Symmetry constraints on generalizations of Bjorken flow (1006.0006v2)

Abstract: I explain a generalization of Bjorken flow where the medium has finite transverse size and expands both radially and along the beam axis. If one assumes that the equations of viscous hydrodynamics can be used, with p=epsilon/3 and zero bulk viscosity, then the flow I describe can be developed into an exact solution of the relativistic Navier-Stokes equations. The local four-velocity in the flow is entirely determined by the assumption of symmetry under a subgroup of the conformal group.

• The paper introduces a novel hydrodynamic model by applying SO(3)_q symmetry, leading to a precise characterization of radial expansion in heavy-ion collisions.
• It derives an exact solution to the relativistic Navier-Stokes equations that incorporates non-zero radial flow, modifying the classic Bjorken energy density scaling.
• The results pave the way for more realistic simulations and improved phenomenological interpretations of quark-gluon plasma formation in high-energy collisions.

Symmetry Constraints on Generalizations of Bjorken Flow

This paper presents a theoretical exploration of fluid dynamics in high-energy heavy-ion collisions, expanding upon the Bjorken flow model by incorporating additional symmetry considerations. Bjorken flow is characterized by boost invariance along the beam axis and translation invariance in the transverse plane, which simplifies relativistic hydrodynamics by assuming the medium is infinite in transverse size and exhibits no radial flow before thermalization. However, these assumptions are insufficient for accurately modeling collisions that involve finite-sized nuclei with potentially significant initial radial flow.

Background and Motivation

The need for refined theoretical models arises from the unrealistic assumptions of infinite transverse size and absence of initial radial flow in Bjorken’s original framework. This paper proposes a generalized model where the medium expands radially, influenced by assuming symmetries involving special conformal transformations instead of translational symmetry in the transverse plane. The paper introduces the $SO(3)_q$ symmetry, a subgroup of the conformal group $SO(4,2)$, which is more accommodating of realistic conditions in finite-size heavy-ion collisions and central collisions.

Theoretical Framework

The generalized flow model is developed by enforcing symmetry constraints from the conformal group on the equations of relativistic hydrodynamics. Assuming the equation of state $p = \epsilon/3$ and zero bulk viscosity, the author derives an exact solution to the relativistic Navier-Stokes equations that incorporates radial expansion. The derivation leverages symmetry considerations to determine the local four-velocity entirely, thus characterizing the velocity profile in terms of the subgroup $SO(3)_q$.

Results

Numerical analysis within the paper indicates that the corrected symmetry considerations lead to non-zero radial flow, deviating from the classical Bjorken picture. The method yields quantitative predictions concerning the energy density and temperature dynamics, asserting that energy density scales as $1/\tau^{4/3}$, akin to Bjorken flow, but with modifications allowing for integrable falloff across the transverse plane. The modeling predicts non-negligible initial radial flow even in perfectly central collisions, addressing previously noted problematic aspects of the original Bjorken approximations.

Implications and Future Directions

By providing a hydrodynamical model that better conforms to physical characteristics seen in real collisions, this paper has implications for improving phenomenological interpretations of heavy-ion collision experiments. The results suggest that making assumptions of conformal symmetry and manipulating them to better accommodate finite-size geometry can lead to more accurate models of medium dynamics. Future developments may involve analyzing off-center collisions to further exploit the symmetry under certain special conformal transformations, potentially leading to refinements in theoretical predictions with increased realism and precision.

The proposed framework fosters a deeper understanding of early-stage dynamics in quark-gluon plasma formation, offering a robust platform for exploring near-conformal dynamics in quantum chromodynamics (QCD). Further studies may involve simulations incorporating realistic viscosity values to predict final-state observables that can be experimentally tested.