Nuclear dynamics in the dissociative recombination of H 3 + and its isotopologues
Recombination in Nuclear Collisions
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Transcript of Recombination in Nuclear Collisions
Recombination in Nuclear Collisions
Rudolph C. HwaUniversity of Oregon
Critical Examination of RHIC Paradigms
University of Texas at Austin
April 14-17, 2010
Outline
1. Introduction
2. Earlier evidences for recombination
3. Recent development
A. Azimuthal dependence --- ridges
B. High pT jets --- scaling behavior
4. Future possibilities and common ground
1. Introduction
pQCDReCoHydro
Fragmentation
kT > pT
Hadronization
Cooper-Frye
k1+k2=pT
lower ki higher density
TT TS SS
low highintermediate
2 6
Usual domains in pT
pT
GeV/c
Regions in time
(fm/c)
1 8
hadronization
0.6rapid thermalization
hydro
Cronin effect: --- initial-state transverse broadening
What about Cronin effect for proton, larger than for ?
Early-time physics: CGC, P violation, …
Pay nearly no attention to hadronization at late times.
In ReCo: Final-state effect, not hard-scattering+Frag, not hydro.
What about semihard scattering (kT<3GeV/c) at <0.6 fm/c?
2. Earlier evidences for Recombination
A. pT distribution at mid-rapidity
Recombination function
R (k1,k2 , pT ) =k1k2
pT2
δ(k1 +k2
pT
−1)
q and qbar momenta, k1, k2, add to give pion pT
It doesn’t work with transverse rapidity yt
TT F(ki ) =Cki exp(−ki /T )dN
pTdpT
=C2
6exp(−pT /T )
TTTdN p
pT dpT
=NppT
2
mT
exp(−pT /T ) same T for partons, , p
empirical evidence
At low pT
phase space factor in RF for proton formation
Pion at y=0 p0 dN
dpT
=dk1
k1∫
dk2
k2
Fqq(k1,k2 )R (k1,k2 , pT )
Proton at y=0
p0 dN p
dpT
=dk1
k1∫
dk2
k2
dk3
k3
Fuud(k1,k2 ,k3)Rp(k1,k2 ,k3, pT )
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PHENIX, PRC 69, 034909 (04)
went on to mT plot
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Hwa-Zhu (preliminary)
dN p
pT dpT
=NppT
2
mT
exp(−pT /T )
Proton production from reco
Same T for , K, p --- a direct consequence of ReCo.
Slight dependence on centrality --- to revisit later
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B. p/ ratio
At higher pT shower partons enter the problem; TS recombination enters first for pion, and lowers the ratio.
It is hard to get large p/ ratio from fragmentation of hard partons.
Rp / (pT ) =dNp / pTdpT
dN / pTdpT
dominated by thermal partons at low pT
= pT2
mT (pT )
ReCo
C. Revisit very early formulation of recombination
[at the suggestion of organizers: Hwa, PRD22,1593(1980)]
The notion of valon needs to be introduced.
q
q
For p+pp+X we need
Rp (x1, x2 , x3, x)uud
p
Consider the time-reversed processu
ud
p puu
d
p+p+X Feynman x distribution at low pT
xdN
dx=
dx1
x1∫
dx2
x2
Fqq(x1,x2 )R (x1,x2 ,x)
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
A valence quark carries its own cloud of gluons and sea quarks --- valon
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 distribution in valon, whether in proton or in pion
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Rp (x1, x2 , x3, x) =g(x1x2
x2 )2.76 (x3
x)2.05δ(
x1
x+
x2
x+
x3
x−1)
R (x1,x2 ,x) =x1x2
x2 δ(x1
x+
x2
x−1) initiated
DY process
p + p h + X in multiparticle production at low pT
p
U
U
Dvalon distribution collisio
n process
partons
chiral-symmetry breaking quarks gain masses momenta persist
U
D
RF
+
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No adjustable parameters
1979 data (Fermilab E118)
Not sure whether anyone has done any better
Feynman’s original parton model PRL(69)
D. Shower partons in AA collisions
At higher pT Hard scattering calculable in pQCD Hadronization by fragmentation
In between hard scattering and fragmentation is jet quenching.
Fine, at very high pT (> 6GeV/c), but not reliable at intermediate pT
pT
qD
i (
pT
q)
T(q1)S(q2/q)R(q1,q2,pT)
Fragmentation: D(z) => SS recombination, but there can also be TS
recombination at lower pT
dNTS+SS
pTdpT
=1pT
2
dqq
Fi∫ (q)[TS∂i∑ (q, pT ) + SS∂ (q, pT )]pio
n
proton [TTS∑ +TSS∑ + SSS∑ ]
We need shower parton distribution.
∫dk k fi(k) G(k,q)
k
q
Description of fragmentation
known from data (e+e-, p, … )
known from recombination model
can be determined
recombination
xD(x) =dx1x1
∫dx2
x2Fq,q (x1,x2)Rπ (x1,x2,x)
shower partons
hard partonmeson
fragmentation
by recombination
Shower parton distributionsFqq '
(i )(x1,x2) =Siq(x1)Si
q ' x2
1−x1
⎛
⎝ ⎜ ⎞
⎠ ⎟
Sij =
K L Ls
L K Ls
L L Ks
G G Gs
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
u
gs
s
d
du
L L DSea
KNS L DV
GG DG
L Ls DKSea
G Gs DKG
5 SPDs are determined from 5 FFs.
assume factorizable, but constrained kinematically.
Hwa & CB Yang, PRC 70, 024904 (04)
BKK FF(mesons)Using SSS we can calculate baryon FF
DM ⇔ DB
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Hwa-Yang, PRC 73, 064904 (06)
Other topics:
1. Constituent quarks, valons, chiral-symmetry breaking, f
2. Collinear recombination
3. Entropy
4. Hadronization of gluons
5. Dominance of TS over TT at pT>3 GeV/c
6. Single-particle distributions
7. RCPp(pT)> RCP
(pT)
8. Forward-backward asymmetry in dAu collisions
9. Large p/ ratio at large
10. v2 (pT) Quark-number scaling
11. Ridges
12. Correlations
earlier
later
recent
3. Recent developmentAzimuthal dependence
PHENIX 0903.4886
85<<90
30<<45
0<<15
pT
Npart
A. pT < 2 GeV/c
B. pT > 2 GeV/c
A. pT<2 GeV/c
Region where hydro claims relevance --- requires rapid thermalization
0 = 0.6 fm/c
Something else happens even more rapidly
Semi-hard scattering 1<kT<3 GeV/c
Copiously produced, but not reliably calculated in pQCD t < 0.1 fm/c
1. If they occur deep in the interior, they get absorbed and become a part of the bulk.
2. If they occur near the surface, they can get out. --- and they are pervasive.
[Tom Trainor’s minijets (?)]
On the way out of the medium, energy loss enhances the thermal partons --- but only locally.
Recombination of enhanced thermal partons ridge particles
ρ1(pT ,φ,b) = B(pT ,b) + R(pT ,φ,b)
Base, independent of , not hydro bulk
Ridge, dependent on , hadrons formed by TT reco
• Ridge can be associated with a hard parton, which can give a high pT trigger.• But a ridge can also be associated with a semihard parton, and a trigger is not necessary; then, the ridge can be a major component of
ρ1(pT ,φ,b)
Correlated part of two-particle distribution on the near side
ρ2corr (1,2) = ρ2
J (1,2) + ρ2R (1,2)
Putschketrigger
assoc part
JET RIDGE
How are these two ridges related?
BOOM
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Hard parton
Ridge
without trigger
but that is a rare occurrence
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Semihard partons, lots of them in each event
Ridges without triggers --- contribute significantly to single-particle distribution
ratatatatatata
We need an analogy
1
2
1
2
Two events: parton 1 is undetected thermal partons 2 lead to detected hadrons with the same 2
R(φ2 ) ∝ dφ1∫ ρ2R(φ1,φ2 )
Ridge is present whether or not 1 leads to a trigger.
Semihard partons drive the azimuthal asymmetry with a dependence that can be calculated from geometry. Hwa-Zhu, 0909.1542, PRC (2010)
If events are selected by trigger (e.g. Putschke QM06, Feng
QM08), the ridge yield is integrated over all associated particles 2.
Y R (φ1) ∝ dφ2∫ ρ2R(φ1,φ2 )
Enhanced thermal partons on average move mainly in the direction normal to the surface
~|2-1|<~0.33 Correlated emission model
(CEM) Chiu-Hwa, PRC 79 (09)
Geometrical consideration in Ridgeology
For every hadron normal to the surface there is a limited line segment on the surface around 2
through which the semihard parton 1 can be emitted.
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b normalized to RA
Ridge due to enhanced thermal partons near the surface
R(pT,,b) S(,b)nuclear density
S(,b) 2
Base
ρ1(pT ,φ,b) = B(pT ,b) + R(pT ,φ,b)
Ridge
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base
ridge
inclusive
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ridge
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RH-L.Zhu (preliminary)
ρ1 (pT ,φ,b) = B(pT ,b) + R(pT ,φ,b) = N(pT ,b)[e− pT /T0 + e− pT /T1 (b)aD(b)S(φ,b)]
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Single-particle distribution at low pT without elliptic flow, but with Ridge
T0 for base
T1(b) for ridge
a can be determined from v2, since S(,b) is the only place that has dependence.
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ridge bas
e
Azimuthal dependence of ρ1(pT,,b) comes entirely from Ridge ---
In hydro, anisotropic pressure gradient drives the asymmetry
x
y
requiring no rapid thermalization, no pressure gradients.
Since there more semihard partons emerging at ~0 than at ~/2, we get in ReCo anisotropic R(pT,,b),
∝ S(φ1,b)
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Hwa-Zhu, PRC (10)
Y R (φ1) ∝ dφ2∫ ρ2R(φ1,φ2 )
Ridge yield’s dependence on trigger
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Feng QM08
φs ⇒ φ1
Normalization adjusted to fit, since yield depends on exp’tal cuts
Normalization is not readjusted.
s dependence is calculated
S(,b) correctly describes the dependence of correlation
Nuclear modification factor
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art
Summary
dependencies in
Ridge R(pT,,b) v2(pT,b)=<cos 2 > yield YR() RAA(pT,,b)
are all inter-related --- for pT<2 GeV/c
Hwa-Zhu, 0909.1542 PRC (2010)
B. pT>2 GeV/cPHENIX 0903.4886
Need some organizational simplification. and b are obviously related by geometry.
Scaling behavior in --- a dynamical path length
5 centralities and 6 azimuthal angles () in one universal curve for each pTLines are results of calculation in Reco.
Hwa-Yang, PRC 81, 024908 (2010)
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Complications to take into account:
• details in geometry
• dynamical effect of medium
• hadronization
Nuclear medium that hard parton traverses
x0,y0
k
Dynamical path length
=γl (x0 , y0 ,φ,b) γ to be determined
Geometrical path length
l (x0 , y0 ,φ,b) = dtD[x(t),y(t)]
0
t1 (x0 ,y0 ,φ,b)
∫D(x(t),y(t))
Geometrical considerations
Average dynamical path length
(φ,b) = γ dx0dy0∫ l (x0 , y0 ,φ,b)Q(x0 , y0 ,b)
Q(x0 , y0 ,b) =TA(x0 ,y0 ,−b / 2)TB(x0 ,y0 ,b / 2)
d2rsTA(rs+
rb / 2)TB(
rs−
rb / 2)∫
Probability of hard parton creation at x0,y0
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Define
P(,φ,b) = dx0 dy0Q(x0 ,y0∫ ,b)δ[ −γl (x0 ,y0 ,b)] (φ,b) = dξξP(∫ ξ ,φ,b)
KNO scaling
P(,φ,b) =ψ (z) (φ,b)
z = / dzψ (z) =1∫dzzψ (z) =1∫
For every pair of and c:
• we can calculate
• PHENIX data gives
(φ,c)
RAA (φ,c)
We can plot the exp’tal data
RAA ( )
There exist a scaling behavior in the data when plotted in terms of
(φ,c)
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Theoretical calculation in the recombination model Hwa-Yang, PRC 81, 024908 (2010) ( γ = 0.11 )
ρ1
TS +SS (pT ,φ,b) =dq
q∫ Fii
∑ (q,φ,b)H i (q, pT )
b
q
TS+SS recombination
G(k,q,) =qδ(q−ke− )
degradation
hadronization
dNihard
kdkdyy=0
= fi (k)
Fi (q,) = dkkfi∫ (k)G(k,q,)
k probability of hard parton creation with momentum k
geometrical factors due to medium
dNTS
pTdpT
=1pT
2
dqq∫
i∑ Fi (q)TS∂ (q, pT )
TS∂ (q, pT ) =
dq2
q2∫ Si
j (q2
q) dq1∫ Ce−q1 /T R (q1,q2 , pT )
dNSS
pTdpT
=1pT
2
dqq∫
i∑ Fi (q)SS∂ (q, pT )
xDi (x) =
dx1
x1∫
dx2
x2
Sij (x1),Si
j '(x2
1−x1
)⎧⎨⎩
⎫⎬⎭R (x1,x2 ,x)
x =pT / q
Nuclear modification factor
RAA (pT ,φ,c) =
dNAA / dpTdφ
NcolldNpp / dpT
only adjustable parameter γ = 0.11
=γl (x0 , y0 ,φ,b)
4. Future Possibilities
At kT not too large, adjacent jets can be so close that shower partons from two parallel jets can recombine.
H ii '(q,q ', pT ) =1pT
2
dq1
q1∫
dq2
q2
Sij (
q1
q)Si '
j '(q2
q')R
Γ (q1,q2 , pT )
≅ΓRπ (q1,q2 , pT )
Γ - probability for overlap of two shower partons
ρAA2 j ∝ Ncoll
2
RAA2 j (pT ,φ,c) =
ρAA2 j (pT ,φ,c)
Ncollρpp1 j (pT ,c)
At LHC, the densities of hard partons is high.
A. Two-jet recombination at LHC
Two hard partons
dNAA2 j
pT dpT dφ=
dqq∫
dq'q'
Fi (qii '∑ ,φ,b)Fi '(q',φ,b)Hii '(q,q', pT )
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Scaling
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Scaling badly broken
Hwa-Yang, PRC 81, 024908 (2010)
2jet
Pion production at LHC
Observation of large RAA at pT~10 GeV/c will be a clear signature of 2-jet recombination.
>1 !
Proton production due to qqq reco is even higher.
Hwa-Yang, PRL 97 (06)
B. Back-to-back dijets
C. Forward production of p and
D. Large correlation
E. Auto-correlation
F. P violation: hadronization of chirality-flipped quarks
G. CGC: hadronization problem
Common ground with the 2-component model of UW-UTA alliance
B. Two-component model
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T.Trainor, 0710.4504, IJMPE17,1499(08)
Hwa-Yang, PRC70,024905(04)
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Similar to our Base, B ~ exp(-pT/T0), T0 independent of b
minijets
Strong enhancement of hard component at small yt
Similar to our Ridge, R ~ exp(-pT/T1), T1 depends on b
ρAA
npart / 2= SNN (yt ) + ν H AA (yt ,ν )
SNN(yt) is independent of
Ridge due to semihard partons --- minijets?
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Comparison
Recombination 2-component
semihard partons minijets
recombination of enhanced fragmentation thermal partons
Ridges --- TT reco effect of jet on medium low-yt enhancement
Jets --- TS+SS effect of medium on jet high-yt suppressionρ1 = B + R + J ρ1 = S + H
no dependence on depend on b and
B+R accounts for v2 at pT<2GeV/c some quadrupole component without hydro without hydro ρ(η Δ ,φΔ )
ρ ref
In Recombination
averaged over B(pT) R(pT,b)ρ
1
h (pT ,b) = N h (pT ,b)[e− pT /T0 + A(b)e− pT /T1 ]
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In 2D autocorrelation
UW-UTA alliance
dependence
ρ(η Δ ,φΔ )
ρ ref
R(pT ,φ,b)=N(pT ,b)e−pT /T1 (b)aD(b)S(φ,b)
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Scaling in variable that depends on initial-state collision parameters only
ρρref
=Δρnf
ρ ref
(η Δ ,φΔ ) + 2Δρ[m]
ρ refm=1
2
∑ cos(mφΔ )
No hydro
Trainor, Kettler, Ray, Daugherity
minijet contribution
φΔ ηΔ
from the hard comp 2<yt<4
I would like to know how it depends on at each b
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cf. our ridge component
Conclusion
We should seek common grounds as well as recognize differences.
• Has common ground with minijets.
At pT<2GeV/c, ridges due to semihard scattering and TT reco account for various aspects of the data.At pT>2GeV/c, hard scattering and TS+SS reco account for the scaling behavior observed.
• Recombination can accommodate fragmentation.
• Has thermal distribution at late times, though not thermalization and hydro expansion at early times.