Covariant diffusion tensor for jet momentum broadening out of equilibrium

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.

💡 Research Summary

The paper addresses the problem of describing jet momentum broadening in the very early stages of ultra‑relativistic heavy‑ion collisions, when the quark‑gluon plasma is still far from local equilibrium. In the standard treatment, the jet transport coefficient (\hat q) is defined as a scalar rate of transverse‑momentum diffusion measured in the local rest frame of the medium. This scalar description implicitly assumes isotropy and equilibrium, and therefore it cannot capture directional dependencies or correlations that arise in a non‑equilibrium, flowing medium.

To overcome this limitation, the authors introduce a Lorentz‑covariant diffusion tensor (\hat q^{\mu\nu}). Starting from the relativistic Boltzmann equation for elastic (2\leftrightarrow2) scattering, they define
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