scott prattmichigan state university femtoscopy: theory...

Scott PrattScott Pratt Michigan State UniversityMichigan State University

Femtoscopy: Theory____________________________________________________ Scott Pratt, Michigan State University


Deriving the Fundamental Formula

1)()(1),( 23 rrSrdqPC qP

2/)(

)()(),(),(

ba

ba

ba

ba

ppqppP

pNpNppNqPC



),(

)2/()2/(),(

)(

43

4

243

xpxsdpd

dN

xxTxxTxdxps

exTxdpd

dN

aaa

aa

xipaa

a

aa

Step 1: Define the source function



xqiqqq

baqbbaa

baqxxiP

baba

exxuxxuxdxqw

xxqwxqPsxqPsqd

xxuexTxTpdpd

dN ba

~*4

4

2)(

33

)2/()2/(),~(

),~(),~2/(),~2/(~

)()()(

Step 2: Write 2-particle probability

),~( xqwq = probability relative momentum q and separation x evolves to q asymptotically


Deriving… Identical particles

)~()~(),~(,2/)( 4 qeqqxqweexu xiqq

iqxiqxq

)(2cos),2/(),2/(),(),(

2121

222

41

433 xxqxPsxPs

xpsxpsxdxdpdpd

dN ba

ba

),2/),2/((),2/),2/((),(),(),2/(),2/(

21

2121xPPEsxPPEs

xpsxpsxPsxPs ba

Smoothness approximation

)2cos()(1),( 3 rqrSrdqPC P

)(),2/(),2/()( ''344 rxxxPsxPsxdxdrS babbaabaP


With final-state interactions

)(),( xxtu qq

),2/),2/((),2/),2/((),~2/(),~2/( 2121 xPPEsxPPEsxqPsxqPs

Smoothness approximation


Approximate (in frame of pair),


Deriving… Summary


Assumptions

Identical Particles1. Symmetrize pairwise2. Independent emission3. Smoothness

Strong/Coulomb1. Independent

emission2. Ignore time

differencefor evolution

3. Smoothness

*

*

* Tested

*


Femtoscopy – Theory

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


Hadronic Interferometry – Theory

Theories predict SP(r) C(P,q)

Correlations provide stringent test of space-time evolution


Using Identical Particles

•Examples: , KK, …•Easy to invert•3-dimensional information

• Rout, Rside, Rlong are functions of P

2

2

2

2

2

2

222exp~)(

long

long

side

side

out

out

Rr

Rr

Rrrs

rQrq cos1),(2

222222exp1)( longlongsidesideoutout RQRQRQQC


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

2222)( sidewardoutwardsource RRvv


Strong Interactions

Peak height determined by scattering length or resonance width

•Examples: pp, p, nn, p, Kp, p, d, …

d Correlations

E (MeV)

G. Verde / MSU Miniball Group


Coulomb Interactions

•Can be calculated classically for larger fragmentsKim et al., PRC45 p. 387 (92)


Proton-protonCorrelations

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:Rx=Ry=4, Rz=8 fm


Detailed Shape Information

),()(4)(

)(cos1)cos,,(cos21),(

),()()(),(1)()(

)(1)cos,,(1)(

,2

,

2,,

,,

23

rqKrSdrrqC

PrqdrqK

YrSdrSYqCdqC

rSrqrdqC

mm

qrqrqr

rmrm

qmqm

qr

Standard formalism:

Defining,

Using identities for Ylms,

Simple correspondence! Danielewicz and Brown


Moments• L=0

• L=1, M=1

• L=2, M=0,2

• L=3, M=1,3

Angle-integrated shape

Lednicky offsets

Shape (Rout/Rside, Rlong/Rside)

Boomerang distortion


Blast Wave Model

• (z -z) CL+M=even(q) = 0

• (y -y) Imag CL,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

• 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 sizesData 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

33

2/3

)2/(32/33

3

1)12(

)(

)2(1

)12()2(),(

22

invC

Rr

inv

RpddN

Jpf

eRpd

dNJ

rpf inv

Any method to extract Rinv is sufficient


Phase space

density

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


HBT and Entropy

)1ln()1(ln)2( 3

33ffffrpddS

Entropy can be determined from averagePhase space density

<f> determined from:• correlations ()• coalescence (KK,ppd)• thermal models…


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

S.Pal and S.P., PLB 2003

4500,1900 dy

dSdydS total

hydroBjorken


SummaryCorrelations CRUCIAL for determining

• Pressure

• Entropy

• Reaction Dynamics

scott prattmichigan state university femtoscopy: theory...

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