The DWEF Model:Refractive Distortions of

HBT

The DWEF Model:Refractive Distortions of

HBT

John G. Cramer (with Gerald A. Miller)University of WashingtonSeattle, Washington, USA

John G. Cramer (with Gerald A. Miller)University of WashingtonSeattle, Washington, USA

Workshop on Particle Correlations & Femtoscopy - 2007Santa Rosa, CAAugust 2, 2007

August 2, 2007 WPCF 2007 2

Since WPCF 2006 …Since WPCF 2006 …1. We discovered in November a convergence vs. integration

step size problem in our calculation of optical model wave functions. This had no effect on the HBT radii, but had a strong effect on the slope of the spectrum. This problem was corrected by changing from Runge-Kutta to Numerov wave function solutions.

2. We discovered in March that the fugacity from the strong pion chemical potential was being applied to the spectrum, but not to the variables for the HBT radii. This error was corrected.

3. The net result, after refitting, is that the “ambiguities” reported last year are gone, and the emission temperature of the model has dropped from T0=193 MeV to T0=161 MeV. The need for a very deep and absorptive optical potential remains.

4. Result: The New Improved DWEF Model (DWEF v.2.1).

August 2, 2007 WPCF 2007 3

Elements of DWEF Approach:

(1) The Nuclear Optical Model

Elements of DWEF Approach:

(1) The Nuclear Optical Model1. Divide the pions into “channels” and focus on pions (Channel

1) that participate in the BE correlation (about 60% of the spectrum pions). Omit “halo” and “resonance” pions and those converted to other particles (Channels 2, 3, etc.).

2. Solve the time-independent Klein-Gordon equation for the wave functions of Channel 1 pions, using a complex potential U. Im(U) accounts for those pions removed from Channel 1.

3. The complex optical potential U does several things:(a) absorbs pions (opacity);(b) deflects pion trajectories (refraction, demagnification);(c) steals kinetic energy from the emerging pions;(d) produces Ramsauer-type resonances in the well, which can modulate apparent source size and emission intensity. In other words, it quantum-mechanically mocks up the effects on pions of passing through the hot dense medium of the fireball.

August 2, 2007 WPCF 2007 4

(2) “Hydro-Inspired”Emission Function

(2) “Hydro-Inspired”Emission Function

30 0( , ) ( , ) ( , ) /(2 )TS x k B b K S

2 20

0 2 22

( )cosh( , ) exp

2 22 ( )

S

1( , ) ( )

exp 1T TB b K M b

K u

T

2 2 2t z

1

2ln

t z

t z

particle momentum 4-vector

flow 4-vector

K

u

(Bose-Einstein thermal function)

(medium density)

(Space-time function)

August 2, 2007 WPCF 2007 5

1 2

41 14 ' ' ( ) ( )*

0 4 2 2

'( , ) ' ( , ') ( ') ( ')

(2 )iK x

p p

d xS x K d K S x K e x x x x

(3) The DWEF Formalism

(3) The DWEF Formalism

We use the Wigner distribution of the pion source current density matrix S0(x,K) (“the emission function”).

The pions interact with the dense medium, producing S(x,K), the distorted wave emission function (DWEF):

The s are distorted (not plane) wave solutions of: , where U is the optical potential.

Gyulassy et al., ‘79

Note: assumes chaotic pion sources.

2( )m U J 2

4

4 41 2

( , , )( , ) 1

( , ) ( , )

d x S x K qC K q

d x S x p d x S x p

Correlationfunction:

Distorted Waves

August 2, 2007 WPCF 2007 6

(4) Potential Consistent (4) Potential Consistent withwith

Chiral Symmetry Chiral Symmetry RestorationRestoration

(4) Potential Consistent (4) Potential Consistent withwith

Chiral Symmetry Chiral Symmetry RestorationRestoration

Both terms of U are negative (attractive)

U(b)=-(w0+w2p2)(b), w0=real, w2=complex

Son & Stephanov (2002):Son & Stephanov (2002):

v2 and v2m2 (T) approach 0 near T = Tc

(We note that this is a low-momentum form of the optical potential that becomes suspect above p~1-2 fm-1 or so.)

August 2, 2007 WPCF 2007 7

Data fitting has led us to a chemical potential near the pion mass. We therefore set = 139.6 MeV = mWe note that the emission temperature favored by the fits (T0~162 MeV) is close to estimates of the temperature for chemical freeze-out, but we leave this as a fit variable.

Parameters of the DWEF Model

Parameters of the DWEF Model

Thermal: T0 (MeV), (MeV) (fixed at m)Space: RWS (fm), aWS (fm)Time: (MeV/c), (MeV/c)Flow: f (#)Optical Pot.: Re(w0) (fm-2), Re(w2) (#), Im(w2) (#)Wave Eqn.: (fixed, Kisslinger term off)

Total number of parameters: 10 (+2)

Note that these parameters describe pion emission at chemical freeze-out, not kinematic freeze-out (e.g., as used in the blast-wave model).

August 2, 2007 WPCF 2007 8

DWEF Fits to STAR DataDWEF Fits to STAR Data We have calculated pion wave functions in a partial wave expansion, applied them to a “hydro-inspired” pion source function, and calculated the HBT radii and spectrum. This DWEF model uses 7 pion source parameters and 3 optical potential parameters, for a total of 10 parameters in the model. The correlation function C near half-maximum (not the 2nd moment of C) is calculated.

We have fitted STAR data at sNN=200 GeV, simultaneously fitting Ro, Rs, Rl, and dNp/dy (fitting both magnitude and shape) at 8 momentum values (i.e., 32 data points), using a Levenberg-Marquardt fitting algorithm. In the resulting fit, the 2 per data point is ~3.6 and the 2 per degree of freedom is ~4.8. Only statistical (not systematic) errors are used in calculating 2.

We remove long-lived “halo” resonance contributions to the spectrum (which are not included in the model) by multiplying the uncorrected spectrum by ½ (the HBT parameter) before fitting, then “un-correcting” the predicted spectrum with ½.

August 2, 2007 WPCF 2007 9

100 200 300 400 500 600 700

50

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8

Components of DWEF Calculations

Components of DWEF Calculations

Red Solid - Full DWEFYellow Dots - Plane wave (W=0, no flow)Green Short Dash - Re(W2) only, no flowAqua Long Dash - Im(W2) only, no flowCyan Dot Dash - Re(W0) only, no flowBlue 2-Dot Dash - Flow onlu, W=0Violet 3-Dot Dash - DWEF with no BE correction

KT (MeV/c)

KT (MeV/c)

RO(fm)

RS(fm)

KT (MeV/c)

SpectrumdN

2/2MTdMTdy

August 2, 2007 WPCF 2007 10

Optical Wave Functions [||2(b)]

Optical Wave Functions [||2(b)]

FullCalculation

KT =197 MeV/c

KT =592 MeV/c

KT =25 MeV/c

ImaginaryOnly

EikonalApprox.

Observer

Wrong!

August 2, 2007 WPCF 2007 11

Optical Wave Functions [||2(b)]

Optical Wave Functions [||2(b)]

KT =250 MeV/c

KT =600 MeV/c

KT =100 MeV/c

EikonalApprox.

Observer

DWEF

August 2, 2007 WPCF 2007 12

100 200 300 400 500 6003

4

5

6

7

8

100 200 300 400 500 6000.95

1

1.05

1.1

1.15

1.2

1.25

100 200 300 400 500 600

33.5

44.5

55.5

66.5

100 200 300 400 500 600

33.5

44.5

55.5

66.5

DWEF Fits toSTAR 200 GeV Pion HBT Radii

DWEF Fits toSTAR 200 GeV Pion HBT Radii

KT (MeV/c)

RO(fm)

RS(fm)

KT (MeV/c)

RL(fm)

RO/RS

KT (MeV/c) KT (MeV/c)

August 2, 2007 WPCF 2007 13

100 200 300 400 500 600 700

20

50

100

200

500

DWEF Fit toSTAR 200 GeV Pion

Spectrum

DWEF Fit toSTAR 200 GeV Pion

Spectrum

Note: accurate predictionof spectrum slope involvessubtle cancellations among wavefunctions, which puts severe demandson the numerical accuracy of wave functioncomputations. => The Numerov algorithm.

KT (MeV/c)

SpectrumdN

2/2MTdMTdy

August 2, 2007 WPCF 2007 14

The ParametersThe ParametersThe ParametersThe Parameters

Temperature: 162 MeV Chemical freezeout at ~ 160 MeV Transverse flow rapidity: 1.215 vmax= 0.84 c, vav= 0.58 c Mean expansion time: 9.1 fm/c system expands at ~ 0.47 c Pions emitted between 7 fm/c and 11 fm/c soft EOS . WS radius: 11.9 fm = R(Au) + 4.3 fm > R @ SPS WS diffuseness: 1.1 fm (a bit larger than LENP experience) Re(U): 0.49 + 1.19 p2 very deep well strong attraction. Im(U): 0.129 p2 mfp 8 fm @ KT=1 fm-1 strong absorption

high density Pion chemical potential: take =m (pions are massless in the

well) We have evidence suggesting a CHIRAL PHASE TRANSITION!

August 2, 2007 WPCF 2007 15

100 200 300 400 500 600

3

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6

7

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100 200 300 400 500 600

3

3.5

4

4.5

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5.5

6

6.5

Centrality: 200 GeV Au+Au

Centrality: 200 GeV Au+Au

RO(fm)Au+Au Fit

Au+Au Predictions

RL(fm)Au+Au Fit

Au+Au Predictions

RS(fm)Au+Au Fit

Au+Au Predictions

Space-time parameters RWS, aWS, are scaled by participant number. Emission duration is constant.

Red: Central Collisions . . .Indigo: Peripheral Collisions

KT (MeV/c)

KT (MeV/c)

KT (MeV/c)

August 2, 2007 WPCF 2007 16

100 200 300 400 500 600

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3

4

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7

100 200 300 400 500 600

2

3

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100 200 300 400 500 600

2

3

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6

Centrality: 200 GeV Cu+Cu

Centrality: 200 GeV Cu+Cu

Cu+Cu Predictions

Cu+Cu Predictions

Cu+Cu Predictions

Space-time parameters RWS, aWS, are scaled by participant number. Emission duration is scaled as A1/3.

Red: Central Collisions . . .Indigo: Peripheral Collisions

RO(fm)Au+Au Fit

RS(fm)Au+Au Fit

RL(fm)Au+Au Fit

KT (MeV/c)

KT (MeV/c)

KT (MeV/c)

August 2, 2007 WPCF 2007 17

10 20 30 40 50 60 70

56789

101112

10 20 30 40 50 60 70200

500

1000

2000

5000

10 20 30 40 50 60 70

510152025303540

10 20 30 40 50 60 70

510152025303540

Low pT Behavior:Ramsauer Resonances in

Well

Low pT Behavior:Ramsauer Resonances in

Well

KT (MeV/c)

Phobos 0-6%

KT (MeV/c)

RO (fm) RS (fm)

RL (fm)Spectrum

dN2/2MTdMTdy

August 2, 2007 WPCF 2007 18

SummarySummaryThe improved DWEF Model allows good fits to RHIC

HBT radii and spectrum data at emission temperatures of about 162 MeV.

We obtain excellent DWEF fits to central STAR sNN=200 GeV data, simultaneously fitting three HBT radii and the pT spectrum, and we can use participant scaling to predict noncentral Au+Au and Cu+Cu with the same optical potential strengths.

The fit parameters are reasonable and indicate strong collective flow, significant opacity, and huge attraction suggesting chiral symmetry restoration.

They describe pion emission in hot, highly dense matter with a soft pion equation of state.

The EndThe End A paper describing this work has been published in Phys. Rev. Lett. 94, 102302 (2005); nucl-th/0411031; A longer paper is published in J. Phys. G; nucl-th/0507004