Future Measurements to Test
Recombination
Rudolph C. Hwa
University of Oregon
Workshop on Future Prospects
in QCD at High Energy
BNL, July 20, 2006
Outline
pT
• Introduction
• Recombination model
• Shower partons
1
xF
• Hadron production at low pT
• Hadron production at large 
• Hadron production at large pT
• Summary
2
I. Introduction
What are the properties of recombination
that we want to know and test?
dN 
dq1 dq2
p

Fqq (q1 , q2 )R (q1 , q2 , p)
dp
q1 q2
0
Fqq (q1,q2 )
R (q1,q2 , p)
probability of finding partons at q1 , q2
What partons?
probability for recombination
to form a pion at p
Same partons? What is that probability?3
Usual strong evidences for recombination
v2
number of constituent quarks scaling
partons

CQ
What about gluons?
Rp /
of order 1 or higher impossible by
fragmentation
Useful to remember in future measurements
4
Quantitative questions about recombination
eventually always become questions about the
nature of partons that are to recombine.
Two-particle correlation
 dqi 
dN
1

F4 (q1,q2 ,q3 ,q4 )R(q1,q3 , p1 )R(q2 ,q4 , p2 )
2  
p1dp1 p2 dp2 ( p1 p2 )  i qi 
q1 , q2 , q3 , q4
Where are the partons from?
Are they independent?
Are they from 1 jet, 2 jets,
or thermal medium?
5
Multiparton distributions in terms of the
thermal and shower parton distributions
Fqq (q1,q2 )  T T +T S +SS
Fuud (q1,q2 ,q3 )  T T T + T T S + T SS + SSS
F4 (q1,q2 ,q3 ,q4 )  (T T + T S + SS)13 (T T + T S + SS)24
6
II. Recombination Model
Recombination depends on the wave function of the hadron.
Constituent quark model describes the bound-state
problem of a static hadron.
What good is it to help us to know about the
distribution of partons in a hadron (proton)?
Valons
Valons are to the scattering problem
what CQs are to the bound-state problem.
7
Deep inelastic scattering
e
e
p
Fq
We need a model to relate Fq to the wave
function of the proton
U
Valon model
Hwa, PRD 22, 759
(1981)
p
U
valons
D
8
Basic assumptions
• valon distribution is
independent of probe
U
• parton distribution in
a valon is independent
of the host hadron
p
U
D
x 2
xuv (x,Q )   dy2GU (y)K NS ( ,Q )
x
y
2
1
x 2
xdv (x,Q )   dyGD (y)K NS ( ,Q )
x
y
2
1
valence quark valon distr
distr in proton in proton,
independent
of Q
valance quark
distribution in
valon, whether
in proton or in
pion
9
Hwa & CB Yang, PRC66(2002)
using CTEQ4LQ
10
Recombination function
It is the time-reversed process of the valon distributions
U
p
U
valon distribution
proton
pion
U
D
Rp (x1 ,x 2 , x3 ,x) 
R (x1 ,x 2 , x) 
From  initiated
Drell-Yan process
p
U
D
recombination
function
x1 x2 x 3 p x1 x2 x3
x1 x2 2.76 x 3 2.05 x1 x2 x3
G
(
,
,
)

g(
( )  (   1)
3
UUD
2 )
x
x x x
x
x
x
x
x
x1 x2  x1 x 2
x1 x 2 x1 x 2
G
(
,
)

  1)
2
2 (
UD
x
x x
x
x
x

xqv (x)  Ax
0.64
1.11
(1 x)
valon model

G (y1 , y2 )   (y1  y2  1)
11
In a pp or AA collision process
U
+
_
D
Is entropy reduced in recombination?
The number of degrees of freedom seems to be reduced.
Soft gluon radiation --- color mutation
without significant change in momentum
The number of degrees of freedom is not reduced.
12
How do gluons hadronize?
In a proton the parton
distributions are
x2u(x)
Gluons carry ~1/2 momentum of proton
but cannot hadronize directly.
x2g(x)
q
Gluon conversion to q-qbar
g
x [log]
q
Sea quark dist. Fq ~ c (1-x)7
Saturated sea quark dist. F’q ~ c’ (1-x)7
Recombination of q q with saturated sea gives pion
distribution in agreement with data.
13
III. Shower Partons from Fragmentation
Functions
The black box of fragmentation
q
1
A QCD process from quark to
pion, not calculable in pQCD
z

Momentum fraction z < 1
14
Description of fragmentation by recombination
hard parton
fragmentation
meson
shower
partons
recombination
dx1 dx2
xD(x)  
Fq,q (x1 , x 2 )R (x1, x 2 , x)
x1 x 2
known from data (e+e-, p, … )
can be
determined
15
Meson fragmentation function
dx1 dx2
xD (x)  
Fq,q (x1 , x2 )RM (x1 , x2 , x)
x1 x2
M
Baryon fragmentation function
S(xi)
dx1 dx2 dx3
xD (x)  
Fqqq (x1 , x2 , x3 )RB (x1 , x2 , x3 , x)
x1 x2 x3
B
DGp and DG can be calculated in the RM
16
Hwa & CB Yang, PRC 73, 064904 (2006)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
D D
M
G
B
G
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Has never been done before in
the 30 years of studying FF.
This is done in the RM with gluon conversion
shower partons  valons  hadrons.
17
IV. Hadron production at low pT
First studied in pp collision.
p
E
dN
dx dx
 H(x)   1 2 Fqq (x1 , x 2 )R (x1 , x2 ,x)
dpL
x1 x 2
p
Parton distributions at low Q2
Fu x1 Fd x 2 
H(x)
x
Hwa, PRD (1980)
.
18
Hadronic collisions
h + p  h’ +X
h
h’
p

K+
+

_
 +

K
Suggested
future
measurement
Hwa & CB Yang, PRC 66, 025205 (2002)
FNAL PL=100 GeV/c (1982)
Better data at higher energy
for p  , K, p, Y
19
Leading and non-leading D production
  p D

(m)
Leading (same valence quark)
Asymmetry
A
X
non-leading (sea quark)
 (leading)   (nonleading)
 (leading)   (nonleading)
Hwa, PRD 51,
85 (1995)
Suggested future
measurement:
x
dN
( p  D )
dx
20
pA collisions
p Ah X
h bears the effect of momentum
degradation --- “baryon stopping”.
NA49 has good data, but never published.
p  Pb  ( p  p)  X
p  Pb  (n  n )  X
(no target fragmentation, only projectile fragmentation)
Shape depends on degradation.
Normalization not adjustable.
Suggested future measurement:
dN
( p  A  h   X)
dx
for all x at higher energy
Measure
Hwa & CB Yang, PRC 65, 034905 (2002)
Need to know well the
momentum degradation effect.
21
Transfragmentation Region (TFR)
A Ah X
Theoretically, can hadrons be produced at xF > 1? (TFR)
It seems to violate momentum conservation, pL > √s/2.
In pB collision the partons that recombine must satisfy
x
i
1
i
x
p
i
1
i
B
A
B
But in AB collision the partons can come from different nucleons
In the recombination model the produced p and  can
have smooth distributions across the xF = 1 boundary.
22
: momentum
degradation factor
proton
proton-to-pion ratio
is very large.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Regeneration of soft
parton has not been
considered.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
pion
Hwa & Yang, PRC 73,044913 (2006)
Suggested
future
measurement
Particles at xF>1 can
be produced only by
recombination.
Determine the xF
distribution in the TFR
23
BRAHMS, PRL 93, 242303 (2004)
V. Large 
BRAHMS data show that in
d+Au collisions there is
suppression at larger .
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Hwa, Yang, Fries, PRC 71,
024902 (2005).
No change in physics
from =0 to 3.2
In the RM the soft parton density decreases,
as  is increased (faster for more central coll).
Suggested
future
measurement
dN
d pT dpT
for  and p
24
AuAu collisions
BRAHMS, nucl-ex/0602018
25
xF = 0.9
xF = 1.0
xF = 0.8
TS
TFR
?
TTT
TT
26
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Hwa & Yang (2006)
pT distribution fitted well by recombination of thermal partons
No jet => no associated particles
Suggested
future
measurement
• Focus on xF>1 region.
• Determine p/ ratio.
• Look for associated particles
27
VI. Hadron production at large pT, small pL
A. Cronin Effect
Cronin et al, Phys.Rev.D (1975)
dN
( pA  hX)  A ,
dpT
 1
for h= both  and p
This is an exp’tal phenomenon.
Not synonymous to initial-state kT broadening.
In the RM we have shown that final-state recombination alone
(without initial-state broadening) is enough to account for CE.
We obtained it for both  and p -- impossible by fragmentation.
Hwa & Yang, PRL 93, 082302 (2004); PRC 70, 037901 (2004).
Suggested
future
measurement
Measure p /  and  / K ratios
in d+Au collisions at all ,
both backward and forward.
28
Backward-forward Asymmetry
B
F
If hadrons at high pT are due to
initial transverse broadening of
parton, then
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
• forward has more
transverse broadening
• backward has no
broadening
Expects more forward particles at
high pT than backward particles
RM has B/F>1, since dN/d of soft partons decrease as  increases.
Suggested
future
measurement
Measure p and  separately
at larger range of ,
and for different centralities.
29
STAR (F.Wang, Hard Probes 06)
Correlation shapes are the
same, yields differ by x2.
d
associated yield
in this case Au
x=0.05
x=0.7
is larger than
associated yield
in that case
Au
d
x=0.7
x=0.05
Degrading of the d valence q?
Soft partons -- less in forward, more in backward
RM => less particles produced forward, more backward
30
B. p/ Ratio
Success of the recombination model
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Measure the ratio
to higher pT
All in recombination/
coalescence model
If it disagrees with prediction, it is not a
breakdown of the RM. On the contrary the
RM can be used to learn about the
distributions of partons that recombine.
31
C. Strange particles
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Hwa & CB Yang, nucl-th/0602024
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
40% lower
30% higher
Data from STAR
nucl-ex/0601042
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
2
4
6
This is not a breakdown of the RM.
We have not taken into account the
different hyperon channels in
competition for the s quark in the
shower.
32
 production
 production
130
GeV
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
small
more suppressed
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
33
We need to do more work to
understand the upbending
of .
We have assumed RFs
for  &  that may
have to be modified.
It is significant to note
that thermal partons can
account for the ratio up to
pT=4 GeV/c.
QGP:
s quarks enhanced
& are thermalized.
34
If  and  are produced mainly by the
recombination of thermal s quarks,
then no jets are involved.
Select events with  or  in the 3<pT<5 region,
and treat them as trigger particles. Look for
associated particles in the 1<pT<3 region.
Predict: no associated particles giving rise to
peaks in , near-side or away-side.
Suggested
future
measurement
Verify or falsify
that prediction
35
D. Jet Correlations
1. Correlation of partons in jets is negative
2. Correlation of pions in jets
Two-particle
distribution
but not directly measurable
 dqi 
dN
1

 q F4 (q1,q2,q3, q4)R(q1,q3, p1)R(q2 ,q4, p2 )
2
p1dp1 p2 dp2 ( p1 p2 )  
 i
i 
F4  (TT + ST + SS)13 (TT + ST + SS) 24
1 (1) 
k
dN 1
p1 dp1
2 (1,2) 
dN1 2
p1 dp1 p2 dp2
C2(1,2)  2 (1,2)  1(1) 1(2)
q1
q3
G2 (1,2) 
q2
C2 (1,2)
1(1)1(2)
1/ 2
q4
Hwa & Tan,
PRC 72,
024908 (2005)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
This can be measured.
36
3. D(zT)
zT 
pT (assoc)
pT (trig)
Trigger-normalized fragmentation function
Trigger-normalized
momentum fraction
X.-N. Wang, Phys. Lett. B 595, 165 (2004)
J. Adams et al., nucl-ex/0604018
STAR claims
universal behavior
in D(zT)
Focus on
this region
fragmentation
violation of universal
behavior due to medium
effect ---thermal37
shower recombination
Suggested future measurement
Study zT ~ 0.5 with pT(trigger) ~ 8-10 GeV/c
pT(assoc) ~ 4-5 GeV/c
Measure p/ ratio of associated particles.
My guess: R(p/) >> 0.1
if so, it can only be explained by
recombination.
Do this for both near and away sides.
38
4. Three-particle correlation
Conical Flow vs Deflected Jets
near
near
π
Medium
Medium
away
deflected jets
away
near
di-jets
Medium
away
mach cone
0
0
π
Ulery’s talk at
Hard Probes 06
39
Signal Strengths
Au+Au Central 0-12% Triggered
d+Au
Δ2
Δ2
Δ1
Δ1
• Evaluate signals by calculating average signals in the boxes.
• Near Side, Away Side, Cone, and Deflected.
40
More studies are needed.
• What is the multiplicity distribution (above background)
on the away side?
• If n=2 is much lower than n=1 events (on away side),
then the Mach-cone type of events is not the
dominant feature on the away side.
• What is the p/ ratio (above background) on the away
side?
• Evolution with higher trigger momentum should settle
the question whether cone events are realistic.
• Whatever the mechanism is, hadronization would be by
recombination for pT<6 GeV/c.
41
5. Using Factorial Moments to suppress statistical
background event by event.
Factorial moment for 1 event
1
fq 
M
M
 n (n
j
j
 1)    (n j  q  1)
j 1
Normalized factorial moment
Fq  fq / f1q
Event averaged NFM
Chiu & Hwa, nucl-th/0605054
Fq
(a) background only
(b) bg + 1jet
(c) bg + 2jets
Try it out, but it is not a way to test recombination.
42
VII. Two-jet Recombination
 and p production at high pT at LHC
New feature at LHC: density of hard partons is high.
High pT jets may be so dense that
neighboring jet cones may overlap.
If so, then the shower partons in two nearby jets
may recombine.
2 hard partons
1 shower parton
from each

p
43
Proton-to-pion ratio at LHC
 -- probability
of overlap of 2
jet cones
If (pT)~pT-7,
then we get
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
single jet
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Hwa & Yang,
PRL (to appear),
nuclth/0603053
44
10  pT  20 GeV/c
The particle detected has some associated partners.
But they are part of the background of an ocean of
hadrons from other jets.
There should be no observable jet structure
distinguishable from the background.
That is very different from a super-high pT jet.
A jet at 30-40 GeV/c would have lots of
observable associated particles.
45
We predict for 10<pT<20 Gev/c at LHC
• Large p/ ratio
• NO associated particles
above the background
Suggested
future
measurement
Verify or falsify these
two predictions
46
Summary
In general, all hadrons produced with pT<6 GeV/c are
by recombination.
Specifically, many measurements have been suggested.
Good signatures: large Rp/
in some regions no particles associated with
high pT trigger.
After recombination is firmly established,
the hadron spectra can be used to probe the
distributions of partons that recombine.
47
Backup slides
48
Let’s look inside the black box of fragmentation.
q
1
z

fragmentation
gluon radiation
quark pair creation
49
assume factorizable,
but constrained
kinematically.
Shower parton distributions
 x2 

F (x1 , x2 )  S (x1 )S 
1 x1 
q
i
u
d
K L
L K
j
Si  
L L

G G
q'
i
s
valence
Ls  u
Ls  d
No gluon
Ks  column

Gs  g
s
(i )
qq '
K  K NS  L
d,d ,u ,u(sea)
Su
sea
L
Ks  K NS  Ls
5 SPDs are determined from 5 FFs.
LL
KNS L
GG
L Ls
G Gs
R
RK





DSea
DV
DG
DKSea
DKG
50
Shower Parton Distributions
Hwa & CB Yang, PRC 70, 024904 (04)
51
D. Jet Correlations
1. Correlation of partons in jets
a. Two shower partons in a jet in vacuum
Fixed hard parton momentum k
(as in e+e- annihilation)
1(1)  Sij (x1)
r2 (1,2) 

x2 
)
1  x1 
k
x1
2 (1,2)  Sij (x1 ),Sij ' (

2 (1,2)
 1 (1) 1 (2)
x2
x1  x2  1
The two shower partons are correlated.
no correlation
C2 (1, 2)  [r2 (1, 2)  1]1 (1)1 (2)  0
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Hwa & Tan, PRC 72, 024908 (2005)
No way to measure this directly.
52
b. Two shower partons in a jet in HIC
Hard parton momentum k is not fixed.
1 (1)  S j (q1 )     dkkfi (k)Sij (q/ k)
i
 j q1 j ' q2 
2 (1, 2) = (SS) (q1,q2 )     dkkfi (k) Si ( ),Si (
)
k
k

q
i
1 

jj '
2 (1,2)
r2 (1,2) 
 1 (1) 1 (2)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
fi(k)
fi(k) fi(k)
fi(k) is small for 0-10%,
smaller for 80-92%
Also, cannot be
measured directly.
53