the approach initiated in: Anchishkin, Gavrilik, Iorgov, Eur. J .Phys. A , 2000 ;

Intercepts of multi-pion correlation functions within different deformed Bose gas models

A. Gavrilik(BITP, Kiev)

in collab. with Anastasiya Rebesh

based on: A.Gavrilik, SIGMA, 2, paper 74, p.1-12, 2006 [hep-ph/0512357] A.G., A.Rebesh, Mod. Phys. Lett. A, 2008 A.G., I.Kachurik, A.Rebesh, J. Phys. A, 2010

A.G., A.Rebesh, arXiv: 1007.5187 [quant-ph/0512357]

the approach initiated in: Anchishkin, Gavrilik, Iorgov, Eur. J .Phys. A, 2000; Mod. Phys. Lett. A, 2000 Anchishkin, Gavrilik, Panitkin, Ukr. J. Phys. A, 2004

Some next steps in: L.Adamska, A.Gavrilik, J. Phys. A, 2004 [hep-ph/0312390] A.Gavrilik, SIGMA, 2, paper 74, p.1-12, 2006 [hep-ph/0512357] Q.Zhang, S.Padula, Phys. Rev. C, 2004

Data on π+π+ and π– π– correlations( B.I.Abelev et al.(STAR collab.)., PR C80: 024905 (2009); arXiv:0903.1296

Pion Interferometry in Au+Au and Cu+Cu Collisions at √sNN=62.4 and 200 GeV )

NB! ”the trend” is seen:

1) Monotonic increasing; 2) Concavity upwards;

3) Saturation by a constant < 1 (= case of Bosons)

OUR GOLE:● use deformed oscillators(DOs)(=deformed bosons),

develop corresp. version of deformed Bose gas – – and try to model the above “trend”

● we examine a number of variants of DOs from two very different classes:

-- Fibonacci oscillators,

with property of en.levels:

En+1= λ En + ρ En-1

-- quasi-Fibonacci oscillators,

with property of en. levels:

Typical: p,q-oscillators,

for which λ=p+q, ρ= -pq

Typical: μ-oscillator, for which λn=λ(n,μ), ρn=ρ(n,μ) with definite

En+1= λn En + ρn En-1

A.G., I.Kachurik, A.Rebesh, J. Phys. A, 2010

Why deformed q-oscillators, and (ideal) q-Bose gas models?

• (a) finite proper volume of particles; • (b) substructure of particles; • (c) memory effects; • (d) particle-particle, or particle-medium interactions (i.e, non-ideal pion or Bose gas);• (e) possible existence of non-chaotic (partially coherent) components of

emitting sources; effects from long-lived resonanses (e.g., put in the

core-halo picture); • (g) non-Gaussian (effects of) sources; • (h) fireball -- short-lived, highly non-equilibrium, complicated system.

If we use 2-parameter say q,p-deformed Bose gas model, then:1) It gives formulae for 1-param. versions (AC,BM,TD,…) as special cases;

2) q,p-Bose gas model accounts jointly for any two independent reasons (to q-deform) from the above list.

About items (b) & (d) --- on next slide

Important also in

other contexts

(b) QUASIBOSONS VS deformed oscillators:

It is shown (e.g. S.Avancini, G.Krein, J.Ph.A 1995) that algebra ofquasiboson (composite boson) operators,

realized in terms of constituent’s fermionic operators, indeed modifies: If: , (a,a†,b,b† -- fermionic)

then:

where

and Δαβ can be made using deformed oscillator!

(more detailed analysis: A.G. & Yu.Mishchenko, in preparation)

(d) Account of interparticle interactions in deformation, then: non-ideal Bose gas ideal deformed Bose gas

(e.g., A.Scarfone, P.Narayana Swamy, J.Ph.A 2008);

Deformed oscillators of “Fibonacci class” --- q,p-oscillator:

(q,p-bracket)

--- q-oscillators: 1) if p=1 - AC (Arik-Coon) type,

2) if p=q.-1- BM (Bied.-Macfarlane)

type,

3) if p=q - TD (Tamm-Dancoff) type,

and many others in this class: 4)…), e.g., if p = exp (½(q-1)) (A.G., A.Rebesh, Mod.Phys.Lett.A 2008) .

n-Particle correlations in q,p-Bose gas model.Ideal gas of q,p-bosons: thermal averages, one-particle distribution:

(AC type q-Bose)(q,p-Bose) Bose

q,p-Bose →

AC type q-Bose

BM q-Bose

Intercepts of 2-particle correl. Functions& their asymptotics

NB: Asymptotics (βω→∞) of intercepts is given solely by the deformation parameter(s) q or q,p

(BM type, q=exp(iθ

))

Intercepts of 3-particle correl. functions & their asymptotics

NB: Asymptotics (βω→∞) of intercepts are given

solely by the deformation parameter(s) q or q,p

(BM type, q=exp(iθ))

Intercept (maxim.value) of n-particle correlation function for the p,q-Bose gas model

General formula: L.Adamska, A.Gavrilik, J. Phys. A (2004)

Asymptotics (βω→∞) of intercept is given by q (or q & p):

q,p

(A.Gavrilik, SIGMA, 2 (2006), paper 74,1-12, [hep-ph/0512357] )

pure Bose value

Intercepts in pq-Bose gas model VS exper.data on pion correlations (NA44 at CERN: I.G.Bearden et al., Phys.Lett. B, 2001)

section 2

section 1

βω Equating our f-las for λ(2) & λ(3)

to exper.values yields two surfaces

Two sections: 1) set • 2) set •

NB: we have 1σcontours

section 2section 1

λpq(2)

λpq(3)

2 phenom. param.

Csanad (for PHENIX), NP A774 (2006) nucl-ex/0509042

Deformed oscillators of “quasi-Fibonacci” class

,[ ]( )

1 ( )(1 )

N Np qN p q

NN p q N

--- μ-oscillator: structure f-n (μ-bracket)

--- (μ;p,q)-oscillators: 1) μ-AC type, 2) μ-BM type,

3) μ-TD type,

(A.G., I.Kachurik,A.Rebesh, J.Ph. A 2010)

1( 1) ( )

1 ( 1) 1

N Na a a a N N

N N

( ),a a N ( )1

NN

N

2 2 3 3 2 2 3 3

(2)22 2 3 3

( 1)(1 )(1 ( 1) ( 1) ( 1) )1

(1 )

N N N N N N N N

N N N N

8

(2) 123

1

0

,

( 1)

rr

r

s s s

s

N

N

0.8306 0.8332

0.75660.7680

Intercept λμ(2) of 2-pion correl. function VS mean momentum,

with a s y m p t o t i c s:

A. Gavrilik, A. Rebesh, Intercepts of momentum correlation functions in μ-Bose gas

model and their asymptotics, arXiv: 1007.5187

Behaviour of , μ-Bose gas

A. Gavrilik, A. Rebesh,

2 2(3)

2 2 2

6(3 1)(2 1)(9 1)(4 1)

( 1) ( 1)asympt

(3)

Intercepts of momentum correlation functions in μ-Bose gas model and their asymptotics, arXiv: 1007.5187

Intercept λμ(3) of 3-pion correl. function VS particle mean momentum KT:

μ=0.1μ=0.15

3.6097

3.0083

(to μ3 )

(to μ5 )

Asymptotics is given as:

Characteristic function r(3)(KT) made from λ(2) & λ(3) :

μ-Bose gas

BM type q-Bose gas

π/30

π/4 (U.Heinz, Q.Zhang, PRC 1997)

Also, it has sense of cos Φ – spec. phase (H.Heiselberg, A.Vischer, nucl-th/9707036)

Data- Theory

NB: centrality (left) temperature (right)

SUMMARY• The explored models, even from different classes (Fibo. or quasi-

Fibo.), provide KT dependence, and naturally mimic ’the trend’

• We have explicit f-las for 2-, 3-, …, n-particle correlation intercepts

λp,q(n)(K,T), λμ

(n)(K,T) & rp,q(3)(K,T) or rμ

(3)(K,T)

(exact in p,q-case, approximate in μ-case)• Basic fact: asymptotics is given in terms of deform. parameters

solely:

λp,q(n),

asymp. =f(p,q); λμ

(n), asymp. =g(μ), etc.

• Having fixed deform. parameter (from asymptotics, using high KT

bins), the temperature is fixed from lower KT bins

• So, we await much more experim. data! (see next.slide)

What is needed from experiment. side,to make choise from among many models:

• More data on π±π± correl. inters. λ(2)(for higher & lower KT)

• Data on π±π±π± correl. inters. λ(3) (for wide range of KT)

• Data on the function r(3)(KT) (for wide range of KT )

THANK YOU!