Parametric Resonance versus Forced Oscillation in Time-Evolution of Quantum Meson Fields

The time-evolution of quantum meson fields in O(4)-linear sigma model is treated approximately. It is shown that the amplification of the amplitudes of pion modes with low momenta occurs by means of both the parametric resonance and the forced oscillation. 1 typeset using PTPTEX.sty It is interesting to study the dynamics of the chiral phase transition in the context of relativistic heavy-ion collisions, such as the problem of the formation of a disoriented chiral condensate. One of the important theoretical aspects is to investigate the time-evolution of the order parameter of the chiral phase transition. As for the fluctuation modes around the order parameter, the amplification of the amplitudes must occur accompanying with the relaxation of the chiral order parameter. Recently, the present author has investigated the time-evolution of a collective meson field in the context of the dynamical chiral phase transition. 1) It has been seen that the amplitudes of quantum fluctuations with low momenta around the mean field configuration have been amplified with the direction of pion modes in the O(4)-linear sigma model. As is indicated by many authors, this phenomena may be understood in terms of a parametric amplification. 2) 5) However, there may be another mechanism to amplify the fluctuation modes. Actually, it is seen that the amplitude of the pion mode with higher momentum without k = 0 can be amplified in the previous paper. 1) In this paper, it is pointed out that there is another mechanism to amplify the quantum fluctuation modes through the chiral phase transition, namely, that a forced oscillation works as well as a parametric resonance. Before treating the O(4)-linear sigma model, it is instructive to recapitulate the ingredients of the parametric resonance and the forced oscillation. Let us consider the following equation of motion for x(t) : ẍ+ ω 0(1− h cos γt)x = 0 . (1) The parametric resonance occurs around γ = 2ω0/m (m = 1, 2, · · ·). 6) For m = 1, the resonance region of the frequency γ is given as −hω/2 < ǫ < hω/2 , for γ = 2ω0 + ǫ , ǫ ≪ ω0 , (2) where we treated h as a small parameter, h ≪ 1. Similarly, for m = 2, we obtain −(5/24) · hω0 < ǫ < (1/24) · h ω0 , for γ = ω0 + ǫ , ǫ ≪ ω0 . (3) In general, for given integer m, the unstable solution of x(t) is obtained as the form x(t) ∝ e×(oscillation and constant parts), where s is of the order of h. Namely, the amplification becomes slower in time as the integer m becomes larger. The resonance region also becomes narrow with the order of h, that is, −O(h) < ǫ < O(h). If h is not so small, we have to deal with the equation (1) directly. The equation (1) is known as Mathieu’s equation and the properties of this equation are investigated in detail. 7) In the following investigation, h is actually small fortunately. Thus, it is not necessary to take care of the full treatment of