Dynamics of a heavy particle in a Luttinger liquid

We study the dynamics of a heavy particle of mass $M$ moving in a one-dimensional repulsively interacting Fermi gas. The Fermi gas is described using the Luttinger model and bosonization. By transforming to a frame co-moving with the heavy particle, we map the model onto a generalized ``quantum impurity problem". A renormalization group calculation reveals a crossover from strong to weak coupling upon scaling down in temperature. Above the crossover temperature scale $T^*=(m/M) E_F$, the particle's mobility, $\mu$, is found to be (roughly) temperature independent and proportional to the dimensionless conductance, $g$, characterizing the 1d Luttinger liquid. Here $m$($<<M$) is the fermion mass, and $E_F$ is the Fermi energy. Below $T^*$, in the weak coupling regime, the mobility grows and diverges as $\mu(T) \sim T^{-4}$ in $T \to 0$ limit.