Renormalized Quantum Stress-Energy Tensor of a Nonzero Radius Cosmic String

We calculate the effects of quantum fluctuations of a scalar field in the "ballpoint pen" cosmic string geometry. Using the approach to renormalization established previously for the energy density in two space dimensions, we extend those calculations to $3+1$ dimensions, nonzero scalar mass, and the full stress-energy tensor, including its contribution to the null energy condition for radial geodesics. The calculation demonstrates in detail the process of renormalization in curved spacetime, including the effects of the conformal anomaly. This model provides one of the few examples where quantum effects in curved spacetime can be explicitly calculated.