Two point amplitude for closed superstrings

TL;DR

The paper addresses the longstanding question of tree-level two-point amplitudes for closed strings in the pure spinor formalism, where the naive product of open-string structures leads to a vanishing result. It introduces a prescription that fixes the conformal Killing group with a mostly BRST-exact operator and replaces the open-string BRST charge by the closed-string charge $Q_{closed}=Q_L+Q_R$, while employing closed-string vertex operators with nonstandard ghost numbers; crucially, the BRST cohomology for total ghost-number ≥4 vanishes, rendering the prescription essentially unique. The main result is that, by using an mBRST-exact operator together with a ghost-number-three closed-string vertex, one obtains a non-vanishing, finite two-point amplitude; among all contractions only six combinations survive the zero-mode saturation, providing a consistent two-point S-matrix element that reproduces the expected kinematics. This work solidifies the equivalence between pure spinor and RNS formalisms for tree-level amplitudes and provides a robust method to fix vertex normalizations via two-point functions, with potential extensions to loop amplitudes and manifest covariance.

Abstract

We present a prescription for computing the tree-level two-point amplitude of closed strings in the pure spinor superstring formalism, thereby completing the analysis of such superstring amplitudes. The construction relies on fixing the residual conformal Killing group using a mostly BRST-exact operator that has been successfully applied in the open-string case. Earlier attempts at a straightforward extension to closed strings--treating them naïvely as products of open strings--fail. Nevertheless, we show that a consistent prescription can be obtained by replacing the open-string BRST charge with the closed-string BRST charge. The key idea is to employ closed-string vertex operators with nonstandard ghost-number assignments, rather than the conventional ghost-number (1,1) vertices. Furthermore, since the pure spinor BRST cohomology for closed strings vanishes at total (left plus right) ghost number four or higher, we find that the resulting prescription is essentially unique.