^{1}

^{2}

^{3}

^{4}

^{3}.

Hydrodynamic excitations corresponding to sound and shear modes in fluids are characterized by gapless dispersion relations. In the hydrodynamic gradient expansion, their frequencies are represented by power series in spatial momenta. We investigate the analytic structure and convergence properties of the hydrodynamic series by studying the associated spectral curve in the space of complexified frequency and complexified spatial momentum. For the strongly coupled

Hydrodynamics is an established universal language for describing near-equilibrium phenomena in fluids

There are two basic physics principles that constrain possible forms of the constitutive relations: symmetry and the derivative expansion. Symmetry is what distinguishes different types of fluids. The derivative expansion is a reflection of the fact that hydrodynamics is only an effective description on length scales much larger than the microscopic scale (such as the mean free path). Thus, the constitutive relations are schematically written as

The naive expectation is that going to higher orders in the derivative expansion improves the hydrodynamic description of the fluid, similar to how the Navier-Stokes equations improve the perfect-fluid approximation by including the viscous effects. The purpose of this Letter is thus to address the following foundational question: viewed as an expansion in small gradients, does the hydrodynamic derivative expansion in fact converge?

In order to make this question precise, we will choose a specific physical quantity whose exact value can be compared with the prediction of the derivative expansion. For fluids, the characteristic feature of the hydrodynamic description is the existence of gapless modes: small near-equilibrium fluctuations of the fluid whose frequencies

Besides the general physics interest in the foundations of hydrodynamics, our motivation comes from the success of the relativistic hydrodynamic framework to describe the quark-gluon plasma produced in the collisions of heavy nuclei

An important comment has to be made before we proceed. The expansion

In order to understand the origin of the series

In what follows, we focus on relativistic hydrodynamics for concreteness

For example, assuming analyticity of the function

An example of a quantum field theory in which the dispersion relations

To compute the coefficients

The results are shown in Fig.

Coefficients of the expansions

The origin of the critical values

Poles of the retarded two-point function of the energy-momentum tensor in the complex

We have shown that the gradient expansions for the hydrodynamic shear and sound frequencies in the strongly coupled

Returning to the question of the unreasonable effectiveness of hydrodynamics, we note that the derivative expansion in relativistic hydrodynamics has been previously argued to diverge

Finally, we comment on the dependence of the radii of convergence on coupling, limiting the discussion to the sound mode in first-order hydrodynamics, where the first nontrivial critical point occurs at

S. G. was supported by the U.S. DOE Grant No. DE-SC0011090. P. K. was supported in part by NSERC of Canada. The work of P. T. is supported by an Ussher Fellowship from Trinity College Dublin.