Investigations of the torque anomaly in an annular sector. I. Global calculations, scalar case

In an attempt to understand a recently discovered torque anomaly in quantum field theory with boundaries, we calculate the Casimir energy and torque of a scalar field subject to Dirichlet boundary conditions on an annular sector defined by two coaxial cylinders intercut by two planes through the axis. In this model the particularly troublesome divergence at the cylinder axis does not appear, but new divergences associated with the curved boundaries are introduced. All the divergences associated with the volume, the surface area, the corners, and the curvature are regulated by point separation either in the direction of the axis of the cylinder or in the (Euclidean) time; the full divergence structure is isolated, and the remaining finite energy and torque are extracted. Formally, only the regulator based on axis splitting yields the expected balance between energy and torque. Because of the logarithmic curvature divergences, there is an ambiguity in the linear dependence of the energy on the wedge angle; if the terms constant and linear in this angle are removed by a process of renormalization, the expected torque-energy balance is preserved.