Regge spectroscopy of higher twist states in $\mathcal{N}=4$ supersymmetric Yang-Mills theory

TL;DR

This work uses the Quantum Spectral Curve to perform Regge spectroscopy of higher-twist states in planar $\mathcal{N}=4$ SYM, revealing a rich Riemann-surface structure connecting twist-3 and higher-twist trajectories. A key finding is the degeneracy of horizontal trajectories at weak coupling, resolved by allowing odd powers of the coupling in the QSC ansatz, which yields an intercept that scales linearly with $g$ and connects smoothly to strong coupling. The authors develop analytic and numerical tools, including a $P\mu$-system formulation, TQ and Baxter-type equations, and Mellin-transform methods, to trace Regge trajectories beyond twist-2 and to extract non-perturbative data for twist-5 local operators. The results provide a non-perturbative window into Regge dynamics in a conformal theory and set the stage for future studies of Odderon-like trajectories and nonlocal light-ray operator mixing in $\mathcal{N}=4$ SYM.

Abstract

We study a family of higher-twist Regge trajectories in $\mathcal{N}=4$ supersymmetric Yang-Mills theory using the Quantum Spectral Curve. We explore the many-sheeted Riemann surface connecting the different trajectories and show the interplay between the degenerate non-local operators known as horizontal trajectories. We resolve their degeneracy analytically by computing the first non-trivial order of the Regge intercept at weak coupling, which exhibits new behaviour: it depends linearly on the coupling. This is consistent with our numerics, which interpolate all the way to strong coupling.

Figures (8)

• Figure 1: Three sheets of the Riemann surface connecting the different twist-trajectories at $g=1/2$. Dots stand for the branch points. The surface is generated with over 9000 points. The blue, orange and green curves are twist-$3$ and -$5$ trajectories also appearing in FIG. \ref{['fig:spectrumandtraj']} for different values of $g$.
• Figure 2: Regge trajectories $S(\Delta)$ corresponding to the twist-3 operators $\mathcal{O}_0,\mathcal{O}_2$, and $\mathcal{O}_4$ (indicated by red circles, and transparent red circles for their shadows) for different values of $g$. Each trajectory is produced by about 400 points.
• Figure 3: The intersection of the Riemann surface $S(\Delta)$ with the real $(\Delta,S)$ plane for $g=1/100$ (left) and $g=1/10$ (right). The blue curve is the Regge trajectory appearing in FIG. \ref{['fig:leadingtraj']}, the orange and green curves belong to the twist-$5$ trajectories accessed by continuation on the Riemann surface. The dashed lines show the zero-coupling limit of the curves. Local operators are indicated by markers, and shadows by their transparent counterparts.
• Figure 4: The non-perturbative scaling dimension of the twist-$5$ operators $\mathcal{O}_\pm$appearing on the orange and green trajectories of FIG. \ref{['fig:spectrumandtraj']} (with markers here representing the same value of $\Delta_\pm$ in FIG. \ref{['fig:spectrumandtraj']}). Dashed lines are the weak coupling predictions \ref{['Deltapm']}.
• Figure 5: The non-perturbative intercept as a function of the coupling $\lambda=16\pi^2g^2$ (we consider the range $g \in [10^{-3},10]$). The solid line is obtained by our numerical procedure (around 250 points), while the dashed lines are the perturbative predictions.