Imaging in Heavy-Ion Collisions

D.A. Brown, A. Enokizono, M. Heffner, R. Soltz (LLNL)

P. Danielewicz, S. Pratt (MSU)

RHIC/AGS Users Meeting21 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

The pair’s final state wave-function, we can compute this if we know the interaction. This is

the kernel K(q,r).

The source function, it gives the probability of producing a pair a distance r apart in the pair CM

frame

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

Rout/Rside == 1! (in LCMS)Experiment analysis wrong?

• Coulomb correction?Theory screwed up?Interpretation confused?

• Instant freeze-out?• Gaussian fits?

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:

(lab pair CM boost) + == non-Gaussian source!

The Test Problem, part 2

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

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

Can extract the important physics!

Core Halo ModelNickerson, Csörgo˝, Kiang, Phys. Rev. C 57, 3251 (1998).

f is fraction of ’s emitted directly from core, effectively = f2 in source function

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 Rx=Ry=Rz=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 RO 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.

Long (beam)

Out

Side

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 TheoremIf 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.

To reconstruct the source at these collocation points, choose 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 sourceWe 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

0th-order 1st-order 2nd-order

…

Representing the Source FunctionRadial dependence of each term in terms of Basis Splines:

• Generalized Splines:

– Nb=0 is histogram,

– Nb=1 is linear interpolation,

– Nb=3 is equivalent to cubic interpolation.

• Recursion relations == fast to evaluate, differentiate, integrate

• Resolution controlled by knot placement

• Form basis for RR2, but not orthonormal

The inversion processKoonin-Pratt eq. in matrix form:

• kernel not square• kernel possibly singular

• noisy data• error propagation

this is an ill-posed problem

Practical solution to linear inverse problem, minimize :

Most probable source is:

With covariance matrix:

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 ResultsTurn 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 pL pairs, so kills core’s tail in L direction

Simple model, interesting resultsNo flow

Lowers source, but not much else

Simple model, interesting resultsSmall temperature

NO PLOT YET, HARD TO SAMPLE FROMSMALL TEMPERATURES

Simple model, interesting resultsNo flow, small temperature

NO PLOT YET, HARD TO SAMPLE FROMSMALL TEMPERATURES