1. & RIKEN-BNL Research Center
Dynamics of Hadronization: Interplay of Fragmentation and Recombination
Steffen A. Bass
Duke University
& RIKEN-BNL Research Center
The baryon puzzle at RHIC
Recombination + Fragmentation Model
Results: single particle observables, correlations & entropy
R.J. Fries, C. Nonaka, B. Mueller & S.A. Bass, PRL (2003)
R.J. Fries, C. Nonaka, B. Mueller & S.A. Bass, PRC (2003)
C. Nonaka, B. Mueller, M. Asakawa, S.A. Bass & R.J. Fries, PRC (2004)
R.J. Fries, S.A. Bass & B. Mueller, PRL in print (nucl-th/ )
C. Nonaka, B. Mueller, S.A. Bass & M. Asakawa, nucl-th/
Steffen A. Bass

2. Standard Model of Hadronization
initial state
pre-equilibrium
QGP and
hydrodynamic expansion
hadronization
hadronic phase
and freeze-out ?
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:
Steffen A. Bass

3. 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
Steffen A. Bass

4. 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?
Steffen A. Bass

5. 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
Steffen A. Bass

6. Recombination: Pro’s & Con’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:
fragmenting parton:
ph = z p, z<1
recombining partons:
p1+p2=ph
simple recombination may violate entropy conservation
gluons at hadronization need to be converted
Steffen A. Bass

7. 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, (2002)
R. Rapp & E.V. Shuryak, Phys. Rev. D67, (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, (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, (2002)
D. Molnar & S. Voloshin, Phys. Rev. Lett 91, (2003)
Steffen A. Bass

8. Recombination: 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:
similarly for baryons:
Steffen A. Bass

9. 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:
Steffen A. Bass

10. Recombination vs. Fragmentation
… never competes with recombination for a thermal (exponential) spectrum:
… but it wins out at large pT, when the spectrum is a power law ~ (pT)-b :
Steffen A. Bass

11. Hadron Spectra
Steffen A. Bass

12. Hadron Ratios vs. pt
Steffen A. Bass

13. 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
Steffen A. Bass

14. Elliptic Flow
anisotropic or “elliptic” flow is sensitive to initial geometry
low pt domain:
high pt domain:
more flow in collision plane than perpendicular to it
less absorption in collision plane than perpendicular to it
total elliptic flow is the sum of both contributions:
r(pt): relative weight of the fragmentation contribution in spectra
Steffen A. Bass

15. Parton Number Scaling of v2
in leading order of v2, recombination predicts:
smoking gun for recombination
measurement of partonic v2 !
Steffen A. Bass

16. Resonance v2: scaling violations
QGP resonances:
hadronizing QGP, no rescattering
HG resonances:
hadronic phase, h-h rescattering
quarks
n=2 scaling
n=4 scaling
Key:
v2 is additive for composite particles
Total
is determined by experiments and
related to width of particles and cross
section in the hadronic medium.
Steffen A. Bass

17. 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 at
intermediate 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
Steffen A. Bass

18. 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:
Steffen A. Bass

19. 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
Steffen A. Bass

20. Recombination: 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
Steffen A. Bass

21. Entropy in the Hadronic Phase
1) resonance gas model:
massless particles:
massive particles:
2) final state entropy from data:
: bosons
Pal & Pratt, PLB578,310 (2004)
: fermions
STAR: PRC68, (2003)
MeV
: resonances
: ground state
Hydro+UrQMD calculations indicate a 5% increase in multiplicity due to rescattering in the hadronic phase
4.72
5.15
entropy content is larger than often assumed!
Steffen A. Bass

22. Entropy 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, unphysically large quark mass…
Steffen A. Bass

23. Entropy Puzzle resolved?
Quark Phase
Hadron Phase
Recombination may be compatible with the
entropy constraint after all! ?
SH is larger than often
assumed.
SQ near TC is strongly
reduced due to interactions.
No direct comparison of the entropy content of both phases:
Volume at hadronization ?
Number of quarks on the lattice ?
Steffen A. Bass

24. 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
Steffen A. Bass

25. The End
Steffen A. Bass

26. hydrodynamic expansion
Connecting the dots …
initial state
pre-equilibrium
QGP and
hydrodynamic expansion
hadronization
hadronic phase
and freeze-out
transverse momentum pt
time
jet
production
fragmentation
jet
quenching
parton
recombination
HBT ?!
radial
flow
shattered
color-glas
hydrodynamic
evolution
reco/SM?
Steffen A. Bass

27. 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:
neglecting quadratic and cubic terms, one finds a simple scaling law:
Steffen A. Bass