Asymmetric transverse momentum broadening in an inhomogeneous medium

Gradient jet tomography in high-energy heavy-ion collisions utilizes the asymmetric transverse momentum broadening of a propagating parton in an inhomogeneous medium. Such broadening is studied within a path integral description of the evolution of the Wigner distribution for a propagating parton in medium. Going beyond the eikonal approximation of multiple scattering, the evolution operator in the transverse direction can be expressed as the functional integration over all classical trajectories of a massive particle with the light-cone momentum $\omega$ as its mass. With a dipole approximation of the Wilson line correlation function, evolution with the light-cone time $t$ is determined by the jet transport coefficient $\hat q$ that can vary with space and time. In a uniform medium with a constant $\hat q_0$, the analytical solution to the Wigner distribution becomes a typical drifted Gaussian in both transverse momentum and coordinate with the diffusion width $\sqrt{\hat q_0t}$ and $\sqrt{\hat q_0t^3/3\omega^2}$, respectively. In the case of a simple Gaussian-like transverse inhomogeneity with a spatial width $\sigma$ on top of a uniform medium, the final asymmetrical momentum distribution can be calculated semi-analytically. The transverse asymmetry defined for jet gradient tomography that characterizes the asymmetrical distribution is found to linearly correlate with the initial transverse position of the propagating parton within the domain of the inhomogeneity. It decreases with the parton energy $\omega$, increases with the propagation time initially and saturates when the diffusion distance is much larger than the size of the inhomogeneity or $t^3\gg 3\omega^2\sigma^2/\hat q_0$. The transverse momentum broadening due to the inhomogeneity also saturates at late time in contrast to the continued increase with time if the drifted diffusion in space is ignored.