5.11.2013 WPCF Catania, November 5-8, 2013 1

Femtoscopic aspects of FSI, resonances and bound states

In memory of V.L.Lyuboshitz (19.3.1937-4.5.2013)

R. Lednický @ JINR Dubna & IP ASCR Prague

• History• QS correlations• FSI correlations • Coalescence Femtoscopy• Correlation study of strong interaction• Correlation asymmetries• Summary Femtoscopy on marsh in Nantes’94

RL VLL MIP

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

3

Fermi function(k,Z,R) in β-decay

= |-k(r)|2 ~ (kR)-(Z/137)2

Z=83 (Bi)β-

β+

R=84 2 fm

k MeV/c

4

2xGoldhaber, Lee & Pais GGLP’60: enhanced ++ , -- vs +- at small opening angles – interpreted as BE enhancement depending on fireball radius R0

R0 = 0.75 fm

p p 2+ 2 - n0

5

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 at small q*

(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...

6

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 , s2

1

0 |q|

1/R0

total pair spin

2R0

KP’71-75

exp(-ip1x1)

CF → |S-k*(r*)|2 = | [ e-ik*r* +(-1)S eik*r*]/√2 |2

PRF

7

Assumptions to derive KP formula

CF - 1 cos qx

- two-particle approximation (small freeze-out PS density f)

- smoothness approximation: Remitter Rsource |p| |q|peak

- incoherent or independent emission

~ OK, <f> 1 ? low pt

~ OK in HIC, Rsource2 0.1 fm2 pt

2-slope of direct particles

2 and 3 CF data approx. consistent with KP formulae:CF3(123) = 1+|F(12)|2+|F(23)|2+|F(31)|2+2Re[F(12)F(23)F(31)]CF2(12) = 1+|F(12)|2 , F(q)| = eiqx

- neglect of FSIOK for photons, ~ OK for pions up to Coulomb repulsion

Resonances as emitters

most of the produced pions come from resonance decays symmetrization accounting for resonance propagator (M2-kr

2-iM)-1

yields (in the limit << M )

CF - 1 = Reexp(-iqx)/(1+iy) 1-½[q(x+l)]2+(q l)2], y=(krq)/M (q l)

• i.e. the width of BE enhancement is determined by the source size enhanced by the resonance decay length l = kr/M

• -meson as a typical resonance decay length in PRF l*~3.3 fm• in contrast with correlation radii ~1 fm measured from qinv CFs in

p, pp or e+e- collisions• explained by a rapid decrease of the slope of the resonance factor

1/(1+y2) with increasing q 8

Grishin, Kopylov, Podgoretsky’71, Grassberger’77, RL’78, RL,Progulova’92

Resonances as emittersNote: the narrow resonance limit resonance as a classical emitter with

exponential decay law ~ exp(-t)

(used in transport codes)

However, at qinv > 0.1 GeV it may lead to a substantial overestimation of the resonance correlation factor (by ~ 15% for )

9

It may be important for the interpretation of CF data from elementary particle collisions,

though much less important for HIC (due to substantial fireball

size)

classical approx.

Meff(-0p)

-0+X

-0+X

“General” parameterization at |q| 0

Particles on mass shell & azimuthal symmetry 5 variables:q = {qx , qy , qz} {qout , qside , qlong}, pair velocity v = {vx,0,vz}

Rx2 =½ (x-vxt)2 , Ry

2 =½ (y)2 , Rz2 =½ (z-vzt)2

q0 = qp/p0 qv = qxvx+ qzvz

y side

x out transverse pair velocity vt

z long beam

Podgoretsky’83, Bertsch, Pratt’95; so called out-side-long parameterization

Interferometry or correlation radii:

cos qx=1-½(qx)2+.. exp(-Rx2qx

2 -Ry2qy

2 -Rz

2qz2

-2Rxz2qx qz)

Grassberger’77RL’78

Csorgo, Pratt’91: LCMS vz = 0

11

Probing source shape and emission duration

Static Gaussian model with space and time dispersions

R2, R||

2, 2

Rx2 = R

2 +v22

Ry2 = R

2

Rz2 = R||

2 +v||22

Emission duration2 = (Rx

2- Ry2)/v

2

f (degree)

Rsi

de2 fm

2If elliptic shape also in transverse plane RyRside oscillates with pair azimuth f

Rside (f=90°) small

Rside (f=0°) large

z

A

B

Out-of reaction plane

In reaction plane

In-planeCircular

Out

-of

plan

e

KP (71-75) …

Probing source dynamics - expansionDispersion of emitter velocities & limited emission momenta (T)

x-p correlation: interference dominated by pions from nearby emitters

Interferometry radii decrease with pair velocity

Interference probes only a part of the sourceResonances GKP’71 ..

Strings Bowler’85 ..

Hydro

Pt=160 MeV/c Pt=380 MeV/c

Rout Rside

Rout Rside

Collective transverse flow F RsideR/(1+mt F2/T)½

during proper freeze-out (evolution) time

Rlong (T/mt)½/coshy

Pratt, Csörgö, Zimanyi’90

Makhlin-Sinyukov’87

}

1 in LCMS

…..

Bertch, Gong, Tohyama’88Hama, Padula’88

Mayer, Schnedermann, Heinz’92

Pratt’84,86Kolehmainen, Gyulassy’86

Longitudinal boost invariant expansion

pion

Kaon

Proton

, , Flow & Radiiin Blast Wave model

← Emission points at a given tr. velocitypx = 0.15 GeV/c 0.3 GeV/c

px = 0.53 GeV/c 1.07 GeV/c

px = 1.01 GeV/c 2.02 GeV/c

For a Gaussian density profile with a radius RG and linear flow velocity profile F (r) = 0 r/ RG:

0.73c 0.91c

Rz2 2 (T/mt)

Rx2= x’2-2vxx’t’+vx

2t’2

Rz = evolution time Rx = emission duration

Ry2 = y’2

Ry2 = RG

2 / [1+ 02 mt /T]

Rx , Ry 0 = tr. flow velocity pt–spectra T = temperature

t’2 (-)2 ()2

BW: Retiere@LBL’05

x = RG bx 0 /[02+T/mt]

hierarchy x(p) < x(K) < x(p)

BW fit ofAu-Au 200 GeV

T=106 ± 1 MeV<bInPlane> = 0.571 ± 0.004 c<bOutOfPlane> = 0.540 ± 0.004 cRInPlane = 11.1 ± 0.2 fmROutOfPlane = 12.1 ± 0.2 fmLife time (t) = 8.4 ± 0.2 fm/cEmission duration = 1.9 ± 0.2 fm/cc2/dof = 120 / 86

Retiere@LBL’05

R

βz ≈ z/τβx ≈ β0 (r/R)

15

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, LL, ...

Assumptions to derive “Fermi-like” formula

CF = |-k*(r*)|2

tFSI (s-wave) = µf0/k* |k*| = ½|q*| hundreds MeV/c

- same as for KP formula in case of pure QS &

- equal time approximation in PRF

» typical momentum transfer in production

RL, Lyuboshitz’82 eq. time condition |t*| r*2

OK (usually, to several % even for pions) fig.

RL, Lyuboshitz ..’98

same isomultiplet only: + 00, -p 0n, K+K K0K0, ...

& account for coupledchannels within the

- tFSI = d /d dE > tprod

∫d3r {WP(r,k) + WP(r,-kn) |-k(r)|2

+ 2[WP(r,k)WP(r,-kn)]1/2 Re[exp(ikr)-k(r)]}

where -k(r) = exp(-ikr)+-k(r) and n = r/r

The usual moothness approximation: WP(r,-kn) WP(r,k) is valid if one can neglect the k-dependence of WP(r,k), e.g.

for k << 1/r0

Caution: Smoothness approximation is justified for small k << 1/r0

It should be generalized in the region k > ~100 MeV/c

CF(p1,p2) ∫d3r WP(r,k) |-k(r)|2

18

Effect of nonequal times in pair cmsRL, Lyuboshitz SJNP 35 (82) 770; RL nucl-th/0501065

Applicability condition of equal-time approximation: |t*| m1,2r*2 r0=2 fm 0=2 fm/c r0=2 fm v=0.1

OK for heavy

particles & small k*

OK within 5%even for pions if0 ~r0 or lower

→

|k*t*| m1,2r*

Using spherical wave in the outer region (r>) & inner region (r<) correction analytical dependence on scatt. amplitudes fL and source radius r0 LL’81

19

Inner region: W(r) W(0) & integral relation (single channel and no Coulomb) with the phase shifts L and momentum derivative L :

∫d3r[|-k(r)|2 -1]= (2/k3)L(2L+1){kL-½[sin2(k+L)-sin2(k)]+..}

Þ FSI contribution to the CF of nonidentical particles, assuming Gaussian source function W(r)=exp(-r2/4r0

2)/(2 r0) :

for kr0 << 1: CFFSI = ½|f0/r0|2[1-d0/(2r0)]+2f0/(r0) ~ r0-1 or r0

-2

f0 and d0 are the s-wave scatt. length and eff. radius entering in the (L=0) amplitude fL(k) = sinLexp(iL)/k (1/fL+½dLk2 - ik)-1

for kr0 >> 1: CFFSI = (2/k2)W(0)L(2L+1)L ~ r0-3

L=[(2L+1)/2k]sin(2 L) - (dL/k2L)sin2L

L-wave saturated by a resonance:

L= /2, dL= 2k2L+1()-1 L(k=k0) = 2k0()-1

20

Þ for kr0 >> 1: CFr,L(k=k0)= 4W(0)(2L+1) (k0)-1

& resonance yield in the narrow width limit L(k) = (k-k0)

rd3Nr,L/d3pr = (2)3W(0)(2L+1)(12/1*2

*)d6N0/(d3p1 d3p2)

® a problem with π+- * no room for direct production !

See also the talk by Petr Chaloupka on Wednesday

Likely due to violation of eq. time approx. for a part of the coalescence contribution (|k*t*| m1,2r* is valid for m2=m only)

Resonance contribution to CF vs r-k correl. b

Rpeak(STAR)

----------- 0.025

Rpeak(NA49)

---------- 0.10 0.14

Smoothness assumption:

WP(r,-kn) WP(r,k) Exact

WP(r,k) ~ exp[-r2/4r02 + bkrcos]; = angle between r and k

CF suppressed by a factor WP(0,k) ~ exp[-b2r02k2]

To leave a room for a direct production b > 0.3 is required for π+- system; however, BW b ~ 0.2; likely eq.-time approx. not valid for π+- at k*~ 150 MeV/c

*(k=146 MeV/c), r0=5 fm (k=126 MeV/c), r0=5 fm-----------

r-k correl. b 0.2 for K+K- CF indicates ~ ½ direct -meson yield – in qualitative agreement with the difference between the measured

rapidity width in central Pb+Pb collisions at SPS and expected from coalescence: -2 = K+-2 + K--2

-

K+

anti-

p+p

K-K+K- coalescence

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Bound state production

Migdal, Watson, Sakharov, .. hadronic processes

Dominated by FSI provided a small binding energy b

Closely related to production of free particles at k* 0

FSI theory: Fermi -decay

Continuum:d6N/(d3p1 d3p2)= d6N0/(d3p1 d3p2)|-k*(r*)|2

d3N/d3pb= (2)3bd6N0/(d3p1 d3p2)|b(r*)|2 Discrete spectrum:

+x1

x2

x2

x1

p1

p2

p1

p2

pb

r*= x1*- x2

*= distance between particle emitters in pair cms

p1/m1≈ p2/m2 pb/mb

Basis of bound state coalescence femtoscopy

Universal relation between production of free and bound ’s is used to determine pionium lifetime in DIRAC exp. at CERN :

Assume only two types of pion sources Nemenov ‘85Short-Lived Emitters SLE (, , ..) r* |a| = 387 fm both ’s from SLE

Long-Lived Emitters LLE (, Ks, , ..) r* |a| one or both ’s from LLE

Then, neglecting strong FSI

|-k*(r*)|2 = |c-k*(0)|2 + (1- ) |c

-k*()|2

|A(r*)|2 = |cA(0)|2 + (1- ) |c

A()|2

1Ac(Q)

01/(|na|3)

A={n,L=0}

Þ d6N/(d3p1 d3p2)= d6N0/(d3p1 d3p2) [ Ac(Q) + (1- )]

d3N/d3pA= d6N0/(d3p1 d3p2) (2)3A/(|na|3), p1 ≈ p2 ≈ ½pA

= SLE pair fraction

Coalescence: deuterons ..

Edd3N/d3pd = B2 Epd3N/d3pp End3N/d3pn pp pn ½pd

Here is realized the opposite limit compared with r*<<|a| for +- atom: r* > deuteron radius = -1 = (2b)-1/2 |b(r*)|2 W(0) ~ r0

-3

Coalescence factor: B2 = (2)3(mpmn/md)-1t|b(r*)|2 ~ r0-3

Triplet fraction = ¾ unpolarized Ns

Assuming Gaussian r* distribution and accounting for

a boost from LCMS to PRF:

B2 33/2/(2mptr03)

r0(pp) ~ 4 fm from AGS to RHIC

Lyuboshitz’88 ..

B2 d,d-bar at pt=1.3 GeV/c

in central HICs

d d-bar PHENIX

B2 at RHIC energies with pt and centrality

in agreement with B2 ~ R-3

+fragmentation ?

coalescence

Is there double counting in thermal model with FSI ?Global thermal equilibrium in nonrelativistic limit; T,i<<E ~ mi

d3Ni/d3p = Vgi(2)-3exp[-(E-i)/T] gi=2Si+1 = spin factor, i= chemical potential, T = temperature

In the limit kV1/3 >> 1 and simultaneous emission: Jennings, Boal, Shillcock’86

the L-wave resonance contribution due to FSI of spin-0 particles d3Nr,L/(d3pr) = (2)3V-1(2L+1)d3N1/d3p1 d3N2d3p2

= Vgr,L(2)-3exp[-(Er-r)/T] gr.L=2L+1, r= 1+ 2, Er=E1+E2

resonances due to FSI are also in thermal equilibrium in this limit

Similarly, in the limit V1/3 >> 1 and simultaneous emission: d3Nb/d3pb= (2)3V-1d3N1/d3p1 d3N2d3p2

= V(2)-3exp[-(Eb-b)/T] i.e., the (s-wave) coalescence bound state is also in thermal equilibrium

Is there double counting in thermal model with FSI ?

No double counting with respect to direct thermal production since the latter is characterized by local thermal equilibrium (not reproduced by the coalescence states – as revealed e.g. by -meson rapidity width)

besides the coalescence FSI contribution, one may expect essential direct contribution of resonances (with sufficiently long lifetime) and bound states (with sufficiently large binding energy)

28

Correlation femtoscopy with nonid. particles p CFs at AGS & SPS & STAR

Fit using RL-Lyuboshitz’82 with

consistent with estimated impurityr0~ 3-4 fm consistent with the radius from pp CF & mt scaling

Goal: No Coulomb suppression as in pp CF &Wang-Pratt’99 Stronger sensitivity to r0

=0.50.2r0=4.50.7 fm

Scattering lengths, fm: 2.31 1.78Effective radii, fm: 3.04 3.22

singlet triplet

AGS SPS STAR

r0=3.10.30.2 fm

Pair purity problem for p CF @ STARÞ PairPurity ~ 15%Assuming no correlation for misidentified particles and particles from weak decays

Fit using RL-Lyuboshitz’82 (for np)

¬ but, there can be residualcorrelations for particles fromweak decays requiring knowledgeof , p, , , p, , correlations

30

Correlation study of strong 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

But r0(p) < r0(p) ? Residual correlations

pp

Correlation study of strong interaction

-

-

scattering lengths f0 from NA49 correlation data

Fit using RL-Lyuboshitz (82) with fixed Pair Purity =0.16 from feed-down and PID

Data prefer |f0| « f0(NN) ~ 20 fm

-

CF=1+[CFFSI+SS(-1)Sexp(-r0

2Q2)]

0= ¼(1-P2) 1= ¼(3+P2) P=Polar.=0

CFFSI = 20[½|f0(k)/r0|2(1-d00/(2r0))

+2Re(f0(k)/(r0))F1(r0Q)

- 2Im(f0(k)/r0)F2(r0Q)]

fS(k)=(1/f0S+½d0

Sk2 - ik)-1 k=Q/2

F1(z)=0z dx exp(x2-z2)/z F2(z)=[1-exp(-z2)]/z

CF=N{1+[CFFSI -½exp(-r02Q2)]}

N r0 f0

0

d00

fmfmfm

B

33

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

k*x on a given direction x and study CF-ratio CF+x/CFx

34

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

35

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

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)

37

Summary

• Assumptions behind femtoscopy theory in HIC seem OK at k 0. At k > ~ 100 MeV/c, the usual smoothness and equal-time approximations may not be valid.

• Wealth of data on correlations of various particles (,K0,p,,), yields of resonances and bound states 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 correlations

• Info on two-particle strong interaction: & & p scattering lengths from HIC at SPS and RHIC. Good perspective at RHIC & LHC (a problem of residual correlations is to be solved).

38

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

39

Examples of NA49 & STAR data

3-dim fit: CF=1+exp(-Rx2qx

2 –Ry2qy

2 -Rz

2qz2

-2Rxz2qx qz)

z x y

Correlation strength (purity, chaoticity, ..)

NA49

Interferometry or correlation radii

KK STAR

Coulomb corrected

AGSSPSRHIC: radii

STAR Au+Au at 200 AGeV 0-5% central Pb+Pb or Au+Au Clear centrality & mt dependence

Weak energy dependence

R ↑ with centrality & with mt only Rlong slightly ↑ with energy

Rside R/(1+mt F2/T)½

Rlong (T/mt)½

tr. collective flow velocity F Evolution (freeze-out) time

41

hadronization

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronic phaseand freeze-out

PCM & clust. hadronization

NFD

NFD & hadronic TM

PCM & hadronic TM

CYM & LGT

string & hadronic TM

Expected evolution of HI collision vs RHIC data

dN/dt

1 fm/c 5 fm/c 10 fm/c 50 fm/c time

Kinetic freeze out

Chemical freeze out

RHIC side & out radii: 2 fm/c

Rlong & radii vs reaction plane: 10 fm/c

Bass’02

Femto-puzzle I

Contradiction with transport and simple hydro calcul.

- small space-time scales

- their weak energy dep.

- Rout/Rside ~ 1

Basically solved due to the initial flow increasing with energy (likely related to the increase of the initial energy density and partonic energy

fraction)

43

Femtoscopy Puzzle I basically solved due to initial flow appearing in realistic IC

Femtoscopic signature of QGP onset3D 1-fluid Hydrodynamics

Rischke & Gyulassy, NPA 608, 479 (1996)

With 1st order

Phase transition

Initial energy density 0

Long-standing signature of QGP onset:

• increase in , ROUT/RSIDE due to the Phase transition

• hoped-for “turn on” as QGP threshold in 0 is reached

• decreases with decreasing Latent heat & increasing tr. Flow

(high 0 or initial tr. Flow)

Femto-puzzle II

No signal of a bump in Rout near

the QGP threshold (expected at AGS-SPS energies) !?

likely solved due to a dramatic decrease of

partonic phase with decreasing

energye.g. in PHSD, Cassing,

Bratkovkaya

46

Cassing – Bratkovskaya: Parton-Hadron-String-Dynamics

Perspectives at FAIR/NICA energies

Femtoscopy of Pb+Pb at LHC ALICE arXiv:1012.4035All radii increase with

Nch from RHIC to LHC (not from SPS to RHIC)!

Multiplicity scaling of the correlation volume universal freeze-out density

Freezeout time f from Rlong=f (T/mt)1/2

The LHC fireball:

- hotter

- lives longer &

- expands to a larger size

Ridge effect

Dense matter (collective flows)

also in pp collisions at LHC (for high Nch) ?

- pt increases with nch and particle mass

- BE CF vs nch and pt points to expansion at high nch

- Ridge effect observed in angular correlations at high nch

R(kt) at large Nch expansion

CMS

Caution: the strong FSI is enhanced by a logarithmic singularity

for particles from resonance decays & a small source size LL’96

e.g. noticeable strong FSI effect even

on CF of identical charged pions:

CFFSI ~ 2f0/lr* 2ln|B| f0/lr

*

For example, if res = -meson:

00 +-

CFFSI (k=0) -0.10 0.25 0.30

50

Even stronger effect of KK-bar FSI on KsKs correlations in pp-collisions at LHC

ALICE: PLB 717 (2012) 151

e.g. for kt < 0.85 GeV/c, Nch=1-11 the neglect of FSI increases by ~100% and Rinv by ~40%

= 0.64 0.07 1.36 0.15 > 1 !

Rinv= 0.96 0.04 1.35 0.07 fm

Resonance FSI contributions to π+- K+K- CF’s • Complete, inner and outer

contributions of p-wave resonance (*) FSI to π+- CF for two cut parameters 0.4 and 0.8 fm and Gaussian radius of 5 fm FSI contribution overestimates measured * by a factor 4 (3) for r0 = 5 (5.5) fm factor 3 (2) if accounting for out

-6 fm • The same for p-wave resonance

() FSI contributions to K+K-

CF FSI contribution overestimates measured by 20% for r0 = 4.5 fm

• Little or no room for direct production when neglecting r-k correlation!

Rpeak(NA49)

» 0.10 0.14after purity correction

Rpeak(STAR)

0.025 ----------- -----

----------- -----

---------------------

r0 = 5 fm no r-k correlation

References related to resonance formation in final state:

R. Lednicky, V.L. Lyuboshitz, SJNP 35 (1982) 770

R. Lednicky, V.L. Lyuboshitz, V.V. Lyuboshitz, Phys.At.Nucl. 61 (1998) 2050

S. Pratt, S. Petriconi, PRC 68 (2003) 054901

S. Petriconi, PhD Thesis, MSU, 2003

S. Bekele, R. Lednicky, Braz.J.Phys. 37 (2007) 994

B. Kerbikov, R. Lednicky, L.V. Malinina, P. Chaloupka, M. Sumbera, arXiv:0907.061v2 B. Kerbikov, L.V. Malinina, PRC 81 (2010) 034901

R. Lednicky, Phys. Part. Nucl. Lett. 8 (2011) 965

R. Lednicky, P. Chaloupka, M. Sumbera, in preparation

53

Discrete spectrum: A2

n0(r*) = cn0(0) [1- r*/|a|+f(0)/r* + O(f(0)/a) + O(r*2/a2)]

n0(r*) exp(-r*/|na|) 0 at r* >> |na|

-k*(r*) Ac [1 - r*(1+cos*)/|a| + f(0)/r* + O(f(0)/a) + O(r*2/a2)] p-wave

|-k*(r*)| 1 at r* >> |a|

k* 0

Continuum: +-

Universal dependence of the WF on r* and scattering amplitude f RL’04

in continuum (at k* 0) and discrete spectrum at a given orbital angular

momentum and r*<< -1

=(2b)1/2 = virtual momentum (k*=i), b(r*) exp(-r*) at r* > -1

Particularly, for +- atom with the main q.n. n: n=|na|-1, |a|=387 fm

FSI effect on CF of neutral kaons

STAR data on CF(KsKs)

Goal: no Coulomb. But R may go up by ~1 fm if neglected FSI in

l = 1.09 0.22r0 = 4.66 0.46 fm 5.86 0.67 fm

KK (~50% KsKs) f0(980) & a0(980)RL-Lyuboshitz’82 couplings from Martin

or Achasov

t

Achasov’01,03Martin’77

no FSI

Lyuboshitz-Podgoretsky’79: KsKs from KK also showBE enhancement

r0 ↓ Mt Universal expansion !

55

NA49 central Pb+Pb 158 AGeV vs RQMD: FSI theory OKLong 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

56

Correlation study of particle interaction

-

+ scattering length f0 from NA49 CF

Fit CF(+) by RQMD with SI scale: f0 sisca f0

input f0

input = 0.232 fm

sisca = 0.60.1 Compare with ~0.8 from SPT

& BNL E765 K e

+

CF=Norm [Purity RQMD(r* Scaler*)+1-Purity]

57

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

58

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

59

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 coalescencee-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

60

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 bx 0 /[02+T/mt+T/Tr]

radial gradient of T

Þ decreasing asymmetry ~1 ? problem