  In a previous paper it was shown how to calculate the ground-state energy
density $E$ and the $p$-point Green's functions $G_p(x_1,x_2,...,x_p)$ for the
$PT$-symmetric quantum field theory defined by the Hamiltonian density
$H=\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\phi^2(i\phi)^\varepsilon$ in
$D$-dimensional Euclidean spacetime, where $\phi$ is a pseudoscalar field. In
this earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed as
perturbation series in powers of $\varepsilon$ and were calculated to first
order in $\varepsilon$. (The parameter $\varepsilon$ is a measure of the
nonlinearity of the interaction rather than a coupling constant.) This paper
extends these perturbative calculations to the Euclidean Lagrangian $L=
\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\mu^2\phi^2+\frac{1}{2}
g\mu_0^2\phi^2\big(i\mu_0^{1-D/2}\phi\big)^\varepsilon-iv\phi$, which now
includes renormalization counterterms that are linear and quadratic in the
field $\phi$. The parameter $g$ is a dimensionless coupling strength and
$\mu_0$ is a scaling factor having dimensions of mass. Expressions are given
for the one-, two, and three-point Green's functions, and the renormalized
mass, to higher-order in powers of $\varepsilon$ in $D$ dimensions ($0\leq
D\leq2$). Renormalization is performed perturbatively to second order in
$\varepsilon$ and the structure of the Green's functions is analyzed in the
limit $D\to 2$. A sum of the most divergent terms is performed to {\it all}
orders in $\varepsilon$. Like the Cheng-Wu summation of leading logarithms in
electrodynamics, it is found here that leading logarithmic divergences combine
to become mildly algebraic in form. Future work that must be done to complete
the perturbative renormalization procedure is discussed.
