Hadronization at RHIC: Interplay of Fragmentation and Recombination
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Transcript of 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
Steffen A. Bass Dynamics of Hadronization #2
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
?
Steffen A. Bass Dynamics of Hadronization #3
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
Steffen A. Bass Dynamics of Hadronization #4
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!
Steffen A. Bass Dynamics of Hadronization #5
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:
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23 3
1 ( / )(2 )
BB Bw
dN VC O TP
d PP
Steffen A. Bass Dynamics of Hadronization #6
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
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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)
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Reco Success: Single Particle Observables
Reco Success: Single Particle Observables
consistent description of spectra, ratios and RAA
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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
Steffen A. Bass Dynamics of Hadronization #10
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
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/0501028 B. 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 B
ABA
dN d N NY
N d d
0.94
cone
0
AB ABY d Y
Steffen A. Bass Dynamics of Hadronization #13
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
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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
Steffen A. Bass Dynamics of Hadronization #15
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
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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
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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!
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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
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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)
Steffen A. Bass Dynamics of Hadronization #20
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
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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)
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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
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The End