quantum opacity and refractivity in hbt puzzle

$: Quantum Opacity and Refractivity in HBT Puzzle$
2005. 11. 5 Inha Nuclear Physics Group

Quantum Opacity andQuantum Opacity and

Refractivity in HBT Refractivity in HBT

PuzzlePuzzleJin-Hee Yoon

Dept. of Physics, Inha University, Korea

John G. Cramer, Gerald A. Miller, M. S. WuDept. of Physics, University of Washington, US

11/05/2005 HIM@Pohang

Inha Nuclear Physics Group

Correlations in Phase-Space

Space-time structure of fireball can be studied by HBT interferometry

Correlation function

)()(

),(),(

2112

21221 pp

pppp

PP

PNC

is typically parametrized as

2121

22222221

ppq )/2p(pK

1pp

]exp[),( llssoo qRqRqRC

l

s

o

K

TKBeam direction




Hydrodynamic calculation predicts(D. H. Rischke and M. Gyulassy, Nucl. Phys. A608, 479 (1996); P. F. Kolb and U. Heinz, Quark Gluon Plasma 3, World Scientific, Singapore, 2004)

But, experimental results are

(STAR Collaboration, C. Adler et al., Phys. Rev. Lett. 87, 082301 (2001); PHENIX Collaboration, K. Adcox et al., Phys. Rev. Lett. 88, 192302 (2002); PHENIX Collaboration, A. Enokizono, Nucl. Phys. A715, 595 (2003))

HBT Puzzle

2.1~1/ so RR

10~5.1/ so RR




Experimental Data(Au+Au@200 GeV) shows Dense Medium

OPACITY and Refractive Effects

Our Purpose : Quantum mechanical treatment of Opacity & Refractive effects

which reproduces lso RRR ,,




Theoretically, the observables are expressed by

)/()/(*)exp()(

),( 2222 3

4

0 yxJyxJyiKyd

KxS

),(),(

|),(|),(),(

204

104

20

4

0210

1Kqpp

pxSxdpxSxd

eKxSxdCC

xiq

Subscript 0 means no final state interaction (FSI).



Final State Interaction

FSI replaces

)/(*)/( )'exp()(

)',('),( )(p

)(p 21

222 4

4

04 yxΨyxΨyiK

ydKxSKdKxS

),( ),(

|),( |1)K,q()p,p(

24

14

24

21pxSxdpxSxd

KxSxdCC

xipexΨxJxJ )(* )( )( (-)P

: full scattering outgoing wave function*(-)PΨ

Includes two 4-dimensional integration



Wigner Emission Function

Using the hydrodynamic source parameterization (B. Tomasik and U. W. Heinz, Eur. Phys. J. C4, 327 (1998); U. A. Wiedermann and U. W. Heinz, Phys. Rep. 319, 145 (1999))

With boost-invariant longitudinal dynamics

cos)(sinh)(coshcosh

),(ln2

1

)(1exp),(

)(2)(2

)(exp

)(2

)cosh(),(

)2/(),(),(),(

21

22

1

2

2

2

2

0

20

3

00

bKbMuK

xxzt

ztzt

bT

uKMB

YΖ

BΖKxS

tTtT

T

b

Kb

Kb

T

T

of potential chemical:

,222 mKM TT

),( bKT



Wigner Emission Function

)(b

: cylindrically symmetric source density

21/)(exp WSWS aRb

)(bt : transverse flow rapidityWS

f R

b

0ln2

1

LK

LK

KE

KEY Since KL=0 for midrapidity data

Parameters : RWS , aWS , 0 , f , T



Full Scattering Wavefunction

Assumption : Matter is cylindrically symmetric with a long axis in a central collision region

2/2/

)()( 2/0

l

ziqxilp eexΨ

qqKP

bx

T1,2

(-)

P

(-)

P 1,21,2

Reduced 2-dimensional Klein-Gordon Eq.

)(*)(*))(( 22 bb (-)p

(-)p pbU




,1

0 cos))(,(2),()(*m

mm mibpfbpf b(-)

p

Optical Potential )()()( 220 bpwwbU p

0w :real

At p=0, no opacity

Parameters : w0 , w2

Using partial wave expansion,

we can solve K-G Eq. exactly.




ppf )](Im[4

In Impulse Approximation, central optical potential

00 4 fU

Significant opacity

Using

f : complex forward scattering amplitude

For low energy interaction

0 : central density

-30

-1 1.5fm ,fm 1p mb, 1 2

0 fm 15.0]Im[ U



Correlation Function

Now our Emission Function is

)2/(*)2/(

)(~

'),()2(

1),(

)()(

202

0

b'bb'b

b'b,

21 pp

BbdeZKxS ziqtiq l

with ]exp[)(')(~ 2 b''K'Kb,b'b, TT iBKdB T

Large Source Approximation : b’~1/T~1 fm << RWS

/2b'Kpppp

T

2121bb

b'b

b'b

ie)(*)(

2*

2)()()()(




2

00

220

2

0

320

22

0

120

22

)*()(0

2

2211

21222222

)(

)()(

)(

)()3(

~

)(

)()3(

)(cos)(sinh

exp)(

)()()())((

||~1),(

-

Kb,

Kb,b b

Kq

T

Tpp

T

ji

KF

KF

KF

KF

KF

KFR

bMT

bKB

Bbfbd

qRqC

l

Tn

tTn

nn

nij

oll




)(

/)()(2)(

)()()(

/)(4/)(2)(

)(2)(

/1/)(cosh)(

)())(()()(

213

202

2101

10

2

2)(2

i

ntTn

nmnn

m

K

KKf

KKf

KKf

Kf

TbMb

bfBbdKF

b Kb, TKT

Here,

Are Modified Bessel function .



HBT Radii

40/

),(2)(

,

2,

,2,

Tso

so

TsoTso

Kq

q

KqCKR

Then our transverse radii can be calculated by

with



Fitting Parameters

(MeV)0(fm/c)w2w0(fm-2)

(fm/c)aWS(fm)RWS(fm)fT(MeV)

6.1

2.173

025.0

314.1

056.0

782.11

015.0

725.0

067.0

852.2

046.0

137.0

002.0014.0

121.0582.0

i

i

10.0

23.8

032.0

063.1

1.1

2.123

2 /N ~ 7.8

AuGeV AusSTAR@ NN 200



Fitting Parameters Check

Temperature T (173 MeV) ~ Tc (160 MeV)

f=1.31 maximum flow velocity ~ 0.85c

Source Size RWS (11.7 fm) ~ RAu (7.3 fm)+4.4 fm

Expansion time 0 (8.2 fm/c)

average expansion velocity ~ 0.5c

Emission duration (2.9 fm/c) << 0(8.2 fm/c)

Longitudinal length

system’s axial length (17.5 fm)

: large enough for long cilyndrical symmetry

full calculation

no flow

no refraction

no potentialBoltzmann for BE thermal distribution

Thank you.

quantum opacity and refractivity in hbt puzzle

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