# Presentation: 369996

## I show how to express the mean-pt fluctuation measure at    full STAR acceptance scale as an integral of the two-particle    number correlations on pt x pt space.  This relates the measure   defined in STAR paper nucl-ex/0308033, “Event-wise <pt>   fluctuations in Au+Au collisions at sqrt{s_NN} =130 GeV,”   to Eq.(1) in STAR paper nucl-ex/0408012, “Two-particle   correlations on transverse momentum and minijet dissipation   in Au+Au collisions at sqrt{s_NN} =130 GeV.”

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

**Relating Fluctuations and Correlations – PART I**

- The general steps are sketched out first.  For those interested in
-    understanding the precise algebraic steps I present a pedagogical
-    derivation in a following Appendix.

- Tutorial 4

## we need to re-express 2pt:n in terms of sums over pairs of particles:

- we introduced a new measure of non-statistical event-wise fluctuations
- in mean transverse momentum based on the difference between the
- total variance and that expected when there are no dynamical correlations:

*(see Appendix – slide 7 for definition of symbols;*
*overline denotes event-wise average)*

- In order to relate this variance difference quantity to two-particle correlations
- we need to re-express 2*pt:n* in terms of _sums over pairs of particles_:

*(see Appendix*
*for complete*
*algebraic details)*

- This is the first line in Eq.(1) in nucl-ex/0408012.

- Tutorial 4

## Bin k, ptk

- Bin *k, p**tk*

- Bin *l*
*p**tl*

*Each pair of particles from *
*each event is represented by *
*a single point in the pt1 vs pt2 space.*

- The sum of pairs in event *j*,
- averaged over events...

- is approx. by a sum over bins,
- averaged over events, where
*n**sib*=#real pairs in 2D bin *(k,l).*

- The latter, avg. number of sibling pairs
- in bin *(k,l)* is identified with sibling pair
- density times 2D bin area.

*Fill this binned space*
*with all pairs of*
*accepted particles*
*using all events in*
*the centrality bin.*

- Tutorial 4

## Bin k, ptk

- Bin *l*
*p**tl*

- Bin *k, p**tk*

- Similarly, relate the inclusive (mean-*pt)**2* term to a sum over mixed-event pairs and the
- mixed pair density, which serves as the uncorrelated two-particle reference density:

*Fill this binned space*
*with all pairs of*
*accepted particles*
*using mixed-events *
*within the centrality bin.*

- Mixed-event avg. (double overlines)
- of sum over 2D bins, where
*n**j,k*=#particles in bin *k.*

- The latter, avg. number of mixed pairs
- in bin *(k,l)* is identified with mixed pair
- density times 2D bin area.

- Tutorial 4

## Combining these two parts gives the relationship between mean-pt fluctuation variance excess measure 2pt:n and two-particle correlations in Eq.(1) of nucl-ex/0408012 given by:

- All the steps and details are given in the following Appendix.

- Note that we usually do not bin the two-particle densities directly in
*p**t*, but rather use a mapping from *p**t* to *X(p**t**)* in order to achieve approx.
- uniform statistics in the bins.  Also, in the future we plan to use transverse rapidity, *y**t*, which is another mapping, in order to optimally display the transverse string fragmentation dynamics, analogous to that in Lund string fragmentation models along the beam axis.

- Tutorial 4

## Appendix

- Definition of symbols
- Manipulation of 2*pt:n* into sums over pairs of particles
- Derivation of lines 2 and 3 in Eq.(1) of nucl-ex/0408012.

- Tutorial 4

## Definition of Symbols:

- Tutorial 4

## Manipulation of 2pt:n into sums over pairs of particles

- Tutorial 4

## First, I show how the mean-pt variance excess measure in Eq.(2) of STAR paper

- nucl-ex/0308033 can be manipulated into sums over pairs of particles from the
- same events (sibling pairs) and from mixed events (mixed pairs):

- [Eq.(2) in nucl-ex/0308033]

- STAR
- variance
- excess
- measure

*(separate true*
*pairs from “self” pairs)*

*(expand symbols)*

*(write out*
*the squares)*

- Tutorial 4

## continued,

*(then collect*
*them)*

*(write out*
*all the terms)*

*(define)*

- Tutorial 4

## This term vanishes exactly if mean-pt2

- This term vanishes exactly if mean-*p**t**2*
- is not correlated with *n**j*.  For STAR applications
- this term is small compared to differences
- between the first two terms and will be neglected.

- This is the first
- line in Eq.(1) in
- nucl-ex/0408012

- Tutorial 4

## Derivation of lines 2 and 3 in Eq.(1) of nucl-ex/0408012.

- Tutorial 4

## ratio in two-dimensional pt x pt space I will explain how the correlation

- can be related to the two-particle correlation.  Starting with this measured
- ratio in two-dimensional *p**t **x p**t* space I will explain how the correlation
- density and the combination of sums over sibling and mixed pairs on
- the preceding pages can be related.

- Represent event averaged,
- bin-wise sibling pair fraction
- with integral over 2D sibling
- pair density, *sib*.

- Fraction of
- total sibling
- pairs in bin *(k,l)*

- Fraction of
- total mixed
- pairs in bin *(k,l)*

*where ****k** , ****p**t** are the p**t** bin size and acceptance*

- Normalized
- ratio of
- sibling-to-mixed
- particle pair
- densities

- Tutorial 4

## bin sizes pt, and introduce

- average density;
- assume uniform
- bin sizes *pt*, and introduce
- bin momentum *p**t,k*:

- Similarly, represent the event
- averaged, bin-wise mixed pair
- fraction as integral over *mix*
- and define _bin-wise average_.

- _Normalize_ the sibling and
- mixed-event densities
- to the event-averaged
- number of pairs;
- express histogram ratio
- in terms of densities:

- Tutorial 4

## with sums over 2D pt x pt bins we get, 

- with sums over 2D *p**t* x *p**t* bins we get,

- This term is
- of order 1/
- relative to the
- leading term;
- neglect for large
- event samples.

*In the limit of very small p**t** bins*

- This is the second line in Eq.(1) in nucl-ex/0408012

- (factor out
*p**tk**p**tl*)

- (separate sums over different and same events)

- Tutorial 4

## Using the definition of histogram ratio r, evaluating (mean pt)2

- using the mixed density, and using the density normalization
- definition, the final expression is written compactly as follows:

- This is the third
- line in Eq.(1) in
- nucl-ex/0408012

- The last two equations summarize the integral relation between correlations
- and fluctuation measures used in nucl-ex/0408012 for full acceptance scale.
- _Similar derivations can be applied to any other binned quantity._

- Tutorial 4