Kinetic approach to a relativistic BEC with inelastic processes

The phenomenon of Bose-Einstein condensation is investigated in the context of the color-glass-condensate description of the initial state of ultrarelativistic heavy-ion collisions. For the first time, in this paper, we study the influence of particle-number changing $2\leftrightarrow 3$ processes on the transient formation of a Bose-Einstein condensate within an isotropic system of scalar bosons by including $2\leftrightarrow 3$ interactions of massive bosons with constant and isotropic cross sections, following a Boltzmann equation. The one-particle distribution function is decomposed in a condensate part and a nonzero momentum part of excited modes, leading to coupled integro-differential equations for the time evolution of the condensate and phase-space distribution function, which are then solved numerically. Our simulations converge to the expected equilibrium state, and only for ${\sigma }_{23}/{\sigma }_{22}\ll 1$, we find that a Bose-Einstein condensate emerges and decays within a finite lifetime in contrast to the case where only binary scattering processes are taken into account, and the condensate is stable due to particle-number conservation. Our calculations demonstrate that Bose-Einstein condensates in the very early stage of heavy-ion collisions are highly unlikely, if inelastic collisions are significantly participating in the dynamical gluonic evolution.

Figure 1

Differential fraction of the isotropic particle density with respect to the momentum and for four different regimes of the cross section ratio, only binary scattering ($\mathrm{a}+\mathrm{e}$), elastically dominated scattering ($\mathrm{b}+\mathrm{f}$), balanced scattering ($\mathrm{c}+\mathrm{g}$), and inelastically dominated ($\mathrm{d}+\mathrm{h}$), with $m=100\mathrm{MeV}$ at $f_0=0.45$ at various times. The rows separate two sequential time periods. In ($\mathrm{a}+\mathrm{e}$) and ($\mathrm{b}+\mathrm{f}$), a condensate is present, but in (f), it decays. The black dashed lines (bottom row) depict the individual expected equilibrium states.

Figure 2

Time evolution of normalized particle (upper panel) and condensate (lower panel) densities for particles with $m=100\mathrm{MeV}$ and $f_0=0.45$. The bluish curves show runs without the onset of condensation ${\mu }_{\mathrm{eff}}<m$. The scattered dots refer to the time stamps of Fig. 1, and the black lines mark the individual expected equilibrium values.

Figure 3

Time evolution of the effective chemical potential, with $m=100\mathrm{MeV}$ at $f_0=0.45$. Bluish curves (inelastic) represent runs where condensation was not feasible, contrary to the greenish (inelastic) curves and red dashed line (elastic). The black line relates to the equilibrium condition for $2\leftrightarrow 2$ processes ($\mu =m$), and the dashed line relates to inelastic processes ($\mu =0\mathrm{GeV}$).

Figure 4

Time evolution of the effective chemical potential with respect to the masses at strongly overpopulated initial condition, $f_0=2$, for three different masses [100 (left), 300 (middle), and 500 MeV (right)] and for different values of ${\sigma }_{23}/{\sigma }_{22}$. Solid black lines correspond to the equilibrium condition for $2\leftrightarrow 2$ processes, and the dashed lines correspond to the inelastic processes.