  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.
