Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Covariant diffusion tensor for jet momentum broadening out of equilibrium

Isabella Danhoni, Nicki Mullins, Jorge Noronha

Abstract

Jets are produced in the earliest stages of heavy-ion collisions, where they can interact with a medium that is not yet close to local equilibrium. Motivated by this, we generalize the usual jet transport coefficient $\hat q$ to a Lorentz-covariant diffusion tensor $\hat q^{μν}$ within a leading-order elastic (Boltzmann/Fokker--Planck) description of jet--medium interactions. The tensor formulation organizes medium effects in a frame-covariant way and reveals additional information beyond the standard scalar definition, including energy diffusion and off-diagonal components that encode correlations between energy and momentum exchange which are absent (or redundant) in equilibrium. We illustrate the formalism in (tree-level) massless $λ\varphi^4$ theory for isotropic but out-of-equilibrium states. For sufficiently large jet momentum, quantum statistical effects become subleading, so that the non-equilibrium evolution can be studied reliably in the classical (Boltzmann) limit. This allows us to solve the corresponding Boltzmann equation for the medium and determine the time dependence of $\hat q^{μν}$ as the system approaches equilibrium. We find that out-of-equilibrium corrections can either enhance or reduce jet momentum broadening, depending on the initial distribution function.

Covariant diffusion tensor for jet momentum broadening out of equilibrium

Abstract

Jets are produced in the earliest stages of heavy-ion collisions, where they can interact with a medium that is not yet close to local equilibrium. Motivated by this, we generalize the usual jet transport coefficient to a Lorentz-covariant diffusion tensor within a leading-order elastic (Boltzmann/Fokker--Planck) description of jet--medium interactions. The tensor formulation organizes medium effects in a frame-covariant way and reveals additional information beyond the standard scalar definition, including energy diffusion and off-diagonal components that encode correlations between energy and momentum exchange which are absent (or redundant) in equilibrium. We illustrate the formalism in (tree-level) massless theory for isotropic but out-of-equilibrium states. For sufficiently large jet momentum, quantum statistical effects become subleading, so that the non-equilibrium evolution can be studied reliably in the classical (Boltzmann) limit. This allows us to solve the corresponding Boltzmann equation for the medium and determine the time dependence of as the system approaches equilibrium. We find that out-of-equilibrium corrections can either enhance or reduce jet momentum broadening, depending on the initial distribution function.
Paper Structure (13 sections, 105 equations, 3 figures)

This paper contains 13 sections, 105 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Kinetic theory and a covariant definition of jet momentum broadening
  3. Covariant jet momentum broadening as a diffusion tensor
  4. Recovering transverse momentum broadening
  5. Energy broadening and $\hat{q}^{00}$
  6. Energy--momentum correlations and $\hat{q}^{0i}$
  7. Positivity of the broadening tensor $\hat{q}^{\mu\nu}$ and diffusion constraints
  8. Summary of physical content of $\hat{q}^{\mu\nu}$
  9. Covariant jet momentum broadening in $\lambda\varphi^4$ theory
  10. The classical limit
  11. Non-equilibrium medium corrections
  12. Summary and Discussion
  13. Acknowledgments

Figures (3)

  • Figure 1: Schematic view of the elastic $2\leftrightarrow 2$ scattering process described by the Boltzmann equation.
  • Figure 2: Schematic representation of the $\lambda\varphi^4$ elastic scattering.
  • Figure 3: Variation of momentum broadening from equilibrium, as defined in Eq. \ref{['Eq:Delta_q']} for various initial conditions. The blue solid curve is the initial condition defined in Eq. \ref{['Eq:analytic_ic']}, which is under-populated. The red dotted-dashed curve is the initial condition with a Gaussian IR enhancement defined in Eq. \ref{['Eq:Gaussian_IR_enhanced']} with $A=1$ and $\sigma = 1/\sqrt{2}$. Finally, the purple dashed curve is the bithermal initial condition defined in Eq. \ref{['Eq:bithermal_ic']} with $T_1 = 2 T_{\mathrm{eq}}$, $T_2 = T_{\mathrm{eq}} / 3$, and $A = 0.35$.