Precision asymptotics of string amplitudes

TL;DR

This work resolves a tension between the Gross–Mende high-energy saddle prediction and numerical data by uncovering an infinite family of complex saddles that dominate the one-loop amplitude. By formulating a saddle-multiplicity bootstrap that leverages modular symmetries, crossing, and analytic constraints, the authors derive a precise high-energy asymptotic expansion with oscillatory contributions. The key advance is showing that a tail of complex saddles, together with a finite set of elementary saddles, controls the amplitude, yielding a richer structure than previously anticipated and connecting to the non-perturbative amplitude and higher-genus extensions. The results illuminate how unitarity cuts, modular geometry, and high-energy behavior intertwine in string theory, with potential implications for open strings and bootstrap programs.

Abstract

Recent work revealed a tension between the Gross-Mende analysis of the high-energy fixed-angle behavior of string amplitudes and the explicit numerical data. Motivated by this puzzle, we revisit the problem of classifying saddle-point geometries for the one-loop amplitude. We find an infinite family of complex saddles that dominate the high-energy regime. Using general constraints and matching to numerical data, we formulate a bootstrap problem that determines their multiplicities. This procedure yields a precise asymptotic expansion of the one-loop amplitude at high energies. The resulting oscillatory contributions lead to a much richer high-energy behavior than that predicted by the original Gross-Mende analysis.

Figures (9)

• Figure 1: Comparison between numerical data (blue), the Gross--Mende saddle (black), and the sum over all saddles (red). The latter is obtained from the formula \ref{['eq:final formula']} with $\alpha = 0$. The quantity plotted is the amplitude for fixed angle $\theta = \frac{\pi}{2}$ as a function of $s$, with stripped prefactors, $s^{4} \mathrm{e}^{s \mathcal{S}(\frac{\pi}{2})} \sin^2(\pi s) A(s,\frac{\pi}{2})$ where $\mathcal{S}(\tfrac{\pi}{2}) = \log(2)$ and $\alpha'=4$. The real and imaginary parts are plotted at the top and bottom respectively. Dots denote integer values of $s$ that can be more easily calculated.
• Figure 2: A collection of sample saddles for $t=-\frac{s}{2}$. We draw the $z_i$'s as red points in the complex plane and the $\bar{z}_i$'s as blue dots in the complex plane. The red parallelogram is the fundamental domain of the torus described by $\tau$ (i.e. has vertices $0$, $1$, $\tau$ and $\tau+1$) and the blue parallelogram is the fundamental domain of the torus described by $\bar{\tau}$ (i.e. has vertices $0$, $1$, $\bar{\tau}$ and $\bar{\tau}+1$). The saddle point actions are in the order of display $\mathop{\text{Re}} S(\boldsymbol{m}_*,\theta)\approx 0.693,\, 0.693, \, 0.996,\, 1.046,\, 1.109,\, 1.124$. The first saddle is the Gross--Mende saddle and is the only saddle we could find for which the right-moving moduli are the complex conjugates of the left-moving moduli. The second saddle has the same real of the action as the first saddle and is a complex version of the Gross--Mende saddle, where the locations of the vertex operators are still located at the half-periods of the parallelogram. The other four saddles have larger real part of the action, and are much more irregular. There are still some patterns, for example the punctures in the fourth saddle form a parallelogram, but this is not true for all saddles. The saddle that we displayed doesn't seem to have any nice features at all.
• Figure 3: The Gross--Mende saddle as a double cover.
• Figure 4: The contour employed in \ref{['amplitude toy with hol split contour']} when deformed to the steepest descent contour in the $\tau$ (top, orange) and $\tilde{\tau}$ (bottom, blue) upper-half planes. The background shows level curves of the absolute value of the integrand for $s=5$ and $\theta=\frac{\pi}{2}$ on a logarithmic scale. Saddles are marked with red dots.
• Figure 5: Comparison of the direct evaluation of the toy integral \ref{['eq: definition of toy amplitude']} (in red) vs the saddle point expression (in blue). We chose the angle $\theta=\frac{\pi}{4}$ in these plots. The plot on the left is the real part, and on the right the imaginary part. The imaginary part can be computed exactly as described in Appendix \ref{['app: im part exact toy']}, while the numerical curve for the real part was generated as described in Appendix \ref{['subapp:numerical evaluation']}. The relevant mathematica notebook is also attached to the Zenodo submission Zenodo.