This paper computes the vacuum polarization energy (VPE) — the energy coming from quantum fluctuations of a field — in the presence of a smooth, non‑singular cosmic string in 2+1 spacetime dimensions. The authors study a geometry with a deficit angle far from the core and a core of nonzero radius r0. In higher dimensions this geometry would correspond to a cosmic string. To make the problem tractable they focus on a specific “ballpoint pen” model of the core: a region of constant curvature joined smoothly to flat space, which is the curvature analogue of a square well.

Rather than summing mode energies directly, the team rewrites the VPE as a renormalized sum and integral over scattering data. They connect local quantities (the Green’s function, which gives a local density of states) to global scattering data (the Jost function and the S‑matrix, which encode how waves scatter from the string). For the ballpoint‑pen model those scattering functions can be written in terms of Legendre and Bessel functions. For a generic smooth profile the same approach can be implemented numerically, and the authors describe how to obtain the necessary scattering data.

The method also makes renormalization transparent. The calculation requires standard local counterterms. One subtraction accounts for the change in area (a free‑space or cosmological constant subtraction). The only other potential divergence in two dimensions is the tadpole term, proportional to the integral of the curvature R, but the authors show that after taking the required derivative with respect to wave number this term does not contribute to the total energy in 2+1 dimensions. They avoid a numerically difficult spatial integral across the step boundary by using an analytic relationship between the space integral of the Green’s function and the Jost function.

As an example, the paper presents results for a massless scalar field (mass µ = 0). Because r0 is the only length scale, the total energy must scale like 1/r0, so the dimensionless combination r0E depends only on the deficit‑angle parameter σ and on the curvature coupling ξ. The authors show results for both minimal coupling (ξ = 0) and conformal coupling (ξ = 1/8). They also note there are no bound states for the configurations they consider, and they compare their nonsingular ballpoint‑pen model with singular limits: the pointstring limit (r0 → 0) concentrates curvature at the origin and has divergent energy, and the “flowerpot” model, which concentrates curvature on a ring, also gives divergent total energy.