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Peter M. Lavrov
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The renormalization of general gauge theories on flat and curved space-time backgrounds is considered within the Sp(2)-covariant quantization method. We assume the existence of a gauge-invariant and diffeomorphism invariant regularization. Using the Sp(2)-covariant formalism one can show that the theory possesses gauge invariant and diffeomorphism invariant renormalizability to all orders in the loop expansion and the extended BRST symmetry after renormalization is preserved. The advantage of the Sp(2)-method compared to the standard Batalin-Vilkovisky approach is that, in reducible theories, the structure of ghosts and ghosts for ghosts and auxiliary fields is described in terms of irreducible representations of the Sp(2) group. This makes the presentation of solutions to the master equations in more simple and systematic way because they are Sp(2)- scalars.
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In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Greens functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=frac{1}{2}( ablaphi)^2+frac{1}{2}phi^2(iphi)^varepsilon$ 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 $L= frac{1}{2}( ablaphi)^2+frac{1}{2}mu^2phi^2+frac{1}{2} gmu_0^2phi^2big(imu_0^{1-D/2}phibig)^varepsilon-ivphi$, 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 Greens functions, and the renormalized mass, to higher-order in powers of $varepsilon$ in $D$ dimensions ($0leq Dleq2$). Renormalization is performed perturbatively to second order in $varepsilon$ and the structure of the Greens functions is analyzed in the limit $Dto 2$. 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.
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