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In a previous paper it was shown how to calculate the ground-state energydensity $E$ 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. Inthis earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed asperturbation series in powers of $\varepsilon$ and were calculated to firstorder in $\varepsilon$. (The parameter $\varepsilon$ is a measure of thenonlinearity of the interaction rather than a coupling constant.) This paperextends 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 nowincludes renormalization counterterms that are linear and quadratic in thefield $\phi$. The parameter $g$ is a dimensionless coupling strength and$\mu_0$ is a scaling factor having dimensions of mass. Expressions are givenfor the one-, two, and three-point Green's functions, and the renormalizedmass, to higher-order in powers of $\varepsilon$ in $D$ dimensions ($0\leqD\leq2$). Renormalization is performed perturbatively to second order in$\varepsilon$ and the structure of the Green's functions is analyzed in thelimit $D\to 2$. A sum of the most divergent terms is performed to {\it all}orders in $\varepsilon$. Like the Cheng-Wu summation of leading logarithms inelectrodynamics, it is found here that leading logarithmic divergences combineto become mildly algebraic in form. Future work that must be done to completethe perturbative renormalization procedure is discussed.

Towards perturbative renormalization of $\phi^2(i\phi)^\varepsilon$ quantum field theory