hadronization of partons by recombination rudolph c. hwa university of oregon summer school on rhic...
TRANSCRIPT
Hadronization of Partons by Recombination
Rudolph C. HwaUniversity of Oregon
Summer School on RHIC Physics
Wuhan, China, June 2005
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Outline
An overview of the recombination model
Some questions and answers on the basics
Shower partons initiated by hard partons
Hadronization in heavy-ion collisions
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Parton Recombination
First studied for low-pT production in pp collision
Das & Hwa, Phys. Lett. 68B, 459 (1977)
EdNπ
dpL
≡H(x) =dx1x1
∫dx2
x2
Fqq (x1,x2)Rπ (x1,x2,x)
p pFu x1( )Fd x2( )
x
H(x)
Ochs observation: H(x) is very similar to the valence quark distribution in a proton.
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Valon model -- to get the proton wave function Hwa, PRD (1980a)
Valon-recombination model -- better formulation of recombination Hwa,
PRD (1980b)
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p+A collisions Hwa & CB Yang (2002a)
We studied the centrality dependence (or the number of collisions) in the valon-recombination model
good data from NA49
Hadronic collisions Hwa & CB Yang
(2002b) h + p h’ +X
h h’
p
K+ + K
m
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Hadron production at high pT pp collision: mainly by fragmentation
AA collision: there were puzzles according to fragmentation
Recombination solved those puzzlesHwa & Yang, PRC 67, 034902 (2003); 70, 024905 (2004)
Greco, Ko, Levai, PRL 90, 202302 (2003); PRC 68, 034904 (2003)
Fries, Muller, Nonaka, Bass, PRL 90,202303(03); PRC 68, 044902 (03)
More recent developments -- 2004, 2005Correlations in jets
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Closer examination of the recombination formulas
pdN
dp=
dq1
q1∫
dq2
q2
Fqq(q1,q2 )R (q1,q2 , p)Pion :
Proton :
pdN p
dp=
dq1
q1∫
dq2
q2
dq3
q3
Fuud(q1,q2 ,q3)Rp(q1,q2 ,q3, p)
Fqq (q1,q2 )
Questions:
1. What is the two-parton distribution ?
2. What are the recombination functions ?
R (q1,q2 , p)
Especially in heavy-ion collisions
Rp (q1,q2 ,q3, p)
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3. What about the gluons?
4. Does entropy decrease?
5. What about the spatial considerations?
6. Isn’t the pion a Goldstone boson?
7. Recombination versus fragmentation:
Which is more important?
8. What is wrong with string fragmentation?
More questions :
Answer in reverse order.
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recombination
8. String fragmentation
• String model may be relevant for pp collisions,
• String/fragmentation has no phenomenological support in heavy-ion collisions.
but not for AA collisions.
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High pT physics in pp collisions is well
understood.What was a discovery yesterday is now used for calibration today.
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7. Recombination versus Fragmentation
Parton distribution (log scale)
p
p
hadron momentum
q1+q2
(recombine)
higher yield
q
(fragment)
heavy penalty
suppressed by power-law
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6. Pion is a Goldstone boson
Is it a boson due to spontaneous symmetry breaking? Or a bound state of quark-antiquark?
Both are aspects of the pion.
No theory exists that can continuously transform one to the other.
In Drell-Yan process in -p collisions, the quark contents of pion and proton are probed.
p
+
-
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5. Spatial considerationsWe have formulated recombination in momentum space only so far.
Shouldn’t the spatial coordinates be important also? Isn’t hadron size relevant?
In heavy-ion collisions there are two sizes:
1. nuclear transverse size RA2. hadron transverse size rh
• If partons are parallel, but far apart, they cannot recombine
• If parton trajectories intersect, they must cross at the same space-time region --- relative momentum suppress recombination.
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Soft partons are restricted to the small spatial spread around the point where hard
parton emerges from the nuclear medium.
Groups at Duke University and Texas A&M University have Monte Carlo codes to implement space & momentum constraints on recombination.
Our approach:
We consider only collinear partons. Hard parton defines the direction of the hadron.
We do not use Monte Carlo code to generate the soft partons throughout the expanding medium. We infer from the soft pion spectrum at low pT what the soft parton
distribution is.Momentum space consideration is sufficient, and that is where observation is made.
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4. Entropy
u + d → +
color: 3 X 3 1
spin: 2 X 2 1
degrees of freedom decreased
R (q1,q2 , p) = f(q1,q2 , p)δ(q1 + q2 −p)
depends on wave function momentum conservation
g
q1 + q2 =pd
+
uSoft gluon radiation: mutates color & carries away spin without changing
RM cannot account for low momentum partons
Entropy: a global quantity that should take into account expanding volume.
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3. How do gluons hadronize?
In pp collisions the parton distributions are
Gluons carry ~1/2 momentum of proton but cannot hadronize directly.
Sea quark dist. Fq ~ c (1-x)7
Saturated sea quark dist. F’q ~ c’
(1-x)7
Gluon conversion to q-qbarq
q
g
Recombination of with saturated sea gives pion distribution in agreement with data.
x2u(x)
x2g(x)
x [log]
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2. Recombination functions
It depends on the wave function.
q
q
uud
p
Consider the time-reversed processuud
p puu
d
What are the distributions of the quarks in momentum fractions in the infinite momentum frame?
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Deep inelastic scattering
ee
p
Fq
We need a model to relate to the wave function of the proton
Fq
Valon modelp
U
U
Dvalons
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p
U
U
D
Basic assumptions
• valon distribution is independent
of probe
• parton distribution in a valon is independent of the hadron
xuv (x,Q2 ) = dy2GUx
1
∫ (y)KNS(xy,Q2 )
xdv (x,Q2 ) = dyGDx
1
∫ (y)KNS(xy,Q2 )
valence quark distr in proton
valon distr in proton, independent of Q
valance quark distr in valon, in proton or in pion
%uv (n,,Q2 ) =2 %GU (n) %KNS(n,Q
2 )
%dv (n,,Q
2 ) = %GD(n) %KNS(n,Q2 )
Moments by convolution theorem
known from CTEQ param
cancel in the ratio
the ratio can be determined
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3-valon exclusive distribution
Recombination function
α =1.76 β =1.05
proton Rp(x1,x2,x3,x) =x1x2x3
x3 GUUDp (
x1
x,x2
x,x3
x)=g(
x1x2
x2 )2.76(x3
x)2.05δ(
x1
x+
x2
x+
x3
x−1)
pion Rπ (x1,x2,x) =x1x2
x2 GUD π (
x1
x,x2
x)=
x1x2
x2 δ(x1
x+
x2
x−1)
From initiated Drell-Yan process
xqvπ(x) =Ax0.64(1−x)1.11 valon
model Gπ (y1,y2) =δ(y1 +y2 −1)
GU (y) = dy2∫ dy3∫ GUUD(y1,y2 ,y3)
GD(y) = dy1∫ dy2∫ GUUD(y1,y2 ,y3)
Single-valon inclusive distribution
Hwa & CB Yang, PRC66(2002)
GUUD(y1,y2,y3) =g(y1y2)α y3
βδ(y1 +y2 +y3 −1)
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1. Two-parton distributions
Fqq (q1,q2 ) Fuud (q1,q2 ,q3)
pp collisions: low pT and large xF
Fud
(x1, x2 ) =Fu(x1)Fd(x2 )
Heavy-ion collisions:
Low pL (mid-rapidity), large
pTThat is high pT physics.
Traditionally, hadronization at high pT is by
fragmentation.However, fragmentation model has met some difficulties, most notably in p/ ratio at intermediate pT in nuclear collisions.
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Before describing what the two-parton distribution should be at high pT in heavy-ion
collisions, we must first
• discuss why fragmentation does not work phenomenologically
• what are the shower partons in fragmentation?
• how does the nuclear medium affect hadronization?
Which parton recombines which parton is the core problem in the recombination model.
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The black box of fragmentation
q
A QCD process from quark to pion, not calculable in pQCD
z1
Momentum fraction z < 1
Phenomenological fragmentation function
D/q
z1
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Let’s look inside the black box of fragmentation.
q
fragmentation
z1
gluon radiation
quark pair creation
Although not calculable in pQCD (especially when Q2 gets low), gluon radiation and quark-pair creation and subsequent hadronization nevertheless take place to form pions and other hadrons.
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Description of fragmentation by recombination
known from data (e+e-, p, … )
known from recombination model
can be determined
hard partonmeson
fragmentationshower partons recombination
xD(x) =dx1x1
∫dx2
x2Fq,q (x1,x2)Rπ (x1,x2,x)
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Shower parton distributions
Fqq '(i )(x1,x2) =Si
q(x1)Siq ' x2
1−x1
⎛
⎝ ⎜ ⎞
⎠ ⎟
Sij =
K L Ls
L K Ls
L L Ks
G G Gs
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
u
gs
s
d
du
K =KNS+L
Ks =KNS +Ls
Sud,d ,u ,u(sea) =L
valence
sea
L L DSea
KNS L DV
G G DG L
Ls DKSea
G Gs DKG
R
RK
5 SPDs are determined from 5 FFs.
assume factorizable, but constrained kinematically.
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If our shower parton distributions are reliable, based on the dynamical independence of the shower partons except for kinematical constraints, then we should be able to calculate the quark fragmentation function into a proton.
Data on Dup(z) not well
determined. KKP parametrization has an error.
Nevertheless, there is only a discrepancy of less than a factor of 2 over 4 order of magnitude.
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Once the shower parton distributions are known, they can be applied to heavy-ion collisions.
The recombination of thermal partons with shower partons becomes conceptually unavoidable.
D(z)
h
qA A
Conventional approach
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Once the shower parton distributions are known, they can be applied to heavy-ion collisions.
The recombination of thermal partons with shower partons becomes conceptually unavoidable.
hNow, a new component
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Pion formation: qq distribution
thermal
shower
soft component
soft semi-hard components
usual fragmentation
(by means of recombination)
T
SFqq =TT+TS+SS
Proton formation: uud distribution
Fuud =TTT +TTS +TSS +SSS
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T(p1)=p1dNq
th
dp1=Cp1exp(−p1/T)
Thermal distribution
Fit low-pT data to determine C & T.
Shower distribution in AuAu collisions
S(p2)=ξ∑i ∫dkkfi(k)Sij(p2 /k)
hard parton momentum
distribution of hard parton i in AuAu collisions
Contains hydrodynamical properties, not included in our model.
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fi (k) =dNi
hard
kdkdyy=0density of hard partons with pT = k
Input: parton distributions CTEQ5L nuclear shadowing EKS98 hard scattering pQCD
Srivastava, Gale, Fries, PRC 67, 034903 (2003)
dNjet
d2 pTdyy=0
=K C(1+pT / B)β
C, B, are tabulated for i=u, d, s, u, d, g K=2.5
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T(p1)=p1dNq
th
dp1=Cp1exp(−p1/T)
Thermal distribution
Fit low-pT data to determine C & T.
Shower distribution in AuAu collisions
S(p2)=ξ∑i ∫dkkfi(k)Sij(p2 /k)
hard parton momentum
distribution of hard parton i in AuAu collisions
SPD of parton j in shower of hard parton i
fraction of hard partons that get out of medium to produce shower
calculable
Contains hydrodynamical properties, not included in our model.
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thermal
fragmentation
soft
hard
TS Pion distribution (log scale)
Transverse momentum
TT
SS
Now, we go to REAL DATA, and real theoretical results.
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production in AuAu central collision at 200 GeV
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Hwa & CB Yang, PRC70, 024905 (2004)
TS
fragmentation
thermal
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Proton production in AuAu collisions
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
TTS+TSS
TSS
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QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.All in recombination/ coalescence model
Compilation of Rp/ obtained by 3 groups
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Puzzle in pA or dA collisions
kT broadening by multiple
scattering in the initial state.
Unchallenged for ~30 years.
If the medium effect is before fragmentation, then should be independent of h= or p
Cronin Effect Cronin et al, Phys.Rev.D (1975)
p
q
hdNdpT
(pA→ πX)∝ Aα , α >1
A
RCPp >RCP
πSTAR, PHENIX (2003)
Cronin et al, Phys.Rev.D (1975)
p >
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RHIC data from dAu collisions at 200 GeV per NN pair
Ratio of central to peripheral collisions:
RCP
RCPh (pT ) =
dNh
dpT
1NColl
central( )
dNh
dpT
1NColl
peripheral( )
PHENIX and STAR experiments found (2002)
RCPp (pT )>RCP
π (pT )
Can’t be explained by fragmentation.
=geometrical factor(central) × Dh
geometrical factor(peripheral) × Dh
(in fragmentation model)
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d
d
central peripheral
more T more TS
less T less TS
RCPh (pT ) =
dNh
dpT
1NColl
central( )
dNh
dpT
1NColl
peripheral( ) ⇒
more TSless TS
>1
d+Au collisions (to study the Cronin Effect)
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d+Au collisions
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Pions
Hwa & CB Yang, PRL 93, 082302 (2004)
No pT broadening by multiple scattering in the initial state.Medium effect is due to thermal (soft)-shower
recombination in the final state.
soft-soft
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QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Proton
Thermal-shower recombination is negligible.
Hwa & Yang, PRC 70, 037901 (2004)
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Nuclear Modification Factor
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
RCPp >RCP
π
This is the most important result that validates parton recombination.
2q , each quark has ~1/2 of momentum
3q p, each quark has ~1/3 of p momentum
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Molnar and Voloshin, PRL 91, 092301 (2003).
Parton coalescence implies that v2(pT)
scales with the number of constituents
STAR data
Azimuthal anisotropy
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Forward-backward asymmetry in d+Au collisions
Expects more forward particles at high pT than backward particles
If initial transverse broadening of parton gives hadrons at high pT, then
• backward has no broadening
• forward has more transverse broadening
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QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Backward-forward asymmetry at intermediate pT
backwardforward
in d+Au collisions
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More interesting behavior found in large pT and large pL region.
It is natural for parton recombination to result in forward-backward asymmetry
Less soft partons in forward (d) direction than backward (Au) direction.
Less TS recombination in forward than in backward direction.
Forward-backward asymmetry by recombination
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QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Hwa, Yang, Fries, PRC 71, 024902 (2005)
Forward production in d+Au collisions
Underlying physics for hadron production is not changed from backward to forward rapidity.
BRAHMS data