hadronization at rhic: interplay of fragmentation and ... · dn mp rdrm mdm tt ⎛⎞ρ ⎛ρ⎞...

Steffen A. Bass Dynamics of Hadronization #1

Hadronization at RHIC: Interplay of Fragmentation and Recombination

Hadronization at RHIC: Interplay of Fragmentation and Recombination

Steffen A. BassDuke University

& RIKEN-BNL Research Center

• The baryon puzzle at RHIC• Recombination + Fragmentation Model• Results: spectra, ratios and elliptic flow• Challenges: correlations, entropy balance & gluons


Standard Model of HadronizationStandard Model of Hadronization

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronization

hadronic phaseand freeze-out

( )1

h3 2 3

0h/( )adNdN dz EE

d P z zz

zD

d P α→= ∫

• low pt:hadron yields & ratios fit with a SM:

spectra via Hydro:

• high pt: pQCD is applicable, hadronization via fragmentationa fast parton fragments via a color string: a → h+Xhadron spectrum is given by:

2

( ) /20

( , )2 1i i s ii

ii E B S TT g p dpn

e µµπµ

∞

− −=±∫

( ) ( )1 0

0

cosh sinhK I

f

R

o

T TT

T T fo

m pdN r dr mm dm T T

ρ ρ⎛ ⎞ ⎛ ⎞∝ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∫

?


The baryon puzzle @ RHICThe baryon puzzle @ RHIC• where does the large proton over pionratio at high pt come from?• why do protons not exhibit the same jet- suppression as pions?• species dependence of v2 saturation?

fragmentation yields Np/Nπ<<1fragmentation starts with a single fast

parton: energy loss affects pions and protons in the same way!

(GeV/c)TTransverse Momentum p0 2 4 6

0

0.1

0.2

0.3

0SK Λ + Λ

2

-+h+h

Hydro calculationsπKpΛ

STAR Preliminary (Au+Au; 200 GeV; |y|<1.0)

v2


Recombination+Fragmentation ModelRecombination+Fragmentation Modelbasic assumptions:• at low pt, the quarks and antiquark spectrum is thermal and

they recombine into hadrons locally “at an instant”:

features of the parton spectrum are shifted to higher pt in the hadron spectrum

• at high pt, the parton spectrum is given by a pQCD power law, partons suffer jet energy loss and hadrons are formed via fragmentation of quarks and gluons

qq M→ qqq B→

• for exponential parton spectrum, recombination is more effective than fragmentation• baryons are shifted to higher ptthan mesons, for same quark distribution

understand behavior of baryons!


Recombination: nonrelativistic formalismRecombination: nonrelativistic formalism

• 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:

( ) ( )

( )

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 qd P

VCP

PT

w

qπ π

π

ϕ− +

⎡ ⎤

=

⎛ ⎞Λ= − +⎣ ⎜ ⎟

⎝ ⎠⎦

∫

L

• similarly for baryons:

( ) ( )313

23 3 1 ( / )

(2 )B

B BwdN VC O TPd P

Pπ

⎡ − Λ⎤⎣ ⎦=


Recombination vs. FragmentationRecombination vs. Fragmentation

Fragmentation…1

h 1h3 3 2

0

( , ) ( )(2 ) z

dN P u dzE d w R P D zd P z α α

απ →

⋅= Σ ∑∫ ∫

… never competes with recombination for a thermal (exponential) spectrum:

( ) ( )/ ex expp( / ) ( / ) /n

w P n P u T P u zT w P z= − ⋅⎡ ⎤⎣ => − ⋅⎦

… but it wins out at large pT, when the spectrum is a power law ~ (pT)-b :

rec 2bdN Pπ−frag bdN Pπ

−


Recombination: single particle observables• hadron spectra• hadron ratios• RAA

• elliptic flow

• R.J. Fries, C. Nonaka, B. Mueller & S.A. Bass, PRL 90 202303 (2003) • R.J. Fries, C. Nonaka, B. Mueller & S.A. Bass, PRC 68 044902 (2003)• C. Nonaka, R.J. Fries & S.A. Bass, Phys. Lett. B583 73-78 (2004) • C. Nonaka, B. Mueller, M. Asakawa, S.A. Bass & R.J. Fries, PRC 69 031902 (2004)


Reco Success: Single Particle ObservablesReco Success: Single Particle Observables

consistent description of spectra, ratios and RAA


Parton Number Scaling of v2Parton Number Scaling of v2

( )

( )

2

2

2 2

3

2 2

2 2

2

2

2

3

2

1 2

and

1 6

33

3

3

p t

p t

p pt t

p t

M

Bt

t

pv

pv

p p

v p

pv

pv

v

v

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞+

⎛

⎜

⎞⎜ ⎟⎝

=⎛ ⎞

+ ⎜ ⎟⎝

⎟⎝ ⎠=

⎛

⎠

⎠

⎞+ ⎜ ⎟

⎝ ⎠

•in leading order of v2, recombination predicts:

( )

( )

2 2

2 2

22

33

pM

p

t

B t

t

tpv

p

v

p v

p

v ⎛ ⎞≈ ⎜ ⎟

⎛ ⎞≈ ⎜ ⎟⎝

⎝

⎠

⎠

smoking gun for recombination

measurement of partonic v2 !


Resonance v2: scaling violationsResonance v2: scaling violationsQGP resonances:hadronizing QGP, no rescattering

HG resonances:hadronic phase, h-h rescattering

Key:v2 is additive for composite particles

*0KK →+ −+ π

d s, quarks*0K

*0K

n=2 scaling

n=4 scaling

⎟⎠⎞

⎜⎝⎛≈ TT

h Pn

nP 1)( 22 vv

TotalHG2

QGP2

full2 ))(1()( vPrvPrv TT −+= )( TPr )( TPr)( TPr is determined by experiments and

related to width of particles and cross section in the hadronic medium.

+K −π

d

s

*0K

7.0:)( TPr

7.0:)( TPr


Challenges:• dynamical two-particle correlations• balancing the entropy• treatment of gluons & sea-quarks

R.J. Fries, S.A. Bass & B. Mueller, PRL 94 122301 (2005)C. Nonaka, B. Mueller, S.A. Bass & M. Asakawa, PRC Rapid Communication

in print, nucl-th/0501028B. Mueller, S.A. Bass & R.J. Fries, Phys. Lett. B in print, nucl-th/0503003


Two-Particle Correlations: a Challenge?Two-Particle Correlations: a Challenge?• PHENIX & STAR measure associated yields in pT windows of a few GeV/c.• trigger hadron A, associated hadron B: associated yield as a function of

relative azimuthal angle

clear jet-like structure observed atintermediate pT

very similar to p+p; jet fragmentation?• analyze as function of integrated yield:

simple recombination of uncorrelated thermal quarks cannot reproduce two particle correlations

( ) ( )1( ) ( )

AB A BAB

A

dN d N NYN d d

φφ φ

⎛ ⎞∆ = −⎜ ⎟∆ ∆⎝ ⎠

( ) ( )0.94

cone

0AB ABY d Yφ φ= ∆ ∆∫


Recombination: Inclusion of CorrelationsRecombination: Inclusion of Correlations

• Recombination approach allows for two particle correlations, provided they are contained in the parton source distributions

• Three distinct types are conceivable: F-F, SH-F and SS-SS

• Ansatz for SS-SS: for two mesons, use product of correlated parton distributions:

Which results in a correlated two hadron yield:

21234 1 3 4 1 iji j

wW w Cw w<

⎛ ⎞+⎜ ⎟

⎝=

⎠∑

1234

6

3 3 A A BBA

BA

B

ABC dd dN

d P d PWσ ϕ ϕσ

Σ

⊗ ⊗= ∫


Correlations: Proof of PrincipleCorrelations: Proof of PrincipleMeson-trigger Baryon-trigger

strong correlations from fragmentation, but suppressed by soft triggerscombination of hard fragmentation and soft recombination correlations with a fixed correlation volume is compatible with data


Recombination: Entropy PuzzleRecombination: Entropy Puzzle

quark recombination

hadronization

• Does recombination violate the 2nd law of thermodynamics ?– particle number decreases drastically in hadronization via reco…

restrict reco approach to intermediate momenta, ignore bulk…decay of hadronic resonances as possible solution (Greco et al.)

need estimate of entropy at hadronization


Entropy in the Hadronic PhaseEntropy in the Hadronic Phase1) resonance gas model:

– massless particles:

– massive particles:

2) final state entropy from data:

Pal & Pratt, PLB578,310 (2004)

• Hydro+UrQMD calculations indicate a 5% increase in multiplicity due to rescattering in the hadronic phase

: resonances: ground state

MeV

: bosons: fermions

4.725.15

STAR: PRC68, 044905 (2003)

entropy content is larger than often assumed!


Entropy in the Deconfined PhaseEntropy in the Deconfined Phase• Lattice QCD [CP-PACS with Nt=6 & Nf=2]

entropy content of the deconfined phase near TC is strongly reduced due to interactions!

systematic uncertainties include: thermodynamic limit, continuum limit, unphysicallylarge quark mass…


Entropy Puzzle resolved?Entropy Puzzle resolved?

• SH is larger than often assumed.

• SQ near TC is strongly reduced due to interactions.

?

Quark Phase Hadron Phase

Recombination may be compatible with the entropy constraint after all!

• No direct comparison of the entropy content of both phases:– Volume at hadronization ?– Number of quarks on the lattice ?


Thermal Recombination beyond the Valence Quark Approximation

Thermal Recombination beyond the Valence Quark Approximation

investigate effects of more sophisticated internal hadron structure• use light-cone frame• write hadron wavefunction as expansion in terms of Fock-States:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1

101

201

30

1 ,

1 , ,

1 , , ,

a b a b a b a b

a b c a b c a b c a b c

a b c d a b c d a b c d a b c d

M dx dx x x c x x q x q x

dx dx dx x x x c x x x q x q x g x

dx dx dx dx x x x x c x x x x q x q x q x q x

α β

α β

α β γ γ

δ

δ

δ

= + −

+ + + −

+ + + + − +

∫∫∫ K

• use probabilistic normalization:• completeness relation takes form:

( )( ) ( )a b a bq x q x x xα β αβδ δ= −

( ) ( )

( )

1 2 31 2

10

1 220

1

1 ,

1 , ,

a b a b a b

a b c i a b ci

C C C

dx dx x x c x x

dx dx dx x c x x x

δ

δ

= + + +

≡ + −

⎛ ⎞+ − +⎜ ⎟⎝ ⎠

∫

∑∫

K

K

Effects of dynamical quark mass m(Q2):assumption of quasi-free partons with current quark masses ∼10 MeVvalid for resolution scale Q2 > 1 GeV2

for Q2∼(πTC)2≈0.5 GeV2 degrees of freedom likely dominated by lowest Fock state (i.e. valence quark state)

Effects of dynamical quark mass m(Q2):assumption of quasi-free partons with current quark masses ∼10 MeVvalid for resolution scale Q2 > 1 GeV2

for Q2∼(πTC)2≈0.5 GeV2 degrees of freedom likely dominated by lowest Fock state (i.e. valence quark state)


Egalitarian Nature of Thermal RecombinationEgalitarian Nature of Thermal RecombinationBoltzmann approximation:• probability of finding a quark with momentum k=xP and energy Ek in a

thermal medium is given by:with ρ the thermal density matrix

• for a state with n partons one obtains:

emission probability for a single Fock state reads:

combining all contributions from all Fock states, one obtains:

emission probability of complex state from a thermal ensemble isindependent of degree of complexity of the structure of the state

( ) / /ˆ( ) ( ) kE T xP Tqw x q x q x e eρ − −= = =

( ) /ˆ( ) ( ) ( ) ( ) ( ) ( ) a bx x P Ta b a b q a q bq x q x q x q x w x w x eρ − + += = KK K K

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 /1 10

1 22 20

ˆ1 ,

ˆ1 , ,

P Tqq a b a b a b a b a b

/P T−qqg a b c a b c a b c a b c a b c

W dx dx x x c x x q x q x q x q x C e

W dx dx dx x x x c x x x q x q x g x q x q x g x C e

δ ρ

δ ρ

−= + − =

= + + − =

∫∫

( ) / /1 2 3( ) P T P T

qq qqg qqqqW P W W W C C C e e− −= + + + = + + + =K K


Higher Fock States: v2 Scaling ViolationsHigher Fock States: v2 Scaling ViolationsGeneralization of scaling law to higher Fock states:• assume all partons carry roughly equal momentum xi≈1/nν

with nν the number of partons in the Fock state

• valence quark approximation: ν=1, n1=2,3 and C1=1scaled v2 for mesons and baryons:

• for valence quark approx:

( ) ( )( )2 2 /Hv P C n v P nν ν ν

ν

≈ ∑

( ) ( )

( ) ( )

( )( ) ( ) ( )2 2

( )( ) ( ) ( )2 2

2 /2

3 /3

MM M M

BB B B

nv p C v p n

nv p C v p n

νν ν

ν

νν ν

ν

=

=

∑

∑

%

%

( ) ( ) ( )( ) ( )2 2 2M Bv p v p v p= =% % scaling violations ≈ 5%

(scaled v2 identical to parton v2)


Summary & OutlookSummary & Outlook

The Recombination + Fragmentation Model:• provides a natural solution to the baryon puzzle at RHIC• describes the intermediate and high pt range of

hadron ratios & spectrajet-quenching phenomenaelliptic flowleading / next-to-leading particle correlations (work in progress)treatment of gluons & higher Fock states

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


The End

hadronization at rhic: interplay of fragmentation and ... · dn mp rdrm mdm tt ⎛⎞ρ ⎛ρ⎞...

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