# Presentation: 543651

## Femtoscopy: Theory

**____________________________________________________**
**                    ****Scott Pratt, Michigan State University**

## Deriving the Fundamental Formula

## Deriving the Fundamental Formula

**Step 1: Define the source function**

## Deriving the Fundamental Formula

**Step 2: Write 2-particle probability**

**= probability relative momentum q and separation x evolves to q asymptotically **

## Deriving... Identical particles

**Smoothness approximation**

## With final-state interactions

**Smoothness approximation**

**Approximate (in frame of pair),**

## Deriving... Summary

**Assumptions**

**Identical Particles**
**Symmetrize pairwise**
**Independent emission**
**Smoothness**

**Strong/Coulomb**
**Independent emission**
**Ignore time difference****for evolution**
**Smoothness**

*****

*****

*** Tested**

*****

## Femtoscopy – Theory 

**Femtoscopy – Theory **

**Measures phase space cloud for fixed velocity**
**Overall source can be larger**
**Inversion depends on**** *****|**********(q,r)|******2***

## Hadronic Interferometry – Theory 

**Hadronic Interferometry – Theory **

**Theories predict **
** *****S******P******(r)************ C(P,q)***

**Correlations provide ****stringent test of ****space-time evolution**

## Using Identical Particles 

**Examples: ********, ****KK, ...**
**Easy to invert**
**3-dimensional information**** **

*** ******R******out******, R******side******, R******long****** *****are functions of***** P***

## Identical Particles: Measuring Lifetime 

**Has been studied for ********, KK, pp, nn**
**Source function *****S(p,r,t)***** is 7-dimensional****  ****–**** requires one dimension of common sense**

## Strong Interactions 

**Peak height determined by ****	****	****scattering length or resonance width**
**Examples: pp, p********, nn, p********, Kp, p********, d********, ...**

******E (MeV)**

**G. Verde / MSU **
**Miniball Group**

## Coulomb Interactions 

## Proton-protonCorrelations

**Deconvoluting *****C(q)***** provides**
**detailed source shape**

**S.Panitkin and D.Brown, PRC61 021901 (2000)**

## Measuring shape without identical particles

## Example: pK+ correlations

**Gaussian Sources:**
**R****x****=R****y****=4, R****z****=8 fm**

## Detailed Shape Information

**Standard formalism:**

**Defining,**

**Using identities for Y****lm****s,**

## Moments

** ****L=0**** **
** ****L=1, M=1**** **
** ****L=2, M=0,2**** **
** ****L=3, M=1,3**

**Angle-integrated shape**

**Lednicky offsets**

**Shape (R****out****/R****side****, R****long****/R****side****)**

***Boomerang *****distortion**

## Blast Wave Model

**Blast Wave Model**

** ****(z ******** -z)****  C****L+M=even****(q) = 0**** **
** ****(y ******** -y)****  Imag C****L,M**** = 0**

**S.P. and S.Petriconi, PRC 2003**

## Liquid-Gas Phase Transition

**Definition of Gas:**
***“******Expands to fill available volume”***
**Liquid = Evaporation**
**	******** ****Long lifetimes**
**Gas = Explosion**
**	******** ****Short lifetimes**

## Change to Explosive Behavior (GAS) at ~ 50 AMeV

## Experimental Signatures

**	****Dramatic change in *****nn***** correlations**

****** ****~ 500 fm/c**

****** ****~ 50 fm/c**

## Phase Transition at RHIC

**Phase Transition at RHIC**

** ****Transparency complicates the problem**
** ****For complete stopping, times could be ~ 100 fm/c**
** ****For Bjorken, strong first-order EOS leads to ****   ******** ~ 20 fm/c**

## Phase Transition at RHIC?

**Stiffer EOS -> Smaller source sizes**
**Data demonstrate no latent heat or significant softness**

## THE HBT PUZZLE AT RHIC

**To fit data:**
**a) Stiff (but not too stiff) EOS**
**b) ****	****Reduce emissivity from surface**
**c) Not that much different than SPS**

## Phase space density

## Phase space density

**<f> rises until threshold of chemical equilibrium**
** ************ ****~ 80 MeV at break-up**

## HBT and Entropy

**Entropy can be determined from average**
**Phase space density**

**<f> determined from:**
** ****correlations (********)**
** ****coalescence (KK********,pp********d)**
** ****thermal models...**

## Entropy for 130 GeV Au+Au at = 1 fm/c

- S.Pal and S.P., PLB 2003

## Summary

**Correlations CRUCIAL for determining**
** ****Pressure**
** ****Entropy**
** ****Reaction Dynamics**