Conservation laws & femtoscopy of small systems
Zbigniew Chajeçki & Mike Lisa
Ohio State University
nucl-th/0612080
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
1
Outline
• Introduction
– RHI, femtoscopy, & collectivity in bulk matter
– including: new representation of correlation functions
• the p+p “reference”
– intriguing features pp versus AA
– non-femtoscopic effects
• Global conservation effects (EMCICs)
– Restricted phasespace calculations: GenBod (FOWL)
– Analytic EMCIC calculations
– Experimentalists’ recipe:
– Fitting correlation functions [in progress]
• Summary
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Nuclear
Particle
PHOBOS
PHENIX
RHIC
BRAHMS
STAR
AGS
TANDEMS
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Nuclear
Why heavy ion collisions?
Particle
STAR ~500 Collaborators
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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2 types (?) of collisions...
looks like fun...
looks like a mess...
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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A dynamical but crude view of the collision
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
c/o UrQMD Collaboration, Frankfurt
In this model, insufficient re-interactions
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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A dynamical but crude view of the collision
In this model, insufficient re-interactions
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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R.H.I.C. defined...
• collision of nuclei sufficiently large that nuclear details unimportant
– distinct from nuclear or particle physics
– “Geranium on Linoleum”
• sufficiently large for meaningful bulk
& thermodynamic quantities
• non-trivial spatial scales & geometry
drive bulk dynamics (e.g. flow)
• how big is “sufficient” ?
– important reference issue
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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phase structure of bulk system:
• driving symmetries
• long-range collective behaviour
• “new” physics [superfluidity in l-He]
• relevance ofma
meaningful
EoS
lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Generating a deconfined state
Present understanding of
Quantum Chromodynamics (QCD)
• heating
• compression
 deconfined color matter !
Hadronic
Nuclear
Matter
Matter
Quark
Gluon
Plasma
(confined)!
deconfined
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Expectations from Lattice QCD
/T4 ~ # degrees of freedom
deconfined:
many d.o.f.
confined:
few d.o.f.
TC ≈ 173 MeV ≈ 21012 K ≈ 130,000T[Sun’s core]
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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The four elements from 400 BC to 2000AD
400 BC : all creation
Air
Water
?
Earth
Fire
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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RHIC energies: the first quantitative success of hydro
• direct access to EoS (phase transitions, lattice, etc.)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
EoS: P versus  versus n
(Heinz et al)
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Microexplosions
Femtoexplosions
• energy quickly
deposited
17 J/m3

10
5 GeV/fm3 = 1036 J/m3
• enter plasma phase
•sexpand hydrodynamically
0.1 J
1 J
•Tcool back to
phase 200 MeV = 1012 K
6 K
10original
• do geometric “postmortem” & infer momentum
rate
1018 K/sec
1035 K/s
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Microexplosions
Femtoexplosions
• energy quickly deposited
0.1phase
J
1 J
•senter plasma
•expand hydrodynamically
1017 J/m3
5 GeV/fm3 = 1036 J/m3
•Tcool back to
phase 200 MeV = 1012 K
6 K
10original
• do geometric “postmortem” & infer momentum
rate
1018 K/sec
1035 K/s
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Impact parameter & Reaction plane
Impact parameter vector b :
 beam direction
connects centers of colliding nuclei
b = 0  “central collision”
many particles produced
“peripheral collision”
fewer particles produced
b
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Impact parameter & Reaction plane
Impact parameter vector b :
 beam direction
connects centers of colliding nuclei
Reaction plane:
spanned by beam direction and b
b = 0  “central collision”
many particles produced
“peripheral collision”
fewer particles produced
b
b
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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How do semi-central collisions evolve?
1) Superposition of independent p+p:
momenta pointed at random
relative to reaction plane
b
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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How do semi-central collisions evolve?
1) Superposition of independent p+p:
high
density / pressure
at center
momenta pointed at random
relative to reaction plane
2) Evolution as a bulk system
Pressure gradients (larger in-plane)
push bulk “out”  “flow”
“zero” pressure
in surrounding vacuum
more, faster particles
seen in-plane
b
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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How do semi-central collisions evolve?
1) Superposition of independent p+p: N
momenta pointed at random
relative to reaction plane
0
/4
/2
0
/4
/2
3/4

3/4

-RP (rad)
2) Evolution as a bulk system
Pressure gradients (larger in-plane)
push bulk “out”  “flow”
more, faster particles
seen in-plane
N
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
-RP (rad)
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Azimuthal distributions at RHIC
STAR, PRL90 032301 (2003)
b ≈ 6.5 fm
b ≈ 4 fm
“central” collisions
midcentral
collisions
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Azimuthal distributions at RHIC
STAR, PRL90 032301 (2003)
b ≈ 10 fm
b ≈ 6.5 fm
b ≈ 4 fm
peripheral collisions
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Beyond press releases
Nature of EoS under
investigation ; agreement with
data may be accidental ;
viscous hydro under
development ; assumption
of thermalization in question
sensitive to modeling of
initial state,
presently under study
The detailed work now underway is what can probe & constrain sQGP properties
It is probably not press-release material...
...but, hey, you’ve already got your coffee mug
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Beyond press releases:
Access to the dynamically-generated geometric substructure?
The feature of collectivity:
space
is globally correlated with
momentum
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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probing source geometry through interferometry
The Bottom line…
p1
r1
x1
  
  to
  i( r  x )p
if a pion is emitted, itis more
i ( r emit
x )p another
1 { likely
U(x1, p1)e
U(x 2 , p2 )e
 source
T
2
source
  
 
1 m momentum
(x) pion with very similar
 if the
is small
i ( r  x )p
i ( r  x )p
 U(x 2 , p1)e
p2
5 fm
r2
1
1
1
2
1
2
U(x1, p2 )e

2
2
1
*TT  U1*U1  U*2 U 2  1  eiq( x1  x 2 )
experimentally measuring
this enhanced
Creation probability
(x,p) =probability:
U*U
quite challenging
P(p1, p 2 )
2
C(p1, p 2 ) 
 1 ~
 (q )
P(p1 )P(p 2 )
C (Qinv)
x2
1
2
2
}

Width ~ 1/R
2
1
Measurable!
F.T. of pion source
  
q  p 2  p1
0.05
0.10
Qinv (GeV/c)
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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C2(Qinv)
Correlation functions for different colliding systems
Au+Au: central collisions
STAR preliminary
p+p
R ~ 1 fm
C(Qout)
d+Au
R ~ 2 fm
Au+Au
R ~ 6 fm
C(Qside)
C(Qlong)
Qinv (GeV/c)
Different colliding systems studied at RHIC
Interferometry probes the smallest
scales ever measured !
3 “radii” by using
3-D vector q
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Femtoscopic information
Sab
( r )
P
= x a - x b  distribution
Au+Au: central collisions
C(Qout)
 (q, r ) = (a,b) relative wavefctn
pa
pb
xa
xa
xb
C(Qside)
xb
C (q) 
ab
P
pa
pb
 d r S ( r)  (q, r )
3
ab
P
2
• femtoscopic correlation at low |q|
• must vanish at high |q|. [indep “direction”]
C(Qlong)
3 “radii” by using
3-D vector q
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Femtoscopic information - Spherical harmonic representation

Al ,m (| Q |) 
 cos  
4
all.bins

 Yl ,m ( i ,i )C(| Q |, cos  i ,i )
i
nucl-ex/0505009
This new Q
method
represents a
LONG of analysis
Q
real breakthrough. ...(should) become a
standard tool in all experiments.
QOUT
- A. Bialas, ISMD2005
QSIDE
Au+Au: central collisions
C(Qout)
C(Qside)

• femtoscopic correlation at low |q|
• must vanish at high |q|. [indep “direction”]
C(Qlong)
3 “radii” by using
3-D vector q
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Femtoscopic information - Spherical harmonic representation

Al ,m (| Q |) 
 cos  
4
all.bins

 Yl ,m ( i ,i )C(| Q |, cos  i ,i )
Au+Au: central collisions
i
nucl-ex/0505009
C(Qout)
L=0
L=2
M=0
L=2
M=2
• femtoscopic correlation at low |q|
• must vanish at high |q|. [indep “direction”]
•ALM(Q) = L,0
C(Qside)
C(Qlong)
3 “radii” by using
3-D vector q
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Why do the radii fall
with increasing momentum ??
(3 "radii" corresponding to the three components of
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
q)
32
Why do the radii fall
with increasing momentum ??
It’s collective flow !!
Direct geometrical/dynamical evidence
for bulk behaviour!!!
Amount of flow consistent with p-space nucl-th/0312024
Huge, diverse systematics consistent with this substructure
nucl-ex/0505014
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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p+p: A clear reference system?
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
35
Z. Chajecki QM05
nucl-ex/0510014
femtoscopy in p+p @ STAR
• Decades of femtoscopy in p+p and in A+A, but...
• for the first time: femtoscopy in p+p and A+A in same experiment, same
analysis definitions...
• unique opportunity to compare physics
• ~ 1 fm makes sense, but...
• pT-dependence in p+p?
• (same cause as in A+A?)
STAR preliminary
mT (GeV)
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
mT (GeV)
36
Surprising („puzzling”) scaling
A. Bialasz (ISMD05):
I personally feel that its solution may provide new
insight into the hadronization
process
of QCD
Ratio
of (AuAu,
CuCu, dAu) HBT
radii by pp
HBT radii scale with pp
Scary coincidence
or something deeper?
On the face: same geometric
substructure
pp, dAu, CuCu - STAR preliminary
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
37
BUT... 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 - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
38
Decomposition of CF onto Spherical Harmonics
STAR preliminary
d+Au peripheral collisions
non-femtoscopic structure
(not just “non-Gaussian”)
Gaussian fit

Al ,m (| Q |) 
 cos 
4
all.bins

i

Yl ,m ( i , i )C (| Q |, cos i , i )
Z.Ch., Gutierrez, MAL,
Lopez-Noriega, nucl-ex/0505009
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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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...)
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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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
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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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)
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University
- January
2007
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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
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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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 - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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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
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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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

ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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Effect of varying frame
& kinematic cuts
Watch the green squares -- 
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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N=18 <K>=0.9 GeV; LabCMS Frame - no cuts
Watch the green squares
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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N=18 <K>=0.9 GeV; LabCMS Frame - ||<0.5
Watch the green squares
kinematic cuts have strong effect!
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
50
N=18 <K>=0.9 GeV, LCMS - no cuts
Watch the green squares
kinematic cuts have strong effect!
as does choice of frame!
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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N=18 <K>=0.9 GeV; LCMS - ||<0.5
Watch the green squares
kinematic cuts have strong effect!
as does choice of frame!
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
52
N=18 <K>=0.9 GeV; PRF - no cuts
Watch the green squares
kinematic cuts have strong effect!
as does choice of frame!
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
53
N=18 <K>=0.9 GeV; PRF - ||<0.5
Watch the green squares
kinematic cuts have strong effect!
as does choice of frame!
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
54
Effect of varying
multiplicity & total energy
Watch the green squares -- 
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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GenBod : 6 pions, <K>=0.5 GeV/c
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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increasing mult reduces P.S. constraint
GenBod : 9 pions, <K>=0.5 GeV/c
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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increasing mult reduces P.S. constraint
GenBod : 15 pions, <K>=0.5 GeV/c
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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increasing mult reduces P.S. constraint
GenBod : 18 pions, <K>=0.5 GeV/c
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
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increasing mult reduces P.S. constraint
GenBod : 18 pions, <K>=0.7 GeV/c
increasing s reduces P.S. constraint
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
60
increasing mult reduces P.S. constraint
GenBod : 18 pions, <K>=0.9 GeV/c
increasing s reduces P.S. constraint
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
61
So...
• Energy & Momentum Conservation Induced Correlations (EMCICs)
“resemble” our data
• So... 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 EMCICs using data!!
– Danielewicz et al, PRC38 120 (1988)
– Borghini, Dinh, & Ollitraut PRC62 034902 (2000)
– Chajecki & MAL, nucl-th/0612080 - generalization of the above
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
62
Distributions w/ phasespace constraints
˜f ( p )  2E f ( p )  2E dN
i
i
i
i 3
d pi
single-particle distribution
w/o P.S. restriction

ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
63

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
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
64
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
otherwise
ma lisa - Femtoscopy
in small systems
EMCICs
- Kent State notation
University
- January 2007
65
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



ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
66
k-particle correlation function
f˜c (p1,...,p k )
C(p1,...,p k )  ˜
fc (p1)....f˜c (p k )
2
2
2
2 

 k
k
k
k
















i1 p x,i  i1 p y,i  i1 p z,i  i1 E i  E  
1
exp





2
2
2
2
2
 N 2
2(N

k)
p
p
p
E

E

x
y
z






N  k 


2 

2
k  2
 N 2k
2
E

E
p
p
p


1


 x,i  y,i  z,i  i

exp

2 
N 1
2
2
2
2

 2(N 1) i1 p x
py
pz
E  E 



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

ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
67
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
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
68
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
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
69
GenBod : 6 pions, <K>=0.5 GeV/c
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
70
GenBod : 9 pions, <K>=0.5 GeV/c
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
71
GenBod : 15 pions, <K>=0.5 GeV/c
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
72
GenBod : 18 pions, <K>=0.5 GeV/c
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
73
GenBod : 18 pions, <K>=0.7 GeV/c
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
74
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 - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
75
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

ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
76
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.
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
77
18 pions, <K>=0.9 GeV
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
78
Highly correlated
EMCIC parameters
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
79
The COMPLETE experimentalist’s recipe
fit this...
femtoscopic
function of
choice
  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
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
80
Summary
• H.I.C raison d’etre & distinction from p+p
– bulk, thermalized, collective matter
• Momentum-space and femtoscopic probes support bulk collectivity
• The p+p “reference” - [femtoscopic apples-to-apples for the first time]
– is it understood?
– is it really different than a “small A+A” ?
– full understanding must be high priority
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
81
Famous picture from famous article by famous guy
Energy loss of energetic partons in quark-gluon plasma:
Possible extinction of high pT jets in hadron-hadron collisions
J.D. Bjorken, 1982
b
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
82
Summary
• H.I.C raison d’etre & distinction from p+p
– bulk, thermalized, collective matter
• Momentum-space and femtoscopic probes support bulk collectivity
• The p+p “reference” - [femtoscopic apples-to-apples for the first time]
– is it understood?
– is it really different than a “small A+A” ?
– full understanding must be high priority
• Interpretation complicated by non-femtoscopic correlations
• May be largely due to P.S. restrictions (EMCICs)
– numerical studies track data
– analytic formulation of EMCIC projection onto femtoscopic analysis
presented
– highly non-trivial interplay between cuts, frames, and conserved
components (pz, pT, E)
– first-order expansion --> experimentalists’ formula
– application to STAR underway
• Hotter spotlight due to impending LHC (December 2007!!)
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
83
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
84
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; resonance effects in +- -]
• Ulrich Heinz (Columbus)
[validating energy constraint in CLT]
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
85
Extra Slides
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
86
ZEUS DIS results: hadronic “v2” w/ known RP
ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 2007
87
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)

• 
N inNsmall systems & EMCICs - Kent State University - January 2007
ma lisa - Femtoscopy
88