Upload
leverett-reynaud
View
30
Download
1
Tags:
Embed Size (px)
DESCRIPTION
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] - PowerPoint PPT Presentation
Citation preview
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Femtoscopic information - Spherical harmonic representation
• 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
QOUT
QSIDE
QLONG Q
∑→→ ΔΔ
=binsall
iiiiimlml QCYQA
.
,
cos
, ),cos|,(|),(|)(| φθφθπ
φθ
4nucl-ex/0505009
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Femtoscopic information - Spherical harmonic representation
• femtoscopic correlation at low |q|• must vanish at high |q|. [indep “direction”]
•ALM(Q) = L,0
Au+Au: central collisions
C(Qout)
C(Qside)
C(Qlong)
3 “radii” by using3-D vector q
∑→→ ΔΔ
=binsall
iiiiimlml QCYQA
.
,
cos
, ),cos|,(|),(|)(| φθφθπ
φθ
4
L=0
L=2M=0
L=2M=2
nucl-ex/0505009
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
p+p: A clear reference system?
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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?)
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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?
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
STAR preliminary d+Au peripheral collisions
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”)
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Baseline problems with small systems: previous treatmentsBaseline 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
STAR preliminary d+Au peripheral collisions
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
€
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)
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
“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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
CF from GenBodCF from GenBod
Varying frame and Varying frame and kinematic cutskinematic cuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.9 GeV, LabCMS Frame - no N=18, <K>=0.9 GeV, LabCMS Frame - no cutscuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.9 GeV, LCMS Frame - no cutsN=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 - |N=18, <K>=0.9 GeV, LCMS Frame - ||<0.5|<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 cutsN=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 - |N=18, <K>=0.9 GeV, PR Frame - ||<0.5|<0.5
The shape of the CF is sensitive to
• kinematic cuts
• frame
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
GenBodGenBod
Varying multiplicity Varying multiplicity and total energyand total energy
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=6, <K>=0.5 GeV, LCMS Frame - no cutsN=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 cutsN=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 cutsN=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 cutsN=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 cutsN=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 cutsN=18, <K>=0.9 GeV, LCMS Frame - no cuts
The shape of the CF is sensitive to
• kinematic cuts
• frame
• particle multiplicity
• total energy : √s
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
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 ( pi) = 2E i f ( pi) = 2E i
dN
d3 pi
single-particle distributionw/o P.S. restriction
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Distributions w/ phasespace constraints
€
˜ f ( pi) = 2E i f ( pi) = 2E i
dN
d3 pi
single-particle distributionw/o P.S. restriction
€
˜ 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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Distributions w/ phasespace constraints
€
˜ f ( pi) = 2E i f ( pi) = 2E i
dN
d3 pi
single-particle distributionw/o P.S. restriction
€
˜ 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
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
€
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
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
CLT & ∑E - ∑p correlations
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
€
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=6, <K>=0.5 GeV, LabCMS Frame - no N=6, <K>=0.5 GeV, LabCMS Frame - no cutscuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=9, <K>=0.5 GeV, LabCMS Frame - no N=9, <K>=0.5 GeV, LabCMS Frame - no cutscuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=15, <K>=0.5 GeV, LabCMS Frame - no N=15, <K>=0.5 GeV, LabCMS Frame - no cutscuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.5 GeV, LabCMS Frame - no N=18, <K>=0.5 GeV, LabCMS Frame - no cutscuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.7 GeV, LabCMS Frame - no N=18, <K>=0.7 GeV, LabCMS Frame - no cutscuts
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
N=18, <K>=0.9 GeV, LabCMS Frame - no N=18, <K>=0.9 GeV, LabCMS Frame - no cutscuts
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...
€
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
€
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
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
18 pions, <K>=0.9 GeV
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
€
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...
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
“Full” fit to min-bias d+Au - work in progress
dataEMCIC
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, 1982b
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
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•
•
€
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)
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
What is an Alm ?
QOUT
QSIDE
QLONG Q
€
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
,)
ma lisa - ALICE week 12-16 Feb 2007 - Muenster
Multiplicity dependence of the baselineMultiplicity 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)
€
Γ pp → πππ( )Γ pp → ππππ( )
=R3 1.876;π ,π ,π( )
R4 1.876;π ,π ,π ,π( )
€
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”:
F. James, CERN REPORT 68-15 (1968)