EMCICs and the
femtoscopy of small systems
Mike Lisa & Zbigniew Chajecki
Ohio State University
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Outline
• LHC predictions
– H.I. see SPHIC06 talk (nucl-th/0701058)
– p+p: see talk of T. Humanic
• 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 EMCIC
• Experimentalists’ recipe: Fitting correlation functions [in progress]
• Conclusion
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster
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, 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 - ALICE week 12-16 Feb 2007 - Muenster
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 (q) 
ab
P
pa
pb
C(Qside)
xb
 d r S ( r)  (q, r )
3
ab
P
2
• femtoscopic correlation at low |q|
• must vanish at high |q|. [indep “direction”]
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
C(Qlong)
3 “radii” by using
3-D vector q
Femtoscopic information - Spherical harmonic representation

Al ,m (| Q |) 
 cos  
4
 Yl ,m ( i ,i )C(| Q |, cos  i ,i )
Au+Au: central collisions
i
nucl-ex/0505009
QLONG
C(Qout)
C(Qside)
Q

QSIDE

all.bins
QOUT

C(Qlong)
• femtoscopic correlation at low |q|
• must vanish at high |q|. [indep “direction”]
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
3 “radii” by using
3-D vector q
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
C(Qside)
C(Qlong)
3 “radii” by using
3-D vector q
Kinematic dependence of femtoscopy:
Geometrical/dynamical evidence of bulk behaviour
(3 "radii" corresponding to the three components of
q)

Amount of flow consistent with p-space nucl-th/0312024
Huge, diverse systematics consistent with this substructure
nucl-ex/0505014
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
p+p: A clear reference system?
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster
mT (GeV)
Surprising („puzzling”) scaling
HBT radii scale with pp
Ratio of (AuAu, CuCu, dAu) HBT
radii by pp
Scary coincidence
or something deeper?
On the face: same geometric
substructure
A. Bialasz (ISMD05):
I personally feel that its solution may provide new
insight into the hadronization process of QCD
pp, dAu, CuCu - STAR preliminary
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster
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 )
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Z.Ch., Gutierrez, MAL,
Lopez-Noriega, nucl-ex/0505009
Baseline problems with small systems: previous treatments
STAR preliminary
d+Au peripheral collisions
Gaussian fit
ad hoc, but try it...
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster
L =2
M=2
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 - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster
“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 - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster
CF from GenBod
Varying frame and
kinematic cuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.9 GeV, LabCMS Frame - no cuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.9 GeV, LabCMS Frame - ||<0.5
The shape of the CF is
sensitive to
• kinematic cuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.9 GeV, LCMS Frame - no cuts
The shape of the CF is
sensitive to
• kinematic cuts
• frame
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.9 GeV, LCMS Frame - ||<0.5
The shape of the CF is
sensitive to
• kinematic cuts
• frame
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.9 GeV, PR Frame - no cuts
The shape of the CF is
sensitive to
• kinematic cuts
• frame
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.9 GeV, PR Frame - ||<0.5
The shape of the CF is
sensitive to
• kinematic cuts
• frame
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
GenBod
Varying multiplicity
and total energy
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=6, <K>=0.5 GeV, LCMS Frame - no cuts
The shape of the CF is
sensitive to
• kinematic cuts
• frame
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=9, <K>=0.5 GeV, LCMS Frame - no cuts
The shape of the CF is
sensitive to
• kinematic cuts
• frame
• particle
multiplicity
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=15, <K>=0.5 GeV, LCMS Frame - no cuts
The shape of the CF is
sensitive to
• kinematic cuts
• frame
• particle
multiplicity
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.5 GeV, LCMS Frame - no cuts
The shape of the CF is
sensitive to
• kinematic cuts
• frame
• particle
multiplicity
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.7 GeV, LCMS Frame - no cuts
The shape of the CF is
sensitive to
• kinematic cuts
• frame
• particle
multiplicity
• total energy : √s
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.9 GeV, LCMS Frame - no cuts
The shape of the CF is sensitive to
• kinematic cuts
The shape of the CF is
•sensitive
frameto
• kinematic cuts
• particle
• frame
• particle
multiplicity
multiplicity
•• total
total
energy
: √s
energy
: √s
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
So...
• Energy & Momentum Conservation Induced Correlations (EMCICs)
“resemble” our data
– ... 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!!
– pT conservation and v2
• Danielewicz et al, PRC38 120 (1988)
• Borghini, Dinh, & Ollitraut PRC62 034902 (2000)
– D spatial dimensions and M-cumulants
• Borghini, Euro. Phys. C30 381 (2003)
– 3+1 (p+E) conservation and femtoscopy
• Chajecki & MAL, nucl-th/0612080 - [WPCF06]
– pT conservation and 3-particle correlations
• Borghini ,nucl-th/0612093
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster

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 - ALICE week 12-16 Feb 2007 - Muenster

k
N




N


4
2
2
4
˜
w/ dphasespace
FNkDistributions
p i (p i  mi )f (p i ) constraints
P   p i   
 p i  P

i
k
1


 i1 
i1



N
k
 single-particle

distribution
 f˜ (Np ) 4 2E 2f ( p2) ˜ 2E
 4 dN
   i d p i  (pii  mii )f (p i ) i 
p

P

p



 iP.S.
3 i
d

i k 1
w/o
p

 restriction

 i k 1 i
i1
g(p i )


3
 4  N
FNk is just the distribution of thesumNof adlarge
N- k 
p i ˜ number

f (p i )  p i  P
i k 1

N 
k i 
k
2E
 i1


˜f (p ,...,p )  
˜


f
(p
)


c 1
k
of uncorrelated
momenta
3 p i    N
 i1 i  p i  PN d

p
4
i
˜
 i1 f (p
 i )  p i  P
i k 1 
i1 2E

 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 - ALICE week 12-16 Feb 2007 - Muenster
k
N




N


4
2
2 ˜
4
FNk P   p i   
d p i (p i  mi )f (p i )  p i  P
i
k
1

 
 i1 

i1


k
 N


 N

4
2
2 ˜
4
  
d p i  (p i  mi )f (p i ) 
  p i  P   p i 

i k 1

 i k 1  i1 
g(p i )


FNk is just the distribution of the sum of a large number
N - k
k


of uncorrelated momenta  p i  P   p i 

i k 1 
i1
N
IF g(p i )  g 0 (p 0,i )  g x (p x,i )  g y (p y,i )  g z (p z,i ) then
k


FNk P   p i 
 i1 
k
 N


N


 i k 1dp0,ig0 p0,i  p0,i  P0   p0,i   



i k 1
i1
k
 N


N


 i k 1dpZ,igZ pZ,i  pZ,i  PZ   pZ,i 



i k 1
(note 1- dimensional  - functions, etc)
then, use CLT on each of 4 1D integrals
But... Energy conservation coupled to on-shell constraint
 huge correlations
between
E12-16
, P- Y,tot
, PZ,tot ???
tot, P
X,tot
ma lisa
- ALICE week
Feb
2007
Muenster
i1
CLT & ∑E - ∑p correlations
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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

 d p  p  f˜ p
unmeasured
parent distrib

ma lisa - ALICE week 12-16 Feb 2007 - Muenster
3
2
c
measured
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 - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster
EMCICs
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 - ALICE week 12-16 Feb 2007 - Muenster
N=6, <K>=0.5 GeV, LabCMS Frame - no cuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=9, <K>=0.5 GeV, LabCMS Frame - no cuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=15, <K>=0.5 GeV, LabCMS Frame - no cuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.5 GeV, LabCMS Frame - no cuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.7 GeV, LabCMS Frame - no cuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.9 GeV, LabCMS Frame - no cuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster
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 - ALICE week 12-16 Feb 2007 - Muenster
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

Ratio of parameters 3, 4
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,
momenta and energy are measured in the lab frame.
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
from the data
18 pions, <K>=0.9 GeV
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
The COMPLETE experimentalist’s recipe
femtoscopic
function of
choice
fit this...
C(p1,p 2 ) 
  R 2 q 2 

M 24
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 "
M3




...or image this...
M 24
C(q)  M1  p1,T  p 2,T  M 2  p1,z  p 2,z M 3  E1  E 2  M 4  E1  E 2 
M3
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
“Full” fit to min-bias d+Au - work in progress
data
EMCIC
Femto (gauss)
full
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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 EMCIC
– sampled by GenBod event generator
– stronger effects for small mult and/or s
• Analytic calculation of EMCIC
–
–
–
–
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|
– parameters are “physical” - values may be guessed
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
b
Thanks to...
• Alexy Stavinsky & Konstantin Mikhaylov (Moscow)
[original suggestion to use Genbod]
• Nicolas Borghini (Bielefeld) & Jean-Yves Ollitrault (Saclay)
[helpful guidance and explanation of previous work]
• Adam Kisiel (Warsaw)
[emphasize energy conservation; resonance effects in +- -]
• Ulrich Heinz (Columbus)
[suggestions on validating CLT in 3+1 case]
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Extra Slides
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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 ma lisa
N - ALICE week 12-16 Feb 2007 - Muenster

Al,m (| Q |) 
all.bins
Y
l,m
 cos  
4


( i ,  i )C(| Q |,cos  i ,  i )
i
QLONG
Q

QSIDE
nucl-ex/0505009
C(|Q|=0.39,cos,)
What is an Alm ?
QOUT

ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Multiplicity dependence of the baseline
Baseline problem is increasing
with decreasing multiplicity
STAR preliminary
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
d+Au
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Schematic: How Genbod works 1/3
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
flow chart,
in text
F. James, CERN REPORT 68-15 (1968)
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
F. James, CERN REPORT 68-15 (1968)
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Schematic: How Genbod works 2/3
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Schematic: How Genbod works 3/3
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster F. James, CERN REPORT 68-15 (1968)