Classical string profile for a class of DDF amplitudes

TL;DR

The paper analyzes the classical limit of a tree-level three-point string amplitude involving DDF states and a photon, deriving an exact large-spin expression in terms of modified Bessel functions and mapping it to a classical string profile. It shows that states on the leading Regge trajectory reproduce a rigid, charged-rotating string, while departures from Regge (finite $\\alpha$) yield a floppy, mostly planar string with a transverse kink, whose endpoint dynamics probe the disk interior as excitations grow. Numerically, the classical profile remains largely planar with a localized kink, and higher DDF excitations smear the profile across multiple modes, suggesting a path toward Kerr-like disk distributions. Overall, the work connects string-theoretic amplitudes to concrete classical string geometries and offers a potential bridge to black hole microstate descriptions via extended DDF states.

Abstract

In the critical bosonic string theory, we explicitly evaluate the three point scattering amplitude at tree level, of a photon with two massive higher spins. The massive excitations belong to states of the form $A_{-r_1}^{s_1} A_{-r_2}^{s_2}$ where $A_{-n}$ is a DDF creation operator. Next, we take the infinite ``spin'' limit to arrive at the classical string dynamics. We find a rotating ``floppy'' string lying mostly on a plane which develops a transverse kink.

Figures (7)

• Figure 1: Snapshots of the time evolution of the classical string profile, for $\alpha = 0.5, r_1 = 1, r_2 = 2$ with $E_{\gamma} =1, a = 1$. The red dot denotes the position of the charge, at $\sigma = \pi a$. The orange parts correspond to the regions $(t \pm \sigma) = 0, 2\pi$, where the string solution extends into the $z$ direction.
• Figure 2: Projection of the trajectory of the point charge on the $X_1- X_2$ plane for $E_{\gamma} =1, a = 1$.
• Figure 3: A cartoon of the classical string profile. The charge is indicated with the red endpoint. The orange dots indicate location of the kink. The slightly transparent afterimage indicates the string-charge configuration in a future instant. The lower left indicates the $|\sqrt{\text{Kerr}}\rangle$ configuration of charged rotating disk.
• Figure 4: Details of the reordering of the summation indices. The red dots denote the $(m,k)$ indices of the summation. From the first to the second diagram, one counts along the $m$ axis instead of the $k$ axis. In the last figure, one counts the different values of $n = m-k$ for a fixed $k$.
• Figure 5: l.h.s. and r.h.s. of eq\ref{['lhs-rhs']} with numerical solutions of $f_1(\phi)$ for $\alpha = 0.5, r_1 = 1, r_2 = 2$. Without introducing non-zero $f_3(\phi)$ there is a mismatch at $\phi=0$ and $\phi=2 \pi$.