Two-Point String Amplitudes

TL;DR

The paper addresses the puzzle that tree-level string theory two-point amplitudes are traditionally thought to vanish due to the infinite volume of residual worldsheet symmetry. It presents two concrete approaches—the path-integral argument and a Fadeev-Popov gauge-fixing derivation—to regularize this quantity and fix the normalization so that the amplitude reproduces the free-particle form $\mathcal{A}_2 = 2 k^0 (2\pi)^{D-1} \delta^{D-1}(\vec{\mathbf{k}}' - \vec{\mathbf{k}})$. For both open and closed strings, the authors show that proper treatment of zero modes and residual symmetry yields the same nonzero result, with normalization fixed by pole-factorization or sphere normalization. The work clarifies the structure of two-point functions in string theory and has implications for AdS correlators and potential nonzero lower-point functions in nontrivial backgrounds.

Abstract

We show that the two-point tree level amplitude in string theory in flat space is given by the standard free particle expression.