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11-14. 09. 2008 WPCF'08 Krakow 1
Femtoscopic Correlations of Nonidentical Particles
R. Lednický, JINR Dubna & IP ASCR Prague
• History
• Assumptions
• Correlation asymmetries
• Conclusion
2
History
Fermi’34: e± Nucleus Coulomb FSI in β-decay modifies the relative momentum (k) distribution → Fermi (correlation) function F(k,Z,R) is sensitive to Nucleus radius R if charge Z » 1
measurement of space-time characteristics R, c ~ fm
Correlation femtoscopy :
of particle production using particle correlations
4
Modern correlation femtoscopy formulated by Kopylov & Podgoretsky
KP’71-75: settled basics of correlation femtoscopyin > 20 papers
• proposed CF= Ncorr /Nuncorr &
• showed that sufficiently smooth momentum spectrum allows one to neglect space-time coherence:
(for non-interacting identical particles)
mixing techniques to construct Nuncorr
• clarified role of space-time characteristics in various models
|∫d4x1d4x2p1p2(x1,x2)...|2 → ∫d4x1d4x2p1p2(x1,x2)|2...
5
QS symmetrization of production amplitude momentum correlations of identical particles are
sensitive to space-time structure of the source
CF=1+(-1)Scos qx
p1
p2
x1
x2
q = p1- p2 → {0,2k} x = x1- x2 → {t,r}
nnt , t
, nns , s
2
1
0 |q|
1/R0
total pair spin
2R0
KP’71-75
exp(-ip1x1)
CF → |S-k(r)|2 = | [ e-ikr +(-1)S eikr]/√2 |2
6
Final State InteractionSimilar to Coulomb distortion of -decay Fermi’34:
e-ikr -k(r) [ e-ikr +f(k)eikr/r ]
eicAc
F=1+ _______ + …kr+krka
Coulomb
s-wavestrong FSIFSI
fcAc(G0+iF0)
}
}
Bohr radius}
Point-likeCoulomb factor k=|q|/2
CF nnpp
Coulomb only
|1+f/r|2
FSI is sensitive to source size r and scattering amplitude fIt complicates CF analysis but makes possible
Femtoscopy with nonidentical particles K, p, .. &
Study relative space-time asymmetries delays, flow
Study “exotic” scattering , K, KK, , p, , ..Coalescence deuterons, ..
|-k(r)|2Migdal, Watson, Sakharov, … Koonin, GKW, ...
Assumptions to derive “Fermi” formula for CF in multiparticle production
CF = |-k*(r*)|2
- tFSI tprod |k*| = ½|q*| hundreds MeV/c
- equal time approximation in PRF
typical momentum transfer
RL, Lyuboshitz’82 eq. time condition |t*| r*2 OK fig.
RL, Lyuboshitz ..’98
+ 00, -p 0n, K+K K0K0, ...& account for coupled channels within the same isomultiplet only:
- two-particle approximation (small freezeout PS density f)~ OK, <f> 1 ? low pt fig.
- smoothness approximation: |p| |q|peak Remitter Rsource
~ OK in HIC, Rsource2 0.1 fm2 pt
2-slope of direct particles
8
Phase space density from CFs and spectra
Bertsch’94
May be high phase space density at low pt ?
? Pion condensate or laser
? Multiboson effects on CFsspectra & multiplicities
<f> rises up to SPSLisa ..’05
BS-amplitude
For free particles relate p to x through Fourier transform:
Then for interacting particles:Product of plane waves -> BS-amplitude :
Production probability W(p1,p2|Τ(p1,p2;)|2
10
Smoothness approximation: rA « r0
p1
p2
x1
x2
2r0
W(p1,p2|∫d4x1d4x2 p1p2(x1,x2) Τ(x1,x2;)|2
x1’x2’≈ ∫d4x1d4x2 |p1p2(x1,x2)|2 G(x1,p1;x2,p2)
r0 - Source radius
rA - Emitter radius p1p2(x1,x2)p1p2*(x1’,x2’)
Τ(x1,x2 ;)Τ*(x1’,x2’ ;)
G(x1,p1;x2,p2)
= ∫d4ε1d4ε2 exp(ip1ε1+ip2ε2)
Τ(x1+½ε1,x2 +½ε2;)Τ*(x1-½ε1,x2-½ε2;)
Source function
= ∫d4x1d4x1’d4x2d4x2’
11
Effect of nonequal times in pair cmsRL, Lyuboshitz SJNP 35 (82) 770; RL nucl-th/0501065
Applicability condition of equal-time approximation: |t*| r*2 r0=2 fm 0=2 fm/c r0=2 fm v=0.1
OK for heavy
particles
OK within 5%even for pions if0 ~r0 or lower
→
12
NA49 central Pb+Pb 158 AGeV vs RQMDLong tails in RQMD: r* = 21 fm for r* < 50 fm
29 fm for r* < 500 fm
Fit CF=Norm [Purity RQMD(r* Scaler*)+1-Purity]
Scale=0.76 Scale=0.92 Scale=0.83
RQMD overestimates r* by 10-20% at SPS cf ~ OK at AGS worse at RHIC
p
13
Correlation asymmetries
CF of identical particles sensitive to terms even in k*r* (e.g. through cos 2k*r*) measures only
dispersion of the components of relative separation r* = r1
*- r2* in pair cms
CF of nonidentical particles sensitive also to terms odd in k*r* measures also relative space-time asymmetries - shifts r*
RL, Lyuboshitz, Erazmus, Nouais PLB 373 (1996) 30
Construct CF+x and CF-x with positive and negative k*-projection
kx* on a given direction x and study CF-ratio CF+x/CFx
SUBATECH-94-20 (1994)
14
Simplified idea of CF asymmetry(valid for Coulomb FSI)
x
x
v
v
v1
v2
v1
v2
k*/= v1-v2
p
p
k*x > 0v > vp
k*x < 0v < vp
Assume emitted later than p or closer to the center
p
p
Longer tint
Stronger CF
Shorter tint Weaker CF
CF
CF
LLEN’94 (also Gelderloos et al’94, Cornell et al'96)
15
CF-asymmetry for charged particlesAsymmetry arises mainly from Coulomb FSI
CF Ac() |F(-i,1,i)|2 =(k*a)-1, =k*r*+k*r*
F 1+ = 1+r*/a+k*r*/(k*a)r*|a|
k*1/r* Bohr radius
}
±226 fm for ±p±388 fm for +±
CF+x/CFx 1+2 x* /ak* 0
x* = x1*-x2* rx* Projection of the relative separation r* in pair cms on the direction x
In LCMS (vz=0) or x || v: x* = t(x - vtt)
CF asymmetry is determined by space and time asymmetries
16
Large lifetimes evaporation or phase transitionx || v |x| |t| CF-asymmetry yields time delay
Ghisalberti’95 GANILPb+Nb p+d+X
CF+(pd)
CF(pd)
CF+/CF< 1
Deuterons earlier than protonsin agreement with coalescence
e-tp/ e-tn/ e-td/(/2) since tp tn td
Two-phase thermodynamic
model
CF+/CF< 11 2 3
1
2
3
Strangeness distillation: K earlier than K in baryon rich QGP
Ardouin et al.’99
17
ad hoc time shift t = –10 fm/c
CF+/CF
Sensitivity test for ALICEa, fm
84
226
249CF+/CF 1+2 x* /a
k* 0
Here x*= - vt
CF-asymmetry scales as - t/a
Erazmus et al.’95
Delays of several fm/ccan be easily detected
18
Usually: x and t comparable
RQMD Pb+Pb p +X central 158 AGeV : x = -5.2 fmt = 2.9 fm/cx* = -8.5 fm+p-asymmetry effect 2x*/a -8%
Shift x in out direction is due to collective transverse flow
RL’99-01 xp > xK > x > 0
& higher thermal velocity of lighter particles
rt
y
x
F
tT
t
F= flow velocity tT = transverse thermal velocity
t = F + tT = observed transverse velocity
x rx = rt cos = rt (t2+F2- t
T2)/(2tF) y ry = rt sin = 0 mass dependence
z rz sinh = 0 in LCMS & Bjorken long. exp.
out
side
measures edge effect at yCMS 0
pion
Kaon
Proton
BW Retiere@LBL’05
Distribution of emissionpoints at a given equal velocity: - Left, x = 0.73c, y = 0 - Right, x = 0.91c, y = 0
Dash lines: average emission Rx
Rx() < Rx(K) < Rx(p)
px = 0.15 GeV/c px = 0.3 GeV/c
px = 0.53 GeV/c px = 1.07 GeV/c
px = 1.01 GeV/c px = 2.02 GeV/c
For a Gaussian density profile with a radius RG and linear flow velocity profile F (r) = 0 r/ RG
RL’04, Akkelin-Sinyukov’96 :
x = RG x 0 /[02+T/mt]
0.73c 0.91c
NA49 & STAR out-asymmetriesPb+Pb central 158 AGeV not corrected for ~ 25% impurityr* RQMD scaled by 0.8
Au+Au central sNN=130 GeV corrected for impurity
Mirror symmetry (~ same mechanism for and mesons) RQMD, BW ~ OK points to strong transverse flow
pp K
(t yields ~ ¼ of CF asymmetry)
21
Decreasing R(pt): x-p correlation
• usually attributed to collective flow
• taken for granted
• femtoscopy the only way to confirm x-p correlations
x2-p correlation: yesx-p correlation: yes
Non-flow possibility• hot core surrounded by cool shell
• important ingredient of Buda-Lund hydro pictureCsörgő & Lörstad’96
x2-p correlation: yesx-p correlation: no
x = RG x 0 /[02+T/mt+T/Tr]
radial gradient of T
decreasing asymmetry ~1 ? problem
M. Lisa’07
22
Summary• Assumptions behind femtoscopy theory in HIC seem OK• Wealth of data on correlations of various particle species
(,K0,p,,) is available & gives unique space-time info on production characteristics including collective flows
• Rather direct evidence for strong transverse flow in HIC at SPS & RHIC comes from nonidentical particle correlation asymmetries (lower thermal velocities of heavier particles lead to mass hierarchy of the shifts arising due to flow)
• For femtoscopic correlations of nonidentical particles, there is no problem with two-track resolution though, in the case of very different masses a large detector acceptance is required
• Info on two-particle strong interaction: & & p scattering lengths from HIC at SPS and RHIC. Good perspective at RHIC and LHC
23
Beta decay rate
d6N/(d3p1 d3p2) = d6N0/(d3p1 d3p2) |-k(r*)|2
d5w = ∑spins ∫ d3p d3q d3k δ4(p0-p-q-k) |∫d3x Τ(x;spins) eiqx -k*(x)|2
= ∑spins ∫ d3p d3q d3k δ4(p0-p-q-k)
∫ d3x d3x’ Τ(x;spins) Τ*(x’;spins) eiq(x-x’) -k*(x) -k(x’)
p0 (A) = p(A’) - q(ν) - k(e)
∑spins ∫ d3k ∫d3x |Τ(x;spins)|2 |-k*(x)|2
Neglecting the limitation due to energy-momentum conservation
cf classical sources approximation valid in multiparticle production
∫ d3q →
24
Fermi theory of beta decay
d5w = ∑spins ∫ d3p d3q d3k δ4(p0-p-q-k) |∫d3x Τ(x;spins) eiqx -k*(x)|2
≈ ∑spins ∫ d3p d3q d3k δ4(p0-p-q-k) |-k(x)|2 |∫d3x Τ(x;spins)|2
ρ
25
Conclusion from beta decay
• Formally (FSI) correlations in beta decay and multiparticle production are determined by the same (Fermi) function |-k(x)|2
• But this factor appears for different reasons in beta decay (a weak r-dependence of -k(r) within the nucleus volume + point like emission) and in multiparticle production (valid classical particle sources approximation + small source sizes compared to their separation in HIC)
26
Grassberger’77: fire sausage
Dispersion of emitter velocities & limited emission momenta (T) x-p correlation: correlation dominated by pions from nearby emitters
besides geometry, femtoscopy probes source dynamics - expansion
27
Correlation study of particle interaction
-
+& & p scattering lengths f0 from NA49 and STAR
NA49 CF(+) vs RQMD with SI scale: f0 sisca f0 (=0.232fm)
sisca = 0.60.1 compare ~0.8 fromSPT & BNL data E765 K e
Fits using RL-Lyuboshitz’82
NA49 CF() data prefer
|f0()| f0(NN) ~ 20 fm
STAR CF(p) data point to
Ref0(p) < Ref0(pp) 0
Imf0(p) ~ Imf0(pp) ~ 1 fm
pp