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

The dip-bump structure in the away-side distribution

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.

Jia (PHENIX nucl-ex/0510019)Au+Au, 0-5%

(2.5-4) (1-2.5) GeV/cDip-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.

Exit

x

x

x

x

x

Trigger

Recoil

Part I. Simulation based on a parton multiple-scattering model

(Chiu and Hwa, preliminary)

Simulation results: ptrigger=4.5

Sample tracks: Superposition of many events, 1 track per event.

(a) Exit tracks: When successive steps are bending away from the center, the

track length is shorter, is likely to get out. (b) Absorbed tracks: When successive steps swing back and forth, the track

length is longer, more energy loss. The track is likely to be absorbed. (c) 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.

(c)(a) (b)

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.

STAR nucl -exp 0604108

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

Factorial moment of order 1 is the avg-multiplicity-per-bin:

f1= N/M = (1/M) j nj (red line).

An event: N pcles in M bins

Fq’s & event averaged <Fq>’s are basic bg-suppressed measures

A toy model to illustrate the use of FM-method

Signal is defined as 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 beI: <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 for pts:M=20,30,40,50

<D> vs <S> plots can be used to distinguish: bg+1j parton-ms modelbg+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.

-c c

I IIII

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.

(b)

Event-Avrage over i=1,2,..Nevt

<Fq > = i Fq(i) /Nevt

Background Events

(a)

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+1jBg+1j

Bg+1j

1j

Signal/Noise ratios of [i], [j] and [k]:

Bg=20, 2, 0.2, S/N ~ 1% , ~10%, ~100%.