to define the solutions in the portion of the complex plane sampled by the QPV calculation.

Results
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### Time-like $q^2$ and bound states

Shown in Fig.\[fig5\] are the solutions to Eq.(\[2.16\]) for the five transverse coefficient functions in the time-like region of photon momentum squared. Two different values of the infrared scale, $R_0$, are reported, from which it is observed that the spectrum is quite sensitive to changes in this parameter. In particular, an increase in $R_0$ leads to an increase in the density of bound-state poles. This situation is, for example, analogous to a potential well, where $R_0$ plays the role of the range while the barrier is provided by the repulsive interaction with the vacuum. With the value of $R_0$ set to $1$ fm, the ground-state mass occurs at $880$ MeV. This is in reasonable agreement with the observed $\rho $ mass, considering that pion dressing is expected to lower this value[@rhomass], yet the excitation spectrum does not coincide with experimentally observed quantities. Comparison with the measured excitation spectrum can be improved by decreasing the range, $R_0$, however this also leads to an increase in the ground-state mass, and away from the known value. As illustrated previously 