Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation

TL;DR

This Letter considers the Müller-Israel-Stewart theory applied to a longitudinally expanding quark-gluon plasma system and identifies hydrodynamics as a universal attractor without invoking the gradient expansion, giving strong evidence for the existence of this attractor.

Abstract

Consistent formulations of relativistic viscous hydrodynamics involve short lived modes, leading to asymptotic rather than convergent gradient expansions. In this Letter we consider the Mueller-Israel-Stewart theory applied to a longitudinally expanding quark-gluon plasma system and identify hydrodynamics as a universal attractor without invoking the gradient expansion. We give strong evidence for the existence of this attractor and then show that it can be recovered from the divergent gradient expansion by Borel summation. This requires careful accounting for the short-lived modes which leads to an intricate mathematical structure known from the theory of resurgence.

Figures (3)

• Figure 1: The blue lines are numerical solutions of Eq. (\ref{['fdef']}) the for various initial conditions; the thick, magenta line is the numerically determined attractor. The red, dashed and green, dotted lines represents first and second order order hydrodynamics. The upper plot was made with parameter values appropriate for ${\mathcal{N}}=4$ SYM theory, while the lower plot has both $\eta/s$ and $C_{\tau \Pi }$ increased by a factor of 3. Note that in the latter case the hydrodynamic attractor is attained at larger values of $w$, as expected.
• Figure 2: The large order behaviour of the hydrodynamic series. The slope is consistent with location of the singularity nearest to the origin as given by Eq. (\ref{['ratio']}).
• Figure 3: The hydrodynamic attractor (magenta), compared with the resummation result (cyan, dot-dashed) and the gradient expansion of order 1 (red, dashed) and 2 (green, dotted).