hbt'04q. h. zhang1 a short introduction to hbt qinghui zhang

HBT'04HBT'04 Q. H. ZhangQ. H. Zhang 11

A Short introduction to A Short introduction to HBTHBT

Qinghui ZhangQinghui Zhang


Heavy Ion Collisions requires understanding Heavy Ion Collisions requires understanding particle distributions in coordinate particle distributions in coordinate and and

momentum spacemomentum space


GGLPGGLP Goldhaber, Goldhaber, Lee and Pais Goldhaber, Goldhaber, Lee and Pais

(1960)(1960) First application of “intensity First application of “intensity

interferometry” in particle physicsinterferometry” in particle physics HBTHBT

Hanbury-Brown and Twiss (1950’s)Hanbury-Brown and Twiss (1950’s) Revolutionary development of Revolutionary development of

intensity interferometry in astronomyintensity interferometry in astronomy B-EB-E

Bose-Einstein (1920’s)Bose-Einstein (1920’s) Role of Bose statistics in fluctuationsRole of Bose statistics in fluctuations


NeglectsNeglects Momentum dependence of sourceMomentum dependence of source Quantum mechanics Quantum mechanics up to up to x and yx and y Final State Interactions Final State Interactions afterafter x and y x and yC2(q) contains shape informationC2(q) contains shape information

y

X

1

2

Source


qOUT R

Sphere vs hemisphere

That is, squared Fourier Transforms measure (only) relative separations of source coordinates

Very different sources (x) can give very similar distributions in relative separation


)(

),(),(

)()(,|)(|1)(

0

0212121

422

rqtqxq

qqppEEppq

xdexqqqC iqx

Fourier Transform provides one time and three space extensions of source

BUT:

That is, q0 is not an independent quantity in the F.T.(e.g., note that )

Fourier Transform has support in only half of the q0-q plane


xdexVqq

tVrqrqtVqxq

Vqq

xdexqqqC

tVrqiPAIR

PAIRPAIR

PAIR

iqx

PAIR 4)(

0

422

)();()(

)()(

)()(,|)(|1)(

So Fourier transform (+ two-particle kinematics) already provides conclusion:

22

22 )();(|)(|1)( tVrVqCqqC PAIRPAIR

Additional dynamic effects (expansion, thermal source, …) will also lead to systematic dependence of extracted “radii” on VPAIR (or mT or kT or …)


There are other decompositions...There are other decompositions...


)])((2

1exp[);(

)](2

1exp[~),(

22222222

2

2

2

2

2

2

2

2

PAIRzzyyxxPAIR

zyx

VqRqRqRqVq

t

R

z

R

y

R

xtr

Simplest possible (yet still very useful) case:Simplest possible (yet still very useful) case:

)])((2

1exp[);(

)](2

1exp[~),(

22222222

2

2

2

2

2

2

2

2

PAIRzzyyxxPAIR

zyx

VqRqRqRqVq

t

R

z

R

y

R

xtr

Choose frame so that x = “Out” (that is, parallel to VPAIR) y = “Side” z = “Long”:

y

z x

222222

2222222222

22222222

""

))((

ySIDEPAIRxOUTxy

PAIRxOUTOUTOUTLONGLONGSIDESIDE

PAIRzzyyxx

RRVRRRRSymmetry

VRRwithRqRqRq

VqRqRqRq


Most experiments with charged Most experiments with charged tracking and particle identification have tracking and particle identification have HBT results:HBT results:

BNL AGS:BNL AGS: E859, E866E859, E866

E877E877

E895E895 CERN SPS:CERN SPS:

NA44NA44

NA49NA49

WA98WA98

RHICRHIC

PHENIX STARPHENIX STAR


The Great Puzzle: Why so little variation with ECM ? Why are lifetimes () ~ 0 ??

( especially at RHIC )


systematic errors in experimental data is systematic errors in experimental data is required:required: Definition of CDefinition of C22

Coulomb correctionsCoulomb corrections Role of Role of , dependence of R on same, dependence of R on same Contributions from resonancesContributions from resonances

(especially when comparing between (especially when comparing between different experiments)different experiments)

A better understanding of theoretical A better understanding of theoretical assumptions and uncertainties is assumptions and uncertainties is required:required: Resonance contributionsResonance contributions Lorentz effectsLorentz effects Modeling of expansion, velocity profilesModeling of expansion, velocity profiles


Our favorite (charged) bosons never Our favorite (charged) bosons never have pure plane-wave stateshave pure plane-wave states Even our ideal-case Fourier transform Even our ideal-case Fourier transform

becomes a “Coulomb transform”becomes a “Coulomb transform” This particular case is easy to treat This particular case is easy to treat

analytically via “Gamow” or “Coulomb” analytically via “Gamow” or “Coulomb” corrections:corrections:


Some indication from both AGS and SPS Some indication from both AGS and SPS that Rthat RSideSide(--) > R(--) > RSideSide(++)(++) E877 (AGS): RE877 (AGS): RSideSide(--) ~ (1.5 +/- 0.4)*R(--) ~ (1.5 +/- 0.4)*RSideSide(+(+

+)+) NA44 (SPS): NA44 (SPS):

RRTSideTSide(--) > R(--) > RTOutTOut(--)(--)

RRTSideTSide(++) < R(++) < RTOutTOut(++)(++)

- Ze

Ze-

Note external Coulomb

•Systematic in RSIDE

•Cancels in ROUT


2)0(

1)0()()(

2

4

qC

qxdexq iqx

This is parameterized away via C2(q) = 1+| (q)|2 , ~0.1-0.5

“Understood” via assumed resonance contributions:

versus observed value of 1.1-1.5


How do resonances affect How do resonances affect ? ? By creating a “halo” around the source By creating a “halo” around the source

“core”:“core”: Some culprits:Some culprits:

(( = 150 MeV = 150 MeV S ~ S ~ 1.3 fm )1.3 fm )

KKKK ( ( = 50 MeV = 50 MeV S ~ 4 S ~ 4 fm ) fm )

(( = 8.4 MeV = 8.4 MeV S ~ 20 S ~ 20 fm ) fm )

(( = 0.2 MeV = 0.2 MeV S ~ 200 S ~ 200 fm ) fm )

(and many more…)(and many more…)

Effect: CEffect: C2 2 develops structure near q=0 develops structure near q=0

to describe the various sources:to describe the various sources:

fCORE fCORE 2(1-fCORE )fCORE (1-fCORE )(1-fCORE )

0

1

2

0 50 100

q (MeV/c)

C2(

q)

Core

Halo

Combination

]GeV/c[%)1(~%)1(~ pp

p

Momentum resolution


Use ~this in cascade codes

Using Wigner function:Using Wigner function: See for example:See for example:

Chao, Gao and ZhangChao, Gao and Zhang

Phys. Rev. C 49,3224 (1994)Phys. Rev. C 49,3224 (1994)


Expect CExpect C22 to depend on q to depend on q andand K K

B.R. Schlei and N. Xu:B.R. Schlei and N. Xu:


Final state phase space points xi are

weighted by 1 + cos[(pa-pb)(xa-xb)]

RQMD vs. Data for HBTRQMD vs. Data for HBT


This approach isThis approach is PlausiblePlausible WRONGWRONG

Final state phase space points xi are

weighted by 1 + cos[(pa-pb)(xa-xb)]

Advantage: Uses precisely what codes produce

Disadvantage Can lead to non-physical oscillations

in C2 First noted empirically by Weiner et al

and Later matin et al. Theoretical analysis by Q.H. Zhang et

al. Phys. Lett. B 407, 33 (1997)

Quantitatively, is it significant for heavy ion collisions? Answer 1: No (Empirical) Answer 2: Not yet? (see next slide)

with


Study these effects with a source that Study these effects with a source that explicitly parameterizes x-p explicitly parameterizes x-p correlations:correlations:

Examples for R=5fm, P=200 MeV/c

-10 -8 -6 -4 -2 0 2 4 6 8 10-500

-400

-300

-200

-100

0

100

200

300

400

500

R (fm)

0

1

2

0 100 200q (MeV/c)

C2(

q)

Naive Correct Usual (used in MC's)

S=0.33

-10 -8 -6 -4 -2 0 2 4 6 8 10-500

-400

-300

-200

-100

0

100

200

300

400

500

R (fm)

0

1

2

0 100 200q (MeV/c)

C2(

q)


S=0.9

-10 -8 -6 -4 -2 0 2 4 6 8 10-500

-400

-300

-200

-100

0

100

200

300

400

500

R (fm)

0

1

2

0 100 200q (MeV/c)

C2(

q)


S=0

x

K


Q. Can one remove these pathologies Q. Can one remove these pathologies from ~classical predictions?from ~classical predictions?

A. Yes (Q.H. Zhang A. Yes (Q.H. Zhang et alet al.): .): Smear the phase space points (x,p) with Smear the phase space points (x,p) with

minimum uncertainty wave-packets:minimum uncertainty wave-packets:

with with ~ 1 fm ~ 1 fm Pictorially:Pictorially:

-10 -8 -6 -4 -2 0 2 4 6 8 10-500

-400

-300

-200

-100

0

100

200

300

400

500

R (fm)-10 -8 -6 -4 -2 0 2 4 6 8 10

-500

-400

-300

-200

-100

0

100

200

300

400

500

R (fm)

=

- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0

- 5 0 0

- 4 0 0

- 3 0 0

- 2 0 0

- 1 0 0

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

R ( f m )x This can be done analytically with

parameterization on previous slide:


What are the Lorentz properties of CWhat are the Lorentz properties of C22(q)?(q)?

How to write our favorite practice Gaussian in an explicitly Lorentz invariant way?

Answered in F.B. Yano and S.E. Koonin, Phys. Lett. B78, 556 (1978).


Can Lorentz effects “distort” information?Can Lorentz effects “distort” information? Apply Yano-Koonin-Podgoretzsky result to Apply Yano-Koonin-Podgoretzsky result to

another toy model:another toy model:

p1

p2

VPAI

RBeamaxis

uS

Apply this to ~1-d motion in “Out” direction:

Nota Bene: This last case allows ROUT<RSIDE

even for~ RSIDE


The chief lesson from last 10-15 years The chief lesson from last 10-15 years of (theoretical) HBT work:of (theoretical) HBT work:

The “radii” are a (potentially) The “radii” are a (potentially) complicated mixture ofcomplicated mixture of Space-time distributionSpace-time distribution Spatial flow gradients Spatial flow gradients Temperature gradients Temperature gradients

More correct terminology:“radii” “lengths of homogeneity”

In the presence of source dynamics, “radii” depend on mean pair(energy, momentum, transverse mass, …)


But : ROUT/RSIDE trend completely eliminates one “hydro” calculation

Soff, Bass, Dumitru, Phys. Rev. Lett. 86, 3981

(2001)


Plotted versus KPlotted versus KTT, there is an amazing , there is an amazing

consistency between data from AGS, SPS, and consistency between data from AGS, SPS, and RHIC, spanning a factor of 100 in CM RHIC, spanning a factor of 100 in CM energy(!):energy(!):

SPSAGS

RHIC


There is now an extensive There is now an extensive experimental and theoretical experimental and theoretical literature built on 20+ years of literature built on 20+ years of “HBT” studies“HBT” studies

We’re now ready to start doing We’re now ready to start doing things right:things right: ExperimentallyExperimentally

Measure resonance contributionsMeasure resonance contributions Measure like and unlike particle effectsMeasure like and unlike particle effects Understand systematic errorsUnderstand systematic errors

TheoreticallyTheoretically Cascade codes for RHICCascade codes for RHIC Right parametrizationRight parametrization Coulomb systematicsCoulomb systematics


Multipion correlationsMultipion correlations

Each events has large number of Each events has large number of pions, therefore, multi-pion pions, therefore, multi-pion correlation will affect correlation will affect

Two-pion interferometryTwo-pion interferometry

(1): If the pion correlation formula (1): If the pion correlation formula is right?is right?

(2): If it is not right, which kind of (2): If it is not right, which kind of formula should we use!formula should we use!


Seems right!Seems right!

Zhang, Phys. Rev. C58, 22 (1998)Zhang, Phys. Rev. C58, 22 (1998) Proved that for a class model, the Proved that for a class model, the

present pion correlation function present pion correlation function formula is right.formula is right.

However generally, the formula is However generally, the formula is not right!not right!

Zhang, Phys. Rev. C59,1646 (1999).Zhang, Phys. Rev. C59,1646 (1999).


The formula is :The formula is :

C(p1,p2)=C(p1,p2)=

A(p1,p2)[1+R(p1,p2)]A(p1,p2)[1+R(p1,p2)]

A(p1,p2) is a function of p1,p2.A(p1,p2) is a function of p1,p2.

R(p1,p2) can be written as the same R(p1,p2) can be written as the same as above.as above.


Three-pions correlationsThree-pions correlations


Some of slides taken from Some of slides taken from

W. A. Zajc, R. Wilson. W. A. Zajc, R. Wilson.

Thank you!Thank you!

hbt'04q. h. zhang1 a short introduction to hbt qinghui zhang

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