Pion Number Fluctuations and Correlations in the

Statistical System with Fixed Isospin

V.B., M.I. Gorenstein, O.A. Mogilevsky,

Phys. Rev. C 82 (2010)

Viktor BegunBogolyubov Institute for Theoretical Physics, Kiev, Ukraine

Frankfurt Institute for Advanced Studies, Frankfurt, Germany

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Motivation

It was found in 2004 (V.B., Gazdzicki, Gorenstein, Becattini, Ferroni…) that exact conservation of Abelian (additive) charges suppress multiplicity fluctuations even in thermodynamic limit

The aim is to study some aspects of non-Abelian symmetries in the statistical models

Bose – Einstein condensation of pions (V.B., Gorenstein) Phys. Lett. B 2007

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Partition functions for gas in GCE and CE with Q=0

where z is the

one-particle

partition function

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CE & GCE

V.B., Gaździcki, Gorenstein, Zozulya, PRC 2004

Mean values, fluctuations and correlations are obta-ined as the derivatives of the corresponding partition function

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The prediction of hadron gas

V.B., Gazdzicki,Gorenstein, Hauer,

Konchakovski, LungwitzPRC 2006

Small acceptance

< 1 ?!

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Comparison with the NA49 data

+-

V.B., Gazdzicki, Gorenstein, Hauer, Konchakovski, Lungwitz, PRC 2006

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The partition function of pion gas with total isospin I=0

Pions are transformed under vector

(adjoint) representation of the SU(2)

group. This group has three para-

meters which can be chosen as

Euler angles. In this case the dia-

gonal matrix elements and the partition function have the following form

Turko and Rafelski, Eur. Phys. J. 2001; Turko, Acta Phys. Polon. 2002

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The ratio RN and the scaled

variance for total pion number

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The ratios and Fluctuations of mesons

Isospin conservation generates correlations between the number of neutral and charged pions.

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Correlation coefficient:

has the value

In the GCE all correlation coefficients

are equal to zero

In the CE

Correlations of mesons

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Bose Statistics

In the thermodynamic

limitfor Boltzmann statistics:

and for Bose statistics:

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Bose-Einstein Condensation of pions

We propose to search for the BEC in high multiplicity events of high-energy particle and/or nuclei collisions

Anomalous fluctuation of neutral and charged pions will indicate BEC

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One way or another, an increase of leads to a strong increase of the fluctuationsof and numbers due to the BEC effects

Fluctuation of 0 – mesons for Q=0, Nπ= const, E=const

Nikitin, Kokoulina,

IHEP, Protvino (Russia)

V.B., Gorenstein, PLB 2007, PRC 2008

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Summary:

1. For neutral pions there is the enhancement of the fluctuations. For charged pions the isospin conservation suppresses fluctuations similar to that in the CE with Q=0.

2. The positive correlations between the numbers of neutral and positive (negative) pions are observed for I=0.

3. The colorless system of SU(2)-color gluons corresponds to the I=0 pion gas. We thus conclude that fluctuations of the number of gluons with different colors are different. This difference survives in the thermodynamic limit.

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Bose-Einstein Corelation

Condensation ?

arXiv:1007.0516v1

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Phase diagram

V.B., M.I. Gorenstein, Phys. Lett. B (2007), Phys. Rev C (2008)

Project THERMALIZATION

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Pais, Annals Phys. 1960

The degeneracy factors

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The partition function of pion gas with total isospin I=0

The change of variables and integration gives:

The final expressions for the partition

functions correspond to .

Taking the derivatives of Z over T

and V one obtains the thermody-

namical functions of the system.