Conservation laws

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Conservation laws femtoscopy of small systems

• Zbigniew Chajecki Mike Lisa
• Ohio State University

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Outline

• 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 MCIC
• Experimentalists recipe Fitting correlation
  functions in progress
• Conclusion

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Introduction Motivation
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femtoscopy in pp _at_ STAR
Z. Chajecki WPCF05

• pp and AA measured in same experiment
• great opportunity to compare physics
• what causes pT-dependence in pp?
• same cause as in AA?

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Surprising (puzzling) scaling

• pp and AA measured in same experiment
• great opportunity to compare physics
• what causes pT-dependence in pp?
• same cause as in AA?

HBT radii scale with pp Scary coincidence or
something deeper?
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Surprising (puzzling) scaling

• pp and AA measured in same experiment
• great opportunity to compare physics
• what causes pT-dependence in pp?
• same cause as in AA?

A. Bialasz (ISMD05) I personally feel that its
solution may provide new insight into the
hadronization process of QCD
HBT radii scale with pp Scary coincidence or
something deeper?
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Clear interpretation clouded by data features
Non-femtoscopic q-anisotropic behaviour at large
q does this structure affect femtoscopic
region as well?
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SHD
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Spherical harmonic decomposition of CF

• Cartesian-space (out-side-long) naturally encodes
  physics, but is inefficient representation
• Harmonic Moments -- 11 connection to source
  geometry Danielewicz,Pratt nucl-th/0501003
• immune to acceptance
• full information content at a glancethanks to
  symmetries

This new method of analysis represents a real
breakthrough. ...(should) become a standard
tool in all experiments. - A. Bialas, ISMD 2005
Chajecki., Gutierrez, MAL, Lopez-Noriega,
nucl-ex/0505009
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Decomposition of CF onto Spherical Harmonics
AuAu central collisions
C(Qout)
C(Qside)
C(Qlong)
Z.Ch., Gutierrez, MAL, Lopez-Noriega,
nucl-ex/0505009 Pratt, Danielewicz
nucl-th/0501003
STAR preliminary
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Decomposition of CF onto Spherical Harmonics
non-femtoscopic structure (not just
non-Gaussian)
Z.Ch., Gutierrez, MAL, Lopez-Noriega,
nucl-ex/0505009 Pratt, Danielewicz
nucl-th/0501003
STAR preliminary
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Just push on....?

• ... no!
• Irresponsible to ad-hoc fit (often the practice)
  or ignore (!!) interpret without understanding
  data
• no particular reason to expect non-femtoscopic
  effect to be limited to non-femtoscopic (large-q)
  region
• not-understood or -controlled contaminating
  correlated effectsat low q ?
• A possibility energy-momentum conservation?
• must be there somewhere!
• but how to calculate / model ?(Upon
  consideration, non-trivial...)

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Genbod
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energy-momentum conservation in n-body states
spectrum of kinematic quantity ? (angle,
momentum) given by
n-body Phasespace factor Rn
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Example of use of total phase space integral

• In absence of physics in M (i.e. phase-space
  dominated)

• single-particle spectrum of ?

• spectrum of events

F. James, CERN REPORT 68-15 (1968)
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Genbodphasespace sampling w/ P-conservation

• 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 probability of event to occur
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Rounder events higher WT
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Rounder events higher WT
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Genbodphasespace sampling w/ P-conservation

• Treat identical to measured events
• use WT directly
• MC sample WT
• Form CF and SHD

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Effect of varying frame kinematic cuts

• Watch the green squares -- ?

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N18 ltKgt0.9 GeV LabCMS Frame - no cuts
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N18 ltKgt0.9 GeV LabCMS Frame - ?lt0.5
kinematic cuts have strong effect!
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N18 ltKgt0.9 GeV, LCMS - no cuts
kinematic cuts have strong effect!
as does choice of frame!
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N18 ltKgt0.9 GeV LCMS - ?lt0.5
kinematic cuts have strong effect!
as does choice of frame!
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N18 ltKgt0.9 GeV PRF - no cuts
kinematic cuts have strong effect!
as does choice of frame!
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N18 ltKgt0.9 GeV PRF - ?lt0.5
kinematic cuts have strong effect!
as does choice of frame!
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Effect of varying multiplicity total energy

• Watch the green squares -- ?

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GenBod 6 pions, ltKgt0.5 GeV/c
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GenBod 9 pions, ltKgt0.5 GeV/c
increasing mult reduces P.S. constraint
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GenBod 15 pions, ltKgt0.5 GeV/c
increasing mult reduces P.S. constraint
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GenBod 18 pions, ltKgt0.5 GeV/c
increasing mult reduces P.S. constraint
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GenBod 18 pions, ltKgt0.7 GeV/c
increasing mult reduces P.S. constraint
increasing ?s reduces P.S. constraint
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GenBod 18 pions, ltKgt0.9 GeV/c
increasing mult reduces P.S. constraint
increasing ?s reduces P.S. constraint
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So...

• Momentum Conservation Induced Correlations (MCIC)
  resemble our data
• So, MCIC... on the right track...
• But what to do with that?
• Sensitivity to ?s, Mult 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 MCICs 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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Distributions w/ phasespace constraints
single-particle distribution w/o P.S. restriction
k-particle distribution (kltN) with P.S.
restriction
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Using central limit theorem (large N-k)
k-particle distribution in N-particle system
() For simplicity, I from now on assume
identical particles (e.g. pions). I.e. all
particles have the same average energy and RMSs
of energy and momentum. Similar results (esp
experimentalist recipe) but more cumbersome
notation otherwise
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Effects on single-particle distribution
? What if all events had the same parent
distribution f, and all centrality dependence of
spectra was due just to loosening of P.S.
restrictions as N increased?
in this case, the index i is only keeping track
of particle type, really
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k-particle correlation function
2-particle correlation function (1st term in 1/N
expansion)
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2-particle correlation function (1st term in 1/N
expansion)
The pT term
The E term
The pZ term
Names used in the following plots
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Effect of varying 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

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GenBod 6 pions, ltKgt0.5 GeV/c
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GenBod 9 pions, ltKgt0.5 GeV/c
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GenBod 15 pions, ltKgt0.5 GeV/c
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GenBod 18 pions, ltKgt0.5 GeV/c
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GenBod 18 pions, ltKgt0.7 GeV/c
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GenBod 18 pions, ltKgt0.9 GeV/c
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Findings

• first-order and full calculations agree well for
  Ngt9
• will be important for experimentalists recipe
• Non-trivial competition/cooperation between pT,
  pz, E terms
• all three important
• pT1pT2 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, but dont nail
  it. Why?
• neither (N-k) nor ?s is infinite
• however, probably more important... next
  slide...

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Remember...
of course, the experimentalist never measures all
particles (including neutrinos) or ltpT2gt anyway,
so maybe not a big loss
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The experimentalists recipe

• Treat the not-precisely-known factors as fit
  parameters (4 of them)
• values determined mostly by large-Q should not
  cause fitting hell
• look, you will either ignore it or fit it ad-hoc
  anyway (both wrong)
• this recipe provides physically meaningful,
  justified form

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18 pions, ltKgt0.9 GeV
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The COMPLETE experimentalists recipe
femtoscopic function of choice
fit this...
...or image this...
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Summary

• 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 MCICs quantified by Alms
• stronger effects for small mult and/or ?s
• Analytic calculation of MCIC
• k-th order CF given by ratio of correction
  factors
• parent only relevant in momentum variances
• first-order expansion works well for Ngt9
• non-trivial interaction b/t pT, pz, E
  conservation effects
• Physically correct recipe to fit/remove MCIC
• 4 parameters, determined _at_ large Q
• parameter are physical - values may be guessed

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Thanks to...

• Alexy Stavinsky Konstantin Mikhaylov (Moscow)
  suggestion to use Genbod
• Jean-Yves Ollitrault (Saclay)original
  correlation formula
• Adam Kisiel (Warsaw) dont forget energy
  conservation
• Ulrich Heinz (Columbus)validating energy
  constraint in CLT

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Extra Slides
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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 N4 (but ?2/dof2.5)
• Gaussian by N10

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Baseline problems with smallest systems
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Try NA22 empirical form
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Schematic How Genbod works 1/3
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flow chart, in text
F. James, CERN REPORT 68-15 (1968)
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F. James, CERN REPORT 68-15 (1968)
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Schematic How Genbod works 2/3
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Schematic How Genbod works 3/3