the approach initiated in: anchishkin, gavrilik, iorgov, eur. j .phys. a , 2000 ;
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Intercepts of multi-pion correlation functions within different deformed Bose gas models A. Gavrilik (BITP, Kiev) in collab. with Anastasiya Rebesh. - PowerPoint PPT PresentationTRANSCRIPT
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!