Attractive polaron in a Dirac system within the ladder approximation

We investigate the properties of the attractive polaron formed by a single impurity dressed with the particle-hole excitations in a Dirac system at zero-temperature limit. Base on the single particle-hole variational ansatz, we deduce the expressions of the pair propagator, self-energy (negative), and the non-self-consistent medium $T$-matrix. Different to the self-consistent $T$-matrix which contains two channels due to the many-body effect, the non-self-consistent $T$-matrix discussed in this paper contains only the closed channel (i.e., the bare one) since we consider only a finite number of the majority particles. The chiral factor is also discussed within the pair propagator due to the scattering feature of the Dirac system (with a certain scattering angle $a$), and we found that the fluctuation of the pair propagator is proportional to the term ${\rm cos}\ a$. Finally, we also discuss the particle spectral function of the polaron in the presence of quantum many-body effect as well as the pair-propagator at finite temperature.