (85). The existenceof a sizable fourth harmonic v = h cos(4 φ ) i had been anticipated in hydrodynamiccalculations (86). Coalescence predictions for the relative v of baryons andmesons provide further checks for the recombination picture. Such relations havebeen first worked out (70). Concrete computations were later performed in theGKL model (78), where it was found that the difference between baryon andmeson v is much more pronounced than for v . This might lead to valuableconstraints for coalescence models in future high-statistics runs at RHIC. Fitshave been performed for identified particle v from 62.4 GeV Au+Au collisions.These studies report good agreement with data for quark v approximately 2 × quark v (83). Coalescence has also been applied to study hadrons involving heavy quarks, inparticular for D and B mesons (60,87,88). Such studies have attracted increasinginterest due to the surprisingly strong interaction of heavy quarks in the medium adronization by Coalescence R AA (92,91) and v of single electrons coming from semi-leptonicdecays of D and B mesons. While the main challenge is to understand the originof this strong interaction with the medium, the hadronization mechanism playsa significant role in the interpretation of the data (89, 90). We show this inFigs. 9 and 10 where the R AA and the v of single electrons from semi-leptonicdecays is shown together with experimental data from PHENIX and STAR (91,92). Comparing the solid (coalescence plus fragmentation) and dashed band(fragmentation only) one notices a significant effect from coalescence. It manifestsitself in an increase of both R AA and v up to p T ∼ c for single electrons(which corresponds to about p T ∼ R AA and v and so allows for a better agreement with the data. We note that the non-photonic electron spectrum can also be effected by coalescence if the Λ c /D ratiois enhanced in Au+Au collisions compared to p+p collisions (93). This is becausethe branching ratios to electrons are much smaller for charm baryons than forcharm mesons.An important development is the impact of coalescence on the physics ofquarkonia in a quark-gluon plasma. Even though coalescence has been applied tothe J/ Ψ for many years, the present implementations can be used to check notonly the yield but also the spectra and the elliptic flow as a function of trans-verse momentum. This makes it possible to perform consistency checks betweenthe spectra observed for open charm mesons and for J/ Ψs. Such studies will beof particular interest at LHC where the J/ Ψ should be dominated by regenera-tion in the plasma (94, 95). In addition, recent studies have found that even ifthe binding of a J/ Ψ is screened in a quark-gluon plasma, the spectral function adronization by Coalescence
Single particle observables and elliptic flow motivated coalescence models andwere a success story throughout the history of RHIC data taking. Later, mea-surements of hadron correlations challenged this picture. At RHIC it has beenpossible to measure the correlation between a trigger particle with momentum p trigT and an associated particle with momentum p T , typically smaller than p trigT .The experimental observable is usually the associated yield, which is the yieldof correlated pairs divided by the trigger yield. Associated yields have beenmeasured as a function of relative azimuthal angle ∆ φ , and both trigger andassociated p T (97, 98). This observable is ideal to detect correlations typical forjets. Jets give signals at ∆ φ = 0 (near-side jet) and ∆ φ = π (away-side jet). Itwas at first very surprising that such jet-like correlations were found with bothtrigger and associated p T in the recombination domain below 4 to 6 GeV/ c .It was then quickly realized that correlations among hadrons in this kinematicregime can come about through two mechanisms. First, mixed soft-hard recom-bination or thermal-shower recombination naturally leads to correlated hadrons.This was first explored by Hwa and Yang (99). A shower parton coalescing tobecome part of a hadron at intermediate p T will provide a correlation of thishadron with all those hadrons coming from fragmentation of the same jet, or theassociated away-side jet.A second possibility was pointed out in a work by Fries, M¨uller and Bass in adronization by Coalescence W ab ( p a , p b ) = f th ( p a ) f th ( p b ) (1 + C ab ( p a , p b )) (17)for two partons a , b . Under these specific assumptions they obtained correla-tions for the coalescing hadrons that are amplified by the product of valencequarks numbers (4, 6 and 9 for meson-meson, baryon-meson and baryon-baryonpairs resp.), similar to the enhancement of elliptic flow by the number of valencequarks. While it is not clear that the specific assumptions (very weak 2-particlecorrelations) hold at RHIC a reasonable result for associated yields as a functionof centrality was obtained.Fig. 11 shows the associated yield of near-side hadrons for trigger baryons (rightpanel) and trigger mesons (left panel) calculated in (62) together with PHENIXresults from (102). A scaling law for correlations between different pairs of hadronspecies has not been observed in data so far. This is compatible with the factthat correlations from jet fragmentation are strong and have to be added evenat intermediate p T , even though fragmentation is suppressed in single inclusiveobservables at the same p T (62). The authors of this study argued that the phasespace relevant for recombination at intermediate momentum is not necessarilycompletely thermalized. Rather, remnants of quenched jets, so-called hot spotscould be an important component, leading to some residual jet-like correlationsamong partons through simple momentum conservation. Independent of themodeling in detail, one can conclude that recombination has been shown to be adronization by Coalescence p T .Charge fluctuations (103) have been shown to be consistent with the recom-bination process as well. They are considered to be a good probe for QGPformation. General expectations from coalescence are in fairly good agreementwith data (105, 104). A recent, more specific study shows that consistency withcoalescence is obtained if the number of quarks and antiquarks is approximately dN/dy ∼ = 1300 for central collisions (106). This is in agreement with the partonmultiplicity estimated in the GKL implementation (53) and with the ALCORmodel (107). This is a valuable consistency test for coalescence models. Most of the work published in the context of coalescence models focuses onAu+Au collisions at RHIC energies of 130 or 200 GeV. Of course, it is impor-tant to understand if the models can predict the correct behavior of observables,e.g. baryon-to-meson ratios, as a function of collision energy √ s . Before the lowenergy Au+Au run at RHIC with √ s = 62 GeV was completed, a predictionwas presented within the GKL approach, utilizing a simple extrapolation of themodel parameters (108). It was found that the p/π ratio increases compared to √ s = 200 GeV while the p/π ratio decreases. This is exactly what was measuredwhen the lower energy data was analyzed (109). The predictions for scenarioswith and without coalescence are shown together with the data in Fig. 12. Thedata are clearly favoring the scenario with quark coalescence. The discrepancyfound for p/π at p T > adronization by Coalescence √ s = 5 . x . Naively, one would expect thewindow in p T where coalescence is dominating to increase. However, the estimatesfor this region depend delicately on the radial flow (which pushes the coalescinghadrons to higher p T ) and jet quenching (which leads to less fragmented hadronsat high p T ).Possible scenarios have been explored in the FMNB framework using differentassumptions for the radial flow (110). These estimates are shown in Fig. 13.Recently, a more systematic study of elliptic flow as a function of collision energywas published (111). The RHIC program has provided remarkable evidence that coalescence of quarksis the dominating mechanism for hadronization from a deconfined plasma. Nonethe-less, some problems remain unsolved and several new questions are raised by theformalism itself. E.g. it is appealing to apply recombination at low momentawhere the phase space is more dense. Some of these problems have been touchedupon briefly in previous sections. We will discuss them in more detail below.
A basic issue that involves all approaches based on instantaneous projection isthat of energy conservation. The underlying kinematics of the projection is ef- adronization by Coalescence → →
1, which makes it impossible to conserve 4-momentum.This is somewhat mediated at intermediate transverse momenta, p T > m , wherethe kinematics is essentially collinear and violations of energy conservation aresuppressed by factors m/p T or k T /p T where k T is the intrinsic transverse mo-mentum of a parton inside the hadron. In principle, this is not really acceptable,since the formalism should be easily extendable to low p T where collinearity ismissing. In fact, a smooth matching with bulk coalescence models like ALCORshould be possible, which describe multiplicities and related observables at low p T successfully (30,112). Interestingly, a naive extension of the GKL approach tolow momenta does not lead to striking disagreement with the experimental data(53,89). However from the theoretical point of view the issue of imperfect energyconservation is clearly unsatisfying.Energy conservation has to be achieved through interactions with the surround-ing medium. Naturally, approximations to this multi-particle dynamics have tobe applied to make the problem tractable. One way is to introduce an effectivemass distribution for the quarks as a way to incorporate some in-medium effects(113). This allows to enforce both momentum and energy conservation and onefinds fairly good agreement with data for p T -spectra.A promising new and very powerful approach has recently been developed byRavagli and Rapp (RR) (48). They replace the instantaneous projection of quarkstates onto hadron states by a procedure which solves the Boltzmann equationfor an ensemble of quarks which are allowed to scatter through hadronic states.Thus the hadrons are given through cross sections with a certain width. Thisimplementation naturally conserves 4-momentum. Ravagli and Rapp find quitegood agreement with data for p T -spectra. They also confirm v scaling (neglecting adronization by Coalescence v vs m T − m ) is in even better agreement withexperimental data, cf. Fig. 7. The RR formalism with energy conservation is theonly one really suited to address the question of kinetic energy scaling.A related issues is entropy conservation. Coalescence through instantaneousprojection seems to reduce the number of particles by about a factor two, whichunderstandably rises the question whether the second law of thermodynamics isviolated. However, strictly speaking this formalism should only be applied atintermediate p T where only a small fraction of the total particle number ( < p T . adronization by Coalescence An important open question is the relation between space-momentum correlationsand v scaling. The valence quark number scaling of elliptic flow was derived ina pure momentum-space picture. This means that the scaling has been explicitlyproven only if the coalescence probability is homogeneous in space. GKL havegone one step beyond by including correlations of radial flow with the radialcoordinate r . They find that scaling still holds to a good approximation withsome small violations (53, 76, 89).However, the situation could be very different if more realistic correlations offlow with the spatial azimuthal angle ϕ are taken into account. One should expecta strong correlation between the spatial azimuthal angle ϕ and the momentumazimuth φ . A detailed discussion of effects coming from space-momentum corre-lations can be found in the work by Pratt and Pal (66). They also map out aclass of phase space distributions that lead to approximate scaling.Parton cascade studies that calculate the time evolution of the phase spacedistributions find that approximate scaling between baryons and mesons stillpersists even if strong deviations of v at the quark level are seen (67). Howeverit is not clear how this depends on the freeze-out criteria, on the width of thewave functions and on the interplay with jet fragmentation. Another study onthe effect of phase space distribution can be found in Ref. (68).Small violations of v scaling have been observed, but as discussed in detail inSec. 3 they can be explained solely by wave function effects, resonance contribu-tions and contributions from higher Fock states in hadrons (51, 108, 115). If thescaling feature were accidental and strongly dependent on details of the phasespace distribution, the very different dynamical evolution at LHC might lead to adronization by Coalescence Quark coalescence models for heavy ion collisions have reached a certain level ofmaturity, but it has also become clear that there are limitations. We hope thatseveral issues will attract attention in the future.Within the established projection formalism several open questions can beaddressed. A huge amount of data on 2- and 3-hadron correlations has beencollected. While preliminary studies have shown that correlations are in prin-ciple compatible with recombination, a comprehensive effort to understand thedata in a picture that contains jets, jet quenching, and coalescing partons atintermediate p T still has to be developed. It would have to include a realisticmicroscopic modeling of the coupling between the medium and jets and how jet-like correlations can be conferred to the medium. A second issue concerns therole of resonance production. Little is know about the relative probabilities ofcoalescence into stable hadrons and unstable resonances. As we have seen abovethis is an important issue for multiplicities and entropy production as well as v scaling violations (in particular for pions).Dynamical transport implementations like the one developed by Ravagli andRapp are very promising candidates to investigate more fundamental open ques-tions. E.g., it would be straight forward to implement resonances and stablehadrons in a realistic fashion. Progress could be made on the issues of kinetic adronization by Coalescence v , the role of space-momentum cor-relations for elliptic flow scaling, and entropy production. There is also a needto explore dynamical coalescence coupled to realistic transport models for theparton and hadron phase.There is a list of more profound questions which we have not touched uponyet at all. Coalescence of particles can be found in systems which do not exhibitconfinement (e.g. in plasmas of electrons and protons). Confinement does notplay a big role in any of the current implementations of coalescence. (In partoncascades, non-coalescing partons are usually fragmented, the only tribute to thefact that there are no free partons allowed in the vacuum.) Nevertheless thereshould be a fundamental difference between confining and non-confining theories.Transport implementations need to explore this difference in the future.It is also not clear what the role of chiral symmetry breaking during the co-alescence process is. Most implementations give constituent-like masses to thequarks, but no direct connection to chiral or thermal masses is made. Unfortu-nately the current observables do not seem to be sensitive to the nature of thequark masses. We would hope that improved implementations together with newhigh-statistics data might allow us to address this question. The first stage of the RHIC program has provided clear evidence that hadroniza-tion at transverse momenta of several GeV/ c is modified when compared to p + p collisions in the light quark sector. The available data is only compatible with ahadronization process through coalescence of quarks. The baryon enhancementand the robust scaling of the elliptic flow with the number of valence quarks are adronization by Coalescence v is a collective effect (comingfrom the hydrodynamic expansion due to pressure gradients), and that this col-lectivity seems to happen on the parton level, leading to a universal elliptic flowfor quarks just above T c . In other words, elliptic flow of hadrons at intermediate p T did not emerge from hadronic interactions.This is indeed very remarkable and it is a strong argument for deconfinement.All signatures for deconfinement use indirect arguments and need some kind oftheoretical input to reach this conclusion. Coalescence, and in particular the v scaling, appear to be convincing because almost no additional assumptions seemto be needed. We hope that this argument is solidified with future improvementsin our understanding of data and of the mechanism of recombination. We like to thank our numerous colleagues who worked with us on the topic ofquark recombination over the recent years. We want to thank the editors of adronization by Coalescence adronization by Coalescence Literature Cited
1. Karsch F.
Lect. Notes Phys.
Phys. Rev. D
JHEP
JHEP
Nucl. Phys.
A757:102 (2005)Adcox K, et al (PHENIX Collaboration).
Nucl. Phys.
A757:184 (2005)4. Gyulassy M, McLerran L.
Nucl. Phys.
A750:30 (2005)5. Adler SS, et al (PHENIX Collaboration).
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. C
Nucl. Phys.
A715:379 (2003)9. Hwa RC, Yang CB.
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Lett.
B443:45 (1998)Vitev I, Gyulassy M.
Phys. Rev. C
Phys. Rev. C
Phys. Lett.
B503:58 (2001)Kolb PF,
AIP Conf. Proc.
Quark Gluon Plasma 3 . ed. RC Hwa, XNWang, 634. Singapore: World Scientific. preprint arXiv:nucl-th/0305084
16. Hirano T, Nara Y.
Phys. Rev. C
Nucl. Phys.
B246:52 (1984) adronization by Coalescence
Adv. Ser. Direct. High Energy Phys.
Phys. Rev. Lett.
Phys. Lett.
B508:279 (2001). erra-tum
Phys. Lett.
B590:309 (2004)20. Collins JC, Soper DE.
Nucl. Phys.
B194:445 (1982)21. Owens JF.
Rev. Mod. Phys.
Adv. Ser. Direct. High Energy Phys. arXiv:hep-ph/0409313
23. Kniehl BA, Kramer G, Potter B.
Nucl. Phys.
B582:514 (2000)S. Albino, B. A. Kniehl and G. Kramer,
Nucl. Phys.
B725:181 (2005)24. Das KP, Hwa RC.
Phys. Lett.
B68:459 (1977). erratum
Phys. Lett.
B73:504(1978)25. Adamovich M, et al (WA82 Collaboration).
Phys. Lett.
B305:402 (1993)26. Aitala EM, et al (E791 Collaboration).
Phys. Lett.
B371:157 (1996)27. Braaten E, Jia Y, Mehen T.
Phys. Rev. Lett.
Nuovo Cim. A
Phys. Rev. Lett.
Phys. Lett.
B347:6 (1995)30. Biro TS, Levai P, Zimanyi J.
J. Phys. G
Phys. Lett.
B472:243 (2000)32. Adler SS, et al (PHENIX Collaboration).
Phys. Rev. Lett. preprint arXiv:nucl-ex/0601042
Long H (STAR Collaboration).
J. Phys. G
Phys. Rev. Lett.
Phys. Rev. Lett. adronization by Coalescence
Eur. Phys. J. C
Nucl. Phys.
B100:237(1975)38. Adams J, et al (STAR Collaboration).
Phys. Rev. Lett.
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. Lett.
Z. Phys. C
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rev. Lett.
Nucl. Phys.
A697:495 (2002). erratum
Nucl. Phys.
A703:893 (2002)47. Baier R, Schiff D, Zakharov BG.
Ann. Rev. Nucl. Part. Sci. arXiv:nucl-th/0302077
48. Ravagli L, R. Rapp R.
Phys. Lett.
B655:126 (2007)49. Fries RJ, Muller B, Nonaka C, Bass SA.
Phys. Rev. C
J. Phys. G
Phys. Lett.
B618:77 (2005)52. Fries RJ.
J. Phys. G
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C adronization by Coalescence
Phys. Rev. D
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
53: 362 (1996)Mattiello R, et al.
Phys. Rev. C
J. Phys. G
Phys. Lett.
B595:202 (2004)61. Pal S, Pratt S.
Phys. Lett.
B578:310 (2004)62. Fries RJ, Bass SA, Muller B.
Phys. Rev. Lett.
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rev. C
Phys. Rev. C
Nucl. Phys.
A749:268 (2005)67. Molnar D. preprint arXiv:nucl-th/0408044
68. Greco V, Ko CM. preprint arXiv:nucl-th/0505061
69. Lin ZW, Molnar D.
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Lett.
B644:336 (2007)73. Sollfrank J, Koch P, Heinz U.
Phys. Lett.
B252:256 (1990)74. Hirano T.
Phys. Rev. Lett.
Phys. Lett.
B597:328 (2004)76. Greco V, Ko CM.
Phys. Rev. C
Phys. Rev. C adronization by Coalescence
J. Phys. G
Phys. Lett.
B583:73 (2004)80. Sorensen P (STAR Collaboration).
J. Phys. G
Phys. Rev. Lett. arXiv:0801.3466 [nucl-ex]
83. Abelev BI, et al (STAR Collaboration).
Phys. Rev.
C 75:054906 (2007)84. Sorensen PR.
Nucl. Phys.
A774:247 (2006)85. Adams J, et al (STAR Collaboration).
Phys. Rev. Lett.
Phys. Rev. C
Phys. Rev. C arXiv:0709.2884 [hep-ph]
89. Greco V.
Eur. Phys. J. Special Topics
AIP Conf. Proc.
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. C
Nucl. Phys.
A709:415 (2002)95. Andronic A, Braun-Munzinger P, Redlich K, Stachel J.
Phys. Lett.
B652:259(2007)96. Mocsy A, Petreczky P.
Phys. Rev. Lett.
Phys. Rev. D
Phys. Rev. Lett.
Phys. Rev. C
Phys. Rev. C adronization by Coalescence
J. Phys. G
J. Phys. Conf. Ser.
J. Phys. G
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Lett.
B532:249 (2001)105. Mitchell J.
J. Phys G
Phys. Rev. C
Heavy Ion Phys.
Phys. Rev. C
Phys. Lett.
B655:104 (2007)110. Fries RJ, Muller B.
Eur. Phys. J. C arXiv:0708.3015 [nucl-th]
Acta Phys. Hung. A
J. Phys. G
Phys. Lett.
B650:193 (2007)115. Sorensen P.
J. Phys. G adronization by Coalescence Central Au+Au: PHENIXCentral Au+Au: STARp+p NSD: STAR ggg: ARGUS fi - +e + e : ARGUSq q fi - +e + e - p p S0 L Central Au+Au: STAR40%-60% central: STAR200 GeV p+p: STAR B a r y on t o M e s on R a t i o s (GeV/c) T Transverse Momentum p
Figure 1: Left: p/π − ratios measured in central Au+Au collisions at √ s NN =200 GeV at RHIC, compared to measurements from e + + e − and p + p collisions.Right: The ratio Λ/2 K S for central and mid-central Au+Au collisions at √ s NN =200 GeV measured by STAR. The p/π − ratio from p+p collisions from STAR isshown for comparison. GKL FMNB HYInstantaneous coal. Yes Yes YesOverlap integr. Full 6-D Long. momentum Long. momentumSoft-hard coal. Yes No Soft-ShowerMassive quark Yes Yes NoResonances Yes No NoTable 1: A summary of key differences between the most popular implementa-tions: GKL = Greco, Ko, L´evai; FMNB = Fries, M¨uller, Nonaka, Bass; HY =Hwa, Yang. adronization by Coalescence CP R – K K f L + L X + X W + W (GeV/c) T Transverse Momentum p
Figure 2: Nuclear modification factors ( R CP ) for various identified particles mea-sured in Au+Au collisions at √ s NN = 200 GeV by the STAR collaboration. The K S and Λ + Λ R CP values demonstrate that strange baryon yields are enhancedin central Au+Au collisions compared to strange meson yields. Later, measure-ments of the φ , Ξ + Ξ and Ω + Ω showed that the rate of increase of the particleyields with collision centrality depended strongly on whether the particle was abaryon or meson with the mass dependence being sub-dominant: the baryon andmeson R CP values fall into two separate bands (indicated by lines to guide theeye) with the baryon R CP larger than the meson R CP . adronization by Coalescence (GeV/c) T Transverse Momentum p v - p + + p - p + + p K - +K + K pp+ pp+ L + L X + X STARPHENIX
Hydro model p Kp L Figure 3: v for a variety of particles from a minimum-bias sample of Au+Aucollisions at √ s NN = 200 GeV measured by the STAR (6) and PHENIX (35) col-laborations. Curves show the results from hydrodynamic model calculations (14). v values also show that baryon production at intermediate p T is enhanced in thein-plane direction, leading to larger baryon v . This observation is incompatiblewith expectation of v coming from parton energy loss. adronization by Coalescence ) / G e V d y ( c T dp T n / p ) d EV N p ( / (GeV/c) T Transverse Momentum p -7 -5 -3 -1 p (a) -7 -5 -3 -1 S0 (b) K (c) protons L + L (d) CoalescenceFragmentationTotal
Figure 4: Hadron p T -spectra at midrapidity from 200 GeV central Au+Aucollisions. The curves show the recombination and fragmentation components ofthe spectra obtained in the FMNB formalism along with the total which compareswell with the data. adronization by Coalescence Central Au+Au: PHENIXCentral Au+Au: STAR - p p L Central Au+Au: STARGKL ModelFMNB Model B a r y on t o M e s on R a t i o s (GeV/c) T Transverse Momentum p
Figure 5: Ratios of baryon yields to meson yields for central Au+Au collisionsat 200 GeV. The GKL and FMNB calculations for ¯ p/π − (left) and Λ / K s arecompared to STAR and PHENIX data. adronization by Coalescence / n v D a t a / F i t /n (GeV/c) T p Polynomial Fit - p + + p K - +K + K pp+ L + L X + X Figure 6: Top panel: The elliptic anisotropy parameter v scaled by quarknumber n and plotted vs p T /n . A polynomial curve is fit to all the data. Theratio of v /n to the fit function is shown in the bottom panel. adronization by Coalescence / n v D a t a / F i t )/n (GeV/c) -m T (m Fit to Mesons - p + + p K - +K + K pp+ L + L X + X Figure 7: Quark number scaled elliptic flow vs ( m T − m ) /n . In the low m T − m region, the scaling is improved by plotting vs m T − m . All data is fit by apolynomial curve and the ratio of v /n to the fit function is shown in the bottompanel. adronization by Coalescence / n v ( B - M ) / ( B + M ) /n (GeV/c) T p Polynomial Fit - p + + p K - +K + K f pp+ L + L X + X W + W Coalescence+ wavefunctionHigher fock=0.5 states: C Figure 8: Top panel: Quark number scaled v showing violation of ideal scal-ing. A polynomial is fit to all the available data. Bottom panel: The differencebetween quark number scaled baryon v and quark number scaled meson v di-vided by the sum: ( B − M ) / ( B + M ). The ratio is formed using hyperons andkaons. The solid curve shows model predictions from FMNB using realistic wavefunctions and a 50% admixture of a higher Fock state containing an additionalgluon. The dashed line shows calculations in the GKL model with realistic wavefunctions. adronization by Coalescence p T [ GeV/c ] R AA PHENIX [ ]STAR [ ] ◆ Au-Au @ 200 AGeV
D+B D+B [no coal.]D [no coal.]D
Figure 9: Nuclear modification factor R AA of single electrons from semi-leptonicdecays in Au+Au collisions at 200 GeV. The solid line represents the predictionsfrom a coalescence plus fragmentation model (89) for electrons from D and B mesons (shaded bands) and from D mesons only (lines). The shaded band reflectsthe theoretical uncertainty in the heavy quark diffusion coefficients (87). Thedashed lines are the results without coalescence. The data are taken from (91,92). adronization by Coalescence p T [GeV/c] v [ % ] PHENIX [ mbias ] Au-Au @ 200 AGeVD+BD D+B [ no coal. ]D [ no coal. ] Figure 10: Elliptic flow v of single electrons from semi-leptonic decays in Au+Aucollisions at 200 GeV. The lines represent the same calculations as in Fig. 9. Thedata are taken from Ref. (91) adronization by Coalescence Y AB N part
50 100 150 200 250 300 350 N part No correlations from recoWith correlationsPHENIX data
Figure 11: Associated hadron yields on the near-side as a function of number ofparticipants for meson triggers (left) and baryon triggers (right) from (62). Thediamonds represent the expected hadron correlations if fragmentation is the onlysource of correlations and recombination is correlation-free. Triangles show thesame calculation with small 2-particle correlations among coalescing partons. adronization by Coalescence p T [GeV/c] R a t i o ● ❍ p/ p + p/ p - p /p [coal.+frag.] p /p [coal.+frag.] p /p [frag. only] p /p [frag. only] Figure 12: p/π + ratio and p/π − ratio in central Au+Au collisions at 62.4 GeV.The predictions of the GKL model (coalescence plus fragmentation) (108) areshown by thick solid lines for p/π + and by thick dashed lines for p/π − ; theprediction from fragmentation only are the corresponding thinner lines. Thedata from STAR are taken from Ref. (109) adronization by Coalescence -6 -5 -4 -3 -2 -1 / P T d N / d P T p T (GeV/c) Pb+Pb @ 5.5 TeV p T (GeV/c) pReco ( =0.65,0.75,0.85)FragReco+Frag ( =0.75) Figure 13: Predictions for π (left) and proton (right) spectra in central Pb+Pbcollisions at LHC using radial flow parameters β = 0.65,0.75 and 0.85 resp. Thelarger the radial flow the more the recombination region extends to higher p T ,possibly up to 10 GeV/ cc