Regge trajectories for UV completions of graviton scattering from polynomial boundedness

TL;DR

The paper proves that any weakly coupled UV completion of graviton scattering with a meromorphic amplitude and standard S-matrix axioms must contain infinitely many Regge trajectories; a finite number leads to nonpolynomial growth and violates causality. It develops a dispersive crossing-based argument to bound graviton–higher-spin couplings and demonstrates that a single trajectory cannot satisfy polynomial boundedness. Through a primal bootstrap using the Häring–Zhiboedov ansatz, it reveals a dominant sister trajectory that emerges in extremal spectra but shows this single-trajectory picture is an artefact of truncation, with amplitudes diverging as the ansatz expands. The findings reinforce the string-like nature of UV completions of gravity and discuss implications for weak coupling, string resolvability, and connections to large-N gauge theories and N=4 SYM.

Abstract

We study graviton scattering amplitudes. Assuming they are UV completed by a theory of weakly coupled massive higher spins, we demonstrate that the UV completion must possess infinitely many Regge trajectories, and thus they are forced to have a stringy spectrum. We extend and simplify a previous proof by some of us for open-string like states to the case of external gravitons. In the present new proof, we trace the need for infinitely many trajectories to the constraint of polynomial boundedness, ultimately tied to causality. We further present numerical results based on the stringy ansatz of Häring-Zhiboedov, which illustrates how single-trajectory-like solutions actually emerge as extremal solutions of numerical bootstrap. In our numerics, these trajectories curiously show up as numerically very large \textit{sister} trajectories. We provide solid evidence that the solutions are spurious as they appear to admit a divergent limit for infinite ansatz size.

Figures (13)

• Figure 1: Blue dots: Numerical evaluation of \ref{['eq:single-traj-sol']} for $t=0$. Saddle point approximation, $e^{0.76s}$.
• Figure 2: We show in the left panel the first couplings $c_{n,J}^{++}$ obtained by maximizing the value of $c_{1,4}^{++}$ at $N_{\rm max} = 20$. We clearly identify an emerging sister trajectory that dominates the couplings by many orders of magnitude (note the log-scale of the couplings). The right panel visualizes a horizontal slice through the coupling matrix at the level $n=15$ at $N_{\rm max} = 20$ (green, see also the box in the left panel) and the corresponding values of the Virasoro-Shapiro amplitude's couplings (orange). Note that the upper plot is in log-scale, whereas the lower plot uses a linear scale for the couplings' values. We observe the growth of the couplings related to the emergent sister trajectory around $J\sim 14$.
• Figure 3: We show the emergence of the sister trajectory with unit slope also when maximizing the coefficients $c^{++}_{2,4}$, $c^{++}_{12,4}$ and $c^{++}_{17,30}$ as an indication of the robustness of the phenomenon.
• Figure 4: Here we display the evolution of the first four coefficients of the amplitude. As $N_{\rm max}$ increases, these coefficients acquire a clear, power-law-like growth, which demonstrates that the resulting amplitude does not admit a finite limit. Orange data points represent positive values, while blue points refer to negative values of the respective parameter. This is complemented with fig. \ref{['fig:amplitude-growth']}, which shows the growth of the amplitude itself.
• Figure 5: Plot of the amplitude itself slightly above the real axis, in the fixed angle limit, as a function of the $\cos(\theta)$. We have $s=100(1+I \epsilon)$ and $\cos(\theta) = 1+\tfrac{2t}{s}$. As $N_{\rm max}$ increases, the amplitude appears to diverge exponentially.