Pion Number Fluctuations and Correlations in the Statistical System with Fixed Isospin

1. Pion Number Fluctuations and Correlations in the Statistical System with Fixed Isospin • Viktor Begun • Bogolyubov Institute for Theoretical Physics, • Kiev, Ukraine • Frankfurt Institute for Advanced Studies, • Frankfurt, Germany V.B., M.I. Gorenstein, O.A. Mogilevsky, Phys. Rev. C 82 (2010)

2. Motivation • It was found in 2004(V.B., Gazdzicki, Gorenstein, Becattini, Ferroni…) that exact conservation of Abelian (additive) chargessuppress multiplicity fluctuations even in thermodynamic limit • The aim is to study some aspects of non-Abelian symmetries in the statistical models • Bose – Einsteincondensation of pions (V.B., Gorenstein) Phys. Lett. B 2007 Viktor Begun

3. Partition functions for gas in GCE and CE with Q=0 where zis the one-particle partition function Viktor Begun

4. CE & GCE Mean values, fluctuations and correlations are obta-ined as the derivatives of the corresponding partition function V.B., Gaździcki, Gorenstein, Zozulya, PRC 2004 Viktor Begun

5. The prediction of hadron gas < 1 ?! Small acceptance V.B., Gazdzicki, Gorenstein, Hauer, Konchakovski, Lungwitz PRC 2006 Viktor Begun

6. Comparison with theNA49data - + V.B., Gazdzicki, Gorenstein, Hauer, Konchakovski, Lungwitz, PRC2006 Viktor Begun

7. 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 Viktor Begun

8. The ratio RN and the scaled variance for total pion number Viktor Begun

9. The ratios and Fluctuations of mesons Isospinconservation generatescorrelations between the number of neutral and charged pions. Viktor Begun

10. Correlations of mesons Correlation coefficient: has the value In the GCE all correlation coefficients are equal to zero In the CE Viktor Begun

11. Bose Statistics In the thermodynamic limit for Boltzmann statistics: and for Bose statistics: Viktor Begun

12. Bose-Einstein Condensation of pions We propose to search for the BEC in high multiplicity events of high-energy particle and/or nuclei collisions Viktor Begun Anomalous fluctuation of neutral and charged pions will indicate BEC

13. Fluctuation of0–mesons forQ=0, Nπ=const, E=const One way or another, an increase of leads to a strong increase of the fluctuations of and numbers due to the BEC effects Nikitin, Kokoulina, IHEP,Protvino(Russia) V.B., Gorenstein, PLB2007, PRC 2008 Viktor Begun

14. Summary: • 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. • The positive correlations between the numbers of neutral and positive (negative) pions are observed for I=0. • 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. Viktor Begun

15. Bose-Einstein Corelation Condensation? Viktor Begun arXiv:1007.0516v1

16. Phase diagram Project THERMALIZATION Viktor Begun V.B., M.I. Gorenstein, Phys. Lett. B(2007), Phys. Rev C (2008)

17. The degeneracy factors Pais, Annals Phys. 1960 Viktor Begun

18. 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. Viktor Begun