Regge Spectroscopy of Higher-Twist States in $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory

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 nonlocal operators known as (near-)horizontal trajectories. We resolve their degeneracy analytically by computing the first nontrivial order of the Regge intercept at weak coupling, which exhibits new behavior: It depends linearly on the coupling. This is consistent with our numerics, which interpolate all the way to strong coupling.

Figure 1

Three sheets of the Riemann surface connecting the different twist trajectories at $g=1/2$. Large colored dots indicate the branch points. The surface is generated with over 9000 points. The blue, orange, and green curves are twist-three and -five trajectories also appearing in Fig. 3 for different values of $g$.

Figure 2

Regge trajectories $S(\mathrm{\Delta })$ corresponding to the twist-three operators ${\mathcal{O}}_{0,2,4}$ (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 $S(\mathrm{\Delta })$ with the real $(\mathrm{\Delta },S)$ plane for $g=1/100$ (left) and $g=1/10$ (right). The blue curve is the Regge trajectory appearing in Fig. 2; the orange and green curves belong to the twist-five 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 scaling dimension of ${\mathcal{O}}_{\pm }$ appearing on the orange and green trajectories in Fig. 3 (markers represent the same value of ${\mathrm{\Delta }}_{\pm }$ in Fig. 3), with their weak-coupling predictions obtained with [31] (dashed line).

Figure 5

The nonperturbative intercept as a function of the coupling $\lambda =16{\pi }^2{g}^2$ (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.