# Presentation: 546617

## Measuring Histograms of the Number of Pairs of Particles, Constructing Ratios of Histograms, and the relationship to Correlations – PART 1

- Define two-particle correlations and density ratios of the number of pairs of particles.
- Describe sibling and mixed pairs of particles and number of pair densities and histograms.
- Application to  space

**Technical issues including:**
-     - Kinematic projections
-     - Normalization
-     - Charge dependence
-     - Notation conventions

*Lanny Ray*
- STAR Event Structure Physics Working Group
- August 2004

_**OUTLINE**_

- Tutorial 2

## Correlations & Density Ratios for the Number of Pairs of Particles:

**Correlations & Density Ratios for the Number of Pairs of Particles:**

- Two-particle
- correlation in
- momentum space

- Two-particle
- density

- Single-particle densities:
- product equals
- _uncorrelated _
- two-particle density

- density of “sibling” pairs - each pair
- of particles is taken from the same
- event, hence the term *sibling*; pairs
- from many events are summed.

- density of “mixed” pairs - one particle for a mixed-
- event pair is taken from one event and the second
- particle is taken from a different, but similar event;
- corresponding pairs from many events are summed.

**mixed pair**

**sibling pair**

- Tutorial 2

## Correlations & Density Ratios for the Number of Pairs of Particles:

*Define ratio of *
*densities of the*
*number of pairs*
*of particles:*

**Correlations & Density Ratios for the Number of Pairs of Particles:**
**(continued)**

- Tutorial 2

## 

**Projections onto ******** - or – **_***Axial***_** momentum space**

- Sort events within a centrality bin according to:
-  (1) Location of primary vertex along beam line (z-axis)
-  (2) Number of primary particles (*N**ch*) within *p**t**,***acceptance.
- For all accepted particles in this subset of events compute
- () and () for every sibling and mixed pair.

*beam *
*axis*

- Where we define:
- 
- 
- (see last slide in this tutorial)

- Compute histograms* *for the number of sibling and mixed pairs
- of particles using only those subsets of events within 7.5 cm along the
- z-axis and *N**ch* within 50.  Repeat for all event subsets in a centrality bin.

- Tutorial 2

## Measuring the Ratio of Histograms of Number of Pairs of Particles:

**Projections onto ******** – or - **_***Axial***_** momentum space**
**(continued)**

- Azimuth angle 

- Pseudorapidity 

*Event 1*

*Event 2*

- Sum the number of sibling pairs
- in each bin in (vs
- for all events in all subsets

- Similarly, sum the number of
- mixed-event pairs in each bin in
- (vsfor all events
- in all subsets

- This is an example of projecting the six-dimensional correlation distribution
- onto two-dimensional, relative pair-wise momentum coordinates.
- We refer to such projections as the **joint **(*i.e.* two things) – **autocorrelation**
- (*i.e.* on relative coordinates)

- Define the
- bin-by-bin ratio:

- Tutorial 2

## Normalized Ratio of Histograms - 

- Total number of sibling pairs for all
- events in the centrality bin.
- Total number of mixed-event pairs used among
- the “similar” events within each *Z**vertex* and *N**ch*
- bin, summed over all such subsets of events
- in the centrality bin.

- Tutorial 2

## Charge Independent and Charge Dependent Ratios:

- Define normalized ratios of histograms for the four combinations
- of charge signs (++,--,+-,-+), for like-sign (LS) pairs and unlike-sign
- (US) pairs, and for all charge-sign pairs as in the preceding discussion.
- We have verified that                                          within statistics.

- Charge independent (CI) normalized ratios of histograms are
- computed exactly as described in the preceding slides using
- all charged particles.
- Charge dependent (CD) normalized ratios of histograms are
- computed by applying the steps described in the preceding
- slides for all like-sign pairs of charged particles (++ and --)
- and for all unlike-sign pairs of charged particles (+- and -+)
- and forming the difference:

- Tutorial 2

## Why does the Event Structure PWG use x, rather than

- Why does the Event Structure PWG use ***x*********, rather than
- the more common *********x*** for difference quantity ***x******1******-x******2***?
- The reason is related to the fact that we often refer to
- rotated coordinate systems which take advantage of the
- observed symmetries in the two-point correlation data.
- Rotated axes lie along the sum **(*****x******1******+x******2*****)**
- and difference ***(x******1******-x******2******)*** directions.
- Notation ***x************,x********* follow naturally.
- On the other hand definition
*********x = x******1******-x******2*** calls for the correspond-
- ing notation** “***********x”*** which would be
- confused with summation.  For
- example the *********1*** versus *********2*** two-point
- ratio data are shown here along with
- the ************************coordinate system.

************

************

************

************

- Tutorial 2

## Application of these methods in STAR papers:

-  “**Hadronization geometry and charge-dependent number **
**   ****autocorrelations on axial momentum space in Au-Au **
**   ****collisions at sqrt(s****NN****) = 130 GeV,”**   *Submitted June 30, 2004*   e-Print Archives (nucl-ex/0406035).
-  “**Minijet deformation and charge-independent particle **
**   ****correlations on axial momentum space in Au-Au **
**   ****collisions at sqrt(s****NN****) = 130 GeV,”**
**   **Presently on starpapers-l** ***(August, 2004).*
-  “**Transverse momentum correlations and minijet dissipation **
**    ****in Au-Au collisions at sqrt(sNN) = 130 GeV,”**    *Submitted August 11, 2004*    e-Print Archives (nucl-ex/0408012)

- Tutorial 2