General Charge Balance Functions, A Tool for Studying the Chemical Evolution of the Quark-Gluon Plasma
GGeneral Charge Balance Functions, A Tool for Studying the Chemical Evolution ofthe Quark-Gluon Plasma
Scott Pratt
Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State UniversityEast Lansing, Michigan 48824 (Dated: May 22, 2018)In the canonical picture of the evolution of the quark-gluon plasma during a high-energy heavy-ion collision, quarks are produced in two waves. The first is during the first fm/c of the collision,when gluons thermalize into the quark-gluon plasma (QGP). After a roughly isentropic expansionthat roughly conserves the number of quarks, a second wave ensues at hadronization, 5-10 fm/cinto the collision. Since entropy conservation requires the number of quasi-particles to stay roughlyequal, and since each hadron contains at least two quarks, the majority of quark production occursat this later time. For each quark produced in a heavy-ion collision, an anti-quark of the sameflavor is created at the same point in space-time. Charge balance functions identify, on a statisticalbasis, the location of balancing charges for a given hadron, and given the picture above one expectsthe distribution in relative rapidity of balancing charges to be characterized by two scales. Afterfirst demonstrating how charge balance functions can be defined using any pair of hadronic states,it will be shown how one can identify and study both processes of quark production. Balancefunction observables will also be shown to be sensitive to the charge-charge correlation function inthe QGP. By considering balance functions of several hadronic species, and by performing illustrativecalculations, this class of measurement appears to hold the prospect of providing the field’s mostquantitative insight into the chemical evolution of the QGP. a r X i v : . [ nu c l - t h ] J a n INTRODUCTION AND THEORY
In a central heavy ion collision at RHIC (Relativistic Heavy Ion Collider) or at the LHC (Large Hadron Collider),several thousand hadrons are created from the initial collision of a few hundred incoming nucleons. In the centralunit of rapidity, aside from a few dozen extra baryons, the roughly one thousand hadrons are created and evolve fromquark-antiquark creation processes. For every up, down or strange quark observed in the final state, one usually findsone extra anti-up, anti-down or anti-strange antiquark within roughly a unit of rapidity. For quark-antiquark pairscreated early in the collision, the balancing pairs might be pulled apart by the initial tunneling process by a fraction ofa fm in distance, and are then pulled further apart by collective longitudinal flow and diffusion. If the quarks are pulleda half fm apart at a time 1.0 fm/ c , collective flow would pull them apart by 7.5 fm by the time breakup occurs ( ∼ c ), and diffusion would spread them apart even further. In the canonical view of the quark-gluon plasma (QGP),a first wave of quark production occurs when the quark-gluon plasma is created during the first fm/ c . The number isthen roughly conserved in an semi-isentropic expansion until hadronization, when a second wave of production ensues.Due to entropy conservation, a thousand partons would be expected to convert to roughly a thousand hadrons, andsince each hadron has multiple quarks, and since the gluonic entropy has no quarks, the number of quarks shouldmore than double during hadronization. If hadronization were to occur later in the process, perhaps at 5-10 fm/ c ,the balancing quark anti-quark pairs created at hadronization would be unlikely to separate by more than a few fmbefore breakup.Charge balance functions were proposed as a means for identifying and quantifying the separation of balancingcharges [1]. They represent the conditional probability of observing a balancing charge in bin p given the observationof a charge in bin p , and are defined by: B + − ( p | p ) ≡ (cid:104) [ n + ( p ) − n − ( p )][ n − ( p ) − n + ( p )] (cid:105)(cid:104) n + ( p ) + n − ( p ) (cid:105) , (1)where (cid:104) n + / − ( p ) (cid:105) is the probability density for observing a positive/negative particle in bin p , and (cid:104) n + ( p ) n − ( p ) (cid:105) is the probability density for observing a positive particle in bin p and a negative particle in p . If the number ofpositives and negatives are equal, and if the detector was perfectly efficient for all p , integrating the balance functionover all p would give unity. The label p i can refer to any measure of momentum, including rapidity or pseudo-rapidity.The observable can be modified to more appropriately treat the case where the net charge significantly differs from zero[2]. In short, the balance function is simply the application of a like-sign subtraction with the purpose of statisticallyisolating the opposite balancing charge.More generally, balance functions can be analyzed for any two set of hadrons and antihadrons, B αβ ( p | p ) ≡ (cid:10) [ n α ( p ) − n ¯ α ( p )][ n β ( p ) − n ¯ β ( p )] (cid:11)(cid:10) n β ( p ) + n ¯ β ( p ) (cid:11) , (2)For instance, one could consider balance functions where α were protons and β were negative kaons. The antiparticlesare noted by ¯ α and ¯ β . For this study we will confine ourselves to the situation where the net charges are zero, whichis certainly a good assumption at LHC energies. For the case where the net charges are not equal, one might wishto follow the example in [2] and define the denominator using the lesser of the two charges, n ¯ α or n α , followed by amixed-event subtraction.Balance functions can be analyzed in six dimensions as a function of p and p , though statistics make thatprospect unlikely. Instead, the condition p is usually the observation of a particle anywhere in the detector, while p refers to the observation of the second particle with relative rapidity ∆ y , or relative azimuthal angle ∆ φ , or relativeinvariant momentum Q inv . Balance functions were used at the CERN ISR to study hadronization dynamics in pp and e + e − collisions in the 1980s [3–7], while their use in heavy ion collisions was motivated by the desire to distinguishbetween early vs. late production of charges [1]. For more central collisions, balance functions have been observed tosignificantly narrow when binned in relative rapidity [8–10]. This behavior is quantitatively consistent with the ideathat a good fraction of the charge is created late in the collision, as expected from delayed hadronization with theexistence of a long-lived quark-gluon plasma [11]. Narrowing is also predicted and observed as a function of relativeazimuthal angle [11, 12], though this paper will focus on the behavior in relative rapidity.Our first goal is to understand how to calculate balance functions between any two hadronic species α and β .We especially wish to know what happens if charge production comes in two waves, the first wave being the initialthermalization of the QGP, which is where the quark number rises quickly from zero in the first ∼ c , and thesecond wave at hadronization, which may be in the 5-10 fm/ c window. This goal is complicated by the fact thathadrons carry three charges (one for each light flavor of quark or equivalently strangeness, electric charge and baryonnumber), which makes the problem rather entangled. Charge balance functions binned in relative spatial rapidity arecharacterized by a scale σ (QGP) before hadronization, which might well be the greater part of a unit of rapidity. Duringhadronization a second group of balancing charges is created with relative coordinates characterized by σ (had) ∼
0. Inthe next section, it will be shown how one can use local charge conservation overlaid onto an assumption that extracharge within a volume is distributed thermally to derive expressions for both components of the balance functions.From this perspective, all balance functions in relative spatial rapidity will be determined in terms of σ (had) and σ (QGP) , the number of quark species per unit rapidity before hadronization, dN u,d,s /dy , and the number of hadronsper unit rapidity, dN α /dy in the final state. The hadronic yields are experimentally measured and the rapidity densityof quarks can be estimated from the total entropy, thus leaving the widths as the least understood quantities. Thebalance function in terms of the relative spatial coordinate along the beam axis translates into a balance functionin relative rapidity after convoluting with a thermal kinetic distribution and including decays. In the final section,we present illustrative predictions for balance functions in relative rapidity for several species by using a thermalblast-wave model to map between spatial rapidity and momentum space rapidity. THEORY
Balance functions are related to charge correlations. For the purposes of this derivation the charge densities areconsidered as a function of the coordinate η , which describes the longitudinal position in Bjorken Coordinates. z = τ sinh( η ) , t = τ cos( η ) , (3) τ = (cid:112) t − z , η = tanh − ( z/τ ) . In the absence of longitudinal acceleration a particle moving with the fluid has fixed η , and aside from diffusion theseparation of balancing charges would be fixed in ∆ η .Before progressing, we define the charge correlation function, g ab ( η , η ) ≡ (cid:104) ρ a ( η ) ρ b ( η ) (cid:105) (cid:48) (4)= (cid:88) i (cid:54) = j q i,a δ ( η − η i ) q j,b ( η − η j ) , where the sum over i (cid:54) = j covers all particles i and j , and the prime notes that the correlation of a charge with itselfis subtracted. The indices a and b refer to the specific charge, e.g. net strangeness. For this paper Roman indices willrefer to charges, e.g., the net number of up, down or strange quarks, while Greek indices denote specific species, e.g., π + , p, K − . Since a chargeless plasma is being considered, (cid:104) ρ a (cid:105) = 0, and one need not subtract the terms (cid:104) ρ a (cid:105)(cid:104) ρ b (cid:105) . Fora hadronic system conserving baryon number, electric charge and strangeness, the index can equivalently sum overthe net number of up, down and strange quarks. The charge-charge correlation can be expressed as: g ++ ( η , η ) = (cid:104) [ n + ( η ) − n − ( η )][ n + ( η ) − n − ( η )] (cid:105) (cid:48) (5)= − B + − ( η | η ) (cid:104) n + ( η ) + n − ( η ) (cid:105) . Here, the positive and negative subscripts refer to the sets of all positive, or all negative particles. The relationbetween the balance function, B αβ and g ab becomes complicated if particles have more varied charges, which is thecase for a hadronic system, e.g., the Σ − carries baryon number, electric charge and strangeness. These cases will bediscussed in the next few paragraphs.The reason we switch from balance functions to correlations is that the correlation does not change suddenly athadronization, except for where η = η . This follows from local charge conservation. It can be understood byconsidering the addition of a pair with η (cid:39) η during hadronization. The contribution of this single pair to the sumin Eq. (4) where either i or j points to any other particle besides those from the created pair vanishes, because oneis considering the creation of a pair with equal but opposite charges at the same point. The only contribution comesfrom the element of the sum where both particles come from the pair, which then shows up at η = η . Assuminghadronization is sudden, and assuming one understands the charge correlation before hadronization, one would alsoknow g ( η , η ) immediately after hadronization, except for the region η ≈ η . To determine the correlation in theregion of small relative coordinate, one can use the sum-rule for charge correlations, which follows from integratingthe definition of g in Eq. (4), − (cid:90) dη g ab ( η , η ) = − (cid:88) i (cid:54) = j q j,a δ ( η − η j ) q i,b (6)= (cid:88) j q j,a q j,b δ ( η − η j ) (7)= χ ab ( η ) ≡ (cid:88) α (cid:104) n α ( η ) (cid:105) q α,a q α,b . The first step used charge conservation, (cid:80) i q i = 0. The average number of particles of a given species α within dη is (cid:104) n α ( η ) (cid:105) dη . Assuming instantaneous hadronization, in order to satisfy the sum rule of Eq. (6), the charge correlationimmediately after hadronization must be: g ab ( η , η ) = g (QGP) ab ( η , η ) + g (had) ab ( η , η ) , (8) g (had) ab ( η , η ) = − (cid:104) χ (had) ab ( η ) − χ (QGP) ab ( η ) (cid:105) δ ( η − η ) ,χ (had) ab ( η ) = (cid:88) α ∈ had q α,a q α,b (cid:104) n α ( η ) (cid:105) χ (QGP) ab ( η ) = (cid:88) α ∈ QGP q α,a q α,b (cid:104) n α ( η ) (cid:105) , where g (QGP) describes the correlations both immediately before and immediately after hadronization, but neglectsthe hadronization component created at η = η . The sums over α cover the species for each state, i.e., over partonicspecies for the QGP state and over hadronic species for the hadronic state. The value χ ab , when multiplied by thedelta function, represents the charge-charge correlation that would ensue from independent particles, i.e., when theonly correlations come from a particle with itself. Here the values (cid:104) n α (cid:105) are the densities per unit η of the species α , soif one measures the final-state yields χ (had) ab can be considered as known. The values of χ ab can also be extracted froma one-body treatment such as hydrodynamics. The matrix χ (QGP) ab is diagonal in a QGP if the charges refer to thenet number of up, down and strange quarks. In contrast, hadrons have multiple charges and χ (had) ab has off-diagonalelements. Since hadronization is sudden, but not instantaneous, one would expect to replace the delta function withsome function of finite but narrow width, normalized to unity.Our next goal is to determine the balance function for any hadronic species just after hadronization, given g ab in theQGP phase. Eq. (8) describes how to extract g ab just after hadronization. However, once there are multiple chargesspread across a variety of species, it is not easy to understand how the correlation functions, g ab ( η , η ), determinethe balance functions, B αβ ( η | η ). Here, a and b refer to any conserved charges, while α and β refer to the chargecarried by a specific species, where the particle and anti-particles of each species are denoted by α and ¯ α , B αβ ( η | η ) = (cid:104) [ n α ( η ) − n ¯ α ( η )][ n β ( η ) − n ¯ β ( η )] (cid:105)(cid:104) n β ( η ) (cid:105) + (cid:104) n ¯ β ( η ) (cid:105) (9)= g αβ ( η , η ) (cid:104) n β ( η ) (cid:105) + (cid:104) n ¯ β ( η ) (cid:105) . Here, n α is the density (number per unit η ) of particles of species α . Thus, g αβ is the correlation of the effectivecharge defined by the number of a specific species minus the number of its antiparticle. With this definition, one cansee that g α ¯ β = − g α,β , g ¯ αβ = − g αβ , g ¯ α ¯ β = g αβ , (10) B α ¯ β = − B α ¯ β , B ¯ α,β = − B αβ , B ¯ α ¯ β = B αβ . As an example, one can consider the proton- K − balance function. In this example, the index α would refer to protonsand β would refer to negative kaons. The corresponding charge correlation function would be g pK − ( η , η ) = (cid:104) [ n p ( η ) − n ¯ p ( η )][ n K − ( η ) − n K + ( η )] (cid:105) . (11)The suffixes α and β can also refer to a subset of species, with ¯ α and ¯ β referring to the equivalent subset ofantiparticles. For instance, α could refer to the set of all positive particles, while β could refer to the set of allantiparticles. Switching the indices leads to the relations: g αβ = g βα , B αβ n β = B βα n α . (12)Determining the balance functions for arbitrary species requires making the jump from g ab to g αβ . There are threeconserved charges, which we will consider to be the net numbers of up, down and strange quarks. Although one couldhave equivalently used baryon number, electric charge and strangeness, the quark numbers are more convenient sinceone does not expect any off-diagonal elements to g ab in this basis for the QGP. For the species-labeled correlations, g αβ , there are many more possibilities in the hadronic state. Even for the final state, one might wish to considercharged pions, charged kaons, protons or lambdas. Neutral kaons must also be taken into account for absorbingstrangeness, but because they oscillate into K s and K l , cannot be easily used for balance functions. Since g αβ hasmore elements than g ab , additional assumptions are required if g αβ is to be determined from g ab .Observing a hadronic species α at position η infers one has observed the three charges q α,a , which is the numberof up, down and strange quarks in the resonance α . The correlation g ab ( η , η ) should then provide the probability offinding the balancing charges at position η . In order to deterine g αβ one then needs a model to determine how anextra charge q b at position η influences the probability of finding a hadronic species β at the same position.By assuming that the local distribution of hadrons is determined by a thermal distribution constrained by thelocal charge density, one can determine g αβ from g ab . To show this, we express the two particle correlation as beingdetermined by a grand canonical ensemble with Lagrange multipliers applied to constrain reproduction of the averagetwo-particle correlation function, i.e., (cid:104) AB (cid:105) = 1 Z Tr (cid:40) ABe − (cid:82) dηH /T ( η ) exp (cid:34)(cid:90) dη dη (cid:88) ab ρ a ( η ) µ ab ( η , η ) ρ b ( η ) (cid:35)(cid:41) , (13) Z = Tr (cid:40) e − (cid:82) dηH /T ( η ) exp (cid:34)(cid:90) dη dη (cid:88) ab ρ a ( η ) µ ab ( η , η ) ρ b ( η ) (cid:35)(cid:41) . Here, H is the Hamiltonian or relevant free energy density, T is the temperature, and µ a,b ( η , η ) plays the role of aLagrange multiplier chosen to enforce that g ab ( η , η ) is reproduced. The strategy will be first to find µ ab in terms of g ab , then to use µ ab to determine g αβ . The correlation function g ab ( η , η ) is found by replacing the operators A and B above with A = ρ a ( η ) = (cid:88) α n α ( η ) q α,a , B = ρ b ( η ) = (cid:88) β n β ( η ) q β,b , (14)where α and β are summed over all hadronic species. By assuming that the weighting is proportional to an exponentialof the constraint (fixing g ab ), this is essentially a thermal ansatz.Since the correlation would be zero if not for µ , we can expand the expression for small µ and find: g ab ( η , η ) = (cid:88) αβ (cid:104) n α ( η ) (cid:105) q α,a q β,b (cid:104) n β ( η ) (cid:105) exp (cid:40)(cid:88) cd q α,c µ cd ( η , η ) q β,d (cid:41) (15) ≈ (cid:88) αβcd (cid:104) n α ( η ) (cid:105) q α,a q α,c µ cd ( η , η ) q β,d q β,b (cid:104) n β ( η ) (cid:105) , = (cid:88) cd χ ac ( η ) µ cd ( η , η ) χ db ( η ) , where χ was defined in Eq. (6). The assumption of small µ is warranted given that charge-conservation correlationsare small (at least for central collisions). Inverting the equation, one can then find µ ab in terms of g ab , µ ab ( η , η ) = (cid:88) cd χ ( − ac ( η ) g cd ( η , η ) χ ( − db ( η ) . (16)One can now find g αβ by inserting A = n α ( η ) − n ¯ α ( η ) , B = n β ( η ) − n ¯ β ( η ) , (17)into Eq. (13). Here n ¯ α is the density of the anti-particles to α . Again, assuming equal numbers of particles andantiparticles, (cid:104) n α (cid:105) = (cid:104) n ¯ α (cid:105) , and assuming that µ ab is small, g αβ ( η , η ) = (cid:10) [ n α ( η ) − n ¯ α ( η )] (cid:2) n β ( η ) − n ¯ β ( η ) (cid:3)(cid:11) (18)= (cid:104) n α ( η ) (cid:105)(cid:104) n β ( η ) (cid:105) exp (cid:40)(cid:88) ab q α,a µ ab ( η , η ) q β,b (cid:41) + (cid:104) n ¯ α ( η ) (cid:105)(cid:104) n ¯ β ( η ) (cid:105) exp (cid:40)(cid:88) ab q α,a µ ab ( η , η ) q β,b (cid:41) −(cid:104) n α ( η ) (cid:105)(cid:104) n ¯ β ( η ) (cid:105) exp (cid:40) − (cid:88) ab q α,a µ ab ( η , η ) q β,b (cid:41) − (cid:104) n ¯ α ( η ) (cid:105)(cid:104) n β ( η ) (cid:105) exp (cid:40) − (cid:88) ab q α,a µ ab ( η , η ) q β,b (cid:41) (cid:39) (cid:104) n α ( η ) (cid:105) q α,a µ ab ( η , η ) q β,b (cid:104) n β ( η ) (cid:105) . From Eq. (9), one then finds an expression for the balance function, B αβ ( η | η ) = 2 (cid:88) ab (cid:104) n α ( η ) (cid:105) q α,a µ ab ( η , η ) q β,b (19)= 2 (cid:88) abcd (cid:104) n α ( η ) (cid:105) q α,a χ ( − ac ( η ) g cd ( η , η ) χ ( − db ( η ) q β,b . One test of this result is to see whether integrating the balance function over all η , summing over all α , and weightingwith q α,a , one should get the net amount of charge a found in other particles due to the condition of having observeda particle of species β at position η . Performing these operations from the expression for B in Eq. (19), (cid:88) α (cid:90) dη q α,a B αβ ( η | η ) = 2 (cid:90) dη (cid:88) αbcd q α,a (cid:104) n α ( η ) (cid:105) q α,b χ ( − bc ( η ) g cd ( η , η ) χ ( − db ( η ) q β,b (20)= 2 (cid:90) dη χ ab ( η ) χ ( − bc ( η ) g cd ( η , η ) χ ( − db ( η ) q β,b = (cid:90) dη g ac ( η , η ) χ ( − cb ( η ) q β,b = − (cid:88) cd χ ac ( η ) χ ( − cd ( η ) q β,d = − q β,a . The second-to-last step used the sum rule for integrating g in Eq. (6). The factor of two comes from the fact thatthe sum over all species, α , double-counted the contributions. For instance, the term for which α = π + also includesthe contribution from π − , and the term for α = π − also includes the contribution from the π + . CALCULATING WEIGHTS FOR BOTH COMPONENTS FOR ALL HADRONIC SPECIES
From Eq. (8), one expects two components to the charge correlation g ab ( η , η ). Assuming a boost-invariant system,one can assume a dependence on ∆ η = η − η , rather than on η and η individually. This expectation inspires oneto write the balance function for all species B αβ (∆ η ) in terms of two components, B αβ (∆ η ) = w (QGP) αβ b (QGP) (∆ η ) + w (had) αβ b (had) (∆ η ) , (21)where b (QGP) and b (had) are both normalized so that (cid:82) d ∆ ηb (∆ η ) = 1.The weights, w (QGP) and w (had) , can be determined from the charge correlations, which in turn depend on thematrices χ ab . From Eq. (8), − g ab (∆ η ) = χ (QGP) ab b (QGP) (∆ η ) + (cid:104) χ (had) ab − χ (QGP) ab (cid:105) b (had) (∆ η ) . (22)Here, the delta function in Eq. (8) was replaced by a Gaussian of finite width, where the width is determined bythe charge diffusion between hadronization and breakup. The correlation before hadronization, g (QGP) ab , should bediagonal if quarks are good quasi-particles, χ (QGP) ab = (cid:104) n a + n ¯ a (cid:105) δ ab , (23)where n a is the density of up, down or strange quarks, and n ¯ a is the density of the antiquarks. In this formulationthere is an explicit assumption that the diffusive widths of the charge correlation before hadronization are independentof flavor. Whereas the form of χ (QGP) is model dependent, χ (had) ab = (cid:104) n α (cid:105) q α,a q β,b is determined from final-state yields.After inserting the above expression for g ab into Eq. (19), one obtains B αβ , from which one can read off the weightsin Eq. (21), w (QGP) αβ = − (cid:88) abcd (cid:104) n α (cid:105) q α,a χ − ab χ (QGP) bc χ − cd q β,d , (24) w (had) αβ = − (cid:88) ab (cid:104) n α (cid:105) q α,a χ − ab q β,b − w (QGP) αβ . The characteristic width of b (QGP) is determined by the charge correlation before hadronization, g (QGP) ab (∆ η ), andone might expect it to be of the order > ∼ .
5. In contrast, b (had) is characterized by a narrow width describing thediffusion of charge after hadronization and might have a width (cid:39) . − .
2. Although the derivations assumed that thespecies were locally populated according to local thermal equilibrium, the weights are completely determined giventhe populations for quarks just before hadronization, and the rapidity density for hadronic species (cid:104) n α (cid:105) .Whereas the hadronic populations can be taken from experiment (or from a thermal model tuned to experiment),the number density of quarks just before hadronization is dependent on model assumptions. Even if one uses en-tropy arguments to infer the number of quarks, neglecting the entropy created during hadronization, the number ofquarks can depend on how much entropy was carried by gluons. For that reason the ratio of the rapidity densityof quarks before hadronization to the rapidity density of final state hadrons was varied. Three ratios were explored: n quarks /n had =0.7, 0.85 and 1.0. The hadron density included neutral hadrons, and the decay products of strangebaryons and the K s .Despite the wide coverage and detailed analysis of RHIC data, the uncertainty in the yields of particular speciesat RHIC can be rather large. Whereas the yields of pions are known to better than the 10% level, yields of protonsand anti-protons are uncertain at the 25% level. Given these uncertainties, we use yields from a thermal calculationbased on a temperature of 165 MeV. The calculation involved generating particles thermally from a hydrodynamicevolution. Particles of all hadronic species were then evolved through a hadronic cascade, whose main purpose wasto model the hadronic decays. Since only the yields were sought, the dynamical evolution of the cascade was ratherinconsequential. Weak decays were not performed. The remaining species and their yields for central collisions aregiven in Table I. The yields given in the table were then modified by an additional factor f B , which reduced the yieldsof all baryons by the same factor. Given that the number of anti-baryons is less than the number of baryons at RHICby a factor of 0.7, and given that only anti-baryons are accompanied by an additional charge, one might expect to afactor of f B ≈ .
85. Comparing the numbers below to proton yields from PHENIX [13], one would expect f B ≈ . f B = 0 . , . . + / − , Ξ − / , Ω − and the corresponding antibaryons.The included mesons are π + / / − and K + , − , . The thermal model provides yields, (cid:104) n (cid:105) α , at midrapidity, and are listedin Table I. The “thermal” model was a hydrodynamic model followed by a cascade simulation, where hadronizationwas performed thermally with a temperature of 165 MeV.The resulting weights for the default calculation ( f B = 0 . , n quarks /n had = 0 .
85) are shown in Table II. Theweights for the π + π − balance functions were not surprising. In the default calculation the hadronization process isresponsible for nearly two thirds of the final quarks (2 for a meson plus three for a baryon). Pions represent ∼ ππ balance functions integrated to 0.636, while the QGP component integrated to 0.239. The sum did not integrate tounity because observing a charged pion does not ensure that the remainder of the system has one extra pion of theopposite charge due to the possibility of the charge being balanced by other species. If one could measure the balancefunction in coordinate space, i.e. as a function of η , there would be a large narrow peak from the hadronizationcomponent and a smaller broader structure from the QGP component.The default results for the K + K − balance functions were also in line with expectations. Since rather few addi-tional strange quarks are produced during hadronization, the hadronization component turned out to be quite small.Observing the lack of a narrow peak in K + K − balance functions would confirm the notion that the QGP was indeedrich in strangeness. hadron species yields, n α p, n, ¯ p, ¯ n , ¯Λ 8.5Σ + , Σ − , ¯Σ − , ¯Σ + − , Ξ , ¯Ξ , ¯Ξ + , ¯Ω 0.35 π + , π , π − K + , K − f B , was applied to the baryon yields listed here to account for the experimental uncertainties andfor greater consistency with experimental observations. p Λ Σ + Σ − Ξ Ξ − Ω − π + K + ¯ p − + + + π − K − -0.175, 0.384 -0.627, 0.352 -0.603, 0.417 -0.651, 0.288 -1.055, 0.385 -1.079, 0.321 -1.507, 0.354 0.024, 0.064 0.452, 0.031TABLE II. Default results for the weights w (QGP) αβ , w (had) αβ resulting from the thermal model described in the text. The weightsdescribe the contribution to the balance functions B αβ (∆ η ) from the correlations driven by the charge correlations just beforehadronization and the additional correlation that appears during hadronization and is local, ∆ η ∼ The p ¯ p balance function came out contrary to expectations expressed in previous papers [1]. Even though protonsare composed entirely of up and down quarks, and even though a large fraction of up and down quarks are produced athadronization, the hadronization component is small, or perhaps negative. This comes from the fact that the strengthof the hadronization component, as determined by the sum rule in Eq. (8), depends on the density of observedbaryons. Since the observed number of baryons is rather small, the sum rule can be saturated by the number ofbaryons in the QGP component. If one were to consider baryons alone, the sign of the hadronization component inthe baryon-baryon correlation depends on the sign of χ (had) bb − χ (QGP) bb , where bb would refer to the baryon charge.Since the baryon number of a single quark is 1/3, the sign switches when the number of quarks is more than ninetimes the number of baryons.Another surprising result in Table II concerns the pK − balance function. Even though the K − meson has ananti-up quark, the QGP component is negative. This derives from µ ab being larger for us than for uu . For the rangeof parameters explored here, the hadronization component of the pK − balance function was always positive. Thismakes it easy to recognize the existence of both the QGP and hadronization components, and if such a structure wereobserved experimentally, it would be difficult to explain without a two-component picture of quark production.The upper two tables in Table III show the dependence of the weights for variations of the baryon suppression f B , which scales the final baryon yields relative to thermal yields. The values of f B roughly span the range ofuncertainties from the experimental measurement. Whereas the default value of f B was set to 0.6, Table III showsresults for f B = 0 . f B = 0 .
7. The number of quarks per unit rapidity in the QGP just before hadronization is alsouncertain, hence a range of quark numbers is explored. Bracketing the default ratio of quarks before hadronization tofinal-state hadrons of 0.85, results for n quarks /n had = 0 . n quarks /n had = 1 . p ¯ p balance function is especially sensitive to both numbers. The hadronization peak is strengthened by raising f B , or by lowering n quarks /n had . For lower baryon yields, or higher quark densities, the hadronization peak becomessmaller, and can even become negative. These would lead to a dip in the p ¯ p balance function at small relative rapidity,which would both provide striking evidence of the two-wave nature of quark production, and suggest that the QGPwas rather quark-rich. This latter conclusion could be better strengthened by better measurements of baryon yields,which differ from collaborations by several tens of percent. BLAST-WAVE PREDICTIONS FOR BALANCE FUNCTIONS
Once one has calculated the weights described in the previous section, one can calculate the balance function betweenany two species in coordinate space given the characteristic widths of the distributions, σ (QGP) and σ (had) . There is nofirm understanding of the scale σ (QGP) , as the value depends on the microscopic details of how quark-antiquark pairsare created in the pre-thermalization stage. In a flux-tube picture, the quarks are pulled apart longitudinally, usingthe tube’s energy to create the particles. From balance functions of pp collisions, one would estimate σ (QGP) > ∼ . σ (had) should be determined by the diffusion that occurs after hadronization. Although the time fromhadronization ( ∼ c ) to breakup ( ∼
14 fm/ c ) is similar as the time from creation to hadronization, the diffusionwidth grows logarithmically with time, and the post hadronization diffusion should be < ∼ . σ (had) , the choice of 0.1 vs 0.2 for the width shouldnot strongly impact the results. Since the purpose of this section is to provide an example providing a crude idea ofwhat one might suspect, the values are picked with some arbitrariness to be σ (QGP) = 0 . , σ (had) = 0 .
2, i.e., B α,β (∆ η ) = w (had) αβ (2 π ) / σ (had) e − (∆ η ) / (2 σ ) + w (QGP) αβ (2 π ) / σ (QGP) e − (∆ η ) / (2 σ ) . (25)Unfortunately, the balance function is not measured in coordinate space. The mapping of η → y has a spread fromthe thermal motion of the particles at breakup. For pions this can be a half unit of rapidity, whereas for protonsthe thermal spread is only a few tenths. Additionally, particles decay. To include both decays and the thermalspread, the correlations in coordinate space were overlaid onto a simple blast-wave parameterization. The blast-waveparameterization models the collective and thermal motion by assuming that the radial flow grows linearly in radius, u i = u max r i /r max , with u max = 1 .
0, and that the breakup temperature is 100 MeV. Decays of unstable particles(lambdas, neutral kaons, Sigmas, Cascades and Omegas) were also accounted for by a Monte-Carlo simulation.Three of the resulting balance functions are presented in Fig. 1, and are broken down by components. The π + π − balance function is dominated by the hadronization component, with the QGP component contributing to the tail.The contribution from final-state decays is small, but non-negligible. Due to large thermal spread for pions, it isdifficult to distinguish the two components.The p ¯ p balance function, displayed in the middle panel of Fig. 1, is dominated by the QGP component, with thehadronization component being small or negative. The calculation includes the contribution to protons from weakdecays, and the hadronization component from hyperon balance functions makes the hadronization component morenegative. Higher quark densities in the QGP or lower final-state baryon yields push this component toward beingnegative. If it is negative, as in the case of the default calculation in Fig. 1, the resulting balance function has aplateau or perhaps a dip at small ∆ y . The existence of such a dip would provide striking evidence for the two-wavenature of charge production. If the hadronization component were zero, one could still see evidence of two componentsby comparing with the π + π − balance functions. Since the p ¯ p balance function is dominated by the QGP componentwhile π + π − balance function is driven by the hadronization component, one could perform a single-wave fit to thewidth of the balance function in coordinate space σ η . One would expect the width for p ¯ p to be significantly largerthan that for π + π − . The width, σ η for π + π − in coordinate space has been determined by a blast-wave analysis in[11]. By using blast-wave parameters fit to spectra and elliptic flow observables, the analysis determined that thewidth of the balance function in coordinate space, assuming a single scale for the charge correlation, was σ η ∼ . p ¯ p balance function one might see σ η stay roughly constant with centrality.The pK − balance function, from the lower panel of Fig. (1), offers yet more promise for demonstrating the two-component nature of the balance function. Since the weights of the two components have different signs with thestronger component having the smaller width, one finds both positive and negative regions of the balance function. Ifthere were only one component, or if the two components had similar widths, this behavior would not ensue. Further,since protons and kaons are more massive, the thermal spread is reduced and the reduction in smearing allows moreresolving power into the correlations in coordinate space. The negative dip for ∆ y ∼ . f B = 0 . , n quarks /n had = 0 . p Λ Σ + Σ − Ξ Ξ − Ω − π + K + ¯ p − + + + π − K − -0.248, 0.459 -0.718, 0.437 -0.694, 0.502 -0.742, 0.373 -1.164, 0.480 -1.188, 0.415 -1.634, 0.458 0.024, 0.064 0.470, 0.022 f B = 0 . , n quarks /n had = 0 . p Λ Σ + Σ − Ξ Ξ − Ω − π + K + ¯ p − + + + π − K − -0.126, 0.331 -0.560, 0.291 -0.536, 0.355 -0.584, 0.227 -0.971, 0.315 -0.994, 0.251 -1.405, 0.275 0.024, 0.064 0.434, 0.040 f B = 0 . , n quarks /n had = 0 . p Λ Σ + Σ − Ξ Ξ − Ω − π + K + ¯ p − + + + π − K − -0.144, 0.353 -0.517, 0.242 -0.497, 0.310 -0.536, 0.173 -0.869, 0.199 -0.889, 0.131 -1.241, 0.088 0.020, 0.068 0.372, 0.111 f B = 0 . , n quarks /n had = 1 . p Λ Σ + Σ − Ξ Ξ − Ω − π + K + ¯ p − + + + π − K − -0.206, 0.415 -0.738, 0.463 -0.710, 0.523 -0.766, 0.403 -1.241, 0.571 -1.269, 0.512 -1.773, 0.620 0.028, 0.060 0.532,-0.048TABLE III. Weights w (QGP) αβ and w (had) αβ as in Table II except for a higher and lower yields of baryon number (adjusted by thescaling factor f B ), and quark densities before hadronization, n quarks . The upper two tables show weights for variations of f B from the default value of 0.6 to 0.5 and 0.7, while the lower two tables display results for varying n quarks /n had from the defaultvalue of 0.85 to 0.7 and 1.0. The p ¯ p weights are considerably sensitive to both the baryon yield and the input baryon density.Either higher final-state baryon yields, or lower quark densities strengthen the hadronization peak in the p ¯ p balance function. pK − balance function would be narrowed by the secondcomponent. Thus, the signal for two waves of charge production which are only semi-distinct would be a narrow pK − balance function, whose width might even be narrower that what one would predict from a single-wave model withzero width after one corrects for thermal broadening. The difficulty with pK − balance functions comes from thefact that they are smaller, by nearly an order of magnitude, than the π + π − balance functions, and thus requirehigh-statistics data sets. Fortunately, both STAR at RHIC and ALICE at the LHC provide both high statistics, anddue to the installation of large-coverage time-of-flight detectors, large acceptances for identified particles. SUMMARY
A central feature of the canonical picture of the chemical evolution of the quark-gluon plasma is the two-wave natureof quark production. Investigating balance functions over a large range of species pairs provide the means to test thishypothesis in great detail. Once baryon production is better understood, the only parameter affecting the calculationof weights is the quark density in the QGP. The blast-wave parameters used to model the thermal broadening ofthe balance-function structures are already well determined by spectra. This leaves three parameters, n quarks /n had , σ (had) and σ (QGP) for fitting the entire array of balance functions. If one were to also question the assumption thatthe strange quark density was close ( (cid:39) • The width of the π + π − balance function in ∆ η (coordinate space rapidity) should be small for central collisions.This has been reported in [11]. • In central collisions the width of the p ¯ p and K + K − balance functions in ∆ η should be larger than that of the π + π − balance function. Whereas the width for pions has been observed to shrink with centrality, these widthsmay well stay fixed, or even broaden for increasing centrality. • The p ¯ p balance function could have a plateau or even a dip at small ∆ y . • The pK − balance function should be narrower than can be fit with a single-wave picture, and might dip negativefor ∆ y ∼ . n quarks , from experiment. Determining the width, σ (QGP) , would provideinsight into the dynamical mechanism for the creation and diffusion of quarks in the plasma.Potentially, the most important implication of charge balance function would be to quantitatively constrain thecharge correlations in the QGP. For this study, the density of quarks was varied, which then determined the magnitudeof the diagonal components of g ab (∆ η ) in the QGP. Several of the hadronic balance functions were then found to besensitive to this number. Additionally, there was an explicit assumption that the off-diagonal elements were zero in theQGP. This would not be the case if quark-antiquark pairs, such as pionic fluctuations, made significant contributionsto the entropy of the QGP. Observing that the off-diagonal elements were small or zero, would make a strong casethat quarks are the dominant quasi-particles in the QGP. In principle, one could extend the ideas presented here andvary the off-diagonal elements to determine the ranges to which they are constrained by experiment.The calculations presented here are somewhat schematic in nature, and can be improved during the coming years.Most immediately, the list of resonances considered was small, and omitted the short-lived hadronic states such as the ρ or ∆, which should have some measurable effect [16]. Although almost all particles have a decay in their history,on the order of 10% of the charged particles produced at RHIC come from the decays of neutral resonances, otherthan the weak decays accounted for here, where both charges escape untouched from the decay. Thus, several of theweights might be affected at the 10% level in a more thorough calculation. If the data indeed seems addressable withthis schematic model, one could consider more sophisticated models of quark production, diffusion, hadronization andemission. [1] S. A. Bass, P. Danielewicz, S. Pratt, Phys. Rev. Lett. , 2689-2692 (2000). [nucl-th/0005044].[2] S. Pratt, S. Cheng, Phys. Rev. C68 , 014907 (2003). [nucl-th/0303025].
0 0.5 1 1.5 2 p + p -
0 0.5 1 1.5 2 p + p -
0 0.5 1 1.5 2 p + p -
0 0.5 1 1.5 2 p + p - -0.25 0 0.25 0.5 0.75
0 0.5 1 1.5 2 B ( D y ) ppbar -0.25 0 0.25 0.5 0.75
0 0.5 1 1.5 2 B ( D y ) ppbar -0.25 0 0.25 0.5 0.75
0 0.5 1 1.5 2 B ( D y ) ppbar -0.1 0 0.1 0.2 0 0.5 1 1.5 2 D y pK - -0.1 0 0.1 0.2 0 0.5 1 1.5 2 D y pK - -0.1 0 0.1 0.2 0 0.5 1 1.5 2 D y pK - FIG. 1. (color online) Balance functions for π + π − are shown as a function of relative rapidity (black circles) in the upperpanel as calculated with the default values given in Table II. The widths assumed are σ (had) = 0 . , σ (QGP) = 0 .
6. The blast-wave model is used to map B (∆ η ) to B (∆ y ). The hadronization component (green squares) is larger and narrower than theQGP component (red upward triangles), due to the fact that most quark-antiquark pairs are created at hadronization. Thecontribution from weak decays (blue downward triangles) to two pions is also shown. Since the QGP contribution is rather smalland since the thermal spread for pions is large, one can not distinguish the two components, but must be content with the widthbeing consistent with being dominated by a single narrow contribution. In contrast, the p ¯ p and pK − balance functions shownin the middle and lower panels clearly illustrate the two-component nature. Because the p ¯ p balance function has contributionswith opposite signs, one can expect to see a plateau or even a dip at small relative ∆ y . Since the p ¯ p balance function isdominated by the QGP contribution while the π + π − balance function is largely driven by the hadronization contribution, onewould expect the apparent width σ η for a single-wave fit to be much broader for the p ¯ p case than for the π + π − case. For the pK − balance functions, adding the two contributions results in a balance function that is narrower than what one could getfrom a model with a single wave. Further, one may even see the balance function become negative for ∆ y ∼ . , 357 (1981).[4] M. Althoff et al., Z. Phys. C , 5 (1983).[5] H. Aihara et al., Phys. Rev. Lett. , 2199 (1984).[6] H. Aihara et al., Phys. Rev. Lett. , 3140 (1986).[7] P. D. Acton et al., Phys. Lett. B , 415 (1993).[8] M. M. Aggarwal et al. [ STAR Collaboration ], Phys. Rev. C82 , 024905 (2010). [arXiv:1005.2307 [nucl-ex]].[9] J. Adams et al. [ STAR Collaboration ], Phys. Rev. Lett. , 172301 (2003). [nucl-ex/0301014].[10] G. D. Westfall [ STAR Collaboration ], J. Phys. G G30 , S345-S349 (2004). [11] S. Schlichting, S. Pratt, Phys. Rev. C83 , 014913 (2011). [arXiv:1009.4283 [nucl-th]].[12] P. Bozek, Phys. Lett.
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C69 , 034909 (2004). [nucl-ex/0307022].[14] J. Adams et al. [ STAR Collaboration ], Phys. Rev. Lett. , 112301 (2004). [nucl-ex/0310004].[15] B. I. Abelev et al. [ STAR Collaboration ], Phys. Rev. C79 , 034909 (2009). [arXiv:0808.2041 [nucl-ex]].[16] W. Florkowski, W. Broniowski, P. Bozek, J. Phys. G