Away-side distribution ina parton multiple-scattering model and background-suppressed measures Charles B. Chiu Center for Particle Physics and Department of Physics University of Texas at Austin Hardprobes, Asilomar, June 9-16, 2006
Collective response of medium: Cherenkov radiation of gluon, Mach Cone structure … Sonic boom, (Casadelerrey-Solana05, Koch05, Dremin05,Shurryak…) Our work:This structure is due to the effect of parton multiple-scattering. The dip-bump structure in the away-side distribution • Jia (PHENIX nucl-ex/0510019) • Au+Au, 0-5% • (2.5-4) (1-2.5) GeV/c • Dip-bump structure • Dip (= - ) ~ 0 • Bumps: ~ 1 rad
Parton multiple scattering: In the plane the beam. p ~P ~ E, in units of GeV. In 1-5 GeV region pQCD not reliable. We use a simple model to simulate effect of multiple scattering. Process is carried out in an expanding medium. At each point, a random angle is selected fom a gaussian distribution of the forward cone. There is successive energy loss and the decrease in step size. There is a cutoff in energy: If parton energy decreases below the cutoff, it is absorbed by the medium. Parton with a sufficient energy exits the medium. Part I. Simulation based on a parton multiple-scattering model (Chiu and Hwa, preliminary) Exit x x x x Recoil x Trigger
Simulation results: ptrigger=4.5 (a) (b) (c) Sample tracks: Superposition of many events, 1 track per event. • Exit tracks: When successive steps are bending away from the center, the track length is shorter, is likely to get out. • Absorbed tracks: When successive steps swing back and forth, the track length is longer, more energy loss. The track is likely to be absorbed. • Comparison with the data: • Parameters are adjusted to qualitatively reproduce the dip-bump structure. • Dashed line indicates the thermal bg related to the parton energy-loss.
STAR nucl -exp 0604108 Model prediction for parton Ptrig=9.5; and Passoc: 4-6. For momenta specified, our model predicts a negligible thermal bg. To display comparison with experimental peak, model curve is plotted above the bg line.
So far we have compared event-averaged data. Next we must also look at the implication of the event by event description of the model. Parton multiple-scattering: • In a given event, there is only one-jet of associated particles. • It takes large event-to-event fluctuations about =0 to build up the dip-bump structure. Mach-cone-type models: • Collective medium response suggests a simultaneous production of particles in <0 and >0 regions. • Less event-by-event fluctuation about =0 is expected. This leads to the second part my talk, where the implication of these two event- by-event descriptions will be explored.
Part II. Use of background-suppressed measures to analyze away-side distribution (Chiu&Hwa nucl-th/0605054) Factorial Moment (FM) FM of order q: fq= (1/M)j nj(nj-1)..(nj -q+1), • only terms with positive last factor contribute to the sum. NFM: Fq= fq / (f1)q. Theorem: Ideal statistical limit (Poisson-like fluctuation, large N limit) • Fq’s 1, for all relevant q’s and M’s. A sample bg-event An event: N pcles in M bins Factorial moment of order 1 is the avg-multiplicity-per-bin: f1= N/M = (1/M) j nj (red line). Fq’s & event averaged <Fq>’s are basic bg-suppressed measures
A toy model to illustrate the use of FM-method Signal is definedas a cluster of several particles spread over a small -interval. We will loosely refer it as a “jet”. 3 types of events • bg: Particles randomly distributed in the full -range of interest. • bg+1j: 1j is randomly distributed over the range indicated. It mimics parton-ms model, i.e. it takes large fluctuations about =0 to build the 1j-spectrum. • bg+2j: The 2j-spectrum shown is symmetric about =0. It meant to mimic Mach–cone-type models. 1j: 5pcles, bg: 60 pcles bg+1j : 65 pcles bg+2j: 70 pcles
<Fq> vs M plots for q= 2, 3, and 4. • Bg events: <Fq>~1, independent on M and q values. • bg+1j, bg+2j events: For q>2, deviations from unity becomes noticeable. Increase of M and q, lead to further increase in <Fq>.
Measurement of fluctuations between two -regions The 2 regions could be I: <0, and II: >0. Difference: FI-FII measures fluctuation. Introduce <D(p,q)> =<(FqI-Fq||)p>. Here raising to the pth power further enhances the measure. To track the relative normalization, one also needs the corresponding sums: <S(p,q)> =<(Fq| +Fq||)p>. Now one can look at features in D vs S plots.
<D(p,q)> vs <S(p,q)> plots Common pattern: • bg: well localized and suppressed. • bg+1j, bg+2j: fanning out with distinct slopes forpts:M=20,30,40,50 <D> vs <S> plotscan be used to distinguish: bg+1j parton-ms model bg+2j Mach-cone-type models These plots are obtained without bg subtraction!
FM-measures which contain -dependent information can also be constructed using the 2-regions approach. Use parameter c to setup two regions: region I(c): <|c| region II(c): >|c | Determine Bq=<Dq >/<Sq. The curve of Bq vs c contains information on -dependence of the signal. II I II -c c
Conclusion (part II) We have investigated FM-method to analyze away-side -distribution. Advantages in using FM-measures. • They are insensitive to statistical fluctuation of bg. • Sensitive to “jet” (localized cluster)-signal. • No explicit bg subtraction is needed. We suggest that FM-method has the potential to provide a common framework to compare results from different experiments and various subtraction schemes.
Event-average of NFM: <Fq> Fq of the bg example (a): F3 vs i, for 500 events. • Event-avg line: <F3> ~ 1 • Fluctuations about the line (b): Distributions of Fq’s • dN/F3 vs F3 (red) • dN/F2 vs F2 (blue) • Width of the dispersion curve increases with q. • In Poissonian large N limit the width 0. Background Events (b) (a) Event-Avrage over i=1,2,..Nevt <Fq > = iFq(i) /Nevt
Bq of bg+1j case for different -peak structure (a) [i], [j], [k] cases: 1j+bg Only 1j part is shown. bg: [i]=20, [j]=2,[k]=0.2 (b) B4 for [i], [j], [k] Case [j]: Red Curve (c): Bg+1j: low plateau on a high bg. (d) Corresponding 1-B4 vs c curve has the features of broad peak in (a) and large background in (c). Bg+1j 1j Bg+1j Bg+1j Signal/Noise ratios of [i], [j] and [k]: Bg=20, 2, 0.2, S/N ~ 1% , ~10%, ~100%.