Lecture course by Harald Appelshäuser Script by Simone Schuchmann.
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Transcript of Lecture course by Harald Appelshäuser Script by Simone Schuchmann.
Lecture course by Harald Appelshäuser
Script by Simone Schuchmann
2
Content:
1. Correlations in heavy ion- collisions
2. HBT- interferometry1. Foundations2. Stellar interferometer by R. Hanbury- Brown and R.Q. Twiss
3. HBT in hadronic systems1. Foundations2. HBT3. Excursion
4. Résumé
5. Literature
3
1. Correlations in heavy – ion collisions
(CERES-TPC)
4
Characteristics:
• For global, event averaged, observables (see particle ratios, energy
spectra, collective flow) we find
large degree of thermalisation
• Nevertheless, correlations do occur due to :
kinematical conservation laws :
energy, momentum, e.g. in particle decays: 0 -> +- , -> p -
dynamical conservation laws:
quantum numbers (charge, strangeness, baryon number)
final state interactions: coulomb
collectivity: flow
quantum statistics: Bose- Einstein- Correlations
5
2. HBT- interferometry
Goal:
The range of correlation in momentum space allows extraction of
spatio-temporal extension in configuration space:
• reaction volume density
• reaction duration
• collectivity
• velocity profile
It is important to constrain models which connect to lifetime or source size
Key words: interference, Fourier transform, coherence, correlation
6
2.1. The foundations
2.1.a) Interference and Fourier transform
Example : Fraunhofer refraction
• constructive interference in forward direction
• first minimum at:
• if is known, the size of the slit can be determined from
• for > x (i.e. sin() >1) there is no minimum
this yields:
the wave length must be smaller than the size of the object
xx
2/
2/)sin(
7
For electrons (particle waves) we obtain with DeBroglie :
and
to resolve spatial structures on the femtometer scale (nucleus),
typical momenta of a few 100 MeV/c are required
the refraction pattern is the Fourier- transform of the slit (or
nucleus) geometry
R. Hofstadter (1957)
: form factor
: charge distribution
fm
8
Example 1: Michelson’s experiment
screen
beam splitter
mirror
d1
mirror
d2
monochromatic
light source
• ring- shaped interference pattern on the screen, depends on d=d2-d1
• interference pattern disappears if d exceeds a certain value , because:
there is only interference if the waves are coherent, e.g. if they have a well defined phase relation
As the light source emits light waves with random phases, there is no interference pattern (averaged over time)
Interference can only happen if the same wave gets split at the mirror
This implies that the optical retardation has to be smaller than the so-called coherence length Sc , which depends on the wave or the band width of the source:
2.1.b) Coherence
1
c
St
cc
the pattern disappears if d > Sc (Sc is between mm (thermal light) and km (lasers))
“temporal coherence”
9
Example 2: Young‘s double slit experiment
• Minima and maxima on the screen depend on the difference of the path length d2 – d1
• In general: waves which are emitted from different points of the source have different path length differences => no interference
• unless , the coherence condition, is
fulfilled for all points in the sourced L>>d
S1
S2
212
dd
4
)²(²²1
Radd
4
)²(²²2
Radd
d
aR
dd
aRdd
21212
=>
22
d
aR
R
da
2-D case:F
dAc
²²
c
F
dSAV ccc
²²3-D case:
F
dc
²
²³
increase a until interference disappears => the angular size R/d of the source, is determined
=>
double slit
screen
(coherence Volume)
10
In case of photons with momentum p we obtain:
If passed slit S1: slit S2:
Using and it yields
for 2 dimensions and for 3-D
2
)( Ra
d
ppx
2
)( Ra
d
ppx
d
Rpppp xxx 21
c
E
c
hp
cd
Rhpx
²²
²²²
dc
Rhpp yx
²³
²³
dc
Fhpppp zyx
with this knowledge we can construct an interferometer....
³hVp c
c
hhp z
= py
=
py
px
p
11
• Optical path length distance:
s = d*sin() ≈ d*for small
• Interference pattern only visible as long as:
s ≤
d*≤ (to be precise d*≤ 1.22 *
• d can be varied until interference disappears
determination of , the angular size of a star
• small requires large d
• d is limited by atmospheric fluctuation (index of refraction)
d*sin()
lenses
2.1.c) Michelson’s stellar interferometer
Example:
for a star: = 5*10-7 m a = 0.1*10-6 => d ≈ 5m
d
12
• Idea: Intensity measurement with 2 separated detectors
for coherent light not only amplitudes should
interfere (phase relation) but also intensities
(because of BE- statistics, see later)
• Advantages:
intensity measurement is much more robust (just
counting the photons)
large distances d possible => higher resolution
• Method
Measure the photo- currents I1 (t) and I2(t) in short
time intervals (~ 10 – 100 MHz i.e. 100- 10 ns)
Calculate the correlator: (next slide)
C
d
photo multipliers
correlator
I1 I2
2.2 Stellar interferometer by R. Hanbury- Brown and R.Q. Twiss (1956)
k kk‘ k‘
13
by definition:
IfI1, I2 are
• uncorrelated: C2 = 1 (coherent source: <I1I2>=<I1><I2>)
• correlated: C2 > 1
• anticorrelated: C2 < 1
because....
14
uncorrelated correlated anti - correlated
x
y
= 0 > 0 < 0
.... we obtain for the product of deviations:
x x
yy
with <>a indicating an average over a variable a,
x= x –x , y= y –y , x, y mean values
a aa
15
Results:
Hanbury- Brown and Twiss observed a
positive correlation (C2-1>0). By measuring
the reduction of the correlation strength as a
function of d, they could determine the
angular size of Sirius:
3.1∙10-8 rad
R. Hanbury-Brown and R.Q. Twiss, Nature 178, 1046 (1956)
But: The observed correlation is only 10-6 – because:
• Currents are measured over a time window of 1/ = 10-8 s. The coherence
time of a star is only 10-14s => signal is “diluted” by 10-6
• => the undiluted signal is, as expected, 1:1
16
3. HBT in hadronic systems
3.1 Foundations
The coherence condition is:
d = distance source – detector
a = distance between the detector
R = source size
Examples:
• for stars: R =109 m , d = 1016 m (≈ 1ly) , = 10-7 m
• for hadronic systems: a = 1m , d = 1m,
If ≈ R (≈ 1- 10 fm) => works for p ≈ 100 MeV/c
Note: Instead of photons we now use hadrons with integer spin: pions
or
≈ 1
17
3.1.a) GGLP- Effect : Goldhaber, Goldhaber, Lee, Pais (1959)
The picture shows the opening angle distribution of
pions from - annihilation at 1.05 GeV/c in a
bubble chamber:
• The goal was the search for the 0 -> +-
• An unexpected difference between like- sign
(identical) and unlike- sign pions was observed
• GGLP interpreted this as being due to BE-
correlations
•The connection to the original HBT- experiment
was found only a few years later
• In the 1970’s HBT was proposed to be a technique to determine source sizes in
nuclear collisions (Podgoretskii and Kopylov, Shuryak, Cocconi)
G.Goldhaber, S.Goldhaber, W.Lee, A.Pais, Physical Review 120 300 (1960).
18
3.1.b) Symmetry of wave functions
Consider two identical particles. Quantum mechanics requires that the square of the wave function does not change if the two particles are exchanged (as you do not know which one is which):
|12|² = |21|²
12 = 21 or 12 = - 21
bosons fermions
)]()()()([2
11221221112 xxxx )]()()()([
2
11221221112 xxxx
19
3.2 HBT
Now consider two identical pions emitted in r1 and r2 , detected in x1 with momentum p1
and in x2 with p2 respectively:
has to be symmetric:
Assuming plane waves
)],(),(),(),([2
11222111221112112 rprprprp
)(2
112 ba
)(exp)(exp 2211 rp
irp
ia
)(exp)(exp 1221 rp
irp
ib
20
... we obtain for the intensity I = |12|² with:
Note: The experimental quantity is the correlation function...
*]***[2
1*²|| 211212 bbbaabaa
)exp(* iDab
)exp(* iDba
)]cos(22[2
1²|| 12 D
)cos(1
rp
aa* = 1 = bb*
D = (p1-p2)(r1-r2)
|12|² 2 for pr 0
only Interference term remains
21
3.2.a) Correlation function
• Expressed in (relative) momentum space, it requires integration over configuration space
• Consider a source emission function S(r, p) which can be factorised:
S(r, p) = (r) ∙ f(p)
=> P1 (p) = = f(p)
• This yields for the two- particle probability P2:
)()(
)()(
2111
212
212 pPpP
ppPppC
42
41221112212 ),(),(²||),( drdrprSprSppP
42
4121
42
412121 )()()cos()()()()( drdrrr
rpdrdrrrpfpf
one- particle probability
distribution
22
The correlation function is connected to the Fourier
transform of the spatial distribution function r) –
analogue to the Fraunhofer refraction, electron
scattering.
Note: The relation between C2 and r) is only correct if S can be factorized.
2~
212 )(1)( pppC
42
41
2121212 exp)()(1)( drdr
pri
prirrppC
... and finally:
Δp~1/R
Again:Ifp1, p2 are
• uncorrelated: C2 = 1
• correlated: C2 > 1
• anticorrelated: C2 < 1
q=p
23
3.2.b) 2- Pion correlation function- experimentally
• Generating the distribution of the momentum
difference q = pi – pj of pairs of identical pions
from each event: the signal distribution S
• Calculating the background B by using the
same procedure for pions of different events.
• Normalizing the spectra and dividing the signal
by the background:
with N: normalisation,
F: other correlations (coulomb, detector)
FB
SNC 2
Signal
Background
(mixed events)
÷
i
j
q
24
Conclusions
• large source size R => narrow width of C2
• experimental requirements:
good two- track resolution (granularity)
good momentum resolution
Small sources are easier to measure then large
For a quantitative analysis of C2 a reasonable
parameterisation is required, which
• describes the data well
• is physically motivated
Gauss seems to be reasonable:
Fit- parameter: R, the source size
2
2
2exp)(
R
rAr 22
2 exp1)( RqqC =>
Note: In general, the HBT- parameter R is not the real the size of the particle source,
one has to consider that the source may be expanding!
25
1
3.2.c) Expanding sources
• Consider one- dimensional collective
expansion:
• Now consider 3 source elements with local
temperature Tf with the following velocity
distribution:
• Pions emitted from different regions of the
source have different velocity:
source distribution S(r, p) can no longer be factorised
(note: factorization only works for static sources)
space- momentum correlations
• Pions with similar momenta must come from close- by regions of the source.
Otherwise the coherence relation: cannot be fulfilled (i.e. q ≈ 0 , large r )
z
20
v
v0 = 0 v1 v2 v
only pions with small r can contribute to the enhancement of C2
26
Consequences:
In case of expanding systems, HBT does not measure the full geometric size of the
source. The measured radius RHBT is interpreted as the length of homogeneity,
which is determined by:
• the collective velocity gradient
• the average thermal velocity
• temperature gradients
• In the presence of source dynamics, “radii” depend on mean pair energy,
momentum, transverse mass, ...
27
1 2
3.2.d) Thermal length scale: a basic parameterisation
Still: consider a one- dimensional expansion in z:
If the velocity of a source element is coherent
in time:
velocity gradient, which decreases
with time (analogue to Hubble-
Expansion of the universe)
What HBT would measure:
The HBT- length of homogeneity will correspond to the spatial distance z, over which the collective velocity difference vz = z / f is equal to the
average thermal velocity:
0
z
t
t=0
t
tzvz
)(
ttdz
dvz 1
)(
f
z
fdz
dv
1
t=0 t1 t2 t=f
v
zRHBT
thfzfHBT vvzR
28
• For thermal velocity in one dimension we get with
• If T ≈ m we have to do a relativistic calculation:
• Assuming pz << p┴ with p┴ (or pt) perpendicular to the beam (z-axis) it yields:
(Makhlin and Sinyukov, 1988)
m
Tv f
th
22 pmmrel
mpmmrel22
thermfHBT Rm
TR
=> k = 1
=>
This is only correct if Rgeo ∞ ...
fth kTvm2
1
2
1 2
29
... because in general we have:
The smaller one of the two scales determines RHBT
• The measured RHBT will depend on Tf and on p┴ due to relativistic effects:
Pions with high p┴ have a higher m┴ in the rest frame of the source at
given Tf. However, their thermal motion is slower.
smaller thermal length scale
• From pair momentum dependent measurement we obtain information about
the expansion profile (Tf , f ….).
C2 (q) -> C2(q,k) with
Usually: and the pair transverse mass
GeothermHBT RRR 222
111
22 kmm
p┴1
p┴2
k┴
2 k┴=p┴2+ p┴1 = p┴
30
In longitudinal direction we calculate the pair rapidity:
This is used to “scan” the source in longitudinal direction.
2121
2121ln2
1
zz
zz
ppEE
ppEEY
31
Parameterisation of Cc(q):
A Gaussian fit to C2 may probably be suitable,
but in case of a non spherical size and
collectivity, it makes sense to split q in into its
components:
with , : chaotisity parameter (see ) and , the emission time from:
Exploiting the symmetry of the system (beam axis, azimuthally symmetric in central
collisions) this leads us to
q
220
2222222 exp1)( qRqRqRqqC zzyyxx
210 EEq
the most popular parameterisation, which was invented by G. Bertsch and S. Pratt :
tf
(t)
32
3.2.e) The BP- parameterisation of the correlation function
momentum parameterisation:
• longitudinal: qlong = qz ,usually in the LCMS, where pz1 = -pz2
• transverse:
for p┴1≈ p┴2 :
qside : difference in azimuthal direction
qout : difference in absolute value of pt => reveals energy difference
p┴1
p┴2 k┴
qout
qside
longoutlongoutlonglongoutoutsideside RqqRqRqRqqC 22222222 exp1)(
space-time corr.
beam-axis
33
CERES 158A GeV/c Pb-Au Nucl. Phys. A714 (2003) 124
|q| < 0.03 GeV/c
STAR Au-Au 200 GeV
Projections of the 3 + 1- dimensional
C2(q) for qlong, qside and qout
34
strong kt dependence
longitudinal expansion
radius parameterisation:
Collective radial expansion is closely connected to thermalisation:
• longitudinal: Rlong || z
GeothermHBT RRR 222
111
35
Hubble-Expansion
Hubble constant <-> lifetime
t
fflong m
TR
lifetime
(Y. Sinyukov)
thermalvelocity
Mpcs
km1041H 42
LB ,
s102fm/c86H
1 23
LB
for Tf = 160 – 120 MeV
CERES Pb-Au Nucl. Phys. A714 (2003) 124
>15% 10-15% 5-10% 0-5%
1/√mt (1/√GeV)
36
Rlong proportional to (mean thermal velocity)
Rlong dominated by thermal length scale (R²Geo >> R²therm)
=>
If RGeo,long >> 8 fm => Hubble- diagram of the “little bang”
Conclusions concerning Rlong:
Longitudinal “flow” is difficult:
• incomplete stopping leads to initial “flow”
• different scenarios lead to similar asymptotic flow pattern
37 weaker kt dependence then Rlong
radius parameterisation:
• transverse: Rside and Rout
38
CERES Pb-Au Nucl. Phys. A714 (2003) 124
ft2f
geoside
Tm1
RR
/
f2 : strength of
transverse expansion
(U. Heinz, B. Tomasik, U. Wiedemann)
< vt > = 0.5-0.6c for Tf = 160 – 120 MeV
>15% 10-15% 5-10% 0-5%
1/√mt (1/√GeV) Instead of Hubble- expansion, we now see a saturation for lower m┴
39
• Assume R(t=0) ≈ 0 or at least << RGeo and R(t=f) = RGeo
average transverse flow velocity
finite size effect
RGeo ≈ 6fm -> 2* Rinitial
significant transverse expansion
ff
Geo
Geo
T
mR
R
2
2
2
1
f
Geoside
T
vm
RR
2
1
m
TR f
ftherm22
22
2
111
Geoff
side R
m
TR
ft2f
geoside
Tm1
RR
/
Conclusions concerning Rside:
• R²Geo ≈ R²therm
• with and we obtain: GeothermHBT RRR 222
111
or
=> R2side
vR
f
Geo
=>
40
• Rout and Rside
Rout
Rside ┴
Rside
222
2 1sideout RR
<=
(approximately)
Goal: determination of Tf , ┴ and RGeo from Rside
41
3.3 Excursion: How can the freeze- out conditions be determined?
Freeze- out – kinematical:
hadronisation (phase transition) temperature Tc
≥
chemical freeze- out temperature Tch
≥
kinematical (thermal) freeze- out temperature Tf
TcTch
Tf
beambeam
42
• From pt (mt) – spectra we obtain:
• In pp collisions all particle species have
T ≈ 150 MeV
thermalisation is questionable
( uncertainty relation)
• In AA collisions there can also be collective
transverse expansion
T
mm
dm
dNexp~
all particles move in a common velocity field
heavier particles pick up more kinetic energy
43
3.3.a) Temperature
• A fit of will yield different temperatures. An approximation is:
• In principle: obtain Tf and vt from fit
of T to spectra of different species
• In practice this is difficult, because:
T
mm
dm
dNexp~
2
2
1 vmTT f
Tf
v┴
complementary approach: HBT
44
3.3.b) HBT
• Remember:
now Tf , v┴² from m┴ - dependence of Rside :
positive correlation:
“Tokyo Subway map”
HBTSpectraTf
v┴
f
Geoside
T
vm
RR
2
1
Spectra
HBT
≈
Tf ≈ 120 MeV < Tch (≈160 MeV)≈ Thad and
t ≈ 0.5-0.6 (≈ speed of sound in the
corresponding
ideal gas)
45
3.3.c) HBT and QGP: further observables
Ideally: increase by increasing
where the phase transition is hit,
observables show discontinuities:
( )
( ) ( )
s
s s
s
46
Volume
• Assume QGP is produced, subsequent evolution does not produce entropy:
dS = 0 = SH – SQGP = sH VH - sQGPVQGP
• We have for the entropy density s:
• Hadrons: dH = 3
• Quarks and Gluons: (Fermi- Dirac- statistics)
3
0
2
Tq
4πds
qgQGP d8
7dd
dT
dPs
3
1p
dg = 2spin + 8colour = 16
dq = 2spin + 2part-antipart. + Nc + Nf = 24 (36, for u,d,s)
dQGP = 37
grand-canon. ensemble
Factor 12 between VH and VQGP (more than factor 2 in each dimension!)
Measurement of V(√s)
47
Lifetime
• measure and
c H
f
T
t
sR
R
side
out)( sRlong
no unusual structures observed
no pronounced √s dependence at all!
WHY?- Important questions:
What is actually the condition for
freeze- out?
When do particles decouple?
What is the relevant critical condition?
48
Pion freeze out:
Suggestions for possible freeze- out
conditions:
• (mean free path) ≥ Rsource
• itself
use HBT to measure f (at freeze- out)
N
V f
ff
1
22
3
2 sidelongf RRV
mean free path
non-monotonic behaviour-
how can this be understood?
...
D. Adamova et al. (CERES), PRL 90, 022301 (2003)
49
N can be measured from particle spectra. Only abundant particles (see figure) are considered:
total multiplicity increases
monotonically with energy
pions start to dominate at higher AGS energies
protons only dominate in the AGS
regime
D. Adamova et al. (CERES), PRL 90, 022301 (2003)
50
Problem:
HBT does not measure the full source size (volume), therefore we cannot use
4 yields to calculate the density.
assume Rside of small kt is about RGeo
estimate the extension of longitudinal length of homogeneity in rapidity
space
yHBT (kt≈160 MeV/c) ≈ 0.87 (r.m.s.)
use cross sections for pion-nucleon and pion-pion interaction since they
are the dominant processes
is a good estimate for 1/f
as we have different particle species
ymid
f
dy
dN
V
287.0
... NNNN NNi
ii
Ymid
ppN dy
dNN 87.022
Ymiddy
dNN
87.023 mb13mbN 72
51
√s(GeV)
Weighted with cross section,
pions start to dominate at SPS
Nshows also non-monotonic
behaviour
together with the Volume Vf we obtain:
D. Adamova et al. (CERES), PRL 90, 022301 (2003)
52
√s(GeV)
the ratio is constant:
the mean free path at
freeze-out is ~1 fm
D. Adamova et al. (CERES), PRL 90, 022301 (2003)
53
Freeze- out Volume versus N
Compilation: S. Schuchmann
the universality holds also for all system sizes!
pp@ 17GeV
pp@200GeV
dAu@200GeV
SS@19GeV
PbPb, PbAu, AuAu
for √sNN 2-200GeV
f=1.08±0.16 fm
54Compilation: S. Schuchmann
Mean free path at freeze- out
55
4. Résumé
• C2(q) does contain a lot of shape information
however, we are confronted with the
• HBT- Puzzle:
- Fairly increasing radii with √s (see slide 44)!
- general trend: the lifetime is overestimated, Rside underestimated
- The failure of models, especially Hydro models, to describe the phenomena properly:
56
M. Lisa, S. Pratt, R. Soltz, U. Wiedemann, nucl-ex/0505014
An ‘expanding fireball, undergoing phase transition, should at
least cause: Rout> Rside ‘
(D. Magestro, QM 04)
57
58
5. Literature
• ‘Introduction to Bose- Einstein Correlations and Subatomic interferometry’,
Richard M. Weiner (Wiley, 2000)
• ‘Introduction to High-Energy Heavy-Ion Collisions’,
C. Y. Wong (World Scientific)