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Physics Letters B 597 (2004) 328–332

www.elsevier.com/locate/physlet

Resonance decay effects on anisotropy parameters

X. Donga,b, S. Esumic, P. Sorensenb, N. Xub, Z. Xud

a Department of Modern Physics, University of Science and Technology of China, Hefei 230026, Chinab Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

c Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japand Physics Division, Brookhaven National Laboratory, Upton, NY 11973, USA

Received 8 March 2004; received in revised form 23 May 2004; accepted 19 June 2004

Available online 2 August 2004

Editor: J.-P. Blaizot

Abstract

We present the elliptic flowv2 of pions produced from resonance decays. The transverse momentumpT spectra of the parenparticles are taken from thermal model fits and theirv2 are fit under the assumption that they follow a number-of-constituentquark (NCQ) scaling law expected from quark-coalescence models. Thev2 of pions from resonance particle decays is foundbe similar to the measured pionv2. We also propose the measurement of electronv2 as a means to extract open-charmv2 andinvestigate whether a thermalized system of quasi-free quarks and gluons (a quark–gluon plasma) is created in collisions of Anuclei at RHIC. 2004 Elsevier B.V. All rights reserved.

PACS: 25.75.Ld

Keywords: Elliptic flow; Resonance decays; Coalescence

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1. Introduction

One of the surprising observations made at RHis the measurement of a number-of-constituent-qu(NCQ) dependence for both elliptic flowv2 and thenuclear modification factorRCP at intermediatepT

(1.5 < pT < 5 GeV/c) [1]. Models of hadron formation by constituent-quark coalescence provide a via

E-mail address: xdong@lbl.gov(X. Dong).

0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2004.06.110

explanation for these observations whereas expetions based on conventional fragmentation approacare inconsistent with the data[2–4]. In coalescencemodels an NCQ-scaling ofv2 arises as a consequenof hadrons coalescing out of a thermal distributionpartons and reveals the flow developed during atonic epoch at RHIC. Pionv2, however, appears tviolate NCQ-scaling. In this Letter, we study the effeof resonance decays on pionv2. We show that whendecays are taken into account, the measured piov2may become consistent with the NCQ-scaling dem

.

X. Dong et al. / Physics Letters B 597 (2004) 328–332 329

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strated by kaon(K+,K−,K0S), proton,Λ, andΞ v2

[1,5].The particle azimuthal distribution with respect

the reaction plane at rapidityy can be described byFourier expansion

(1)dN

d�φ∝ 1+ Σn2vn cos(n�φ),

where�φ is the difference in azimuth angle betwethe particle and the reaction plane. The first and secFourier coefficients,v1 andv2, historically are calleddirected and elliptic flow, respectively. All coefficiencan be calculated from the relation:vn = 〈cos(n�φ)〉.

As the volume of the system created in an oaxis nucleus–nucleus collision expands, its spaanisotropy quenches. The momentum-space antropies represented by the Fourier coefficientsvn

preserve information aboutthe early collision dynamics when the spatial anisotropy was largest[2,6–8].Since the initial overlap region is elliptical in shapthe second harmonic coefficientv2 is the largest andmost studied.

2. NCQ-scaling of v2

Fig. 1shows theπ+ + π−, K0S , p + p̄, andΛ + Λ̄

v2 from minimum-bias197Au + 197Au collisions at√sNN = 200 GeV[1,5]. In the lowerpT region (pT <

1.0 GeV/c), the values ofv2 are lower for higher mashadrons. Hydrodynamic calculations[9] predict theobserved mass dependence ofv2—perhaps implyingthat a thermalized system has been created in csions at RHIC energy. At higherpT (pT � 2 GeV/c),thev2 measurements saturate at values below thedrodynamic model predictions. The saturated valuv2 and thepT scale where the saturation sets in dpends on the particle-type: the baryonv2 saturates ahigherpT and at larger values than that of mesons.

According to coalescence models[4], after scalingboth v2 and pT with the number of the constituenquarks (NCQ) in the corresponding hadron, all pacles at intermediatepT should fall onto one universacurve. The NCQ-scaledv2 measurements inFig. 1(b)show thatv2(pT /nq)/nq for pT /nq > 0.6 GeV/c

is similar for all particlesexcept pions. This observation, coupled with the NCQ-dependence obserat intermediatepT in the nuclear modification facto

Fig. 1. (a) Measurements of thepT dependence of the evenanisotropy parameters forπ , K0

S, p, Λ. Dot-dashed lines are th

results of fits. (b) Number-of-constituent-quark (NCQ) scaledv2.All particles except the pions follow the NCQ scaling.

RCP is evidence of hadron formation by coalescenor recombination. In this case,v2(pT /nq)/nq repre-sents a constituent quark momentum-space anisotv

q

2 that arises as a consequence of collectivity ipartonic stage. Based on coalescence models, Nscaling suggests the creation of a quark–gluon pla(QGP) with v

q

2 characterizing the properties of thQGP. For this reason, understanding the source odiscrepancy in the NCQ-scaled pionv2 is imperative.

With this goal in mind, we study the effect of seondary pions (from particle decays) on the measupion v2. We assume that NCQ scaling is valid forhadrons other than pions and use the publishedv2measurements[1,5] to parameterizev2(pT /nq)/nq .The pT distributions are assumed to follow an expnential form with slope parameters taken from msurements when available. We use chemical fitsfix the relative hadron abundances[10,11]. Since thepion mass is much smaller than the sum of its cstituent quarks masses, direct pions are not necessassumed to follow the scaling predicted from co

330 X. Dong et al. / Physics Letters B 597 (2004) 328–332

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lescence models. As such, we do not consider dipions, and instead choose to study thev2 of the sec-ondary pions. Given the model uncertainties, extrtion of the direct pionv2 is difficult and remains anopen question. Finally we will discuss how to extraopen-charmv2 based on the decayed electrons.

3. Simulation results

Thev2 values of the simulated resonances are pmeterized by fittingK0

S andΛ v2 [1] with the equation

(2)fv2(n) = an

1+ exp(−(pT /n − b)/c)− dn,

wherea, b, c and d are the fit parameters,n is theconstituent-quark number, andpT is in the unit ofGeV/c. The fit results are shown as dot-dashed liin Fig. 1, where the fitting parameters area = 0.1,b = 0.35, c = 0.2 andd = 0.03. The NCQ-scaling ov2 works well for kaons, protons and Lambdas with0.6 � pT /nq � 1.5 GeV/c whereas pionv2 deviatesfrom NCQ scaling for allpT . The parameters fromchemical fits are listed inTable 1.

In high-energy collisions, a large fraction of harons are produced through resonance decays. Thparticularly true for pions in high-energy heavy-iocollisions. At mid-rapidity, in collisions at RHIC, amany as 80% of pions are from resonance dec[12]. The dominant decays areρ → ππ , ω → 3π ,K∗(892) → Kπ , K0

S → ππ and∆ → Nπ . With sucha potentially large fraction of pions arising from dcays, accounting for their effect on the observed pv2 is very important.

The pT distributions of pions from resonance dcays are shown inFig. 2. Many pions at lowpT aregenerated from resonances at largerpT where pre-

Table 1Parameters for the input resonances: the units for slope parameteT are GeV. The fraction of the hadrons are fixed from the measabundances

T 1 T 2 T 3 Fraction (%)

ρ 0.5 0.4 0.3 60±10ω 0.5 0.4 0.3 30±10K0

S 0.3 0.3 0.3 6±5

K∗ 0.5 0.4 0.3 2±1∆ 0.55 0.55 0.55 2±2

sumably the parent particlev2 is larger. As a resulthe decayed pions take on a relatively largev2 value.The decays from theρ- andω-mesons dominate thsecondary pionpT spectrum. In this plot, a slope prameter ofT = 0.4 GeV is used for theρ distributions.In peripheral collisions, the STAR measured sloperameter is 319± 4(stat.) ± 32(syst.) MeV [13]. Thesimulated results are in a good agreement withPHENIX π0 data from minimum bias197Au + 197Aucollisions[14].

In Fig. 3, the v2 values for the simulated decapions are shown as dashed-lines. The resonancecluded in this study are theρ, ω, K∗, K0

S and ∆.The decayρ → ππ with a 100% branching ratiodominates the production of secondary-pions. Tsimulated resonance particles are restricted to mrapidity |y| < 0.5. Increasing the rapidity windowdoes not change the results. The pionv2 in the re-gionpT � 1.5 GeV/c is sensitive to the shape of theρ

pT spectrum. Dashed-, dotted-, and dot-dashed-lcorrespond to thev2 results from the different slopparameters listed inTable 1. For the smaller slope parameterT = 300 MeV, the decayed pionv2 is belowthe data, leaving room for other contributions[15].

Fig. 2. Resonance decayed pion distributions. The summed strum is shown as a dashed-line. In this simulation, a slope paramof T = 0.4 GeV is used forρ,ω andK∗ . ForK0

Sand∆ the respec-

tive slope parametersT = 0.3 GeV and 0.55 GeV are used. Threlative fraction of the hadrons are listed inTable 1. For compari-son, PHENIXπ0 results are shown as open-crosses.

X. Dong et al. / Physics Letters B 597 (2004) 328–332 331

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Fig. 3. The measured pionv2 (symbols) is compared to the simulated v2 for pions from resonance decays (dashed lines).assumedv2 of mesons and baryons are represented by the soliddot-dashed lines, respectively.

The effects of resonance decays on proton and kv2 were also investigated. As in Ref.[15], they werefound to increase the observedv2 above the assumeNCQ-scaled inputv2 but by a much smaller degrethan that of pions. Since thev2 of kaons is used in oufits, resonance decays can change our assumedv2. These changes, however, will be small. In this Lter we concentrate on the pionv2.

4. D-meson v2

NCQ scaling suggests that hadrons at intermedpT are formed from a thermal partonic phase creain heavy-ion collisions at RHIC. In this system, thigh initial matter density gradient and copiousteractions among partons leads to a collective motioof partons. The largev2 values measured for multstrange hadrons also indicate that partonic collectidevelops in collisions at RHIC[16]. The demonstration of collectivity is necessary but not sufficientshow that local thermal equilibrium is establisheIf collectivity is developed by much heavier charquarks, however, it suggests that interactions betwcharm quarks andu-, d-, or s-quarks must have beefrequent enough and strong enough for the lighquarks to have thermalized. For this reason, the m

t

Fig. 4. Thev2 of electrons fromD-meson decays (top) andπ0 de-cays (bottom). An inputD-mesonv2 from a coalescence model witzero charm quarkv2 [17] is shown as a solid line (top). The inpmesonv2 curve taken from our fit to the NCQ-scaledK0

SandΛ+ Λ̄

v2 is shown as a dot-dashed line in both plots.

surement of heavy flavor (open-charm)v2 can probethe degree to which the lighteru-, d-, and s-quarksthermalize.

Since a large fraction of heavy-flavor hadronscay through leptonic modes, electronv2 measure-ments may offer a convenient way to study opcharm v2. In Fig. 4(top), thev2 for electrons fromD-meson decays is shown. Dot-dashed lines reprethe v2 of the parent meson taken from our fit to tNCQ-scaledK0

S andΛ + Λ̄ v2. The solid line represents aD-mesonv2 whereD-mesons are assumedcoalesce from light quarks withvu,d

2 > 0 and charmquarks withvc

2 = 0 [17]. We simulate theD-mesonpT distribution using the PYTHIA event generat[18]. The shape of the calculated spectrum is wrepresented by a power-law function. Approximat50 M D-meson events are used in this simulatiFor electronpT > 1.5 GeV/c, and a saturatedv2 athigherpT the v2 of electrons fromD-meson decaybecomes similar to the parentD-mesonv2. The de-cay electronv2 is found to be sensitive to changin the assumedD-mesonv2. As such, the measurement of electronv2 can distinguish betweenvc

2 ≈v

u,d2 andvc

2 ≈ 0. The degree of heavy flavor therm

332 X. Dong et al. / Physics Letters B 597 (2004) 328–332

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equilibrium can be assessed by measuring elecv2 within 1 � pT (e) � 3 GeV/c, a region whichcorresponds to 2� pT (D) � 5 GeV/c. The authorsof Ref. [19] come to a similar conclusion but dnot evaluate background contributions to the electspectrum.

Neutral pion decays are the dominant source“background” electrons. While the two-photon decprocess,

π0 ∼100%−→ γ + γfew%−→ e+ + e− + e+ + e−,

can be identified by its decay topology[20], the pionDalitz decay can only be subtracted statistically.Fig. 4(bottom) we show thev2 of electrons from sim-ulated pion Dalitz decays. The pion distribution cbe obtained from measurements at RHIC[14]. Forthese simulations a 100% conversion probability issumed. In STAR, however, the probability is closer5%. The decayed electrons predominantly havepT �0.5 GeV/c [20].

Electrons from heavy flavor decays begin to domnate the electron spectrum abovepT ∼ 3 GeV/c. Withknowledge of the pion yield,D-meson yield, the pionv2, and the electronv2 it will be possible to extracthe D-mesonv2. These measurements can be mby both the PHENIX and STAR CollaborationsRHIC. Direct photonv2 can also be measured withis method.

5. Summary

We have studied the effect of resonance decaythe pionv2 in Au + Au collisions at RHIC. When thev2 values for resonances are assumed to follow Nscaling, the pions generated in their decays take ov2values similar to those measured at RHIC. The donant source of secondary pions isρ decays. We haveshown that when decays are accounted for, the msured pionv2 valuesmay become consistent with thNCQ scaling law that suggests the developmenpartonic collectivity in collisions at RHIC. Model uncertainties, however, make it difficult to extract tdirect pionv2. In addition, we propose the measument of electronv2 as a means to study open-cha

v2 and hence the degree of thermalization reacheRHIC.

Acknowledgements

We appreciate fruitful discussions with V. GrecP. Huovinen, C. Ko, H.G. Ritter, E.V. Shuryak anS. Voloshin. This work was supported in part by tNSFC under the Project No. 10275027 and theDepartment of Energy under Contract No. DE-AC076SF00098.

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