Transcript Mean pt fluctuations - Universitetet i Bergen

Anisotropic Flow and Phase transitions,
…and a little bit on fluctuations/correlations
Sergei Voloshin
Wayne State University, Detroit
Outline:
- Anisotropic flow: where to look for a phase transition
- v1(y) - directed flow “wiggle”
- v2(pt) – constituent quark number scaling
- v2(pt) – “mass splitting” and QGP
- v2(energy,centrality) – approaching “hydro limit”
- v2/ vs dN/dy/S, any “wiggle/step”?
- Correlation functions and fluctuations.
- Centrality dependence of <dpt dpt> and radial flow.
- Conclusions
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
Anisotropic flow
Anisotropic flow  correlations
with respect to the reaction plane
Picture: © UrQMD
X
Term “flow” does not mean
necessarily “hydro” flow – used
only to emphasize the collective
behavior  multiparticle
azimuthal correlation.
Z
Note large orbital angular momentum in the system.
- Parity violation
- Orbital momentum  particle spin.
d 3N
d 2N 1

( 1  2v1 cos(φ )  2v2 cos( 2φ)  ...)
dpt dy dφ dpt dy 2
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2
b
XZ – the reaction plane
Fourier decomposition of single particle inclusive spectra:
Directed flow

Elliptic flow
2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
v1
Hydro: “antiflow”, “third flow component”
Csernai, Rohrich, PLB 458 (1999) 454.
Magas, Csernai, Strottman, hep-ph/0010307
rapidity
Brachmann, Soff, Dumitru, Stocker, Maruhn, Greiner
Bravina, Rischke , PRC 61 (2000) 024909
Net baryon
density
flow
antiflow
- Strongest at the softest point?
- The same for pions and protons ?
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Third flow component as the QGP signal
L.P. Csernai, D. Rohrich
PRL 458 (1999) 454
px  v1 pt
“Wiggle is present only
for the QGP EoS.
This calculations have been
done at 11 AGeV. Would the
results change for RHIC?
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Wiggle from anti-flow: development in time.
J. Brachmann Soff, Dumitru,
Stocker, Maruhn, Greiner
Bravina, Rischke,
PRC 61, 024909 (2000)
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Wiggle from uRQMD
“Rich” dependence on the particle type: baryons, antibaryons, mesons
Marcus Bleicher, Horst Stocker
PLB 526, (2002) 309-314
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Anti-flow from shadowing
Anti-flow is developing
in more peripheral collisions
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Directed flow “wiggle” in cascade models
R. Snellings, A. Poskanzer, S.V., nucl-ex/9904003
R. Snellings, H. Sorge, S.V., F. Wang, Nu Xu,
PRL 84 (2000) 2803
x
z
x
Baryon stopping
rapidity
x

Radial flow
px
The wiggle is pronounced only at high energies
Does the picture contradict FOPI results
on different isotope collisions?
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
 <x px>
px, v1
“wiggle”
rapidity
2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
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>
0
QM2002
Warning: Large
systematic errors!
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Laszlo’s slide from BNL Flow workshop ‘03
The slope of v1(eta) at eta=0
is indeed as in antiflow
scenario, … but also the same
as always for pions at lower
energies
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PHOBOS, v1(eta)
Qualitatively the same
picture from SPS energies
to highest RHIC energy.
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STAR: ZDC-SMD
SMD is an 8 channel by
7 channel hodoscope
that sits directly on
the face of the 2nd
ZDC module
What about ALICE, CMS, do they have something like that?
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v1(eta), v1(pt), AuAu@62 GeV,different centralities
Qualitatively the picture is
very similar at different
centralities
STAR preliminary
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Comparison with models. Centrality dependence
STAR preliminary
STAR preliminary
Neither model describes v1(eta) close to midrapidty
- In order to prove the “wiggle” one needs identified particle measurements
and look for the change of sign of the slope with energy/centrality. At 62 GeV the
errorbars are too large, we hope to have it such results for 200 GeV data.
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Elliptic Flow.
XZ-plane - the reaction plane
y 2  x 2 
ε 2
y  x 2 
Tr ans ve r s e Plane
Y
X
v2  
px 2  py 2
px  py
2
v2 > 0,
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2
Sensitive to “early” times.
(Free streaming kills  )
   cos( 2φ )
E877, PRL 73 (1994) 2532
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Elliptic flow as function of …
It is measured vs:
- collision energy
- transverse momentum
- centrality
- rapidity
- particle ID
- Integrated values of
v2 noticeably increase
with energy
- The slope of v2(pt)
increase slowly
 Most of the
increase in integrated
v2 comes from the
increase in mean pt.
In mid and more
central collisions
elliptic flow is rather
well described by
hydro model
PHOBOS
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Integrated v2 at different energies
(0-40% central)
(0-40% central)
We still have to analyze carefully the centrality dependence
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Constituent quark model + coalescence
coalescence
fragmentation
Low pt quarks
High pt quarks
S.V., QM2002
D. Molnar, S.V., PRL 2003
Only in the intermediate region (rare processes) coalescence can be
described by:
2
3

d 3nM  d nq
 pq  pM / 2

d 3 pM  d 3 pq


v2,M ( pt )  2 v2, q ( pt / 2)
v2,B ( pt )  3 v2, q ( pt / 3)
R. Fries
In the low pt region density is large and most quarks coalesce:
N
hadron
~N
quark
e Bpt
2
(e Bpt
2
/4 2
)
In the high pt region fragmentation eventually wins:
pt n
(( pt / 2) n )2
Taking into account that in coalescence pt ,quark  pt ,meson / 2
and in fragmentation
pt ,quark  pt ,meson / z
,
there could be a region in quark pt where only few quarks coalesce, but give hadrons
in the hadron pt region where most hadrons are produced via coalescence.
Side-notes:
a) more particles produced via coalescence rather than parton
fragmentation  larger mean pt…
b)
 higher baryon/meson ratio
c)
 lower multiplicity per “participant”
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-> D. Molnar, QM2004, in progress
-> Bass, Fries, Mueller. Nonaka; Levai, Ko; …
-> Eremin, S.V.
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Constiuent quark scaling: v2 and RCP
AuAu@62 GeV
STAR Preliminary
- Constituent quark scaling holds well.
Deviations are where expected.
- Elliptic flow saturates at pt~ 1 GeV, just at
constituent quark scale. An accident?
Gas of constituent quarks – deconfinement !?
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PHENIX: const. quark scaling, v2 saturates at RHIC energy
AuAu@62 GeV
STAR Preliminary
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Are they thermalized?
S. Pratt, S. Pal , nucl-th/0409038
Two pictures correspond to the same v2 of quarks, but
a)
v2(B) = 3/2 v2(M)
(no thermalization ?)
b)
v2(B) = v2(M)
(freeze-out at constant phase space density)
My conclusion: constituent quark scaling 
- Deconfinement!
- No thermalization (at least in this region of pt)
(Freeze-out at constant density in the configuration space)
The same mechanism at sqrt(s_NN) 200 and 62 GeV.
If thermalized, disappear at LHC??
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v2(pt) at 200 GeV. “Mass splitting”.
Data: PHENIX, Nucl. Phys. A715, 599, 2003
Hydro: P. Huovinen, P. Kolb, U. Heinz, P.
Ruuskanen, S.V., Phys. Lett. B503, 58, 2001;
Mass dependence is rather well reproduced by
hydrodynamical model calculations.
Note dependence on the EoS.
But qualitatively such a mass dependence will be
present in any model, for example, in the
constituent quark coalescence picture
(heavier particle  larger difference in
constituent quark momenta)
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v2(pt) @ 200 and 62 GeV
pion
Y. Bai (STAR), DNP ‘04
min. bias 0 ~
80%
0 ~ 80 %
star preliminary
Pt
STAR expects good identified particle v2 measurements up to relatively high pt.
Need detailed/tuned hydro calculations for different centralities and identified particles.
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
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Centrality dependence. Hydro and Low Density limits
Hydro: v2~ 
Low Density Limit: v2~  dN/dy / A
Ollitrault, PRD 46 (1992) 229
Heiselberg & Levy, PRC C59 (1999) 2716
Hydro: P.F. Kolb, et al
SV & A. Poskanzer, PLB 474 (2000) 27
v2 / 
SPS 40 GeV/A
LDL
SPS
hydro
RHIC 160 GeV/A
(pts are RQMD v2.4)
Suppressed scale!
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b (fm)
5
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b (fm)
S.A. Voloshin
v2/ and phase transitions
Centrality dependence: Sorge, PRL 82 2048 (’99), Heiselberg & Levy, PRC 59 2716 (’99)
Dependence on the particle density in the transverse
plane: S.V. & A. Poskanzer, PLB 474 (2000) 27
Uncertainties:
Hydro limits: slightly depend
on initial conditions
Data: no systematic errors,
shaded area –uncertainty in
centrality determinations.
Curves: “hand made”
“Cold” deconfinement?
E877 NA49
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v2 / 
Hydro limits
SPS 40 GeV/A
SPS
RHIC 160 GeV/A
Hydro: P.F. Kolb, et al
Suppressed scale!
Hydro: v2~ 
Ollitrault, PRD 46 (1992) 229
26
Heinz, Kolb, Sollfrank
Low Density Limit: v2~  dN/dy / S
Heiselberg & Levy, PRC C59 (1999) 2716
Questions to address:
- is it saturating?
- what happens at SPS energies? Any ‘wiggle’?
page
b (fm)
v 2  0.04 0.04*
2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
dN
/ 3000
dy
S.A. Voloshin
“Cold” deconfinement, color percolation?
There is a need for the “next generation”
of this plot: better estimates of epsilon,
adding more data (in particular 62 GeV)
It is a real pity that NA49 measurements
have so large systematic uncertainty.
Need detector with better azimuthal
acceptance (could be just a simple extra
detector used to determine the RP) .
Percolation point by H. Satz, QM2002
CERN SPS energies
RHIC:
b ~ 7 fm
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FT RHIC?
b ~ 4 fm
2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
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Charm flow (via electron measurements)
STAR SQM04
But is it surprising?:
v2 ~
p
;
p
m  ~ 5m;
p  ~ 2p; p  ~ 5 p
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v2 stays the same?
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Correlations/fluctuations
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2-particle correlation functions
“Inclusive”
 dy  ( y)  n;
1
 dy  dy  ( y , y )  n(n  1)
1
2
2
1
2
Distribution of “correlated” pairs:
C( y1 , y2 )  2 ( y1 , y2 )  1 ( y1 ) 1 ( y2 )
Distribution of “associated”
C ( y1 , y2 )
particles (2) per “trigger” B( y1 , y2 ) 
1 ( y1 )
particle (1)
“Probability” to find a
“correlated” pair
R( y1 , y2 ) 
C ( y1 , y2 )
1 ( y1 ) 1 ( y2 )
ISR data. Filled circles – sqrt(s) = 63 GeV
RHIC: PHOBOS?
Production via Nc clusters [e.g. independent NN collisions]
1{ Nc} ( y )  N c 1{1} ( y );
R
{ Nc }
 2{ Nc} ( y1 , y2 )  N c  2{1} ( y1 , y2 )  N c ( N c  1) 1{1} ( y1 ) 1{1} ( y2 )
N c  2{1} ( y1 , y2 )  N c ( N c  1) 1{1} ( y1 ) 1{1} ( y2 )  N c2 1{1} ( y1 ) 1{1} ( y2 ) R{1}


2 {1}
{1}
N c 1 ( y1 ) 1 ( y2 )
Nc
Relation to fluctuations
σ n2 n 2   n 2 n(n  1)  n 2  n   const  1  C   (y)
ωn 


 n
 n
 n
~
n(n  1)  n 2
 1   n

1


n

R
 n 2
~
R
 dy  dy C ( y , y )
 dy  dy  ( y )  ( y )
1
1
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2
2 1
1
1
2
1
2
“Fluctuations” are
determined by the
“average“ value
of the correlation
function over momentum
region under study.
2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
<pt> fluctuations: observables and observables.
What are the main requirements for a
good observable?
-- be sensitive to the physics under
study
-- be defined at the “theoretical
level”, be detector/experiment
independent
-- have clear physical meaning
-- not to be limited in scope, provide
new venues for further study
 pt ,1  pt , 2 ,
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δpt,i  pt,i  pt ;
Possibilities:
- test scaling with Nch, Npart, Nbin, etc.
-Particles “1” and “2” could be of different type
(e.g. same/opposite charge),
- taken from different rapidity/azimuthal angle
regions (e.g. “same-side” , |y1-y2|>1 correlations as
mostly “free” from jet contribution).
2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
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Multiplicity fluctuations
r12 
n1
n2
 - Free from “volume” fluctuations
 - Fails at small < n2 >
Particle ratios:
( r12 )2  ! (n1 )2  (n2 )2 
δ n1δ n2 



2
 r12  2
 n1  2
 n2  2
 n1  n2 
D   n1  n2  (r  )2 
“Charge” fluctuations
2 (a ,b )da db
n (n  1) n (n  1) 2n n  ~
~
~
~

Y
 , dyn 


 R  R  2R , Rab 
1
n  2
n  2
n  n 
 1 (a )da  1 (b )db
Y
Y
σ n2
~
ωn 
 1   n R
 n
~
~
~
D
( R   R  2R  ) n  n 
 1
4
4
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 - < n++ n-> - “used” multiplicity, subject
to cuts and acceptance
2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
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Comparison to PHENIX, Fpt
200 GeV Au+Au
STAR Cuts
(slide from G. Westfall (STAR), QM’04)
T ,real  T ,mixed  T ,real   T ,mixed
T
T 
Fpt 

T
T ,mixed
 T ,mixed
|| < 1.0
 = 360
0.1 < pt < 2 GeV
200 GeV Au+Au
STAR with
PHENIX Cuts
|| < 0.35
 = 2x90
0.2 < pt < 2 GeV
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Elliptic flow contribution to <dpt dpt>
Shengli Huang (STAR)
USTC RHIC Workshop,
Hefei, China , Oct. 2004
y
x
In-plane
STAR Preliminary
Out-of-plane
Could be better to plot <dpt dpt> /<pt>^2
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pt ,1pt , 2
correlations: elongation in 
“Inclusive”
All data on <dpt dpt> are
STAR preliminary, taken from talks
of G. Westwall (STAR) at QM2004
and Nuclear Dynamics WSs ‘04 and ‘05
R() ~ 1 - ||

<R> (Y) ~ 1 - 4/3  Y,
where Y= ()max/2
Blue dotted lines assume the same .
Note difference in slopes (red vs blue) –
broadening of R() with centrality
ISR data. Filled circles – sqrt(s) = 63 GeV
A way to do it better  study directly
as function of y1 and y2
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pt ,1 pt , 2
2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005 max S.A. Voloshin
pt ,1pt , 2 : centrality dependence
Data: G. Westfall (STAR), QM2004
Rcc(0)0.66
~ n(n  1)
R
1
2
n
Production via Nc clusters (Nc~Npart/2)
[e.g. independent NN collisions]
 pt ,1 pt , 2
DN coll 
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36
AA
 DNcoll  pt ,1 pt , 2
n(n  1)
( N coll  1) n
2
NN
NN
NN
 n(n  1)
NN
At midrapidity, the probability to find a
particle is about 60% larger if one particle
has been already detected.
In a superposition of two independent collisions,
the ratio of the probability that in a randomly
chosen pair both particles are from the same
collision to the probability that two particles
are from different collisions is about 1.66
2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
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“Elementary” NN-collision. Correlation functions.
 dy  ( y)  n
 dy  dy  ( y , y )  n(n 1)
y
1
1
2
2
1
2
Distribution of “correlated” pairs:
C( y1 , y2 )  2 ( y1 , y2 )  1 ( y1 ) 1 ( y2 )
Distribution of “associated”
C ( y1 , y2 )
particles (2) per “trigger” B( y1 , y2 ) 
1 ( y1 )
particle (1)
x
Correlations are due to local charge(s)
conservation, resonances, due to fluctuations
in number of produced strings, e.g. number of
qq-collisions.
Rcc(0)0.66
ISR
“Probability” to find a
“correlated” pair
n( n  1)  1.6 n
R( y1 , y2 ) 
C ( y1 , y2 )
1 ( y1 ) 1 ( y2 )
2
At midrapidity, the probability to find a
particle is about 60% larger if one particle
has been already detected.
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
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Radial flow  2- particle correlations
All particles produced in the same NN-collision
(qq-string) experience the transverse radial “push”
that is
(a) in the same direction (leads to correlations in phi)
(b) the same in magnitude ( correlations in pt)
 Position-momentum correlations caused by transverse
expansion “brings” totally new mechanism for momentum
correlations, not present in NN-collisions
y
pp collision
x
AA collision

-Long range rapidity correlations (“bump”- narrow in
phi and wide in rapidity, charge independent)
-Stronger 2-particle pt correlation in narrow phi bins
-Narrowing of the charge balance function
( pz  mt sinh(y) -- increase in mt  decrease
in rapidity separation) [same as in S. Pratt et al,
in “late hadronization scenario”]
- Charge correlations in phi. Azimuthal Balance
function
Everything evolving with centrality (radial flow)
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
Transverse radial expansion
STAR Collaboration, PRL 92, 112301 (2004)
y
AA collision
n=1 x
Blast wave parameterization (Schnedermann, Sollfrank, Heinz, PRC 48, 2462 (1993), d3n/d3p ~ e-E/T)
of the source at freeze-out:
m cosh(y  ys ) cosh(t )  pt sinh(t ) cos(   s )
Ed 3n
  rdr  dys d s mt cosh(y  ys ) exp[ t
]
3
d p 0
T
R
R
d 3n
 rdr mt K1 ( t ) I 0 ( t );
dy d 2 pt 0
t 
mt
p
cosh(t );  t  t sinh(t )
T
T
Parameters: T-temperature, velocity profile
page
39
t r n
Note: uniform source density
at r < R has been assumed
2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
Azimuthal correlations
Figures are shown for particles from the same NN collision. Dilution factor to be applied!
n=1, T=110 MeV
First and second harmonics of the
distribution on the left
! - the large values of transverse
flow, > 0.25, would contradict “non-flow”
estimates in elliptic flow measurements
No momentum conservation effects has been included. Those would be important
for the charge independent first harmonic correlations.
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
 x  correlations
- Charge independent correlations: particles at large rapidities, initially uncorrelated,
become correlated, as all of them are pushed by radial flow in the same direction.
For those, one needs 2d correlations (rapidity X azimuth)
Shown below – hand drawn sketch.
Peripheral
Central


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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
Extracting Near-Side Jet Yields
D. Magestro (STAR) –
Hard Probes 2004
d+Au, 40-100%
In Au+Au, jet-like correlation sits on top of an
additional, approximately flat correlation in 
STAR preliminary

Au+Au, 0-5%



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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
3 < pT(trig) < 6 GeV
Voloshin < p (trig)
2 <S.A.
pT(assoc)
T
Brief comparison to data:
pt ,1pt , 2 centrality dependence
n=1
n=0.5
Possible reasons for discrepancy:
- diffusion, thermalization time
- spatial source profile (not
uniform density in transverse plane,
e.g. cylinder shell)
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
correlation summary
~
~
~
1. Avoid using ratios (n+/n-, K+/K0,…), use
to get rid of “volume”
Raa  Rbb  2Rab
fluctuations and be free from problems related to low multiplicities.
2. If use normalized variance – correct for the efficiency.
page
44
1.
Transverse radial flow leads to strong space-momentum correlation.
In combination with space correlations between particles created in
the same NN collision, it leads to characteristic two (and many) particle
rapidity, transverse momentum, and azimuthal correlations.
2.
This phenomenon provides a natural (at present, qualitative) explanation of
the centrality dependence of mean pt pseudorapidity/azimuthal angle
correlations. It can be further used to study the details of the system
equilibration/thermalization and evolution (e.g. thermalization time, velocity
profile, etc.)
2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
EXTRA SLIDES
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
Rapidity correlations
How to disentangle “initial” correlations at the parton production stage and obtained
due to the transverse expansion? - Charge dependent and charge independent correlations.
- Correlation of conserved charges (Balance Functions). In this case the correlations
existed already at the production moment would be modified (narrowed) by radial flow.
- Charge independent correlations: particles at large rapidities, initially uncorrelated,
become correlated, as all of them are pushed by radial flow in the same direction.
Charge Balance function
pz  mt sinh(y)  mt y
As <mt> increases due to the transverse
radial flow, the balance function gets narrower.
For the BW parameters used above,
<mt> indeed increases for about 15-20%,
but the centrality dependence is somewhat
different from what is observed in the narrowing
of the Balance Function.
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
Initial and freeze-out configurations
Uncertainty: particles are at the same position
at the moment of production, but the blast
wave parameterization is done at freeze-out
Final
initial
Smearing would depend on the
- thermalization time (which is supposedly small)
- diffusion during the system evolution before freeze-out
- non-zero “expansion velocity” in pp
Should we take it as a possibility to study all the above effects?
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
AA collision. “Single jet tomography”.
AA collision
In this picture, the transverse
momentum of the (same side, large )
associated particles would be a measure
of the space position the hard scattering
occurred
The plot on the right shows particle azimuthal
distribution (integrated over all pt’s) with respect
to the boost direction.
In order to compare with data it should be also
convoluted with jet azimuthal distribution relative
to radial direction.
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
Sensitivity to the velocity profile
t  r n
Results for n=0.5 and n=2 are shown
t2  t (2  n) 2 /(4n  4)
2
Mean pt is almost insensitive to the actual
velocity profile.
The correlations are.
In general, mean pt is sensitive to the first moment
of the respective transverse rapidity distribution
while the two particle correlation are measuring
the second moment.
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
Parity violation study via 3-particle correlations

L
hep-ph/0406311
dN
 1  2a sin 
d
Looking for the effect of
D. Kharzeev, hep-ph/0406125
projections onto reaction plane
a > 0  preferential emission along the angular momentum
The sign can vary event by event, a~Q/N, where Q is
the topological charge, |Q|=1,2,…
at dN/dy~100, |a|~1%.
Projections on the direction
of angular momentum
cos( a  2 ) cos(b  2 )  sin( a  2 ) sin(b  2 ) 
 cos( a  b  22 )  (v1,a v1,b  aa ab ) cos(22  2RP )
All effects non
sensitive to the RP
cancel out!
Possible systematics:
clusters that flow
And using only one particle instead of the event flow vector
cos( a  c ) cos(b   c )  sin( a   c ) sin(b   c ) 
 cos( a  b  2 c )  (v1,a v1,b  aa ab )v2,c
note that for a rapidity region symmetric with respect to the midrapidity v1=0
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin
Ebye and inclusive approaches
 pt  
1
M
 pt ,i
  pt   
i
1
 pt  k

N ev k
Most of the present measurements
are done this way
 δpt,i δpt, j  i  j  ; δpt,i  pt,i   pt .
 d dp  d
1
 pt ,1 pt , 2  
t ,1
dpt , 2  pt ,1 pt , 2  2 ( pt ,1 ,1 , pt , 2 , 2 )
2
1
 d dp  d
1
t ,1
1

2
2
dpt , 2  2 ( pt ,1 ,1 , pt , 2 , 2 )

2
 d1 dpt ,1  d2 dpt ,2  pt ,1 pt ,2 [  2 ( pt ,1 ,1 , pt ,2 ,2 )  1 ( pt ,1 ,1 ) 1 ( pt ,2 ,2 )]
1
2
 d dp  d
1
t ,1
1
δpt,i  pt,i  pt ; pt
2
Would be better, easier to
analyze theoretically.
(! Numerically both are very
close)
dpt , 2  2 ( pt ,1 ,1 , pt , 2 , 2 )
2
 p

M
t ,i
k
i
C (1,2)
k
k
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2nd Int. Workshop on the Critical Point and Onset of Deconfinement, Bergen, 2005
S.A. Voloshin