<p>In this work we study the relativistic kinetic theory of a boost-invariant conformal gas on a static, maximally symmetric background <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><msub><mi>S</mi><mn>3</mn></msub><mo>×</mo><mi mathvariant="double-struck">R</mi></math>, considering all constant-curvature slicings of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><msub><mi>S</mi><mn>3</mn></msub></math>—flat, spherical, or hyperbolic—and their associated symmetry groups. Using a symmetry-driven cotangent bundle approach, we show that the isometry group of each slicing acts on phase space in such a way that only its Casimir invariants and the timelike coordinate are unconstrained, so the distribution function depends solely on these quantities. This yields a unified boost-invariant exact solution of the Boltzmann equation valid for each constant–curvature foliation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><msub><mi>S</mi><mn>3</mn></msub><mo>×</mo><mi mathvariant="double-struck">R</mi></math>. Specializing this general solution to the flat and spherical foliations reproduces the Bjorken and Gubser flows, respectively, while its restriction to the hyperbolic foliation produces a genuinely new analytic solution (“Grozdanov flow”). Hydrodynamics and free streaming emerge naturally as limiting regimes of this novel exact solution. We further comment on several relevant aspects of the new boost-invariant solution on the hyperbolic slicing and on their interpretation once mapped back to Minkowski space.</p>

United States

Mauricio Martinez

Christopher Plumberg

Maximally symmetric boost-invariant solutions of the Boltzmann equation in foliated geometries

We formulate the first-order dissipative anisotropic hydrodynamical theory for a relativistic conformal uncharged fluid, which generalizes the Bemfica-Disconzi-Noronha-Kovtun first-order viscous fluid framework. Our approach maintains causal behavior in the nonlinear regime with or without general relativity coupling, and we derive and analyze the constraints on transport coefficients imposed by causality. We demonstrate the causal and stable behavior of our theory in specific cases, including the discussion of nonlinear causality as well as stability for linearized perturbations. We apply our newly developed first-order anisotropic theory to the Bjorken flow and show how causality and stability impose constraints on the behavior of the early-time attractor.

Brazil

Fábio S. Bemfica

Germany

Masoud Shokri

Causality and stability in first-order conformal anisotropic hydrodynamics

Nonequilibrium Green’s functions provide an efficient way to describe the evolution of the energy-momentum tensor during the early-time preequilibrium stage of high-energy heavy-ion collisions. Besides their practical relevance they also provide a meaningful way to address the question when and to what extent a hydrodynamic description of the system becomes applicable. Within the kinetic theory framework we derive a new method to calculate time-dependent nonequilibrium Green’s functions describing the evolution of energy and momentum perturbations on top of an evolving far-from-equilibrium background. We discuss the approach toward viscous hydrodynamics along with the emergence of various scaling phenomena for conformal systems. By comparing our results obtained in the relaxation time approximation to previous calculations in Yang-Mills kinetic theory, we further address the question which macroscopic features of the energy-momentum tensor are sensitive to the underlying microscopic dynamics.

China

Syo Kamata

Philip Plaschke

Stephan Ochsenfeld

Sören Schlichting

Hydrodynamization and nonequilibrium Green’s functions in kinetic theory

In this work we introduce the generic conditions for the existence of a non-equilibrium attractor that is an invariant manifold determined by the long-wavelength modes of the physical system. We investigate the topological properties of the global flow structure of the Gubser flow for the Israel-Stewart theory and a kinetic model for the Boltzmann equation by employing Morse-Smale theory. We present a complete classification of the invariant submanifolds of the flow and determine all the possible flow lines connecting any pair of UV/IR fixed points. The formal transseries solutions to the Gubser dynamical system around the early-time (UV) and late-time (IR) fixed points are constructed and analyzed. It is proven that these solutions are purely perturbative (or power-law asymptotic) series with a finite radius of convergence. Based on these analyses, we find that Gubser-like expanding kinetic systems do not hydrodynamize owing to the failure of the hydrodynamization process which heavily relies on the classification of (non)hydrodynamic modes in the IR regime. This is in contrast to longitudinal boost-invariant plasmas where the asymptotic dynamics is described by a few terms of the hydrodynamic gradient expansion. We finally compare our results for both Bjorken and Gubser conformal kinetic models.

Alireza Behtash

Haosheng Shi

Global flow structure and exact formal transseries of the Gubser flow in kinetic theory

We construct analytical models to study the critical phenomena in gravitational collapse of the Husain-Martinez-Nunez massless scalar field. We first use the cut-and-paste technique to match the conformally flat solution ($$c=0$$ <math><mrow><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math>  ) onto an outgoing Vaidya solution. To guarantee the continuity of the metric and the extrinsic curvature, we prove that the two solutions must be joined at a null hypersurface and the metric function in Vaidya spacetime must satisfy certain constraints. We find that the mass of the black hole in the resulting spacetime takes the form $$M\propto (p-p^*)^\gamma $$ <math><mrow><mi>M</mi><mo>∝</mo><msup><mrow><mo>(</mo><mi>p</mi><mo>-</mo><msup><mi>p</mi><mo>∗</mo></msup><mo>)</mo></mrow><mi>γ</mi></msup></mrow></math> , where the critical exponent $$\gamma $$ <math><mi>γ</mi></math>  is equal to 0.5. For the case $$c\ne 0$$ <math><mrow><mi>c</mi><mo>≠</mo><mn>0</mn></mrow></math> , we show that the scalar field must be joined onto two pieces of Vaidya spacetimes to avoid a naked singularity. We also derive the power-law mass formula with $$\gamma =0.5$$ <math><mrow><mi>γ</mi><mo>=</mo><mn>0.5</mn></mrow></math> . Compared with previous analytical models which were constructed from a different scalar field with continuous self-similarity, we obtain the same value of $$\gamma $$ <math><mi>γ</mi></math> . However, we show that the solution with $$c\ne 0$$ <math><mrow><mi>c</mi><mo>≠</mo><mn>0</mn></mrow></math>  is not self-similar. Therefore, we provide a rare example that a scalar field without self-similarity also possesses the features of critical collapse.

Xiaobao Wang

Xiaoning Wu

Sijie Gao

Critical phenomena in gravitational collapse of Husain–Martinez–Nunez scalar field

In relativistic kinetic theory, the one-particle distribution function is approximated by an asymptotic perturbative power series in the Knudsen number which is divergent. For the Bjorken flow, we expand the distribution function in terms of its moments and study their nonlinear evolution equations. The resulting coupled dynamical system can be solved for each moment consistently using a multiparameter transseries which makes the constitutive relations inherit the same structure. A new nonperturbative dynamical renormalization scheme is born out of this formalism that goes beyond the linear response theory. We show that there is a Lyapunov function, also known as dynamical potential, which is, in general, a function of the moments and time satisfying Lyapunov stability conditions along renormalization group flows connected to the asymptotic hydrodynamic fixed point. As a result, the transport coefficients get dynamically renormalized at every order in the time-dependent perturbative expansion by receiving nonperturbative corrections present in the transseries. The connection between the integration constants and the UV data is discussed using the language of dynamical systems. Furthermore, we show that the first dissipative correction in the Knudsen number to the distribution function is not only determined by the known effective shear viscous term but also a new high-energy nonhydrodynamic mode. It is demonstrated that the survival of this new mode is intrinsically related to the nonlinear mode-to-mode coupling with the shear viscous term. Finally, we comment on some possible phenomenological applications of the proposed nonhydrodynamic transport theory.

Dynamical systems and nonlinear transient rheology of the far-from-equilibrium Bjorken flow

In this paper, we compute a number of cross sections for the production of multiple particles at mid-rapidity in the semi-dilute / dense regime of the color-glass condensate (CGC) effective field theory. In particular, we present new results for the production of two quark-antiquark pairs (whether the same or different flavors) and for the production of one quark-antiquark pair and a gluon. We also demonstrate the existence of a simple mapping which transforms the cross section to produce a quark-antiquark pair into the corresponding cross section to produce a gluon, which we use to obtain various results and to cross-check them against the literature. We also discuss hadronization effects in the heavy flavor sector, writing explicit expressions for the production of various combinations of D and   <math> <mover> <mi>D</mi> <mo>¯</mo> </mover> </math>  $$ \overline{D} $$  mesons, J/ψ mesons, and light hadrons. The various multiparticle cross sections presented here contain a wealth of information and can be used to study heavy flavor production, charge-dependent correlations, and “collective” flow phenomena arising from initial-state dynamics.

Matthew Sievert

Spain

Douglas Wertepny

Multiparticle production at mid-rapidity in the color-glass condensate

In this paper, we study the spatial correlations among quarks and antiquarks produced at mid-rapidity by gluon pair production in the color glass condensate framework. This paper is the first part of a series in which we calculate a complete set of quark/quark, quark/antiquark, and antiquark/antiquark spatial correlation functions in heavy-light ion collisions, with the goal of incorporating their conserved charges into the initial conditions of hydrodynamics. The physical mechanisms captured in this calculation include geometric, entanglement, and interaction-mediated correlations. In this first paper, we construct the building blocks for the correlations arising from single- and double-pair production, studying in detail the single-pair case and the general features of the double-pair case. We find a rich correlation structure in transverse coordinate space, with different mechanisms dominating over different length scales, and we present explicit results for the quark-antiquark correlations in the single-pair production regime. We reserve a detailed discussion of the double-pair production regime for the next paper in this sequence.