The leading trajectory in the 2+1D Ising CFT

TL;DR

This work studies high-energy scattering in the 2+1D Ising CFT by applying the Lorentzian inversion formula to precision bootstrap data. It identifies a leading Regge intercept $j_*\,\approx\ 0.8<1$, implying asymptotic transparency and a negative Lyapunov exponent, and benchmarks the inversion by accurately reconstructing the spin-2 stress tensor data to within ${\,10^{-4}}$. The authors extend the analysis to the leading $Z_2$-odd trajectory, find evidence for the absence of a spin-1 current in $[\sigma\epsilon]_0$, and explore analytic continuation to spin 0 with a possible shadow-operator interpretation. They connect these results to large-$N$ $O(N)$ analytics, which display analogous Regge patterns and shadow structure, and discuss implications for the regularity of the heavy spectrum and for conformal Regge theory. Overall, the paper links real-time CFT dynamics, Regge theory, and numerical bootstrap in a quantitative framework for the 3D Ising CFT.

Abstract

We study the scattering of lumps in the 2+1-dimensional Ising CFT, indirectly, by analytically continuing its spectrum using the Lorentzian inversion formula. We find evidence that the intercept of the model is below unity: $j_*\approx 0.8$, indicating that scattering is asymptotically transparent corresponding to a negative Lyapunov exponent. We use as input the precise spectrum obtained from the numerical conformal bootstrap. We show that the truncated spectrum allows the inversion formula to reproduce the properties of the spin-two stress tensor to $10^{-4}$ accuracy and we address the question of whether the spin-0 operators of the model lie on Regge trajectories. This hypothesis is further supported by analytics in the large-N O(N) model. Finally, we show that anomalous dimensions of heavy operators decrease with energy at a rate controlled by $(j_*-1)$, implying regularity of the heavy spectrum.

Figures (25)

• Figure 1: Scattering of lumps. We probe this process by correlating four local measurements.
• Figure 2: Integrand of the inversion formula (\ref{['eq:generating_int']}) with $z=10^{-2}$, comparing the exact cross-channel block for $\epsilon$-exchange (using the 3d to 2d series) with its collinear series in $(1-\bar{z})$. For $\beta=5$, the collinear limit in eq. (\ref{['collinear limit block']}) approximates the dominant region well but underestimates the integrand at small $\bar{z}$. At larger values of $\beta$ this region becomes negligible. Note that we rescaled the integrand by $2^\beta \beta\sqrt{1-\bar{z}}$ to make features more visible.
• Figure 3: The effect of different $t$-channel truncations in eq. (\ref{['eq:generating_int']}) on the extracted stress-tensor anomalous dimension $\gamma_T=\tau_T-2\Delta_\phi$. Including more operators enables us to reach lower values of $z$ where we find a stable $z$-independent plateau.
• Figure 4: Partial sum contributions to $C(z,\beta=5$) for the first three leading families as a function of the maximal spin. The $[\sigma\sigma]_0$ family converges for any $z$, but other families, especially $[\epsilon\epsilon]_0$, are very sensitive to large spins. Figure (d) shows how the exchange of $[\epsilon\epsilon]_0$ can contribute as $\log^2 z$
• Figure 5: Partial sums contributing to $C(z{=}10^{-4.5},\beta{=}5)$, extrapolated to very large spins.