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]


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


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?


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


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


Spherical Harmonic Spherical Harmonic Decomposition Decomposition

of the Correlation of the Correlation FunctionFunction


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


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


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


Multiplicity dependence of the Multiplicity dependence of the baselinebaseline

Baseline problem is increasing

with decreasing multiplicity

STAR preliminary


GenBodGenBodPhase-Space Event

Generator


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


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


CF from GenBodCF from GenBod

Varying frame and Varying frame and kinematic cutskinematic cuts


N=18, <K>=0.9 GeV, LabCMS Frame - no N=18, <K>=0.9 GeV, LabCMS Frame - no cutscuts


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


N=18, <K>=0.9 GeV, LCMS Frame - no cutsN=18, <K>=0.9 GeV, LCMS Frame - no cuts


• kinematic cuts

• frame


N=18, <K>=0.9 GeV, LCMS Frame - |N=18, <K>=0.9 GeV, LCMS Frame - ||<0.5|<0.5


• kinematic cuts

• frame


GenBodGenBod

Varying multiplicity Varying multiplicity and total energyand total energy




• kinematic cuts

• frame




• kinematic cuts

• frame

• particle multiplicity




• kinematic cuts

• frame





• kinematic cuts

• frame


• total energy : √s




• kinematic cuts

• frame



The shape of the CF is sensitive to:

• kinematic cuts

• frame




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


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


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 ?


E - p correlations?E - p correlations?


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


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

∑ ⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟


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

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


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


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


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


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


EMCIC’s FIT: N=18, <K>=0.9GeV, EMCIC’s FIT: N=18, <K>=0.9GeV, LCMSLCMS


Fit contoursFit contours


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{ }


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


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]


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


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


Simple, Gaussian source calculations

~acceptance freeRL < RT

RL > RT

RO < RS

RO > RS


NA22 parametrization of CFNA22 parametrization of CF

?


NA22: 1D projections of 3D CFNA22: 1D projections of 3D CF

NA22, Z. Phys. C71 (1996) 405


NA22 parametrization of CFNA22 parametrization of CF

STAR preliminary d+Au peripheral collisions

NA22 fit

d+Au peripheral collisions


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)


Schematic: How GenBod works Schematic: How GenBod works 1/31/3


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

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟


Reconstruction of CF from Reconstruction of CF from Alm’sAlm’s

conservation laws and femtoscopy of small systems

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