Duke University Chiho NONAKA. in Collaboration with R. J. Fries (Duke), S. A. Bass (Duke & RIKEN), B. Muller (Duke) nucl-th/ to appear in PRL

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1 Duke University Chiho NONAKA in Collaboration with R. J. Fries (Duke), S. A. Bass (Duke & RIKEN), B. Muller (Duke) nucl-th/00108 to appear in PRL May 1, University of Washington, Seattle

2 Introduction Experimental data Elliptic Flow, Proton Puzzle, RAA Hadronization mechanism Recombination + Fragmentation Model Comparison with Experimental data Hadron Spectra, Hadron ratio, RAA, Centrality dependence of hadron spectra Elliptic Flow Summary

3 Au+Au at sqrt(s NN )=00GeV v min. bias At Low low PT PT Hydro works well r.p. =~4 pi+ p Positives h+,pi+,k+,p PHENIX Preliminary p T (GeV/c) PHENIX : nucl-ex/01001 dn d Parameter v v ( 1v cos( ) v cos( )) STAR Preliminary (Au+Au; 00 GeV; y <1.0) + 0 KS Hydro calculations π K p Λ Λ + Λ h +h Transverse Momentum p T (GeV/c) Saturation in v of baryon occurs at higher PT than one of meson. -

4 Proton Puzzle at RHIC p/ ratio ~ 1 (PT> GeV) at central collisions p Why is it interesting? 4

5 Hadronization from Fragmentation At high PT g qq Fast parton in the vacuum with a color string : a h X ratio of KKP fragmentation functions for p and π from u quarks dn dn E a D h ( z) dp z zd P/ z 1 h dz E 0 p/ Energy loss should affect pions and protons in the same way. RHIC Data 5

6 geom R AA N/( πp t dydp t ) d v KS 0-5% 60-80% Λ + Λ Scaling binary participant + - h +h 10 0 Λ + Λ K S % 60-80% 0 KS Hydro calculations π K p Λ Λ + Λ Suppression in RAA of baryon occurs at higher PT than one of meson. There are some correlations between RAA and v. There is the hadronization mechanism which describes these experimental data Transverse Momentum p t (GeV/c) Sorensen@SQM00 Recombination + fragmentation Model PT > GeV 6

7 Recombination at low PT Recombination occurs at an instant qq M qqq B The parton spectrum is shifted to higher p T in the hadron spectrum. Fragmentation at high PT The parton spectrum has thermal part (quarks) and a power law tail (quarks and gluons) from pqcd. The parton spectrum is recombining partons: p 1 +p =p h shifted to lower p T in the hadron spectrum. Recomb fragmenting parton: p h = z p, z<1 Frag 7

8 Recombination a a non-relativistic model (1) Quarks and antiquarks in volume V quarks mesons 1 i( p ), x M, P 1 x 1 p x x q p 1 p V e R x x, y ( x ) 1 / 1 x q V ( ) q, p1p M, P ( P p ) / 1 p V Total number of mesons N a, b V d q ( ) d r ( ) Degeneracy factor :CM d P ( ) w a ( 1 e i( PR) ( r) M ( y) p p, r ( p ) 1 / 1 p p 1) wb ( p) q, p1p M M Quark distribution M, P Non-relativistic wave function r / M ( r) N e 8

9 Recombination-a nonrelativistic model () Use thermal quark spectrum: w(p) = exp(-p/t) for a Gaussian meson wave function with momentum width ΛM Meson spectrum P r M QCD dnm V d q 1 1 C ˆ M w Pq w Pq M ( q) dp ( ) ( ) V 1 M CM w P 1 ( ) TP Baryon spectrum dn dp V ( ) 1 P B B C B w 1 O( / TP) 9

10 E Spectrum of parton pv( )/ w (, p) e a Recombination dn P dp T rec B,M T a T e / a : fugacity factor, rapidity width f(transverse distribution T=175 MeV, fm, vt=0.55 Fragmentation dn d p pert a T PT AT CB,M dyd cosh( y) ( ) dy y0 1 dnh dz E D ( z) ah d P z a Spectrum of parton 0 C K (1 p / B) T Ea dn a w n a f (, ) Energy Loss : P p, D h (z) :KKP fragmentation functions d P a a K=1.5 :roughly account for higher order corrections C, B, are taken from a leading order pqcd calculation ) (0) p ( 0) 0.8 GeV 1/ 10 T ( T T (, P / n)

11 Comparison with Experimental Data I Hadron spectra, Hadron ratio, Central dependence of hadron spectra

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14 Thermal Recomb Frag t 14

15 Fragmentation dn T ( b) pert AuAu pert a ( b) dna TAuAu(0) N N coll coll ( b) dn (0) The values of Ncoll(b) are given by PHENIX collaboration Energy loss L : average length, Recombination P ( b, p ) ( b) T T 1 e 1 e p T pert a (R b)/ ( b) Transverse area of the overlap zone A l( b) w( b) b) R T ( A T A (0) L R A R A A l(b) w(b) 15

16 R+F model reproduces the centrality dependence of and p ratio 16

17 High-p t Suppression RAA as a function of PT is determined by contribution of recombination and Fragmentation. We can reproduce Rcp of proton and pion as a function of PT. 17

18 Comparison with Experimental data II Elliptic Flow

19 Elliptic flow is sensitive to initial geometry At low PT At high PT recomb frag Pressure gradient Collision plane > Perpendicular plane Energy loss Perpendicular plane > Collision plane Total elliptic flow recomb p 1rp frag v t r p v p t v p t t t r(p t ): relative weight of the fragmentation contribution in spectra 19

20 azimuthal anisotropy of parton spectra is determined by elliptic flow: d N p dp d 1 dn pdp t t p t t 1v cos with Blastwave parametrization for parton spectra: p (Ф p : azimuthal angle in p-space) s mt cosh s p sinh t dscoss I K1 0 T T v pt cos p ptsinh s mt cosh s ds I0 K1 0 T T azimuthal anisotropy is parameterized in coordinate space and is damped as a function of p 1 1 t : ln t s 1 ppt cos s and ppt 1 t s t p / p t 0 0

21 t azimuthal anisotropy is driven by parton energy/momentum loss Δp t RA b p 1 exp t p L 1 p cos R A L: average thickness of the medium α(p t ): damping function for low p t s t t p s the unquenched parton p t distribution is defined according to: R d N 1 t t dr t t 0 t t dn p p pdpd R pdp v is then calculated via: v p cos t p 1

22 From v of Parton At low PT (recombination) meson and baryon v can be obtained from the parton v p pt p pt p pt v v v M B v pt and v pt p pt p pt 1 v 1 6 v neglecting quadratic and cubic terms, one finds a simple scaling law: M p pt B p pt v pt v and v pt v v of Baryon and meson

23 Input : Parton Output: Hadron frag recomb p 1rp v t r p v p t v p t t t Parton elliptic flow Relative weight of recombination Gray region shows the uncertainty of limiting of R and F.

24 Recombination only Recombination describes the flavor dependence. This behavior is consistent with experimental data. 4

25 Recombination + Fragmentation Pt~6 GeV, There are deviation between p and pion. At high Pt, all hadron merges, because energy loss does not have the flavor dependence. At high Pt, parton number scaling breaks down. We need the experimental data at high Pt. 5

26 We can reproduce the experimental data. Hadron spectra Hadron ratio Centrality dependence High PT suppression of RAA and RCP Elliptic Flow Recombination + Fragmentation Model provides the solution of understanding RHIC data! Hydro, thermal model Recombination Fragmentation Future plan d+au Collisions PT 6

27 Back up

28 Hdronization occurs on hypersurface Σ use local light cone coordinates (hadron defining the + axis) w a (r,p): single particle Wigner function for quarks at hadronization Ф M & Ф B : light-cone wave-functions for the meson & baryon respectively x, x & (1-x): momentum fractions carried by the quarks integrating out transverse degrees of freedom yields: dn M Pu E d dxw ( R, xp ) w ( R,(1 x) P ) M() x dp ( ), dn Pu E d dxdx' w( R, xp ) w( R, x' P ) w( R, (1 xx') P ) B(, x x' ) dp B ( ),, 8

29 Anisotropy v /n Ratio S K (n=) Λ+Λ (n=) in leading order of v, recombination predicts: v M p t v p p B p t pt v pt v Transverse Momentum p t /n (GeV/c) P. Soerensen, UCLA & SQM00 9