1. 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

2. 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

3. 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

4. m=134.9 MeV Pion gas in heavy-ion collisions T=167 MeV m=126 MeV
SPS data: 200 GeV/A
LHC data: Pb+Pb
[Kataja, Ruuskanen, PLB 243(1990)181]
[Begun, Florkowski, Rybczynski, PRC90 (2104) ]
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

5. 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 50 GeV [Ryadovikov et al PAN75 (2012)]
SPS data: 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

6. … 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 …

7. 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

8. 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)

9. 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

10. 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

11. 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

12. 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]

13. 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.

14. 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, ] [Peshier, Giovannoni, JPCS668 (2016) ]
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]

15. 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

16. 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

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

18. 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