One-loop Renormalization of BPS String Masses in Pseudo-anomalous Heterotic String

Compactification of heterotic string on a Calabi-Yau threefold can lead to a four-dimensional low-energy effective theory which contains a $U(1)$ gauge theory which is pseudo-anomalous, meaning that the fermion content is anomalous, but that the fermion anomaly is cancelled by a four-dimensional version of the Green-Schwarz mechanism involving a shift of the model-independent axion field. It is also well known that a field-dependent Fayet-Iliopoulos like term is induced at one-loop in string perturbation theory in such compactifications and that this leads to a mass for the $U(1)$ gauge field. Explicit one-loop computations of the mass shifts of massless charged scalars and fermions implied by this mechanism were originally carried out in the 1980's and have been revisited more recently using techniques of string field theory and super Riemann surfaces. We consider such theories further compactified on a circle of radius $R$. There is then an infinite tower of BPS states with winding and momenta on the circle. BPS strings which wind the circle can, at large $R$, be viewed as macroscopic or cosmic strings in four spacetime dimensions, or, at generic $R$, as BPS states in three dimensions. We compute the one-loop mass renormalization of such states and show that it is nonzero and proportional to their pseudo-anomalous $U(1)$ charge.