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CorAL and the Future of Imaging

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.

Outline: What is CorAL Tour of components: A test problem Imaging New bases

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What is CorAL?• The Correlation

Algorithm Library– Source & correlation data

structures• Various 1d• Cartesian harmonics• Spherical harmonics• 3d histograms

– OSCAR formatted input– Various model sources– Imaging tools– Fitting tools in development– Kernels– Oodles of wavefunctions– Sample codes

• Status:– It (sortof) runs, but we’re

still cleaning it up – Documentation in progress– Code collaboration

agreements available (see me later)

– Official GPL release “any time now”

• Platforms:– Linux, Unix, MacOSX– Written in C++– Requires Gnu Scientific

Library

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CorAL components• Core libraries

– libcoral: core CorAL library

– libcoralutils: utility codes CorAL uses

• Binaries– CHUM: freeze-out point (emission function)

generator

– SHARK: precompute the kernels for CRAB and DIVER

– CRAB: constructs correlations and sources from OSCAR data

– DIVER: imaging code

– plotting codes (bfplot, scplot, converts2c)

•Various component tests

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Core Halo Ur-Model (CHUM)Variation of the Core-Halo model of Nickerson, Csörgo˝, Kiang, Phys. Rev. C 57, 3251 (1998), etc.

• Blast-wave like flow profile• Gaussian source with finite source lifetime• production from source and resonance decay

The emission function:

f is fraction of ’s emitted directly from core, i.e. = f 2 in source.

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CHUM cont.• From exploding core, with Gaussian shape:

• Use full 3-body decay kinematics and lifetime of =23 fm/c:

• Blast-wave like flow profile:

where , and

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CHUM source is non-Gaussian in 3d

Set Rx=Ry=Rz=4 fm, f/o=10 fm/c, T=165 MeV, f=0.5

Difference in side, out and long directions mean tail in higher lm terms as well as the 00 term. The out-long tail due to lifetime of source.

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Source function is related to emission function:

Work in Bertsch-Pratt coordinates in pair Centre of Mass (CM) frame

The Koonin-Pratt equation:

Pair final state relative wave-function, q(r), defines the kernel:

K(q,r)=|q(r)|2-1.

CoRrelation AfterBurner (CRAB)

CRAB’s role has increased: it now computes sources and correlations from OSCAR data

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Breaking Problem into 1d Problems

Where

Expand in Ylm’s and Legendre polynomials:

Cartesian harmonics give analogous expressions

• 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

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Use CRAB to generate Slm(r)

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Use CRAB to generate Clm(q)

Note: Coulomb turned off

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Demonstration InVERter (DIVER)

• kernel not square & may be singular• noisy data• error propagation

imaging is an ill-posed problem

Practical solution to linear inverse problem, minimize :

Most probable source is:

With covariance matrix:

Convert Koonin-Pratt equation to matrix form:

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What about radial basis?

• Radial dependence of each term in terms of basis functions:

• Basis choice affects sensitivity to correlation. Characterize with l=0 kernel:

A crappy basis gives bad source with small uncertainty

• Many bases to test:– Orthogonal polynomials on

interval (-1,1):• Legendre polynomials• Chebyshev polynomials

– Orthogonal polynomials on interval (0,∞):

• Laguerre polynomials• Hermite polynomials

– Basis Splines

• Others possible:– Spherical Bessel functions– Coulomb wave functions

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Basis Spline basis• Previous versions of CorAL (and

HBTprogs) used Basis Splines:

– Nb=0 is histogram,

– Nb=1 is linear interpolation,

– Nb=3 is equivalent to cubic interpolation.

• Resolution controlled by knot placement

• Knot placement controlled by – qscale parameter

– Moving Nc-Nb-1 knots by hand

• Local in r-space means non-local in q-space:– High q wiggles

– Tails in r too sensitive to high q

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Optimizing Basis Spline basis tough

Imaging for l=2 terms is problematic with Basis Spline basis: two length scales important in source (core & tail)

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Legendre polynomial basis• Orthonormal polynomial

over fit range• Only 2 parameters: fit

range and number of coefficients

• Non-local in r-space, local in q-space– wiggles in q reduced

• Chebyshev polynomials have very similar behavior

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Laguerre function basis• Polynomial times exponential

weight– 0th term is exponential – Higher order terms encode

deviation from exponential

• Only can tune exponential weight radius, fit range, and number coefficients

• May have trouble with non- exponential tails

• Highly non-local in r-space so localized to low-q

• Hermite basis has similar behavior

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First results are promising…

r (fm)

S00

(r)

(fm

-3)

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Other terms in the imaged source

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CorAL developers and testers

SCOTT PRATTLI YANGPAWEL

DANIELEWICZ

DAVE BROWN JASON NEWBY

RON SOLTZAKITOMO

ENOKIZONOMIKE HEFFNER

NATHANIELBROWN-PEREZ

Lawrence Livermore National Laboratory