Hadronization at RHIC: Interplay of Fragmentation and Recombination

Steffen A. Bass Dynamics of Hadronization #1

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

Hadronization at RHIC: Interplay of Fragmentation and

Recombination

Hadronization at RHIC: Interplay of Fragmentation and

Recombination


Standard Model of Hadronization

Standard Model of Hadronization

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronization

hadronic phaseand freeze-out

1

h3 2 3

0

h/( )adNdN dz E

Ed P z z

zzD

d P

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

spectra via Hydro:

• high pt: pQCD is applicable, hadronization via fragmentation a fast parton fragments via a color string: a h+X hadron spectrum is given by:

2

( ) /20

( , )2 1i i s ii

ii E B S TT

g p dpn

e

1 0

0

cosh sinhK I

f

R

o

T TT

T T fo

m pdNr dr m

m dm T T

?


The baryon puzzle @ RHIC

The baryon puzzle @ RHIC• where does the large proton over pion ratio 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<<1 fragmentation starts with a single fast parton: energy loss affects pions and protons in the same way!

v2


Recombination+Fragmentation Model

Recombination+Fragmentation Model

basic 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 pt than mesons, for same quark distribution understand behavior of baryons!


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


Recombination vs. Fragmentation

Recombination vs. Fragmentation

1h 1

h3 3 20

( , ) ( )(2 ) z

dN P u dzE d w R P D zd P z

Fragmentation…

frag bdN P rec 2bdN P

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

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

/ ex expp( / ) ( / ) /n

w P n P u T P u zT w P z


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 Observables

Reco Success: Single Particle Observables

consistent description of spectra, ratios and RAA


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

p

v

pv

v

v

Parton Number Scaling of v2Parton Number Scaling of v2

smoking gun for recombination

measurement of partonic v2 !

•in leading order of v2, recombination predicts:

2 2

2 2

22

33

pM

p

t

B t

t

t

pv

p

v

p v

p

v


Resonance v2: scaling violations

Resonance v2: scaling violations

QGP resonances: hadronizing QGP, no rescattering

HG resonances: hadronic phase, h-h rescattering

Key: v2 is additive for composite particles

*0KK

d s, quarks*0K

*0K

n=2 scaling

n=4 scaling

TT

h Pn

nP1

)( 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

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/0501028 B. 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 B

ABA

dN d N NY

N d d

0.94

cone

0

AB ABY d Y


Recombination: Inclusion of Correlations

Recombination: 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 d

dd

N

d P d PW


Correlations: Proof of Principle

Correlations: Proof of Principle

Meson-trigger Baryon-trigger

strong correlations from fragmentation, but suppressed by soft triggers

combination of hard fragmentation and soft recombination correlations with a fixed correlation volume is compatible with data


Recombination: Entropy Puzzle

Recombination: Entropy Puzzle

• 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

quark recombination

hadronization


1) resonance gas model:– massless particles:

– massive particles:

Entropy in the Hadronic Phase

Entropy in the Hadronic Phase

entropy content is larger than often assumed!

: resonances

: ground state

MeV

: bosons: fermions

4.725.15

2) final state entropy from data:

Pal & Pratt, PLB578,310 (2004)

STAR: PRC68, 044905 (2003)

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


Entropy in the Deconfined Phase

Entropy in the Deconfined Phase

• Lattice QCD [CP-PACS with Nt=6 & Nf=2]

systematic uncertainties include: thermodynamic limit, continuum limit, unphysically large quark mass…

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


Entropy Puzzle resolved?

Entropy Puzzle resolved?

• No direct comparison of the entropy content of both phases:

– Volume at hadronization ?– Number of quarks on the lattice ?

• SH is larger than often assumed.

• SQ near TC is strongly reduced due to interactions.

Recombination may be compatible with the entropy constraint after all!

?

Quark Phase Hadron Phase


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

10

1

20

1

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

• use probabilistic normalization:• completeness relation takes form:

( ) ( )a b a bq x q x x x

1 2 3

1 2

10

1 2

20

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

Effects of dynamical quark mass m(Q2): assumption of quasi-free partons with current quark masses 10

MeV valid for resolution scale Q2 > 1 GeV2

for Q2(πTC)20.5 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

MeV valid for resolution scale Q2 > 1 GeV2

for Q2(πTC)20.5 GeV2 degrees of freedom likely dominated by lowest Fock state (i.e. valence quark state)


Egalitarian Nature of Thermal Recombination

Egalitarian Nature of Thermal Recombination

Boltzmann 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 is independent of degree of complexity of the structure of the state

/ /ˆ( ) ( ) kE T xP Tqw x q x q x e e

( ) /ˆ( ) ( ) ( ) ( ) ( ) ( ) a bx x P Ta b a b q a q bq x q x q x q x w x w x e

1 2 /1 10

1 2 /2 20

ˆ1 ,

ˆ1 , ,

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

P Tqqg 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

/ /1 2 3( ) P T P T

qq qqg qqqqW P W W W C C C e e


Higher Fock States: v2 Scaling Violations

Higher Fock States: v2 Scaling Violations

Generalization of scaling law to higher Fock states:• assume all partons carry roughly equal momentum

xi1/nν

with nν the number of partons in the Fock state

• valence quark approximation: ν=1, n1=2,3 and C1=1

scaled v2 for mesons and baryons:

• for valence quark approx:

( )2 2 /Hv P C n v P n

( )( ) ( ) ( )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

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

(scaled v2 identical to parton v2)


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 of hadron ratios & spectra jet-quenching phenomena elliptic flow leading / 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

Documents

Hadronization at RHIC: Interplay of Fragmentation and Recombination