History Assumptions & Technicalities

16. 9. 2010 WPCF Kiev‘10 1

Femtoscopic Correlations and Final State Resonance Formation

R. Lednický, JINR Dubna & IP ASCR Prague

• History

• Assumptions & Technicalities

• Narrow resonance FSI contributions to π+- K+K- CF’s

• Conclusions

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 F(k,Z,R) in β-decay

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

Z=83 (Bi)β-

β+

R=84 2 fm

k MeV/c

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 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...

Assumptions to derive “Fermi” formula for CF

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

- tFSI ddE tprod

- equal time approximation in PRF

typical momentum transfer

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

usually OK

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

- smoothness approximation: p qcorrel Remitter Rsource

~ OK in HIC, Rsource2 0.1 fm2 pt

2-slope of direct particles

tFSI (s-wave) = µf0/k* k* = ½q*

hundreds MeV/c

tFSI (resonance in any L-wave) = 2/ hundreds MeV/c

in the production process

to several %

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

∫d4x1d4x2 G(x1,p1;x2,p2) |p1p2(x1,x2)|2 ∫d3r WP(r,k) |-k(r)|2

∫d3r {WP(r,k) + WP(r,½(k-kn)) 2Re[exp(ikr)-k(r)]

+WP(r,-kn) |-k(r)|2 }

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

The smoothness approximation WP(r,½(k-kn)) 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

Technicalities – 1: spin & isospin equilibration

Technicalities – 2: treating the spin & angular dependence

Assuming that one of the two particles is spinless and the other one is either spin-1/2 or spin-1/2, we have:

Angular dependence enters only through the angle between the vectors k and r in the argument of the Wigner (rotation) d-functions.

Technicalities – 3: contribution of the outer region

For r > d = the range of the strong interaction potential,the radial functions:

Technicalities – 4: projecting pair spin & isospin

=π+-

=K+K-

Technicalities – 5: resonance dominance in the JT-wave

Technicalities – 6: contribution of the inner region

Technicalities – 8: simple Gaussian emission functions

In the following we use the simple one-parameter Gaussianemission function:

WP(r,k) = (8π3/2r03)-1 exp(-r2/4r0

2)

and account for the k-dependence in the form( = angle between r and k)

WP(r,k) = (8π3/2r03)-1 exp(-b2r0

2k2) exp(-r2/4r02 + bkrcos)

Note the additional suppression of WP(0,k) if out 0:

WP(0,k) ~exp[-(out/2r0)2] (~30% suppression for π+-)

References related to resonance formation in final state:

R. Lednicky, V.L. Lyuboshitz, SJNP 35 (1982) 770R. Lednicky, V.L. Lyuboshitz, V.V. Lyuboshitz, Phys.At.Nucl. 61 (1998) 2050S. Pratt, S. Petriconi, PRC 68 (2003) 054901S. Petriconi, PhD Thesis, MSU ,2003S. Bekele, R. Lednicky, Braz.J.Phys. 37 (2007) 994B. Kerbikov, R. Lednicky, L.V. Malinina, P. Chaloupka, M. Sumbera, arXiv:0907.061v2 B. Kerbikov, L.V. Malinina, PRC 81 (2010) 034901 R. Lednicky, P. Chaloupka, M. Sumbera, in preperation

correlations in Au+Au (STAR)

• Coulomb and strong FSI present *1530, k*=146 MeV/c, =9.1 MeV

• No energy dependence seen

• Centrality dependence observed, quite strong in the * region; 0-10% CF peak value CF-1 0.025

• Gaussian fit of 0-10% CF’s at k* < 0.1 GeV/c: r0=4.8±0.7 fm, out = -5.6±1.0 fm

r0 =[½(rπ2+r2)]½ of ~5 fm is in

agreement with the dominant rπ of 7 fm

P. Chaloupka, JPG 32 (2006) S537; M. Sumbera, Braz.J.Phys. 37(2007)925

correlations in Pb+Pb (NA49)

• Coulomb and strong FSI present 1020, k*=126 MeV/c, =4.3 MeV

• Centrality dependence observed, particularly strong in the region; 0-5% CF peak value CF-1 0.10

• 3D-Gaussian fit of 0-5% CF’s: out-side-long radii of 4-5 fm

PLB 557 (2003) 157

Resonance FSI contributions to π+- K+K- CF’s • Complete and corresponding 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 strongly overestimated

• The same for p-wave resonance () FSI contributions to K+K- CF FSI contribution well described but no room for direct production !

Rpeak(NA49) 0.10

Rpeak(STAR) 0.025 ----------- -----

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

Peak values of resonance FSI contributions to π+- K+K- CF’s vs cut parameter

• Complete and corresponding inner and outer contributions of p-wave resonance (*) FSI to peak value of π+- CF for Gaussian radius of 5 fm

• The same for p-wave resonance () FSI contributions to K+K- CF

for < r0, CF is practically -independent

at ~1 fm, CFCF(inner)

Rpeak(NA49) 0.10

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

Rpeak(STAR) 0.025

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

Angular dependence in the *-resonance region (k*=140-160 MeV/c)

r* < 1 fm

r* < 0.5 fm

0-10% 200 GeV Au+AuFASTMC-codefrom L. Malinina

= angle between r and k

WP(r,k) ~ exp[-r2/4r02 + b krcos]

b 0.5

Angular dependence – example parametrization

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

Suppressing WP(0,k)by a factor of exp[-b2r0

2k2]

Similar suppression of resonance contribution to CF

Rpeak(STAR) -------------- 0.025

b

k=146 MeV/c, r0=5 fmR(π+-)

Smoothness assumption:WP(r,½(k-kn)) WP(r,-kn) WP(r,k)

Exact

b= 0.3 0.4

Accounting for angular dependence:FASTMC-code (implying smoothness assumption)

close to the STAR peak but too low at small k* (L.M.)

OK at small k* but much higher peak (P. Ch.)

Rpeak(STAR)-------------- 0.025

b < 0.4

Indicating that theparametrization of the angular dependence~ exp(b krcos), b0.5is oversimplified

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Summary• Assumptions behind femtoscopy theory in HIC seem OK,

including both short-range s-wave and narrow resonance FSI up to a problem of the angular dependence in the resonance region.

• The effect of narrow resonance FSI scales as inverse emission volume r0

-3, compared to r0-1 or r0

-2 scaling of the short-range s-wave FSI, thus being more sensitive to the space-time extent of the source. The higher sensitivity may be however disfavoured by the theoretical uncertainty in the case of a strong angular asymmetry.

• The NA49 (? STAR) correlation data from the most central collisions seem to leave a little or no room for a direct (thermal) production of near threshold narrow resonances.

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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

PRF

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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, ...

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

Smoothness approximation: rA « r0 (q « p)

p1

p2

x1

x2

2r0

W(p1,p2|∫d4x1d4x2 p1p2(x1,x2) Τ(x1,x2;)|2

x1’x2’

≈ ∫d4x1d4x2 G(x1,p1;x2,p2) |p1p2(x1,x2)|2

r0 - Source radius

rA - Emitter radiusp1p2(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’

For non-interacting identical spin-0 particles – exact result (p=½(p1+p2) ):W(p1,p2 ∫ d4x1d4x2 [G(x1,p1;x2,p2)+G(x1,p;x2,p) cos(qx)]

approx. result: ≈ ∫d4x1d4x2 G(x1,p1;x2,p2) [1+cos(qx)]

= ∫ d4x1d4x2 G(x1,p1;x2,p2) |p1p2(x1,x2)|2

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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

→

Technicalities – 8: simple Gaussian emission functions

Technicalities – 7: volume integral I(,k)

In the single flavor case & no Coulomb FSI

For s & p-waves it recovers the results of Wigner’55 & Luders’55