One-loop mass corrections of interacting string states

TL;DR

This paper calculates one-loop mass corrections for string states in Type-II theories, revealing decay widths and regularizing divergences using elliptic functions and the $i\, ext{epsilon}$-prescription.

Contribution

It provides explicit formulas and numerical results for one-loop mass corrections of first Regge trajectory states, including regularization techniques.

Findings

Derived closed-form expressions for mass corrections.

Regularized IR divergences using $i\, ext{epsilon}$-prescription.

Numerical results obtained up to level N=4.

Abstract

The free string spectrum is highly degenerate, with a degeneracy that grows exponentially with the mass. Turning on a non-vanishing string coupling $g_s$ introduces interactions, rendering massive string states unstable and allowing them to decay into lower-mass states, with mixing constrained by Lorentz invariance. This behavior is expected already at one-loop level. The imaginary part of the one-loop mass correction is related to the width of the decay into two lower-mass states at tree level, whereas the real part is generally IR-divergent and needs regularization and renormalization. The analysis simplifies for states in the first Regge trajectory. In particular, we consider the one-loop mass corrections for these states in the NS-NS sector of Type-II string theories. We explicitly construct the related vertex operators and exploit the properties of elliptic functions in order to obtain a closed form expression for the integral over the insertion point. We further regularize the divergences of the integral over the modular parameter of the torus by means of the $i\varepsilon$-prescription in string theory. Finally we extract numerical results for the mass correction up to level $N=4$.