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The Re-Combinatorics of Thermal Quarks
LBNL School on
Twenty Years of Collective Expansion
Berkeley, 19-27 May 2005
Berndt Müller
Duke University
Re-Combinatorics of Thermal Quarks
• M. Asakawa• S.A. Bass• R.J. Fries• C. Nonaka
• PRL 90, 202303 • PRC 68, 044902• PLB 583, 73• PRC 69, 031902• PRL 94, 122301• PLB (in print)
Special thanks to: The Duke QCD theory group
Jet quenching seen in Au+Au, not in d+Au
PHENIX Data: Identified 0
d+Au
Au+Au
Suppression Pattern: Baryons vs. Mesons
What makes baryons different from mesons ?
…or what really came as a complete surprise…
Suppression: Baryons vs. mesons
baryons mesons
RC
P
pT (GeV/c)
behaves like meson ?
(also -meson)
Hadronization Mechanisms
q
Baryon1
Meson
Fragmentation
q q
q q q
Baryon1
Meson
Recombination
M Q B Q2 3p p p p
Recombination was predicted in the 1980’s – Hwa, Ochiai, …
S. Voloshin
QM2002
Recombination is favored …
… for a thermal source
Baryons compete with mesons
Fragmentation wins out for a power law tail
Instead of a History ….
• Recombination as explanation for the “leading particle effect”:– K.P. Das & R.C. Hwa: Phys. Lett. B68, 459 (1977)
– Braaten, Jia, Mehen: Phys. Rev. Lett. 89, 122002 (2002)
• Fragmentation as recombination of fragmented partons:– R.C. Hwa, C.B. Yang, Phys. Rev. 024904 + 024905 (2004)
• Relativistic coalescence model– C.B. Dover, U.W. Heinz, E. Schnedermann, J. Zimanyi,
Phys. Rev. C44, 1636 (1991)
• Statistical recombination– ALCOR model (see T.S. Biro’s lecture)
• Quark recombination / coalescence– Greco, Ko, Levai, Chen, Rapp / Lin, Molnar / Duke group
– A. Majumder, E. Wang & X.N. Wang (in progress)
Recombination: The Concept
“fragmentation”
“recombination”REC
DER
1 2 p p P
“freeze-out”
Sudden recombination picture
q
qM
d
Transition time from QGP into vacuum (in rest frame of produced hadron) is:
/fT
md d
p
Not gradual coalescence from dilute system !!!
pT m
Allows to ignore complex dynamics in hadronization region; corrections O(m/pT)2
QGP
Tutorial: Non-Relativistic Recombination
Consider system of quarks and antiquarks (no gluons!) of volume V and phase-space distribution wa(p) = p,ap,a.
Quark-antiquark state vs. meson state:
1 1 2 2( )11 2
1
,
, ( )
i p x p x
iP RM
x p p V e
x P V r
Q
M e
11 22
1 2
( )
)
R x x
r x x
with
Probability for finding a meson with P and q = (p1-p2)/2 :3
2 231 2 1 22
(2 )ˆ, , ( ) ( )MQ p p M P P p p q
V
Number of produced mesons:3 2
3
3 3323 1 2
1 1 1 23 3 3
, ,(2 )
( ) ( ) , ,(2 ) (2 ) (2 )
M
abM a b
ab
d PN V M P M P
d p d pd PC V w p w p Q p p M P
Tutorial – page 2
3
21 12 23 3 3
ˆ ( )(2 ) (2 )
abMM a b M
ab
dN V d qC w P q w P q q
d P
The meson spectrum is given by:
Consider case P q, where q is of order M, and expand (for wa = wb):
21 1 1 1 1 1 12 2 2 2 2 2 2i j i j i j
ij
w P q w P q w P q q w P w P w P w P
Using only lowest order term:
3
2 221 12 23 3 3 3
ˆ ( )(2 ) (2 ) (2 )
MM M M
dN V d q VC w P q C w P
d P
Corrections are of order M2w/Pw = M
2/PT for thermal quarks.
Tutorial – page 3
The same for baryons:3
2 231 2 3 1 2 32
(2 )ˆ, , ( ) ( , )BQ p p p B P P p p p q s
V
where q, s are conjugate to internal coordinates
11 2 3 1 22 ( ) a d 'nr x x x r x x
The baryon spectrum is then:
3 321 1 1 1 1
3 2 3 2 33 3 3 3
3 2133
ˆ ( , )(2 ) (2 ) (2 )
1 ( / )(2 )
abcBM a b c B
abc
B B
dN V d q d sC w P q s w P q s w P q q s
d P
VC w P O PT
Tutorial - page 4
For a thermal Boltzmann distribution ( ) exp ( ) /w p E p T
we get
21 12 2
31 13 3
exp 2 / exp /
exp 3 / exp /
M
B
w P E P T E T
w P E P T E T
and therefore: 2( )1 ( / )
( )B B
M M
dN E CO PT
dN E C
Wigner function formulation
General formulation relies on Wigner functions:
1 2 2
3 31 21 1 1 1
1 1 2 2 1 1 2 2 1 2 1 22 2 2 2 3 3, , ( , ; , )
(2 ) (2 )p r p r
ab
d p d pr r r r r r r r e W r r p p
Meson number becomes:
3 33 3 3 1 1 1 1
2 2 2 23 3
*1 12 2
, ; ,(2 ) (2 )M ab
ab
iq rM M
d P d qN d Rd rd r W R r R r P q P q
e r r r r
Relativistic generalization (u = time-like normal of volume):
3 3 3P ud P d R d Pd
E
Relativistic formulation
2M
M3 3,
2B
B3 3, ,
( , ) ( , (1 ) ) ( )(2 )
' ( , ) ( , ' ) ( , (1 ') ) ( , ')(2 )
dN P uE d dxw R xP w R x P x
d P
dN P uE d dxdx w R xP w R x P w R x x P x x
d p
Relativistic formulation using hadron light-cone frame (P = P):
For a thermal distribution,
the hadron wavefunctions can be integrated out, eliminating the model dependence of predictions. This is true even if higher Fock space states are included!
( , ) exp( v / ) w r p p T
0
3 2 012
with and
k
d k dk d k k k k k xPk
Beyond the lowest Fock state1
2 21
0
12
2
0
; ( 1) ( , ; ) ( ) ( )
( 1) ( , , ; ) ( ) ( ) ( )
a b a b a b a b
a b c a b c a b c a b c
M Q dx dx x x x x Q q x q x
dx dx dx x x x x x x Q q x q x g x
12
2
0
12/
2
0
( ) ( ) (( 1) ( , , )
( , ,1 )
)q a qqqg a b c a b c a b c
P Ta b a b a b
a g cW dx dx dx x x x x x x
e
w x w x w
dx dx x x x x
x
/ / / ( ) /a b c a b cx P T x P T x P T x x x P Te e e e
1 12 2/
1 2
0
/
0
( ,1 ) ( , ,1 )P Tqq qqg a a
P Tb a b a a bM bW W e dx x x dx dx x x x xW e
12
1
0
1 12 2/
1 1
0 0
( 1) ( , )
( )
( ) ( ) ( ) ( )
( 1) ( , ) ( ,( ) 1 )
qq a b a b a b a b a b
P Ta b q a q aa b a b a a aw x w x
W dx dx x x x x q x q x q x q x
dx dx x x x x e dx x x
/ / ( ) /a b a bx P T x P T x x P Te e e For thermal medium
Statistical model vs. recombination
3
1
3
3 exp / 12
with
B s
i
ii i i iI
id NE P d
df P u
gf P u P u B S I T
P
In the stat. model, the hadron distribution at freeze-out is given by:
For pt , hadron ratios in SM are identical to those in recombination!
(only determined by hadron degeneracy factors & chem. pot.) recombination provides microscopic basis for apparent chemical equilibrium among hadrons at large pt BUT: Elliptic flow pattern is approximately additive in valence quarks, reflecting partonic, rather than hadronic origin of flow.
Recombination vs. Fragmentation
1h 1
h3 3 30
( , ) ( )(2 ) z
dN P u dzE d w r P D z
d P z
Fragmentation:
…always wins over fragmentation for an exponential spectrum (z<1):
… but loses at large pT, where the spectrum is a power law ~ (pT) -b
ee xxp p(v / )) v /( PP zTT
Recombination…
( , ) ( , (1 ) ) exp v /
( , ) ( , ' ) ( , (1 ') ) exp v /
w r xP w r x P P T
w r xP w r x P w r x x P P T
Meson
Baryon
2M
M3 3,
( , ) ( , (1 ) ) ( )(2 )
dN P uE d dxw R xP w R x P x
d P
Recombination:
Model fit to hadron spectrum
Teff = 350 MeV blue-shifted temperature
pQCD spectrum shifted by 2.2 GeV
R.J. Fries, BM, C. Nonaka, S.A. Bass (PRL )
Corresponds to ≥ 0.6 !!!
Recall:
G 0.5 ln(…)
Q 0.25 ln(…)
Hadron Spectra I For more details – see S.A. Bass’ talk tomorrow
Hadron Spectra II
Hadron dependence of high-pt suppression
• R+F model describes different RAA behavior of protons and pions
• Jet-quenching becomes universal in the fragmentation region
Hadron production at the LHC
r = 0.75
r = 0.85
r = 0,65
R.J. Fries
includes parton energy loss
Conclusions (1)
• Evidence for dominance of hadronization by quark recombination from a thermal, deconfined phase comes from:
– Large baryon/meson ratios at moderately large pT;
– Compatibility of measured abundances with statistical model predictions at rather large pT;
– Collective radial flow still visible at large pT.
• -meson is an excellent test case (if not from KK).
Parton Number Scaling of Elliptic Flow
In the recombination regime, meson and baryon v2 can be obtained from the parton v2 (using xi = 1/n):
3
2
2 2
2
2
22
2
2
3v 32v2
v
1 2 v
v3 3
v
1 6 v32
and
p pt t
B
p t
Mt t
p tp t
p p
p
p
pp p
Neglecting quadratic and cubic terms, a simple scaling law holds:
2 2 2 2an vv23
vd2
v 3M B ptt
tp tpp
pp
Originally poposed by S. Voloshin
Hadron v2 reflects quark flow !
Higher Fock states don’t spoil the fun
1 2
1 2
M C qq C qqg
B C qqq C qqqg
2
2
( )1
( ) 22
( )1
( ) 22
( , , )
( , ,
(
,
, )
( , , )
)
Ba b c a b c
Ba b c g
Ma b a b
Ma b g
a b c g
a b g
x x x
x x x x
x x x x x
x x x
x x x x x x x x
x
Conclusions (2)
• Recombination model works nicely for v2(p):
– v2(pT) curves for different hadrons collapse to universal curve for constituent quarks;
– Saturation value of v2 for large pT is universal for quarks and agrees with expectations from anisotropic energy loss;
– Vector mesons (, K*) permit test for influence of mass versus constituent number (but note the effects of hadronic rescattering on resonances!);
– Higher Fock space components can be accommodated.
Enough of the Successes…
Give us some Challenges!
Dihadron correlations
Data: A. Sickles et al. (PHENIX) Hadrons created by reco from a thermal medium should not be correlated.
But jet-like correlations between hadrons persist in the momentum range (pT 4 GeV/c) where recombination is thought to dominate!
( STAR + PHENIX data)
Hadron-hadron correlations
Near-side dihadron correlations are larger than in d+Au !!!
Far-side correlations disappear for central collisions.
d+Au
A. Sickles et al. (PHENIX)
Sources of correlations
• Standard fragmentation
• Fragmentation followed by recombination with medium particles
• Recombination from (incompletely) thermalized, correlated medium
• But how to explain the baryon excess?
• “Soft-hard” recombi-nation (Hwa & Yang). Requires microscopic fragmentation picture
• Requires assumptions about two-body cor-relations (Fries et al.)
How serious is this?
• Original recombination model is based on the assumption of a one-body quark density. Two-hadron correlations are determined by quark correlations, which are not included in pure thermal model.
• Two- and multi-quark correlations are a natural result of jet quenching by energy loss of fast partons.
• Incorporation of quark correlations is straightforward, but introduces new parameters: C(p1, p2).
Diparton correlations
• Parton correlations naturally translate into hadron correlations.
• Parton correlations likely to exist even in the "thermal" regime, created as the result of stopping of suprathermal partons.
A plausible explanation?
Two-point velocity correlations among 1-2 GeV/c hadrons
away-side same-side
T. Trainor
Dihadron formation mechanisms
(1 )
( )(1 )
A Aa
A A A
A BA
A A
dz Pg E
z z z
z PD
F
z
F
z DP
12
1eff2
( )
exp2
a A B
A B
A B
g P P E
P PD
P P
S F
T
H
eff
exp A BSSP
TS
PS
A
A
A
B
B
B
Correlations - formalism
222211114
2
2
2
123
13
6
2
23
13 2
1,
2
1,
2
1,
2
1)()(
)2(qPqPqPqPWqqqdqd
V
PdPd
dNMM
Di-meson production:
)(),,( 1 in
nn pwppW 1 ,qq i ji j
C p p
Partons with pairwise
correlations2
2 21 2 0 1 23 3 6
1 2
1 1 1 11 2 4 ,
(2 ) 2 2 2 2MM
dN Vw P w P C C P P
d Pd P
Meson-meson, baryon-baryon, baryon-meson correlations
9 , 6 , 4BB qq MB qq MM qqC C C C C C
First results of model studies are encouraging →
Dihadron correlations - results
Meson triggers Baryon triggers
by 100/Npart
Fixed correlation
volume
Comparison with Data
R.J. Fries, S.A. Bass, BM, nucl-th/0407102, acct’d in PRL
Conclusions – at last!
Evidence for the formation of a deconfined phase of QCD matter at RHIC:
Hadrons are emitted in universal equilibrium abundances;
Most hadrons are produced by recombination of quarks;
Hadrons show evidence of collective flow (v0 and v2);
Flow pattern (v2) is not universal for hadrons, but universal for the (constituent) quarks.
Hadron correlations from quasithermal quark correlations.