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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
Steffen A. Bass Dynamics of Hadronization #2
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⎛ ⎞ ⎛ ⎞∝ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠∫
?
Steffen A. Bass Dynamics of Hadronization #3
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
Steffen A. Bass Dynamics of Hadronization #4
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!
Steffen A. Bass Dynamics of Hadronization #5
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⎤⎣ ⎦=
Steffen A. Bass Dynamics of Hadronization #6
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
−
Steffen A. Bass Dynamics of Hadronization #7
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)
Steffen A. Bass Dynamics of Hadronization #8
Reco Success: Single Particle ObservablesReco Success: Single Particle Observables
consistent description of spectra, ratios and RAA
Steffen A. Bass Dynamics of Hadronization #9
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 !
Steffen A. Bass Dynamics of Hadronization #10
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
Steffen A. Bass Dynamics of Hadronization #11
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
Steffen A. Bass Dynamics of Hadronization #12
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= ∆ ∆∫
Steffen A. Bass Dynamics of Hadronization #13
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
⊗ ⊗= ∫
Steffen A. Bass Dynamics of Hadronization #14
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
Steffen A. Bass Dynamics of Hadronization #15
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
Steffen A. Bass Dynamics of Hadronization #16
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!
Steffen A. Bass Dynamics of Hadronization #17
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…
Steffen A. Bass Dynamics of Hadronization #18
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 ?
Steffen A. Bass Dynamics of Hadronization #19
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)
Steffen A. Bass Dynamics of Hadronization #20
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
Steffen A. Bass Dynamics of Hadronization #21
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)
Steffen A. Bass Dynamics of Hadronization #22
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
Steffen A. Bass Dynamics of Hadronization #23
The End