Relations Between Closed String Amplitudes at Higher-order Tree Level and Open String Amplitudes

TL;DR

This paper explores how closed string amplitudes relate to open string amplitudes on non-orientable surfaces, revealing new relations that differ from traditional KLT factorization, especially in low-energy limits.

Contribution

It demonstrates that KLT relations do not hold on D2 and RP2 surfaces and introduces new amplitude relations connecting closed and open strings in these cases.

Findings

KLT relations fail on D2 and RP2 surfaces.

New relations connect closed and open string amplitudes with phase factors.

Graviton amplitudes require these new relations in low-energy limits.

Abstract

KLT relations almost factorize closed string amplitudes on $S_2$ by two open string tree amplitudes which correspond to the left- and the right- moving sectors. In this paper, we investigate string amplitudes on $D_2$ and $RP_2$. We find that KLT factorization relations do not hold in these two cases. The relations between closed and open string amplitudes have new forms. On $D_2$ and $RP_2$, the left- and the right- moving sectors are connected into a single sector. Then an amplitude with closed strings on $D_2$ or $RP_2$ can be given by one open string tree amplitude except for a phase factor. The relations depends on the topologies of the world-sheets.Under T-duality, the relations on $D_2$ and $RP_2$ give the amplitudes between closed strings scattering from D-brane and O-plane respectively by open string partial amplitudes.In the low energy limits of these two cases, the factorization relations for graviton amplitudes do not hold. The amplitudes for gravitons must be given by the new relations instead.