Conformal Bjorken flow in the general frame and its attractor: Similarities and discrepancies with the M\"uller-Israel-Stewart formalism

Abstract

We investigate the implications of the general frame approach for conformal Bjorken flow beyond the earlier studies. We show that the power series solution at late times is not unique and is accompanied by an exact solution of the form $1/\tau$, which becomes unphysical if taken on shell. In contrast to the M\"uller-Israel-Stewart formalism, a matching between $\NFSYM$ results and the hydro expansion is only possible up to the first order, which gives rise to $\eta/s=1/4\pi$. Matching the results to the next order gives rise to causality/stability-violating values. Furthermore, we show that the pressure anisotropy in the general frame cannot capture the hydrodynamization, and we introduce an alternative measure to find the attractor. Using slow-roll expansion, we find an analytical approximation form for the attractor. We also show that the early-time behavior of attractors is related to stability and causality conditions. The attractor solutions outside the stable and causal regime give rise to reheating and negative longitudinal pressures in early times, in contrast to the stable and causal ones. We also comment on the violation of the second law of thermodynamics by the off-shell parameters. We show that for the stable and causal choice of parameters, the off-shell canonical entropy of the attractors, which is not a physical quantity, has a negative divergence in early times before tending to its on-shell limit. On the other hand, the unstable and acausal attractors have non-negative entropy divergence. We speculate that the violation of the second law by stable and causal off-shell parameters is required for stability of the first-order hydrodynamics. We investigate the analytical structure of the Borel-transformed series and find the proper relation between the poles and nonhydro modes.