Dirac Impurity in a Luttinger liquid

We consider a linearly-dispersing quantum impurity interacting through a contact density-density term with a one-dimensional (1D) superfluid described by the Tomonaga-Luttinger liquid theory. Using a linked cluster expansion we characterize the impurity dynamics by calculating approximate expressions for the single-particle Green's function and for the time evolution of the density profile. We show the existence of two different dynamical regimes: (i) a $\textit{quasiparticle}$ regime, dominated by the presence of a finite lifetime for the host impurity, and (ii) an $\textit{infrared-dominated}$ regime, where the impurity causes a Anderson's orthogonality catastrophe of the 1D bath. We discuss the possible experimental consequences of these findings for cold atoms experiments.