Upload
lysandra-jarvis
View
37
Download
0
Embed Size (px)
DESCRIPTION
D.A. Brown , A. Enokizono, M. Heffner, R. Soltz (LLNL) P. Danielewicz, S. Pratt (MSU). Imaging in Heavy-Ion Collisions. Talk Outline: Background Nontrivial test problem Inverting a 3d correlation Implications for HBT Puzzle CorAL Status. RHIC/AGS Users Meeting 21 June 2005. - PowerPoint PPT Presentation
Citation preview
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