Hydro+ Formalism in Critical QCD

• Hydro+ formalism is a deterministic extension of relativistic hydrodynamics that integrates slow, non-conserved fluctuation modes near critical points.
• It systematically extends traditional energy-momentum conservation to include dynamic fluctuations, yielding frequency-dependent bulk viscosity and modified constitutive relations.
• The approach employs advanced numerical schemes, such as operator splitting, to accurately simulate heavy-ion collisions in regimes with critical slowing down.

The Hydro+ formalism is a deterministic, local extension of relativistic hydrodynamics that incorporates the effects of parametrically slow, non-conserved fluctuations—especially those arising near critical points such as the QCD critical point. By promoting slow fluctuation modes to macroscopic dynamical variables, Hydro+ provides a unified framework to describe both the bulk evolution of energy-momentum and charge, as well as the non-equilibrium dynamics of fluctuations whose relaxation times can compete with hydrodynamic expansion rates. This formalism enables causally consistent simulations of heavy-ion collisions near criticality and clarifies the emergence of frequency-dependent bulk viscosity and dynamic equation of state stiffening.

1. Theoretical Foundations and Motivation

Standard relativistic hydrodynamics describes low-frequency, long-wavelength phenomena governed solely by conserved densities: energy-momentum and (optionally) a $U(1)$ charge. However, near a second-order critical point—such as the conjectured QCD critical point—equilibration of certain fluctuation modes becomes exceedingly slow ("critical slowing down"), invalidating the rapid local equilibration assumption. The Hydro+ formalism explicitly includes one or more such non-conserved but slow relaxation modes, denoted $\phi$ (or a set $\{\phi_Q\}$ parameterized by wavenumber), into the set of hydrodynamic fields. These modes typically correspond to order parameter fluctuations, e.g., baryon number fluctuations ($M$) or entropy/charge ratios $m \equiv s/n$.

Hydro+ thus systematically extends the domain of hydrodynamic simulation to regimes where the relaxation time $\tau_\phi$ of slow modes becomes comparable to (or exceeds) the macroscopic expansion scale (Stephanov et al., 2017).

2. Dynamical Fields and Generalized Entropy Structure

The fundamental dynamical fields of Hydro+ are

$y^A = \{\varepsilon, n, u^\mu, \phi\}\,,$

where $\varepsilon$ is energy density, $n$ is charge/baryon density, $u^\mu$ is the fluid 4-velocity, and $\phi$ encodes the slow, non-hydrodynamic degree of freedom (Gavassino et al., 2023). For fluctuations at finite wavenumber, a continuum of fields $\phi_Q$ (Fourier coefficients of the unequal-time correlator) is introduced.

The thermodynamic state is described by a non-equilibrium (partial equilibrium) entropy density,

$$s_{(+)} = s(\varepsilon, n) + \frac{1}{2} \int \frac{d^3 Q}{(2\pi)^3} \Bigl[\ln \frac{\phi_Q}{\bar{\phi}_Q} - \frac{\phi_Q}{\bar{\phi}_Q} + 1\Bigr],$$

where $\bar{\phi}_Q$ is the equilibrium two-point function. This entropy functional is a Legendre transform of the two-particle-irreducible (2PI) action, closely paralleling the effective action approach in quantum field theory (Stephanov et al., 2017).

The generalized thermodynamic conjugate to $\phi_Q$ is

$$\pi_Q(x) = -\frac{\delta s_{(+)}}{\delta \phi_Q(x)} = \frac{1}{2} \left[ \bar{\phi}_Q^{-1} - \phi_Q^{-1} \right].$$

Expansion around equilibrium yields modified expressions for temperature, chemical potential, and pressure that depend explicitly on the out-of-equilibrium slow mode variables (Rajagopal et al., 2019).

3. Equations of Motion and Constitutive Relations

The full Hydro+ equations consist of:

• Conservation of energy-momentum:

$\partial_\mu T^{\mu\nu} = 0,$

• Conservation of charge:

$\partial_\mu J^\mu = 0,$

• Kinetic evolution equation for the slow mode(s):

$$u^\mu \partial_\mu \phi + \Xi\,\theta = 0,\qquad \theta = -\left.\frac{\partial s}{\partial\phi}\right|_{\varepsilon, n},$$

or in the multi-mode case,

$D \phi_Q = -\gamma_\pi(Q) \pi_Q + \dots,$

where $D = u^\mu \partial_\mu$ and $\gamma_\pi(Q)$ is a $Q$-dependent relaxation rate (Stephanov et al., 2017, Gavassino et al., 2023).

Constitutive relations for the stress tensor generalize the ideal fluid result: $$T^{\mu\nu} = \left(\varepsilon + P\right) u^\mu u^\nu + P\,g^{\mu\nu},$$ with a pressure $P(\varepsilon, n, \phi)$ or, for the multi-mode formalism, $p_{(+)}(\varepsilon, n, \{\phi_Q\})$ (Gavassino et al., 2023, Rajagopal et al., 2019). Bulk and shear viscosities, as well as conductivity, receive nontrivial corrections from the out-of-equilibrium slow modes.

4. Frequency-Dependent Bulk Viscosity and Critical Dynamics

A defining feature of Hydro+ is the emergence of a frequency-dependent bulk viscosity, encoding the non-instantaneous response of slow modes to expansion: $$\zeta_{\rm eff}(\omega) = \frac{\zeta}{1 - i \omega \tau_\Pi}$$ in the single-mode limit, with relaxation time $\tau_\Pi$ determined by the slow mode's properties (Gavassino et al., 2023). For a spectrum of modes: $$\zeta(\omega) = \beta \int_Q \frac{p_\pi(Q)^2}{\phi_\pi(Q)} \frac{\Gamma(Q)}{\Gamma(Q)^2 + \omega^2},$$ and at low frequencies the static bulk viscosity diverges near a critical point as $\zeta(0) \sim \xi^3$ due to critical slowing down of the relaxation rate $\Gamma \sim \xi^{-z}$ ($z$ is the dynamic critical exponent, typically $z=2$ for QCD) (Stephanov et al., 2017). This frequency dependence is essential for capturing the breakdown of standard hydrodynamics in the critical region.

5. Extension to Multi-Mode (Hydro+) Formalism and Critical Region

In a full critical regime, Hydro+ generalizes the additional variable $\phi$ to a set $\{\phi_Q\}$ labeling fluctuation modes across wavenumbers $Q$. Each mode evolves toward its local-equilibrium two-point function $\bar{\phi}_Q$ with a $Q$-dependent relaxation rate, incorporating both critical slowing down and spatial dispersion. Near the QCD critical point, $\phi_Q$ embodies the spectrum of order parameter fluctuations (typically m-mode, $m = s/n$) (Stephanov et al., 2017, An et al., 2019).

The energy-momentum tensor and pressure become functionals of $\{\phi_Q\}$, and the entropy production and causality structure remain strictly local and deterministic—an advantage over approaches reliant on stochastic noise terms. Renormalization procedures, based on the 2PI analogy, render all UV divergences finite and cutoff-independent (An et al., 2019).

When wavenumber-dependent relaxation of further, subleading fluctuating modes (such as $N_{mi}$, $N_{ij}$) also becomes critical, Hydro+ breaks down and must be systematically extended ("Hydro++") by introducing additional coupled kinetic equations for these further modes (An et al., 2019).

6. Numerical Implementation and Practical Considerations

Hydro+ evolution requires integration of both the usual conservation equations and the relaxation equations for all relevant $\phi_Q$ fields (or their discretized representations). Key aspects include:

• Precomputation of equilibrium entropy, pressure, and fluctuation correlators for the equation of state tables.
• Time integration of $\phi_Q$ relaxation equations, often requiring implicit or semi-implicit schemes given critical slowing down and potentially stiff systems.
• Use of operator splitting and explicit restraining (cap on correlation length $\xi$) to control numerical stiffness (Stephanov et al., 2017).
• Monitoring of non-negativity for all kinetic coefficients and entropy production, ensuring second law compliance.
• Demonstrated capability to capture memory effects, advection of nonequilibrium structure, and hydrodynamic backreaction in idealized fireball and realistic heavy-ion collision setups. In a test scenario, the relative magnitude of backreaction on flow observables was found to be $\mathcal{O}(1-5\%)$ for realistic ranges of relaxation parameters (Rajagopal et al., 2019).

7. Relation to Israel–Stewart Theory, Universality, and Causality

Hydro+ is linearly indistinguishable from the Israel–Stewart (IS) theory of bulk viscosity: the slow mode $\phi$ maps directly onto the bulk viscous stress via an explicit field redefinition, yielding the canonical IS relaxation equation for bulk pressure $\Pi$: $$\tau_\Pi \partial_t \delta\Pi + \delta\Pi = -\zeta\,\partial_i\delta u^i,$$ in addition to the standard conservation laws. This mathematical equivalence applies for any theory in the same universality class $(3,1,0)\to(2,1,0)$ and implies all second-order theories near equilibrium are identical in the linear regime (Gavassino et al., 2023). Causality and linear stability are guaranteed by the symmetric-hyperbolic structure of the linearized system, rooted in thermodynamic stability conditions (future-directed "information current" and non-negative entropy production) (Gavassino et al., 2023).

Hydro+ thus unifies the description of hydrodynamics with slow, non-conserved but locally equilibrating fluctuation modes, essential for modeling bulk QCD dynamics near a critical endpoint.

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References:

• "Universality Classes of Relativistic Fluid Dynamics: Applications" (Gavassino et al., 2023)
• "Hydro+: hydrodynamics with parametric slowing down and fluctuations near the critical point" (Stephanov et al., 2017)
• "Hydro+ in Action: Understanding the Out-of-Equilibrium Dynamics Near a Critical Point in the QCD Phase Diagram" (Rajagopal et al., 2019)
• "Fluctuation dynamics in a relativistic fluid with a critical point" (An et al., 2019)