On-shell Recursion Relations for Tree-level Closed String Amplitudes

TL;DR

This paper develops a general on-shell recursion relation for tree-level closed string amplitudes by analyzing their pole structure using Schwinger's parametrization and residue factorization, extending techniques from open string theory.

Contribution

It introduces a novel method to derive recursion relations for closed string amplitudes, including handling delta-function poles and factorization into lower-point amplitudes.

Findings

Derived a general recursion relation for closed string tree amplitudes.

Utilized Schwinger's parametrization to analyze pole structure.

Confirmed consistency with previous results.

Abstract

We derive a general expression for on-shell recursion relations of closed string tree-level amplitudes. Starting with the string amplitudes written in the form of the Koba-Nielsen integral, we apply the BCFW shift to deform them. In contrast to open string amplitudes, where poles are explicitly determined by the integration over vertex positions, we utilize Schwinger's parametrization to handle the pole structure in closed strings. Our analysis reveals that the shifted amplitudes contain $\delta$-function poles, which yield simple poles upon taking residues. This allows us to present a general expression for the on-shell recursion relation for closed strings. Additionally, we offer an alternative method for computing the residue of the shifted amplitudes by factorizing an $n$-point closed string amplitude into two lower-point amplitudes. This is achieved by inserting a completeness relation that includes all possible closed string states in the Fock space. Our results are consistent with those previously obtained.