in this initial investigation is not necessarily to reproduce the known bound-state spectrum, but rather to correlate trends in the spectrum with changes in the parameterization of the infrared form of the gluon two-point function. The sensitivity of the spectrum to these changes then provides a constraint on the calculation of the QPV in the space-like region. To this end, a sufficient parametrization of the gluon two-point function is given by $$g^2D(P^2)=\frac{3\pi ^4\eta ^2}{16}R_0^4e^{-R_0^2P^2/4}+\frac{16\pi
^2}{11P^2\mbox{ln}\left( e+P^2/\Lambda ^2\right) },\label{4.1}$$ where the first term (which has the coordinate space representation $e^{-r^2/R_0^2}$) provides a length scale, $R_0$, characterizing the infrared behavior while the second term models the known ultraviolet form. The parameter $\Lambda $, which is associated with the QCD scale, is fixed at $1$ fm$^{-1}$ throughout this investigation. As $R_0$ approaches infinity the first term reduces to the delta-function model of Eq.(\[3.1\]). The gluon two-point function (\[4.1\]) has been employed previously in the study of Bethe-Salpeter and Schwinger-Dyson equations[@PCR89], and is expected to provide an acceptable accounting of the infrared effects for the purposes of the investigation conducted here.

### Quark self energy ansatz

The solution of the Bethe-Salpeter equation in the time-like region requires knowledge of the quark Green’s functions, and 