https://doi.org/10.1140/epjc/s10052-025-14858-8
Regular Article - Theoretical Physics
On-shell recursion relations for tree-level closed string amplitudes
1
The Institute for Fundamental Study, Naresuan University, 65000, Phitsanulok, Thailand
2
Quantum Computing and Information Research Centre (QX), Department of Physics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok, Thailand
Received:
22
October
2024
Accepted:
24
September
2025
Published online:
8
October
2025
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 
-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.
© The Author(s) 2025
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