Conservation laws & femtoscopy of small systems
Zbigniew Chajecki & Mike Lisa
Ohio State University
ma lisa - Momentum conservation effects - WPCF 2006
1
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
ma lisa - Momentum conservation effects - WPCF 2006
2
Z. Chajecki WPCF05
femtoscopy in p+p @ STAR
• p+p and A+A measured in same
experiment
• great opportunity to compare
physics
• what causes pT-dependence in
p+p?
• same cause as in A+A?
STAR preliminary
mT (GeV)
ma lisa - Momentum conservation effects - WPCF 2006
mT (GeV)
4
Surprising („puzzling”) scaling
• p+p and A+A measured in same
experiment
• great opportunity to compare
physics
Ratio of (AuAu, CuCu, dAu) HBT
radii by pp
• what causes pT-dependence in
p+p?
• same cause as in A+A?
HBT radii scale with pp
Scary coincidence
or something deeper?
pp, dAu, CuCu - STAR preliminary
ma lisa - Momentum conservation effects - WPCF 2006
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Surprising („puzzling”) scaling
A. Bialaszin(ISMD05):
• p+p and A+A measured
same
I personally feel that its solution may provide new
experiment
insight into the hadronization
process
of QCD
Ratio
of (AuAu,
CuCu, dAu) HBT
radii by pp
• great opportunity to compare
physics
• what causes pT-dependence in
p+p?
• same cause as in A+A?
HBT radii scale with pp
Scary coincidence
or something deeper?
pp, dAu, CuCu - STAR preliminary
ma lisa - Momentum conservation effects - WPCF 2006
6
Clear interpretation clouded by data features
STAR preliminary
d+Au peripheral collisions
Gaussian fit
Non-femtoscopic q-anisotropic
behaviour at large |q|
does this structure affect
femtoscopic region as well?
ma lisa - Momentum conservation effects - WPCF 2006
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Spherical harmonic decomposition of CF
• Cartesian-space (out-side-long) naturally
encodes physics, but is “inefficient”
representation
2
2
2
Q  QOUT
 QSIDE
 QLONG
cos( ) 
QLONG
QTOT
Q
• Harmonic Moments -- 1::1 connection to source
  arctan SIDE
geometry
This new method of analysisQOUT
represents a
[Danielewicz,Pratt: nucl-th/0501003]
• ~immune to acceptance
real breakthrough. ...(should) become a
standard tool in all experiments.
- A. Bialas, ISMD 2005
• full information content at a glance
[thanks to symmetries]

Al ,m (| Q |) 
 cos  
4
all.bins
QLONG

 Yl ,m ( i ,i )C(| Q |, cos  i ,i )
i
QSIDE
Q

QOUT

Chajecki., Gutierrez, MAL, Lopez-Noriega,
ma lisanucl-ex/0505009
- Momentum conservation effects - WPCF 2006
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Decomposition of CF onto Spherical Harmonics
Au+Au: central collisions
C(Qout)
C(Qside)

Al ,m (| Q |) 
 cos 
4
all.bins


Yl ,m ( i , i )C (| Q |, cos i , i )
i
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”)
d+Au:
peripheral collisions

Al ,m (| Q |) 
STAR preliminary
 cos 
4
all.bins

i

Yl ,m ( i , i )C (| Q |, cos i , i )
Z.Ch., Gutierrez, MAL,
Lopez-Noriega, nucl-ex/0505009
Pratt, Danielewicz [nucl-th/0501003]
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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 effects
at low q ?
• A possibility: energy-momentum conservation?
– must be there somewhere!
– but how to calculate / model ?
(Upon consideration, non-trivial...)
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energy-momentum conservation in n-body states
spectrum of kinematic quantity 
(angle, momentum) given by
f   

d
2
M  Rn
d

where
M  matrix element describing interaction
(M = 1  all spectra given by phasespace)

n-body Phasespace factor Rn
Rn 

4n
n

 n
2
2
4

 
P

p

p

m
d
pi




j
i
i


 j1 i1
statistics: “density of states”
2
p
p  m d p i  i d p i  dcos i  d i
Ei
2
i
4
where
P  total 4 - momentum of n - particle system 
p i  4 - momentum of particle i
mi  mass of particle i
2
i
4
larger particle momentum more available states
P conservation
n

 Induces “trivial” correlations
 
P   p j 
 (i.e. even for M=1)
 j1 
4
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Example of use of total phase space integral
• In absence of “physics” in M : (i.e. phase-space dominated)
pp   
R 3 1.876; ,  ,  

pp    R 4 1.876; ,  ,  ,  
• single-particle spectrum of :

d
f   
Rn
d
• “spectrum of events”:

In limit where " "="event" = collection of momenta p i
d
"spectrum of events" = f   
Rn
d
d 3n
 Pr ob event   n
Rn
 dp3i
i1
F. James, CERN REPORT 68-15 (1968)
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Genbod:phasespace sampling w/ Pconservation
• 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
1 n1
WT 
M i1R 2 M i1;M i ,mi1

M m i1
WT ~ probability of event to occur
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“Rounder” events: higher WT
Rn 

4n
n

 n
2
2
4

 
P

p

p

m
d
pi




j
i
i


 j1 i1
4
2
p
 p  m d p i  i d p i  dcos i   d i
Ei
2
i
2
i
4
larger particle momentum more available states

6 particles
ma lisa - Momentum conservation effects - WPCF 2006
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“Rounder” events: higher WT
Rn 

4n
n

 n
2
2
4

 
P

p

p

m
d
pi




j
i
i


 j1 i1
4
2
p
 p  m d p i  i d p i  dcos i   d i
Ei
2
i
2
i
4
larger particle momentum more available states

30 particles
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Genbod:phasespace sampling w/ Pconservation
• Treat identical to
measured events
• use WT directly
• MC sample WT
• Form CF and SHD
1 n1
WT 
M i1R 2 M i1;M i ,mi1
M m i1

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Effect of varying frame
& kinematic cuts
Watch the green squares -- 
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N=18 <K>=0.9 GeV; LabCMS Frame - no cuts
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N=18 <K>=0.9 GeV; LabCMS Frame - ||<0.5
kinematic cuts have strong effect!
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N=18 <K>=0.9 GeV, LCMS - no cuts
kinematic cuts have strong effect!
as does choice of frame!
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N=18 <K>=0.9 GeV; LCMS - ||<0.5
kinematic cuts have strong effect!
as does choice of frame!
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N=18 <K>=0.9 GeV; PRF - no cuts
kinematic cuts have strong effect!
as does choice of frame!
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N=18 <K>=0.9 GeV; PRF - ||<0.5
kinematic cuts have strong effect!
as does choice of frame!
ma lisa - Momentum conservation effects - WPCF 2006
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Effect of varying
multiplicity & total energy
Watch the green squares -- 
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GenBod : 6 pions, <K>=0.5 GeV/c
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increasing mult reduces P.S. constraint
GenBod : 9 pions, <K>=0.5 GeV/c
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increasing mult reduces P.S. constraint
GenBod : 15 pions, <K>=0.5 GeV/c
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increasing mult reduces P.S. constraint
GenBod : 18 pions, <K>=0.5 GeV/c
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increasing mult reduces P.S. constraint
GenBod : 18 pions, <K>=0.7 GeV/c
increasing s reduces P.S. constraint
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increasing mult reduces P.S. constraint
GenBod : 18 pions, <K>=0.9 GeV/c
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
˜f ( p )  2E f ( p )  2E dN
i
i
i
i 3
d pi
k
˜f (p ,...,p )  
˜ (p )


f

c 1
k
 i1 i 
single-particle distribution
w/o P.S. restriction

 N d 3p i
 4  N
˜
 i k 1 2E f (pi )  pi  P
i1

i

 N d 3p i
 4  N
˜
 i1 2E f (pi )  pi  P
i1

i
 N

N


4
2
2 ˜
4

d p i (p i  mi )f (p i )  p i  P

i
k
1

 
k



i1
˜
  f (p i )
 i1

 N

N


4
2
2 ˜
4
 i1d pi(pi  mi )f (pi )  pi  P
i1

k-particle distribution (k<N) with P.S. restriction
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Using central limit theorem (“large N-k”)
k-particle distribution in N-particle system

 k
2 

p i,  p   
2
3 

i1
 
 k
 N 
f˜c (p1,...,p k )   f˜ (p i )
exp

  2(N  k) 2 
 i1
N  k 


   0





where

 2  p 2  p 
p  0


2
for   1,2,3
N.B.
relevant later
p2 
 d p  p  f˜p
3
2
unmeasured
parent distrib

 d p  p  f˜ p
3
2
c
measured

(*) For simplicity, I from now on assume identical particles (e.g. pions). I.e. all particles have
the same average energy and RMS’s of energy and momentum. Similar results (esp
“experimentalist
recipe)
but more cumbersome
notation
ma lisa - Momentum
conservation
effects - WPCF
2006otherwise
36
Effects on single-particle distribution
2 
 3
p i,  p  
N 

˜f (p )  f˜ (p )
 exp
c
i
i 
2 
N 1
  0 2(N 1) 


2 

 2
2
2
2
E

E
p


p z,i  i
 
N
1  p x,i
y,i
 f˜ (p i )



 exp
2 
2
2
2
 2(N 1)  p 2x

N 1
p
p
E

E
y
z



2



? What if all events had the same “parent” distribution f,
and all centrality dependence of spectra was due just to
in this case, the index i is only keeping
loosening of P.S. restrictions as
ma N
lisaincreased?
- Momentum conservation effects - WPCFtrack
2006of particle type, really
37
k-particle correlation function
f˜c (p1,...,p k )
C(p1,...,p k )  ˜
fc (p1 )....f˜c (p k )
 N 2


N  k 

 N 2k


N 1
2
2
2
2 

 k
k
k
k








k  p x,i 
p  
p  
E  E  

1
 i1  i1 y,i  i1 z,i  i1  i

exp






2
2
2
2
2
2(N

k)
p
p
p
E  E

i1 
x
y
z




2 

2
k  2
2
E

E
p
 
1
 p x,i  y,i  p z,i   i
exp

2 
2
2
2
 2(N 1) i1  p 2x

p
p
E

E
y
z



Dependence on “parent” distrib f vanishes,
except for energy/momentum means and RMS
2-particle correlation function (1st term in 1/N expansion)


1  pT,1  pT,2 pz,1  pz,2 E1  E  E 2  E 
C(p1,p2 )  1 2


2
2
2
2


N 
pT
pz
E  E

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2-particle correlation function (1st term in 1/N expansion)


E

E

E

E



1  pT,1  pT,2 pz,1  pz,2  1
2

C(p1,p2 )  1 2


2
2
2
2

N 
p
p
E

E
T
z


“The pT term”
“The pZ term”
“The E 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, <K>=0.5 GeV/c
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GenBod : 9 pions, <K>=0.5 GeV/c
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GenBod : 15 pions, <K>=0.5 GeV/c
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GenBod : 18 pions, <K>=0.5 GeV/c
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GenBod : 18 pions, <K>=0.7 GeV/c
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GenBod : 18 pions, <K>=0.9 GeV/c
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Findings
• 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, but don’t nail it. Why?
– neither (N-k) nor s is infinite
– however, probably more important... [next slide]...
ma lisa - Momentum conservation effects - WPCF 2006
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Remember...


E

E

E

E
p

p
p

p




1
1
2

C(p1,p2 )  1 2 T,1 2 T,2  z,1 2 z,2 
2
2

N 
p
p
E

E
T
z


p2   d3p p2  f˜p  p2   d3p p2  f˜c p
c
unmeasured
parent distrib
measured
relevant quantities are average over the (unmeasured) “parent” distribution,
not the physical distribution

expect
p2  p2
c
of course, the experimentalist never measures all particles
2> anyway, so maybe not a big loss
(including neutrinos)
or
<p
T

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The experimentalist’s 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
C(p1,p 2 )  1
2
N p 2T
NEW PARAM 1


1
N p 2Z
 p1,T  p 2,T 
NEW PARAM 2
1
N E  E
2
NEW PARAM 3
 p1,z  p 2,z 
2
 E1  E 2  


E
N E  E
2
2
 E1  E 2 

NEW PARAM 4

E
2
N E2  E
2

UNIMPORTANT
"NORMALIZED AWAY"
where
X denotes the average of X over the (p 1,p 2 ) bin. (or q - bin or whatever we are binning in)
I.e. it is just another histogram which the experimentalist makes,
from the data
momenta and energy are measured in the lab frame.
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18 pions, <K>=0.9 GeV
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51
The COMPLETE experimentalist’s recipe
femtoscopic
function of
choice
fit this...
  R 2 q 2 

C(p1,p 2 )  Norm1 M1  p1,T  p 2,T  M 2  p1,z  p 2,z  M 3  E1  E 2   M 4  E1  E 2 "   e  o,s,l "




...or image this...
C(q)  M1  p1,T  p2,T  M2  p1,z  p2,z M3  E1  E 2  M4  E1  E 2
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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 Alm’s
– 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 N>9
non-trivial interaction b/t pT, pz, E conservation effects
• Physically correct “recipe” to fit/remove MCIC
– 4 parameters, determined @ 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)
[don’t forget energy conservation]
• Ulrich Heinz (Columbus)
[validating energy constraint in CLT]
ma lisa - Momentum conservation effects - WPCF 2006
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Extra Slides
ma lisa - Momentum conservation effects - WPCF 2006
55
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
N
• x  x N x (remember plots scaled by N)


i
i1
 
2
2


N




N

(

remember plots scaled by N)

• 
Nma lisaN
- Momentum conservation effects - WPCF 2006
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Baseline problems with smallest systems
STAR preliminary
d+Au peripheral collisions
Gaussian fit
ad hoc, but try it...
ma lisa - Momentum conservation effects - WPCF 2006
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Try NA22 empirical form
STAR preliminary
d+Au peripheral collisions
Spherical harmonics
NA22 fit
data
L =1
M=0
L =2
M=0
NA22 fit
L =1
M=1
ma lisa - Momentum conservation effects - WPCF 2006
L =2
M=2
58
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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
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