dynamics of hadronization: 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: single particle observables, correlations & entropy

Dynamics of Hadronization: Interplay of Fragmentation and

Recombination

Dynamics of Hadronization: Interplay of Fragmentation and

Recombination

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, B. Mueller, M. Asakawa, S.A. Bass & R.J. Fries, PRC 69 031902 (2004) R.J. Fries, S.A. Bass & B. Mueller, PRL in print (nucl-th/0407102) C. Nonaka, B. Mueller, S.A. Bass & M. Asakawa, nucl-th/0501028


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

zz

Dd 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 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


Species dependent saturation of elliptic flow

Species dependent saturation of elliptic flow

• hyperon v2 saturates later and higher than kaon v2.

• same effect observed for protons and pions.

• at low pT the phenomenology seems better described in mT – m0 than pT , indicating hydro scaling, yet scaling breaks down for high pT

• what drives the different pT scales for KS and Λ v2?

novel mechanism of baryon formation?


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


Recombination: Pro’s & Con’s

Recombination: Pro’s & Con’s

Pro’s:• 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 protons!Con’s:

• simple recombination may violate entropy conservation• gluons at hadronization need to be converted

recombining partons:p1+p2=ph

fragmenting parton:ph = z p, z<1


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. Phys. A715, 379 (2003) • Z.W. Lin & C.M. Ko, Phys. Rev. Lett 89, 202302 (2002)• D. Molnar & S. Voloshin, Phys. Rev. Lett 91, 092301 (2003)


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: relativistic formalism

Recombination: relativistic formalism

• choose a hypersurface Σ for hadronization

• use local light cone coordinates (hadron defining the + axis)

• wa(r,p): single particle Wigner function for quarks at hadronization

• ФM & ФB: 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:2

M

2

M3 3

,

B3 3

, ,B

( , ) ( , (1 ) )

( , ) ( , ' ) ( ,

( )

( ,

(

(1 ') '

2 )

'(2 )

) )

dN P uE d dx

d P

dN P

w R xP w R x P

w R xP w R x Pu

E d dx dxd p

w x P xR x

x

x


Recombination vs. Fragmentation

Recombination vs. Fragmentation

1h 1

h3 3 20

( , ) ( )(2 ) z

dN P u dzE d w R P D z

d 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


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


• 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


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


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


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

issues to be addressed in the future:• treatment of gluons & higher Fock states (work in progress)• realistic space-time dynamics of parton source

• need improved data of identified hadrons at high pt


The End


transv

ers

e m

om

entu

m p

t

time

Connecting the dots …

Connecting the dots …

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronization

hadronic phaseand freeze-out

shattered

color-glas

jetproductio

n

hydrodynamicevolution

jetquenching

partonrecombination

fragmentation

reco/SM?radialflow

HBT?!


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

dynamics of hadronization: interplay of fragmentation and recombination

Documents

higher pt

fragmentation of quarks

hadron spectrumat high

different pt scales

interplay of fragmentation

fragmentation baryons

antiquark spectrum

meson spectrum