wpcf 2007 - aug. 1-3, 2007 1 conservation laws in low-multiplicity collisions zbigniew chajęcki and...
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WPCF 2007 - Aug. 1-3, 2007 1
Conservation LawsConservation Lawsin low-multiplicity collisions in low-multiplicity collisions
Zbigniew Chajęcki and Michael A. Lisa
The Ohio State University
WPCF 2007 - Aug. 1-3, 2007 2
OutlineOutline
Introduction / Motivation– Non-femtoscopic correlations in low-multiplicity collisions :
OPAL, NA22, STAR, … * data features not under control: Energy-momentum conservation?
Analytic calculation of Energy and Momentum Conservation Induced Correlations for– single particle spectra– two-particle correlations
• Experimentalists’ recipe: Fitting correlation functions
– Minv correlation function & background subtraction
– V2
– Two-particle correlations– Resonance contribution to non-femtoscopic correlations - (π+,π-)– (π+,π-) correlations in p+p(p) at 200 GeV collisions from PYTHIA
Conclusion
WPCF 2007 - Aug. 1-3, 2007 3
Non-femtoscopic correlations : Non-femtoscopic correlations : OPALOPAL
OPAL, CERN-PH-EP/2007-025(submitted to Eur. Phys. J. C.)
1D projections of 3D CF
Femtoscopic correlations should go to the constant number at large Q(no directional dependence!)
Qx<0.2 GeV/c
WPCF 2007 - Aug. 1-3, 2007 4
Non-femtoscopic correlations : Non-femtoscopic correlations : NA22NA22
NA22, Z. Phys. C71 (1996) 405 1D projections of 3D CF
WPCF 2007 - Aug. 1-3, 2007 5
Non-femtoscopic correlations : Non-femtoscopic correlations : STARSTAR
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
STAR, NPA 774 (2006) 599
Clear interpretation clouded by data features
WPCF 2007 - Aug. 1-3, 2007 6
Spherical harmonic decompositionSpherical harmonic decomposition
∑→→ ΔΔ
=binsall
iiiiimlml QCYQA
.
,
cos
, ),cos|,(|),(|)(| φθφθπ
φθ
4 QOUT
QSIDE
QLONG Q
: [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
WPCF 2007 - Aug. 1-3, 2007 7
Non-femtoscopic correlations : Non-femtoscopic correlations : STARSTAR
Baseline problem is increasing
with decreasing multiplicity
STAR preliminary
WPCF 2007 - Aug. 1-3, 2007 8
€
C(qo,qs,ql ) = C femto(qo,qs,ql ) ⋅F(qo,qs,ql )
€
F(qo,qs,ql ) = 1+ δo qo + δs qs + δl ql
€
F(qo,qs,ql ) = 1+ δoqo + δsqs + δlql
• MC simulations
• ‘ad-hoc’ parameterizations
• OPAL, NA22, …
Common approaches to „remove” Common approaches to „remove” non-femtoscopic correlationsnon-femtoscopic correlations
•A possibility: energy-momentum conservation?
–must be there somewhere!–but how to calculate / model ?(Upon consideration, non-trivial...)
• “zeta-beta” fit by STAR [parameterization of non-femtoscopic correlations in Alm’s]
WPCF 2007 - Aug. 1-3, 2007 10
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
WPCF 2007 - Aug. 1-3, 2007 11
N=9, N=9, KK=0.5 GeV, LCMS Frame - no cuts=0.5 GeV, LCMS Frame - no cuts
The shape of the CF is sensitive to:
• kinematic cuts
• frame
• particle multiplicity
• total energy : √s
WPCF 2007 - Aug. 1-3, 2007 12
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
WPCF 2007 - Aug. 1-3, 2007 13
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
observed
P - total 4-momentum
WPCF 2007 - Aug. 1-3, 2007 14
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) and that they share the same parent distribution (same RMS 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 ?
WPCF 2007 - Aug. 1-3, 2007 16
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(p),and all multiplicity (centrality) dependence of spectra was due just to loosening of P.S. restrictions as N increased?
WPCF 2007 - Aug. 1-3, 2007 17
EMCIC’s in spectraEMCIC’s in spectra
€
˜ f c (pT , i) = ˜ f (pT , i)N
N −1
⎛
⎝ ⎜
⎞
⎠ ⎟
3 / 2
exp −1
2(N −1)
2 pT, i2
pT2
+E i − E( )
2
E 2 − E2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
€
fc (pT , i) → f (pT , i)
For N
€
~
€
~
WPCF 2007 - Aug. 1-3, 2007 18
EMCICs: Ratio of particle EMCICs: Ratio of particle spectra spectra
€
˜ f c (pT , i) = ˜ f (pT , i)N
N −1
⎛
⎝ ⎜
⎞
⎠ ⎟
3 / 2
exp −1
2(N −1)
2 pT, i2
pT2
+E i − E( )
2
E 2 − E2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
€
˜ f c (pT , i) =
˜ f (pT , i)C(N, pT
2 , E 2 , E )
€
˜ f c (pT , i)N small
˜ f c (pT , i)N larg e
=
˜ f ( pT, i)Nsmall
C(Nsmall , pT2 , E 2 , E )
˜ f (pT , i)Nlarge
C(Nlarge , pT2 , E 2 , E )
=C(Nsmall , pT
2 , E 2 , E )
C(N l arg e, pT2 , E 2 , E )
Ph
ys. Rev. D
74
(20
06) 0
320
06
p+p @ 200GeV, STARpT spectra from GenBodSimulations: Ratio of pT spectra for N=9 and N=18.
Ratio of pT spectra in p+p@STAR for the lowest and the highest multiplicity events
WPCF 2007 - Aug. 1-3, 2007 19
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
∑ ⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
€
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
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2-particle correlation 2-particle correlation functionfunction
WPCF 2007 - Aug. 1-3, 2007 20
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
WPCF 2007 - Aug. 1-3, 2007 21
EMCICsEMCICs
An example of EMCICs:An example of EMCICs:Effect of varying Effect of varying
multiplicitymultiplicity
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
WPCF 2007 - Aug. 1-3, 2007 22
N=9, N=9, KK=0.9 GeV, LabCMS Frame - no cuts=0.9 GeV, LabCMS Frame - no cuts
WPCF 2007 - Aug. 1-3, 2007 23
N=18, N=18, KK=0.9 GeV, LabCMS Frame - no =0.9 GeV, LabCMS Frame - no cutscuts
WPCF 2007 - Aug. 1-3, 2007 24
FindingsFindings
CF from GenBod (as well as EMCICs) depends on – multiplicity– frame– energy of the collisions
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
WPCF 2007 - Aug. 1-3, 2007 25
NN=12,N=12,NKK=3,N=3,Npp=3, =3, KK=0.9 GeV, LCMS Frame - no cuts=0.9 GeV, LCMS Frame - no cuts
WPCF 2007 - Aug. 1-3, 2007 26
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}(Q) - average of X over # of pairs for each Q-bin
WPCF 2007 - Aug. 1-3, 2007 28
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[ ]{ } ×
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{ }
WPCF 2007 - Aug. 1-3, 2007 29
MMinvinv distribution w/ background distribution w/ background subtraction subtraction
N=18
WPCF 2007 - Aug. 1-3, 2007 30
EMCICs contribution to vEMCICs contribution to v22
€
C( p1, 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
∑ ⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
€
(cos mΔφ) cos( nΔφ)dΔφ = δmnπ∫ for v2 n=2
€
1N
: − 2r p T,1 ⋅
r p T ,2
pT2
+E1 − E( ) E2 − E( )
E 2 − E2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
€
rp T,1 ⋅
r p T ,2 ~ cos( Δφ) no contribution to v2 from 1/N term
€
1N2
: 2r p T ,1 ⋅
r p T,2
pT2
+E1 − E( ) E2 − E( )
E 2 − E2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
2
€
rp T ,1 ⋅
r p T,2( )
2~ cos 2 (Δφ) ~ cos( 2Δφ)
contribution to v2 from 1/N2 term
€
1N3
: 2r p T ,1 ⋅
r p T,2
pT2
+pz,1 ⋅ pz,2
pz2
+E1 − E( ) E2 − E( )
E 2 − E2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
3
€
rp T ,1 ⋅
r p T,2( )
2~ cos 2 (Δφ) ~ cos( 2Δφ)
contribution to v2 from 1/N3 term
WPCF 2007 - Aug. 1-3, 2007 31
Non-id correlations (Resonance Non-id correlations (Resonance contrib.)contrib.)
WPCF 2007 - Aug. 1-3, 2007 32
Non-id correlations (PYTHIA@200 GeV)Non-id correlations (PYTHIA@200 GeV)
WPCF 2007 - Aug. 1-3, 2007 33
SummarySummary• understanding particle spectra, two-particle correlations,
v2, resonances in small systems– important physics-wise
– should not be attempted until data fully under control
• Restricted P.S. due to energy-momentum conservation– sampled by GenBod event generator
– generates EMCICs [femtoscopy : quantified by Alm’s]
– stronger effects for small multiplicities 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 [femtoscopy]– 4 new parameters, determined @ large |Q|
WPCF 2007 - Aug. 1-3, 2007 34
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]
• Mark Baker (BNL)
[local momentum conservation]• Dariusz Miskowiec (GSI)
[multiply (don’t add) correlations]