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 9

GenBodGenBodPhase-Space Event

Generator

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 15

E - p correlations?E - p correlations?

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 27

EMCIC’s FIT: N=18, EMCIC’s FIT: N=18, KK=0.9GeV, =0.9GeV, LCMSLCMS

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]