Longitudinal nonlocality in the string S-matrix

TL;DR

This work provides a coherent, wavepacket-based analysis of four- and five-point open string amplitudes in the Regge limit, showing that longitudinal nonlocality is needed to reconcile phase-derived peak trajectories and time shifts with causality. By constructing explicit intermediate long-string (yo-yo) solutions and analyzing five-point radiation, the authors demonstrate that longitudinal spreading must occur at scales set by α' E, beyond what transverse spreading alone would permit. The results connect Reggeon dynamics, Bremsstrahlung patterns, and precise S-matrix data to a spacetime picture that supports longitudinal nonlocality and has implications for horizon dynamics in related contexts. The work lays groundwork for further explorations in higher-point amplitudes and AdS/CFT extensions, with companion studies applying string spreading to black hole horizons.

Abstract

We analyze four and five-point tree-level open string S-matrix amplitudes in the Regge limit, exhibiting some basic features which indicate longitudinal nonlocality, as suggested by light cone gauge calculations of string spreading. Using wavepackets to localize the asymptotic states, we compute the peak trajectories followed by the incoming and outgoing strings, determined by the phases in the amplitudes. These trajectories trace back in all dimensions such that the incoming strings deflect directly into corresponding outgoing ones, as expected from a Reggeon analysis. Bremsstrahlung radiation at five points emerges from the deflection point, corroborating this picture. An explicit solution for the intermediate state produced at four points in the $s$-channel exists, with endpoints precisely following the corresponding geometry and a periodicity which matches the series of time delays predicted by the amplitude. We find a nonzero peak impact parameter for this process, and show that it admits an interpretation in terms of longitudinal-spreading induced string joining, at the scale expected from light cone calculations, and does not appear to admit a straightforward interpretation purely in terms of the well-established transverse spreading. At five points, we exhibit a regime with advanced emission of one of the deflected outgoing strings. This strongly suggests early interaction induced by longitudinal nonlocality. In a companion paper, we apply string spreading to horizon dynamics.

Figures (10)

• Figure 1: A schematic diagram of the scattering process for the first few oscillations at $\theta=0$. The region before the center of masses collide that is within the longitudinal spreading radius is shaded in gray. The oscillating long string that is produced by the collision is shaded in black.
• Figure 2: The three independent open string orderings. We have drawn arrows to signify the spacetime direction of the momenta of each of the strings at $\theta=0$.
• Figure 3: $(a)$ The asymptotic trajectories of the incoming and outgoing states, traced back to the collision at $T=0$. The endpoints of the intermediate state follow the rhombus in the center of the diagram. $(b)$ The process is suppressed at $b=0$, since the endpoints of the intermediate state at a given snapshot of time do not hit the red and blue lines. $(c)$ The intermediate state corresponding to the backscattering ordering $A_{su}$.
• Figure 4: At five points, a low energy string 2 is emitted from String B at the point where it turns.
• Figure 5: The worldsheet diagram in light-cone gauge for the ordering $A_{st}$ in the transverse brick wall frame, where the length of each of the strings is conserved. The interaction occurs during a short time in the Regge limit.