Post on 18-Jan-2018
description
Scott PrattScott Pratt Michigan State UniversityMichigan State University
Femtoscopy: Theory____________________________________________________ Scott Pratt, Michigan State University
Scott PrattScott Pratt Michigan State UniversityMichigan State University
Deriving the Fundamental Formula
1)()(1),( 23 rrSrdqPC qP
2/)(
)()(),(),(
ba
ba
ba
ba
ppqppP
pNpNppNqPC
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
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
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
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),
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
*
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
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
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
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
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
Scott PrattScott Pratt Michigan State UniversityMichigan State University
Coulomb Interactions
•Can be calculated classically for larger fragmentsKim et al., PRC45 p. 387 (92)
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)
Scott PrattScott Pratt Michigan State UniversityMichigan State University
Measuring shape without identical particles
Scott PrattScott Pratt Michigan State UniversityMichigan State University
Example: pK+ correlations
Gaussian Sources:Rx=Ry=4, Rz=8 fm
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
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
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
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
Scott PrattScott Pratt Michigan State UniversityMichigan State University
Change to Explosive Behavior (GAS) at ~ 50 AMeV
Scott PrattScott Pratt Michigan State UniversityMichigan State University
Experimental Signatures
Dramatic change in nn correlations
~ 500 fm/c ~ 50 fm/c
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
Scott PrattScott Pratt Michigan State UniversityMichigan State University
Phase Transition at RHIC?
Stiffer EOS -> Smaller source sizesData demonstrate no latent heat or significant softness
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
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
Scott PrattScott Pratt Michigan State UniversityMichigan State University
Phase space
density
<f> rises until threshold of chemical equilibrium ~ 80 MeV at break-up
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…
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
Scott PrattScott Pratt Michigan State UniversityMichigan State University
SummaryCorrelations CRUCIAL for determining
• Pressure
• Entropy
• Reaction Dynamics