Open quantum system approach to the transverse momentum broadening of a colour dipole
Using the open quantum systems formalism, we study the propagation of a quark-antiquark pair propagating through a dense QCD plasma of size $L$ and transverse momentum broadening transport coefficient $\hat q$, and we derive the Lindblad evolution equation for the density matrix of the system. We focus on the boosted regime where the opening angle $θ_{q\bar q}$ of this effective colour dipole satisfies $θ_{q\bar q}\ll 1$. In the correlation limit where the quark-antiquark relative transverse momentum $p_\perp$ is much larger than the imbalance $q_\perp$ as well as the medium typical transverse momentum scale $Q_s=\sqrt{\hat q L}$, we demonstrate that the resulting Wigner distribution displays quasi factorisation between a hard factor describing the hard splitting producing the $q\bar q$ pair and the transverse momentum imbalance $q_\perp$-distribution encoding the broadening induced by the medium. The factorisation is violated by a "colour decoherence" factor that controls the $θ_{q\bar q}$ dependence of the $q_\perp$-distribution through the ratio $θ_{q\bar q}/θ_c$, with $θ_c \sim (\hat q L^3)^{-1/2}$. The open quantum systems approach enables us to clarify the role of this critical angle $θ_c$ and its associated critical time $t_c$ in the genuine quantum decoherence of the density matrix in colour and kinematic space: in particular, $t_c$ controls both the suppression of the off-diagonal elements of the density matrix in colour space and the transition between singlet and octet states. We find, however, that colour decoherence sets in earlier than the full decoherence of the density matrix, thereby marking the onset of classical behaviour in the system. Finally, we investigate the corrections beyond the quasi-factorised picture due to the quantum diffusion term in $p_\perp$ of the Lindblad equation and we find that these corrections are mild.