The Reggeon Vertex for DDF States

TL;DR

This paper derives a compact generating function for correlators of arbitrary numbers of bosonic open string DDF states, enabling explicit computation of correlators and verification of Lorentz invariance for various amplitudes.

Contribution

It provides the first explicit evaluation of the generating function for DDF state correlators, connecting DDF and covariant formalisms and verifying Lorentz invariance.

Findings

Explicit generating function for DDF correlators derived

Validated amplitudes for 3 and 4 DDF states match Lorentz invariance

Reassembled DDF amplitudes into Lorentz-covariant form

Abstract

We provide a compact expression for the generating function of correlators involving an arbitrary number of bosonic open string DDF states. The explicit correlators for $M$ DDF states can then be obtained by differentiating this generating function with respect to the DDF polarization tensors. The generating function depends on single and double complex integrals centred around the punctures corresponding to the insertion points of the DDF vertices on the real axis in the upper half-plane. We have explicitly evaluated these integrals for an arbitrary number of DDF states. To check our results, we computed some massive scalars and spin-2 amplitudes for $M=3$ and $M=4$, verifying that these amplitudes have the form expected from Lorentz invariance. Moreover, for the first excited levels, we also show that the DDF amplitudes with generic polarizations can be reassembled into Lorentz-covariant amplitudes, with emergent covariant polarizations that match the expressions obtained by comparing the string states in both the DDF and covariant formalisms.