Closed string self-energy on the lightcone worldsheet lattice

Abstract

We study the one loop correction to the closed bosonic string propagator, including the possible presence of D-branes, by discretizing the lightcone worldsheet on an $M\ifmmode\times\else\texttimes\fi{}N$ rectangular lattice, with $M\ensuremath{\propto}{P}^{+}$ and $N+1\ensuremath{\propto}i{x}^{+}$. The integrals over the moduli then become sums which we evaluate numerically. The main purpose of this study is to assess the reliability of the worldsheet lattice as a regulator of the divergences in string perturbation theory. There are two natural geometrical counterterms for the lightcone worldsheet, one proportional to the area of the worldsheet and the other proportional to the length of worldsheet boundaries, tracing the ends of open strings. We show that the divergences in the closed string self-energy can be cancelled by the area counterterm and a renormalization of the Regge slope parameter. The residual finite part is compatible with Lorentz invariance, provided a novel regularization, natural to the lightcone worldsheet lattice and described in this article, is employed.