emcics and the femtoscopy of small systems

ma lisa - ALICE week 12-16 Feb 2007 - Muenster

EMCICs and the femtoscopy of small systems

Mike Lisa & Zbigniew Chajecki

Ohio State University


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


Microexplosions Femtoexplosions

1017 J/m3 5 GeV/fm3 = 1036 J/m3

s 0.1 J 1 J

T 106 K 200 MeV = 1012 K

rate 1018 K/sec 1035 K/s

• energy quickly deposited• enter plasma phase• expand hydrodynamically• cool back to original phase• do geometric “postmortem” & infer momentum


Microexplosions Femtoexplosions

s 0.1 J 1 J

1017 J/m3 5 GeV/fm3 = 1036 J/m3

T 106 K 200 MeV = 1012 K

rate 1018 K/sec 1035 K/s

• energy quickly deposited• enter plasma phase• expand hydrodynamically• cool back to original phase• do geometric “postmortem” & infer momentum


Beyond press releases

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

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


Femtoscopic information

€

C r P

ab(r q ) = d3 ′

r r ⋅Sr

P

ab( ′ r r )∫ ⋅ φ(

r ′ q ,r ′ r )

2

xaxb

pa

pbxa

xb

pa

pb

€

Sr P

ab( ′ r r ) =

r x a -

r x b( ) distribution

φ(r ′ q ,r ′ r ) = (a,b) relative wavefctn

• femtoscopic correlation at low |q|• must vanish at high |q|. [indep “direction”]

Au+Au: central collisions

C(Qout)

C(Qside)

C(Qlong)

3 “radii” by using3-D vector q


Femtoscopic information - Spherical harmonic representation



C(Qout)

C(Qside)

C(Qlong)


QOUT

QSIDE

QLONG Q

∑→→ ΔΔ

=binsall

iiiiimlml QCYQA

.

,

cos

, ),cos|,(|),(|)(| φθφθπ

φθ

4nucl-ex/0505009


Femtoscopic information - Spherical harmonic representation


•ALM(Q) = L,0


C(Qout)

C(Qside)

C(Qlong)


∑→→ ΔΔ

=binsall

iiiiimlml QCYQA

.

,

cos

, ),cos|,(|),(|)(| φθφθπ

φθ

4

L=0

L=2M=0

L=2M=2

nucl-ex/0505009


Kinematic dependence of femtoscopy:Geometrical/dynamical evidence of bulk behaviour

€

(3 "radii" corresponding to the three components of r q )

Amount of flow consistent with p-space nucl-th/0312024Huge, diverse systematics consistent with this substructure

nucl-ex/0505014


p+p: A clear reference system?


STAR preliminary

mT (GeV) mT (GeV)

Z. Chajecki QM05nucl-ex/0510014femtoscopy 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?)


Surprising („puzzling”) scaling

HBT radii scale with pp

Scary coincidence or something deeper?

On the face: same geometric substructure

pp, dAu, CuCu - STAR preliminary

Ratio of (AuAu, CuCu, dAu) HBT radii by pp

A. Bialasz (ISMD05):I personally feel that its solution may provide new

insight into the hadronization process of QCD


BUT... Clear BUT... Clear interpretationinterpretation clouded by clouded by datadata features features

STAR preliminary d+Au peripheral collisions

Gaussian fitNon-femtoscopic q-anisotropicbehaviour at large |q|

does this structure affect femtoscopic region as well?



Gaussian fit

∑→→ ΔΔ

=binsall

iiiiimlml QCYQA

.

,cos

, ),cos|,(|),(4

|)(| φθφθπ

φθ

Decomposition of CF onto Spherical Harmonics

Z.Ch., Gutierrez, MAL, Lopez-Noriega, nucl-ex/0505009

non-femtoscopic structure(not just “non-Gaussian”)


Baseline problems with small systems: previous treatmentsBaseline problems with small systems: previous treatments


Gaussian fit

ad hoc, but try it...



NA22 fit

Try NA22 empirical formTry NA22 empirical form

NA22 fit

data

Spherical harmonics

L =1M=0

L =2M=0

L =1M=1

L =2M=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...)


energy-momentum conservation in n-body states

€

f α( ) =d

dαM

2⋅Rn( )

where

M = matrix element describing interaction

(M =1 → all spectra given by phasespace)

spectrum of kinematic quantity (angle, momentum) given by

€

Rn = δ 4 P − p j

j=1

n

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ δ pi

2 − mi2

( )d4pi

i=1

n

∏4n

∫

where

P = total 4 - momentum of n - particle system

pi = 4 - momentum of particle i

mi = mass of particle i

n-body Phasespace factor Rn

€

pi2 − mi

2( )d

4pi =r p i

2

E i

dr p i ⋅d cosθ i( ) ⋅dφi

statistics: “density of states”

larger particle momentum more available states

P conservation

€

4 P − p j

j=1

n

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Induces “trivial” correlations(i.e. even for M=1)


Genbod:phasespace 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, buteach 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

ALL EVENTS ARE

EQUAL, BUT SOME EVENTS ARE

MORE EQUAL THAN OTHERS


“Rounder” events: higher WT

ALL EVENTS ARE

EQUAL, BUT SOME EVENTS ARE

MORE EQUAL THAN OTHERS

larger particle momentum more available states

€

Rn = δ 4 P − p j

j=1

n

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ δ pi

2 − mi2

( )d4pi

i=1

n

∏4n

∫

δ pi2 − mi

2( )d

4pi =r p i

2

E i

dr p i ⋅d cosθ i( ) ⋅dφi

30 particles


Genbod:phasespace sampling w/ P-conservation

€

WT =1

Mm

M i+1R2 M i+1;M i,mi+1( ){ }i=1

n−1

∏

• Treat identical to measured events

• use WT directly• MC sample WT

• Form CF and SHD


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


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


• kinematic cuts

• frame


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




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


Distributions w/ phasespace constraints

€

˜ f ( pi) = 2E i f ( pi) = 2E i

dN

d3 pi

single-particle distributionw/o P.S. restriction



€

˜ f ( pi) = 2E i f ( pi) = 2E i

dN

d3 pi


€

˜ 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



€

˜ f ( pi) = 2E i f ( pi) = 2E i

dN

d3 pi


€

˜ 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

€

FN−k P − pi

i=1

k

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟≡ d4piδ(pi

2 − mi2)˜ f (pi)i= k +1

N

∏ ⎛ ⎝ ⎜ ⎞

⎠ ⎟∫ δ 4 pi

i=1

N

∑ − P ⎛

⎝ ⎜

⎞

⎠ ⎟

= d4piδ(pi2 − mi

2)˜ f (pi)

g(pi)1 2 4 4 3 4 4 i= k +1

N

∏ ⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

∫ δ 4 pi

i= k +1

N

∑ − P − pi

i=1

k

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

FN−k is just the distribution of the sum of a large number N - k( )

of uncorrelated momenta pi

i= k +1

N

∑ = P − pi

i=1

k

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟


€

IF g(pi) = g0(p0,i) ⋅gx (px,i) ⋅gy (py,i) ⋅gz(pz,i) then

FN−k P − pi

i=1

k

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟=

dp0,ii= k +1

N

∏ g0 p0,i( ) ⎛ ⎝ ⎜ ⎞

⎠ ⎟∫ δ p0,i

i= k +1

N

∑ − P0 − p0,i

i=1

k

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟⋅⋅⋅ dpZ,ii= k +1

N

∏ gZ pZ,i( ) ⎛ ⎝ ⎜ ⎞

⎠ ⎟∫ δ pZ,i

i= k +1

N

∑ − PZ − pZ,i

i=1

k

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

(note 1- dimensional δ - functions, etc)

then, use CLT on each of 4 1D integrals

But... Energy conservation coupled to on-shell constraint huge correlations between Etot, PX,tot, PY,tot, PZ,tot ???

€

FN−k P − pi

i=1

k

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟≡ d4piδ(pi

2 − mi2)˜ f (pi)i= k +1

N

∏ ⎛ ⎝ ⎜ ⎞

⎠ ⎟∫ δ 4 pi

i=1

N

∑ − P ⎛

⎝ ⎜

⎞

⎠ ⎟

= d4piδ(pi2 − mi

2)˜ f (pi)

g(pi)1 2 4 4 3 4 4 i= k +1

N

∏ ⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

∫ δ 4 pi

i= k +1

N

∑ − P − pi

i=1

k

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

FN−k is just the distribution of the sum of a large number N - k( )

of uncorrelated momenta pi

i= k +1

N

∑ = P − pi

i=1

k

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟


CLT & ∑E - ∑p correlations


Using central limit theorem (“large N-k”)

€

˜ 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

€

pμ2 ≡ d3p ⋅pμ

2 ⋅ ˜ f p( )unmeasuredparent distrib

{∫ ≠ d3p ⋅pμ2 ⋅ ˜ f c p( )

measured{∫N.B.

relevant later


k-particle correlation function

€

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

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

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

∑ ⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

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

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


€

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 correlation function (1st term in 1/N expansion)

“The pT term” “The pZ term” “The E term”

Names used in the following plots


EMCICsEMCICs

Effect of varying multiplicity & total Effect of varying multiplicity & total energy 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


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]...


Remember...

€

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

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

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

of course, the experimentalist never measures all particles(including neutrinos) or <pT

2> anyway, so maybe not a big loss


€

C(p1,p2) ≅ 1−2

N pT2

NEW PARAM 11 2 3

⋅r p 1,T ⋅

r p 2,T{ } −

1

N pZ2

NEW PARAM 21 2 3

⋅ p1,z ⋅p2,z{ }

−1

N E 2 − E2

( )

NEW PARAM 31 2 4 4 3 4 4

⋅ E1 ⋅E 2{ } +E

N E 2 − E2

( )

NEW PARAM 41 2 4 4 3 4 4

⋅ E1 + E 2{ } +E

2

N E 2 − E2

( )

Ratio of parameters 3,41 2 4 4 3 4 4

where

X{ } denotes the average of X over the (p1,p2) bin. (or r 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.

The experimentalist’s recipeTreat 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


18 pions, <K>=0.9 GeV


€

C(p1,p2) =

Norm 1− M1 ⋅r p 1,T ⋅

r p 2,T{ } − M2 ⋅ p1,z ⋅p2,z{ } − M3 ⋅ E1 ⋅E 2{ } + M4 ⋅ E1 + E 2{ } −

M42

M3

+"λ ⋅e− Rα

2 qα2

α =o,s ,l

∑"

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

The COMPLETE experimentalist’s recipe

€

C(q) + M1 ⋅r p 1,T ⋅

r p 2,T{ } + M2 ⋅ p1,z ⋅p2,z{ } + M3 ⋅ E1 ⋅E 2{ } − M4 ⋅ E1 + E 2{ } +

M42

M3

femtoscopicfunction ofchoicefit this...

...or image this...


“Full” fit to min-bias d+Au - work in progress

dataEMCIC

Femto (gauss)full


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


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, 1982b


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]


Extra Slides


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•

•

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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)


What is an Alm ?

QOUT

QSIDE

QLONG Q

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Al,m (|Q→

|) =Δcosθ Δφ

4π×

Yl,m (θ i,φi)C(|Q→

|,cosθ i,φi)i

all.bins

∑

nucl-ex/0505009

C(|

Q|=

0.39

,cos

,)


Multiplicity dependence of the baselineMultiplicity dependence of the baseline

Baseline problem is increasing

with decreasing multiplicity

STAR preliminary


d+Au


Schematic: How Genbod works 1/3


flow chart,in text

F. James, CERN REPORT 68-15 (1968)


Example of use of total phase space integral

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

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Γ pp → πππ( )Γ pp → ππππ( )

=R3 1.876;π ,π ,π( )

R4 1.876;π ,π ,π ,π( )

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In limit where "α "="event" = collection of momenta r p i

"spectrum of events" = f α( ) =d

dαRn

→ Probevent α ∝d3n

dpi3

i=1

n

∏Rn

• single-particle spectrum of :

€

f α( ) =d

dαRn

• “spectrum of events”:


emcics and the femtoscopy of small systems

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