# Presentation: 842206

## 3D Two-pion Source Function extraction from SPS to RHIC energies: a probe of source breakup time & emission duration

- Paul Chung
- SUNY Stony Brook

- ISMD07, Berkeley, California, August 5 2007

## PHENIX nucl-ex/0605032

1D Pion-Pion Correl Func
Au+Au sqrt(s)=200AGeV

Imaging source function vs 3D HBT source function

Discrepancy for r>20fm
Corresponds to q<10MeV 

30% more pions in tail above 3D HBT Gaussian source
20% in <r> compared to 3D HBT

- PHENIX nucl-ex/0605032
- 1D Pion-Pion Correl Func
- Au+Au sqrt(s)=200AGeV
- Imaging source function vs 3D HBT source function
- Discrepancy for r>20fm
- Corresponds to q<10MeV
- 30% more pions in tail above 3D HBT Gaussian source
- 20% in <r> compared to 3D HBT

- Long range structure in pion source @ RHIC

- ISMD07, Berkeley, California, August 5 2007

## Outline

- 1D Imaging technique & application to NA49 data @SPS 17.3 AGeV Pb+Pb collisions
- Overview of 3D source function extraction method : Cartesian Spherical Harmonic decomposition
- 3D source function extraction via Moment Imaging: PHENIX data @ RHIC 200 AGeV Au+Au collisions
- 3D source function extraction by Moment  Fitting @ RHIC & SPS
- Therminator model comparison - Source breakup time & emission duration extraction from 3D source function shape @ SPS & RHIC

- ISMD07, Berkeley, California, August 5 2007

## Source function

**Technique Devised by:**
** **
**D. Brown, P. Danielewicz,**
**PLB 398:252 (1997).** **PRC 57:2474 (1998).**

**Inversion of Linear integral equation to obtain source function**

**Source function**
**(Distribution of pair separations)**

- Encodes FSI

**Correlation**
**function**

**Inversion of this integral equation**
**==******** ****Source Function**

- 1D Koonin Pratt Eqn.

- Extracted S(r) in pair CM frame

- Hence Model-independent i.e Kernel independent of freeze-out conditions

- No Shape assumption for S(r)

- ISMD07, Berkeley, California, August 5 2007

## 158 AGeV -Angle-averaged C(q) & S(r)

- ISMD07, Berkeley, California, August 5 2007

## 3D Analysis

- (3)

**3D Koonin**
**Pratt**

- Plug in (1) and (2) into (3)

- (1)

- (2)

**Expansion of R(q) and S(r) in Cartesian Harmonic basis**

***Basis of Analysis***

**(Danielewicz and Pratt nucl-th/0501003 (v1)  2005)**

- X=out-direction

- Y=side-direction

- Z=long-direction

- ISMD07, Berkeley, California, August 5 2007

## Moments restoredfrom Imaging

- Moments decrease   with increasing       order for x, y and z :
-     truncate at l=6
- Restored moments      in good agreement   with data moments :
-     Imaging under control

- ISMD07, Berkeley, California, August 5 2007

## Fitting functions: Ellipsoid & 2-Gaussian 

- Ellipsoid : S(x,y,z) = λ G                                        G = exp[ (x/Rx)2 + (y/Ry)2 + (z/Rz)2 ]
- 2-Gaussian : S(x,y,z) = λ Gs . Gl
-    Gs  = exp[-fs ( (x/Rxs)2 + (y/Rys)2 + (z/Rzs)2 ) ]
-    Gl    = exp[-fl ( (x/Rxl)2 + (y/Ryl)2 + (z/Rzl)2 ) ]
-     fs    = 1/( 1 + (r/r0)2 )
-     fl    =  (r/r0)2 /( 1 + (r/r0)2 )

- ISMD07, Berkeley, California, August 5 2007

## Moment Fitting

- ISMD07, Berkeley, California, August 5 2007

## 3D C(q) & S(r) – 17.3 AGeV Pb+Pb

- Non-Gaussian tails in x
- x-y assymetric indicates
- non-zero relative emission
- time in LCMS

- Long-range exponential-like
- Tail in longitudinal direction

- Signature of longitudinal expansion
- S.Pratt PRD 33, 5, 1314 (1986)

- ISMD07, Berkeley, California, August 5 2007

## 3D C(q) & S(r) – 200 AGeV Au+Au

- 2-Gaussian Fit function in good
- Agreement with source image

- X-Y assymetry reflects Lorentz
- Transformation boost effect as
- well as lifetime effects

- ISMD07, Berkeley, California, August 5 2007

## Therminator model Event Generator

- A.Kisiel et al. Comput.Phys.Commun.174, 669 (2006)
- Thermal model with Bjorken assumption of boost invariance for longitudinal expansion
- Blast Wave Transverse Flow vt (r) semilinear in r
- Includes all known resonance decays
- Emission from infinitely long cylinder transverse radius ρmax : controls sideward extent
- Proper breakup time in source rest frame, tau : controls longitudinal extent
- Proper emission duration in rest frame Δtau : controls tails in long and out directions

- ISMD07, Berkeley, California, August 5 2007

## Therminator model comparison

- ISMD07, Berkeley, California, August 5 2007

## Conclusions

- First 3D source imaging analysis for mid-rapidity pions at SPS & RHIC
- Prominent non-Gaussian tails in source image in longitudinal and outward directions @ sqrt_s=17.3AGeV & 200AGeV
- 3D pion source function well described by 2-Gaussian function in good agreement with source image @ SPS & RHIC
- Extracted 3D source functions consistent with 3D dynamics incorporated in Therminator Model: Bjorken longitudinal boost invariance and transverse blast-wave
- sqrt_s=17.3AGeV : pion source breakup time  tau=7.3fm/c, pion emission duration Δtau=3.7fm/c
- sqrt_s=200AGeV : tau=8.5fm/c, Δtau=2fm/c

- ISMD07, Berkeley, California, August 5 2007