Causal UV completions of relativistic hydrodynamics

Abstract: Relativistic hydrodynamics successfully provides an effective field theory description for the low energy regime of many out-of-equilibrium systems. On the other hand, in this paper we proof that any stand-alone hydrodynamic EFT is inherently acausal and therefore requires the addition of transient UV modes in order to restore causality. This is made possible by the exponential decay of dissipative hydrodynamics in a majority of the lightcone, allowing the possibility of a causal description that still reduces to the hydrodynamic one at late timescales. We then investigate the emergence and possible restrictions of the non-hydrodynamic modes in these causal UV completions.

• The paper shows that isolated hydrodynamic modes induce acausal (superluminal) propagation unless corrected by UV modes.
• It employs rigorous theorems and saddle-point analysis to reveal exponential decay rates concentrating hydrodynamic contributions along the sound trajectory.
• The study constructs causal retarded correlators by integrating UV completions, thus setting constraints on non-hydrodynamic modes and transport coefficients.

Summary of "Causal UV completions of relativistic hydrodynamics" (2605.21377)

Overview

The paper offers a rigorous examination of the intrinsic acausality of standalone (dissipative) hydrodynamic Effective Field Theories (EFTs) in relativistic settings, establishing that such theories necessarily admit superluminal signal propagation unless supplemented by non-hydrodynamic (UV) modes. It demonstrates—through explicit mathematical theorems—that restoration of causality demands UV completions which feature transient modes in the correlation structure. The analysis systematically investigates the decay properties of hydrodynamic modes, the conditions under which causal completions are possible, and the emergent constraints on non-hydrodynamic sectors.

Mathematical Foundations and Core Results

The paper provides a formal basis for why any isolated hydrodynamic mode, illustrated by paradigmatic cases like Fick's diffusion, leads to acausal behavior. The theorem on the acausality of hydrodynamics rigorously proves that correlation functions from purely hydrodynamic dispersion relations $G(t,x)$ possess non-vanishing support outside the lightcone, violating microcausality requirements.

Further, the causality constraint on pole locations,

$\Im \omega(k) \leq |\Im k|,$

is emphasized as central for any retarded correlator in relativistic QFTs. The discussion carefully distinguishes hydrodynamic modes from non-hydrodynamic (transient) degrees, explicitly contrasting their analytic structures, and showing that the inclusion of UV modes is essential for removing acausal support from the correlation function.

Hydrodynamic Decay and Late-Time Behavior

By employing steepest descent methods and rigorous saddle-point analysis, the study shows that hydrodynamic modes exhibit exponential decay across nearly the entire lightcone except along the sound speed trajectory, $v = c_s$. The generalized decay rate $\Gamma(v)$, increasing monotonically with $|v - c_s|$, characterizes the suppression of hydrodynamic contributions away from the sound propagation direction. This exponential decay is formally shown to be a robust feature given that the hydrodynamic dispersion relation maintains a finite imaginary gap for $k \neq 0$, enforcing that hydrodynamic late-time dynamics are restricted to a narrow sector of the lightcone.

The implications are twofold:

• Hydrodynamics cannot provide a universal late-time description across the lightcone unless supported by suitable UV modes.
• The minimal thermalization timescale for non-hydrodynamic modes (e.g., for pure diffusion, $\tau_R \geq 4D$) can be directly bootstrapped from the hydrodynamic decay rate.

Construction and Constraints of Causal UV Completions

A key result is the theorem demonstrating that for any hydrodynamic dispersion relation with subluminal sound speed and finite radius of convergence, there always exists a causal UV completion. Specifically, the paper constructs causal retarded correlation functions in such a way that the hydrodynamic pole dominates the mode structure within a region $\mathcal{K}$ of $k$-space, while UV (non-hydrodynamic) corrections enforce the vanishing of $G(t,x)$ outside the lightcone.

Practical realization involves:

• Compact support for $\Im \omega(k) \leq |\Im k|,$0 in $\Im \omega(k) \leq |\Im k|,$1,
• Exponential suppression and redistribution of hydrodynamic tails outside the lightcone,
• Conservation-consistent addition of UV modes, realized as branch cuts or peaks (e.g., for diffusion, ballistic transport contributions localized at the lightcone edge).

These results imply that no causality constraints can universally restrict hydrodynamic transport coefficients; rather, the regime of hydrodynamic applicability is modulated by the underlying UV completion.

Emergence and Structure of Non-Hydrodynamic Modes

The analysis bootstraps constraints on the non-hydrodynamic sector, showing that causality enforcement requires transient modes with decay rates matching or exceeding hydrodynamic decay rates outside the lightcone. For pure diffusion, the minimal gap of the non-hydrodynamic sector is $\Im \omega(k) \leq |\Im k|,$2, strictly bounding the thermalization timescale.

Notably, in the minimal (conservative) causal completion, all non-hydrodynamic contributions are uniquely fixed by the requirement of causality and conservation. More complex completions may admit additional, long-lived modes, but such scenarios obscure the EFT interpretation of hydrodynamics dominating late-time behavior.

Implications and Future Directions

The theoretical implications are substantial. The paper unambiguously shows that hydrodynamics as an EFT must be understood as embedded within a broader, causally-complete framework—standalone hydrodynamics inherently fails microcausality, and the space of allowed transport coefficients is not restricted by causality per se, but by the physics of the UV sector.

Practically, this implies that:

• Hydrodynamic descriptions in relativistic systems are conditional, only valid within a $\Im \omega(k) \leq |\Im k|,$3-space regime where hydrodynamic modes decay most slowly.
• Bounds such as those given by the "hydrohedron" cannot constrain hydrodynamic transport coefficients directly; instead, they restrict the scales where hydrodynamics applies.
• Exponential decay rates provide sharpened causal bounds on allowed pole locations, offering new tools for constraining transport and relaxation in QFT and many-body theory.

Future developments may focus on:

• Extending the causal bootstrap to incorporate stochastic fluctuations and higher-order corrections,
• Refining hydrohedron bounds via decay rate arguments,
• Systematically constructing causal completions for more intricate hydrodynamic dispersion relations, especially in multi-dimensional and non-linear settings.

Conclusion

The paper delivers a mathematically rigorous foundation for the necessity and construction of causal UV completions in relativistic hydrodynamics EFTs. The results redefine how causality shapes the applicability and interpretation of hydrodynamics, compelling recognition that full microcausality and physical conservation can only be achieved in theories where hydrodynamic modes coexist with suitable transient UV sectors. The formalism provided is poised to guide future investigations of transport, relaxation, and causality in quantum many-body systems and emergent gravitating media.