Scott PrattMichigan State University Femtoscopy: Theory...
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![Page 1: Scott PrattMichigan State University Femtoscopy: Theory ____________________________________________________ Scott Pratt, Michigan State University.](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b877f8b9ab0599bd273/html5/thumbnails/1.jpg)
Scott PrattScott Pratt Michigan State UniversityMichigan State University
Femtoscopy: Theory____________________________________________________ Scott Pratt, Michigan State University
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Deriving the Fundamental Formula
1)()(1),( 23 rrSrdqPC qP
2/)(
)()(),(),(
ba
ba
ba
ba
ppqppP
pNpNppNqPC
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Deriving the Fundamental Formula
),(
)2/()2/(),(
)(
43
4
243
xpxsdpd
dN
xxTxxTxdxps
exTxdpd
dN
aaa
aa
xipaa
a
aa
Step 1: Define the source function
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Deriving the Fundamental Formula
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
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
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
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
With final-state interactions
)(),( xxtu qq
),2/),2/((),2/),2/((),~2/(),~2/( 2121 xPPEsxPPEsxqPsxqPs
Smoothness approximation
1)()(1),( 23 rrSrdqPC qP
Approximate (in frame of pair),
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Deriving… Summary
1)()(1),( 23 rrSrdqPC qP
Assumptions
Identical Particles1. Symmetrize pairwise2. Independent emission3. Smoothness
Strong/Coulomb1. Independent
emission2. Ignore time
differencefor evolution
3. Smoothness
*
*
* Tested
*
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Femtoscopy – Theory
•Measures phase space cloud for fixed velocity•Overall source can be larger•Inversion depends on |(q,r)|2
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Hadronic Interferometry – Theory
Theories predict SP(r) C(P,q)
Correlations provide stringent test of space-time evolution
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
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
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
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
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
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
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Coulomb Interactions
•Can be calculated classically for larger fragmentsKim et al., PRC45 p. 387 (92)
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Proton-protonCorrelations
Deconvoluting C(q) provides
detailed source shape
S.Panitkin and D.Brown, PRC61 021901 (2000)
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Measuring shape without identical particles
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Example: pK+ correlations
Gaussian Sources:Rx=Ry=4, Rz=8 fm
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
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
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
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
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Blast Wave Model
• (z -z) CL+M=even(q) = 0
• (y -y) Imag CL,M = 0
S.P. and S.Petriconi, PRC 2003
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Liquid-Gas Phase Transition
Definition of Gas:“Expands to fill
available volume”
Liquid = Evaporation Long lifetimes
Gas = Explosion Short lifetimes
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Change to Explosive Behavior (GAS) at ~ 50 AMeV
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Experimental Signatures
Dramatic change in nn correlations
~ 500 fm/c ~ 50 fm/c
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
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
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Phase Transition at RHIC?
Stiffer EOS -> Smaller source sizesData demonstrate no latent heat or significant softness
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
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
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
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
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Phase space
density
<f> rises until threshold of chemical equilibrium ~ 80 MeV at break-up
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
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…
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
Entropy for 130 GeV Au+Au at = 1 fm/c
S.Pal and S.P., PLB 2003
4500,1900 dy
dSdydS total
hydroBjorken
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Scott PrattScott Pratt Michigan State UniversityMichigan State University
SummaryCorrelations CRUCIAL for determining
• Pressure
• Entropy
• Reaction Dynamics