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)