# Presentation: 417818

## Imaging in Heavy-Ion Collisions      

**D.A. Brown**, A. Enokizono,
- M. Heffner, R. Soltz (LLNL)
- P. Danielewicz, S. Pratt (MSU)

- RHIC/AGS Users Meeting
- 21 June 2005

- This work was performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.

-  _**Talk Outline:**_
-  Background
-  Nontrivial test problem
-  Inverting a 3d correlation
-  Implications for HBT Puzzle
-  CorAL Status

## The Koonin-Pratt Formalism

*We will invert to get **S(r)** directly*

- Source function related to emission function:

- We work in Bertsch-Pratt coords., in pCM

## 3d HBT at RHIC

- Boost may have interesting observable consequence (Rischke, Gyulassy Nucl. Phys. A **608**, 479 (1996)):
-  If there is a phase-transition, hydro evolution will slow in mixed phase.
-  Will lead to long-lived source
-  Huge difference in Outward/Sideward radii

## The Test Problem, part 1

-  T = 50 MeV -- source temperature
-  Rx= Ry= Rz= 4 fm
-   = 20 fm/c, ballpark of  lifetime (23 fm/c)

- Emission function is Gaussian with finite lifetime, in lab frame:

## The Test Problem, part 2

- Combine with ++ kernel to get correlation using Scott Pratt’s CRAB code

## Breaking Problem into 1d Problems

**Breaking Problem into 1d Problems**

- Where

- Expand in Ylm’s and Legendre polynomials:

## Breaking Up Problem, cont. 

## Imaging Summary

- Use CorAL version 0.3
- Source radial terms written in Basis Spline representation
- knots set using Sampling Theorem from Fourier Theory
- use 3rd degree splines
- Use full Coulomb wavefunction, symmetrized
- Use generalized least-square for inversion
- Use constraints to stabilize inversion
- Cut off at finite *l*, *q* in input correlation

## Comparison to 1d Source Fits

-  Fits & imaging extract same core parameters:
-  Rinv = 5.38 +/- 0.13 fm
-  = 0.647 +/- 0.029

-  Same 30% undershoot for fits and images
-  Gaussian fits can’t get halo
-  CorAL Gaussian fit == “best” Coulomb correction

## 3d Results

-  Nail radii
-  Height low by 30%
-  Nail integral of source
-  Resolve halo in
-   L & O directions

- Measured Gaussian parameters (from source!):
-    RS: 3.42 +/- 0.21
-    RO: 7.02 +/- 0.25
-    RL: 5.04 +/- 0.31
-    S(r=0): 10 +/- 1x10-5
-    : 0.442 +/- 0.067

## Core Halo Model

- Nickerson, Csörgo ̋, Kiang, Phys. Rev. C **57**, 3251 (1998).

*f* is fraction of ’s emitted directly from core,
- effectively ***= f**2* in source function

## Core Halo Model, cont.

**Core Halo Model, cont.**

-  From exploding core, with Gaussian shape:
-  Flow profile simple, but adjustable:
-  From decay of emitted resonances,   has most likely lifetime (23 fm/c)

## Simple Model, Interesting Results

- Set *R**x**=R**y**=R**z**=4* fm, ***f/o**=10* fm/c, *T=175* MeV, *f=0.56*

- High velocity ’s get bigger boost.  Boost + finite **  tail,
*but only modest core increase*, in L,O directions.

## Focus on Outwards Direction

- Lifetime effects have dramatic affect
- on non-Gaussian halo in outwards direction.

## Focus on Outwards Direction

- ... but only* modest core changes*.

_**Gaussian R**__**O **__**fits:**_
- Baseline:	5.63
- Instant f/o:	4.78
- No :		5.35
- No lifetime:	4.48
- max=0.35:	6.51
- radii in fm,
- fit w/ r<15 fm,
- fit tolerance 1%

## Focus on Sidewards Direction

- Minor changes deep in tail reflecting
- geometry of  induced halo.
- All curves have core radii ~ 4 fm.

## Focus on Longitudinal Direction

- Rapidity cut eliminates tail,
- otherwise similar to outwards direction.

## Take Home Messages from Model

- Need emission duration + particle motion to get lifetime effect
- Finite lifetime can create tail, w/o changing core radii substantially
- RHIC HBT puzzle could be Gaussian fit missing tail from long lifetime
- Resonance effects detectable, but similar to freeze-out duration effects
- Don’t be fooled by acceptance effects

- Since source tails hide in Coulomb hole of correlation, imaging RHIC data should shed light on puzzle

## CorAL Features

- Variety of kernels:
- Coulomb, NN interactions, asymptotic forms
- Any combination of p, n, , K+-, , plus some exotic pairs
- Fit a 1d or 3d correlation w/ variety of Gaussian sources
- Directly image a correlation, in 1d or 3d
- Build model correlations/sources from
- OSCAR formatted output,
- Blast Wave,
- variety of simple models
- Build model correlations/sources in Spherical or Cartesian harmonics

## CorAL Status

- Merging of three related projects:
- original CorAL by M. Heffner, fitting correlations w/ source convoluted w/ full kernel
- HBTprogs in 3d by D. Brown, P. Danielewicz, imaging sources from correlation data w/ full kernels
- CRAB (++) by S. Pratt, use models to generate correlations, sources w/ full kernels
-  Rewritten in C++, open source, nearly stand alone (depends on GSL only)
- Developed on MacOS X, linux, cygwin
- Time scale for 1.0 release: end of summerish (sorry, probably not in time for QM2005)

## Extra Slides

## Low Relative Momentum Correlations

- Signal, pairs from same event (these guys are entangled)

- Background, pairs from different event, same cuts
- (looks like signal, but no entanglement)

-  When C=1, there is no correlation
-  Removes features in spectra not caused by entangled particles

*A versatile observable: can use nearly any type of pair*

## 3d HBT at RHIC

- Work in Bertsch-Pratt coordinates in pair CM frame:

- Boost from lab  pair CM means lifetime effects transformed into Outwards/Longitudinal direction.

## What do the terms mean?

-  l = 0 : Angle averaged
-   correlation, get access to Rinv
-  l = 1 : Access to Lednicky offset,
-   i.e. who emitted first (unlike only)
-  l = 2 : Shape information:
-   access to RO, RS, RL:
- 	C20  RL
- 	C00-(C20±C22)  RS ,RO
-  l = 3 : Boomerang/triaxial
-   deformation (unlike only)
-  l = 4 : Squares off shape

## Sampling Theorem

- If we can ignore FSI, the 1-dimensional kernels are:
- The l = 0 term is just a Fourier Transform.  Sampling Theorem says for a given binning in *q*-space, source uniquely determined at certain *r* points (termed collocation points) spaced by:
- Sampling Theorem may be generalized to the l > 0 terms giving
- Here the ln are the zeros of the Spherical Bessel function.

## Setting knots

- In general the spin-zero like pair meson kernel is much more complicated than this:
- and these knots are only an approximation to the best knots.

**Setting knots**

## Representing terms in source

- We use basis splines to represent the radial dependence of each source term:

- Basis splines are piece-wise continuous polynomials:

*Setting knots for the splines is crucial for representing source well*

## Representing the Source Function

- Radial dependence of each term in terms of Basis Splines:

## The inversion process

- Koonin-Pratt eq. in matrix form:

## Constraints for 3d problem(s)

- Inversion can be stabilized with constraints.  Constraints we use:

## Outwards/Sidewards Ratio

- Gaussian fits driven by r < 10 fm where ratio close to unity.

## Simple Model, Interesting Results

- Turn off resonance production (*f=1*)

- L,O tail reduced somewhat, S tail gone.
- Core lifetime causes tail same size as resonances.

## Simple Model, Interesting Results

- Instant freeze-out (***f/o**=0* fm/c)

- Kills core-induced non-Gaussian tail,
- but resonance induced tail of similar magnitude.

## Simple Model, Interesting Results

- No resonances (*f=1*), instant freeze-out (***f/o**=0* fm/c)

- Essentially kills non-Gaussian tail.
- Obviously need finite time effect for tail.

## Simple Model, Interesting Results

- Apply PHENIX acceptance cut of ***max**=0.35*

- Kills contribution from high *p**L* pairs,
- so kills core’s tail in L direction

## Simple model, interesting results

- No flow

- Lowers source, but not much else

## Simple model, interesting results

- Small temperature

- NO PLOT YET, HARD TO SAMPLE FROM
- SMALL TEMPERATURES

## Simple model, interesting results

- No flow, small temperature

- NO PLOT YET, HARD TO SAMPLE FROM
- SMALL TEMPERATURES