on the quark diminishes and the mass rises as in (\[3.3\]). The quarks are thus confined to the region of nonzero mean field by virtue of their interaction with the vacuum.

The situation is similar for the QPV of interest here. The use of the gluon two-point function (\[3.1\]) reduces the equations in (\[2.16\]) to algebraic form, which can then be solved by numerical or symbolic techniques. Shown in Fig.3 is the solutions for the five transverse coefficient functions in both the time-like and space-like regions of photon momentum $q^2$ for $P^2=C_{Pq}=0$. The single parameter, and the only dimensionful constant, $\eta $, is given the value $5$ fm$^{-1}$. It appears from Fig.3a that there is a pole for $q^2=-\eta ^2/2$. Closer analysis of the $q^2-P^2$ plane shows, however, that the singularity structure is not a discrete pole, but is instead a continuous spectrum. For the region $P_{\pm}^2\leq \eta ^2/4$, the analytic solutions have the simple form $$\begin{aligned}
\lambda ^T_1&=&\frac{4\eta \left[ B(P^2_{+})+B(P^2_{-})\right]}{\eta
^2-4P^2+5q^2+B(P^2_{+})B(P^2_{-})}\nonumber \\
\lambda ^T_2&=&-\frac{q^2}{\eta ^2}\lambda ^T_4\nonumber \\
\lambda ^T_3&=&\frac{2\left[ 2\eta
^2-4P^2+q^2+B(P^2_{+})B(P^2_{-})\right] }{3\eta ^2-8P^2+6q^2}\label{3.4} \\
\lambda ^T_4&=&\frac{-32\eta ^4}{\left[ \eta
^2-4P^2+5q^2+B(P^2_{+})B(P^2_{-})\right] \left[ 3\eta
^2-8P^2+6q^2\right] }\nonumber \\
\lambda ^T_5&=&\frac{8\eta ^2}{3\eta ^2-8P^2+6q^2}\nonumber\end{aligned}$$ from which the singularity structure is visible. The self-energy function $B$ 