Fluctuation in a non-ideal pion gas with the dynamically fixed number of particles E.E. Kolomeitsev, D.N. Voskresensky and M.E. Borisov Matej Bel University, Banska Bystrica, Slovakia JINR, Dubna, Russia National Research Nuclear University “MEPhI”, Moscow, Russia COST ACTION

BEC in ideal Bose gases Ideal gas of boson with mass m Compton wave length: Critical temperature Tc is given by equation prediction: Bose and Einstein (1924) realization: Cornell, Ketterle, Wieman (1995) Academic interest P.T. Landsberg, Thermodynamics with Quantum Statistical Illustrations (1961) Particle-antiparticle system with charge conservation Haber, Weldon PRL 46 (1981) 1497 Standard model, Higgs J.I. Kapusta, PRD 24 (1981) 426

Pion gas in heavy-ion collisions central (midrapidity) regions in HIC kinematical separation of heavier baryon component QGP hadronization meson-resonance-enriched medium resonance decays pion-enriched medium for [Goity, Leutwyler PLB 228(89)517] pion absorption rate (inelastic collision) is smaller than rescattering rate (elastic collisions) equilibration time Can a quantum regime be reached? Possible signals? time T m On some time interval we deal with a pion gas with a approximately constant number of particles

m=134.9 MeV Pion gas in heavy-ion collisions T=167 MeV m=126 MeV SPS data: O+Au @ 200 GeV/A LHC data: Pb+Pb [Kataja, Ruuskanen, PLB 243(1990)181] [Begun, Florkowski, Rybczynski, PRC90 (2104) 014906] m=134.9 MeV T=167 MeV m=126 MeV Mishustin et al, PLB 276 (1992) 403; PRC 51 (1995) 2099 … effects of quantum statistics could be important

Pion gas in heavy-ion collisions The appearance of a pion BEC and its consequences were discussed by Voskresensky, JETP 78 (1994) 793 pion BEC from resonance decays [Ornik Plumer, Strottman, PLB 314 (1993) 401] fluctuations of particle number and relative isospin composition calculated by Begun, Gorenstein [PLB 653 (2007) 190] for an ideal gas divergent at Tc SVD-2 Collaboration observed an enhancement in variance of #charged pion/#total in events with high pion multiplicity in p+p@ 50 GeV [Ryadovikov et al PAN75 (2012)] SPS data: O+Au @ 200 GeV/A T=167 MeV, m=126 MeV LHC data: Pb+Pb T=138 MeV, m=134.9 MeV pion interaction is to be included

… study influence of a pion interaction on fluctuations … We will use a simplified lf4 model to analyze -- how to describe the system of p+,p-,p0 meson with dynamically fixed number of particles? -- general properties of pion spectra -- equation of state (which isospin configuration is preferable) -- can BEC be formed in HIC? … study influence of a pion interaction on fluctuations …

Lagrangian for a system with dynamical fixed number of pions Weinberg’s chiral Lagrangian Introduce charged and neutral pion fields Introduce creation and annihilation operators Write the Lagrangian in terms of and keep terms with equal number of c.o. and a.o

only 2 chemical potentials (3 complex fields) keeps the number of each pion species fixed equilibration among pion species (fast reactions) in equilibrium only 2 chemical potentials The system is described by 2 numbers: total number of pions N and total charge Q We drop (slow reactions) collision term in kinetic equation (gain, loss terms)

Pion spectrum in Hartree approximation (only tadpole term) replacement: fi is a momentum distribution function Divergent vacuum parts are small after renormalization effective mass The partial densities of pions are determined as

Pion gas with Q=0 electric charge different compositions: (i) symmetric gas p+,p-, p0; (ii) gas of p+,p- ; (iii) gas of p0 main expressions are the same only coefficients are different [Ivanov, Knoll, Voskresensky, NPA672 (2000) 313 Voskresnesky, NPA812 (2008) 158] can be used for non-eq. distributions In thermal equilibrium free-energy density

1st order p.t.? pion effective mass increases with a temperature decrease at pion chemical potential at “induced” Bose condensation a pole in momentum distribution function

The smaller is the number of components, the higher is the value Tcind. Account of interaction decreases Tc Strong difference from the non-relativistic case. In non-relativistic boson gas mean-field HF terms do not shift Tc [Baym et al, EPJB 24 (2001) 107]

At fixed total density and Q=0 the symmetric pion gas has a lower free-energy than the gas of other isospin compositions Thus the system with Q=0 having initially an asymmetric composition will after a while come to the symmetric state with the equal number of pions of each species.

Creation of a pion BEC? non-equilibrium overcooling effects [Voskresensky, JETP 78 (1994) 793] non-equilib. pions from resonance decays [Ornik et al., PLB 314 (1993) 401] decomposition of a “blurred phase" of hot baryonless matter [Voskresensky NPA 744 (2004) 378] sudden hadronization of supercooled QGP [Csorgo Csernai PLB 333 (1994) 494] decay of gluon BEC preformed at the initial stage of heavy-ion collision [Blaizot NPA 873(2012) 68] [Xu, Zhoum, Zhuang, Greiner, PRL114, 182301] [Peshier, Giovannoni, JPCS668 (2016) 012076] glueball condensation [Kochelev, Phys.Part.Nulc.Lett 13 (2016) 149] The critical condition m=mp* cannot be reached in an isentropic expansion [Greiner,Gong, Mueller, PLB316, 226]

BEC from non-equilibrium pion gas Equilibration is much faster than expansion Initial non-equilibrium distribution of pions is characterized by Ein and nin After a while the system riches the thermal equilibrium characterized by m and T Bose-enhancement formation time non-eq. distributions [Semikoz, Tkachev PRL74 (1995) 3093] symmetric pion gas

Susceptibilities and fluctuations W is the grand thermodynamic potential free gas l=0 Divergent! cs can be related to observed particle number fluctuations normalized variance Divergent? skewness

Particle number fluctuations final expression These integrals diverge at Tc! perturbation theory does not work! Dispersion and skewness remain finite at Tc!

Conclusions Pion-enriched gas can be created in HIC The number of pions is approximately fixed on a final stage of collision collision rate >> absorption rate At freeze-out the pion gas can be close to a quantum limit mp~mp Interaction lead to an increase of pion mass and a decrease of Tc Pion BEC can be created in equilibration of some initial non-equilibrium pion component Pion number fluctuation remains finite at Tc if a pion interaction is take into account non-perturbatively Dedicated analysis of selected multi-pion events is needed