Remarks on the two-point string amplitudes

The two-point string amplitude at tree level in flat spacetime reproduces the expected expression for free particles. This has been proven by Erbin et al. in [J. High Energy Phys. 07 (2019) 139.] by two methods. Here, we provide an alternative proof of this result. Our method consists in considering a timelike Liouville direction as a regularization parameter, which suffices to break time translation invariance and, at the same time, to stabilize the residual conformal symmetry in the worldsheet. This allows taking a limit and restoring Poincare ' symmetry in a controlled way, thus reproducing the correct expression of the two-point amplitude.