UCRL-PRES-?????? 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

UCRL-PRES-?????? 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

UCRL-PRES-?????? 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

UCRL-PRES-?????? 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.

UCRL-PRES-?????? 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

UCRL-PRES-?????? CHUM source is non-Gaussian in 3d Set R x =R y =R z =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.

UCRL-PRES-?????? 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)| CoRrelation AfterBurner (CRAB) CRAB ’s role has increased: it now computes sources and correlations from OSCAR data

UCRL-PRES-?????? 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 R inv l = 1 : Access to Lednicky offset, i.e. who emitted first (unlike only) l = 2 : Shape information, access to R O, R S, R L : C 20  R L  C 00 -(C 20 ±C 22 )  R S, R O l = 3 : Boomerang/triaxial deformation (unlike only) l = 4 : Squares off shape

UCRL-PRES-?????? Use CRAB to generate S lm (r)

UCRL-PRES-?????? Use CRAB to generate C lm (q) Note: Coulomb turned off

UCRL-PRES-?????? 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:

UCRL-PRES-?????? 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

UCRL-PRES-?????? Basis Spline basis Previous versions of CorAL (and HBTprogs) used Basis Splines: –N b =0 is histogram, –N b =1 is linear interpolation, –N b =3 is equivalent to cubic interpolation. Resolution controlled by knot placement Knot placement controlled by –q scale parameter –Moving N c -N b -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

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

UCRL-PRES-?????? 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

UCRL-PRES-?????? Laguerre function basis Polynomial times exponential weight –0 th 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

UCRL-PRES-?????? First results are promising… r (fm) S 00 (r) (fm -3 )

UCRL-PRES-?????? Other terms in the imaged source

UCRL-PRES-?????? CorAL developers and  testers SCOTT PRATT LI YANG PAWEL DANIELEWICZ DAVE BROWN JASON NEWBY RON SOLTZ AKITOMO ENOKIZONO MIKE HEFFNER NATHANIEL BROWN-PEREZ Lawrence Livermore National Laboratory