Elliptic flow of hadrons in equal-velocity quark combination mechanism in relativistic heavy-ion collisions
aa r X i v : . [ nu c l - t h ] A ug Revealing the elliptic flow of quarks in relativistic heavy-ion collisions
Jun Song, Hai-hong Li, and Feng-lan Shao ∗ Department of Physics, Jining University, Shandong 273155, China School of Physics and Physical Engineering, Qufu Normal University, Shandong 273165, China
We apply a quark combination model with equal-velocity combination (EVC) approximationto study the elliptic flow ( v ) of hadrons in heavy-ion collisions in a wide collision energy range( √ s NN =
27 - 5020 GeV). Taking advantage of the simple relationship between v of hadrons andthose of quarks and antiquarks under EVC, we extract v of up/down quarks by the experimentaldata of proton and find it is consistent with that obtained by the data of Λ and Ξ . We extract v of strange quarks by the data of Ω and find it is consistent with that obtained by the data of Λ and Ξ , and at RHIC energies we find it is also consistent with that obtained by the data of φ . Thismeans that v of these light-flavor hadrons have a common quark-level source at hadronization.We further extract v of charm quarks by the data of D in Pb+Pb collisions at √ s NN = √ s NN =
200 GeV. We find that charm quark dominates v of D mesons at low p T but light-flavor quarks significantly contribute to v of D mesons in theintermediate range . p T . GeV/c. We predict v of charmed baryons Λ + c and Ξ c which showa significant enhancement at intermediate p T due to the double contribution of light-flavor quarks.The properties of the obtained quark v at hadronization are studied and a new regularity for v ofquarks as the function of p T /m is found. I. INTRODUCTION
In non-central heavy-ion collisions, momentum distri-butions of the produced hadrons are anisotropic in thetransverse plane perpendicular to the beam direction [1].The elliptic flow ( v ) is the second harmonic coefficient ofFourier expansion for the azimuthal distribution of par-ticle transverse momentum and denotes the asymmetrybetween x and y components of particle transverse mo-mentum. v of hadron carries important information onthe degree of initial thermalization, the pressure gradi-ents, the equation of state, and the hadronization for thecreated quark matter [1–5].In non-central heavy-ion collisions at RHIC and LHCenergies, the data of v for light-flavor hadrons as thefunction of transverse momentum ( p T ) exhibit a number-of-constituent quark scaling (NCQ) property [5–9]. As v and p T of identified hadrons are divided by the num-ber of constituent quarks (2 for meson, 3 for baryon),the scaled data of different hadrons approximately fol-low a common tendency. Replacing p T by the transversekinetic energy KET = p p T + m − m in the horizon-tal axis, NCQ looks better. Such a NCQ is expectedin quark recombination/coalescence model [10, 11]. Thepreliminary data of D in Au+Au collisions at √ s NN =
200 GeV [12] seemingly also follow this NCQ.In our recent works, we found a constituent quarknumber scaling property exhibited in the p T spectra ofidentified hadrons in high multiplicity pp , p -Pb and AAcollisions [13–15]. This property is the direct conse-quence of the equal-velocity combination (EVC) of con-stituent quarks and antiquarks at hadronization. We fur-ther demonstrated that EVC can self-consistently explain ∗ shaofl@mail.sdu.edu.cn the data of p T spectra for different identified hadrons inhigh multiplicity pp and p -Pb collisions at LHC energies[14, 16–18] and in heavy-ion collisions in a wide colli-sion energy range [19]. Taking advantage of rich datafor hadronic v in heavy-ion collisions, in particular, atRHIC BES energies [6–8, 20, 21], it is interesting to fur-ther study whether this EVC mechanism works for v ofhadrons or not.Because of different constituent mass such as m d ≈ m u ≈ . GeV, m s ≈ . GeV and m c = 1 . GeV, quarksof different flavors under EVC will contribute differentmomenta (proportional to quark mass) to the formedhadron. Therefore, p T / for meson and p T / for baryonin the aforementioned NCQ operation on v data of light-flavor hadrons can not exactly denote the transverse mo-mentum of each quark in the hadron containing differentquark flavors. Instead, we should divide the momentumof hadron into different pieces so as to obtain the actualmomentum of the quark at hadronization and the actualrelationship between quark transverse momentum and its v . Therefore, applying the EVC to study hadronic v data will bring new results for v of quarks at hadroniza-tion.In this paper, we apply the quark combination model(QCM) with EVC to systematically study the v of iden-tified hadrons and extract the v of quarks at hadroniza-tion using the available data of identified hadrons inheavy-ion collisions in a wide collision energy range( √ s NN = v of strange quarks from that of up/downquarks. We extract them separately from the data of dif-ferent hadrons and study the consistency of results fromdifferent extraction channels as the test of EVC mecha-nism. With the obtained v of light-flavor quarks, we fur-ther obtain the v of charm quarks from the data of D .We study the dominant ingredient for v of D mesons atdifferent p T and predict v of charmed baryons Λ + c and Ξ c . Finally, we discuss the properties for the extracted v of up/down, strange and charm quarks.The paper is organized as follows. In Sec. II, we derivethe v of hadrons in QCM with EVC. In Sec. III and Sec.IV, we apply formulas of EVC to decompose the data ofhadron v at RHIC and LHC energies into v of up/downquarks and that of strange quarks at hadronization. Westudy the consistency for results obtained from differentextraction channels. In Sec. V, we study the v of charmquarks extracted from D data at LHC and RHIC ener-gies. In Sec. VI, we study properties of the extracted v of quarks. The summary and outlook are given at last. II. HADRONIC v IN QCM WITH EVC
In this section, we apply a quark combination modelwith equal-velocity combination (EVC) approximation[13, 16] to study the production of identified hadronsin the two-dimension transverse plane at mid-rapidity.Here, we define the distribution function f h ( p T , ϕ ) ≡ dN h /dp T dϕ where ϕ is the azimuthal angle. The dis-tribution function of hadron under EVC is simply theproduct of those of quarks and/or antiquarks, i.e., f M i ( p T , ϕ ) = κ M i f q ( x p T , ϕ ) f ¯ q ( x p T , ϕ ) , (1) f B i ( p T , ϕ ) = κ B i f q ( x p T , ϕ ) f q ( x p T , ϕ ) f q ( x p T , ϕ ) . (2)Under EVC, the quark and/or antiquark have the samedirection ( ϕ ) as the hadron and take a given frac-tion of p T of the hadron. Because of p i = m i γv ∝ m i at equal velocity, the momentum fraction x , = m , / ( m + m ) for meson with x + x = 1 , and x , , = m , , / ( m + m + m ) for baryon with x + x + x = 1 . m i is the constituent mass of quark q i . κ M i and κ B i are coefficients independent of p T and ϕ but can be depen-dent on the hadron species and system size. Their ex-pressions can be found in [16] and are not shown heresince κ M i and κ B i are irrelevant to the derivation of v .The quark distribution function can be written in thefollowing form f q ( p T , ϕ ) = f q ( p T ) " ∞ X n =1 v n,q ( p T ) cos nϕ , (3)where we denote f q ( p T ) ≡ dN q /dp T as the ϕ -independent p T distribution function. The ϕ dependencepart is expressed as usual in terms of the Fourier seriesand the harmonic coefficient is defined as v n,q ( p T ) = R f q ( p T , ϕ ) cos nϕ dϕ R f q ( p T , ϕ ) dϕ . (4)In this paper, we study the second harmonic coefficient v of hadrons. Using Eqs. (1)-(3), we obtain for meson M i ( q ¯ q ) v ,M i ( p T ) = R dϕ cos 2 ϕf M i ( p T , ϕ ) R dϕf M i ( p T , ϕ )= 1 N M i h v ,q + v , ¯ q + ∞ X n,m =1 v n,q v m, ¯ q ( δ ,m + n + δ n, m + δ m, n ) i (5)with N M i = 1 + 2 ∞ X n =1 v n,q v n, ¯ q . (6)Here, we use the abbreviation v ,q for v ,q ( x p T ) and v , ¯ q for v , ¯ q ( x p T ) . For baryon B i ( q q q ) , we have v ,B i ( p T ) = 1 N B i ( v ,q + v ,q + v ,q + ∞ X n,m =1 ( v n,q v m,q + v n,q v m,q + v n,q v m,q ) ( δ m,n +2 + δ n,m +2 + δ m + n, )+ ∞ X n,m,k =1 v n,q v m,q v k,q ( δ k,m + n +2 + δ k,m + n − + δ n,k + m − + δ n,k + m +2 + δ m,k + n − + δ m,k + n +2 ) ) (7)with N B i = 1 + 2 ∞ X n =1 ( v n,q v n,q + v n,q v n,q + v n,q v n,q ) + 2 ∞ X n,m,k =1 v n,q v m,q v k,q ( δ k,m + n + δ n,k + m + δ m,k + n ) , (8)where we use the abbreviation v ,q j for v ,q j ( x j p T ) with j = 1 , , .Since the data of hadronic v at mid-rapidity [22, 23] are only about v ,h . − , v of quarks should be verysmall and therefore can be safely neglected. In addition,according to NCQ estimation of the 2-4th flow of quarks v / / ,q ∼ − [5–9], we can neglect the influence of highorder terms ( v n,q ) , in N M i and N B i , and obtain v ,M i ( p T ) ≈ v ,q ∞ X n =2 v n,q v ,q v n +2 , ¯ q ! + v , ¯ q ∞ X n =2 v n, ¯ q v , ¯ q v n +2 ,q ! , (9)and v ,B i ( p T ) ≈ v ,q " ∞ X n =2 v n,q v ,q ( v n +2 ,q + v n +2 ,q ) + v ,q " ∞ X n =2 v n,q v ,q ( v n +2 ,q + v n +2 ,q ) + v ,q " ∞ X n =2 v n,q v ,q ( v n +2 ,q + v n +2 ,q ) . (10)Here we split the v of meson into two parts and that ofbaryon into three parts. Each part is v of constituentquark multiplying a term containing the small correctionfrom higher-order harmonic flows.The magnitude of the correction is a few percentagesbecause of v ,q ∼ − as mentioned above [5–9]. Higher-order harmonic flows are often unavailable at present andtheir influence is usually expected to be not larger thanthese lower-order harmonic flows. In addition, the differ-ence among v n,q of different quark flavors is usually much(about one order of magnitude) lower than absolute valueof v n,q . Therefore, the relative difference among terms inbrackets in Eq. (9) and that among these in Eq. (10)should be very small ( . − ). Eqs.(9) and (10) can bethus expressed approximately as the simplest form v ,M i ( p T ) = v ,q ( x p T ) + v , ¯ q ( x p T ) , (11) v ,B i ( p T ) = v ,q ( x p T ) + v ,q ( x p T ) + v ,q ( x p T ) . (12) III. QUARK v AT RHIC
In this section, we apply the EVC model to study thedata of hadronic v in heavy-ion collisions at RHIC en-ergies. Here, we focus on proton, Λ , Ξ , Ω , and φ . Thesehadrons can be properly explained by constituent quarkmodel with constituent masses m u = m d ≈ . − . GeV and m s ≈ . − . GeV. Therefore, their pro-duction can be suitably described by EVC of constituent quarks at hadronization. However, pion and kaon, be-cause of their significantly small masses, can not be di-rectly described in the same way. We discuss their pro-duction in Appendix A.Using Eqs. (11) and (12), we obtain v , Ω ( p T ) = 3 v ,s ( p T / , (13) v ,p ( p T ) = 3 v ,u ( p T / , (14) v ,φ ( p T ) = v ,s ( p T /
2) + v , ¯ s ( p T / , (15) v , Λ ( p T ) = 2 v ,u (cid:18)
12 + r p T (cid:19) + v ,s (cid:18) r r p T (cid:19) , (16) v , Ξ ( p T ) = v ,u (cid:18)
11 + 2 r p T (cid:19) + 2 v ,s (cid:18) r r p T (cid:19) (17)with the factor r = m s /m u =1.667. Here, we use v ,u = v ,d .We can reversely obtain v of u quarks from proton orthat from hyperons v ,u ( p T ) = 13 v ,p (3 p T ) , (18) v ,u ( p T ) = 13 [2 v , Λ ((2 + r ) p T ) − v , Ξ ((1 + 2 r ) p T )] . (19)We can obtain v of s quarks from φ or hyperons v ,s ( p T ) = 13 v , Ω (3 p T ) , (20) v ,s ( p T ) = 12 v ,φ (2 p T ) , (21) v ,s ( p T ) = 13 (cid:20) v , Ξ (cid:18) rr p T (cid:19) − v , Λ (cid:18) rr p T (cid:19)(cid:21) . (22)Here, we use v ,s = v , ¯ s in Eq. (21). (GeV/c) T p p Λ ΞΩ φ (GeV/c) T p v Ξ , Λ by v φ by v Ξ , Λ by v Ω by v v Figure 1. (a) Data for v of identified hadrons at midrapidityin Au+Au collisions at √ s NN =
200 GeV for 30-80% centrality[6]; (b) v ,u ( p T ) and v ,s ( p T ) extracted from these hadrons. As an example, we first test Eqs. (18-22) by exper-imental data of proton, Λ ( Λ + ¯Λ) , Ξ ( Ξ − + ¯Ξ + ) , Ω ( Ω − + ¯Ω + ), and φ in Au+Au collisions at √ s NN = q Au 62.4GeV 10-40% by p v Ξ , Λ by v φ by v Ω by v Ξ , Λ by v Au 62.4GeV 10-40%q p by u2, v Ξ , Λ by u2, v Ω by s2, v Ξ , Λ by s2, v 0.5 1 1.500.05 q Au 62.4GeV 0-80% Au 62.4GeV 0-80%q q Au 39GeV 10-40%
Au 39GeV 10-40%q q Au 39GeV 0-80%
Au 39GeV 0-80%q q Au 27GeV 10-40%
Au 27GeV 10-40%q q Au 27GeV 0-80%
Au 27GeV 0-80%q (GeV/c) T p (GeV/c) T p (GeV/c) T p (GeV/c) T p v v v Figure 2. v ,u ( p T ) and v ,s ( p T ) extracted from data of identified hadrons in Au+Au collisions in different centralities and atdifferent collision energies [20]. causes little influence on v of it’s daughter baryon due todecay kinematics, we neglect the decay influence and di-rectly apply Eqs. (18-22) to these experimental data. Wealso neglect the possible rescattering effect in hadronicstage for the moment until we find its necessity in follow-ing analysis such as in φ study at LHC energies in nextsection IV.Panel (a) in Fig. 1 shows the original data for v ofthese hadrons which are different for different hadronspecies. Panel (b) shows the extracted v ,u ( p T ) and v ,s ( p T ) according to Eqs. (18-22). We see that v ,s ( p T ) extracted from Ω using Eq. (20) is very close to that from φ using Eq. (21) and is also very close to that from Λ and Ξ data using Eq. (22). For u quarks, we see that v ,u ( p T ) extracted from proton data using Eq. (18) is very close tothat from Λ and Ξ data using Eq.(19). Therefore, v dataof these hadrons can be reasonably attributed to a com-mon v ,u ( p T ) and a common v ,s ( p T ) at hadronizationwithin the experimental uncertainties. In addition, wesee that the extracted v ,u ( p T ) is obviously larger thanthe extracted v ,s ( p T ) in the available range p T . . GeV/c, suggesting a flavor hierarchy property at quarklevel.Furthermore, in Fig. 2, we carry out a systematic anal-ysis for STAR BES data in Au+Au collisions at √ s NN = p T range and haverelatively poor statistics, their results are not shown inthis paper. Here, data of hadrons and those of anti-hadrons are separately analyzed to obtain v ,u ( p T ) and v , ¯ u ( p T ) . In the figure, v ,s ( p T ) obtained by φ usingEq. (21) is compared with those obtained from baryonsand also with those from anti-baryons. At these threecollision energies each with two centralities, we see that v ,u ( p T ) obtained from p and that from Λ and Ξ are con-sistent with each other. The same case is for v , ¯ u ( p T ) .Compared with v ,u ( p T ) and v , ¯ u ( p T ) data, v ,s ( p T ) ob-tained from different strange hadrons are limited by finitestatistics but are also close to each other. IV. QUARK v AT LHC AND φ SPECIFICITY
We further study the property of v under EVC byexperimental data in Pb+Pb collisions at LHC energies.In Fig. 3, we present v ,u ( p T ) and v ,s ( p T ) extractedfrom midrapidity data of proton, Λ , Ξ and Ω in Pb+Pbcollisions at √ s NN = v ,u ( p T ) extracted from proton and that from Λ and Ξ are very consistent. v ,s ( p T ) extracted from Ω v Ξ , Λ by v Ω by v Ξ , Λ by v Pb+Pb 5.02 TeV 10-20%
Pb+Pb 5.02 TeV 30-40%
Pb+Pb 2.76 TeV 10-20%
Pb+Pb 2.76 TeV 30-40% (GeV/c) T p (GeV/c) T p v v (a) (b)(c) (d) Figure 3. v ,u ( p T ) and v ,s ( p T ) extracted from experimentaldata for v of identified hadrons at midrapidity in Pb+Pbcollisions at LHC energies [8, 21]. and that from Λ and Ξ are also very close to each other.We further consider the φ data [7, 20, 21] to extract v ,s ( p T ) and compare with those obtained from strangebaryons. The results are shown in top panels in Fig.4. Surprisingly, even though statistical uncertainties arerelatively large, we see that the v ,s ( p T ) extracted from φ seems to be higher than those from strange baryons toa certain extent as p T,s & GeV/c. This is different fromthe case at RHIC energies in Figs. 1 and 2.Results in top panels in Fig. 4 imply that φ may receivethe contribution of other production channels in heavy-ion collisions at LHC energies. Here, we consider a possi-ble contribution, that is, two-kaon coalescence KK → φ in hadron scattering stage [24–26]. In this case, the dis-tribution of final-state φ has two contributions f ( final ) φ ( p T , ϕ ) = f φ,s ¯ s ( p T , ϕ ) + f φ,KK ( p T , ϕ ) . (23)The elliptic flow is v ( final )2 ,φ ( p T ) = (1 − z ) v ,s ¯ s ( p T ) + z v ,KK ( p T ) (24)with the fraction z = f φ,KK ( p T ) f φ,s ¯ s ( p T ) + f φ,KK ( p T ) . (25)Using the relation Eq. (15) for s ¯ s combination and thesimilar one for the coalescence of two kaons also withequal velocity (because the mass m φ ≈ m K ), the el-liptic flow of φ after considering the possible two-kaoncoalescence has v ( final )2 ,φ ( p T ) ≈ h (1 − z ) v ,s (cid:16) p T (cid:17) + z v ,K (cid:16) p T (cid:17)i . (26) Here, we have taken v , ¯ s = v ,s at LHC energies.In bottom panels in Fig. 4, we calculate the ellipticflow of φ and compare with experimental data [7, 20, 21].We firstly calculate the elliptic flow of pure s ¯ s combi-nation. The results, marked as s ¯ s coal, are shown asthe dashed lines in bottom panels in Fig. 4. The ac-tual v ,s ( p T ) at hadronization is identified as that by fit-ting data of strange baryons, see the dashed lines in toppanels in Fig. 4. We see that pure s ¯ s combination candescribe φ data in low p T range ( p T . . GeV/c) butunder-estimates the v of φ at intermediate p T ( p T & . GeV/c). We then consider the contribution of two-kaoncoalescence in a simple case that a p T -independent z istaken. Using data of elliptic flow for kaons [7, 8], wecalculate the elliptic flow of final-state φ by Eq. (26)and compare with the data. We find that data of φ at p T & . GeV/c at two LHC energies can be roughly de-scribed by z = 0 . , see solid lines in bottom panels inFig. 4. This implies that there is about 20% of φ with p T & . GeV/c coming from two-kaon coalescence inthe hadronic stage. We note that, compared with pure s ¯ s combination, two-kaon coalescence does not signifi-cantly increase elliptic flow of φ in low p T range ( p T . GeV/c) because here v of participant kaons is small as p T,K = p T / . GeV/c. Therefore, we emphasize that v data of φ in the low p T range do not necessarily containthe contribution of two-kaon coalescence. In addition, wenote that two-kaon coalescence will influence slightly the p T distribution function of φ and thus will slightly influ-ence the quark number scaling property for p T spectra of Ω and φ [13–15]. This influence is discussed in AppendixB. V. CHARM QUARK v FROM D MESONS
The EVC can be applied not only to light-flavor quarksbut also to heavy-flavor quarks at hadronization [27]. In[17, 18, 28], we show the EVC of charm quarks and softlight-flavor quarks provides good description for p T spec-tra of single-charm hadrons in high energy collisions. Ap-plying EVC to elliptic flows of D , ± mesons, we obtain v ,D ( p T ) = v ,u (cid:18)
11 + r cu p T (cid:19) + v ,c (cid:18) r cu r cu p T (cid:19) (27)with r cu = m c /m u = 5 . Since we have obtained v ,u inthe previous sections, v of charm quarks can be reverselyextracted by the data of D mesons, v ,c ( p T ) = v ,D (cid:18) r cu r cu p T (cid:19) − v ,u (cid:18) r cu p T (cid:19) . (28)In Fig. 5 (a), we show the extracted v ,c ( p T ) in Pb+Pbcollisions at √ s NN = v ,c ( p T ) extractedfrom latest preliminary and previously published data of D [29, 30], respectively. Data of D are also presented inthe figure as open circles and squares, respectively. The φ by v Ω by v Ξ , Λ by v polynomial fit Pb+Pb 5.02 TeV 10-20%
Pb+Pb 5.02 TeV 30-40%
Pb+Pb 2.76 TeV 10-20%
Pb+Pb 2.76 TeV 30-40% data φ coalss + 20% KK coals80% sPb+Pb 5.02 TeV 10-20% Pb+Pb 5.02 TeV 30-40%
Pb+Pb 2.76 TeV 10-20%
Pb+Pb 2.76 TeV 30-40% (GeV/c) T p (GeV/c) T p (GeV/c) T p (GeV/c) T p v v (a) (b) (c) (d)(e) (f) (g) (h) Figure 4. Top panels: v ,s ( p T ) extracted from data of strange hadrons [7, 20, 21].The dashed lines are polynomial fits of v ,s ( p T ) from strange baryons. Bottom panels: The v of φ . Symbols are data of φ [7, 20, 21]. The dashed lines, markedas s ¯ s coal, are results of strange quark antiquark combination. The solid lines are results considering the mixture of quarkcombination and two-kaon coalescence. See text for details. contribution of up quarks to the v of D at different p T is shown as the dashed line. Under EVC, v of D at aspecific p T absorbs the contribution of u quark at a muchsmaller momentum p T / (1 + r cu ) . Therefore, v of D inthe low p T range ( p T . GeV/c) only receives the smallcontribution of u quark with p T,u . . GeV/c, see Fig.3. However, v of D at p T & GeV/c contains the largecontribution of u quark with p T,u & . GeV/c which isabout 0.1 reading from Fig. 3. Subtracting the u quarkcontribution from D , v of charm quarks is obtained asthe solid circles and squares in Fig. 5 (a). A smooth fitof these discrete points of charm quarks is shown as thesolid line. We see that the v of charm quarks is closeto that of D as p T . GeV/c and is obviously smallerthan the latter as p T & v ,c , v ,u and v ,s are obtained, we can predict v of D + s , Λ + c and Ξ c , v ,D s ( p T ) = v ,s (cid:18)
11 + r cs p T (cid:19) + v ,c (cid:18) r cs r cs p T (cid:19) , (29) v , Λ c ( p T ) = 2 v ,u (cid:18)
12 + r cu p T (cid:19) + v ,c (cid:18) r cu r cu p T (cid:19) , (30) v , Ξ c ( p T ) = v ,u (cid:18)
11 + r + r cu p T (cid:19) + v ,s (cid:18) r r + r cu p T (cid:19) , + v ,c (cid:18) r cu r + r cu p T (cid:19) (31)with r cs = m c /m s = 3 . Here, we neglect the statisti-cal uncertainties of the extracted datum points for v of exp data D )) cu /(1+r T = p T,u (p vc quark QCMfit of c quark Pb+Pb 5.02 TeV 30-50% exp data +s D QCM +s D QCM +c Λ exp data D QCM Ξ QCM +s D (GeV/c) T p (GeV/c) T p(a) (b)(c) (d) v v Figure 5. (a) Charm quark v extracted from D meson inPb+Pb collisions at √ s NN = D + s , Λ + c and Ξ c . Data of D and D + s are taken from Ref. [29–31]. quarks and use their smooth fits, i.e., the solid line for v ,c ( p T ) in Fig. 5 (a) and the dashed line for v ,s ( p T ) in Fig 4(b), to calculate v of D + s , Λ + c and Ξ c by Eqs.(29)-(31). Results are shown in Fig. 5 (b)-(d) as differenttypes of lines.Result of D + s is compared with the preliminary dataof ALICE collaboration [31]. Results of Λ + c and Ξ c arecompared with those of D and D + s . We see that v ofcharmed baryons are close to those of charmed mesonsas p T . p T & v of charmed baryons com-pared with those of charmed mesons. This is becausesingle-charm baryons absorb the v of two light-flavorquarks at hadronization, see Eqs. (30) and (31), and thecontribution of light-flavor quarks becomes large as p T & v of charm quarks inAu+Au collisions at √ s NN =
200 GeV for 0-80% cen-trality. The result is similar to that in Pb+Pb collisionsat √ s NN = p T . GeV/c,charm quark dominates the v of D meson while at in-termediate p T the u quark contributes significantly tothe v of D meson. Using the smooth fit of discretepoints of charm quark v in panel (a) and those of light-flavor quarks in the corresponding centrality extractedfrom data of light-flavor hadrons [6], we predict in panel(b) the v of D + s meson, Λ + c and Ξ c baryons. We seethat in low p T range v of D + s is close to those of Λ + c and Ξ c and at intermediate p T it is smaller than the latter.We also see that v of Λ + c is slightly smaller than that of Ξ c in the range . p T . GeV/c, which is the kineticeffect caused by the mass difference of u and s quarks incombination with charm quark. exp data D )) cu /(1+r T = p T,u (p vc quark QCM fit of c quark QCM +c Λ QCM Ξ QCM +s D AuAu 200 GeV 0-80% (GeV/c) T p (GeV/c) T p(a) (b) v Figure 6. Panel (a) Charm quark v extracted from midra-pidity data of D in Au+Au collisions at √ s NN =
200 GeVfor 0-80% centrality [12], and (b) predictions for D + s , Λ + c and Ξ c in QCM. VI. PROPERTIES FOR v OF QUARK
In this section, we study the property for v of up,strange and charm quarks obtained in previous sections.We focus on two main properties, i.e., flavor dependenceand the quark-antiquark split, which are discussed as fol-lows. A. compare v of u , s and c quarks As an example, in Fig. 7 (a), we present v ofup, strange and charm quarks in Pb+Pb collisions at √ s NN = v of up and strange quarks increase with p T as p T . p T . In par-ticular, we see that v of up quarks is higher than thatof strange quarks in the range p T . p T by p p T + m − m in the horizontal axis, thesplit between up and strange quarks becomes small butdoes not disappear. v of charm quarks at small p T . v continues to increase and reachesthe peak value about 0.13 at p T ≈ . GeV/c, which isobviously higher than those of light-flavor quarks (about0.09) at p T ≈ . GeV/c. usc b=9fm =400/fm AZHYDRO sm=0.3 GeVm=0.5 GeVm=1.5 GeV (GeV/c) T p /m T p(a) (b) v Pb+Pb 5.02 TeV 30-50%
Figure 7. v of up, strange and charm quarks in Pb+Pb colli-sions at √ s NN = p T (a) and of p T /m (b). Lines are results of AZHYDROcode with initial entropy density s = 400 /fm , impact pa-rameter b = 9 fm, and hadronization temperature T = 0 . GeV.
For the increase of v at small p T for up and strangequarks, we can qualitatively understand it in generalby the hydrodynamic evolution of quark gluon plasma(QGP) [32]. The v for charm quarks is the result of thediffusion in QGP [33], which is related to many evolutiondynamics such as the large collective flow, quench effectsof the background medium and possible thermalizationof charm quarks [33–39]. To explain quark v at small p T , we run AZHYDRO code [32] for 2+1-dimensionalhydrodynamic and show results as lines in Fig. 7. Theinitial entropy density is set to be s = 400 /f m , im-pact parameter b = 9 fm, and hadronization temperature T = 0 . GeV. The inelastic pp cross section is set to be70 mb. We see AZHYDRO simulations, the dashed anddotted lines in Fig. 7(a), can well fit the extracted v ofup and strange quarks as p T . v of charmquarks in the case of thermal equilibrium is also shownas the dot-dashed line. It is below the extracted charm v at small p T and intersects the latter at p T ≈ p T /m = γv may be an alternative kineticvariable to reveal the regularity for v of three quarkflavors. In panel (b), we show quark v as the functionof p T /m . Here, we see a clear property relating to quarkmass: as p T /m . the quark with heavier mass haslarger v while the reverse behavior appears as p T /m & .We observe the same property in Au+Au collisions at √ s NN = 200 GeV for 0-80% centrality. This regularityof quark v is interesting and is worthwhile to be studiedin the future work. B. v split between quark and antiquark STAR experiments observed the elliptic flow split be-tween hadrons and their anti-hadrons at low collision en-ergies [20]. In this paper, as shown in Fig. 2, we canapply the EVC model to successfully decompose v ofhadrons and their anti-hadrons into v of quarks and an-tiquarks. Therefore, we can extract the v split betweenquark and antiquark. Here, we take the data in Au+Aucollisions at √ s NN = 39 GeV for 10-40% centrality as anexample.In Fig. 8 (a), we first present experimental data for thedifference in v between baryons and their anti-hadrons.We see that v ,p − v , ¯ p show clearly positive values. v , Λ − v , ¯Λ is also positive and is slightly smaller than v ,p − v , ¯ p . Data of v , Ξ − v , ¯Ξ have relatively poor statisticsand still show a positive tendency and smaller magnitudecompared to v ,p − v , ¯ p and v , Λ − v , ¯Λ . Data of Ω − − ¯Ω + are not shown here because of bad statistics.Using v of quarks and that of antiquarks obtainedin Fig. 2, we calculate v ,u − v , ¯ u and show results inFig. 8 (b). We obtain a good agreement between resultsobtained from p − ¯ p via Eq. (18) and those from Λ − ¯Λ and Ξ − ¯Ξ via Eq. (19). We see clearly the positive v ,u − v , ¯ u with weak p T dependence in the low p T range ( p T < GeV/c).Results of v ,s − v , ¯ s from Λ − ¯Λ and Ξ − ¯Ξ channelare shown in Fig. 8 (c). Because of large statisticaluncertainties, the p T dependence of v ,s − v , ¯ s is not con-clusive. For the overall sign of v ,s − v , ¯ s , we can roughlyestimate by averaging seven datum points and obtain . ± . , which might imply the equal v for s and ¯ s in Au+Au collisions at √ s NN = 39 GeV. Resultsof v ,s − v , ¯ s at lower collision energies have poorer statis-tics and therefore we cannot draw further conclusion atpresent. VII. SUMMARY AND OUTLOOK
We applied a quark combination model (QCM) withequal-velocity combination (EVC) approximation to study the elliptic flow ( v ) of identified hadrons in rel-ativistic heavy-ion collisions at √ s NN = ∼ v of hadron consisting of different con-stituent quarks is therefore the sum of v of quarks withdifferent transverse momenta. This is different from thenumber-of-constituent quark scaling (NCQ) operation inexperimental study of hadronic v [5–9].Under EVC, we obtained simple formulas of reverselyextracting v of quarks and antiquarks from the exper-imental data of identified hadrons. By the combinationof data for Λ and Ξ , we obtained v of up/down quarkswhich is consistent with that from data of proton; wealso obtained v of strange quarks which is consistentwith that from data of Ω . At RHIC energies, v ofstrange quarks extracted from hyperons is also consis-tent with that from φ meson. This means that v of theselight-flavor hadrons have a common quark-level source athadronization. At LHC energies, however, v of strangequarks extracted from hyperons is somewhat lower thanthat from φ . This indicates the possible contribution oftwo-kaon coalescence to φ production at LHC energies.Using results for v of light flavor quarks, we furtherextracted v of charm quarks from the data of D mesonin Pb+Pb collisions at √ s NN = v of charmquark and that of D meson, we found that v of D meson at low p T ( p T . p T ( . p T . GeV/c)is significantly contributed by v of light-flavor quarks.We predicted v of D + s meson, Λ + c and Ξ c baryons. Wefound that v of charmed baryons is significantly en-hanced at intermediate p T ( . p T . GeV/c), com-pared to those of D mesons, which is due to the doublecontribution of light-flavor quarks.We finally studied the properties of the extracted v ofquarks and antiquarks at hadronization. We first com-pared v of up, strange and charm quarks. We found that v of up quarks is always higher than that of strangequarks at low p T ( p T . v can bereasonably understood by hydrodynamics. v of charmquark at small p T ( p T . . GeV/c) is roughly con-sistent with those of light-flavor quarks within the sta-tistical uncertainty. However, differing from light-flavorquarks, v of charm quarks continues to increase with p T and reaches larger value at p T ≈ . GeV/c. Interest-ingly, by plotting quark v as the function of transversevelocity p T /m , we found a regularity relating to quarkmass, i.e., as p T /m . the quark with heavier masshas larger v while as p T /m & reverse property holds.We further studied the difference in v between quarksand antiquarks at low RHIC energies. We found that v ,u − v , ¯ u extracted from hyperons and anti-hyperonscoincides with that from proton and anti-proton. Resultsof v ,s − v , ¯ s have large statistical uncertainties, and theaverage value of all datum points implies the symmetry − − Λ - Λ Ξ - Ξ − − Ξ - Ξ , Λ - Λ via uu- 0 0.5 10.01 − − Ξ - Ξ , Λ - Λ via ss- (GeV/c) T p (GeV/c) T p (GeV/c) T p v v v (a) (b) (c) Figure 8. Panel(a): The difference in v between hadrons and their anti-hadrons in Au+Au collisions at √ s NN = 39 GeV for10-40% centrality. Data are from [20]; panel (b) the difference in v between u and ¯ u ; panel (c) that between s and ¯ s . in v between strange quarks and strange antiquarks inAu+Au collisions at 39 GeV.These results suggest that QCM with EVC is quite ef-fective and self-consistent in understanding v of hadronsin heavy-ion collisions. In addition, this provides a con-venient method to extract v of quarks and antiquarksat hadronization, by which we can obtain deeper insightsinto the information of partonic stage evolution in rela-tivistic heavy-ion collisions. Studies along this directionare deserved with the help of more precise experimental data in the future. VIII. ACKNOWLEDGMENTS
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Appendix A: v of pion and kaon Because the masses of pion and kaon are small, theproduction of pion and kaon is not suitably describedby the direct combination of constituent quarks and an-tiquarks. To reconcile the mass mismatch, we adopt anaive treatment [16] which provides the good descriptionfor the p T distribution of pions and that of kaons. Weconsider the processes such as u + ¯ d → π + X for pionproduction and u + ¯ s → K + X for kaon production.Here X is some soft degrees of freedom. For simplicity,we identify X as soft pions. As an example, using theextracted v of u and s quarks in Au+Au collisions at √ s NN = 200 GeV for 30-80% centrality, we calculate v of the directly-produced pions and kaons by above pro-cesses and then consider the decay contribution of otherhadrons. We show results as solid lines, marked as “final π/K QCM”, in Fig. 9 and compare with experimentaldata [6]. We see that v of pion and kaon in the low p T range ( p T . GeV/c) can be described by v of quarksthat is extracted from baryons and φ .As a contrast, we also calculate v of pionsand that of kaons by direct EVC formulas, i.e., v ,π ( p T ) = 2 v ,u ( p T / and v ,K ( p T ) = v ,u (cid:16) r p T (cid:17) + v ,s (cid:16) r r p T (cid:17) . We present results as dashed lines in Fig.19, marked as “direct π/K QCM”. Comparing to solidlines, we see the important effect of extra X in pion andkaon production and that of resonance decays. (GeV/c) T p v exp data π QCM π direct QCM π final (GeV/c) T p v K exp datadirect K QCMfinal K QCM (a) (b)
Au+Au 200GeV 30-80%
Figure 9. v of pion and kaon in Au+Au collisions at √ s NN =200 GeV for 30-80% centrality. Symbols are experimentaldata [6]. Dashed lines are results of v directly applying EVCformulas. Solid lines are results of u + ¯ d/ ¯ s → π/K + X processafter considering the decay contribution of other hadrons. Appendix B: the influence of two-kaon coalescenceon p T spectrum of φ Starting from Eqs. (1) and (2), the p T distributions of Ω and φ under equal-velocity combination have f Ω ( p T ) = κ Ω f s ( p T / , (B1) f φ ( p T ) = κ φ f s ( p T / (B2)from which we get a quark number scaling property f / (3 p T ) = κ φ, Ω f / φ (2 p T ) , (B3)where κ φ, Ω = κ / /κ / φ is independent of p T . We take f s ( p T ) = f ¯ s ( p T ) at LHC energies.As the discussion of φ elliptic flow at LHC energies inSec. IV, the coalescence of two kaons may be anothercontribution to φ production. The p T spectrum of φ bythe coalescence of two charged kaons with equal velocityis f φ,KK ( p T ) ∝ f K + ( p T / f K − ( p T / (B4)or by that of two neutral kaons is f φ,KK ( p T ) ∝ f K ( p T / f ¯ K ( p T / ∝ (cid:2) f K s ( p T / (cid:3) . (B5)In Fig. 10, we compare results of two-kaon coalescenceby Eqs. (B4) and (B5) with data of φ in heavy-ion colli-sions at four collision energies. We see that the spectraof two-kaon coalescence are almost parallel to those of φ for p T,φ . GeV/c. This indicates that two-kaon coales-cence does not change the shape of φ distribution in this p T range. Therefore, it also does not break the quarknumber scaling property Eq. (B3) in the range p T,φ . p T,s . GeV/c domi-nated by soft or thermal strange quarks. ) T (p φ f /2)] T (p K [f ∝ /2) T (p - K f × /2) T (p + K f ∝ Pb+Pb 5.02 TeV 10-20% Pb+Pb 2.76 TeV 10-20% Au+Au 200 GeV 0-5% Au+Au 39 GeV 10-20% − − − −
10 110 − − − −
10 110 (GeV/c) T p (GeV/c) T p - ( G e V / c ) T d N / dp - ( G e V / c ) T d N / dp Figure 10. The p T spectra by the coalescence of two kaonswith equal velocity, which are compared with those of φ . Dataof φ are taken from Refs. [40–43] and those of kaons are takenfrom Refs. [44–50]. In the range p T & GeV/c, see top panels in Fig.10, the spectra of two-kaon coalescence are steeper thanthose of φ to a certain extent. Therefore includingthis contribution will make φ spectrum softer than thatformed purely by the strange quark combination. Here,we take a simple case as an illustration, i.e., 80% of final-state φ comes from the direct strange quark combinationand the remaining 20% comes from two-kaon coalescence.Fig. 11 shows our calculations and the comparison withdata of φ in Pb+Pb collisions at √ s NN = Ω [51] by the scaling property Eq. (B3), f φ,s ¯ s ( p T ) = κ − φ, Ω f / (3 p T / . (B6)We see that they are in good agreement with data of φ for p T . p T = 4 . GeV/cis higher than the φ datum to a certain extent. Then, weconsider the contribution of two-kaon coalescence and re-sults are shown as up-triangles. Data of kaons are takenfrom Ref. [46]. We see that the results for p T . p T = φ .2 (GeV/c) T p − − −
10 110 - ( G e V / c ) T d N / dp Pb+Pb 2.76 TeV 10-20% exp data φ /2) T (3p Ω f ∝ coal ss + 20% KK coals80% s Figure 11. The p T spectrum of φ calculated by pure s ¯ s com-bination and that by mixture of s ¯ s combination and KK coa-lescence in Pb+Pb collisions at √ s NN = φφ