Conservation Laws and Femtoscopy of Small Systems
description
Transcript of Conservation Laws and Femtoscopy of Small Systems
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 1
Conservation Laws Conservation Laws and Femtoscopy and Femtoscopy of Small Systemsof Small Systems
Zbigniew Chajęcki and Michael A. Lisa
The Ohio State University
[nucl-th/0612080]
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 2
OutlineOutline
• Introduction / Motivation– intriguing pp versus AA [reminder]– data features not under control: Energy-momentum
conservation?
• SHD as a diagnostic tool [reminder]• Phase-space event generation: GenBod• Analytic calculation of Energy and Momentum
Conservation Induced Correlations• Experimentalists’ recipe:
Fitting correlation functions [in progress]• Conclusion
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 3
Id. pion femtoscopy in p+p @ Id. pion femtoscopy in p+p @ STARSTAR
STAR preliminary
mT (GeV) mT (GeV)
Z. Ch. (for STAR) QM05, NP A774:599-602,2006
• For the first time: femtoscopy in p+p and A+A measured in same experiment, same analysis definitions, ….
• great opportunity to compare physics
• what causes mT-dependence in p+p?
• same cause as in A+A?
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 4
Ratio of femtoscopic radii Ratio of femtoscopic radii
All pT(mT) dependences of HBT radii observed by STAR scale with pp although it’s expected that different origins drive these dependences
Femtoscopic radii scale with pp
• Scary coincidence or something deeper?
pp, dAu, CuCu - STAR preliminary
Ratio of (AuAu, CuCu, dAu) HBT radii by pp
Z. Ch. (for STAR) QM05, NP A774:599-602,2006
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 5
Clear interpretation clouded by data Clear interpretation clouded by data featuresfeatures
d+Au: peripheral collisions
STAR preliminary
Non-femtoscopic q-anisotropicbehaviour at large |q|
does this structure affect femtoscopic region as well?
Qx<0.12 GeV/c
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Spherical Harmonic Spherical Harmonic Decomposition Decomposition
of the Correlation of the Correlation FunctionFunction
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 7
Spherical Harmonic Decomposition Spherical Harmonic Decomposition of CFof CF
∑→→ ΔΔ
=binsall
iiiiimlml QCYQA
.
,
cos
, ),cos|,(|),(|)(| φθφθπ
φθ
4
QOUT
QSIDE
QLONG Q
• Cartesian-space (out-side-long) naturally encodes physics, but is poor/inefficient representation
• Recognize symmetries of Q-space -- decompose by spherical harmonics!
• Direct connection to source shapes [Danielewicz,Pratt: nucl-th/0501003] – decomposition of CF on cartesian harmonics
• ~immune to acceptance
• full information content at a glance[thanks to symmetries]
: [0,2] : [0,]
OUT
SIDE
TOT
LONG
LONGSIDEOUT
Q
Q
Q
Q
QQQQ
arctan
)cos(
222
=
=
++=
φ
Z.Ch., Gutierrez, Lisa, Lopez-Noriega, nucl-ex/0505009
This new method of analysis represents, in my opinion, a real breakthrough. […] it has a good chance to become a standard tool in all experiments.
A. Bialas, ISMD 2005
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Decomposition of CF onto Spherical Decomposition of CF onto Spherical HarmonicsHarmonics
Au+Au: central collisions
C(Qout)
C(Qside)
C(Qlong)
∑→→ ΔΔ
=binsall
iiiiimlml QCYQA
.
,cos
, ),cos|,(|),(4
|)(| φθφθπ
φθ
Z.Ch., Gutierrez, Lisa, Lopez-Noriega, [nucl-ex/0505009]
Pratt, Danielewicz [nucl-th/0501003]
Qx<0.03 GeV/c
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Z.Ch., Gutierrez, Lisa, Lopez-Noriega, [nucl-ex/0505009]
Pratt, Danielewicz [nucl-th/0501003]
Decomposition of CF onto Spherical Decomposition of CF onto Spherical HarmonicsHarmonics
d+Au: peripheral collisions
STAR preliminary
∑→→ ΔΔ
=binsall
iiiiimlml QCYQA
.
,cos
, ),cos|,(|),(4
|)(| φθφθπ
φθ
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Multiplicity dependence of the Multiplicity dependence of the baselinebaseline
Baseline problem is increasing
with decreasing multiplicity
STAR preliminary
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GenBodGenBodPhase-Space Event
Generator
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GenBod: Phase-space sampling GenBod: Phase-space sampling with energy/momentum with energy/momentum
conservationconservation• F. James, Monte Carlo Phase Space CERN REPORT 68-15 (1 May 1968)• Sampling a parent phasespace, conserves energy & momentum explicitly
– no other correlations between particles !
Events generated randomly, but each has an Event Weight
€
WT =1
Mm
M i+1R2 M i+1;M i,mi+1( ){ }i=1
n−1
∏
WT ~ probability of event to occur
€
Rn = δ 4 P − p j
j=1
n
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ δ pi
2 − mi2
( )d4pi
i=1
n
∏4 n
∫
where
P = total 4 - momentum of n - particle system
pi = 4 - momentum of particle i
mi = mass of particle i
P conservation
€
δ 4 P − p j
j=1
n
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Induces “trivial” correlations(i.e. even for M=1)
Energy-momentum conservation in n-body systemEnergy-momentum conservation in n-body system
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Sampling MC EventsSampling MC EventsLow probability (PS weight)
High probability (PS weight)
30 particles
€
WT =1
Mm
M i+1R2 M i+1;M i,mi+1( ){ }i=1
n−1
∏
To treat MC events identical to measured events we have to sample them according to WT (PS weight)
Then we can construct CF
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CF from GenBodCF from GenBod
Varying frame and Varying frame and kinematic cutskinematic cuts
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N=18, <K>=0.9 GeV, LabCMS Frame - no N=18, <K>=0.9 GeV, LabCMS Frame - no cutscuts
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 16
N=18, <K>=0.9 GeV, LabCMS Frame - |N=18, <K>=0.9 GeV, LabCMS Frame - |||<0.5<0.5
The shape of the CF is sensitive to
• kinematic cuts
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N=18, <K>=0.9 GeV, LCMS Frame - no cutsN=18, <K>=0.9 GeV, LCMS Frame - no cuts
The shape of the CF is sensitive to
• kinematic cuts
• frame
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N=18, <K>=0.9 GeV, LCMS Frame - |N=18, <K>=0.9 GeV, LCMS Frame - ||<0.5|<0.5
The shape of the CF is sensitive to
• kinematic cuts
• frame
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GenBodGenBod
Varying multiplicity Varying multiplicity and total energyand total energy
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N=6, <K>=0.5 GeV, LCMS Frame - no cutsN=6, <K>=0.5 GeV, LCMS Frame - no cuts
The shape of the CF is sensitive to
• kinematic cuts
• frame
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N=9, <K>=0.5 GeV, LCMS Frame - no cutsN=9, <K>=0.5 GeV, LCMS Frame - no cuts
The shape of the CF is sensitive to
• kinematic cuts
• frame
• particle multiplicity
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N=15, <K>=0.5 GeV, LCMS Frame - no cutsN=15, <K>=0.5 GeV, LCMS Frame - no cuts
The shape of the CF is sensitive to
• kinematic cuts
• frame
• particle multiplicity
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N=18, <K>=0.5 GeV, LCMS Frame - no cutsN=18, <K>=0.5 GeV, LCMS Frame - no cuts
The shape of the CF is sensitive to
• kinematic cuts
• frame
• particle multiplicity
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N=18, <K>=0.7 GeV, LCMS Frame - no cutsN=18, <K>=0.7 GeV, LCMS Frame - no cuts
The shape of the CF is sensitive to
• kinematic cuts
• frame
• particle multiplicity
• total energy : √s
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N=18, <K>=0.9 GeV, LCMS Frame - no cutsN=18, <K>=0.9 GeV, LCMS Frame - no cuts
The shape of the CF is sensitive to
• kinematic cuts
• frame
• particle multiplicity
• total energy : √s
The shape of the CF is sensitive to:
• kinematic cuts
• frame
• particle multiplicity
• total energy : √s
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FindingsFindings
• Energy and Momentum Conservation Induced Correlations (EMCICs) “resemble” our data
so, EMCICs... on the right track...
• But what to do with that?– Sensitivity to s, multiplicity of particles of interest and other particles
– will depend on p1 and p2 of particles forming pairs in |Q| bins
risky to “correct” data with Genbod...
• Solution: calculate EMCICs using data!!– Danielewicz et al, PRC38 120 (1988)– Borghini, Dinh, & Ollitraut PRC62 034902 (2000)
we generalize their 2D pT considerations to 4-vectors
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k-particle distributions w/ phase-space k-particle distributions w/ phase-space constraintsconstraints
€
˜ f ( pi) = 2E i f ( pi) = 2E i
dN
d3 pi
single-particle distributionw/o P.S. restriction
€
˜ f c(p1,...,pk ) ≡ ˜ f (pi)i=1
k
∏ ⎛ ⎝ ⎜ ⎞
⎠ ⎟⋅
d3pi
2E i
˜ f (pi)i= k +1
N
∏ ⎛
⎝ ⎜
⎞
⎠ ⎟∫ δ 4 pi
i=1
N
∑ − P ⎛
⎝ ⎜
⎞
⎠ ⎟
d3pi
2E i
˜ f (pi)i=1
N
∏ ⎛
⎝ ⎜
⎞
⎠ ⎟∫ δ 4 pi
i=1
N
∑ − P ⎛
⎝ ⎜
⎞
⎠ ⎟
= ˜ f (pi)i=1
k
∏ ⎛ ⎝ ⎜ ⎞
⎠ ⎟⋅
d4piδ(pi2 − mi
2)˜ f (pi)i= k +1
N
∏ ⎛ ⎝ ⎜ ⎞
⎠ ⎟∫ δ 4 pi
i=1
N
∑ − P ⎛
⎝ ⎜
⎞
⎠ ⎟
d4piδ(pi2 − mi
2)˜ f (pi)i=1
N
∏ ⎛ ⎝ ⎜ ⎞
⎠ ⎟∫ δ 4 pi
i=1
N
∑ − P ⎛
⎝ ⎜
⎞
⎠ ⎟
k-particle distribution (k<N) with P.S. restriction
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Central Limit TheoremCentral Limit Theorem
€
˜ f c(p1,...,pk ) = ˜ f (pi)i=1
k
∏ ⎛ ⎝ ⎜ ⎞
⎠ ⎟ N
N − k
⎛
⎝ ⎜
⎞
⎠ ⎟2
exp −
pi,μ − pμ( )i=1
k
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
2(N − k)σ μ2
μ = 0
3
∑
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
where
σ μ2 = pμ
2 − pμ
2
pμ = 0 for μ =1,2,3
k-particle distribution in N-particle system
For simplicity we will assume that all particles are identical (e.g. pions)
-> they have the same average energy and RMS’s of energy/momentum
Then, we can apply CLT (the distribution of averages from any distribution approaches Gaussian with increase of N)
€
˜ f c (p1,..., pk ) ∝ exp
pi,n
i=1
k
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
2(N − k)σ n2
n=1
3
∑
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
exp
E i − E( )i=1
k
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
2(N − k)σ E2
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
Can we assume that E and p are not correlated ?
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E - p correlations?E - p correlations?
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EMCICs in single-particle EMCICs in single-particle distributiondistribution
€
˜ f c(pi) = ˜ f (pi)N
N −1
⎛
⎝ ⎜
⎞
⎠ ⎟2
exp −pi,μ − pμ( )
2
2(N −1)σ μ2
μ = 0
3
∑ ⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
= ˜ f (pi)N
N −1
⎛
⎝ ⎜
⎞
⎠ ⎟2
exp −1
2(N −1)
px,i2
px2
+py,i
2
py2
+pz,i
2
pz2
+E i − E( )
2
E 2 − E2
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
? What if all events had the same “parent” distribution f,and all centrality dependence of spectra was due just toloosening of P.S. restrictions as N increased?
in this case, the index i is only keepingtrack of particle type
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k-particle correlation k-particle correlation functionfunction
€
C(p1,...,pk ) ≡˜ f c(p1,...,pk )
˜ f c(p1)....̃ f c(pk )
=
N
N − k
⎛
⎝ ⎜
⎞
⎠ ⎟2
N
N −1
⎛
⎝ ⎜
⎞
⎠ ⎟2k
exp −1
2(N − k)
px,ii=1
k
∑ ⎛ ⎝ ⎜ ⎞
⎠ ⎟2
px2
+py,ii=1
k
∑ ⎛ ⎝ ⎜ ⎞
⎠ ⎟2
py2
+pz,ii=1
k
∑ ⎛ ⎝ ⎜ ⎞
⎠ ⎟2
pz2
+E i − E( )
i=1
k
∑ ⎛ ⎝ ⎜ ⎞
⎠ ⎟2
E 2 − E2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟i=1
k
∑
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
exp −1
2(N −1)
px,i2
px2
+py,i
2
py2
+pz,i
2
pz2
+E i − E( )
2
E 2 − E2
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
i=1
k
∑ ⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 32
2-particle correlation 2-particle correlation functionfunction
€
C( p1, p2 ) ≡˜ f c ( p1, p2 )
˜ f c (p1) ˜ f c (p2 )
=
N
N − 2
⎛
⎝ ⎜
⎞
⎠ ⎟
2
N
N −1
⎛
⎝ ⎜
⎞
⎠ ⎟
4
exp −1
2(N − 2)
px, ii=1
2
∑ ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
px2
+py, ii=1
2
∑ ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
py2
+pz, ii=1
2
∑ ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
pz2
+E i − E( )i=1
2
∑ ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
E 2 − E2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟i=1
2
∑
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
exp −1
2(N −1)
px, i2
px2
+py, i
2
py2
+pz, i
2
pz2
+E i − E( )
2
E 2 − E2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟i=1
2
∑ ⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
Dependence on “parent” distrib f vanishes,except for energy/momentum means and RMS
2-particle correlation function (1st term in 1/N expansion)
€
C(p1,p2) ≅1−1
N2
r p T,1 ⋅
r p T,2
pT2
+pz,1 ⋅pz,2
pz2
+E1 − E( ) ⋅ E 2 − E( )
E 2 − E2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 33
2-particle CF (1st term in 1/N 2-particle CF (1st term in 1/N expansion)expansion)
€
C(p1,p2) ≅1−1
N2
r p T,1 ⋅
r p T,2
pT2
+pz,1 ⋅pz,2
pz2
+E1 − E( ) ⋅ E 2 − E( )
E 2 − E2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
“The pT term” “The pZ term” “The E term”
Names used in the following plots
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 34
EMCICsEMCICs
Effect of varying Effect of varying multiplicity & total energy multiplicity & total energy
Same plots as before, but now we look at:
• pT (), pz () and E () first-order terms
• full () versus first-order () calculation
• simulation () versus first-order () calculation
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 35
N=6, <K>=0.5 GeV, LabCMS Frame - no N=6, <K>=0.5 GeV, LabCMS Frame - no cutscuts
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 36
N=9, <K>=0.5 GeV, LabCMS Frame - no N=9, <K>=0.5 GeV, LabCMS Frame - no cutscuts
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 37
N=15, <K>=0.5 GeV, LabCMS Frame - no N=15, <K>=0.5 GeV, LabCMS Frame - no cutscuts
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 38
N=18, <K>=0.5 GeV, LabCMS Frame - no N=18, <K>=0.5 GeV, LabCMS Frame - no cutscuts
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 39
N=18, <K>=0.7 GeV, LabCMS Frame - no N=18, <K>=0.7 GeV, LabCMS Frame - no cutscuts
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 40
N=18, <K>=0.9 GeV, LabCMS Frame - no N=18, <K>=0.9 GeV, LabCMS Frame - no cutscuts
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 41
FindingsFindings
• first-order and full calculations agree well for N>9– will be important for “experimentalist’s recipe”
• Non-trivial competition/cooperation between pT, pz, E terms– all three important
• pT1•pT2 term does affect “out-versus-side” (A22)
• pz term has finite contribution to A22 (“out-versus-side”)
• calculations come close to reproducing simulation for reasonable (N-2) and energy
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 42
The Experimentalist’s RecipeThe Experimentalist’s Recipe
€
C( p1, p2 ) = 1−2
N pT2
r p 1,T ⋅
r p 2,T{ } −
1
N pZ2
p1,Z ⋅ p2,Z{ }
−1
N E 2 − E2 ⎛
⎝ ⎜
⎞ ⎠ ⎟
E1 ⋅E2{ } +E
N E 2 − E2 ⎛
⎝ ⎜
⎞ ⎠ ⎟
E1 + E2{ } −E
2
N E 2 − E2 ⎛
⎝ ⎜
⎞ ⎠ ⎟
€
C( p1, p2 ) = 1− M1
r p 1,T ⋅
r p 2,T{ } − M2 p1,Z ⋅ p2,Z{ } − M3 E1 ⋅E2{ } + M4 E1 + E2{ } −
M4( )2
M3
Fitting formula:
€
{X} - average of X over # of pairs for each Q-bin
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 43
EMCIC’s FIT: N=18, <K>=0.9GeV, EMCIC’s FIT: N=18, <K>=0.9GeV, LCMSLCMS
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 44
Fit contoursFit contours
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 45
The Complete Experimentalist’s The Complete Experimentalist’s RecipeRecipe
€
C( p1, p2 ) = Norm ⋅(1+ λ ⋅ Kcoul (Qinv ) 1+ exp −Rout2 Qout
2 − Rside2 Qside
2 − Rlong2 Qlong
2( )( ) −1[ ]
−M1
r p 1,T ⋅
r p 2,T{ } − M2 p1,Z ⋅ p2,Z{ } − M3 E1 ⋅E2{ } + M4 E1 + E2{ } −
M4( )2
M3
)
or any other parameterization of CF
9 fit parameters
- 4 femtoscopic
- normalization
- 4 EMCICs
Fit this ….
or image this …
€
C(q) + M1
r p 1,T ⋅
r p 2,T{ } + M2 p1,Z ⋅ p2,Z{ } + M3 E1 ⋅E2{ } − M4 E1 + E2{ }
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 46
SummarySummary• understanding the femtoscopy of small systems
– important physics-wise
– should not be attempted until data fully under control
• SHD: “efficient” tool to study 3D structure• Restricted P.S. due to energy-momentum conservation
– sampled by GenBod event generator
– generates EMCICs quantified by Alm’s
– stronger effects for small mult and/or s
• Analytic calculation of EMCICs– k-th order CF given by ratio of correction factors
– “parent” only relevant in momentum variances
– first-order expansion works well for N>9
– non-trivial interaction b/t pT, pz, E conservation effects
• Physically correct “recipe” to fit/remove EMCICs– 4 new parameters, determined @ large |Q|
– parameters are “physical” - values may be guessed
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 47
Thanks to:Thanks to:
• Alexy Stavinsky & Konstantin Mikhaylov (Moscow) [suggestion to use Genbod]
• Jean-Yves Ollitrault (Saclay) & Nicolas Borghini (Bielefeld)[original correlation formula]
• Adam Kisiel (Warsaw) [don’t forget energy conservation]
• Ulrich Heinz (Columbus)[validating energy constraint in CLT]
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 48
Some properties of ASome properties of Almlm coefficientscoefficients
Alm = 0 for l or m odd – identical particle correlations (for non-id particles, odd l encodes shift information)
A00(Q) ≈ one-dimensional “CF(Qinv)” (bump ~ 1/R)
Alm(Q) = δl,0 where correlations vanish
Al≠0,m(Q) ≠ 0 anisotropy in Q space
Im[Alm] = 0
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 49
LongLong--range correlations : range correlations : JETS ?JETS ?
Jets as a origin of the baseline problem ??
The idea was to try to eliminate pions coming from jet fragmentation from data sample. It can be done by applying an event cut which accepts only events that have no high-pt tracks (jets).
HBT analyses where done for three classes of events
all - all events accepted – as a reference
soft – only events without high-pT tracks ( highest-pT < 1.2 GeV/c was chosen)
hard - only events with least one track with pT > 2 GeV/c
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 50
Simple, Gaussian source calculations
~acceptance freeRL < RT
RL > RT
RO < RS
RO > RS
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NA22 parametrization of CFNA22 parametrization of CF
?
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 52
NA22: 1D projections of 3D CFNA22: 1D projections of 3D CF
NA22, Z. Phys. C71 (1996) 405
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 53
NA22 parametrization of CFNA22 parametrization of CF
STAR preliminary d+Au peripheral collisions
NA22 fit
d+Au peripheral collisions
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 54
CLT?
distribution of N uncorrelated numbers
(and then scaled by N, for convenience)
• Note we are not starting with a very Gaussian distribution!!
• “pretty Gaussian” for N=4 (but 2/dof~2.5)
• “Gaussian” by N=10
€
xΣ = xi =i=1
N
∑ N x (remember plots scaled by N)
σ Σ2 = Nσ 2 → σ Σ = Nσ (→
σ Σ
N=
σ
N remember plots scaled by N)
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 55
Schematic: How GenBod works Schematic: How GenBod works 1/31/3
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 56
Schematic: How GenBod works Schematic: How GenBod works 2/32/3
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 57
Schematic: How GenBod works Schematic: How GenBod works 3/33/3
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 58
RememberRemember
€
pμ2 ≡ d3p ⋅pμ
2 ⋅ ˜ f p( )unmeasuredparent distrib
{∫ ≠ pμ2
c≡ d3p ⋅pμ
2 ⋅ ˜ f c p( )measured{∫
relevant quantities are average over the (unmeasured) “parent” distribution,not the physical distribution
€
expect pμ2
c< pμ
2
€
C( p1, p2 ) ≅ 1−1
N2
r p 1,T ⋅
r p 2,T
pT2
+p1,Z ⋅ p2,Z
pZ2
+E1 − E( ) ⋅ E2 − E( )
E 2 − E2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
23rd WWND, Big Sky, MT - Feb. 12-17, 2007 59
Reconstruction of CF from Reconstruction of CF from Alm’sAlm’s