Evidence of Coalescence Sum Rule in Elliptic Flow of Identified Particles in High-energy Heavy-ion Collisions
EExtraction of Elliptic Flow for Quarks in High-energy Heavy-ion Collisions
Amir Goudarzi, Gang Wang, ∗ and Huan Zhong Huang
1, 2 Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),and Institute of Modern Physics, Fudan University, Shanghai-200433, Peoples Republic of China
The major goal of high-energy heavy-ion collisions is to study the properties of the deconfinedquark gluon plasma (QGP), such as partonic collectivity. The collective motion of constituent quarkscan be derived from the anisotropic flow measurements of identified hadrons within the coalescenceframework. Based on published results of elliptic flow ( v ), we shall test the coalescence sum ruleusing K ± , p , ¯ p , Λ and ¯Λ, and further extract v values for produced u ( d , ¯ u , ¯ d ), s and ¯ s quarks, aswell as transported u ( d ) quarks in 10-40% Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39and 62.4 GeV. We also attempt to link the v difference between π − and π + to the different numbersof u and d quarks in the initial gold ions, and to relate the v measurements of multi-strange hadronsto the formation times of φ , Ω ± and Ξ + . Keywords: heavy-ion collision, elliptic flow, coalescence, transported quark
I. INTRODUCTION
High-energy heavy-ion collisions have been performedat experimental facilities such as the Relativistic HeavyIon Collider (RHIC) and the Large Hadron Collider(LHC), to create a deconfined partonic matter, quarkgluon plasma (QGP), and to probe its properties. Oneof the QGP signatures is partonic collectivity, in whichscenario, the spatial anisotropies of the initial partici-pating zone are converted by the pressure gradients ofthe QGP into the momentum anisotropies of final-stateparticles. Accordingly, experimental data analyses oftenexpress the azimuthal distributions of emitted particleswith a Fourier expansion [1, 2] dNdϕ ∝ ∞ (cid:88) n =1 v n cos[ n ( ϕ − Ψ RP )] , (1)where ϕ denotes the azimuthal angle of the particle andΨ RP is the reaction plane azimuth (defined by the impact parameter vector). The Fourier coefficients, v n = (cid:104) cos[ n ( ϕ − Ψ RP )] (cid:105) , (2)are referred to as anisotropic flow of the n th harmonic.Here the average is taken over all particles and over allevents. By convention, v and v are called “directedflow” and “elliptic flow”, respectively. They reflect thehydrodynamic response of the QGP fluid to the initialgeometry of the collision system [3].The coalescence mechanism intuitively describeshadronization in heavy-ion collisions, assuming quarkand gluon constituents join into a hadron when they areclose to each other in space and traveling with similarvelocities [4, 5]. Besides particle spectra, coalescence isalso applicable to anisotropic flow. Taking π + ( u ¯ d ) as anexample, without loss of generality, we set Ψ RP to zero,and ignore the normalization: v π + n = (cid:90) (cid:90) (cid:90) dϕ u dϕ ¯ d dϕ π + · cos (cid:16) ϕ π + (cid:17) [1 + 2 v un cos(2 ϕ u )][1 + 2 v ¯ dn cos (cid:16) ϕ ¯ d (cid:17) ] δ ( ϕ π + − ϕ u ) δ ( ϕ π + − ϕ ¯ d )= (cid:90) dϕ π + · cos (cid:16) ϕ π + (cid:17) [1 + 2 v un cos (cid:16) ϕ π + (cid:17) ][1 + 2 v ¯ dn cos (cid:16) ϕ π + (cid:17) ] ≈ (cid:90) dϕ π + · cos (cid:16) ϕ π + (cid:17) [1 + (2 v un + 2 v ¯ dn ) cos (cid:16) ϕ π + (cid:17) ]= v un + v ¯ dn . (3)The δ functions are there to enforce coalescence, and theapproximation ignores the higher-order term, v un v ¯ dn , as-suming v n (cid:28)
1. Equation (3) demonstrates the coales- ∗ [email protected] cence sum rule [6]: the v n of the resulting mesons orbaryons is the summed v n of their constituent quarks.If n q constituent quarks have the same transverse mo-mentum ( p T ) and the same v n , we have the number-of-constituent-quark (NCQ) scaling [7]: v hn ( p hT ) = n q · v q ( p hT /n q ) = n q · v q ( p qT ) . (4) a r X i v : . [ nu c l - e x ] J u l The approximate NCQ scaling of v has been observedexperimentally at RHIC and LHC energies [8–14], evi-dencing partonic collectivity in these heavy-ion collisions.While the same- p T condition is roughly another expres-sion of “traveling with similar velocities” in the coales-cence process, the same- v n requirement could be releasedto reveal anisotropic flow of different quarks. Signifi-cant v differences have been discovered between particlesand corresponding antiparticles at the RHIC Beam En-ergy Scan (BES) [13, 15], and the observed v orderingscould be explained by a coalescence picture with differ-ent v values for (anti)quarks produced in pair and for u ( d ) quarks transported from initial-state nuclei towardsmidrapidity ( y ≈
0) [16]. Furthermore, p T -integrated dv /dy values for K ± , p (¯ p ) and Λ(¯Λ) have been used totest the coalescence sum rule for both produced quarksand transported quarks [6], and v slopes have been ex-tracted for produced u ( d , ¯ u and ¯ d ), s and ¯ s quarks, as wellas transported u ( d ) quarks as functions of collision en-ergy [17]. In the following, we shall extend this method-ology to p T -integrated v with statistical uncertaintiesmuch smaller than those of v . Therefore, the coales-cence sum rule will be examined with better precision,and v values will be estimated for constituent quarks inAu+Au collisions at BES. II. COALESCENCE SUM RULE
Figure 1(a) illustrates the beam-energy dependence of p T -integrated v for K ± , p , ¯ p , Λ and ¯Λ in 10-40% Au+Aucollisions, obtained from the corresponding v ( p T ) datapublished by the STAR Collaboration [15]. The lower p T bound is 0.2 GeV/ c for π ± , K ± , p and ¯ p , and 0.4GeV/ c for all other hadrons, such that the majority ofthe particle yields are included, and the tracking effi-ciency correction can be largely ignored. The centralityrange is selected in consideration of nonflow . The non-flow correlations [18] are unrelated to the reaction planeorientation or the initial geometry, and originate fromtransverse momentum conservation, Coulomb and Bose-Einstein correlations, resonance decays, inter- and intra-jet correlations, etc. Although the pseudorapidity ( η )gap of 0.05 between the particles of interest and the eventplane suppresses some short-range nonflow contributionsin the STAR measurements, it is unlikely to eliminatelonger-range correlations due to, e.g., transverse momen-tum conservation and back-to-back jet pairs. Nonflow isless influential in the 10-40% centrality, where v itself islarge, and nonflow is diluted by multiplicity [19].At each beam energy under study, Λ(¯Λ) shows larger p T -integrated v values than p (¯ p ), though they have verysimilar v ( p T ) functions [15]. This is because Λ(¯Λ) hashigher mean p T values than p (¯ p ) [20, 21], and v increaseswith p T for most of the accessible p T range. The v dif-ference between particles ( K + , p and Λ) and their an-tiparticles ( K − , ¯ p and ¯Λ) warrants the effort to sepa-rate transported quarks and produced quarks. The num- v + K - K pp LL (a) Au + Au: 10 - 40% (b)net Knet p L net Center of Mass Collision Energy (GeV)
FIG. 1. (Color online) Elliptic flow ( v ) for K ± , p , ¯ p , Λ and ¯Λ(a), and for net K , net p and net Λ (b), in 10-40% Au+Au col-lisions as function of beam energy, based on STAR data [15].Quoted errors are statistical uncertainties only. Some datapoints are staggered horizontally to improve visibility. ber of transported quarks is conserved, and transportedquarks experience the whole system evolution. In con-trast, the total number of produced quarks is not con-served, and produced quarks are presumably created indifferent stages [22]. In experiments, produced quarkscan be studied with purely “produced” particles, such as K − , ¯ p and ¯Λ, whereas transported quarks can be betterprobed with net particles that represent the excess yieldof a particle species over its antiparticle. For example, v net p = ( v p − r ¯ p/p v ¯ p ) / (1 − r ¯ p/p ) , (5)where r ¯ p/p is the ratio of observed ¯ p to p yield at eachbeam energy. Similar expressions for net K and net Λcan be written by replacing p (¯ p ) with K + ( K − ) and Λ(¯Λ),respectively. v for net particles are shown in Fig. 1(b),and will be discussed later.The test of the coalescence sum rule for producedquarks is straightforward via produced particles. Fig- v )uds ( L )uud (p 31s) + u ( - K (a)
Au + Au: 10 - 40% (b) L net + sp 31net p - net p + s31net p - Center of Mass Collision Energy (GeV)
FIG. 2. (Color online) v versus beam energy forintermediate-centrality (10-40%) Au+Au collisions. Panel (a)compares ¯Λ with the prediction of the coalescence sum rulefor produced quarks. Panel (b) shows two further sum-ruletests, based on comparisons with net Λ. ure 2(a) compares the observed v for ¯Λ(¯ u ¯ d ¯ s ) with thecalculations for K − (¯ us )+ ¯ p (¯ u ¯ u ¯ d ). This comparison onlyinvolves produced quarks, and we assume that ¯ u and ¯ d quarks have the same flow, and that s and ¯ s have thesame flow. A close agreement appears at √ s NN = 14 . s and ¯ s quarksmay by lifted by the associated strangeness production, pp → p Λ(1115) K + [23], whose role has also been mani-fested in other physics observables [6, 24].The test of the collective behavior of u and d quarksis complicated by the extra component of transportedquarks. In the limit of high √ s NN , most u and d areproduced, whereas in the limit of low √ s NN , most ofthem are presumably transported. Transported quarksare more concentrated in net particles than in particles,roughly in proportion to N particle /N net particle [6]. There-fore we employ net-Λ and net- p v in these tests. Fig-ure 2(b) compares net-Λ( uds ) v with calculations in twoscenarios. The first (red diamond markers) consists of net p ( uud ) minus ¯ u plus s , with ¯ u estimated from ¯ p , and s from K − (¯ us ) − ¯ u . Here we assume that a produced u quark in net p is replaced with an s quark. This sum-rulecalculation agrees reasonably well with the net-Λ data at √ s NN = 14 . net p plus s (blue circle markers). In this scenario, weassume that the constituent quarks of net p are domi-nated by transported quarks in the limit of low beam en-ergy, and that one of the transported quarks is replacedwith s . This approximation seems to work well at 7.7and 11.5 GeV, and breaks down as the beam energy in-creases, with disagreement between the black stars andblue circles above 11.5 GeV. Overall, at each beam en-ergy, net-Λ v can always be explained by one of the twocoalescence scenarios. This suggests that the coalescencesum rule is valid for all the beam energies under study,as long as the difference between different quark speciesis taken into account. III. ELLIPTIC FLOW FOR QUARKS
Assuming the coalescence sum rule, we shall furtherextract elliptic flow for constituent quarks. v for pro-duced u ( d, ¯ u, ¯ d ) quarks is approximated with ¯ p (¯ u ¯ u ¯ d ), asdisplayed in Fig. 3(a). With decreased beam energy, col-lectivity of produced u ( d, ¯ u, ¯ d ) quarks becomes weaker,and approaches zero at 7.7 GeV, indicating a graduallack of the QGP state. The collective behaviors of s and¯ s quarks are estimated from K − (¯ us ) − ¯ u and ¯Λ(¯ u ¯ d ¯ s ) − u ,respectively, as depicted in Fig. 3(b). In general, s and¯ s quarks show larger p T -integrated v values than pro- duced u ( d, ¯ u, ¯ d ) quarks, for the same reason why Λ(¯Λ)acquires larger p T -integrated v values than p (¯ p ): theformer bears higher mean p T values. The consistencybetween s and ¯ s holds well at 14.5 GeV and above, anda seeming split appears at 7.7 and 11.5 GeV. The afore-mentioned associated strangeness production effectivelyconverts the excess of p (over ¯ p ) into Λ( uds ) and K + ( u ¯ s ).The mass difference between Λ and K + will be reflectedin the amount of collectivity they inherit from p , andcauses asymmetry between s and ¯ s quarks inside them.Later, Λ and K + are melted in the QGP, and then s and ¯ s quarks participate in the coalescence to form otherstrange hadrons. This intuitive picture not only explainsthe v difference between s and ¯ s , but also expects alarger difference at lower energies, where baryon chemi-cal potential is higher.Elliptic flow for transported quarks can be obtainedwith several approaches: by removing ¯ s and produced u from K + ( u ¯ s ), and by removing produced u ( d ) quarksfrom p or net p . v trans .u ( d )2 = [ v K + − v ¯ s − (1 − f u ) · v ¯ u ] /f u (6)= [ v p / − (1 − f u ( d ) ) · v ¯ u ( ¯ d )2 ] /f u ( d ) (7)= v net p − (3 − N net p trans .u + d ) · v ¯ u ( ¯ d )2 N net p trans .u + d , (8)where f u ( d ) represents the fraction of transported u ( d ) inall u ( d ) quarks, and N net p trans .u + d is the number of trans-ported quarks per net p , N net p trans .u + d = (2 f u N p + f d N p ) / ( N p − N ¯ p )= (2 f u + f d ) / (1 − r ¯ p/p ) . (9)Following Boltzmann statistics, we have f u ( d ) = e µ u ( d ) /T ch − e − µ u ( d ) /T ch e µ u ( d ) /T ch = 1 − e − µ u ( d ) /T ch , (10) v p 31) = d(u + trans. u(d) from Ktrans. u(d) from p (a) Au + Au: 10 - 40% (b)p 31 - - s = K p 32 - L = s Center of Mass Collision Energy (GeV)
FIG. 3. (Color online) v for produced ¯ u ( ¯ d ) and transported u ( d ) quarks (a), and for s and ¯ s quarks (b), in 10-40% Au+Aucollisions as function of beam energy. TABLE I. Chemical freeze-out parameters ( µ B and T ch ) forStrangeness Canonical Ensemble [20], f u ( d ) and N net p trans .u + d in10-40% Au+Au collisions at RHIC BES. Errors in parenthesisare systematic uncertainties. The values at 14.5 GeV areinterpolated using results from other beam energies. √ s NN (GeV) µ B (MeV) T ch (MeV) f u ( d ) N net p trans .u + d where µ u ( d ) is chemical potential for u ( d ) quarks, and T ch is chemical freeze-out temperature. In the currentscope, we always ignore the difference between u and d ,and take µ u ( d ) = µ B /
3, where µ B is baryon chemicalpotential. Therefore, Eq.(9) becomes [25] N net p trans .u + d = 3[1 − e − µ B / (3 T ch ) ] / (1 − r ¯ p/p ) . (11)A similar idea as Eq. (10) maintains that r ¯ p/p is roughly e − µ B /T ch . Hence, in the limit of low √ s NN or high µ B , N net p trans .u + d is close to three, whereas in the limitof high √ s NN or low µ B , N net p trans .u + d approaches unity.These features are confirmed by Tab. I that lists f u ( d ) and N net p trans .u + d as functions of √ s NN in 10 - 40% Au+Aucollisions, based on STAR data of µ B and T ch [20].The v values for transported quarks from K + and p (or net p ) corroborate each other, as exhibited inFig. 3(a). Results from p and net p are in good agree-ment, with different statistical uncertainties owing to ourconservative error propagation. We opt to present thepoints with the smaller error bars in Fig. 3(a). Ref [16] ar-gues that the v orderings between particles and antipar-ticles observed at RHIC BES stem from the v differencebetween transported and produced quarks. Supposedly,transported quarks undergo the entire evolution, receivemore scatterings, and thus attain larger v values. Thisspeculation is supported by our v extraction for con-stituent quarks. Unlike dv /dy for transported quarksthat reveals a clear non-monotonic dependence on beamenergy, evidencing the first-order phase transition [17], v for transported quarks does not demonstrate an appar-ent non-monotonic trend. A possible explanation is that v for transported quarks is mostly imparted at the earlystage after the initial impact, whereas v for transportedquarks is built up gradually by the pressure gradient ofthe QGP. Therefore, v and v have different sensitivitiesto the pertinent physics. v Au + Au: 10 - 40% (a) - p + p R a t i o (b) u2 - 2v + p v u2 - 2v - p v Center of Mass Collision Energy (GeV)
FIG. 4. (Color online) (a) v for π − and π + in 10-40%Au+Au collisions as function of beam energy, based on STARdata [15]. Quoted errors are statistical uncertainties only. (b)The ratio of ( v π − − v ¯ u ) over ( v π + − v ¯ u ), in comparisonwith the ratio between the numbers of constituent d and u quarks in Au, 315 / v values for π − and π + are ar-tificially scaled down by relative 5% in view of the trackinginefficiency. IV. PIONS AND MULTI-STRANGE HADRONS
Pions are the most abundant hadrons produced inheavy-ion collisions under study. π − (¯ ud ) and π + ( u ¯ d ) arealmost symmetric, with their constituent quarks com-ing from similar sources: ¯ u and ¯ d quarks are produced,while u and d could be either produced or transported.Figure 4(a) shows v for π − and π + in 10-40% Au+Aucollisions as function of beam energy, based on STARdata [15]. π − and π + are close to each other at higher col-lision energies, and deviate towards lower energies. Theordering between π − and π + could be explained by thenumbers of constituent u and d quarks inside the initialnuclei [16]: there are 315 d quarks and only 276 u quarksin Au. In consideration of this slight asymmetry, v for pions can be decomposed as v π − = N π − trans .d · v trans .d + (2 − N π − trans .d ) v ¯ u (12) v π + = N π + trans .u · v trans .u + (2 − N π + trans .u ) v ¯ d , (13)where N π − trans .d and N π + trans .u denote the numbers of trans-ported d and u quarks per π − and π + , respectively. Stillassuming v trans .u = v trans .d and v ¯ u = v ¯ d , we reach N π − trans .d N π + trans .u = v π − − v ¯ u v π + − v ¯ u . (14)Figure 4(b) presents the ratio of ( v π − − v ¯ u ) over ( v π + − v ¯ u ), with a reasonable consistency with 315 / p T than other hadrons,and thus are more affected by the lower p T bound of 0.2GeV/ c and by the detector inefficiency, which is typically v )s (s φ p - Λ + - K )/3 + Ω + - Ω ( (a) Au + Au: 10 - 40% (b))dss ( + Ξ p - Λ Center of Mass Collision Energy (GeV)
FIG. 5. (Color online) v versus beam energy in intermediate-centrality (10-40%) Au+Au collisions, for φ mesons (a) andΞ + hyperons (b), in comparison with the corresponding pre-dictions of the coalescence sum rule for produced quarks,based on STAR data [15]. Some data points are staggeredhorizontally to improve visibility. more severe at lower p T . The shaded boxes representthe speculated results where the v values for π − and π + are artificially scaled down by relative 5% in viewof the lower p T bound and the tracking inefficiency. Ingeneral, such a manipulation would bring up the centralvalues of the ratios close to 315 / p T and hence larger v values, so that the daughter pi-ons would inherit larger v than expected by coalescence.Therefore, we refrain from deriving transported-quark v from pion data.The coalescence production of multi-strange hadrons,such as φ mesons and Ξ and Ω hyperons, relies on thelocal strange quark density, and their formation timecan be reflected in the corresponding flow measurements.Figure 5(a) shows that v for φ ( s ¯ s ) is significantly smallerthan the coalescence prediction with K − + ¯Λ − ¯ p , whichcould be attributed to the early freeze-out of φ mesons.At later stages of the system expansion, s and ¯ s mayhave sufficiently scattered inside the QGP to acquire flow,but they have a small chance to meet again or meetanother ¯ s and s to coalesce into φ , owing to the lowerstrangeness abundance at lower collision energies. There-fore, φ mesons are not expected to flow as much as the co-alescence prediction using hadrons with later formationtimes. A previous STAR publication [26] has demon- strated that the productions of Ω ± and φ are closelyrelated through coalescence at BES energies. Indeed,Fig. 5(a) suggests that v values for φ can be reproducedby the prediction using (Ω − +Ω + ) /
3, especially when thestatistical uncertainties are smaller, implying similar for-mation times for φ and Ω ± . The penalty due to the lowstrangeness abundance also applies to Ξ + ( ¯ d ¯ s ¯ s ), when itis compared with 2¯Λ − ¯ p in Fig. 5(b). However, as anopen-strange hadron, Ξ + could be formed later than Ω + (and φ ), and flow closer to the coalescence prediction. V. SUMMARY
The coalescence mechanism is important for particleproduction in high-energy heavy-ion collisions, and wehave extensively tested the coalescence sum rule in the el-liptic flow measurements using published STAR data [15].The tests involving ¯Λ and net Λ support the coales-cence sum rule at all the beam energies under study, aslong as different quark species are separated. Follow-ing the idea of differentiating produced and transportedquarks [6, 16, 17], we have extracted elliptic flow for pro-duced u ( d , ¯ u , ¯ d ), s and ¯ s quarks, as well as transported u ( d ) quarks in 10-40% Au+Au collisions at √ s NN = 7.7,11.5, 14.5, 19.6, 27, 39 and 62.4 GeV. We have confirmedthe speculation that transported u ( d ) quarks bear larger v values than produced ones [16]. A possible break-down of degeneracy between s and ¯ s hints at the sig-nificant role of the associated strangeness production atlower collision energies. Even though π − and π + are al-most symmetric in production and flow, we have relatedtheir v difference to the different numbers of constituent u and d quarks in a gold ion. The v measurements ofmulti-strange hadrons indicate early formation times for φ , Ω ± and Ξ + . The high-statistics data from the RHICBES-II program are anticipated to improve the precisionfor the analyses in this article. ACKNOWLEDGEMENTS
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