imaging in heavy-ion collisions
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 PresentationTRANSCRIPT
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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
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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
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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?
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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!
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The Test Problem, part 2
Combine with ++ kernel to get correlation using Scott Pratt’s CRAB code
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Breaking Problem into 1d Problems
Where
Expand in Ylm’s and Legendre polynomials:
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Breaking Up Problem, cont.
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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
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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
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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!
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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
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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)
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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.
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Focus on Outwards Direction
Lifetime effects have dramatic affect on non-Gaussian halo in outwards direction.
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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%
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Focus on Sidewards Direction
Minor changes deep in tail reflecting geometry of induced halo.
All curves have core radii ~ 4 fm.
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Focus on Longitudinal Direction
Rapidity cut eliminates tail, otherwise similar to outwards direction.
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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
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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
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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)
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Extra Slides
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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
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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
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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
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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.
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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
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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
…
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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
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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:
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Constraints for 3d problem(s)
Inversion can be stabilized with constraints. Constraints we use:
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Outwards/Sidewards Ratio
Gaussian fits driven by r < 10 fm where ratio close to unity.
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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.
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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.
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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.
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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
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Simple model, interesting resultsNo flow
Lowers source, but not much else
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Simple model, interesting resultsSmall temperature
NO PLOT YET, HARD TO SAMPLE FROMSMALL TEMPERATURES
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Simple model, interesting resultsNo flow, small temperature
NO PLOT YET, HARD TO SAMPLE FROMSMALL TEMPERATURES