Hadronization @ RHIC: Interplay of Fragmentation and Recombination

Steffen A. Bass Hadronization @ RHIC #1

Steffen A. Bass

Duke University& RIKEN-BNL Research Center

• The Protons Puzzle at RHIC - the demise of pQCD?• Recombination + Fragmentation Model• Results and Predictions

Hadronization @ RHIC: Interplay of Fragmentation and

Recombination

Hadronization @ RHIC: Interplay of Fragmentation and

Recombination

R.J. Fries, C. Nonaka, B. Mueller & S.A. Bass, PRL 90 in print


The proton puzzle @ RHIC

The proton puzzle @ RHIC• where does the large proton over pion ratio at high pt come from?

• why do protons not exhibit the same suppression as pions? fragmentation yields Np/Nπ<<1 fragmentation starts with a single fast parton: energy loss affects pions and protons in the same way!

ratio of KKP fragmentation functions for p and π from u quarks


Recombination+Fragmentation Model

Recombination+Fragmentation Model

basic assumptions:

• at low pt, quarks and antiquarks recombine into hadrons locally “at an instant”:

• hadron momentum P is much larger than average momentum Δp2 of the internal quark wave function of the hadron; features of the parton spectrum are shifted to higher pt

in the hadron spectrum

• parton spectrum has thermal part (quarks) and a power law tail (quarks and gluons) from pQCD.

qq M qqq B


Recombination: new life for an old idea

Recombination: new life for an old idea

High Energy Physics Phenomenology:• K.P. Das & R.C. Hwa, Phys. Lett. B68, 459 (1977)

Quark-Antiquark Recombination in the Fragmentation Region description of leading particle effect• T. Ochiai, Prog. Theo. Phys. 75, 1184 (1986)• E. Braaten, Y. Jia & T. Mehen, Phys. Rev. Lett. 89, 122002 (2002)• R. Rapp & E.V. Shuryak, Phys. Rev. D67, 074036 (2003)

Heavy-Ion Phenomenology:• T. S. Biro, P. Levai & J. Zimanyi, Phys. Lett. B347, 6 (1995)

ALCOR: a dynamical model for hadronization yields and ratios via counting of constituent quarks• R.C. Hwa & C.B. Yang, Phys. Rev. C66, 025205 (2002) • R. Fries, B. Mueller, C. Nonaka & S.A. Bass, Phys. Rev. Lett. 90• V. Greco, C.M. Ko and P. Levai, Phys. Rev. Lett. 90

Anisotropic flow:• S. Voloshin, QM2002, nucl-ex/020014• Z.W. Lin & C.M. Ko, Phys. Rev. Lett 89, 202302 (2002)• D. Molnar & S. Voloshin, nucl-th/0302014


pt range of parton recombination

pt range of parton recombination

• quark recombination (coalescence) may dominate for all pt < p0 . Combinatorical models (ALCOR, etc.) work well for total particle yields at SPS and RHIC

• low pt is not calculable, but calculation at moderate pt (few GeV) may be possible using hadron light-cone formalism

• transition from dense medium to dilute medium appears very rapid for fast partons (Δt=Δx/γ), validating sudden approximation

• focus on “high” pt evades problems of energy and entropy conservation in recombination: E = (p2+m2)1/2 p


Recombination: nonrelativistic formalism

Recombination: nonrelativistic formalism

3

3 31 12 2

212

3

2

3

2

(2 ) (2 )

21

(

ˆ ( )

2 )

MM

M

MM

dN V d qC w P q w P q

d P

VC

PP

Tw

q

• use thermal quark spectrum given by: w(p) = exp(-p/T)

• for a Gaussian meson wave function with momentum width ΛM, the meson spectrum is obtained as:

• similarly for baryons:

313

23 3

1 ( / )(2 )

BB Bw

dN VC O TP

d PP


• anisotropic or “elliptic” flow is sensitive to initial geometry

Elliptic FlowElliptic Flow

more flow in collision plane than perpendicular to it

Force P

less absorption in collision plane than perpendicular to it

E L

low pt domain: high pt domain:

r fecomb g222ra1t t ttt v pv r p v pr pp

•total elliptic flow is the sum of both contributions:

r(pt): relative weight of the fragmentation contribution in spectra


Parton Number Scaling of Elliptic Flow

Parton Number Scaling of Elliptic Flow

•in the recombination regime, meson and baryon v2 can be obtained from the parton v2 in the following way:

3

2

2

2

2

2 2

2

2

2

3 322

1

an

2

d

3

12

3

63

p pt tp t

Mt

p pt t

Bt

p pp v vv

p pv

p pv v

v

neglecting quadratic and cubic terms, one finds a simple scaling law:

2 2 2 2an22 3

d 3p pM tt

B tt

pv p vv

pv p


Input and Model Parameters

Input and Model Parameters

Input for the model is the momentum distributions of constituent quarks and anti-quarks at the time of hadronization

• the quark distribution is assumed to have a low pt thermal component and a high pt pQCD mini-jet component

• the thermal component is parameterized as:

with a flavor dependent fugacity ga, temperature T,

rapidity width Δ and transverse distribution f(ρ,ф)

• the pQCD component is parameterized as:

with parameters C, B and β taken from a lo pQCD calculation

2 2/ / 2, ,Tp va agw e fp e

0 1 /a

t t y t

dN

p dp y

CK

d Bp


Hadron Spectra Hadron Spectra


Hadron Ratios vs. pt

Hadron Ratios vs. pt


Flavor Dependence of high-pt Suppression

Flavor Dependence of high-pt Suppression

•R+F model describes different RAA behavior of protons and pions

•in the fragmentation region all hadron flavors exhibit jet-quenching


Parton Number Scaling of v2

Parton Number Scaling of v2

smoking gun for recombination

measurement of partonic v2 !

P. Soerensen, UCLA & STAR @ SQM2003

2 2

2 2

22

33

pM

p

t

B t

t

t

pv

p

v

p v

p

v

•in leading order of v2, recombination predicts:


Elliptic Flow: Recombination vs. Fragmentation

Elliptic Flow: Recombination vs. Fragmentation

•high pt: v2 for all hadrons merge, since v2 from energy-loss is flavor blindquark number scaling breaks down in the fragmentation domain


Summary & Outlook

Summary & Outlook

The Recombination + Fragmentation Model:•provides a natural solution to the baryon puzzle at RHIC

•describes the intermediate and high pt range ofhadron ratios & spectrajet-quenching phenomenaelliptic flow

•provides a microscopic basis for the Statistical Model

issues to be addressed in the future:•entropy production•treatment of gluons•realistic space-time dynamics of parton source•need improved data of identified hadrons at high pt

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Hadronization @ RHIC: Interplay of Fragmentation and Recombination