π-Ξ Correlations in Heavy Ion Collisions and Ξ(1530) Puzzle

P. Chaloupka(NPI ASCR, Czech Republic), B. Kerbikov (ITEP, Russia),R. Lednicky (JINR & NPI ASCR), Malinina (JINR & SINP MSU), and M. Šumbera (NPI ASCR, Czech Republic)

V Workshop on Particle Correlations and Femtoscopy 14-17 Octobre, 2009, CERN

nucl-th 0907.0617

Outline

Motivation for femtoscopy with Ξ

Current experimental results of π-Ξ

FSI calculations

Comparison with data

Conclusions

Evolution of matter in HI collisions

t

~ fm/c

CGC (?)

Mixed phase

Hadron gas

Thermal freeze-out

Chemical freeze-out

Prethermal partonic state ~ fm/c

~ 7fm/c

GLP’60: enhanced ++ , -- vs +- at small opening angles – interpreted as BE

enhancement

Kopylov and Podgortsky ’71-75: settled basics of correlation femtoscopy

Analogy in Astronomy: Hanbury Brown and Twiss (HBT effect)

Correlation femtoscopy : measurement of space-time characteristics R, c ~fm of particle production using

particle correlations due to the effects of QS and FSI

FEMTOSCOPY: Momentum correlations

q = p1- p2 , x = x1- x2

out transverse pair velocity vt

side

long beam

The corresponding correlation widths are usually parameterized in terms of the Gaussian correlation radii R_i:

We choose as the reference frame the longitudinal co-moving system (LCMS)

The idea of the correlation femtoscopy with the help of identical particlesis based on an impossibility to distinguish between registered particles emitted from different points

Correlation strength or chaoticity

CF=1+exp(-Ro2qo

2 –Rs2qs

2 -Rl

2ql2

-2Rol

2qoql)

Weights for QS only: P12=C(q)=1+(-1)Scos qx >

( )

( )inv

inv

S QCF N

B Q

Final State Interaction

CF nnpp

Coulomb only

FSI is sensitive to source size and scattering amplitude. It complicates CF analysis but makes possible:

Femtoscopy with nonidentical particles:K,, p,, Ξ ...Study of the “exotic” scatterings: , K,, KK,, , p,Ξ , ..

Study of the relative space-time asymmetries C+/C-: Lednicky, Lyuboshitz et al. PLB 373 (1996) 30

Spherical harmonics method: Danielewicz, P. and Pratt, S., Phys. Rev C75 03490 (2007) Z.Chajevski and M.Lisa, PRC78 064903 A.Kisiel and D.A. Brown (2009) 0901.3527

k*=|q|/2

CF of identical particles sensitive to terms even in k*r* ( 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*>

+ - K+ K- K0S p p

p

p

K0S

K-

K+

-

+

What particle systems provide us with most interesting and direct information on dynamics of heavy ion collisions ?

Why π-Ξ ?models predict an early decoupling of multi-strange

hadrons like Ξ (ΔS=2) dueto their small interaction cross section

They provide us with footprints of the early stages of evolution.

A window on π-Ξ scatteringlength

Experimental data (STAR) on π-Ξ correlations

Coulomb and strong ( Ξ*1530 ) final state interaction effects are present.

Ξ* (1530), P13 I(J)=1/2(3/2+), Γ=9.1 MeV

Centrality dependence is observed, particularly strong in the Ξ* region

M

mmMmmMk

2

])([])([ 2/1222/122*

*(1530)

200 GeV AuAu

Braz.J.Phys.37:925-932,2007

M.Sumbera

Braz.J.Phys.37:925-932,2007

M.Sumbera

π-Ξ model comparisonAnalysis of P. Chaloupka and M. Sumbera (Braz.J.Phys.37:925-932,2007) :

Used FSI model:S. Pratt, S. Pertricioni Phys.Rev. C68, 054901 (2003)

+ Emission points from hydro-inspired Blastwave constrained by π-πHBT

Discrepancy in Ξregion, over predicts A

00 and A

11

Coulomb part in qualitative agreement

R = (6.7±1.0) fm ∆rout

= (-5.6±1.0) fm

∆out

<0 Ξ emitted more on the outside –agrees with the flow Ξ scenario

Spherical decomposition

Z. Chajecki , T.D. Gutierrez ,

M.A. Lisa and M. López-Noriega,

nucl-ex/0505009 )

A00 monopole – size -

A11 dipole - shift in out-direction -

Spherical decomposition

, Z. Chajecki , T.D. Gutierrez

,M.A. Lisa and M. López-Noriega

( nucl-ex/0505009

A00 - monopole – size

A11 - dipole - shift in out-direction

The factors which have to be taken into account in π+Ξ- FSI calculations:

- The superposition of strong and Coulomb interactions

- The presence of Ξ*(1530) resonance

- The spin structure of w.f. including spin flip

- The fact that π+Ξ- state is a superposition of I=1/2

and I=3/2 isospin states and π+Ξ- is coupled in π0Ξ0 and that the thresholdes of the two channels are non-degenerate

- The contribution from inner potential region

where the structure of the strong interaction is unknown.

1

1/2

( ) *1 1

1

( , ) ~ ( cos )iik r

ik r ii i i

ek r e f

r

The outgoing multichannel wave functions of π+Ξ- system enter as a building block into CF

i=1-4, π+Ξ-,

2 2( ) ( )( ) ( ) ( , ) ( ) ( , )i i i

i i

C k d rS r k r d rS r k r

k- is a relative momentum of the pair

The out-state w.f.’s have the asymptotic form

π0Ξ0 without and with spin flip

Coulomb

Ξ*(1530)π0Ξ0 π+Ξ-

EStrong S wave Strong P- wave

P-wave is dominated by Ξ*(1530)

Ξ*(1530)

πΞ Ξ*(1530)The low energy region of interaction up to the resonance is dominated by S and P-waves.

Therefore the w.f. contains 2 phase shifts with I = 1/2, 3/2 for S-wave and 4 phase shifts with I = 1/2, 3/2 and J = 1/2, 3/2 for P-wave (J = L ± 1/2 is the total momentum).

To reduce the number of parameters we have assumed that the dominant interaction in P-wave occurs in a state with J = 3/2, I = 1/2 containing Ξ*(1530) resonance. Since the parameters of Ξ*(1530) are known from the experiment we are left with two S-wave phase shifts which are expressed in terms of the two scattering lengths a1/2 and a3/2 with isospin I=1/2 and I=3/2 correspondingly. 

The structure of π+Ξ- Wave Function

The wave function

here is the pure Coulomb w.f., k1 and k2 are the c.m. momenta in and π0Ξ0 channels;spherical harmonics correspond to the reversed direction of the vector k,

2224( ) * 0 0 1*

0 0 0 1 1 1 1 1 11

22

0 0 1*20 0 0 1 1 1 1 1 1

1

2 2( , ) ( , )

33

2 2

33

Couli

k r k r T Y T Y T Y

kR Y R Y R Y

k

Coulπ+Ξ-

( , )m ml lY Y

1

1

( ) ( )1 1 1 1 1

( )1/2 ( )1 2 1 2 2

( ln 2 /2)( )

( , ) 4 ( ) ( , ) /

( , ) ( / ) 4 ( ) (0, ) /

( , )

l

l

l

ill l

ill i l

i ll

i e H

i e H

H e

combination of the regular and singular Coulomb functions Fl and Gl

ρ1 =k1r, ρ2 =k2r, η =(a1k1) -1, a1=-214 fm is a Bohr radius of the π+Ξ-

system taking into account the negative sign of the Coulomb repulsion.

The wave function2224

( ) * 0 0 1*0 0 0 1 1 1 1 1 1

1

22

0 0 1*20 0 0 1 1 1 1 1 1

1

2 2( , ) ( , )

33

2 2

33

Couli

k r k r T Y T Y T Y

kR Y R Y R Y

k

1/21 1 2

1 10 0

1 2 1 2

/ 2 ( ) / 2,

/ 2 / 2

, 2 / 3, / 3

T RE E i E E i

The quantities

contain the elastic 1 → 1 and inelastic 1 → 2 scattering amplitudes fl{J;11} and fl

{J;21}. For the S-waves (l=0, J=1/2), they are expressed through the scattering lengths a±1/2 and a±3/2 in a similar way as in pion-nucleon scattering.

For the resonance P-wave

;11* ;21*1 1 2, ( )J J

l l l lT k f R k k f

Ξ*(1530)

= 2 Γ/3, Γ2=Γ/3

The inner region correctionThe above expression describes the region r > ε ~ 1~fm where the strong potential is assumed to vanish.In the inner region r < ε , we substitute by and take into account the effect of strong interaction in a form of a correction which depends on the strong interaction time (expressed through the phase shift derivatives) and can be calculatedwithout any new parameters unless the S-wave effective radii are extremely large. It is important that the complete CF does not depend on ε provided the sourcefunction is nearly constant in the region r< ε . (MC procedure was checked using exact calculations within Mathematica)

2

Coul

Without introduction of the correction, the resonance region exhibits clear interference of Coulomb and strong interactions,which is not observed in the data !

How to calculate CF numerically

3/2 3 1 2 2( ) (8 ) exp( / 4 )S r R r R

2) HYDJET++ http://cern.ch/lokhtin/hydjet++ I.Lokhtin, L.Malinina, S.Petrushanko, A.Snigirev, I.Arsene, K.Tywoniuk, e-print arXiv:0809.2708, Comput.Phys.Commun.180:779-799,2009. The soft part of HYDJET++ event represents the "thermal" hadronic state FASTMC: Part I: N.S. Amelin et al, PRC 74 (2006) 064901; Part II: N.S. Amelin, et al. PR C 77 (2008) 014903

1) source is approximated with Gaussian in PRF

3) Standard UrQMD (v2.2) output of freeze-out particles http://www.th.physik.uni-frankfurt/~urqmd

FSI code Richard Lednicky's code for calculation of the two particle

correlations due to QS and FSI

Source models:

π-Ξ FSI model comparison

The influence of S-wave scattering lengths parameters on CF in Coulomb region.

source is approximated with Gaussian in PRF

Ro = Rs = Rl = 7 fm

At present experimental errors, the CF at R > 7 fm is practically independent of the S-wave scattering parameters.

Ro = Rs = Rl = 2 fm

source is approximated with Gaussian

π-Ξ FSI model comparison

Similarly to the FSI model S. Pratt's (PRC68, 054901(2003) )our calculations are in agreement with he data in the low-k Coulomb region. Contrary to this model, they are however much closer to the experimental peak in the Ξ*(1530) region though, they still somewhat overestimate this peak (at R=7 fm. Exp. ~1.05, Model ~1.2)The predicted peak is however expected to decrease due to a strong angular asymmetry of a more realisticsource function obtained from Blast-wave like simulations.

Angular asymmetry in Blastwave-like model HYDJET++

Consider Source model: Gaussian in PRF (at R=7 fm. Exp. ~1.05, Model ~1.2, m, Model with angular dependence ~ 1.05)

π-Ξ FSI model comparison

– HYDJET++– UrQMD (v2.2)

Conclustions & Plans•Using a simple Gaussian model for the source function, we have reasonably described the experimental π-Ξ CF.

•Estimated the emission source radius (~7fm) and tested the sensitivity to the low energy parameters of the strong interaction.

•The predicted peak is however expected to decrease due to a strong angular asymmetry of a more realistic source function obtained from the Blast-wave like simulations.

•Blast-wave like model HYDJET++ (soft part) allows to get the

reasonable description of the π-Ξ CF in the whole k* region.

•UrQMD 2.2 model also provides reasonable description of CF.

•Spherical harmonic method will be applied to extract space-time shifts.

Additional slides

Reason of differences withS. Pratt, S. Petriconi (PRC68, 054901(2003) )

1. The approach is of course the same, the resulting CF formula coincides with ours if corrected for misprints.

2. So there is likely a bug in the Pratt-Petriconi code. As for our calculations, they have been checked with the help of MATHEMATICA. Criterium of correcteness is independence of ε

Boost to pair rest frame

Particle 1source

Particle 2Source

Separation between particle1 and 2 andBoost to pairRest frame

r*out = T (rout – T t)

2 free parameters in the Gaussian approximationWidth of the distribution in pair rest frameOffset of the distribution from zero

(slide from F.Retiere QM05)

Space-Time shifts in pair rest frame πΞ, πK, πp, KpπΞ, πK, πp, Kp

Within HYDJET++ model the combined freeze-out scenario

describes better the observed in experiment πΞ πΞ space-time differences;

π, K, p π, K, p freeze-out atfreeze-out at TTthth=100 MeV =100 MeV

HYDJET++: hydro + part related to the partonic states The soft part of HYDJET++ event represents the "thermal" hadronic state FASTMC: Part I: N.S. Amelin, R. Lednisky, T.A. Pocheptsov, I.P. Lokhtin, L.V. Malinina, A.M. Snigirev, Yu.A. Karpenko, Yu.M. Sinyukov, Phys. Rev. C 74 (2006) 064901; Part II: N.S. Amelin, R. Lednisky, I.P. Lokhtin, L.V. Malinina, A.M. Snigirev, Yu.A. Karpenko, Yu.M. Sinyukov, I.C. Arsene, L. Bravina, Phys. Rev. C 77 (2008) 014903 http://uhkm.jinr.ru

The hard, multi-partonic part of HYDJET++ event is identical to the hard part of Fortran-written HYDJET (PYTHIA6.4xx + PYQUEN1.5) : I.P.Lokhtin and A.M.Snigirev, Eur. Phys. J. C 45, 211 (2006), http://cern.ch/lokhtin/pyquen, http://cern.ch/lokhtin/hydro/hydjet.html

Official version of HYDJET++ code and web-page with the documentation: http://cern.ch/lokhtin/hydjet++

The complete manual: I.Lokhtin, L.Malinina, S.Petrushanko, A.Snigirev, I.Arsene, K.Tywoniuk, e-print arXiv:0809.2708, Comput.Phys.Commun.180:779-799,2009.

HYDJET++ is capable of reproducing the bulk properties of multi-particle system created in heavy ion collisions at RHIC (hadron spectra and ratios, radial and elliptic flow, momentum correlations), as well as the main high-pTobservables.

The 2-particle momentum CF is defined as a normalized ratio of correspondingtwo and single particle distributions.

CF q , k = γN 2 p i , p j

N 1 p i N 2 p j

CF p1 , p2 =1 λ exp −R out2 qout

2 −R side2 q side

2 −Rlong2 q long

2 −2R out , long2 qout q long

q= pi− p j

k=1/2 pi p j CF ∞=1

1D CF Most simple parametrization:CF q ,k =1 λ k exp−Rinv

2 qinv2

q inv2 =−q2

strength of correlations

Decompose q into components:QlongLong : in beam directionQoutOut : in direction of pair transverse momentumQsideSide : qLong & qOut

Parametrizations of CF

CF= γA2 pi , p j B 2 p i , p j

Z. Chajecki , T.D. Gutierrez , M.A. Lisa and M. López-Noriega, nucl-ex/0505009 Z. Chajecki , T.D. Gutierrez , M.A. Lisa and M. López-Noriega, nucl-ex/0505009

200GeV AuAu different centralities

Ξ

Spherical decomposition – accessing emission shift

Different Alm

coefficients correspond to different symmetries of the source

A00 - monopole – size

A11 - dipole - shift in out-direction

A11

≠ 0 - shift in the average emission point between and

Simplified idea of CF asymmetry(valid for Coulomb FSI)

x

x

v

v

v1

v2

v1

v2

k*/= v1-v2

Ξ

Ξ

k*x > 0v > vp

k*x < 0v < vp

Assume emitted later than Ξ or closer to the center

Ξ

Ξ

Longer tint

Stronger CF

Shorter tint Weaker CF

CF

CF

Modified slide of R.Lednicky

flow

CF-asymmetry for charged particlesAsymmetry arises mainly from Coulomb FSI

CF Ac() |exp(-ik*r*)F(-i,1,i)|2

=(k*a)-1, =k*r*+k*r*F 1+r*/a+k*r*/(k*a)r*|a|

k*1/r* Bohr radius

}

±226 fm for p±214 fm for Ξ

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

Modified slide of R.Lednicky

Shift <x in out direction is due to collective transverseflow & higher thermal velocity of lighter particles

y

X

1

2

Source

For QS only: P12=C(q)=1+(-1)Scos qx>

pp

Momentum correlations of identical particles QS only

K1

xb

Two plane-waves:

ππ

1/R

C q =N 2 k 1 , k 2

N 1 k 1 N 2 k 2 q=k 1−k 2

C ∞ =1

The 2-particle correlation function C(q) is defined as a normalized ratio of the corresponding two and single particle distributions.

K2K1

K2

Out: direction of the mean transverse momentum of the pairSide: orthogonal to outLong: beam direction

The corresponding correlation widths are usually parameterized in terms of the Gaussian correlation radii:

The above wave function corresponds to r> R< 1 fm. Can we say anything about without knowning the small distance dynamics ? Luders and Wigner solved the problem for us:

We know δ for Ξ*(1530):

Then

The Explanation:

kr

kdk

dRdr

R

(2sin2

11

0

2

2)( )( Rr

0

2/

EEarctg

RrRr

rqrdSrqrSrdqC2)(2)( ),()0(),()()(

4/)(

2/~

220

EE

And see the last figure

à la L-W