Asymptotics of superfluid Bjorken flow

Abstract

We consider the dynamics of an expanding superfluid modeled by Mueller-Israel-Stewart theory coupled to a complex scalar field with a $U(1)$ symmetry that is spontaneously broken. This is a manageable theoretical setting for explorations of the chiral phase transition of expanding quark-gluon plasma. We study the late proper-time behavior of Bjorken flow in this physical system and find that asymptotic solutions can be expressed as a transseries of a novel form, which contains factors like $τ^{-a\ln τ}$. This transseries describes how the information encoded in the initial data is diluted in the course of dissipative evolution. These solutions retain memory of the symmetry-breaking transition and describe two qualitatively different late-time behaviors of the dynamical variables, depending on condensate relaxation rate: either a purely damped fall-off or damped oscillations. The possibility that such oscillations could be imprinted in the observed outcomes of heavy ion collision experiments is the main physical insight that follows from our analysis.

Figures (4)

• Figure 1: The factorial growth of the coefficients in \ref{['eq:toyasym']} of the frozen condensate case, where each curve corresponds to a specific log-sector, with increasing power of $\ln (\Lambda\tau)$ going from left to right.
• Figure 2: The temperature and the condensate evaluated by numerically solving \ref{['eq:eoms']} for the same set of initial conditions and parameters, for three characteristic values of the condensate relaxation rate $C_\kappa$. Note that the critical value of $C_\kappa$ corresponds to $8/3$ with the choice of parameters \ref{['eq:nice-parameters']}.
• Figure 3: The red curve is the numerical solution with $C_\kappa=1/10$ in \ref{['fig:oscillations']}, while the blue dashed curve is the result of fitting $\Lambda$ and the transseries parameters $\sigma_k$ appearing in \ref{['eq:transseries-temp']}. Note that the oscillations are due entirely to the exponential contributions to \ref{['eq:transseries-temp']} and is completely invisible to the perturbative series.
• Figure 4: The poles of the analytically continued Borel transform of the series $t_{k, 0}$ with $k=1,\dots, 58$. The condensation of poles is indicative of branch point singularities. The pattern depends on whether the value of $C_\kappa$ exceeds the critical value defined in the text, see \ref{['eq:crit-kappa']}.