Nonequilibrium meson production in strong fields

Nonequilibrium meson production in strong fields

L. Juchnowski, D. Blaschke, T. Fischer, S.A. Smolyansky

We develop a kinetic equation approach to nonequilibrium pion and sigma meson production in a time-dependent, chiral symmetry breaking field (inertial mechanism). We investigate the question to what extent the low-momentum pion enhancement observed in heavy-ion collisions at CERN - LHC can be addressed within this formalism. In a first step, we consider the inertial mechanism for nonequilibrium production of mesons and their simultaneous decay into pion pairs for two cases of mass evolution. The resulting pion distribution shows a strong low-momentum enhancement which can be approximated by a thermal Bose distribution with a chemical potential that appears as a trace of the nonequilibrium process of its production.

  • Institute of Theoretical Physics, University of Wroclaw, 50-204 Wroclaw, Poland,
    Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, 141980 Dubna, Russia
    National Research Nuclear University (MEPhI), 115409 Moscow, Russia
    Department of Physics, Saratov State University, 410026 Saratov, Russia

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1 Kinetic theory for inertial particle production

The formulation of a kinetic theory approach to the problem of nonequilibrium particle production in strong fields has been advanced recently in studies of the dynamical Schwinger effect for plasma creation in high-intensity lasers [1, 2]. Strong and time-dependent fields govern also the particle production in ultrarelativistic heavy-ion collisions. Here the kinetic theory approach has been developed, e.g., for studying the role of time-dependent masses (the so-called ”inertial mechanism” [3]) in the course of the chiral symmetry breaking transition for pion production [4] and photon production [5]. In the present work we develop the kinetic approach to nonequilibrium pion production in a time-dependent chiral symmetry breaking homogeneous field and investigate the question to what extent the low-momentum pion enhancement observed in heavy-ion collisions at CERN - LHC being discussed as Bose-Einstein condensation of pions [6] can be described within this formalism. To this end we set up a detailed study of the three main processes that are intertwined in this case: (a) the nonequilibrium meson production in the time-dependent external field, (b) the decay and (c) the rescattering and formation of the Bose condensate [7, 8]. The results shall be compared with the effect observed at the LHC. Here we report on steps (a) and (b).

In order to address the question of the and meson production in a heavy-ion collision we consider the simplified situation of one species of pions only, i.e. we neglect the isospin degree of freedom. We describe the evolution of the single-particle distribution functions and as solutions of the coupled Boltzmann equations for these relativistic bosons with the dispersion relations,


For the evolution of the mass of the , we apply the following expression


where MeV is critical temperature for the chiral transition and is the vacuum mass. The time at the begin of the 3-dimensional spherical expansion is fm/c, corresponding to a Hubble flow velocity of c for gold nuclei with radius fm. The initial temperature is taken to be .

Here we assume spatial homogeneity, i.e. the distribution functions have no dependence on position and despite their momentum dependence we neglect derivative terms so . In our simplified model the evolution of and are dominated by production in the evolving chiral condensate (inertial mechanism) and the subsequent decay . The Boltzmann transport equation for where the rescattering of pions is not yet considered at this step, reads


Note that the source term for production occurs just due to the time dependence of the dispersion law (1) and works in the absence of pions. It is the first term at the right hand side of Eq. 3, with the following definitions


The last two terms in Eq. 3 are due to the decay and regeneration . For the pions, the dispersion law is time-independent for so that there is no inertial production mechanism


For the decay and regeneration we assume a constant matrix element so that the momentum dependence of is simply given by the momentum conserving delta-function


. ^

-1 ,  i=π,σ .\@close@row In the subsequent evolution the mass departs from the pion mass and rises towards its vacuum value (chiral symmetry breaking) while the the pion mass keeps the value at the onset of the chiral symmetry breaking due to chiral protection.

According to our model the chiral transition takes place because of very fast 3-dimensional expansion (and thus dilution and cooling) of the fireball. Therefore it is reasonable to assume that the process is strongly suppressed. In such a case one can disregard the terms containing in Eqs. (3) and (5). The resulting system of kinetic equations to be solved is given by




The results for the evolution of the distribution functions due to the coupled kinetics in the evolving scalar background field are presented in Fig. LABEL:fig:evolution. One can notice that the distributions in the middle panels of Fig. LABEL:fig:evolution show oscillatory behaviour which is common for the kinetic approach to particle production and has been discussed also in the context of the dynamical Schwinger effect in lasers [1, 2].

\hb@xt@ . c an be described within this formalism. For simplicity we have neglected the isospin of pions here. Along the lines of this project, we have performed the first two steps (a) and (b) of a detailed study of the three main processes that are intertwined in this case: (a) the nonequilibrium meson production in the time-dependent external field, (b) the decay. The step (c) consisting in the inclusion of rescattering and formation of the Bose condensate [7, 8] as well as the comparison of the obtained results with the effect observed at the LHC is subject to current research. At the present stage we can conclude that the distribution function depends strongly on magnitude, shape and duration of the chiral symmetry breaking (inertial) source term . The distribution function before rescattering can roughly be approximated (for the more realistic case GeV) by a Bose distribution with a nonequilibrium pion chemical potential MeV and a freeze-out temperature MeV.


We are grateful to V. Begun, W. Florkowski, P.M. Lo, G. Röpke, L. Turko and D.N. Voskresensky for their enlightening discussions. We thank also E.-M. Ilgenfritz and A. Tawfik for their continued interest in the progress of this project. This research is supported in part by the Polish Narodowe Centrum Nauki (NCN) under grant number UMO-2014/15/B/ST2/03752 (L.J. and D.B.) and UMO-2013/11/D/ST2/02645 (T.F.).



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  • [3] A.V. Filatov, A.V Prozorkevich, S.A. Smolyansky, and V.D. Toneev, Inertial mechanism: dynamical mass as a source of particle creation Phys. Part. Nucl. 39, 886 (2008).
  • [4] A. V. Filatov, A. V. Prozorkevich, S. A. Smolyansky and D. B. Blaschke, Low momentum -meson production from evolvable quark condensate, Phys. Part. Nucl. 39, 1116 (2008).
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  • [6] V. Begun and W. Florkowski, Bose-Einstein condensation of pions in heavy-ion collisions at the CERN Large Hadron Collider (LHC) energies, Phys. Rev. C 91, 054909 (2015).
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  • [9] M. Kataja and P. V. Ruuskanen, Nonzero Chemical Potential and the Shape of the Distribution of Hadrons in Heavy Ion Collisions, Phys. Lett. B 243, 181 (1990).
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