# DDF operators, open string coherent states and their scattering amplitudes

## TL;DR

This paper investigates open string coherent states using DDF operators, analyzing their classical profiles, scattering amplitudes, and high-energy behavior, with implications for string interactions and potential extensions to D-branes and closed strings.

## Contribution

It provides a detailed study of open string coherent states in the DDF formalism, including amplitude calculations and high-energy limits, offering new insights into string interactions.

## Key findings

- Amplitudes exponentiate in a subtle way.
- Classical profiles become more compact with more harmonics.
- High-energy limit dominated by saddle-point with coherent states.

## Abstract

We study interactions of string coherent states in the DDF (after Di Vecchia, Del Giudice, Fubini) formalism. For simplicity we focus on open bosonic strings. After reviewing basic properties of DDF operators and of excited open strings, we present some classical profiles and show how they become more and more compact as the number of harmonics increases at fixed mass. We then compute various three- and four-point amplitudes with insertions of coherent states, tachyons and vector bosons on the boundary of the disk relying on a convenient choice of reference null momenta. We find that the amplitudes exponentiate in a rather subtle and interesting way. We then study the high-energy fixed-angle limit, dominated by a saddle-point when coherent states are present, and the soft behaviour as the momentum of a vector boson is taken to zero. We briefly comment on generalisation of our analysis to multiple intersecting and magnetised D-branes and to closed strings.